Ordinary least squares

6674: 6335: 6669:{\displaystyle {\begin{aligned}\mathbf {y} &={\begin{bmatrix}\mathbf {X} &\mathbf {K} \end{bmatrix}}{\begin{bmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {\gamma }}}\end{bmatrix}},\\{}\Rightarrow {\begin{bmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\boldsymbol {\gamma }}}\end{bmatrix}}&={\begin{bmatrix}\mathbf {X} &\mathbf {K} \end{bmatrix}}^{-1}\mathbf {y} ={\begin{bmatrix}\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\\\left(\mathbf {K} ^{\top }\mathbf {K} \right)^{-1}\mathbf {K} ^{\top }\end{bmatrix}}\mathbf {y} .\end{aligned}}} 12142: 11903: 12962: 15853: 405: 2516: 449: 5579: 2169: 11790: 13091: 4596: 10394: 5646: 2511:{\displaystyle \mathbf {X} ={\begin{bmatrix}X_{11}&X_{12}&\cdots &X_{1p}\\X_{21}&X_{22}&\cdots &X_{2p}\\\vdots &\vdots &\ddots &\vdots \\X_{n1}&X_{n2}&\cdots &X_{np}\end{bmatrix}},\qquad {\boldsymbol {\beta }}={\begin{bmatrix}\beta _{1}\\\beta _{2}\\\vdots \\\beta _{p}\end{bmatrix}},\qquad \mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{bmatrix}}.} 4989: 5510: 4290: 10156: 10129: 3359: 9426: 4719: 2906: 5248: 13098:

Using either of these equations to predict the weight of a 5' 6" (1.6764 m) woman gives similar values: 62.94 kg with rounding vs. 62.98 kg without rounding. Thus a seemingly small variation in the data has a real effect on the coefficients but a small effect on the results of the equation.

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The theorem can be used to establish a number of theoretical results. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the de-meaned variables but without the

4591:{\displaystyle s^{2}={\frac {{\hat {\varepsilon }}^{\mathrm {T} }{\hat {\varepsilon }}}{n-p}}={\frac {(My)^{\mathrm {T} }My}{n-p}}={\frac {y^{\mathrm {T} }M^{\mathrm {T} }My}{n-p}}={\frac {y^{\mathrm {T} }My}{n-p}}={\frac {S({\hat {\beta }})}{n-p}},\qquad {\hat {\sigma }}^{2}={\frac {n-p}{n}}\;s^{2}} 1815:

As a concrete example where regressors are non-linearly dependent yet estimation may still be consistent, we might suspect the response depends linearly both on a value and its square; in which case we would include one regressor whose value is just the square of another regressor. In that case, the

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This highlights a common error: this example is an abuse of OLS which inherently requires that the errors in the independent variable (in this case height) are zero or at least negligible. The initial rounding to nearest inch plus any actual measurement errors constitute a finite and non-negligible

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An important consideration when carrying out statistical inference using regression models is how the data were sampled. In this example, the data are averages rather than measurements on individual women. The fit of the model is very good, but this does not imply that the weight of an individual

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will suggest the form and strength of the relationship between the dependent variable and regressors. It might also reveal outliers, heteroscedasticity, and other aspects of the data that may complicate the interpretation of a fitted regression model. The scatterplot suggests that the relationship

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Second, for each explanatory variable of interest, one wants to know whether its estimated coefficient differs significantly from zero—that is, whether this particular explanatory variable in fact has explanatory power in predicting the response variable. Here the null hypothesis is that the true

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Regressors do not have to be independent for estimation to be consistent e.g. they may be non-linearly dependent. Short of perfect multicollinearity, parameter estimates may still be consistent; however, as multicollinearity rises the standard error around such estimates increases and reduces the

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Residuals against the explanatory variables in the model. A non-linear relation between these variables suggests that the linearity of the conditional mean function may not hold. Different levels of variability in the residuals for different levels of the explanatory variables suggests possible

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can be cast in order to make the OLS technique applicable. Each of these settings produces the same formulas and same results. The only difference is the interpretation and the assumptions which have to be imposed in order for the method to give meaningful results. The choice of the applicable

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This example also demonstrates that coefficients determined by these calculations are sensitive to how the data is prepared. The heights were originally given rounded to the nearest inch and have been converted and rounded to the nearest centimetre. Since the conversion factor is one inch to

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is used to test whether two subsamples both have the same underlying true coefficient values. The sum of squared residuals of regressions on each of the subsets and on the combined data set are compared by computing an F-statistic; if this exceeds a critical value, the null hypothesis of no

14243: 10389:{\displaystyle {\hat {\beta }}^{c}=R(R^{\operatorname {T} }X^{\operatorname {T} }XR)^{-1}R^{\operatorname {T} }X^{\operatorname {T} }y+{\Big (}I_{p}-R(R^{\operatorname {T} }X^{\operatorname {T} }XR)^{-1}R^{\operatorname {T} }X^{\operatorname {T} }X{\Big )}Q(Q^{\operatorname {T} }Q)^{-1}c,} 9165: 11845:. If the t-statistic is larger than a predetermined value, the null hypothesis is rejected and the variable is found to have explanatory power, with its coefficient significantly different from zero. Otherwise, the null hypothesis of a zero value of the true coefficient is accepted. 3824: 11181: 518:

Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting

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This assumption is not needed for the validity of the OLS method, although certain additional finite-sample properties can be established in case when it does (especially in the area of hypotheses testing). Also when the errors are normal, the OLS estimator is equivalent to the

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Another way of looking at it is to consider the regression line to be a weighted average of the lines passing through the combination of any two points in the dataset. Although this way of calculation is more computationally expensive, it provides a better intuition on OLS.

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Two hypothesis tests are particularly widely used. First, one wants to know if the estimated regression equation is any better than simply predicting that all values of the response variable equal its sample mean (if not, it is said to have no explanatory power). The

5775: 3267: 10706: 9229: 8696: 4984:{\displaystyle R^{2}={\frac {\sum ({\hat {y}}_{i}-{\overline {y}})^{2}}{\sum (y_{i}-{\overline {y}})^{2}}}={\frac {y^{\mathrm {T} }P^{\mathrm {T} }LPy}{y^{\mathrm {T} }Ly}}=1-{\frac {y^{\mathrm {T} }My}{y^{\mathrm {T} }Ly}}=1-{\frac {\rm {RSS}}{\rm {TSS}}}} 3610: 1812:

precision of such estimates. When there is perfect multicollinearity, it is no longer possible to obtain unique estimates for the coefficients to the related regressors; estimation for these parameters cannot converge (thus, it cannot be consistent).

2741: 5505:{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\sum _{i=1}^{n}{(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}}{\sum _{i=1}^{n}{(x_{i}-{\bar {x}})^{2}}}}\\{\widehat {\alpha }}&={\bar {y}}-{\widehat {\beta }}\,{\bar {x}}\ ,\end{aligned}}} 1052: 13679: 8032: 6071: 14323: 1131: 12586: 5956: 1415: 5159:, then this is called the "simple regression model". This case is often considered in the beginner statistics classes, as it provides much simpler formulas even suitable for manual calculation. The parameters are commonly denoted as 12136: 13518: 3172: 5549:

was obtained as a value that minimizes the sum of squared residuals of the model. However it is also possible to derive the same estimator from other approaches. In all cases the formula for OLS estimator remains the same:

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in the model, although usually it is also estimated. If this assumption is violated then the OLS estimates are still valid, but no longer efficient. It is customary to split this assumption into two parts:

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Ordinary least squares analysis often includes the use of diagnostic plots designed to detect departures of the data from the assumed form of the model. These are some of the common diagnostic plots:

12668: 13194: 10124:{\displaystyle {\hat {\beta }}^{c}={\hat {\beta }}-(X^{\operatorname {T} }X)^{-1}Q{\Big (}Q^{\operatorname {T} }(X^{\operatorname {T} }X)^{-1}Q{\Big )}^{-1}(Q^{\operatorname {T} }{\hat {\beta }}-c).} 10993: 2087: 7244: 1614: 9811: 8850:. If we are willing to allow biased estimators, and consider the class of estimators that are proportional to the sum of squared residuals (SSR) of the model, then the best (in the sense of the 6340: 3354:{\displaystyle {\hat {\boldsymbol {\beta }}}={\boldsymbol {\beta }}+\left(\mathbf {X} ^{\operatorname {T} }\mathbf {X} \right)^{-1}\mathbf {X} ^{\operatorname {T} }{\boldsymbol {\varepsilon }}.} 4657: 11265: 9728: 8073: 5697: 10758: 3976: 10590: 8406: 6188: 13110:

error. As a result, the fitted parameters are not the best estimates they are presumed to be. Though not totally spurious the error in the estimation will depend upon relative size of the

9421:{\displaystyle {\hat {y}}_{j}^{(j)}-{\hat {y}}_{j}=x_{j}^{\mathrm {T} }{\hat {\beta }}^{(j)}-x_{j}^{\operatorname {T} }{\hat {\beta }}=-{\frac {h_{j}}{1-h_{j}}}\,{\hat {\varepsilon }}_{j}} 6692:(MLE) under the normality assumption for the error terms. This normality assumption has historical importance, as it provided the basis for the early work in linear regression analysis by 3160: 13898: 13102:

While this may look innocuous in the middle of the data range it could become significant at the extremes or in the case where the fitted model is used to project outside the data range (

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is calculated under the assumption that errors follow normal distribution. Even though the assumption is not very reasonable, this statistic may still find its use in conducting LR tests.

8912: 8581: 8371: 8246: 7915: 7748: 7444:. Importantly, the normality assumption applies only to the error terms; contrary to a popular misconception, the response (dependent) variable is not required to be normally distributed. 1462: 9481:. Usually the observations with high leverage ought to be scrutinized more carefully, in case they are erroneous, or outliers, or in some other way atypical of the rest of the dataset. 8589: 5054: 14431: 14124: 8319: 6816: 5691:), we are looking for a solution that could provide the smallest discrepancy between the right- and left- hand sides. In other words, we are looking for the solution that satisfies 2622: 1848: 1205: 9690: 7038: 3469: 10519: 2901:{\displaystyle S({\boldsymbol {\beta }})=\sum _{i=1}^{n}\left|y_{i}-\sum _{j=1}^{p}X_{ij}\beta _{j}\right|^{2}=\left\|\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right\|^{2}.} 15308: 14469: 12977:

Residuals against explanatory variables not in the model. Any relation of the residuals to these variables would suggest considering these variables for inclusion in the model.

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These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose

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growth should depend linearly on the changes in the unemployment rate. Here the ordinary least squares method is used to construct the regression line describing this law.

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tests whether there is any evidence of serial correlation between the residuals. As a rule of thumb, the value smaller than 2 will be an evidence of positive correlation.

13439: 13339: 13223: 13007: 12326: 12284: 12242: 1760: 1556: 1488: 1231: 11829:. If the calculated F-value is found to be large enough to exceed its critical value for the pre-chosen level of significance, the null hypothesis is rejected and the 2577: 1695: 13288: 12472: 5895: 2113: 1373: 12773: 12746: 11561: 11211: 8512:. This theorem establishes optimality only in the class of linear unbiased estimators, which is quite restrictive. Depending on the distribution of the error terms 5634: 5607: 1336: 913: 733: 13042:

an exact conversion. The original inches can be recovered by Round(x/0.0254) and then re-converted to metric without rounding. If this is done the results become:

7458: 6836:, and thus the system is exactly identified. This is the so-called classical GMM case, when the estimator does not depend on the choice of the weighting matrix. 5081: 14086: 14012: 13965: 13918: 13402: 13382: 13263: 13243: 7367: 5109: 2933: 2733: 2597: 2024: 2001: 1634: 1580: 1508: 1282: 1183: 782: 706: 614: 7465:

which makes all the assumptions listed earlier simpler and easier to interpret. Also this framework allows one to state asymptotic results (as the sample size

6228: 12748:, designed to penalize for the excess number of regressors which do not add to the explanatory power of the regression. This statistic is always smaller than 4696:

It is common to assess the goodness-of-fit of the OLS regression by comparing how much the initial variation in the sample can be reduced by regressing onto

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models, where exogeneity is assumed only with respect to the past shocks but not the future ones), then these estimators will be biased in finite samples.

3251:{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\operatorname {T} }\mathbf {X} \right)^{-1}\mathbf {X} ^{\operatorname {T} }\mathbf {y} .} 15301: 12903:

are both used for model selection. Generally when comparing two alternative models, smaller values of one of these criteria will indicate a better model.

13444: 10442:. However, generally we also want to know how close those estimates might be to the true values of parameters. In other words, we want to construct the 10138:

is invertible. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails,

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The properties listed so far are all valid regardless of the underlying distribution of the error terms. However, if you are willing to assume that the

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One of the lines of difference in interpretation is whether to treat the regressors as random variables, or as predefined constants. In the first case (

3043:{\displaystyle \left(\mathbf {X} ^{\operatorname {T} }\mathbf {X} \right){\hat {\boldsymbol {\beta }}}=\mathbf {X} ^{\operatorname {T} }\mathbf {y} \ .} 12949:

indicates probability that the hypothesis is indeed true. Note that when errors are not normal this statistic becomes invalid, and other tests such as

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is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The

11526:{\displaystyle \left({\hat {y}}_{0}-y_{0}\right)\ {\xrightarrow {d}}\ {\mathcal {N}}\left(0,\;\sigma ^{2}x_{0}^{\mathrm {T} }Q_{xx}^{-1}x_{0}\right),} 7764: 14238:{\displaystyle A={\begin{bmatrix}1&-0.731354\\1&-0.707107\\1&-0.615661\\1&\ 0.052336\\1&0.309017\\1&0.438371\end{bmatrix}}} 9160:{\displaystyle {\hat {\beta }}^{(j)}-{\hat {\beta }}=-{\frac {1}{1-h_{j}}}(X^{\mathrm {T} }X)^{-1}x_{j}^{\mathrm {T} }{\hat {\varepsilon }}_{j}\,,} 8726: 5123:

of data on regressors must contain a column vector of ones to represent the constant whose coefficient is the regression intercept. In that case,

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Sometimes the variables and corresponding parameters in the regression can be logically split into two groups, so that the regression takes form

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We can use the least square mechanism to figure out the equation of a two body orbit in polar base co-ordinates. The equation typically used is

15370: 15294: 1867: 10577: 8709:, this result establishes optimality among both linear and non-linear estimators, but only in the case of normally distributed error terms. 7183: 6084: 3819:{\displaystyle {\hat {\beta }}=\operatorname {argmin} _{b\in \mathbb {R} ^{p}}S(b)=(X^{\operatorname {T} }X)^{-1}X^{\operatorname {T} }y\ .} 11176:{\displaystyle ({\hat {\sigma }}^{2}-\sigma ^{2})\ {\xrightarrow {d}}\ {\mathcal {N}}\left(0,\;\operatorname {E} \left-\sigma ^{4}\right).} 13748: 6319:{\displaystyle {\hat {\mathbf {r} }}:=\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}}=\mathbf {K} {\hat {\boldsymbol {\gamma }}}.} 11274: 9931:. In this case least squares estimation is equivalent to minimizing the sum of squared residuals of the model subject to the constraint 4264:{\displaystyle {\hat {\varepsilon }}=y-{\hat {y}}=y-X{\hat {\beta }}=My=M(X\beta +\varepsilon )=(MX)\beta +M\varepsilon =M\varepsilon .} 15379: 14495: 12784: 8207:{\displaystyle {\widehat {\operatorname {s.\!e.} }}({\hat {\beta }}_{j})={\sqrt {s^{2}\left(X^{\operatorname {T} }X\right)_{jj}^{-1}}}} 2705:{\displaystyle {\hat {\boldsymbol {\beta }}}={\underset {\boldsymbol {\beta }}{\operatorname {arg\,min} }}\,S({\boldsymbol {\beta }}),} 2125: 9858: 9495: 8418: 6957:. For practical purposes, this distinction is often unimportant, since estimation and inference is carried out while conditioning on 6843:

implies a far richer set of moment conditions than stated above. In particular, this assumption implies that for any vector-function

6126: 621: 15384: 10422:. Such a matrix can always be found, although generally it is not unique. The second formula coincides with the first in case when 3098: 3056: 435: 15817: 14330: 12707:

indicating goodness-of-fit of the regression. This statistic will be equal to one if fit is perfect, and to zero when regressors

7344:, cluster samples, hierarchical data, repeated measures data, longitudinal data, and other data with dependencies. In such cases 5964: 5177: 787: 345: 12607: 12024:

is strong and can be approximated as a quadratic function. OLS can handle non-linear relationships by introducing the regressor

17: 13134: 11833:, that the regression has explanatory power, is accepted. Otherwise, the null hypothesis of no explanatory power is accepted. 546:(rank condition), consistent for the variance estimate of the residuals when regressors have finite fourth moments and—by the 15277: 15251: 15232: 15213: 10816: 6700:. From the properties of MLE, we can infer that the OLS estimator is asymptotically efficient (in the sense of attaining the 2033: 15352: 6997: 5130:

The variance in the prediction of the independent variable as a function of the dependent variable is given in the article

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matrix of X. This formulation highlights the point that estimation can be carried out if, and only if, there is no perfect

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indicate that the null hypothesis can be rejected and that the corresponding coefficient is not zero. The second column,

9659: 5770:{\displaystyle {\hat {\beta }}={\rm {arg}}\min _{\beta }\,\lVert \mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\rVert ,} 3627:, denoting the values of all the independent variables associated with a particular value of the dependent variable, are 15692: 15342: 10701:{\displaystyle ({\hat {\beta }}-\beta )\ {\xrightarrow {d}}\ {\mathcal {N}}{\big (}0,\;\sigma ^{2}Q_{xx}^{-1}{\big )},} 8079:-th diagonal element of this matrix. The estimate of this standard error is obtained by replacing the unknown quantity 7162:

is finite and positive semi-definite. When this assumption is violated the regressors are called linearly dependent or

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Usually, it is also assumed that the regressors have finite moments up to at least the second moment. Then the matrix

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The classical model focuses on the "finite sample" estimation and inference, meaning that the number of observations

3922: 8380: 6164: 15418: 14892: 8691:{\displaystyle {\hat {\beta }}\ \sim \ {\mathcal {N}}{\big (}\beta ,\ \sigma ^{2}(X^{\mathrm {T} }X)^{-1}{\big )}.} 7361: 3136: 299: 13835: 8886: 8555: 8345: 8220: 7889: 7722: 1445: 14500: 4610: 3892: 350: 288: 108: 83: 12933:

tries to test the hypothesis that all coefficients (except the intercept) are equal to zero. This statistic has

7301:. If the errors have infinite variance then the OLS estimates will also have infinite variance (although by the 7052:. The exogeneity assumption is critical for the OLS theory. If it holds then the regressor variables are called 15888: 15654: 7690: 6974: 6930: 551: 210: 14519: 3605:{\displaystyle S(b)=\sum _{i=1}^{n}(y_{i}-x_{i}^{\operatorname {T} }b)^{2}=(y-Xb)^{\operatorname {T} }(y-Xb),} 14397: 14091: 11801: 10150:

identifiable, in which case one would like to find the formula for the estimator. The estimator is equal to

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framework depends mostly on the nature of data in hand, and on the inference task which has to be performed.

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which is equivalent to regression on a constant; it simply subtracts the mean from a variable.) In order for

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OLS estimation can be viewed as a projection onto the linear space spanned by the regressors. (Here each of

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values is still possible for new values of the regressors that lie in the same linearly dependent subspace.

6689: 4702: 579: 428: 10488: 15867: 15682: 14485: 12885: 12466: 12393: 8509: 5825: 2625: 371: 14914: 14436: 1047:{\displaystyle y_{i}=\beta _{1}\ x_{i1}+\beta _{2}\ x_{i2}+\cdots +\beta _{p}\ x_{ip}+\varepsilon _{i},} 15362: 13674:{\displaystyle \cos(\theta -\theta _{0})=\cos(\theta )\cos(\theta _{0})+\sin(\theta )\sin(\theta _{0})} 13026: 13012:

Residuals against the preceding residual. This plot may identify serial correlations in the residuals.

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is "large enough" so that the true distribution of the OLS estimator is close to its asymptotic limit.

8027:{\displaystyle \operatorname {Var} =\sigma ^{2}\left(X^{\operatorname {T} }X\right)^{-1}=\sigma ^{2}Q.} 7878: 7472:), which are understood as a theoretical possibility of fetching new independent observations from the 7305:

they will nonetheless tend toward the true values so long as the errors have zero mean). In this case,

7058: 3668: 340: 309: 236: 6066:{\displaystyle (\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})\cdot \mathbf {X} \mathbf {v} } 5789: 1769: 1341: 1287: 1236: 1139: 738: 15707: 15534: 15498: 15467: 14480: 14318:{\displaystyle b={\begin{bmatrix}0.21220\\0.21958\\0.24741\\0.45071\\0.52883\\0.56820\end{bmatrix}}.} 11342: 10784: 10551: 10459: 8997:-th observation and consider how much the estimated quantities are going to change (similarly to the 8945: 7345: 5842: 5523: 4681: 2628: 1126:{\displaystyle y_{i}=\mathbf {x} _{i}^{\operatorname {T} }{\boldsymbol {\beta }}+\varepsilon _{i},\,} 330: 319: 283: 190: 14044: 14017: 13970: 13923: 11569: 11339:. Clearly the predicted response is a random variable, its distribution can be derived from that of 6158:, that is, the variance of the residuals is the minimum possible. This is illustrated at the right. 15687: 15565: 15529: 15457: 15347: 15329: 14490: 13716: 13684: 13527: 13032: 12690: 11866: 8717: 8706: 8326: 7672: 7542: 7502: 7045: 6852: 5669:

For mathematicians, OLS is an approximate solution to an overdetermined system of linear equations

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Suppose it is known that the coefficients in the regression satisfy a system of linear equations

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for the model, and thus is optimal in the class of all unbiased estimators. Note that unlike the

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will always be a number between 0 and 1, with values close to 1 indicating a good degree of fit.

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The following data set gives average heights and weights for American women aged 30–39 (source:

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These moment conditions state that the regressors should be uncorrelated with the errors. Since

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is defined as a ratio of "explained" variance to the "total" variance of the dependent variable

1738: 15893: 15781: 15607: 15472: 12581:{\displaystyle {\hat {\sigma }}_{j}=\left({\hat {\sigma }}^{2}\left_{jj}\right)^{\frac {1}{2}}} 11830: 7298: 6984:. The linear functional form must coincide with the form of the actual data-generating process. 6922: 5951:{\displaystyle (\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})^{\top }\mathbf {X} =0.} 4675:

is used more often, since it is more convenient for the hypothesis testing. The square root of

1535: 1467: 1410:{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\,} 1210: 1058: 520: 278: 215: 14691: 14664: 14637: 14572: 13225:

is the radius of how far the object is from one of the bodies. In the equation the parameters

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goes to infinity. While the sample size is necessarily finite, it is customary to assume that

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First we need to represent e and p in a linear form. So we are going to rewrite the equation

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is just a certain linear combination of the vectors of regressors. Thus, the residual vector

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Such a system usually has no exact solution, so the goal is instead to find the coefficients

1859: 484: 366: 62: 9223:-th observation resulting from omitting that observation from the dataset will be equal to 2092: 15827: 15766: 15753: 15702: 15602: 15524: 15503: 15477: 15223:

Heij, Christiaan; Boer, Paul; Franses, Philip H.; Kloek, Teun; van Dijk, Herman K. (2004).

12775:, can decrease as new regressors are added, and even be negative for poorly fitting models: 12751: 12724: 12697:-values smaller than 0.05 are taken as evidence that the population coefficient is nonzero. 12141: 11902: 11539: 11189: 10545: 7755: 7454: 7302: 7077: 7056:. If it does not, then those regressors that are correlated with the error term are called 6832:-vector, the number of moment conditions is equal to the dimension of the parameter vector 5869: 5658: 5612: 5585: 4996: 4713:, in the cases where the regression sum of squares equals the sum of squares of residuals: 2958: 2116: 1314: 891: 711: 535: 512: 496: 480: 386: 376: 257: 225: 180: 159: 67: 12715:, and will never decrease if additional regressors are added, even if they are irrelevant. 8922:, the fact which comes in useful when constructing the t- and F-tests for the regression. 7040:

The immediate consequence of the exogeneity assumption is that the errors have mean zero:

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between the explanatory variables (which would cause the gram matrix to have no inverse).

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tell us how much of the initial variation in the sample were explained by the regression.

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Using this asymptotic distribution, approximate two-sided confidence intervals for the

9824: 8851: 7596: 7578: 7294: 7270: 5094: 5059: 4668: 4663:. The two estimators are quite similar in large samples; the first estimator is always 2918: 2718: 2582: 2009: 1986: 1619: 1565: 1493: 1267: 1168: 767: 691: 599: 504: 409: 138: 123: 12131:{\displaystyle w_{i}=\beta _{1}+\beta _{2}h_{i}+\beta _{3}h_{i}^{2}+\varepsilon _{i}.} 7758:, meaning that their expected values coincide with the true values of the parameters: 7348:

provides a better alternative than the OLS. Another expression for autocorrelation is

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is the unknown. Assuming the system cannot be solved exactly (the number of equations

15852: 15812: 15539: 15449: 15440: 15273: 15247: 15228: 15209: 15183: 15155: 15075: 15023: 14983: 14972: 14766: 14697: 14670: 14663:

Hofmann-Wellenhof, Bernhard; Lichtenegger, Herbert; Wasle, Elmar (20 November 2007).

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are used to determine the path of the orbit. We have measured the following data.

15508: 15375: 9847: 8998: 8377:(BLUE). Efficiency should be understood as if we were to find some other estimator 7278: 5112: 4066: 616: 220: 149: 13830:. We use the original two-parameter form to represent our observational data as: 13513:{\displaystyle {\frac {1}{r(\theta )}}={\frac {1}{p}}-{\frac {e}{p}}\cos(\theta )} 12711:

have no explanatory power whatsoever. This is a biased estimate of the population

8873:, which even beats the Cramér–Rao bound in case when there is only one regressor ( 2543:

contains information on the data points. The first column is populated with ones,

448: 15791: 15697: 15638: 15633: 14967: 11822: 11031:

is also consistent and asymptotically normal (provided that the fourth moment of

7600: 7314: 7254: 6329:

The equation and solution of linear least squares are thus described as follows:

5961:

A geometrical interpretation of these equations is that the vector of residuals,

5829: 5800: 3437:, and thus assesses the degree of fit between the actual data and the model. The 2003: 563: 492: 381: 88: 12600:

columns are testing whether any of the coefficients might be equal to zero. The

15735: 15201: 12196: 11842: 10435: 7859:{\displaystyle \operatorname {E} =\beta ,\quad \operatorname {E} =\sigma ^{2}.} 5578: 4664: 3662: 456: 133: 11837:

coefficient is zero. This hypothesis is tested by computing the coefficient's

10540:

We can show that under the model assumptions, the least squares estimator for

15882: 15807: 15337: 15317: 14606: 13103: 11214: 10134:

This expression for the constrained estimator is valid as long as the matrix

8975:, meaning that it represents a linear combination of the dependent variables 8801:{\displaystyle s^{2}\ \sim \ {\frac {\sigma ^{2}}{n-p}}\cdot \chi _{n-p}^{2}} 8339: 7632: 7462: 6961:. All results stated in this article are within the random design framework. 3130: 1559: 559: 500: 452: 252: 128: 10142:

will not be identifiable. However it may happen that adding the restriction

8373:

is efficient in the class of linear unbiased estimators. This is called the

1766:. Without the intercept, the fitted line is forced to cross the origin when 503:: minimizing the sum of the squares of the differences between the observed 14666:

GNSS – Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more

12945:) distribution under the null hypothesis and normality assumption, and its 11853:

difference between the two subsets is rejected; otherwise, it is accepted.

10449:

Since we have not made any assumption about the distribution of error term

8986:, and generally are unequal. The observations with high weights are called 8335: 6977:

of OLS, and in which the behavior at a large number of samples is studied.

6697: 6002: 3908: 2027: 543: 118: 14544: 13094:

Residuals to a quadratic fit for correctly and incorrectly converted data.

8990:

because they have a more pronounced effect on the value of the estimator.

7428:{\displaystyle \varepsilon \mid X\sim {\mathcal {N}}(0,\sigma ^{2}I_{n}).} 2915:

below. This minimization problem has a unique solution, provided that the

15395: 14545:"What is a complete list of the usual assumptions for linear regression?" 12592: 12201: 12020: 11906: 11838: 8982:. The weights in this linear combination are functions of the regressors 7870: 7337: 7319: 7085: 6010: 5889:

In other words, the gradient equations at the minimum can be written as:

5832: 3841: 3092: 1973:{\displaystyle \sum _{j=1}^{p}x_{ij}\beta _{j}=y_{i},\ (i=1,2,\dots ,n),} 164: 113: 27:

Method for estimating the unknown parameters in a linear regression model

13090: 11789: 9730:

will be numerically identical to the residuals and the OLS estimate for

7440:(MLE), and therefore it is asymptotically efficient in the class of all 11825:

of no explanatory value of the estimated regression is tested using an

7341: 7265:

is a parameter which determines the variance of each observation. This

6954: 4020: 3406:-th observation, measures the vertical distance between the data point 3163: 468: 7293:

in each observation. When this requirement is violated this is called

6116:{\displaystyle \mathbf {y} -\mathbf {X} {\boldsymbol {\hat {\beta }}}} 5242:

The least squares estimates in this case are given by simple formulas

15015: 12950: 11849: 6693: 3620: 539: 528: 15182:(2nd ed.). New York: Oxford University Press. pp. 48–113. 13823:{\displaystyle \tan \theta _{0}=\sin(\theta _{0})/\cos(\theta _{0})} 13017:

woman can be predicted with high accuracy based only on her height.

11909:

of the data, the relationship is slightly curved but close to linear

11428: 11094: 10626: 8516:, other, non-linear estimators may provide better results than OLS. 6973:

is fixed. This contrasts with the other approaches, which study the

15246:(3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 8–47. 14893:"Assumptions of multiple regression: Correcting two misconceptions" 8993:

To analyze which observations are influential we remove a specific

8549:), then additional properties of the OLS estimators can be stated. 3665:, and therefore this function possesses a unique global minimum at 571: 15227:(1st ed.). Oxford: Oxford University Press. pp. 76–115. 14662: 11332:{\displaystyle {\hat {y}}_{0}=x_{0}^{\mathrm {T} }{\hat {\beta }}} 9219:-th observation. Similarly, the change in the predicted value for 7457:, an additional assumption is imposed — that all observations are 523:

can be expressed by a simple formula, especially in the case of a

13521: 12865:{\displaystyle {\overline {R}}^{2}=1-{\frac {n-1}{n-p}}(1-R^{2})} 12683: 12206: 8583:

is normally distributed, with mean and variance as given before:

7869:

If the strict exogeneity does not hold (as is the case with many

1820:

in the second regressor, but none-the-less is still considered a

508: 8842:. However it was shown that there are no unbiased estimators of 7448: 2153:{\displaystyle \mathbf {X} {\boldsymbol {\beta }}=\mathbf {y} ,} 1639:

Typically, a constant term is included in the set of regressors

15225:

Econometric Methods with Applications in Business and Economics

15022:(Second ed.). New York: J. Wiley & Sons. p. 319. 11826: 11536:

which allows construct confidence intervals for mean response

10456:, it is impossible to infer the distribution of the estimators 9895:{\displaystyle A\colon \quad Q^{\operatorname {T} }\beta =c,\,} 9560:{\displaystyle y=X_{1}\beta _{1}+X_{2}\beta _{2}+\varepsilon ,} 8498:{\displaystyle \operatorname {Var} -\operatorname {Var} \geq 0} 6151:{\displaystyle \mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}} 5152:

contains only two variables, a constant and a scalar regressor

681:{\displaystyle \left\{\mathbf {x} _{i},y_{i}\right\}_{i=1}^{n}} 12028:. The regression model then becomes a multiple linear model: 6241:), the residual vector should satisfy the following equation: 15272:(4th ed.). Mason, OH: Cengage Learning. pp. 22–67. 15242:

Hill, R. Carter; Griffiths, William E.; Lim, Guay C. (2008).

15208:(Fifth ed.). Boston: McGraw-Hill Irwin. pp. 55–96. 7048:), and that the regressors are uncorrelated with the errors: 3122:{\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {y} } 3080:{\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {X} } 14014:

is constructed by the first column being the coefficient of

12674:

follows a Student-t distribution. Under weaker conditions,

9001:). It can be shown that the change in the OLS estimator for 7360:. It is sometimes additionally assumed that the errors have 2624:

which fit the equations "best", in the sense of solving the

14862: 14860: 14823: 14821: 14384:{\displaystyle {\binom {x}{y}}={\binom {0.43478}{0.30435}}} 7617:} is nonstationary, OLS results are often spurious unless { 5994:{\displaystyle \mathbf {y} -X{\hat {\boldsymbol {\beta }}}} 5232:{\displaystyle y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}.} 4053:; this is a projection matrix onto the space orthogonal to 877:{\displaystyle \mathbf {x} _{i}=\left^{\operatorname {T} }} 567: 14890: 14715: 14713: 14636:

Ghilani, Charles D.; Paul r. Wolf, Ph. D. (12 June 2006).

12663:{\displaystyle t={\hat {\beta }}_{j}/{\hat {\sigma }}_{j}} 5570:; the only difference is in how we interpret this result. 15316: 14891:

Williams, M. N; Grajales, C. A. G; Kurkiewicz, D (2013).

7062:, and the OLS estimator becomes biased. In such case the 6713: 4667:, while the second estimator is biased but has a smaller 460: 14857: 14818: 13189:{\displaystyle r(\theta )={\frac {p}{1-e\cos(\theta )}}} 7289:, which means that the error term has the same variance 6704:

for variance) if the normality assumption is satisfied.

2911:

A justification for choosing this criterion is given in

2521:(Note: for a linear model as above, not all elements in 1490:

vectors of the response variables and the errors of the

538:

for the level-one fixed effects when the regressors are

14710: 12452:

column gives the least squares estimates of parameters

10988:{\displaystyle \beta _{j}\in {\bigg _{jj}}}\ {\bigg ]}} 7166:. In such case the value of the regression coefficient 5868:

in this case can be interpreted as the coefficients of

2082:{\displaystyle \beta _{1},\beta _{2},\dots ,\beta _{p}} 1367:. This model can also be written in matrix notation as 14263: 14144: 13121: 9699: 8890: 8559: 8384: 8349: 8224: 8044: 8037:

In particular, the standard error of each coefficient

7893: 7726: 7239:{\displaystyle \operatorname {Var} =\sigma ^{2}I_{n},} 6541: 6496: 6444: 6388: 6360: 5062: 5006: 4628: 2579:. Only the other columns contain actual data. So here 2449: 2376: 2186: 1165:, as introduced previously, is a column vector of the 574:. Under the additional assumption that the errors are 15145: 14439: 14400: 14333: 14251: 14132: 14094: 14074: 14047: 14020: 14000: 13973: 13953: 13926: 13906: 13838: 13751: 13719: 13687: 13562: 13530: 13447: 13418: 13390: 13370: 13318: 13276: 13251: 13231: 13202: 13137: 12986: 12787: 12754: 12727: 12610: 12475: 12307: 12265: 12223: 12037: 12019:

When only one dependent variable is being modeled, a

11572: 11542: 11377: 11345: 11277: 11223: 11192: 11047: 10819: 10787: 10717: 10593: 10554: 10491: 10462: 10159: 9948: 9861: 9746: 9698: 9669: 9498: 9232: 9014: 8948: 8889: 8729: 8592: 8558: 8421: 8383: 8348: 8257: 8223: 8096: 8043: 7926: 7892: 7767: 7725: 7370: 7186: 7094: 7000: 6929:. This approach allows for more natural study of the 6873:, which results in the moment equation posted above. 6729: 6338: 6250: 6200: 6167: 6129: 6087: 6018: 5967: 5898: 5845: 5700: 5615: 5588: 5526: 5251: 5180: 5097: 4722: 4627: 4293: 4131: 3925: 3709: 3671: 3472: 3270: 3175: 3139: 3101: 3059: 2974: 2941: 2921: 2744: 2721: 2640: 2608: 2585: 2549: 2527: 2172: 2128: 2095: 2036: 2012: 1989: 1870: 1834: 1772: 1741: 1703: 1667: 1645: 1622: 1609:{\displaystyle \mathbf {x} _{i}^{\operatorname {T} }} 1588: 1568: 1538: 1516: 1496: 1470: 1448: 1426: 1376: 1344: 1317: 1290: 1270: 1239: 1213: 1191: 1171: 1142: 1070: 924: 894: 790: 770: 741: 714: 694: 624: 602: 566:. Under these conditions, the method of OLS provides 515:. Some sources consider OLS to be linear regression. 507:(values of the variable being observed) in the input 15222: 12689:, expresses the results of the hypothesis test as a 10415:) matrix such that the matrix is non-singular, and 9806:{\displaystyle M_{1}y=M_{1}X_{2}\beta _{2}+\eta \,,} 7461:. This means that all observations are taken from a 7336:. This assumption may be violated in the context of 6964: 6892:

There are several different frameworks in which the

6839:

Note that the original strict exogeneity assumption

5520:

In the previous section the least squares estimator

582:

that outperforms any non-linear unbiased estimator.

14635: 14520:"The Origins of Ordinary Least Squares Assumptions" 13364:We need to find the least-squares approximation of 8248:is uncorrelated with the residuals from the model: 7297:, in such case a more efficient estimator would be 4274:Using these residuals we can estimate the value of 1636:-th observations on all the explanatory variables. 15147: 14971: 14463: 14425: 14383: 14317: 14237: 14118: 14080: 14060: 14033: 14006: 13986: 13959: 13939: 13912: 13892: 13822: 13737: 13705: 13673: 13548: 13512: 13433: 13396: 13376: 13333: 13282: 13257: 13237: 13217: 13188: 13001: 12864: 12767: 12740: 12662: 12580: 12320: 12278: 12236: 12130: 11841:, as the ratio of the coefficient estimate to its 11755: 11555: 11525: 11360: 11331: 11259: 11205: 11175: 10987: 10802: 10752: 10700: 10569: 10513: 10477: 10388: 10123: 9894: 9805: 9722: 9684: 9559: 9420: 9159: 8963: 8906: 8800: 8690: 8575: 8497: 8400: 8365: 8313: 8240: 8206: 8067: 8026: 7909: 7858: 7742: 7427: 7238: 7144: 7032: 6810: 6668: 6318: 6222: 6182: 6150: 6115: 6065: 5993: 5950: 5860: 5769: 5628: 5601: 5541: 5504: 5231: 5103: 5075: 5048: 4983: 4651: 4590: 4263: 3970: 3818: 3692: 3604: 3353: 3250: 3154: 3121: 3079: 3042: 2949: 2927: 2900: 2727: 2704: 2616: 2591: 2571: 2535: 2510: 2152: 2107: 2081: 2018: 1995: 1972: 1842: 1800: 1754: 1727: 1689: 1653: 1628: 1608: 1574: 1550: 1524: 1502: 1482: 1456: 1434: 1409: 1359: 1338:from sources other than the explanatory variables 1330: 1303: 1276: 1252: 1225: 1199: 1185:-th observation of all the explanatory variables; 1177: 1157: 1125: 1046: 907: 876: 776: 756: 727: 700: 680: 608: 552:optimal in the class of linear unbiased estimators 15106: 15094: 15074:. New York: Oxford University Press. p. 33. 15069: 15020:Linear Statistical Inference and its Applications 14375: 14362: 14350: 14337: 13868: 13855: 11673: 10980: 10912: 10835: 10343: 10255: 10070: 10020: 9215:is the vector of regressors corresponding to the 8846:with variance smaller than that of the estimator 8107: 7098: 6107: 5137: 4652:{\displaystyle \scriptstyle {\hat {\sigma }}^{2}} 15880: 15241: 12670:. If the errors ε follow a normal distribution, 11260:{\displaystyle y_{0}=x_{0}^{\mathrm {T} }\beta } 9723:{\displaystyle \scriptstyle {\hat {\beta }}_{2}} 8068:{\displaystyle \scriptstyle {\hat {\beta }}_{j}} 7095: 6933:of the estimators. In the other interpretation ( 6707: 5730: 3373:is a "candidate" value for the parameter vector 2599:is equal to the number of regressors plus one). 1558:matrix of regressors, also sometimes called the 15070:Davidson, Russell; MacKinnon, James G. (1993). 14897:Practical Assessment, Research & Evaluation 14041:and the second column being the coefficient of 10753:{\displaystyle Q_{xx}=X^{\operatorname {T} }X.} 9939:estimator can be given by an explicit formula: 8217:It can also be easily shown that the estimator 6716:case the OLS estimator can also be viewed as a 3971:{\displaystyle {\hat {y}}=X{\hat {\beta }}=Py,} 3162:is the coefficient vector of the least-squares 1311:accounts for the influences upon the responses 511:and the output of the (linear) function of the 15196: 14639:Adjustment Computations: Spatial Data Analysis 8401:{\displaystyle \scriptstyle {\tilde {\beta }}} 6183:{\displaystyle {\hat {\boldsymbol {\gamma }}}} 3700:, which can be given by the explicit formula: 1762:corresponding to this regressor is called the 531:on the right side of the regression equation. 15302: 14950: 14948: 14946: 14745: 10690: 10645: 9663:states that in this regression the residuals 8680: 8623: 7449:Independent and identically distributed (iid) 7131: 7101: 7080:. Mathematically, this means that the matrix 6851:will hold. However it can be shown using the 6797: 6737: 6720:estimator arising from the moment conditions 3155:{\displaystyle {\hat {\boldsymbol {\beta }}}} 1233:vector of unknown parameters; and the scalar 429: 15270:Introductory Econometrics: A Modern Approach 15146:Burnham, Kenneth P.; David Anderson (2002). 15129: 15127: 14581:. New York: John Wiley & Sons. pp. 13893:{\displaystyle A^{T}A{\binom {x}{y}}=A^{T}b} 8907:{\displaystyle \scriptstyle {\hat {\beta }}} 8576:{\displaystyle \scriptstyle {\hat {\beta }}} 8366:{\displaystyle \scriptstyle {\hat {\beta }}} 8241:{\displaystyle \scriptstyle {\hat {\beta }}} 7910:{\displaystyle \scriptstyle {\hat {\beta }}} 7743:{\displaystyle \scriptstyle {\hat {\beta }}} 5761: 5740: 1853: 1457:{\displaystyle {\boldsymbol {\varepsilon }}} 15065: 15063: 11027:Similarly, the least squares estimator for 8925: 8811:The variance of this estimator is equal to 7710: 7476:. The list of assumptions in this case is: 6990:. The errors in the regression should have 5687:is much larger than the number of unknowns 15309: 15295: 15260: 14943: 14797: 14760: 14567: 14496:Numerical methods for linear least squares 13020: 12678:is asymptotically normal. Large values of 11456: 11122: 10656: 10438:of the linear regression model parameters 10429: 8334:assumption (that is, the errors should be 7170:cannot be learned, although prediction of 6949:is sampled conditionally on the values of 5515: 4577: 4024:because it "puts a hat" onto the variable 3463:)) is a measure of the overall model fit: 915:, is a linear function of the regressors: 436: 422: 15177: 15150:Model Selection and Multi-Model Inference 15124: 14763:Standard Mathematical Tables and Formulae 14613:. Princeton University Press. p. 15. 11873:The World Almanac and Book of Facts, 1975 11856: 9891: 9841: 9799: 9484: 9398: 9153: 8485: 8466: 8450: 8431: 8301: 8267: 7955: 7936: 7836: 7819: 7796: 7777: 7693:, with a finite matrix of second moments 7206: 7196: 7128: 7106: 7020: 7010: 6911:are random and sampled together with the 6794: 6742: 5739: 5479: 3738: 2684: 2668: 1406: 1122: 491:model (with fixed level-one effects of a 15072:Estimation and Inference in Econometrics 15060: 14693:GPS: Theory, Algorithms and Applications 13089: 12960: 12140: 11901: 6941:are treated as known constants set by a 6123:is the shortest of all possible vectors 5577: 5049:{\textstyle L=I_{n}-{\frac {1}{n}}J_{n}} 1260:represents unobserved random variables ( 447: 15818:Numerical smoothing and differentiation 15133: 15118: 15054: 15042: 15002: 14966: 14954: 14937: 14878: 14866: 14851: 14839: 14827: 14812: 14785: 14719: 14623: 14605: 14426:{\displaystyle p={\frac {1}{x}}=2.3000} 14119:{\displaystyle {\frac {1}{r(\theta )}}} 11863:Simple linear regression § Example 9431:From the properties of the hat matrix, 9204:-th diagonal element of the hat matrix 8942:As was mentioned before, the estimator 8314:{\displaystyle \operatorname {Cov} =0.} 7459:independent and identically distributed 6466: 6450: 6410: 6394: 6304: 6284: 6171: 6144: 6104: 6038: 5982: 5918: 5757: 5654:This section may need to be cleaned up. 5636:refers to a column of the data matrix.) 3642:which minimizes this sum is called the 3344: 3287: 3274: 3179: 3143: 3005: 2880: 2752: 2692: 2679: 2644: 2610: 2364: 2135: 1836: 1450: 1399: 1391: 1193: 1102: 570:estimation when the errors have finite 14: 15881: 14734:Practical Regression and Anova using R 13075:Converted to metric without rounding. 7453:In some applications, especially with 6811:{\displaystyle \mathrm {E} {\big }=0.} 6688:The OLS estimator is identical to the 3861: 3133:of regressand by regressors. Finally, 2617:{\displaystyle {\boldsymbol {\beta }}} 1843:{\displaystyle {\boldsymbol {\beta }}} 1200:{\displaystyle {\boldsymbol {\beta }}} 15290: 14765:. Chapman&Hall/CRC. p. 626. 12980:Residuals against the fitted values, 11774: 11014:of standard normal distribution, and 9685:{\displaystyle {\hat {\varepsilon }}} 8519: 7033:{\displaystyle \operatorname {E} =0.} 6683: 4028:. Another matrix, closely related to 15353:Iteratively reweighted least squares 14978:. Harvard University Press. p. 13126: 11784: 10514:{\displaystyle {\hat {\sigma }}^{2}} 9925:×1 vector of known constants, where 7571: 6888:Linear regression § Assumptions 6855:that the optimal choice of function 5639: 15014: 14746:Kenney, J.; Keeping, E. S. (1963). 13122:Another example with less real data 13061:Converted to metric with rounding. 12913:, standard error of the error term. 12604:-statistic is calculated simply as 11867:Linear least squares § Example 11038:exists) with limiting distribution 7066:may be used to carry out inference. 5820:will have the smallest length when 24: 15371:Pearson product-moment correlation 15171: 14689: 14464:{\displaystyle e=p\cdot y=0.70001} 14366: 14341: 13859: 12721:is a slightly modified version of 11706: 11651: 11605: 11478: 11440: 11311: 11248: 11123: 11106: 11024:-th diagonal element of a matrix. 10890: 10739: 10638: 10359: 10333: 10323: 10294: 10284: 10242: 10232: 10203: 10193: 10092: 10043: 10030: 9994: 9874: 9343: 9299: 9128: 9095: 8854:) estimator in this class will be 8656: 8616: 8172: 8108: 8101: 7983: 7810: 7768: 7385: 7001: 6781: 6731: 6641: 6610: 6588: 6557: 6194:with the assumption that a matrix 5932: 5724: 5721: 5718: 4975: 4972: 4969: 4964: 4961: 4958: 4931: 4911: 4878: 4855: 4843: 4466: 4428: 4416: 4378: 4325: 3802: 3776: 3576: 3535: 3338: 3307: 3235: 3204: 3109: 3067: 3024: 2987: 2675: 2672: 2669: 2665: 2662: 2659: 1601: 1096: 869: 25: 15905: 14088:is the values for the respective 13520:. Furthermore, one could fit for 10521:. Nevertheless, we can apply the 10434:The least squares estimators are 6965:Classical linear regression model 4093:), and relate to the data matrix 3693:{\displaystyle b={\hat {\beta }}} 2961:, given by solving the so-called 15851: 14798:Akbarzadeh, Vahab (7 May 2014). 13713:and in the extra basis function 11788: 7145:{\displaystyle \Pr \!{\big }=1.} 7064:method of instrumental variables 6655: 6636: 6616: 6605: 6583: 6563: 6552: 6529: 6507: 6500: 6371: 6364: 6344: 6297: 6277: 6269: 6255: 6213: 6205: 6139: 6131: 6097: 6089: 6059: 6054: 6031: 6023: 5969: 5938: 5911: 5903: 5752: 5744: 5644: 4687:standard error of the regression 3333: 3313: 3302: 3241: 3230: 3210: 3199: 3115: 3104: 3073: 3062: 3030: 3019: 2993: 2982: 2943: 2875: 2867: 2529: 2437: 2174: 2143: 2130: 1828:still linear in the parameters ( 1801:{\displaystyle x_{i}={\vec {0}}} 1647: 1591: 1518: 1428: 1386: 1378: 1360:{\displaystyle \mathbf {x} _{i}} 1347: 1304:{\displaystyle \varepsilon _{i}} 1253:{\displaystyle \varepsilon _{i}} 1158:{\displaystyle \mathbf {x} _{i}} 1145: 1086: 793: 757:{\displaystyle \mathbf {x} _{i}} 744: 633: 483:method for choosing the unknown 403: 15178:Dougherty, Christopher (2002). 15139: 15112: 15100: 15088: 15048: 15036: 15008: 14996: 14960: 14931: 14907: 14884: 14872: 14845: 14833: 14806: 14791: 14779: 14754: 14739: 14725: 14690:Xu, Guochang (5 October 2007). 14501:Nonlinear system identification 11361:{\displaystyle {\hat {\beta }}} 10803:{\displaystyle {\hat {\beta }}} 10570:{\displaystyle {\hat {\beta }}} 10478:{\displaystyle {\hat {\beta }}} 9937:constrained least squares (CLS) 9868: 8964:{\displaystyle {\hat {\beta }}} 8075:is equal to square root of the 7809: 5861:{\displaystyle {\hat {\beta }}} 5542:{\displaystyle {\hat {\beta }}} 4536: 3916:) from the regression will be 2912: 2435: 2362: 784:parameters (regressors), i.e., 585: 351:Least-squares spectral analysis 289:Generalized estimating equation 109:Multinomial logistic regression 84:Vector generalized linear model 15107:Davidson & MacKinnon (1993 15095:Davidson & MacKinnon (1993 14683: 14656: 14629: 14617: 14599: 14561: 14537: 14512: 14110: 14104: 14061:{\displaystyle {\frac {e}{p}}} 14034:{\displaystyle {\frac {1}{p}}} 13987:{\displaystyle {\frac {e}{p}}} 13940:{\displaystyle {\frac {1}{p}}} 13817: 13804: 13790: 13777: 13732: 13726: 13700: 13694: 13668: 13655: 13646: 13640: 13628: 13615: 13606: 13600: 13588: 13569: 13543: 13537: 13507: 13501: 13463: 13457: 13428: 13422: 13328: 13322: 13212: 13206: 13180: 13174: 13147: 13141: 12993: 12859: 12840: 12648: 12624: 12511: 12483: 12469:of each coefficient estimate: 11756:{\displaystyle y_{0}\in \left} 11683: 11668: 11656: 11617: 11390: 11352: 11323: 11285: 11083: 11058: 11048: 10922: 10907: 10895: 10850: 10794: 10615: 10603: 10594: 10561: 10499: 10469: 10368: 10351: 10306: 10276: 10215: 10185: 10167: 10115: 10103: 10084: 10052: 10035: 10003: 9986: 9977: 9956: 9707: 9676: 9406: 9354: 9325: 9319: 9312: 9273: 9258: 9252: 9240: 9141: 9105: 9086: 9049: 9035: 9029: 9022: 8955: 8897: 8666: 8647: 8599: 8566: 8486: 8473: 8463: 8451: 8438: 8428: 8391: 8375:best linear unbiased estimator 8356: 8302: 8289: 8274: 8264: 8231: 8143: 8131: 8121: 8052: 7956: 7943: 7933: 7900: 7837: 7816: 7797: 7784: 7774: 7733: 7719:assumption the OLS estimators 7691:martingale difference sequence 7419: 7390: 7364:conditional on the regressors: 7207: 7193: 7119: 7113: 7021: 7007: 6881: 6469: 6453: 6436: 6413: 6397: 6307: 6287: 6259: 6217: 6201: 6174: 6047: 6041: 6019: 5985: 5928: 5921: 5899: 5852: 5707: 5533: 5486: 5458: 5416: 5409: 5387: 5359: 5353: 5331: 5328: 5322: 5300: 5138:Simple linear regression model 4819: 4792: 4778: 4752: 4742: 4691:standard error of the equation 4636: 4611:statistical degrees of freedom 4544: 4516: 4510: 4501: 4373: 4363: 4337: 4317: 4234: 4225: 4219: 4204: 4183: 4159: 4138: 3950: 3932: 3785: 3768: 3762: 3756: 3716: 3684: 3596: 3581: 3572: 3556: 3544: 3509: 3482: 3476: 3277: 3182: 3146: 3008: 2885: 2862: 2756: 2748: 2696: 2688: 2647: 1964: 1934: 1792: 568:minimum-variance mean-unbiased 459:states that in an economy the 13: 1: 15266:"The Simple Regression Model" 14573:"Classical Linear Regression" 14506: 13738:{\displaystyle \sin(\theta )} 13706:{\displaystyle \cos(\theta )} 13549:{\displaystyle \cos(\theta )} 12149:The output from most popular 10584:) and asymptotically normal: 9737:in the following regression: 9470:, and observations with high 6876: 6708:Generalized method of moments 5573: 5119:to be meaningful, the matrix 4018:is also sometimes called the 3364: 2715:where the objective function 596:Suppose the data consists of 527:, in which there is a single 170:Nonlinear mixed-effects model 15841:Regression analysis category 15731:Response surface methodology 15180:Introduction to Econometrics 14750:. van Nostrand. p. 187. 12907:Standard error of regression 12894:Akaike information criterion 12794: 12705:coefficient of determination 10781:-th component of the vector 10762: 8827:, which does not attain the 8716:will be proportional to the 8508:in the sense that this is a 7529:no perfect multicollinearity 7438:maximum likelihood estimator 6690:maximum likelihood estimator 5000:for the dependent variable, 4813: 4772: 4703:coefficient of determination 2950:{\displaystyle \mathbf {X} } 2536:{\displaystyle \mathbf {X} } 1728:{\displaystyle i=1,\dots ,n} 1654:{\displaystyle \mathbf {X} } 1525:{\displaystyle \mathbf {X} } 1435:{\displaystyle \mathbf {y} } 580:maximum likelihood estimator 7: 15713:Frisch–Waugh–Lovell theorem 15683:Mean and predicted response 14474: 13556:with an extra parameter as 13407: 12153:will look similar to this: 9660:Frisch–Waugh–Lovell theorem 8701:This estimator reaches the 8510:nonnegative-definite matrix 7671:is of full rank, and hence 7309:techniques are recommended. 3893:Moore–Penrose pseudoinverse 708:includes a scalar response 578:with zero mean, OLS is the 372:Mean and predicted response 10: 15910: 15363:Correlation and dependence 15244:Principles of Econometrics 15154:(2nd ed.). Springer. 14915:"Memento on EViews Output" 13681:, which is linear in both 13434:{\displaystyle r(\theta )} 13334:{\displaystyle r(\theta )} 13218:{\displaystyle r(\theta )} 13030: 13027:Errors-in-variables models 13024: 13002:{\displaystyle {\hat {y}}} 12321:{\displaystyle \beta _{3}} 12279:{\displaystyle \beta _{2}} 12237:{\displaystyle \beta _{1}} 11860: 11778: 10766: 10529:properties as sample size 9845: 8935: 8929: 7879:variance-covariance matrix 6885: 5141: 4659:, is the MLE estimate for 4617:, is the OLS estimate for 4010:spanned by the columns of 1755:{\displaystyle \beta _{1}} 589: 165:Linear mixed-effects model 15836: 15800: 15749: 15721: 15708:Minimum mean-square error 15675: 15621: 15595:Decomposition of variance 15593: 15558: 15517: 15499:Growth curve (statistics) 15486: 15468:Generalized least squares 15448: 15437: 15404: 15361: 15328: 14748:Mathematics of Statistics 12432: 12416: 12400: 12384: 12370: 12356: 12343: 12212: 12184: 12179: 12171: 12163: 11900: 9917:matrix of full rank, and 8883:Moreover, the estimators 8408:which would be linear in 7346:generalized least squares 5806:. The predicted quantity 4682:regression standard error 3860:, closely related to its 2115:. This can be written in 1854:Matrix/vector formulation 1551:{\displaystyle n\times p} 1483:{\displaystyle n\times 1} 1226:{\displaystyle p\times 1} 888:, the response variable, 331:Least absolute deviations 15566:Generalized linear model 15458:Simple linear regression 15348:Non-linear least squares 15330:Computational statistics 13033:Quantization error model 10578:converges in probability 8926:Influential observations 8718:chi-squared distribution 7715:First of all, under the 7711:Finite sample properties 7543:positive-definite matrix 7164:perfectly multicollinear 7046:law of total expectation 5656:It has been merged from 5144:Simple linear regression 5132:Polynomial least squares 3902:After we have estimated 3440:sum of squared residuals 2572:{\displaystyle X_{i1}=1} 1824:model because the model 1690:{\displaystyle x_{i1}=1} 525:simple linear regression 79:Generalized linear model 14761:Zwillinger, D. (1995). 14732:Julian Faraway (2000), 14491:Nonlinear least squares 14486:Fama–MacBeth regression 13283:{\displaystyle \theta } 13021:Sensitivity to rounding 12925:residual sum of squares 12886:Durbin–Watson statistic 11857:Example with real data 10430:Large sample properties 8932:Influential observation 7501:from, and has the same 7474:data generating process 6894:linear regression model 6847:, the moment condition 5516:Alternative derivations 3661:with positive-definite 3457:residual sum of squares 886:linear regression model 592:Linear regression model 15858:Mathematics portal 15782:Orthogonal polynomials 15608:Analysis of covariance 15473:Weighted least squares 15463:Ordinary least squares 15414:Ordinary least squares 14481:Bayesian least squares 14465: 14427: 14385: 14319: 14239: 14120: 14082: 14062: 14035: 14008: 13988: 13961: 13941: 13914: 13894: 13824: 13739: 13707: 13675: 13550: 13514: 13435: 13398: 13378: 13335: 13284: 13259: 13239: 13219: 13190: 13095: 13003: 12966: 12866: 12769: 12742: 12664: 12582: 12322: 12280: 12238: 12146: 12132: 11910: 11831:alternative hypothesis 11757: 11557: 11527: 11362: 11333: 11261: 11207: 11177: 10989: 10810:can be constructed as 10804: 10754: 10702: 10571: 10515: 10479: 10390: 10125: 9896: 9842:Constrained estimation 9807: 9724: 9686: 9561: 9485:Partitioned regression 9422: 9161: 8965: 8908: 8802: 8692: 8577: 8499: 8402: 8367: 8330:states that under the 8315: 8242: 8208: 8069: 8028: 7911: 7860: 7744: 7429: 7322:between observations: 7299:weighted least squares 7240: 7146: 7034: 6812: 6670: 6320: 6239:Orthogonal projections 6224: 6184: 6152: 6117: 6067: 5995: 5952: 5862: 5826:projected orthogonally 5771: 5637: 5630: 5603: 5543: 5506: 5385: 5298: 5233: 5105: 5077: 5050: 4985: 4653: 4621:, whereas the second, 4613:. The first quantity, 4592: 4265: 3972: 3820: 3694: 3606: 3508: 3355: 3252: 3156: 3123: 3081: 3044: 2951: 2935:columns of the matrix 2929: 2902: 2822: 2782: 2729: 2706: 2618: 2593: 2573: 2537: 2512: 2154: 2109: 2108:{\displaystyle n>p} 2083: 2020: 1997: 1974: 1891: 1844: 1802: 1756: 1729: 1691: 1655: 1630: 1610: 1576: 1552: 1526: 1504: 1484: 1458: 1436: 1411: 1361: 1332: 1305: 1278: 1254: 1227: 1201: 1179: 1159: 1127: 1048: 909: 878: 778: 758: 729: 702: 682: 610: 499:) by the principle of 473:ordinary least squares 464: 410:Mathematics portal 336:Iteratively reweighted 18:Ordinary Least Squares 15889:Parametric statistics 15823:System identification 15787:Chebyshev polynomials 15772:Numerical integration 15723:Design of experiments 15667:Regression validation 15494:Polynomial regression 15419:Partial least squares 14974:Advanced Econometrics 14569:Goldberger, Arthur S. 14466: 14428: 14386: 14320: 14240: 14121: 14083: 14063: 14036: 14009: 13989: 13962: 13942: 13915: 13895: 13825: 13740: 13708: 13676: 13551: 13515: 13436: 13399: 13379: 13336: 13285: 13260: 13240: 13220: 13191: 13093: 13038:2.54 cm this is 13004: 12964: 12867: 12770: 12768:{\displaystyle R^{2}} 12743: 12741:{\displaystyle R^{2}} 12665: 12583: 12323: 12281: 12239: 12144: 12133: 11905: 11758: 11558: 11556:{\displaystyle y_{0}} 11528: 11363: 11334: 11262: 11208: 11206:{\displaystyle x_{0}} 11178: 10990: 10805: 10755: 10703: 10572: 10523:central limit theorem 10516: 10480: 10391: 10126: 9897: 9808: 9725: 9692:and the OLS estimate 9687: 9562: 9446:, so that on average 9442:, and they sum up to 9423: 9162: 8966: 8938:Leverage (statistics) 8909: 8803: 8693: 8578: 8528:holds (that is, that 8500: 8403: 8368: 8316: 8243: 8209: 8070: 8029: 7912: 7861: 7745: 7430: 7241: 7147: 7035: 6982:Correct specification 6931:asymptotic properties 6813: 6671: 6321: 6225: 6185: 6153: 6118: 6073:is equal to zero for 6068: 6001:is orthogonal to the 5996: 5953: 5863: 5772: 5631: 5629:{\displaystyle X_{2}} 5604: 5602:{\displaystyle X_{1}} 5581: 5544: 5507: 5365: 5278: 5234: 5106: 5078: 5051: 4986: 4654: 4593: 4266: 4122:from the regression: 3973: 3821: 3695: 3607: 3488: 3356: 3253: 3157: 3124: 3082: 3045: 2952: 2930: 2903: 2802: 2762: 2730: 2707: 2619: 2594: 2574: 2538: 2513: 2155: 2110: 2084: 2021: 1998: 1975: 1871: 1860:overdetermined system 1845: 1803: 1757: 1730: 1692: 1656: 1631: 1611: 1577: 1553: 1527: 1505: 1485: 1459: 1437: 1412: 1362: 1333: 1331:{\displaystyle y_{i}} 1306: 1279: 1255: 1228: 1202: 1180: 1160: 1128: 1049: 910: 908:{\displaystyle y_{i}} 879: 779: 759: 730: 728:{\displaystyle y_{i}} 703: 683: 611: 564:serially uncorrelated 534:The OLS estimator is 497:explanatory variables 451: 367:Regression validation 346:Bayesian multivariate 63:Polynomial regression 15828:Moving least squares 15767:Approximation theory 15703:Studentized residual 15693:Errors and residuals 15688:Gauss–Markov theorem 15603:Analysis of variance 15525:Nonlinear regression 15504:Segmented regression 15478:General linear model 15396:Confounding variable 15343:Linear least squares 15198:Gujarati, Damodar N. 14437: 14398: 14331: 14249: 14130: 14092: 14072: 14045: 14018: 13998: 13971: 13951: 13924: 13904: 13836: 13749: 13717: 13685: 13560: 13528: 13445: 13416: 13404:for the given data. 13388: 13368: 13316: 13274: 13249: 13229: 13200: 13135: 12984: 12921:model sum of squared 12917:Total sum of squares 12785: 12752: 12725: 12608: 12473: 12305: 12263: 12221: 12151:statistical packages 12035: 11570: 11540: 11375: 11343: 11275: 11221: 11190: 11045: 10817: 10785: 10715: 10591: 10552: 10489: 10460: 10157: 9946: 9928:q < p 9859: 9744: 9696: 9667: 9496: 9230: 9012: 8946: 8887: 8727: 8707:Gauss–Markov theorem 8590: 8556: 8526:normality assumption 8419: 8412:and unbiased, then 8381: 8346: 8327:Gauss–Markov theorem 8255: 8221: 8094: 8041: 7924: 7890: 7765: 7723: 7455:cross-sectional data 7368: 7303:law of large numbers 7184: 7092: 7078:linearly independent 7072:. The regressors in 7070:No linear dependence 6998: 6853:Gauss–Markov theorem 6727: 6336: 6248: 6230:is non-singular and 6198: 6165: 6127: 6085: 6016: 5965: 5896: 5870:vector decomposition 5843: 5839:. The OLS estimator 5698: 5659:Linear least squares 5613: 5586: 5524: 5249: 5178: 5095: 5060: 5004: 4997:total sum of squares 4720: 4625: 4291: 4129: 3923: 3707: 3669: 3470: 3449:error sum of squares 3268: 3173: 3137: 3099: 3057: 2972: 2959:linearly independent 2939: 2919: 2742: 2719: 2638: 2606: 2583: 2547: 2525: 2170: 2126: 2093: 2034: 2010: 1987: 1868: 1832: 1770: 1739: 1701: 1665: 1643: 1620: 1586: 1566: 1536: 1514: 1494: 1468: 1446: 1424: 1374: 1342: 1315: 1288: 1268: 1237: 1211: 1189: 1169: 1140: 1068: 922: 892: 788: 768: 739: 735:and a column vector 712: 692: 622: 600: 576:normally distributed 548:Gauss–Markov theorem 513:independent variable 481:linear least squares 392:Gauss–Markov theorem 387:Studentized residual 377:Errors and residuals 211:Principal components 181:Nonlinear regression 68:General linear model 15846:Statistics category 15777:Gaussian quadrature 15662:Model specification 15629:Stepwise regression 15487:Predictor structure 15424:Total least squares 15406:Regression analysis 15391:Partial correlation 15322:regression analysis 15262:Wooldridge, Jeffrey 12974:heteroscedasticity. 12693:. Conventionally, 12548: 12394:Durbin–Watson stat. 12385:Residual sum-of-sq. 12111: 11732: 11711: 11672: 11610: 11563:to be constructed: 11504: 11483: 11432: 11316: 11253: 11147: 11098: 10959: 10911: 10773:Prediction interval 10769:Confidence interval 10687: 10630: 9459:. These quantities 9347: 9304: 9262: 9133: 8797: 8201: 7638:The regressors are 7362:normal distribution 6975:asymptotic behavior 6927:observational study 6785: 5882:along the basis of 5148:If the data matrix 4281:reduced chi-squared 3619:denotes the matrix 3539: 3447:) (also called the 3424:and the hyperplane 1605: 1100: 688:. Each observation 677: 237:Errors-in-variables 104:Logistic regression 94:Binomial regression 39:Regression analysis 33:Part of a series on 15863:Statistics outline 15762:Numerical analysis 14578:Econometric Theory 14461: 14423: 14381: 14327:On solving we get 14315: 14306: 14235: 14229: 14116: 14078: 14058: 14031: 14004: 13984: 13957: 13937: 13910: 13890: 13820: 13735: 13703: 13671: 13546: 13510: 13431: 13394: 13374: 13331: 13280: 13255: 13235: 13215: 13186: 13096: 12999: 12967: 12909:is an estimate of 12862: 12765: 12738: 12719:Adjusted R-squared 12691:significance level 12660: 12578: 12528: 12357:S.E. of regression 12318: 12276: 12234: 12169:Dependent variable 12147: 12128: 12097: 11911: 11800:. You can help by 11781:Hypothesis testing 11775:Hypothesis testing 11753: 11712: 11695: 11626: 11594: 11553: 11523: 11484: 11467: 11358: 11329: 11300: 11269:predicted response 11257: 11237: 11203: 11173: 11133: 10985: 10939: 10865: 10800: 10750: 10698: 10667: 10567: 10511: 10475: 10444:interval estimates 10386: 10121: 9892: 9825:annihilator matrix 9803: 9720: 9719: 9682: 9557: 9418: 9333: 9288: 9233: 9157: 9117: 9005:will be equal to 8961: 8904: 8903: 8852:mean squared error 8798: 8777: 8688: 8573: 8572: 8520:Assuming normality 8495: 8398: 8397: 8363: 8362: 8311: 8238: 8237: 8204: 8161: 8083:with its estimate 8065: 8064: 8024: 7907: 7906: 7856: 7740: 7739: 7579:stochastic process 7442:regular estimators 7425: 7350:serial correlation 7295:heteroscedasticity 7271:nuisance parameter 7236: 7142: 7030: 6937:), the regressors 6808: 6771: 6684:Maximum likelihood 6666: 6664: 6648: 6513: 6477: 6421: 6377: 6316: 6220: 6180: 6148: 6113: 6081:. This means that 6077:conformal vector, 6063: 5991: 5948: 5858: 5835:by the columns of 5767: 5738: 5638: 5626: 5599: 5539: 5502: 5500: 5229: 5101: 5076:{\textstyle J_{n}} 5073: 5046: 4981: 4669:mean squared error 4649: 4648: 4588: 4261: 3968: 3816: 3690: 3657:) is quadratic in 3644:OLS estimator for 3623:, and the rows of 3602: 3525: 3351: 3248: 3152: 3119: 3077: 3040: 2947: 2925: 2898: 2725: 2702: 2682: 2614: 2589: 2569: 2533: 2508: 2499: 2426: 2353: 2150: 2105: 2079: 2016: 1993: 1970: 1840: 1798: 1752: 1735:. The coefficient 1725: 1687: 1651: 1626: 1606: 1589: 1572: 1548: 1522: 1510:observations, and 1500: 1480: 1454: 1432: 1407: 1357: 1328: 1301: 1274: 1250: 1223: 1197: 1175: 1155: 1123: 1084: 1044: 905: 874: 774: 754: 725: 698: 678: 625: 606: 542:and forms perfect 505:dependent variable 465: 124:Multinomial probit 15876: 15875: 15868:Statistics topics 15813:Calibration curve 15622:Model exploration 15589: 15588: 15559:Non-normal errors 15450:Linear regression 15441:statistical model 15279:978-0-324-58162-1 15253:978-0-471-72360-8 15234:978-0-19-926801-6 15215:978-0-07-337577-9 15206:Basic Econometics 14800:"Line Estimation" 14415: 14373: 14348: 14199: 14114: 14081:{\displaystyle b} 14056: 14029: 14007:{\displaystyle A} 13982: 13960:{\displaystyle y} 13935: 13913:{\displaystyle x} 13866: 13493: 13480: 13467: 13397:{\displaystyle p} 13377:{\displaystyle e} 13362: 13361: 13258:{\displaystyle e} 13238:{\displaystyle p} 13184: 13127:Problem statement 13088: 13087: 12996: 12900:Schwarz criterion 12838: 12797: 12651: 12627: 12575: 12514: 12486: 12440: 12439: 12426:Schwarz criterion 12145:Fitted regression 12015: 12014: 11848:In addition, the 11818: 11817: 11770:confidence level. 11747: 11743: 11686: 11645: 11620: 11593: 11437: 11433: 11422: 11393: 11355: 11326: 11288: 11103: 11099: 11088: 11061: 11012:quantile function 11002:confidence level, 10977: 10973: 10925: 10884: 10853: 10842: 10797: 10635: 10631: 10620: 10606: 10564: 10502: 10472: 10170: 10106: 9980: 9959: 9710: 9679: 9634:×1 vectors, with 9409: 9396: 9357: 9315: 9276: 9243: 9144: 9084: 9052: 9025: 8958: 8900: 8871: + 2) 8772: 8748: 8742: 8636: 8613: 8607: 8602: 8569: 8476: 8441: 8394: 8359: 8292: 8277: 8234: 8202: 8134: 8118: 8055: 7946: 7903: 7884:covariance matrix 7787: 7736: 7717:strict exogeneity 7673:positive-definite 7572:Time series model 7318:: the errors are 7307:robust estimation 6988:Strict exogeneity 6904:) the regressors 6472: 6456: 6416: 6400: 6310: 6290: 6262: 6211: 6177: 6110: 6044: 5988: 5924: 5855: 5729: 5710: 5667: 5666: 5536: 5494: 5489: 5476: 5461: 5442: 5427: 5412: 5356: 5325: 5265: 5104:{\displaystyle L} 5091:matrix of ones. ( 5034: 4994:where TSS is the 4979: 4944: 4891: 4829: 4816: 4775: 4755: 4639: 4601:The denominator, 4575: 4547: 4531: 4513: 4490: 4452: 4402: 4355: 4340: 4320: 4186: 4162: 4141: 4003:projection matrix 3953: 3935: 3897:multicollinearity 3862:covariance matrix 3844:and its inverse, 3812: 3719: 3687: 3280: 3185: 3149: 3036: 3011: 2928:{\displaystyle p} 2728:{\displaystyle S} 2657: 2650: 2592:{\displaystyle p} 2019:{\displaystyle p} 1996:{\displaystyle n} 1933: 1795: 1661:, say, by taking 1629:{\displaystyle i} 1616:and contains the 1575:{\displaystyle i} 1503:{\displaystyle n} 1284:-th observation. 1277:{\displaystyle i} 1178:{\displaystyle i} 1014: 979: 950: 777:{\displaystyle p} 701:{\displaystyle i} 609:{\displaystyle n} 489:linear regression 446: 445: 99:Binary regression 58:Simple regression 53:Linear regression 16:(Redirected from 15901: 15856: 15855: 15613:Multivariate AOV 15509:Local regression 15446: 15445: 15438:Regression as a 15429:Ridge regression 15376:Rank correlation 15311: 15304: 15297: 15288: 15287: 15283: 15257: 15238: 15219: 15193: 15166: 15165: 15153: 15143: 15137: 15131: 15122: 15116: 15110: 15104: 15098: 15092: 15086: 15085: 15067: 15058: 15052: 15046: 15040: 15034: 15033: 15012: 15006: 15000: 14994: 14993: 14977: 14968:Amemiya, Takeshi 14964: 14958: 14952: 14941: 14935: 14929: 14928: 14926: 14924: 14919: 14911: 14905: 14904: 14888: 14882: 14876: 14870: 14864: 14855: 14849: 14843: 14837: 14831: 14825: 14816: 14810: 14804: 14803: 14795: 14789: 14783: 14777: 14776: 14758: 14752: 14751: 14743: 14737: 14729: 14723: 14717: 14708: 14707: 14687: 14681: 14680: 14660: 14654: 14653: 14633: 14627: 14621: 14615: 14614: 14603: 14597: 14596: 14565: 14559: 14558: 14556: 14555: 14541: 14535: 14534: 14532: 14531: 14516: 14470: 14468: 14467: 14462: 14432: 14430: 14429: 14424: 14416: 14408: 14390: 14388: 14387: 14382: 14380: 14379: 14378: 14365: 14355: 14354: 14353: 14340: 14324: 14322: 14321: 14316: 14311: 14310: 14244: 14242: 14241: 14236: 14234: 14233: 14197: 14125: 14123: 14122: 14117: 14115: 14113: 14096: 14087: 14085: 14084: 14079: 14067: 14065: 14064: 14059: 14057: 14049: 14040: 14038: 14037: 14032: 14030: 14022: 14013: 14011: 14010: 14005: 13993: 13991: 13990: 13985: 13983: 13975: 13966: 13964: 13963: 13958: 13946: 13944: 13943: 13938: 13936: 13928: 13919: 13917: 13916: 13911: 13899: 13897: 13896: 13891: 13886: 13885: 13873: 13872: 13871: 13858: 13848: 13847: 13829: 13827: 13826: 13821: 13816: 13815: 13797: 13789: 13788: 13767: 13766: 13745:, used to extra 13744: 13742: 13741: 13736: 13712: 13710: 13709: 13704: 13680: 13678: 13677: 13672: 13667: 13666: 13627: 13626: 13587: 13586: 13555: 13553: 13552: 13547: 13519: 13517: 13516: 13511: 13494: 13486: 13481: 13473: 13468: 13466: 13449: 13440: 13438: 13437: 13432: 13403: 13401: 13400: 13395: 13383: 13381: 13380: 13375: 13340: 13338: 13337: 13332: 13289: 13287: 13286: 13281: 13268: 13267: 13264: 13262: 13261: 13256: 13244: 13242: 13241: 13236: 13224: 13222: 13221: 13216: 13195: 13193: 13192: 13187: 13185: 13183: 13154: 13045: 13044: 13008: 13006: 13005: 13000: 12998: 12997: 12989: 12871: 12869: 12868: 12863: 12858: 12857: 12839: 12837: 12826: 12815: 12804: 12803: 12798: 12790: 12774: 12772: 12771: 12766: 12764: 12763: 12747: 12745: 12744: 12739: 12737: 12736: 12669: 12667: 12666: 12661: 12659: 12658: 12653: 12652: 12644: 12640: 12635: 12634: 12629: 12628: 12620: 12587: 12585: 12584: 12579: 12577: 12576: 12568: 12566: 12562: 12561: 12560: 12552: 12547: 12539: 12522: 12521: 12516: 12515: 12507: 12494: 12493: 12488: 12487: 12479: 12433:p-value (F-stat) 12410:Akaike criterion 12401:Total sum-of-sq. 12371:Model sum-of-sq. 12327: 12325: 12324: 12319: 12317: 12316: 12285: 12283: 12282: 12277: 12275: 12274: 12243: 12241: 12240: 12235: 12233: 12232: 12158: 12157: 12137: 12135: 12134: 12129: 12124: 12123: 12110: 12105: 12096: 12095: 12083: 12082: 12073: 12072: 12060: 12059: 12047: 12046: 11880: 11879: 11813: 11810: 11792: 11785: 11769: 11762: 11760: 11759: 11754: 11752: 11748: 11745: 11744: 11742: 11741: 11731: 11723: 11710: 11709: 11703: 11694: 11693: 11688: 11687: 11679: 11675: 11671: 11655: 11654: 11647: 11646: 11638: 11622: 11621: 11613: 11609: 11608: 11602: 11591: 11582: 11581: 11562: 11560: 11559: 11554: 11552: 11551: 11532: 11530: 11529: 11524: 11519: 11515: 11514: 11513: 11503: 11495: 11482: 11481: 11475: 11466: 11465: 11444: 11443: 11435: 11434: 11424: 11420: 11419: 11415: 11414: 11413: 11401: 11400: 11395: 11394: 11386: 11367: 11365: 11364: 11359: 11357: 11356: 11348: 11338: 11336: 11335: 11330: 11328: 11327: 11319: 11315: 11314: 11308: 11296: 11295: 11290: 11289: 11281: 11266: 11264: 11263: 11258: 11252: 11251: 11245: 11233: 11232: 11217:is the quantity 11212: 11210: 11209: 11204: 11202: 11201: 11182: 11180: 11179: 11174: 11169: 11165: 11164: 11163: 11151: 11146: 11141: 11110: 11109: 11101: 11100: 11090: 11086: 11082: 11081: 11069: 11068: 11063: 11062: 11054: 11001: 10994: 10992: 10991: 10986: 10984: 10983: 10975: 10974: 10972: 10971: 10963: 10958: 10950: 10933: 10932: 10927: 10926: 10918: 10914: 10910: 10894: 10893: 10886: 10885: 10877: 10861: 10860: 10855: 10854: 10846: 10840: 10839: 10838: 10829: 10828: 10809: 10807: 10806: 10801: 10799: 10798: 10790: 10759: 10757: 10756: 10751: 10743: 10742: 10730: 10729: 10707: 10705: 10704: 10699: 10694: 10693: 10686: 10678: 10666: 10665: 10649: 10648: 10642: 10641: 10633: 10632: 10622: 10618: 10608: 10607: 10599: 10576: 10574: 10573: 10568: 10566: 10565: 10557: 10525:to derive their 10520: 10518: 10517: 10512: 10510: 10509: 10504: 10503: 10495: 10484: 10482: 10481: 10476: 10474: 10473: 10465: 10421: 10395: 10393: 10392: 10387: 10379: 10378: 10363: 10362: 10347: 10346: 10337: 10336: 10327: 10326: 10317: 10316: 10298: 10297: 10288: 10287: 10269: 10268: 10259: 10258: 10246: 10245: 10236: 10235: 10226: 10225: 10207: 10206: 10197: 10196: 10178: 10177: 10172: 10171: 10163: 10130: 10128: 10127: 10122: 10108: 10107: 10099: 10096: 10095: 10083: 10082: 10074: 10073: 10063: 10062: 10047: 10046: 10034: 10033: 10024: 10023: 10014: 10013: 9998: 9997: 9982: 9981: 9973: 9967: 9966: 9961: 9960: 9952: 9930: 9901: 9899: 9898: 9893: 9878: 9877: 9848:Ridge regression 9812: 9810: 9809: 9804: 9792: 9791: 9782: 9781: 9772: 9771: 9756: 9755: 9729: 9727: 9726: 9721: 9718: 9717: 9712: 9711: 9703: 9691: 9689: 9688: 9683: 9681: 9680: 9672: 9653: 9584:have dimensions 9566: 9564: 9563: 9558: 9547: 9546: 9537: 9536: 9524: 9523: 9514: 9513: 9458: 9441: 9427: 9425: 9424: 9419: 9417: 9416: 9411: 9410: 9402: 9397: 9395: 9394: 9393: 9377: 9376: 9367: 9359: 9358: 9350: 9346: 9341: 9329: 9328: 9317: 9316: 9308: 9303: 9302: 9296: 9284: 9283: 9278: 9277: 9269: 9261: 9250: 9245: 9244: 9236: 9199: 9166: 9164: 9163: 9158: 9152: 9151: 9146: 9145: 9137: 9132: 9131: 9125: 9116: 9115: 9100: 9099: 9098: 9085: 9083: 9082: 9081: 9062: 9054: 9053: 9045: 9039: 9038: 9027: 9026: 9018: 8999:jackknife method 8970: 8968: 8967: 8962: 8960: 8959: 8951: 8913: 8911: 8910: 8905: 8902: 8901: 8893: 8879: 8872: 8841: 8829:Cramér–Rao bound 8826: 8807: 8805: 8804: 8799: 8796: 8791: 8773: 8771: 8760: 8759: 8750: 8746: 8740: 8739: 8738: 8703:Cramér–Rao bound 8697: 8695: 8694: 8689: 8684: 8683: 8677: 8676: 8661: 8660: 8659: 8646: 8645: 8634: 8627: 8626: 8620: 8619: 8611: 8605: 8604: 8603: 8595: 8582: 8580: 8579: 8574: 8571: 8570: 8562: 8548: 8504: 8502: 8501: 8496: 8478: 8477: 8469: 8443: 8442: 8434: 8407: 8405: 8404: 8399: 8396: 8395: 8387: 8372: 8370: 8369: 8364: 8361: 8360: 8352: 8342:) the estimator 8332:spherical errors 8320: 8318: 8317: 8312: 8294: 8293: 8285: 8279: 8278: 8270: 8247: 8245: 8244: 8239: 8236: 8235: 8227: 8213: 8211: 8210: 8205: 8203: 8200: 8192: 8184: 8180: 8176: 8175: 8160: 8159: 8150: 8142: 8141: 8136: 8135: 8127: 8120: 8119: 8114: 8099: 8074: 8072: 8071: 8066: 8063: 8062: 8057: 8056: 8048: 8033: 8031: 8030: 8025: 8017: 8016: 8004: 8003: 7995: 7991: 7987: 7986: 7971: 7970: 7948: 7947: 7939: 7916: 7914: 7913: 7908: 7905: 7904: 7896: 7865: 7863: 7862: 7857: 7852: 7851: 7829: 7828: 7789: 7788: 7780: 7749: 7747: 7746: 7741: 7738: 7737: 7729: 7705: 7670: 7642:: E = 0 for all 7566: 7558:homoscedasticity 7554: 7540: 7524: 7481:iid observations 7471: 7470: → ∞ 7434: 7432: 7431: 7426: 7418: 7417: 7408: 7407: 7389: 7388: 7335: 7325: 7288: 7279:Homoscedasticity 7269:is considered a 7252: 7245: 7243: 7242: 7237: 7232: 7231: 7222: 7221: 7178:Spherical errors 7161: 7151: 7149: 7148: 7143: 7135: 7134: 7105: 7104: 7051: 7043: 7039: 7037: 7036: 7031: 6992:conditional mean 6920: 6872: 6858: 6850: 6846: 6842: 6817: 6815: 6814: 6809: 6801: 6800: 6793: 6789: 6784: 6779: 6767: 6766: 6752: 6751: 6741: 6740: 6734: 6702:Cramér–Rao bound 6675: 6673: 6672: 6667: 6665: 6658: 6653: 6652: 6645: 6644: 6639: 6633: 6632: 6624: 6620: 6619: 6614: 6613: 6608: 6592: 6591: 6586: 6580: 6579: 6571: 6567: 6566: 6561: 6560: 6555: 6532: 6527: 6526: 6518: 6517: 6510: 6503: 6482: 6481: 6474: 6473: 6465: 6458: 6457: 6449: 6435: 6426: 6425: 6418: 6417: 6409: 6402: 6401: 6393: 6382: 6381: 6374: 6367: 6347: 6325: 6323: 6322: 6317: 6312: 6311: 6303: 6300: 6292: 6291: 6283: 6280: 6272: 6264: 6263: 6258: 6253: 6229: 6227: 6226: 6223:{\displaystyle } 6221: 6216: 6209: 6208: 6189: 6187: 6186: 6181: 6179: 6178: 6170: 6157: 6155: 6154: 6149: 6147: 6142: 6134: 6122: 6120: 6119: 6114: 6112: 6111: 6103: 6100: 6092: 6072: 6070: 6069: 6064: 6062: 6057: 6046: 6045: 6037: 6034: 6026: 6000: 5998: 5997: 5992: 5990: 5989: 5981: 5972: 5957: 5955: 5954: 5949: 5941: 5936: 5935: 5926: 5925: 5917: 5914: 5906: 5881: 5867: 5865: 5864: 5859: 5857: 5856: 5848: 5819: 5788:is the standard 5787: 5785: 5776: 5774: 5773: 5768: 5760: 5755: 5747: 5737: 5728: 5727: 5712: 5711: 5703: 5678: 5648: 5647: 5640: 5635: 5633: 5632: 5627: 5625: 5624: 5608: 5606: 5605: 5600: 5598: 5597: 5569: 5548: 5546: 5545: 5540: 5538: 5537: 5529: 5511: 5509: 5508: 5503: 5501: 5492: 5491: 5490: 5482: 5478: 5477: 5469: 5463: 5462: 5454: 5444: 5443: 5435: 5428: 5426: 5425: 5424: 5423: 5414: 5413: 5405: 5399: 5398: 5384: 5379: 5363: 5362: 5358: 5357: 5349: 5343: 5342: 5327: 5326: 5318: 5312: 5311: 5297: 5292: 5276: 5267: 5266: 5258: 5238: 5236: 5235: 5230: 5225: 5224: 5212: 5211: 5190: 5189: 5170: 5113:centering matrix 5110: 5108: 5107: 5102: 5082: 5080: 5079: 5074: 5072: 5071: 5055: 5053: 5052: 5047: 5045: 5044: 5035: 5027: 5022: 5021: 4990: 4988: 4987: 4982: 4980: 4978: 4967: 4956: 4945: 4943: 4936: 4935: 4934: 4923: 4916: 4915: 4914: 4903: 4892: 4890: 4883: 4882: 4881: 4870: 4860: 4859: 4858: 4848: 4847: 4846: 4835: 4830: 4828: 4827: 4826: 4817: 4809: 4804: 4803: 4787: 4786: 4785: 4776: 4768: 4763: 4762: 4757: 4756: 4748: 4737: 4732: 4731: 4658: 4656: 4655: 4650: 4647: 4646: 4641: 4640: 4632: 4597: 4595: 4594: 4589: 4587: 4586: 4576: 4571: 4560: 4555: 4554: 4549: 4548: 4540: 4532: 4530: 4519: 4515: 4514: 4506: 4496: 4491: 4489: 4478: 4471: 4470: 4469: 4458: 4453: 4451: 4440: 4433: 4432: 4431: 4421: 4420: 4419: 4408: 4403: 4401: 4390: 4383: 4382: 4381: 4361: 4356: 4354: 4343: 4342: 4341: 4333: 4330: 4329: 4328: 4322: 4321: 4313: 4308: 4303: 4302: 4270: 4268: 4267: 4262: 4188: 4187: 4179: 4164: 4163: 4155: 4143: 4142: 4134: 4113: 4106: 4092: 4082: 4057:. Both matrices 4052: 3977: 3975: 3974: 3969: 3955: 3954: 3946: 3937: 3936: 3928: 3914:predicted values 3825: 3823: 3822: 3817: 3810: 3806: 3805: 3796: 3795: 3780: 3779: 3749: 3748: 3747: 3746: 3741: 3721: 3720: 3712: 3699: 3697: 3696: 3691: 3689: 3688: 3680: 3638:. The value of 3611: 3609: 3608: 3603: 3580: 3579: 3552: 3551: 3538: 3533: 3521: 3520: 3507: 3502: 3436: 3423: 3395: 3360: 3358: 3357: 3352: 3347: 3342: 3341: 3336: 3330: 3329: 3321: 3317: 3316: 3311: 3310: 3305: 3290: 3282: 3281: 3273: 3257: 3255: 3254: 3249: 3244: 3239: 3238: 3233: 3227: 3226: 3218: 3214: 3213: 3208: 3207: 3202: 3187: 3186: 3178: 3161: 3159: 3158: 3153: 3151: 3150: 3142: 3129:is known as the 3128: 3126: 3125: 3120: 3118: 3113: 3112: 3107: 3087:is known as the 3086: 3084: 3083: 3078: 3076: 3071: 3070: 3065: 3049: 3047: 3046: 3041: 3034: 3033: 3028: 3027: 3022: 3013: 3012: 3004: 3001: 2997: 2996: 2991: 2990: 2985: 2963:normal equations 2956: 2954: 2953: 2948: 2946: 2934: 2932: 2931: 2926: 2907: 2905: 2904: 2899: 2894: 2893: 2888: 2884: 2883: 2878: 2870: 2856: 2855: 2850: 2846: 2845: 2844: 2835: 2834: 2821: 2816: 2798: 2797: 2781: 2776: 2755: 2734: 2732: 2731: 2726: 2711: 2709: 2708: 2703: 2695: 2683: 2678: 2652: 2651: 2643: 2623: 2621: 2620: 2615: 2613: 2598: 2596: 2595: 2590: 2578: 2576: 2575: 2570: 2562: 2561: 2542: 2540: 2539: 2534: 2532: 2517: 2515: 2514: 2509: 2504: 2503: 2496: 2495: 2475: 2474: 2461: 2460: 2440: 2431: 2430: 2423: 2422: 2402: 2401: 2388: 2387: 2367: 2358: 2357: 2350: 2349: 2330: 2329: 2315: 2314: 2276: 2275: 2256: 2255: 2244: 2243: 2230: 2229: 2210: 2209: 2198: 2197: 2177: 2159: 2157: 2156: 2151: 2146: 2138: 2133: 2114: 2112: 2111: 2106: 2088: 2086: 2085: 2080: 2078: 2077: 2059: 2058: 2046: 2045: 2025: 2023: 2022: 2017: 2004:linear equations 2002: 2000: 1999: 1994: 1979: 1977: 1976: 1971: 1931: 1927: 1926: 1914: 1913: 1904: 1903: 1890: 1885: 1849: 1847: 1846: 1841: 1839: 1807: 1805: 1804: 1799: 1797: 1796: 1788: 1782: 1781: 1761: 1759: 1758: 1753: 1751: 1750: 1734: 1732: 1731: 1726: 1696: 1694: 1693: 1688: 1680: 1679: 1660: 1658: 1657: 1652: 1650: 1635: 1633: 1632: 1627: 1615: 1613: 1612: 1607: 1604: 1599: 1594: 1581: 1579: 1578: 1573: 1557: 1555: 1554: 1549: 1531: 1529: 1528: 1523: 1521: 1509: 1507: 1506: 1501: 1489: 1487: 1486: 1481: 1463: 1461: 1460: 1455: 1453: 1441: 1439: 1438: 1433: 1431: 1416: 1414: 1413: 1408: 1402: 1394: 1389: 1381: 1366: 1364: 1363: 1358: 1356: 1355: 1350: 1337: 1335: 1334: 1329: 1327: 1326: 1310: 1308: 1307: 1302: 1300: 1299: 1283: 1281: 1280: 1275: 1259: 1257: 1256: 1251: 1249: 1248: 1232: 1230: 1229: 1224: 1206: 1204: 1203: 1198: 1196: 1184: 1182: 1181: 1176: 1164: 1162: 1161: 1156: 1154: 1153: 1148: 1132: 1130: 1129: 1124: 1118: 1117: 1105: 1099: 1094: 1089: 1080: 1079: 1053: 1051: 1050: 1045: 1040: 1039: 1027: 1026: 1012: 1011: 1010: 992: 991: 977: 976: 975: 963: 962: 948: 947: 946: 934: 933: 914: 912: 911: 906: 904: 903: 883: 881: 880: 875: 873: 872: 867: 863: 862: 861: 840: 839: 824: 823: 802: 801: 796: 783: 781: 780: 775: 763: 761: 760: 755: 753: 752: 747: 734: 732: 731: 726: 724: 723: 707: 705: 704: 699: 687: 685: 684: 679: 676: 671: 660: 656: 655: 654: 642: 641: 636: 615: 613: 612: 607: 438: 431: 424: 408: 407: 315:Ridge regression 150:Multilevel model 30: 29: 21: 15909: 15908: 15904: 15903: 15902: 15900: 15899: 15898: 15879: 15878: 15877: 15872: 15850: 15832: 15796: 15792:Chebyshev nodes 15745: 15741:Bayesian design 15717: 15698:Goodness of fit 15671: 15644: 15634:Model selection 15617: 15585: 15554: 15513: 15482: 15439: 15433: 15400: 15357: 15324: 15315: 15280: 15254: 15235: 15216: 15202:Porter, Dawn C. 15190: 15174: 15172:Further reading 15169: 15162: 15144: 15140: 15132: 15125: 15117: 15113: 15105: 15101: 15093: 15089: 15082: 15068: 15061: 15053: 15049: 15041: 15037: 15030: 15013: 15009: 15001: 14997: 14990: 14965: 14961: 14953: 14944: 14940:, pages 27, 30) 14936: 14932: 14922: 14920: 14917: 14913: 14912: 14908: 14889: 14885: 14877: 14873: 14865: 14858: 14850: 14846: 14838: 14834: 14826: 14819: 14811: 14807: 14796: 14792: 14784: 14780: 14773: 14759: 14755: 14744: 14740: 14730: 14726: 14718: 14711: 14704: 14688: 14684: 14677: 14661: 14657: 14650: 14634: 14630: 14622: 14618: 14604: 14600: 14593: 14566: 14562: 14553: 14551: 14549:Cross Validated 14543: 14542: 14538: 14529: 14527: 14518: 14517: 14513: 14509: 14477: 14438: 14435: 14434: 14407: 14399: 14396: 14395: 14374: 14361: 14360: 14359: 14349: 14336: 14335: 14334: 14332: 14329: 14328: 14305: 14304: 14298: 14297: 14291: 14290: 14284: 14283: 14277: 14276: 14270: 14269: 14259: 14258: 14250: 14247: 14246: 14228: 14227: 14222: 14216: 14215: 14210: 14204: 14203: 14195: 14189: 14188: 14180: 14174: 14173: 14165: 14159: 14158: 14150: 14140: 14139: 14131: 14128: 14127: 14100: 14095: 14093: 14090: 14089: 14073: 14070: 14069: 14048: 14046: 14043: 14042: 14021: 14019: 14016: 14015: 13999: 13996: 13995: 13974: 13972: 13969: 13968: 13952: 13949: 13948: 13927: 13925: 13922: 13921: 13905: 13902: 13901: 13881: 13877: 13867: 13854: 13853: 13852: 13843: 13839: 13837: 13834: 13833: 13811: 13807: 13793: 13784: 13780: 13762: 13758: 13750: 13747: 13746: 13718: 13715: 13714: 13686: 13683: 13682: 13662: 13658: 13622: 13618: 13582: 13578: 13561: 13558: 13557: 13529: 13526: 13525: 13485: 13472: 13453: 13448: 13446: 13443: 13442: 13417: 13414: 13413: 13410: 13389: 13386: 13385: 13369: 13366: 13365: 13317: 13314: 13313: 13275: 13272: 13271: 13250: 13247: 13246: 13230: 13227: 13226: 13201: 13198: 13197: 13158: 13153: 13136: 13133: 13132: 13129: 13124: 13035: 13029: 13023: 12988: 12987: 12985: 12982: 12981: 12957:should be used. 12853: 12849: 12827: 12816: 12814: 12799: 12789: 12788: 12786: 12783: 12782: 12759: 12755: 12753: 12750: 12749: 12732: 12728: 12726: 12723: 12722: 12654: 12643: 12642: 12641: 12636: 12630: 12619: 12618: 12617: 12609: 12606: 12605: 12567: 12553: 12540: 12532: 12524: 12523: 12517: 12506: 12505: 12504: 12503: 12499: 12498: 12489: 12478: 12477: 12476: 12474: 12471: 12470: 12467:standard errors 12457: 12444:In this table: 12344: 12312: 12308: 12306: 12303: 12302: 12270: 12266: 12264: 12261: 12260: 12228: 12224: 12222: 12219: 12218: 12213: 12185: 12119: 12115: 12106: 12101: 12091: 12087: 12078: 12074: 12068: 12064: 12055: 12051: 12042: 12038: 12036: 12033: 12032: 12027: 11869: 11859: 11823:null hypothesis 11814: 11808: 11805: 11798:needs expansion 11783: 11777: 11764: 11737: 11733: 11724: 11716: 11705: 11704: 11699: 11689: 11678: 11677: 11676: 11674: 11650: 11649: 11648: 11637: 11630: 11612: 11611: 11604: 11603: 11598: 11590: 11586: 11577: 11573: 11571: 11568: 11567: 11547: 11543: 11541: 11538: 11537: 11509: 11505: 11496: 11488: 11477: 11476: 11471: 11461: 11457: 11449: 11445: 11439: 11438: 11423: 11409: 11405: 11396: 11385: 11384: 11383: 11382: 11378: 11376: 11373: 11372: 11347: 11346: 11344: 11341: 11340: 11318: 11317: 11310: 11309: 11304: 11291: 11280: 11279: 11278: 11276: 11273: 11272: 11247: 11246: 11241: 11228: 11224: 11222: 11219: 11218: 11197: 11193: 11191: 11188: 11187: 11159: 11155: 11142: 11137: 11129: 11115: 11111: 11105: 11104: 11089: 11077: 11073: 11064: 11053: 11052: 11051: 11046: 11043: 11042: 11036: 11019: 10996: 10979: 10978: 10964: 10951: 10943: 10935: 10934: 10928: 10917: 10916: 10915: 10913: 10889: 10888: 10887: 10876: 10869: 10856: 10845: 10844: 10843: 10834: 10833: 10824: 10820: 10818: 10815: 10814: 10789: 10788: 10786: 10783: 10782: 10775: 10767:Main articles: 10765: 10738: 10734: 10722: 10718: 10716: 10713: 10712: 10689: 10688: 10679: 10671: 10661: 10657: 10644: 10643: 10637: 10636: 10621: 10598: 10597: 10592: 10589: 10588: 10556: 10555: 10553: 10550: 10549: 10505: 10494: 10493: 10492: 10490: 10487: 10486: 10464: 10463: 10461: 10458: 10457: 10454: 10436:point estimates 10432: 10426:is invertible. 10416: 10371: 10367: 10358: 10354: 10342: 10341: 10332: 10328: 10322: 10318: 10309: 10305: 10293: 10289: 10283: 10279: 10264: 10260: 10254: 10253: 10241: 10237: 10231: 10227: 10218: 10214: 10202: 10198: 10192: 10188: 10173: 10162: 10161: 10160: 10158: 10155: 10154: 10098: 10097: 10091: 10087: 10075: 10069: 10068: 10067: 10055: 10051: 10042: 10038: 10029: 10025: 10019: 10018: 10006: 10002: 9993: 9989: 9972: 9971: 9962: 9951: 9950: 9949: 9947: 9944: 9943: 9926: 9873: 9869: 9860: 9857: 9856: 9850: 9844: 9838:constant term. 9833: 9827:for regressors 9822: 9787: 9783: 9777: 9773: 9767: 9763: 9751: 9747: 9745: 9742: 9741: 9736: 9713: 9702: 9701: 9700: 9697: 9694: 9693: 9671: 9670: 9668: 9665: 9664: 9648: 9641: 9635: 9633: 9626: 9619: 9612: 9605: 9594: 9583: 9576: 9542: 9538: 9532: 9528: 9519: 9515: 9509: 9505: 9497: 9494: 9493: 9487: 9479:leverage points 9475: 9466:are called the 9464: 9452: 9447: 9438: 9432: 9412: 9401: 9400: 9399: 9389: 9385: 9378: 9372: 9368: 9366: 9349: 9348: 9342: 9337: 9318: 9307: 9306: 9305: 9298: 9297: 9292: 9279: 9268: 9267: 9266: 9251: 9246: 9235: 9234: 9231: 9228: 9227: 9213: 9197: 9183: 9176: 9171: 9147: 9136: 9135: 9134: 9127: 9126: 9121: 9108: 9104: 9094: 9093: 9089: 9077: 9073: 9066: 9061: 9044: 9043: 9028: 9017: 9016: 9015: 9013: 9010: 9009: 8980: 8950: 8949: 8947: 8944: 8943: 8940: 8934: 8928: 8892: 8891: 8888: 8885: 8884: 8874: 8867: − 8855: 8832: 8821: − 8812: 8792: 8781: 8761: 8755: 8751: 8749: 8734: 8730: 8728: 8725: 8724: 8679: 8678: 8669: 8665: 8655: 8654: 8650: 8641: 8637: 8622: 8621: 8615: 8614: 8594: 8593: 8591: 8588: 8587: 8561: 8560: 8557: 8554: 8553: 8545: 8529: 8522: 8468: 8467: 8433: 8432: 8420: 8417: 8416: 8386: 8385: 8382: 8379: 8378: 8351: 8350: 8347: 8344: 8343: 8284: 8283: 8269: 8268: 8256: 8253: 8252: 8226: 8225: 8222: 8219: 8218: 8193: 8185: 8171: 8167: 8166: 8162: 8155: 8151: 8149: 8137: 8126: 8125: 8124: 8100: 8098: 8097: 8095: 8092: 8091: 8058: 8047: 8046: 8045: 8042: 8039: 8038: 8012: 8008: 7996: 7982: 7978: 7977: 7973: 7972: 7966: 7962: 7938: 7937: 7925: 7922: 7921: 7895: 7894: 7891: 7888: 7887: 7847: 7843: 7824: 7820: 7779: 7778: 7766: 7763: 7762: 7728: 7727: 7724: 7721: 7720: 7713: 7703: 7694: 7687: 7683: 7667: 7662: 7629: 7622: 7615: 7608: 7593: 7586: 7574: 7561: 7552: 7537: 7532: 7520: 7517: 7510: 7495: 7488: 7466: 7451: 7413: 7409: 7403: 7399: 7384: 7383: 7369: 7366: 7365: 7327: 7323: 7315:autocorrelation 7283: 7255:identity matrix 7251: 7247: 7227: 7223: 7217: 7213: 7185: 7182: 7181: 7158: 7153: 7130: 7129: 7100: 7099: 7093: 7090: 7089: 7084:must have full 7049: 7041: 6999: 6996: 6995: 6967: 6918: 6916: 6909: 6890: 6884: 6879: 6860: 6856: 6848: 6844: 6840: 6826: 6796: 6795: 6780: 6775: 6762: 6758: 6757: 6753: 6747: 6743: 6736: 6735: 6730: 6728: 6725: 6724: 6710: 6686: 6663: 6662: 6654: 6647: 6646: 6640: 6635: 6634: 6625: 6615: 6609: 6604: 6603: 6602: 6598: 6597: 6594: 6593: 6587: 6582: 6581: 6572: 6562: 6556: 6551: 6550: 6549: 6545: 6544: 6537: 6536: 6528: 6519: 6512: 6511: 6506: 6504: 6499: 6492: 6491: 6490: 6483: 6476: 6475: 6464: 6463: 6460: 6459: 6448: 6447: 6440: 6439: 6434: 6431: 6430: 6420: 6419: 6408: 6407: 6404: 6403: 6392: 6391: 6384: 6383: 6376: 6375: 6370: 6368: 6363: 6356: 6355: 6348: 6343: 6339: 6337: 6334: 6333: 6302: 6301: 6296: 6282: 6281: 6276: 6268: 6254: 6252: 6251: 6249: 6246: 6245: 6212: 6204: 6199: 6196: 6195: 6169: 6168: 6166: 6163: 6162: 6143: 6138: 6130: 6128: 6125: 6124: 6102: 6101: 6096: 6088: 6086: 6083: 6082: 6058: 6053: 6036: 6035: 6030: 6022: 6017: 6014: 6013: 5980: 5979: 5968: 5966: 5963: 5962: 5937: 5931: 5927: 5916: 5915: 5910: 5902: 5897: 5894: 5893: 5873: 5847: 5846: 5844: 5841: 5840: 5830:linear subspace 5811: 5801:Euclidean space 5783: 5781: 5756: 5751: 5743: 5733: 5717: 5716: 5702: 5701: 5699: 5696: 5695: 5670: 5663: 5649: 5645: 5620: 5616: 5614: 5611: 5610: 5593: 5589: 5587: 5584: 5583: 5576: 5551: 5528: 5527: 5525: 5522: 5521: 5518: 5499: 5498: 5481: 5480: 5468: 5467: 5453: 5452: 5445: 5434: 5433: 5430: 5429: 5419: 5415: 5404: 5403: 5394: 5390: 5386: 5380: 5369: 5364: 5348: 5347: 5338: 5334: 5317: 5316: 5307: 5303: 5299: 5293: 5282: 5277: 5275: 5268: 5257: 5256: 5252: 5250: 5247: 5246: 5220: 5216: 5207: 5203: 5185: 5181: 5179: 5176: 5175: 5160: 5157: 5146: 5140: 5096: 5093: 5092: 5067: 5063: 5061: 5058: 5057: 5040: 5036: 5026: 5017: 5013: 5005: 5002: 5001: 4968: 4957: 4955: 4930: 4929: 4925: 4924: 4910: 4909: 4905: 4904: 4902: 4877: 4876: 4872: 4871: 4854: 4853: 4849: 4842: 4841: 4837: 4836: 4834: 4822: 4818: 4808: 4799: 4795: 4788: 4781: 4777: 4767: 4758: 4747: 4746: 4745: 4738: 4736: 4727: 4723: 4721: 4718: 4717: 4642: 4631: 4630: 4629: 4626: 4623: 4622: 4582: 4578: 4561: 4559: 4550: 4539: 4538: 4537: 4520: 4505: 4504: 4497: 4495: 4479: 4465: 4464: 4460: 4459: 4457: 4441: 4427: 4426: 4422: 4415: 4414: 4410: 4409: 4407: 4391: 4377: 4376: 4372: 4362: 4360: 4344: 4332: 4331: 4324: 4323: 4312: 4311: 4310: 4309: 4307: 4298: 4294: 4292: 4289: 4288: 4178: 4177: 4154: 4153: 4133: 4132: 4130: 4127: 4126: 4108: 4098: 4097:via identities 4084: 4074: 4046: 4037: 4006:onto the space 3945: 3944: 3927: 3926: 3924: 3921: 3920: 3872: 3854:cofactor matrix 3801: 3797: 3788: 3784: 3775: 3771: 3742: 3737: 3736: 3729: 3725: 3711: 3710: 3708: 3705: 3704: 3679: 3678: 3670: 3667: 3666: 3649:. The function 3636: 3632: 3575: 3571: 3547: 3543: 3534: 3529: 3516: 3512: 3503: 3492: 3471: 3468: 3467: 3425: 3420: 3413: 3407: 3390: 3383: 3378: 3377:. The quantity 3367: 3343: 3337: 3332: 3331: 3322: 3312: 3306: 3301: 3300: 3299: 3295: 3294: 3286: 3272: 3271: 3269: 3266: 3265: 3240: 3234: 3229: 3228: 3219: 3209: 3203: 3198: 3197: 3196: 3192: 3191: 3177: 3176: 3174: 3171: 3170: 3166:, expressed as 3141: 3140: 3138: 3135: 3134: 3114: 3108: 3103: 3102: 3100: 3097: 3096: 3095:and the matrix 3072: 3066: 3061: 3060: 3058: 3055: 3054: 3029: 3023: 3018: 3017: 3003: 3002: 2992: 2986: 2981: 2980: 2979: 2975: 2973: 2970: 2969: 2942: 2940: 2937: 2936: 2920: 2917: 2916: 2889: 2879: 2874: 2866: 2865: 2861: 2860: 2851: 2840: 2836: 2827: 2823: 2817: 2806: 2793: 2789: 2788: 2784: 2783: 2777: 2766: 2751: 2743: 2740: 2739: 2720: 2717: 2716: 2691: 2658: 2656: 2642: 2641: 2639: 2636: 2635: 2609: 2607: 2604: 2603: 2584: 2581: 2580: 2554: 2550: 2548: 2545: 2544: 2528: 2526: 2523: 2522: 2498: 2497: 2491: 2487: 2484: 2483: 2477: 2476: 2470: 2466: 2463: 2462: 2456: 2452: 2445: 2444: 2436: 2425: 2424: 2418: 2414: 2411: 2410: 2404: 2403: 2397: 2393: 2390: 2389: 2383: 2379: 2372: 2371: 2363: 2352: 2351: 2342: 2338: 2336: 2331: 2322: 2318: 2316: 2307: 2303: 2300: 2299: 2294: 2289: 2284: 2278: 2277: 2268: 2264: 2262: 2257: 2251: 2247: 2245: 2239: 2235: 2232: 2231: 2222: 2218: 2216: 2211: 2205: 2201: 2199: 2193: 2189: 2182: 2181: 2173: 2171: 2168: 2167: 2142: 2134: 2129: 2127: 2124: 2123: 2094: 2091: 2090: 2073: 2069: 2054: 2050: 2041: 2037: 2035: 2032: 2031: 2011: 2008: 2007: 1988: 1985: 1984: 1922: 1918: 1909: 1905: 1896: 1892: 1886: 1875: 1869: 1866: 1865: 1856: 1835: 1833: 1830: 1829: 1816:model would be 1787: 1786: 1777: 1773: 1771: 1768: 1767: 1746: 1742: 1740: 1737: 1736: 1702: 1699: 1698: 1672: 1668: 1666: 1663: 1662: 1646: 1644: 1641: 1640: 1621: 1618: 1617: 1600: 1595: 1590: 1587: 1584: 1583: 1567: 1564: 1563: 1537: 1534: 1533: 1517: 1515: 1512: 1511: 1495: 1492: 1491: 1469: 1466: 1465: 1449: 1447: 1444: 1443: 1427: 1425: 1422: 1421: 1398: 1390: 1385: 1377: 1375: 1372: 1371: 1351: 1346: 1345: 1343: 1340: 1339: 1322: 1318: 1316: 1313: 1312: 1295: 1291: 1289: 1286: 1285: 1269: 1266: 1265: 1244: 1240: 1238: 1235: 1234: 1212: 1209: 1208: 1192: 1190: 1187: 1186: 1170: 1167: 1166: 1149: 1144: 1143: 1141: 1138: 1137: 1113: 1109: 1101: 1095: 1090: 1085: 1075: 1071: 1069: 1066: 1065: 1035: 1031: 1019: 1015: 1006: 1002: 984: 980: 971: 967: 955: 951: 942: 938: 929: 925: 923: 920: 919: 899: 895: 893: 890: 889: 868: 854: 850: 832: 828: 816: 812: 811: 807: 806: 797: 792: 791: 789: 786: 785: 769: 766: 765: 748: 743: 742: 740: 737: 736: 719: 715: 713: 710: 709: 693: 690: 689: 672: 661: 650: 646: 637: 632: 631: 630: 626: 623: 620: 619: 601: 598: 597: 594: 588: 493:linear function 479:) is a type of 442: 402: 382:Goodness of fit 89:Discrete choice 28: 23: 22: 15: 12: 11: 5: 15907: 15897: 15896: 15891: 15874: 15873: 15871: 15870: 15865: 15860: 15848: 15843: 15837: 15834: 15833: 15831: 15830: 15825: 15820: 15815: 15810: 15804: 15802: 15798: 15797: 15795: 15794: 15789: 15784: 15779: 15774: 15769: 15764: 15758: 15756: 15747: 15746: 15744: 15743: 15738: 15736:Optimal design 15733: 15727: 15725: 15719: 15718: 15716: 15715: 15710: 15705: 15700: 15695: 15690: 15685: 15679: 15677: 15673: 15672: 15670: 15669: 15664: 15659: 15658: 15657: 15652: 15647: 15642: 15631: 15625: 15623: 15619: 15618: 15616: 15615: 15610: 15605: 15599: 15597: 15591: 15590: 15587: 15586: 15584: 15583: 15578: 15573: 15568: 15562: 15560: 15556: 15555: 15553: 15552: 15547: 15542: 15537: 15535:Semiparametric 15532: 15527: 15521: 15519: 15515: 15514: 15512: 15511: 15506: 15501: 15496: 15490: 15488: 15484: 15483: 15481: 15480: 15475: 15470: 15465: 15460: 15454: 15452: 15443: 15435: 15434: 15432: 15431: 15426: 15421: 15416: 15410: 15408: 15402: 15401: 15399: 15398: 15393: 15388: 15382: 15380:Spearman's rho 15373: 15367: 15365: 15359: 15358: 15356: 15355: 15350: 15345: 15340: 15334: 15332: 15326: 15325: 15314: 15313: 15306: 15299: 15291: 15285: 15284: 15278: 15258: 15252: 15239: 15233: 15220: 15214: 15194: 15188: 15173: 15170: 15168: 15167: 15160: 15138: 15123: 15111: 15099: 15087: 15080: 15059: 15047: 15035: 15028: 15007: 14995: 14988: 14959: 14942: 14930: 14906: 14883: 14871: 14856: 14844: 14832: 14817: 14805: 14790: 14778: 14771: 14753: 14738: 14724: 14709: 14702: 14682: 14675: 14655: 14648: 14628: 14616: 14607:Hayashi, Fumio 14598: 14591: 14560: 14536: 14524:Feature Column 14510: 14508: 14505: 14504: 14503: 14498: 14493: 14488: 14483: 14476: 14473: 14460: 14457: 14454: 14451: 14448: 14445: 14442: 14422: 14419: 14414: 14411: 14406: 14403: 14377: 14372: 14369: 14364: 14358: 14352: 14347: 14344: 14339: 14314: 14309: 14303: 14300: 14299: 14296: 14293: 14292: 14289: 14286: 14285: 14282: 14279: 14278: 14275: 14272: 14271: 14268: 14265: 14264: 14262: 14257: 14254: 14232: 14226: 14223: 14221: 14218: 14217: 14214: 14211: 14209: 14206: 14205: 14202: 14196: 14194: 14191: 14190: 14187: 14184: 14181: 14179: 14176: 14175: 14172: 14169: 14166: 14164: 14161: 14160: 14157: 14154: 14151: 14149: 14146: 14145: 14143: 14138: 14135: 14112: 14109: 14106: 14103: 14099: 14077: 14055: 14052: 14028: 14025: 14003: 13981: 13978: 13956: 13934: 13931: 13909: 13889: 13884: 13880: 13876: 13870: 13865: 13862: 13857: 13851: 13846: 13842: 13819: 13814: 13810: 13806: 13803: 13800: 13796: 13792: 13787: 13783: 13779: 13776: 13773: 13770: 13765: 13761: 13757: 13754: 13734: 13731: 13728: 13725: 13722: 13702: 13699: 13696: 13693: 13690: 13670: 13665: 13661: 13657: 13654: 13651: 13648: 13645: 13642: 13639: 13636: 13633: 13630: 13625: 13621: 13617: 13614: 13611: 13608: 13605: 13602: 13599: 13596: 13593: 13590: 13585: 13581: 13577: 13574: 13571: 13568: 13565: 13545: 13542: 13539: 13536: 13533: 13509: 13506: 13503: 13500: 13497: 13492: 13489: 13484: 13479: 13476: 13471: 13465: 13462: 13459: 13456: 13452: 13430: 13427: 13424: 13421: 13409: 13406: 13393: 13373: 13360: 13359: 13356: 13353: 13350: 13347: 13344: 13341: 13330: 13327: 13324: 13321: 13310: 13309: 13306: 13303: 13300: 13297: 13294: 13291: 13279: 13254: 13234: 13214: 13211: 13208: 13205: 13182: 13179: 13176: 13173: 13170: 13167: 13164: 13161: 13157: 13152: 13149: 13146: 13143: 13140: 13128: 13125: 13123: 13120: 13086: 13085: 13082: 13079: 13076: 13072: 13071: 13068: 13065: 13062: 13058: 13057: 13054: 13051: 13048: 13025:Main article: 13022: 13019: 13014: 13013: 13010: 12995: 12992: 12978: 12975: 12965:Residuals plot 12959: 12958: 12928: 12914: 12904: 12890: 12882: 12879:Log-likelihood 12875: 12874: 12873: 12872: 12861: 12856: 12852: 12848: 12845: 12842: 12836: 12833: 12830: 12825: 12822: 12819: 12813: 12810: 12807: 12802: 12796: 12793: 12777: 12776: 12762: 12758: 12735: 12731: 12716: 12698: 12657: 12650: 12647: 12639: 12633: 12626: 12623: 12616: 12613: 12588: 12574: 12571: 12565: 12559: 12556: 12551: 12546: 12543: 12538: 12535: 12531: 12527: 12520: 12513: 12510: 12502: 12497: 12492: 12485: 12482: 12459: 12455: 12442: 12441: 12438: 12437: 12434: 12431: 12428: 12422: 12421: 12418: 12415: 12412: 12406: 12405: 12402: 12399: 12396: 12390: 12389: 12386: 12383: 12380: 12379:Log-likelihood 12376: 12375: 12372: 12369: 12366: 12362: 12361: 12358: 12355: 12352: 12346: 12345: 12341: 12340: 12337: 12334: 12331: 12328: 12315: 12311: 12299: 12298: 12295: 12292: 12289: 12286: 12273: 12269: 12257: 12256: 12253: 12250: 12247: 12244: 12231: 12227: 12215: 12214: 12210: 12209: 12204: 12199: 12194: 12191: 12187: 12186: 12182: 12181: 12178: 12174: 12173: 12170: 12166: 12165: 12164:Least squares 12162: 12139: 12138: 12127: 12122: 12118: 12114: 12109: 12104: 12100: 12094: 12090: 12086: 12081: 12077: 12071: 12067: 12063: 12058: 12054: 12050: 12045: 12041: 12025: 12017: 12016: 12013: 12012: 12009: 12006: 12003: 12000: 11997: 11993: 11992: 11989: 11986: 11983: 11980: 11977: 11973: 11972: 11969: 11966: 11963: 11960: 11957: 11953: 11952: 11949: 11946: 11943: 11940: 11937: 11933: 11932: 11929: 11926: 11923: 11920: 11917: 11913: 11912: 11899: 11896: 11893: 11890: 11887: 11884: 11858: 11855: 11843:standard error 11816: 11815: 11795: 11793: 11779:Main article: 11776: 11773: 11772: 11771: 11765:1 − 11763: at the 11751: 11740: 11736: 11730: 11727: 11722: 11719: 11715: 11708: 11702: 11698: 11692: 11685: 11682: 11670: 11667: 11664: 11661: 11658: 11653: 11644: 11641: 11636: 11633: 11629: 11625: 11619: 11616: 11607: 11601: 11597: 11589: 11585: 11580: 11576: 11550: 11546: 11534: 11533: 11522: 11518: 11512: 11508: 11502: 11499: 11494: 11491: 11487: 11480: 11474: 11470: 11464: 11460: 11455: 11452: 11448: 11442: 11431: 11427: 11418: 11412: 11408: 11404: 11399: 11392: 11389: 11381: 11354: 11351: 11325: 11322: 11313: 11307: 11303: 11299: 11294: 11287: 11284: 11267:, whereas the 11256: 11250: 11244: 11240: 11236: 11231: 11227: 11200: 11196: 11184: 11183: 11172: 11168: 11162: 11158: 11154: 11150: 11145: 11140: 11136: 11132: 11128: 11125: 11121: 11118: 11114: 11108: 11097: 11093: 11085: 11080: 11076: 11072: 11067: 11060: 11057: 11050: 11034: 11015: 11004: 11003: 10997:1 − 10995: at the 10982: 10970: 10967: 10962: 10957: 10954: 10949: 10946: 10942: 10938: 10931: 10924: 10921: 10909: 10906: 10903: 10900: 10897: 10892: 10883: 10880: 10875: 10872: 10868: 10864: 10859: 10852: 10849: 10837: 10832: 10827: 10823: 10796: 10793: 10764: 10761: 10749: 10746: 10741: 10737: 10733: 10728: 10725: 10721: 10709: 10708: 10697: 10692: 10685: 10682: 10677: 10674: 10670: 10664: 10660: 10655: 10652: 10647: 10640: 10629: 10625: 10617: 10614: 10611: 10605: 10602: 10596: 10563: 10560: 10508: 10501: 10498: 10471: 10468: 10452: 10431: 10428: 10397: 10396: 10385: 10382: 10377: 10374: 10370: 10366: 10361: 10357: 10353: 10350: 10345: 10340: 10335: 10331: 10325: 10321: 10315: 10312: 10308: 10304: 10301: 10296: 10292: 10286: 10282: 10278: 10275: 10272: 10267: 10263: 10257: 10252: 10249: 10244: 10240: 10234: 10230: 10224: 10221: 10217: 10213: 10210: 10205: 10201: 10195: 10191: 10187: 10184: 10181: 10176: 10169: 10166: 10132: 10131: 10120: 10117: 10114: 10111: 10105: 10102: 10094: 10090: 10086: 10081: 10078: 10072: 10066: 10061: 10058: 10054: 10050: 10045: 10041: 10037: 10032: 10028: 10022: 10017: 10012: 10009: 10005: 10001: 9996: 9992: 9988: 9985: 9979: 9976: 9970: 9965: 9958: 9955: 9903: 9902: 9890: 9887: 9884: 9881: 9876: 9872: 9867: 9864: 9846:Main article: 9843: 9840: 9831: 9820: 9814: 9813: 9802: 9798: 9795: 9790: 9786: 9780: 9776: 9770: 9766: 9762: 9759: 9754: 9750: 9734: 9716: 9709: 9706: 9678: 9675: 9646: 9639: 9631: 9624: 9617: 9610: 9603: 9592: 9581: 9574: 9568: 9567: 9556: 9553: 9550: 9545: 9541: 9535: 9531: 9527: 9522: 9518: 9512: 9508: 9504: 9501: 9486: 9483: 9473: 9462: 9450: 9436: 9429: 9428: 9415: 9408: 9405: 9392: 9388: 9384: 9381: 9375: 9371: 9365: 9362: 9356: 9353: 9345: 9340: 9336: 9332: 9327: 9324: 9321: 9314: 9311: 9301: 9295: 9291: 9287: 9282: 9275: 9272: 9265: 9260: 9257: 9254: 9249: 9242: 9239: 9211: 9195: 9181: 9174: 9168: 9167: 9156: 9150: 9143: 9140: 9130: 9124: 9120: 9114: 9111: 9107: 9103: 9097: 9092: 9088: 9080: 9076: 9072: 9069: 9065: 9060: 9057: 9051: 9048: 9042: 9037: 9034: 9031: 9024: 9021: 8978: 8957: 8954: 8930:Main article: 8927: 8924: 8899: 8896: 8809: 8808: 8795: 8790: 8787: 8784: 8780: 8776: 8770: 8767: 8764: 8758: 8754: 8745: 8737: 8733: 8712:The estimator 8699: 8698: 8687: 8682: 8675: 8672: 8668: 8664: 8658: 8653: 8649: 8644: 8640: 8633: 8630: 8625: 8618: 8610: 8601: 8598: 8568: 8565: 8552:The estimator 8543: 8521: 8518: 8506: 8505: 8494: 8491: 8488: 8484: 8481: 8475: 8472: 8465: 8462: 8459: 8456: 8453: 8449: 8446: 8440: 8437: 8430: 8427: 8424: 8393: 8390: 8358: 8355: 8322: 8321: 8310: 8307: 8304: 8300: 8297: 8291: 8288: 8282: 8276: 8273: 8266: 8263: 8260: 8233: 8230: 8215: 8214: 8199: 8196: 8191: 8188: 8183: 8179: 8174: 8170: 8165: 8158: 8154: 8148: 8145: 8140: 8133: 8130: 8123: 8117: 8113: 8110: 8106: 8103: 8061: 8054: 8051: 8035: 8034: 8023: 8020: 8015: 8011: 8007: 8002: 7999: 7994: 7990: 7985: 7981: 7976: 7969: 7965: 7961: 7958: 7954: 7951: 7945: 7942: 7935: 7932: 7929: 7902: 7899: 7867: 7866: 7855: 7850: 7846: 7842: 7839: 7835: 7832: 7827: 7823: 7818: 7815: 7812: 7808: 7805: 7802: 7799: 7795: 7792: 7786: 7783: 7776: 7773: 7770: 7735: 7732: 7712: 7709: 7708: 7707: 7698: 7685: 7681: 7676: 7665: 7651: 7636: 7633:co-integrating 7627: 7620: 7613: 7606: 7591: 7584: 7573: 7570: 7569: 7568: 7555: 7546: 7535: 7526: 7515: 7508: 7493: 7486: 7450: 7447: 7446: 7445: 7424: 7421: 7416: 7412: 7406: 7402: 7398: 7395: 7392: 7387: 7382: 7379: 7376: 7373: 7355: 7354: 7353: 7310: 7249: 7235: 7230: 7226: 7220: 7216: 7212: 7209: 7205: 7202: 7199: 7195: 7192: 7189: 7175: 7156: 7141: 7138: 7133: 7127: 7124: 7121: 7118: 7115: 7112: 7109: 7103: 7097: 7088:almost surely: 7067: 7029: 7026: 7023: 7019: 7016: 7013: 7009: 7006: 7003: 6985: 6966: 6963: 6914: 6907: 6883: 6880: 6878: 6875: 6824: 6819: 6818: 6807: 6804: 6799: 6792: 6788: 6783: 6778: 6774: 6770: 6765: 6761: 6756: 6750: 6746: 6739: 6733: 6709: 6706: 6685: 6682: 6677: 6676: 6661: 6657: 6651: 6643: 6638: 6631: 6628: 6623: 6618: 6612: 6607: 6601: 6596: 6595: 6590: 6585: 6578: 6575: 6570: 6565: 6559: 6554: 6548: 6543: 6542: 6540: 6535: 6531: 6525: 6522: 6516: 6509: 6505: 6502: 6498: 6497: 6495: 6489: 6486: 6484: 6480: 6471: 6468: 6462: 6461: 6455: 6452: 6446: 6445: 6443: 6438: 6433: 6432: 6429: 6424: 6415: 6412: 6406: 6405: 6399: 6396: 6390: 6389: 6387: 6380: 6373: 6369: 6366: 6362: 6361: 6359: 6354: 6351: 6349: 6346: 6342: 6341: 6327: 6326: 6315: 6309: 6306: 6299: 6295: 6289: 6286: 6279: 6275: 6271: 6267: 6261: 6257: 6219: 6215: 6207: 6203: 6176: 6173: 6146: 6141: 6137: 6133: 6109: 6106: 6099: 6095: 6091: 6061: 6056: 6052: 6049: 6043: 6040: 6033: 6029: 6025: 6021: 5987: 5984: 5978: 5975: 5971: 5959: 5958: 5947: 5944: 5940: 5934: 5930: 5923: 5920: 5913: 5909: 5905: 5901: 5854: 5851: 5778: 5777: 5766: 5763: 5759: 5754: 5750: 5746: 5742: 5736: 5732: 5726: 5723: 5720: 5715: 5709: 5706: 5665: 5664: 5652: 5650: 5643: 5623: 5619: 5596: 5592: 5575: 5572: 5535: 5532: 5517: 5514: 5513: 5512: 5497: 5488: 5485: 5475: 5472: 5466: 5460: 5457: 5451: 5448: 5446: 5441: 5438: 5432: 5431: 5422: 5418: 5411: 5408: 5402: 5397: 5393: 5389: 5383: 5378: 5375: 5372: 5368: 5361: 5355: 5352: 5346: 5341: 5337: 5333: 5330: 5324: 5321: 5315: 5310: 5306: 5302: 5296: 5291: 5288: 5285: 5281: 5274: 5271: 5269: 5264: 5261: 5255: 5254: 5240: 5239: 5228: 5223: 5219: 5215: 5210: 5206: 5202: 5199: 5196: 5193: 5188: 5184: 5155: 5142:Main article: 5139: 5136: 5100: 5070: 5066: 5043: 5039: 5033: 5030: 5025: 5020: 5016: 5012: 5009: 4992: 4991: 4977: 4974: 4971: 4966: 4963: 4960: 4954: 4951: 4948: 4942: 4939: 4933: 4928: 4922: 4919: 4913: 4908: 4901: 4898: 4895: 4889: 4886: 4880: 4875: 4869: 4866: 4863: 4857: 4852: 4845: 4840: 4833: 4825: 4821: 4815: 4812: 4807: 4802: 4798: 4794: 4791: 4784: 4780: 4774: 4771: 4766: 4761: 4754: 4751: 4744: 4741: 4735: 4730: 4726: 4679:is called the 4671:. In practice 4645: 4638: 4635: 4599: 4598: 4585: 4581: 4574: 4570: 4567: 4564: 4558: 4553: 4546: 4543: 4535: 4529: 4526: 4523: 4518: 4512: 4509: 4503: 4500: 4494: 4488: 4485: 4482: 4477: 4474: 4468: 4463: 4456: 4450: 4447: 4444: 4439: 4436: 4430: 4425: 4418: 4413: 4406: 4400: 4397: 4394: 4389: 4386: 4380: 4375: 4371: 4368: 4365: 4359: 4353: 4350: 4347: 4339: 4336: 4327: 4319: 4316: 4306: 4301: 4297: 4272: 4271: 4260: 4257: 4254: 4251: 4248: 4245: 4242: 4239: 4236: 4233: 4230: 4227: 4224: 4221: 4218: 4215: 4212: 4209: 4206: 4203: 4200: 4197: 4194: 4191: 4185: 4182: 4176: 4173: 4170: 4167: 4161: 4158: 4152: 4149: 4146: 4140: 4137: 4073:(meaning that 4044: 4014:. This matrix 3979: 3978: 3967: 3964: 3961: 3958: 3952: 3949: 3943: 3940: 3934: 3931: 3891:is called the 3873:. The matrix ( 3868: 3827: 3826: 3815: 3809: 3804: 3800: 3794: 3791: 3787: 3783: 3778: 3774: 3770: 3767: 3764: 3761: 3758: 3755: 3752: 3745: 3740: 3735: 3732: 3728: 3724: 3718: 3715: 3686: 3683: 3677: 3674: 3634: 3630: 3613: 3612: 3601: 3598: 3595: 3592: 3589: 3586: 3583: 3578: 3574: 3570: 3567: 3564: 3561: 3558: 3555: 3550: 3546: 3542: 3537: 3532: 3528: 3524: 3519: 3515: 3511: 3506: 3501: 3498: 3495: 3491: 3487: 3484: 3481: 3478: 3475: 3418: 3411: 3388: 3381: 3366: 3363: 3362: 3361: 3350: 3346: 3340: 3335: 3328: 3325: 3320: 3315: 3309: 3304: 3298: 3293: 3289: 3285: 3279: 3276: 3259: 3258: 3247: 3243: 3237: 3232: 3225: 3222: 3217: 3212: 3206: 3201: 3195: 3190: 3184: 3181: 3148: 3145: 3117: 3111: 3106: 3075: 3069: 3064: 3051: 3050: 3039: 3032: 3026: 3021: 3016: 3010: 3007: 3000: 2995: 2989: 2984: 2978: 2945: 2924: 2909: 2908: 2897: 2892: 2887: 2882: 2877: 2873: 2869: 2864: 2859: 2854: 2849: 2843: 2839: 2833: 2830: 2826: 2820: 2815: 2812: 2809: 2805: 2801: 2796: 2792: 2787: 2780: 2775: 2772: 2769: 2765: 2761: 2758: 2754: 2750: 2747: 2724: 2713: 2712: 2701: 2698: 2694: 2690: 2687: 2681: 2677: 2674: 2671: 2667: 2664: 2661: 2655: 2649: 2646: 2612: 2588: 2568: 2565: 2560: 2557: 2553: 2531: 2519: 2518: 2507: 2502: 2494: 2490: 2486: 2485: 2482: 2479: 2478: 2473: 2469: 2465: 2464: 2459: 2455: 2451: 2450: 2448: 2443: 2439: 2434: 2429: 2421: 2417: 2413: 2412: 2409: 2406: 2405: 2400: 2396: 2392: 2391: 2386: 2382: 2378: 2377: 2375: 2370: 2366: 2361: 2356: 2348: 2345: 2341: 2337: 2335: 2332: 2328: 2325: 2321: 2317: 2313: 2310: 2306: 2302: 2301: 2298: 2295: 2293: 2290: 2288: 2285: 2283: 2280: 2279: 2274: 2271: 2267: 2263: 2261: 2258: 2254: 2250: 2246: 2242: 2238: 2234: 2233: 2228: 2225: 2221: 2217: 2215: 2212: 2208: 2204: 2200: 2196: 2192: 2188: 2187: 2185: 2180: 2176: 2161: 2160: 2149: 2145: 2141: 2137: 2132: 2104: 2101: 2098: 2076: 2072: 2068: 2065: 2062: 2057: 2053: 2049: 2044: 2040: 2015: 1992: 1981: 1980: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1948: 1945: 1942: 1939: 1936: 1930: 1925: 1921: 1917: 1912: 1908: 1902: 1899: 1895: 1889: 1884: 1881: 1878: 1874: 1855: 1852: 1838: 1794: 1791: 1785: 1780: 1776: 1749: 1745: 1724: 1721: 1718: 1715: 1712: 1709: 1706: 1686: 1683: 1678: 1675: 1671: 1649: 1625: 1603: 1598: 1593: 1571: 1547: 1544: 1541: 1520: 1499: 1479: 1476: 1473: 1452: 1430: 1418: 1417: 1405: 1401: 1397: 1393: 1388: 1384: 1380: 1354: 1349: 1325: 1321: 1298: 1294: 1273: 1247: 1243: 1222: 1219: 1216: 1195: 1174: 1152: 1147: 1134: 1133: 1121: 1116: 1112: 1108: 1104: 1098: 1093: 1088: 1083: 1078: 1074: 1055: 1054: 1043: 1038: 1034: 1030: 1025: 1022: 1018: 1009: 1005: 1001: 998: 995: 990: 987: 983: 974: 970: 966: 961: 958: 954: 945: 941: 937: 932: 928: 902: 898: 871: 866: 860: 857: 853: 849: 846: 843: 838: 835: 831: 827: 822: 819: 815: 810: 805: 800: 795: 773: 751: 746: 722: 718: 697: 675: 670: 667: 664: 659: 653: 649: 645: 640: 635: 629: 605: 590:Main article: 587: 584: 457:macroeconomics 444: 443: 441: 440: 433: 426: 418: 415: 414: 413: 412: 397: 396: 395: 394: 389: 384: 379: 374: 369: 361: 360: 356: 355: 354: 353: 348: 343: 338: 333: 325: 324: 323: 322: 317: 312: 307: 302: 294: 293: 292: 291: 286: 281: 276: 268: 267: 266: 265: 260: 255: 247: 246: 242: 241: 240: 239: 231: 230: 229: 228: 223: 218: 213: 208: 203: 198: 193: 191:Semiparametric 188: 183: 175: 174: 173: 172: 167: 162: 160:Random effects 157: 152: 144: 143: 142: 141: 136: 134:Ordered probit 131: 126: 121: 116: 111: 106: 101: 96: 91: 86: 81: 73: 72: 71: 70: 65: 60: 55: 47: 46: 42: 41: 35: 34: 26: 9: 6: 4: 3: 2: 15906: 15895: 15894:Least squares 15892: 15890: 15887: 15886: 15884: 15869: 15866: 15864: 15861: 15859: 15854: 15849: 15847: 15844: 15842: 15839: 15838: 15835: 15829: 15826: 15824: 15821: 15819: 15816: 15814: 15811: 15809: 15808:Curve fitting 15806: 15805: 15803: 15799: 15793: 15790: 15788: 15785: 15783: 15780: 15778: 15775: 15773: 15770: 15768: 15765: 15763: 15760: 15759: 15757: 15755: 15754:approximation 15752: 15748: 15742: 15739: 15737: 15734: 15732: 15729: 15728: 15726: 15724: 15720: 15714: 15711: 15709: 15706: 15704: 15701: 15699: 15696: 15694: 15691: 15689: 15686: 15684: 15681: 15680: 15678: 15674: 15668: 15665: 15663: 15660: 15656: 15653: 15651: 15648: 15646: 15645: 15637: 15636: 15635: 15632: 15630: 15627: 15626: 15624: 15620: 15614: 15611: 15609: 15606: 15604: 15601: 15600: 15598: 15596: 15592: 15582: 15579: 15577: 15574: 15572: 15569: 15567: 15564: 15563: 15561: 15557: 15551: 15548: 15546: 15543: 15541: 15538: 15536: 15533: 15531: 15530:Nonparametric 15528: 15526: 15523: 15522: 15520: 15516: 15510: 15507: 15505: 15502: 15500: 15497: 15495: 15492: 15491: 15489: 15485: 15479: 15476: 15474: 15471: 15469: 15466: 15464: 15461: 15459: 15456: 15455: 15453: 15451: 15447: 15444: 15442: 15436: 15430: 15427: 15425: 15422: 15420: 15417: 15415: 15412: 15411: 15409: 15407: 15403: 15397: 15394: 15392: 15389: 15386: 15385:Kendall's tau 15383: 15381: 15377: 15374: 15372: 15369: 15368: 15366: 15364: 15360: 15354: 15351: 15349: 15346: 15344: 15341: 15339: 15338:Least squares 15336: 15335: 15333: 15331: 15327: 15323: 15319: 15318:Least squares 15312: 15307: 15305: 15300: 15298: 15293: 15292: 15289: 15281: 15275: 15271: 15267: 15263: 15259: 15255: 15249: 15245: 15240: 15236: 15230: 15226: 15221: 15217: 15211: 15207: 15203: 15199: 15195: 15191: 15189:0-19-877643-8 15185: 15181: 15176: 15175: 15163: 15161:0-387-95364-7 15157: 15152: 15151: 15142: 15135: 15134:Amemiya (1985 15130: 15128: 15120: 15119:Amemiya (1985 15115: 15108: 15103: 15096: 15091: 15083: 15081:0-19-506011-3 15077: 15073: 15066: 15064: 15056: 15055:Amemiya (1985 15051: 15044: 15043:Amemiya (1985 15039: 15031: 15029:0-471-70823-2 15025: 15021: 15017: 15011: 15004: 15003:Amemiya (1985 14999: 14991: 14989:9780674005600 14985: 14981: 14976: 14975: 14969: 14963: 14956: 14955:Hayashi (2000 14951: 14949: 14947: 14939: 14938:Hayashi (2000 14934: 14916: 14910: 14902: 14898: 14894: 14887: 14880: 14879:Hayashi (2000 14875: 14868: 14867:Hayashi (2000 14863: 14861: 14853: 14852:Hayashi (2000 14848: 14841: 14840:Hayashi (2000 14836: 14829: 14828:Hayashi (2000 14824: 14822: 14814: 14813:Hayashi (2000 14809: 14801: 14794: 14787: 14786:Hayashi (2000 14782: 14774: 14772:0-8493-2479-3 14768: 14764: 14757: 14749: 14742: 14736: 14735: 14728: 14721: 14720:Hayashi (2000 14716: 14714: 14705: 14703:9783540727156 14699: 14695: 14694: 14686: 14678: 14676:9783211730171 14672: 14668: 14667: 14659: 14651: 14649:9780471697282 14645: 14641: 14640: 14632: 14625: 14624:Hayashi (2000 14620: 14612: 14608: 14602: 14594: 14592:0-471-31101-4 14588: 14584: 14580: 14579: 14574: 14570: 14564: 14550: 14546: 14540: 14525: 14521: 14515: 14511: 14502: 14499: 14497: 14494: 14492: 14489: 14487: 14484: 14482: 14479: 14478: 14472: 14471: 14458: 14455: 14452: 14449: 14446: 14443: 14440: 14420: 14417: 14412: 14409: 14404: 14401: 14391: 14370: 14367: 14356: 14345: 14342: 14325: 14312: 14307: 14301: 14294: 14287: 14280: 14273: 14266: 14260: 14255: 14252: 14230: 14224: 14219: 14212: 14207: 14200: 14192: 14185: 14182: 14177: 14170: 14167: 14162: 14155: 14152: 14147: 14141: 14136: 14133: 14107: 14101: 14097: 14075: 14053: 14050: 14026: 14023: 14001: 13979: 13976: 13954: 13932: 13929: 13907: 13887: 13882: 13878: 13874: 13863: 13860: 13849: 13844: 13840: 13831: 13812: 13808: 13801: 13798: 13794: 13785: 13781: 13774: 13771: 13768: 13763: 13759: 13755: 13752: 13729: 13723: 13720: 13697: 13691: 13688: 13663: 13659: 13652: 13649: 13643: 13637: 13634: 13631: 13623: 13619: 13612: 13609: 13603: 13597: 13594: 13591: 13583: 13579: 13575: 13572: 13566: 13563: 13540: 13534: 13531: 13524:by expanding 13523: 13504: 13498: 13495: 13490: 13487: 13482: 13477: 13474: 13469: 13460: 13454: 13450: 13425: 13419: 13405: 13391: 13371: 13357: 13354: 13351: 13348: 13345: 13342: 13325: 13319: 13312: 13311: 13307: 13304: 13301: 13298: 13295: 13292: 13290:(in degrees) 13277: 13270: 13269: 13266: 13252: 13232: 13209: 13203: 13177: 13171: 13168: 13165: 13162: 13159: 13155: 13150: 13144: 13138: 13119: 13117: 13113: 13107: 13105: 13104:extrapolation 13100: 13092: 13083: 13080: 13077: 13074: 13073: 13069: 13066: 13063: 13060: 13059: 13055: 13052: 13049: 13047: 13046: 13043: 13041: 13034: 13028: 13018: 13011: 12990: 12979: 12976: 12972: 12971: 12970: 12963: 12956: 12952: 12948: 12944: 12940: 12936: 12932: 12929: 12926: 12922: 12918: 12915: 12912: 12908: 12905: 12902: 12901: 12896: 12895: 12891: 12888: 12887: 12883: 12880: 12877: 12876: 12854: 12850: 12846: 12843: 12834: 12831: 12828: 12823: 12820: 12817: 12811: 12808: 12805: 12800: 12791: 12781: 12780: 12779: 12778: 12760: 12756: 12733: 12729: 12720: 12717: 12714: 12710: 12706: 12702: 12699: 12696: 12692: 12688: 12686: 12681: 12677: 12673: 12655: 12645: 12637: 12631: 12621: 12614: 12611: 12603: 12599: 12595: 12594: 12589: 12572: 12569: 12563: 12557: 12554: 12549: 12544: 12541: 12536: 12533: 12529: 12525: 12518: 12508: 12500: 12495: 12490: 12480: 12468: 12465:column shows 12464: 12460: 12458: 12451: 12447: 12446: 12445: 12435: 12429: 12427: 12424: 12423: 12419: 12413: 12411: 12408: 12407: 12403: 12397: 12395: 12392: 12391: 12387: 12381: 12378: 12377: 12373: 12367: 12364: 12363: 12359: 12353: 12351: 12348: 12347: 12342: 12338: 12335: 12332: 12329: 12313: 12309: 12301: 12300: 12296: 12293: 12290: 12287: 12271: 12267: 12259: 12258: 12254: 12251: 12248: 12245: 12229: 12225: 12217: 12216: 12211: 12208: 12205: 12203: 12200: 12198: 12195: 12192: 12189: 12188: 12183: 12176: 12175: 12168: 12167: 12160: 12159: 12156: 12155: 12154: 12152: 12143: 12125: 12120: 12116: 12112: 12107: 12102: 12098: 12092: 12088: 12084: 12079: 12075: 12069: 12065: 12061: 12056: 12052: 12048: 12043: 12039: 12031: 12030: 12029: 12022: 12010: 12007: 12004: 12001: 11998: 11995: 11994: 11990: 11987: 11984: 11981: 11978: 11975: 11974: 11970: 11967: 11964: 11961: 11958: 11955: 11954: 11950: 11947: 11944: 11941: 11938: 11935: 11934: 11930: 11927: 11924: 11921: 11918: 11915: 11914: 11908: 11904: 11897: 11894: 11891: 11888: 11885: 11882: 11881: 11878: 11877: 11876: 11874: 11868: 11864: 11854: 11851: 11846: 11844: 11840: 11834: 11832: 11828: 11824: 11812: 11809:February 2017 11803: 11799: 11796:This section 11794: 11791: 11787: 11786: 11782: 11768: 11749: 11738: 11734: 11728: 11725: 11720: 11717: 11713: 11700: 11696: 11690: 11680: 11665: 11662: 11659: 11642: 11639: 11634: 11631: 11627: 11623: 11614: 11599: 11595: 11587: 11583: 11578: 11574: 11566: 11565: 11564: 11548: 11544: 11520: 11516: 11510: 11506: 11500: 11497: 11492: 11489: 11485: 11472: 11468: 11462: 11458: 11453: 11450: 11446: 11429: 11425: 11416: 11410: 11406: 11402: 11397: 11387: 11379: 11371: 11370: 11369: 11349: 11320: 11305: 11301: 11297: 11292: 11282: 11270: 11254: 11242: 11238: 11234: 11229: 11225: 11216: 11215:mean response 11198: 11194: 11170: 11166: 11160: 11156: 11152: 11148: 11143: 11138: 11134: 11130: 11126: 11119: 11116: 11112: 11095: 11091: 11078: 11074: 11070: 11065: 11055: 11041: 11040: 11039: 11037: 11030: 11025: 11023: 11018: 11013: 11009: 11000: 10968: 10965: 10960: 10955: 10952: 10947: 10944: 10940: 10936: 10929: 10919: 10904: 10901: 10898: 10881: 10878: 10873: 10870: 10866: 10862: 10857: 10847: 10830: 10825: 10821: 10813: 10812: 10811: 10791: 10780: 10774: 10770: 10760: 10747: 10744: 10735: 10731: 10726: 10723: 10719: 10695: 10683: 10680: 10675: 10672: 10668: 10662: 10658: 10653: 10650: 10627: 10623: 10612: 10609: 10600: 10587: 10586: 10585: 10583: 10579: 10558: 10547: 10543: 10538: 10536: 10532: 10528: 10524: 10506: 10496: 10466: 10455: 10447: 10445: 10441: 10437: 10427: 10425: 10419: 10414: 10411: − 10410: 10406: 10402: 10383: 10380: 10375: 10372: 10364: 10355: 10348: 10338: 10329: 10319: 10313: 10310: 10302: 10299: 10290: 10280: 10273: 10270: 10265: 10261: 10250: 10247: 10238: 10228: 10222: 10219: 10211: 10208: 10199: 10189: 10182: 10179: 10174: 10164: 10153: 10152: 10151: 10149: 10145: 10141: 10137: 10118: 10112: 10109: 10100: 10088: 10079: 10076: 10064: 10059: 10056: 10048: 10039: 10026: 10015: 10010: 10007: 9999: 9990: 9983: 9974: 9968: 9963: 9953: 9942: 9941: 9940: 9938: 9934: 9929: 9924: 9920: 9916: 9912: 9908: 9888: 9885: 9882: 9879: 9870: 9865: 9862: 9855: 9854: 9853: 9849: 9839: 9835: 9830: 9826: 9819: 9800: 9796: 9793: 9788: 9784: 9778: 9774: 9768: 9764: 9760: 9757: 9752: 9748: 9740: 9739: 9738: 9733: 9714: 9704: 9673: 9662: 9661: 9655: 9652: 9645: 9638: 9630: 9623: 9616: 9609: 9602: 9598: 9591: 9587: 9580: 9573: 9554: 9551: 9548: 9543: 9539: 9533: 9529: 9525: 9520: 9516: 9510: 9506: 9502: 9499: 9492: 9491: 9490: 9482: 9480: 9476: 9469: 9465: 9457: 9453: 9445: 9439: 9413: 9403: 9390: 9386: 9382: 9379: 9373: 9369: 9363: 9360: 9351: 9338: 9334: 9330: 9322: 9309: 9293: 9289: 9285: 9280: 9270: 9263: 9255: 9247: 9237: 9226: 9225: 9224: 9222: 9218: 9214: 9207: 9203: 9198: 9191: 9188: 9184: 9177: 9154: 9148: 9138: 9122: 9118: 9112: 9109: 9101: 9090: 9078: 9074: 9070: 9067: 9063: 9058: 9055: 9046: 9040: 9032: 9019: 9008: 9007: 9006: 9004: 9000: 8996: 8991: 8989: 8985: 8981: 8974: 8971:is linear in 8952: 8939: 8933: 8923: 8921: 8917: 8894: 8881: 8877: 8870: 8866: 8862: 8858: 8853: 8849: 8845: 8840: 8836: 8830: 8824: 8820: 8816: 8793: 8788: 8785: 8782: 8778: 8774: 8768: 8765: 8762: 8756: 8752: 8743: 8735: 8731: 8723: 8722: 8721: 8719: 8715: 8710: 8708: 8704: 8685: 8673: 8670: 8662: 8651: 8642: 8638: 8631: 8628: 8608: 8596: 8586: 8585: 8584: 8563: 8550: 8546: 8540: 8536: 8532: 8527: 8517: 8515: 8511: 8492: 8489: 8482: 8479: 8470: 8460: 8457: 8454: 8447: 8444: 8435: 8425: 8422: 8415: 8414: 8413: 8411: 8388: 8376: 8353: 8341: 8340:homoscedastic 8337: 8333: 8329: 8328: 8308: 8305: 8298: 8295: 8286: 8280: 8271: 8261: 8258: 8251: 8250: 8249: 8228: 8197: 8194: 8189: 8186: 8181: 8177: 8168: 8163: 8156: 8152: 8146: 8138: 8128: 8115: 8111: 8104: 8090: 8089: 8088: 8086: 8082: 8078: 8059: 8049: 8021: 8018: 8013: 8009: 8005: 8000: 7997: 7992: 7988: 7979: 7974: 7967: 7963: 7959: 7952: 7949: 7940: 7930: 7927: 7920: 7919: 7918: 7897: 7885: 7881: 7880: 7874: 7872: 7853: 7848: 7844: 7840: 7833: 7830: 7825: 7821: 7813: 7806: 7803: 7800: 7793: 7790: 7781: 7771: 7761: 7760: 7759: 7757: 7753: 7730: 7718: 7701: 7697: 7692: 7688: 7677: 7674: 7668: 7660: 7656: 7652: 7649: 7645: 7641: 7640:predetermined 7637: 7634: 7630: 7623: 7616: 7609: 7602: 7598: 7594: 7587: 7580: 7576: 7575: 7565: 7559: 7556: 7550: 7547: 7544: 7538: 7530: 7527: 7523: 7518: 7511: 7504: 7500: 7496: 7489: 7482: 7479: 7478: 7477: 7475: 7469: 7464: 7463:random sample 7460: 7456: 7443: 7439: 7422: 7414: 7410: 7404: 7400: 7396: 7393: 7380: 7377: 7374: 7371: 7363: 7359: 7356: 7351: 7347: 7343: 7339: 7334: 7330: 7321: 7317: 7316: 7311: 7308: 7304: 7300: 7296: 7292: 7287: 7281: 7280: 7276: 7275: 7272: 7268: 7264: 7260: 7257:in dimension 7256: 7233: 7228: 7224: 7218: 7214: 7210: 7203: 7200: 7197: 7190: 7187: 7179: 7176: 7173: 7169: 7165: 7159: 7139: 7136: 7125: 7122: 7116: 7110: 7107: 7087: 7083: 7079: 7075: 7071: 7068: 7065: 7061: 7060: 7055: 7047: 7027: 7024: 7017: 7014: 7011: 7004: 6993: 6989: 6986: 6983: 6980: 6979: 6978: 6976: 6972: 6962: 6960: 6956: 6952: 6948: 6944: 6940: 6936: 6932: 6928: 6924: 6917: 6910: 6903: 6902:random design 6898: 6895: 6889: 6874: 6871: 6867: 6863: 6854: 6837: 6835: 6831: 6827: 6805: 6802: 6790: 6786: 6776: 6772: 6768: 6763: 6759: 6754: 6748: 6744: 6723: 6722: 6721: 6719: 6715: 6705: 6703: 6699: 6695: 6691: 6681: 6659: 6649: 6629: 6626: 6621: 6599: 6576: 6573: 6568: 6546: 6538: 6533: 6523: 6520: 6514: 6493: 6487: 6485: 6478: 6441: 6427: 6422: 6385: 6378: 6357: 6352: 6350: 6332: 6331: 6330: 6313: 6293: 6273: 6265: 6244: 6243: 6242: 6240: 6236: 6233: 6193: 6190:and a matrix 6159: 6135: 6093: 6080: 6076: 6050: 6027: 6012: 6008: 6004: 5976: 5973: 5945: 5942: 5907: 5892: 5891: 5890: 5887: 5885: 5880: 5876: 5871: 5849: 5838: 5834: 5831: 5827: 5823: 5818: 5814: 5809: 5805: 5802: 5799:-dimensional 5798: 5794: 5792: 5764: 5748: 5734: 5713: 5704: 5694: 5693: 5692: 5690: 5686: 5682: 5677: 5673: 5661: 5660: 5655: 5651: 5642: 5641: 5621: 5617: 5594: 5590: 5580: 5571: 5568: 5565: 5561: 5558: 5554: 5530: 5495: 5483: 5473: 5470: 5464: 5455: 5449: 5447: 5439: 5436: 5420: 5406: 5400: 5395: 5391: 5381: 5376: 5373: 5370: 5366: 5350: 5344: 5339: 5335: 5319: 5313: 5308: 5304: 5294: 5289: 5286: 5283: 5279: 5272: 5270: 5262: 5259: 5245: 5244: 5243: 5226: 5221: 5217: 5213: 5208: 5204: 5200: 5197: 5194: 5191: 5186: 5182: 5174: 5173: 5172: 5168: 5164: 5158: 5151: 5145: 5135: 5133: 5128: 5126: 5122: 5118: 5114: 5098: 5090: 5086: 5068: 5064: 5041: 5037: 5031: 5028: 5023: 5018: 5014: 5010: 5007: 4999: 4998: 4952: 4949: 4946: 4940: 4937: 4926: 4920: 4917: 4906: 4899: 4896: 4893: 4887: 4884: 4873: 4867: 4864: 4861: 4850: 4838: 4831: 4823: 4810: 4805: 4800: 4796: 4789: 4782: 4769: 4764: 4759: 4749: 4739: 4733: 4728: 4724: 4716: 4715: 4714: 4712: 4708: 4705: 4704: 4699: 4694: 4692: 4688: 4684: 4683: 4678: 4674: 4670: 4666: 4662: 4643: 4633: 4620: 4616: 4612: 4608: 4604: 4583: 4579: 4572: 4568: 4565: 4562: 4556: 4551: 4541: 4533: 4527: 4524: 4521: 4507: 4498: 4492: 4486: 4483: 4480: 4475: 4472: 4461: 4454: 4448: 4445: 4442: 4437: 4434: 4423: 4411: 4404: 4398: 4395: 4392: 4387: 4384: 4369: 4366: 4357: 4351: 4348: 4345: 4334: 4314: 4304: 4299: 4295: 4287: 4286: 4285: 4283: 4282: 4277: 4258: 4255: 4252: 4249: 4246: 4243: 4240: 4237: 4231: 4228: 4222: 4216: 4213: 4210: 4207: 4201: 4198: 4195: 4192: 4189: 4180: 4174: 4171: 4168: 4165: 4156: 4150: 4147: 4144: 4135: 4125: 4124: 4123: 4121: 4117: 4111: 4105: 4101: 4096: 4091: 4087: 4081: 4077: 4072: 4068: 4064: 4060: 4056: 4051: 4047: 4040: 4035: 4031: 4027: 4023: 4022: 4017: 4013: 4009: 4005: 4004: 3999: 3995: 3992: 3988: 3984: 3965: 3962: 3959: 3956: 3947: 3941: 3938: 3929: 3919: 3918: 3917: 3915: 3911: 3910: 3909:fitted values 3905: 3900: 3898: 3894: 3890: 3887: 3883: 3879: 3876: 3871: 3867: 3863: 3859: 3855: 3851: 3847: 3843: 3839: 3836: 3832: 3813: 3807: 3798: 3792: 3789: 3781: 3772: 3765: 3759: 3753: 3750: 3743: 3733: 3730: 3726: 3722: 3713: 3703: 3702: 3701: 3681: 3675: 3672: 3664: 3660: 3656: 3652: 3648: 3647: 3641: 3637: 3626: 3622: 3618: 3599: 3593: 3590: 3587: 3584: 3568: 3565: 3562: 3559: 3553: 3548: 3540: 3530: 3526: 3522: 3517: 3513: 3504: 3499: 3496: 3493: 3489: 3485: 3479: 3473: 3466: 3465: 3464: 3462: 3458: 3454: 3450: 3446: 3442: 3441: 3435: 3432: 3428: 3421: 3414: 3405: 3401: 3400: 3396:, called the 3394: 3391: 3384: 3376: 3372: 3348: 3326: 3323: 3318: 3296: 3291: 3283: 3264: 3263: 3262: 3245: 3223: 3220: 3215: 3193: 3188: 3169: 3168: 3167: 3165: 3132: 3131:moment matrix 3094: 3090: 3089:normal matrix 3037: 3014: 2998: 2976: 2968: 2967: 2966: 2964: 2960: 2922: 2914: 2895: 2890: 2871: 2857: 2852: 2847: 2841: 2837: 2831: 2828: 2824: 2818: 2813: 2810: 2807: 2803: 2799: 2794: 2790: 2785: 2778: 2773: 2770: 2767: 2763: 2759: 2745: 2738: 2737: 2736: 2722: 2699: 2685: 2653: 2634: 2633: 2632: 2630: 2627: 2600: 2586: 2566: 2563: 2558: 2555: 2551: 2505: 2500: 2492: 2488: 2480: 2471: 2467: 2457: 2453: 2446: 2441: 2432: 2427: 2419: 2415: 2407: 2398: 2394: 2384: 2380: 2373: 2368: 2359: 2354: 2346: 2343: 2339: 2333: 2326: 2323: 2319: 2311: 2308: 2304: 2296: 2291: 2286: 2281: 2272: 2269: 2265: 2259: 2252: 2248: 2240: 2236: 2226: 2223: 2219: 2213: 2206: 2202: 2194: 2190: 2183: 2178: 2166: 2165: 2164: 2147: 2139: 2122: 2121: 2120: 2118: 2102: 2099: 2096: 2074: 2070: 2066: 2063: 2060: 2055: 2051: 2047: 2042: 2038: 2029: 2013: 2005: 1990: 1967: 1961: 1958: 1955: 1952: 1949: 1946: 1943: 1940: 1937: 1928: 1923: 1919: 1915: 1910: 1906: 1900: 1897: 1893: 1887: 1882: 1879: 1876: 1872: 1864: 1863: 1862: 1861: 1851: 1827: 1823: 1819: 1813: 1809: 1789: 1783: 1778: 1774: 1765: 1747: 1743: 1722: 1719: 1716: 1713: 1710: 1707: 1704: 1684: 1681: 1676: 1673: 1669: 1637: 1623: 1596: 1569: 1561: 1560:design matrix 1545: 1542: 1539: 1497: 1477: 1474: 1471: 1403: 1395: 1382: 1370: 1369: 1368: 1352: 1323: 1319: 1296: 1292: 1271: 1263: 1245: 1241: 1220: 1217: 1214: 1172: 1150: 1119: 1114: 1110: 1106: 1091: 1081: 1076: 1072: 1064: 1063: 1062: 1060: 1041: 1036: 1032: 1028: 1023: 1020: 1016: 1007: 1003: 999: 996: 993: 988: 985: 981: 972: 968: 964: 959: 956: 952: 943: 939: 935: 930: 926: 918: 917: 916: 900: 896: 887: 864: 858: 855: 851: 847: 844: 841: 836: 833: 829: 825: 820: 817: 813: 808: 803: 798: 771: 749: 720: 716: 695: 673: 668: 665: 662: 657: 651: 647: 643: 638: 627: 618: 603: 593: 583: 581: 577: 573: 569: 565: 561: 560:homoscedastic 557: 553: 549: 545: 541: 537: 532: 530: 526: 522: 516: 514: 510: 506: 502: 501:least squares 498: 494: 490: 486: 482: 478: 474: 470: 462: 458: 454: 450: 439: 434: 432: 427: 425: 420: 419: 417: 416: 411: 406: 401: 400: 399: 398: 393: 390: 388: 385: 383: 380: 378: 375: 373: 370: 368: 365: 364: 363: 362: 358: 357: 352: 349: 347: 344: 342: 339: 337: 334: 332: 329: 328: 327: 326: 321: 318: 316: 313: 311: 308: 306: 303: 301: 298: 297: 296: 295: 290: 287: 285: 282: 280: 277: 275: 272: 271: 270: 269: 264: 261: 259: 256: 254: 253:Least squares 251: 250: 249: 248: 244: 243: 238: 235: 234: 233: 232: 227: 224: 222: 219: 217: 214: 212: 209: 207: 204: 202: 199: 197: 194: 192: 189: 187: 186:Nonparametric 184: 182: 179: 178: 177: 176: 171: 168: 166: 163: 161: 158: 156: 155:Fixed effects 153: 151: 148: 147: 146: 145: 140: 137: 135: 132: 130: 129:Ordered logit 127: 125: 122: 120: 117: 115: 112: 110: 107: 105: 102: 100: 97: 95: 92: 90: 87: 85: 82: 80: 77: 76: 75: 74: 69: 66: 64: 61: 59: 56: 54: 51: 50: 49: 48: 44: 43: 40: 37: 36: 32: 31: 19: 15801:Applications 15640: 15518:Non-standard 15462: 15413: 15269: 15243: 15224: 15205: 15179: 15149: 15141: 15114: 15102: 15090: 15071: 15050: 15038: 15019: 15010: 14998: 14973: 14962: 14933: 14921:. Retrieved 14909: 14900: 14896: 14886: 14874: 14847: 14835: 14808: 14793: 14781: 14762: 14756: 14747: 14741: 14733: 14727: 14692: 14685: 14665: 14658: 14638: 14631: 14619: 14611:Econometrics 14610: 14601: 14577: 14563: 14552:. Retrieved 14548: 14539: 14528:. Retrieved 14526:. 2022-03-01 14523: 14514: 14394: 14392: 14326: 13832: 13411: 13363: 13130: 13115: 13111: 13108: 13101: 13097: 13039: 13036: 13015: 12968: 12946: 12942: 12938: 12934: 12930: 12924: 12920: 12916: 12910: 12906: 12898: 12892: 12884: 12878: 12718: 12712: 12708: 12700: 12694: 12684: 12679: 12675: 12671: 12601: 12597: 12591: 12462: 12453: 12449: 12443: 12177:Observations 12148: 12018: 11996:Weight (kg) 11956:Weight (kg) 11916:Weight (kg) 11872: 11870: 11847: 11835: 11819: 11806: 11802:adding to it 11797: 11766: 11535: 11185: 11032: 11028: 11026: 11021: 11016: 11010:denotes the 11007: 11005: 10998: 10778: 10776: 10710: 10581: 10541: 10539: 10534: 10530: 10526: 10450: 10448: 10439: 10433: 10423: 10417: 10412: 10408: 10404: 10400: 10398: 10147: 10143: 10139: 10135: 10133: 9936: 9932: 9927: 9922: 9918: 9914: 9910: 9906: 9904: 9851: 9836: 9828: 9817: 9815: 9731: 9658: 9656: 9650: 9643: 9636: 9628: 9621: 9614: 9607: 9600: 9596: 9589: 9585: 9578: 9571: 9569: 9488: 9478: 9471: 9467: 9460: 9455: 9448: 9443: 9434: 9430: 9220: 9216: 9209: 9205: 9201: 9193: 9189: 9186: 9179: 9172: 9169: 9002: 8994: 8992: 8987: 8983: 8976: 8972: 8941: 8915: 8882: 8875: 8868: 8864: 8860: 8859:= SSR 8856: 8847: 8843: 8838: 8834: 8822: 8818: 8814: 8810: 8713: 8711: 8700: 8551: 8541: 8538: 8534: 8530: 8525: 8523: 8513: 8507: 8409: 8374: 8336:uncorrelated 8331: 8325: 8323: 8216: 8084: 8080: 8076: 8036: 7917:is equal to 7883: 7877: 7875: 7868: 7751: 7716: 7714: 7699: 7695: 7679: 7663: 7658: 7654: 7647: 7643: 7639: 7625: 7618: 7611: 7604: 7589: 7582: 7563: 7557: 7548: 7533: 7528: 7521: 7513: 7506: 7503:distribution 7491: 7484: 7480: 7467: 7452: 7357: 7349: 7332: 7328: 7320:uncorrelated 7312: 7290: 7285: 7277: 7266: 7262: 7258: 7177: 7171: 7167: 7154: 7081: 7076:must all be 7073: 7069: 7057: 7053: 6987: 6981: 6970: 6968: 6958: 6950: 6946: 6938: 6935:fixed design 6934: 6921:s from some 6912: 6905: 6901: 6899: 6891: 6869: 6865: 6861: 6838: 6833: 6829: 6822: 6820: 6711: 6687: 6678: 6328: 6234: 6231: 6191: 6161:Introducing 6160: 6078: 6074: 6009:, since the 6006: 6003:column space 5960: 5888: 5883: 5878: 5874: 5836: 5821: 5816: 5812: 5807: 5803: 5796: 5790: 5779: 5688: 5684: 5680: 5675: 5671: 5668: 5657: 5653: 5566: 5563: 5559: 5556: 5552: 5519: 5241: 5166: 5162: 5153: 5149: 5147: 5129: 5124: 5120: 5116: 5088: 5084: 4995: 4993: 4710: 4706: 4701: 4697: 4695: 4690: 4686: 4680: 4676: 4672: 4660: 4618: 4614: 4606: 4602: 4600: 4279: 4275: 4273: 4119: 4118:creates the 4115: 4109: 4103: 4099: 4094: 4089: 4085: 4079: 4075: 4062: 4058: 4054: 4049: 4042: 4038: 4033: 4029: 4025: 4019: 4015: 4011: 4007: 4001: 3997: 3993: 3990: 3986: 3982: 3980: 3913: 3907: 3903: 3901: 3888: 3885: 3881: 3877: 3874: 3869: 3865: 3857: 3853: 3849: 3845: 3837: 3834: 3830: 3829:The product 3828: 3658: 3654: 3650: 3645: 3643: 3639: 3628: 3624: 3616: 3614: 3460: 3456: 3452: 3448: 3444: 3438: 3433: 3430: 3426: 3416: 3409: 3403: 3397: 3392: 3386: 3379: 3374: 3370: 3368: 3260: 3088: 3052: 2962: 2910: 2735:is given by 2714: 2629:minimization 2601: 2520: 2162: 2028:coefficients 1982: 1858:Consider an 1857: 1825: 1821: 1817: 1814: 1810: 1763: 1638: 1562:, whose row 1419: 1135: 1056: 617:observations 595: 586:Linear model 533: 517: 495:of a set of 476: 472: 466: 310:Non-negative 273: 14923:28 December 14854:, page 187) 12931:F-statistic 12593:t-statistic 12417:F-statistic 12202:t-statistic 12021:scatterplot 11976:Height (m) 11936:Height (m) 11907:Scatterplot 11883:Height (m) 11839:t-statistic 9477:are called 8988:influential 8920:independent 7882:(or simply 7871:time series 7499:independent 7338:time series 7086:column rank 6925:, as in an 6882:Assumptions 6859:is to take 6011:dot product 4284:statistic: 4034:annihilator 3842:Gram matrix 3093:Gram matrix 3053:The matrix 544:colinearity 320:Regularized 284:Generalized 216:Least angle 114:Mixed logit 15883:Categories 15676:Background 15639:Mallows's 15136:, page 22) 15121:, page 21) 15109:, page 20) 15097:, page 36) 15057:, page 27) 15045:, page 20) 15016:Rao, C. R. 15005:, page 14) 14957:, page 27) 14881:, page 34) 14869:, page 10) 14830:, page 52) 14815:, page 49) 14788:, page 20) 14722:, page 19) 14626:, page 18) 14554:2022-09-28 14530:2024-05-16 14507:References 13031:See also: 12365:Adjusted R 12190:Parameter 11861:See also: 10546:consistent 10527:asymptotic 8936:See also: 7646:= 1, ..., 7597:stationary 7549:exogeneity 7519:) for all 7342:panel data 7059:endogenous 6955:experiment 6923:population 6886:See also: 6877:Properties 5793: norm 5574:Projection 4278:using the 4071:idempotent 4021:hat matrix 3365:Estimation 3164:hyperplane 2913:Properties 536:consistent 485:parameters 469:statistics 453:Okun's law 359:Background 263:Non-linear 245:Estimation 15751:Numerical 14842:, page 7) 14450:⋅ 14183:− 14168:− 14153:− 14108:θ 13809:θ 13802:⁡ 13782:θ 13775:⁡ 13760:θ 13756:⁡ 13730:θ 13724:⁡ 13698:θ 13692:⁡ 13660:θ 13653:⁡ 13644:θ 13638:⁡ 13620:θ 13613:⁡ 13604:θ 13598:⁡ 13580:θ 13576:− 13573:θ 13567:⁡ 13541:θ 13535:⁡ 13505:θ 13499:⁡ 13483:− 13461:θ 13426:θ 13326:θ 13278:θ 13210:θ 13178:θ 13172:⁡ 13163:− 13145:θ 13081:−131.5076 13070:61.96033 12994:^ 12951:Wald test 12847:− 12832:− 12821:− 12812:− 12795:¯ 12713:R-squared 12701:R-squared 12649:^ 12646:σ 12625:^ 12622:β 12542:− 12512:^ 12509:σ 12484:^ 12481:σ 12463:Std error 12310:β 12288:–143.1620 12268:β 12226:β 12197:Std error 12117:ε 12089:β 12066:β 12053:β 11850:Chow test 11726:− 11684:^ 11681:σ 11640:α 11635:− 11624:± 11618:^ 11615:β 11584:∈ 11498:− 11459:σ 11403:− 11391:^ 11353:^ 11350:β 11324:^ 11321:β 11286:^ 11255:β 11157:σ 11153:− 11135:ε 11127:⁡ 11075:σ 11071:− 11059:^ 11056:σ 10953:− 10923:^ 10920:σ 10879:α 10874:− 10863:± 10851:^ 10848:β 10831:∈ 10822:β 10795:^ 10792:β 10763:Intervals 10681:− 10659:σ 10613:β 10610:− 10604:^ 10601:β 10562:^ 10559:β 10548:(that is 10500:^ 10497:σ 10470:^ 10467:β 10373:− 10311:− 10271:− 10220:− 10168:^ 10165:β 10110:− 10104:^ 10101:β 10077:− 10057:− 10008:− 9984:− 9978:^ 9975:β 9957:^ 9954:β 9880:β 9866:: 9797:η 9785:β 9708:^ 9705:β 9677:^ 9674:ε 9552:ε 9540:β 9517:β 9468:leverages 9407:^ 9404:ε 9383:− 9364:− 9355:^ 9352:β 9331:− 9313:^ 9310:β 9274:^ 9264:− 9241:^ 9142:^ 9139:ε 9110:− 9071:− 9059:− 9050:^ 9047:β 9041:− 9023:^ 9020:β 8956:^ 8953:β 8898:^ 8895:β 8786:− 8779:χ 8775:⋅ 8766:− 8753:σ 8744:∼ 8671:− 8639:σ 8629:β 8609:∼ 8600:^ 8597:β 8567:^ 8564:β 8490:≥ 8480:∣ 8474:^ 8471:β 8461:⁡ 8455:− 8445:∣ 8439:~ 8436:β 8426:⁡ 8392:~ 8389:β 8357:^ 8354:β 8296:∣ 8290:^ 8287:ε 8275:^ 8272:β 8262:⁡ 8232:^ 8229:β 8195:− 8132:^ 8129:β 8116:^ 8053:^ 8050:β 8010:σ 7998:− 7964:σ 7950:∣ 7944:^ 7941:β 7931:⁡ 7901:^ 7898:β 7845:σ 7831:∣ 7814:⁡ 7804:β 7791:∣ 7785:^ 7782:β 7772:⁡ 7734:^ 7731:β 7401:σ 7381:∼ 7375:∣ 7372:ε 7358:Normality 7215:σ 7201:∣ 7198:ε 7191:⁡ 7111:⁡ 7054:exogenous 7044:(for the 7015:∣ 7012:ε 7005:⁡ 6953:as in an 6787:β 6769:− 6642:⊤ 6627:− 6611:⊤ 6589:⊤ 6574:− 6558:⊤ 6521:− 6470:^ 6467:γ 6454:^ 6451:β 6437:⇒ 6414:^ 6411:γ 6398:^ 6395:β 6308:^ 6305:γ 6288:^ 6285:β 6274:− 6260:^ 6237:= 0 (cf. 6175:^ 6172:γ 6145:β 6136:− 6108:^ 6105:β 6094:− 6051:⋅ 6042:^ 6039:β 6028:− 5986:^ 5983:β 5974:− 5933:⊤ 5922:^ 5919:β 5908:− 5853:^ 5850:β 5828:onto the 5762:‖ 5758:β 5749:− 5741:‖ 5735:β 5708:^ 5705:β 5534:^ 5531:β 5487:¯ 5474:^ 5471:β 5465:− 5459:¯ 5440:^ 5437:α 5410:¯ 5401:− 5367:∑ 5354:¯ 5345:− 5323:¯ 5314:− 5280:∑ 5263:^ 5260:β 5218:ε 5201:β 5195:α 5024:− 4953:− 4900:− 4814:¯ 4806:− 4790:∑ 4773:¯ 4765:− 4753:^ 4740:∑ 4637:^ 4634:σ 4609:, is the 4566:− 4545:^ 4542:σ 4525:− 4511:^ 4508:β 4484:− 4446:− 4396:− 4349:− 4338:^ 4335:ε 4318:^ 4315:ε 4256:ε 4247:ε 4238:β 4217:ε 4211:β 4184:^ 4181:β 4172:− 4160:^ 4151:− 4139:^ 4136:ε 4120:residuals 4114:. Matrix 4067:symmetric 3951:^ 3948:β 3933:^ 3852:, is the 3790:− 3751:⁡ 3734:∈ 3717:^ 3714:β 3685:^ 3682:β 3621:transpose 3588:− 3563:− 3523:− 3490:∑ 3345:ε 3324:− 3288:β 3278:^ 3275:β 3221:− 3183:^ 3180:β 3147:^ 3144:β 3009:^ 3006:β 2881:β 2872:− 2838:β 2804:∑ 2800:− 2764:∑ 2753:β 2693:β 2680:β 2648:^ 2645:β 2626:quadratic 2611:β 2481:⋮ 2416:β 2408:⋮ 2395:β 2381:β 2365:β 2334:⋯ 2297:⋮ 2292:⋱ 2287:⋮ 2282:⋮ 2260:⋯ 2214:⋯ 2136:β 2071:β 2064:… 2052:β 2039:β 1956:… 1907:β 1873:∑ 1837:β 1818:quadratic 1793:→ 1764:intercept 1744:β 1717:… 1543:× 1475:× 1451:ε 1400:ε 1392:β 1293:ε 1264:) of the 1242:ε 1218:× 1194:β 1111:ε 1103:β 1033:ε 1004:β 997:⋯ 969:β 940:β 845:… 572:variances 554:when the 540:exogenous 529:regressor 521:estimator 226:Segmented 15581:Logistic 15571:Binomial 15550:Isotonic 15545:Quantile 15264:(2008). 15204:(2009). 15018:(1973). 14970:(1985). 14609:(2000). 14571:(1964). 14475:See also 14225:0.438371 14213:0.309017 14201:0.052336 14186:0.615661 14171:0.707107 14156:0.731354 13408:Solution 13118:errors. 13084:58.5046 13078:119.0205 13067:−143.162 13064:128.8128 12246:128.8128 11426:→ 11092:→ 10624:→ 9185: ( 8863: ( 8087:. Thus, 7756:unbiased 5786:‖ 5782:‖ 5679:, where 4665:unbiased 3402:for the 3399:residual 3369:Suppose 2886:‖ 2863:‖ 2631:problem 2119:form as 2026:unknown 1697:for all 341:Bayesian 279:Weighted 274:Ordinary 206:Isotonic 201:Quantile 15576:Poisson 14459:0.70001 14371:0.30435 14368:0.43478 14302:0.56820 14295:0.52883 14288:0.45071 14281:0.24741 14274:0.21958 14267:0.21220 13522:apsides 13358:1.7599 13355:1.8910 13352:2.2187 13349:4.0419 13346:4.5542 13343:4.7126 13056:Height 12955:LR test 12947:p-value 12703:is the 12598:p-value 12436:0.0000 12430:0.3964 12420:5471.2 12414:0.2548 12404:693.37 12398:2.1013 12388:0.7595 12382:1.0890 12374:692.61 12368:0.9987 12360:0.2516 12354:0.9989 12339:0.0000 12336:10.3122 12330:61.9603 12297:0.0000 12294:–7.2183 12291:19.8332 12255:0.0000 12249:16.3083 12207:p-value 12172:WEIGHT 11020:is the 9823:is the 9627:×1 and 9200:is the 7689:} is a 7661:matrix 7601:ergodic 7253:is the 6698:Pearson 5833:spanned 5795:in the 4036:matrix 4032:is the 4000:is the 3663:Hessian 2089:, with 884:. In a 509:dataset 300:Partial 139:Poisson 15540:Robust 15276: 15250: 15231: 15212: 15186: 15158: 15078: 15026: 14986: 14769: 14700: 14673: 14646: 14589: 14421:2.3000 14198: 13900:where 13196:where 13053:Height 12923:, and 12687:-value 12333:6.0084 12252:7.8986 12193:Value 12161:Method 12026:HEIGHT 12011:74.46 11971:64.47 11931:57.20 11865:, and 11827:F-test 11746: 11592: 11436: 11421: 11102: 11087: 11006:where 10976: 10841: 10711:where 10634: 10619: 10399:where 10146:makes 9935:. The 9905:where 9816:where 9606:, and 9570:where 9208:, and 9170:where 8747: 8741: 8635: 8612: 8606: 7603:; if { 7562:Var = 7553:E = 0; 7340:data, 7261:, and 7246:where 6945:, and 6943:design 6210: 5780:where 5493: 5083:is an 5056:, and 4700:. The 3981:where 3906:, the 3811: 3727:argmin 3615:where 3035: 2163:where 2117:matrix 1932: 1822:linear 1532:is an 1420:where 1262:errors 1136:where 1061:form, 1059:vector 1057:or in 1013: 978: 949: 556:errors 258:Linear 196:Robust 119:Probit 45:Models 14918:(PDF) 14903:(11). 14126:so 13050:Const 12450:Value 12008:72.19 12005:69.92 12002:68.10 11999:66.28 11991:1.83 11968:63.11 11965:61.29 11962:59.93 11959:58.57 11951:1.70 11928:55.84 11925:54.48 11922:53.12 11919:52.21 11898:1.57 10403:is a 9921:is a 9909:is a 7886:) of 7631:} is 7595:} is 7541:is a 7522:i ≠ j 7505:as, ( 7497:) is 7324:E = 0 7050:E = 0 7042:E = 0 6994:zero: 6919:' 6849:E = 0 6841:E = 0 6828:is a 5111:is a 4689:, or 3840:is a 3455:) or 1207:is a 487:in a 305:Total 221:Local 15320:and 15274:ISBN 15248:ISBN 15229:ISBN 15210:ISBN 15184:ISBN 15156:ISBN 15076:ISBN 15024:ISBN 14984:ISBN 14925:2020 14767:ISBN 14698:ISBN 14671:ISBN 14644:ISBN 14587:ISBN 14433:and 14245:and 14068:and 13994:and 13947:and 13384:and 13308:116 13305:108 13245:and 13114:and 12897:and 12596:and 12590:The 12461:The 12448:The 11988:1.80 11985:1.78 11982:1.75 11979:1.73 11948:1.68 11945:1.65 11942:1.63 11939:1.60 11895:1.55 11892:1.52 11889:1.50 11886:1.47 10771:and 10485:and 9657:The 9620:are 9577:and 9433:0 ≤ 8918:are 8914:and 8537:(0, 8338:and 8324:The 7876:The 7754:are 7750:and 7653:The 7599:and 7577:The 7326:for 7284:E = 7108:rank 6868:) = 6696:and 6694:Yule 5609:and 4107:and 4083:and 4069:and 4065:are 4061:and 3912:(or 2957:are 2100:> 1464:are 1442:and 562:and 558:are 15655:BIC 15650:AIC 14583:158 14393:so 13967:is 13920:is 13799:cos 13772:sin 13753:tan 13721:sin 13689:cos 13650:sin 13635:sin 13610:cos 13595:cos 13564:cos 13532:cos 13496:cos 13441:as 13302:93 13299:52 13296:45 13293:43 13169:cos 13106:). 13040:not 12953:or 12943:n–p 12939:p–1 12180:15 11875:). 11804:. 11271:is 10580:to 10544:is 10420:= 0 9456:p/n 9440:≤ 1 8880:). 8878:= 1 8831:of 8458:Var 8423:Var 8259:Cov 7928:Var 7704:= E 7700:xxε 7669:= E 7539:= E 7483:: ( 7313:No 7188:Var 7160:= E 6718:GMM 6714:iid 6712:In 6075:any 6005:of 5872:of 5824:is 5731:min 5555:= ( 4112:= 0 3856:of 3633:= x 3461:RSS 3453:ESS 3445:SSR 3261:or 3091:or 2006:in 1983:of 1850:). 1582:is 764:of 477:OLS 467:In 461:GDP 455:in 15885:: 15268:. 15200:; 15126:^ 15062:^ 14982:. 14980:13 14945:^ 14901:18 14899:. 14895:. 14859:^ 14820:^ 14712:^ 14696:. 14669:. 14642:. 14585:. 14575:. 14547:. 14522:. 12919:, 11368:: 11017:jj 10446:. 10424:XX 10418:RQ 10407:×( 10136:XX 9834:. 9654:. 9649:= 9642:+ 9613:, 9595:, 9454:≈ 9178:= 8817:/( 8720:: 8533:~ 8309:0. 7666:xx 7624:, 7610:, 7588:, 7560:: 7551:: 7536:xx 7531:: 7512:, 7490:, 7331:≠ 7282:: 7157:xx 7140:1. 7096:Pr 7028:0. 6806:0. 6266::= 5946:0. 5886:. 5879:Py 5877:= 5817:Xβ 5815:− 5808:Xβ 5674:≈ 5672:Xβ 5171:: 5165:, 5134:. 4693:. 4685:, 4110:MX 4102:= 4100:PX 4088:= 4078:= 4048:− 4041:= 3985:= 3880:) 3864:, 3429:= 3415:, 3385:− 2965:: 2253:22 2241:21 2207:12 2195:11 2030:, 1826:is 1808:. 471:, 15643:p 15641:C 15387:) 15378:( 15310:e 15303:t 15296:v 15282:. 15256:. 15237:. 15218:. 15192:. 15164:. 15084:. 15032:. 14992:. 14927:. 14802:. 14775:. 14706:. 14679:. 14652:. 14595:. 14557:. 14533:. 14456:= 14453:y 14447:p 14444:= 14441:e 14418:= 14413:x 14410:1 14405:= 14402:p 14376:) 14363:( 14357:= 14351:) 14346:y 14343:x 14338:( 14313:. 14308:] 14261:[ 14256:= 14253:b 14231:] 14220:1 14208:1 14193:1 14178:1 14163:1 14148:1 14142:[ 14137:= 14134:A 14111:) 14105:( 14102:r 14098:1 14076:b 14054:p 14051:e 14027:p 14024:1 14002:A 13980:p 13977:e 13955:y 13933:p 13930:1 13908:x 13888:b 13883:T 13879:A 13875:= 13869:) 13864:y 13861:x 13856:( 13850:A 13845:T 13841:A 13818:) 13813:0 13805:( 13795:/ 13791:) 13786:0 13778:( 13769:= 13764:0 13733:) 13727:( 13701:) 13695:( 13669:) 13664:0 13656:( 13647:) 13641:( 13632:+ 13629:) 13624:0 13616:( 13607:) 13601:( 13592:= 13589:) 13584:0 13570:( 13544:) 13538:( 13508:) 13502:( 13491:p 13488:e 13478:p 13475:1 13470:= 13464:) 13458:( 13455:r 13451:1 13429:) 13423:( 13420:r 13392:p 13372:e 13329:) 13323:( 13320:r 13253:e 13233:p 13213:) 13207:( 13204:r 13181:) 13175:( 13166:e 13160:1 13156:p 13151:= 13148:) 13142:( 13139:r 13116:y 13112:x 13009:. 12991:y 12941:, 12937:( 12935:F 12911:σ 12860:) 12855:2 12851:R 12844:1 12841:( 12835:p 12829:n 12824:1 12818:n 12809:1 12806:= 12801:2 12792:R 12761:2 12757:R 12734:2 12730:R 12709:X 12695:p 12685:p 12680:t 12676:t 12672:t 12656:j 12638:/ 12632:j 12615:= 12612:t 12602:t 12573:2 12570:1 12564:) 12558:j 12555:j 12550:] 12545:1 12537:x 12534:x 12530:Q 12526:[ 12519:2 12501:( 12496:= 12491:j 12456:j 12454:β 12350:R 12314:3 12272:2 12230:1 12126:. 12121:i 12113:+ 12108:2 12103:i 12099:h 12093:3 12085:+ 12080:i 12076:h 12070:2 12062:+ 12057:1 12049:= 12044:i 12040:w 11811:) 11807:( 11767:α 11750:] 11739:0 11735:x 11729:1 11721:x 11718:x 11714:Q 11707:T 11701:0 11697:x 11691:2 11669:) 11666:1 11663:, 11660:0 11657:( 11652:N 11643:2 11632:1 11628:q 11606:T 11600:0 11596:x 11588:[ 11579:0 11575:y 11549:0 11545:y 11521:, 11517:) 11511:0 11507:x 11501:1 11493:x 11490:x 11486:Q 11479:T 11473:0 11469:x 11463:2 11454:, 11451:0 11447:( 11441:N 11430:d 11417:) 11411:0 11407:y 11398:0 11388:y 11380:( 11312:T 11306:0 11302:x 11298:= 11293:0 11283:y 11249:T 11243:0 11239:x 11235:= 11230:0 11226:y 11199:0 11195:x 11171:. 11167:) 11161:4 11149:] 11144:4 11139:i 11131:[ 11124:E 11120:, 11117:0 11113:( 11107:N 11096:d 11084:) 11079:2 11066:2 11049:( 11035:i 11033:ε 11029:σ 11022:j 11008:q 10999:α 10981:] 10969:j 10966:j 10961:] 10956:1 10948:x 10945:x 10941:Q 10937:[ 10930:2 10908:) 10905:1 10902:, 10899:0 10896:( 10891:N 10882:2 10871:1 10867:q 10858:j 10836:[ 10826:j 10779:j 10748:. 10745:X 10740:T 10736:X 10732:= 10727:x 10724:x 10720:Q 10696:, 10691:) 10684:1 10676:x 10673:x 10669:Q 10663:2 10654:, 10651:0 10646:( 10639:N 10628:d 10616:) 10595:( 10582:β 10542:β 10535:n 10531:n 10507:2 10453:i 10451:ε 10440:β 10413:q 10409:p 10405:p 10401:R 10384:, 10381:c 10376:1 10369:) 10365:Q 10360:T 10356:Q 10352:( 10349:Q 10344:) 10339:X 10334:T 10330:X 10324:T 10320:R 10314:1 10307:) 10303:R 10300:X 10295:T 10291:X 10285:T 10281:R 10277:( 10274:R 10266:p 10262:I 10256:( 10251:+ 10248:y 10243:T 10239:X 10233:T 10229:R 10223:1 10216:) 10212:R 10209:X 10204:T 10200:X 10194:T 10190:R 10186:( 10183:R 10180:= 10175:c 10148:β 10144:A 10140:β 10119:. 10116:) 10113:c 10093:T 10089:Q 10085:( 10080:1 10071:) 10065:Q 10060:1 10053:) 10049:X 10044:T 10040:X 10036:( 10031:T 10027:Q 10021:( 10016:Q 10011:1 10004:) 10000:X 9995:T 9991:X 9987:( 9969:= 9964:c 9933:A 9923:q 9919:c 9915:q 9913:× 9911:p 9907:Q 9889:, 9886:c 9883:= 9875:T 9871:Q 9863:A 9832:1 9829:X 9821:1 9818:M 9801:, 9794:+ 9789:2 9779:2 9775:X 9769:1 9765:M 9761:= 9758:y 9753:1 9749:M 9735:2 9732:β 9715:2 9651:p 9647:2 9644:p 9640:1 9637:p 9632:2 9629:p 9625:1 9622:p 9618:2 9615:β 9611:1 9608:β 9604:2 9601:p 9599:× 9597:n 9593:1 9590:p 9588:× 9586:n 9582:2 9579:X 9575:1 9572:X 9555:, 9549:+ 9544:2 9534:2 9530:X 9526:+ 9521:1 9511:1 9507:X 9503:= 9500:y 9474:j 9472:h 9463:j 9461:h 9451:j 9449:h 9444:p 9437:j 9435:h 9414:j 9391:j 9387:h 9380:1 9374:j 9370:h 9361:= 9344:T 9339:j 9335:x 9326:) 9323:j 9320:( 9300:T 9294:j 9290:x 9286:= 9281:j 9271:y 9259:) 9256:j 9253:( 9248:j 9238:y 9221:j 9217:j 9212:j 9210:x 9206:P 9202:j 9196:j 9194:x 9192:) 9190:X 9187:X 9182:j 9180:x 9175:j 9173:h 9155:, 9149:j 9129:T 9123:j 9119:x 9113:1 9106:) 9102:X 9096:T 9091:X 9087:( 9079:j 9075:h 9068:1 9064:1 9056:= 9036:) 9033:j 9030:( 9003:β 8995:j 8984:X 8979:i 8977:y 8973:y 8916:s 8876:p 8869:p 8865:n 8861:/ 8857:σ 8848:s 8844:σ 8839:n 8837:/ 8835:σ 8833:2 8825:) 8823:p 8819:n 8815:σ 8813:2 8794:2 8789:p 8783:n 8769:p 8763:n 8757:2 8736:2 8732:s 8714:s 8686:. 8681:) 8674:1 8667:) 8663:X 8657:T 8652:X 8648:( 8643:2 8632:, 8624:( 8617:N 8547:) 8544:n 8542:I 8539:σ 8535:N 8531:ε 8514:ε 8493:0 8487:] 8483:X 8464:[ 8452:] 8448:X 8429:[ 8410:y 8306:= 8303:] 8299:X 8281:, 8265:[ 8198:1 8190:j 8187:j 8182:) 8178:X 8173:T 8169:X 8164:( 8157:2 8153:s 8147:= 8144:) 8139:j 8122:( 8112:. 8109:e 8105:. 8102:s 8085:s 8081:σ 8077:j 8060:j 8022:. 8019:Q 8014:2 8006:= 8001:1 7993:) 7989:X 7984:T 7980:X 7975:( 7968:2 7960:= 7957:] 7953:X 7934:[ 7854:. 7849:2 7841:= 7838:] 7834:X 7826:2 7822:s 7817:[ 7811:E 7807:, 7801:= 7798:] 7794:X 7775:[ 7769:E 7752:s 7706:. 7702:² 7696:Q 7686:i 7684:ε 7682:i 7680:x 7678:{ 7675:; 7664:Q 7659:p 7657:× 7655:p 7650:; 7648:n 7644:i 7635:. 7628:i 7626:y 7621:i 7619:x 7614:i 7612:y 7607:i 7605:x 7592:i 7590:y 7585:i 7583:x 7581:{ 7567:. 7564:σ 7545:; 7534:Q 7525:; 7516:j 7514:y 7509:j 7507:x 7494:i 7492:y 7487:i 7485:x 7468:n 7423:. 7420:) 7415:n 7411:I 7405:2 7397:, 7394:0 7391:( 7386:N 7378:X 7352:. 7333:j 7329:i 7291:σ 7286:σ 7267:σ 7263:σ 7259:n 7250:n 7248:I 7234:, 7229:n 7225:I 7219:2 7211:= 7208:] 7204:X 7194:[ 7180:: 7172:y 7168:β 7155:Q 7137:= 7132:] 7126:p 7123:= 7120:) 7117:X 7114:( 7102:[ 7082:X 7074:X 7025:= 7022:] 7018:X 7008:[ 7002:E 6971:n 6959:X 6951:X 6947:y 6939:X 6915:i 6913:y 6908:i 6906:x 6870:x 6866:x 6864:( 6862:ƒ 6857:ƒ 6845:ƒ 6834:β 6830:p 6825:i 6823:x 6803:= 6798:] 6791:) 6782:T 6777:i 6773:x 6764:i 6760:y 6755:( 6749:i 6745:x 6738:[ 6732:E 6660:. 6656:y 6650:] 6637:K 6630:1 6622:) 6617:K 6606:K 6600:( 6584:X 6577:1 6569:) 6564:X 6553:X 6547:( 6539:[ 6534:= 6530:y 6524:1 6515:] 6508:K 6501:X 6494:[ 6488:= 6479:] 6442:[ 6428:, 6423:] 6386:[ 6379:] 6372:K 6365:X 6358:[ 6353:= 6345:y 6314:. 6298:K 6294:= 6278:X 6270:y 6256:r 6235:X 6232:K 6218:] 6214:K 6206:X 6202:[ 6192:K 6140:X 6132:y 6098:X 6090:y 6079:v 6060:v 6055:X 6048:) 6032:X 6024:y 6020:( 6007:X 5977:X 5970:y 5943:= 5939:X 5929:) 5912:X 5904:y 5900:( 5884:X 5875:y 5837:X 5822:y 5813:y 5804:R 5797:n 5791:L 5784:· 5765:, 5753:X 5745:y 5725:g 5722:r 5719:a 5714:= 5689:p 5685:n 5681:β 5676:y 5662:. 5622:2 5618:X 5595:1 5591:X 5567:y 5564:X 5562:) 5560:X 5557:X 5553:β 5496:, 5484:x 5456:y 5450:= 5421:2 5417:) 5407:x 5396:i 5392:x 5388:( 5382:n 5377:1 5374:= 5371:i 5360:) 5351:y 5340:i 5336:y 5332:( 5329:) 5320:x 5309:i 5305:x 5301:( 5295:n 5290:1 5287:= 5284:i 5273:= 5227:. 5222:i 5214:+ 5209:i 5205:x 5198:+ 5192:= 5187:i 5183:y 5169:) 5167:β 5163:α 5161:( 5156:i 5154:x 5150:X 5125:R 5121:X 5117:R 5099:L 5089:n 5087:× 5085:n 5069:n 5065:J 5042:n 5038:J 5032:n 5029:1 5019:n 5015:I 5011:= 5008:L 4976:S 4973:S 4970:T 4965:S 4962:S 4959:R 4950:1 4947:= 4941:y 4938:L 4932:T 4927:y 4921:y 4918:M 4912:T 4907:y 4897:1 4894:= 4888:y 4885:L 4879:T 4874:y 4868:y 4865:P 4862:L 4856:T 4851:P 4844:T 4839:y 4832:= 4824:2 4820:) 4811:y 4801:i 4797:y 4793:( 4783:2 4779:) 4770:y 4760:i 4750:y 4743:( 4734:= 4729:2 4725:R 4711:y 4707:R 4698:X 4677:s 4673:s 4661:σ 4644:2 4619:σ 4615:s 4607:p 4605:− 4603:n 4584:2 4580:s 4573:n 4569:p 4563:n 4557:= 4552:2 4534:, 4528:p 4522:n 4517:) 4502:( 4499:S 4493:= 4487:p 4481:n 4476:y 4473:M 4467:T 4462:y 4455:= 4449:p 4443:n 4438:y 4435:M 4429:T 4424:M 4417:T 4412:y 4405:= 4399:p 4393:n 4388:y 4385:M 4379:T 4374:) 4370:y 4367:M 4364:( 4358:= 4352:p 4346:n 4326:T 4305:= 4300:2 4296:s 4276:σ 4259:. 4253:M 4250:= 4244:M 4241:+ 4235:) 4232:X 4229:M 4226:( 4223:= 4220:) 4214:+ 4208:X 4205:( 4202:M 4199:= 4196:y 4193:M 4190:= 4175:X 4169:y 4166:= 4157:y 4148:y 4145:= 4116:M 4104:X 4095:X 4090:M 4086:M 4080:P 4076:P 4063:M 4059:P 4055:V 4050:P 4045:n 4043:I 4039:M 4030:P 4026:y 4016:P 4012:X 4008:V 3998:X 3996:) 3994:X 3991:X 3989:( 3987:X 3983:P 3966:, 3963:y 3960:P 3957:= 3942:X 3939:= 3930:y 3904:β 3889:X 3886:Q 3884:= 3882:X 3878:X 3875:X 3870:β 3866:C 3858:β 3850:N 3848:= 3846:Q 3838:X 3835:X 3833:= 3831:N 3814:. 3808:y 3803:T 3799:X 3793:1 3786:) 3782:X 3777:T 3773:X 3769:( 3766:= 3763:) 3760:b 3757:( 3754:S 3744:p 3739:R 3731:b 3723:= 3676:= 3673:b 3659:b 3655:b 3653:( 3651:S 3646:β 3640:b 3635:i 3631:i 3629:X 3625:X 3617:T 3600:, 3597:) 3594:b 3591:X 3585:y 3582:( 3577:T 3573:) 3569:b 3566:X 3560:y 3557:( 3554:= 3549:2 3545:) 3541:b 3536:T 3531:i 3527:x 3518:i 3514:y 3510:( 3505:n 3500:1 3497:= 3494:i 3486:= 3483:) 3480:b 3477:( 3474:S 3459:( 3451:( 3443:( 3434:b 3431:x 3427:y 3422:) 3419:i 3417:y 3412:i 3410:x 3408:( 3404:i 3393:b 3389:i 3387:x 3382:i 3380:y 3375:β 3371:b 3349:. 3339:T 3334:X 3327:1 3319:) 3314:X 3308:T 3303:X 3297:( 3292:+ 3284:= 3246:. 3242:y 3236:T 3231:X 3224:1 3216:) 3211:X 3205:T 3200:X 3194:( 3189:= 3116:y 3110:T 3105:X 3074:X 3068:T 3063:X 3038:. 3031:y 3025:T 3020:X 3015:= 2999:) 2994:X 2988:T 2983:X 2977:( 2944:X 2923:p 2896:. 2891:2 2876:X 2868:y 2858:= 2853:2 2848:| 2842:j 2832:j 2829:i 2825:X 2819:p 2814:1 2811:= 2808:j 2795:i 2791:y 2786:| 2779:n 2774:1 2771:= 2768:i 2760:= 2757:) 2749:( 2746:S 2723:S 2700:, 2697:) 2689:( 2686:S 2676:n 2673:i 2670:m 2666:g 2663:r 2660:a 2654:= 2587:p 2567:1 2564:= 2559:1 2556:i 2552:X 2530:X 2506:. 2501:] 2493:n 2489:y 2472:2 2468:y 2458:1 2454:y 2447:[ 2442:= 2438:y 2433:, 2428:] 2420:p 2399:2 2385:1 2374:[ 2369:= 2360:, 2355:] 2347:p 2344:n 2340:X 2327:2 2324:n 2320:X 2312:1 2309:n 2305:X 2273:p 2270:2 2266:X 2249:X 2237:X 2227:p 2224:1 2220:X 2203:X 2191:X 2184:[ 2179:= 2175:X 2148:, 2144:y 2140:= 2131:X 2103:p 2097:n 2075:p 2067:, 2061:, 2056:2 2048:, 2043:1 2014:p 1991:n 1968:, 1965:) 1962:n 1959:, 1953:, 1950:2 1947:, 1944:1 1941:= 1938:i 1935:( 1929:, 1924:i 1920:y 1916:= 1911:j 1901:j 1898:i 1894:x 1888:p 1883:1 1880:= 1877:j 1790:0 1784:= 1779:i 1775:x 1748:1 1723:n 1720:, 1714:, 1711:1 1708:= 1705:i 1685:1 1682:= 1677:1 1674:i 1670:x 1648:X 1624:i 1602:T 1597:i 1592:x 1570:i 1546:p 1540:n 1519:X 1498:n 1478:1 1472:n 1429:y 1404:, 1396:+ 1387:X 1383:= 1379:y 1353:i 1348:x 1324:i 1320:y 1297:i 1272:i 1246:i 1221:1 1215:p 1173:i 1151:i 1146:x 1120:, 1115:i 1107:+ 1097:T 1092:i 1087:x 1082:= 1077:i 1073:y 1042:, 1037:i 1029:+ 1024:p 1021:i 1017:x 1008:p 1000:+ 994:+ 989:2 986:i 982:x 973:2 965:+ 960:1 957:i 953:x 944:1 936:= 931:i 927:y 901:i 897:y 870:T 865:] 859:p 856:i 852:x 848:, 842:, 837:2 834:i 830:x 826:, 821:1 818:i 814:x 809:[ 804:= 799:i 794:x 772:p 750:i 745:x 721:i 717:y 696:i 674:n 669:1 666:= 663:i 658:} 652:i 648:y 644:, 639:i 634:x 628:{ 604:n 550:— 475:( 437:e 430:t 423:v 20:)

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Knowledge

Ordinary least squares

Index