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Linear Regression: Coefficients Analysis

Last Update: February 21, 2022

Linear Regression: Coefficients Analysis is used to analyze linear relationship between one dependent variable y and two or more independent variables x_{1}...x_{p}. Variable y is also known as target or response feature and variables x_{1}...x_{p} are also known as predictor features. It is also used to evaluate whether adding independent variables individually improved linear regression model.

As example, we can fit a three-variable multiple linear regression with formula \hat{y}_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}x_{1i}+\hat{\beta}_{2}x_{2i}\;(1). Regression fitted values \hat{y}_{i} are the estimated y_{i} values. Estimated constant coefficient \hat{\beta}_{0} is the \hat{y} value when x_{1}=0 and x_{2}=0. Estimated partial regression coefficient \hat{\beta}_{1} is the estimated change in y when x_{1} changes in one unit while holding x_{2} constant. Similarly, estimated partial regression coefficient \hat{\beta}_{2} is the estimated change in y when x_{2} changes in one unit while holding x_{1} constant.

Then, we can estimate coefficient k standard error with formula se_{\hat{\beta}_{k}}=\sqrt{ms_{res}diag((x'x)^{-1})_{k}}\;(2) as squared root of residual mean squared error ms_{res} multiplied by k element of matrix (x'x)^{-1} principal diagonal.

Residual mean squared error ms_{res} with formula ms_{res}=\frac{ss_{res}}{df_{res}}\;(3) is estimated as residual sum of squares ss_{res} divided by residual degrees of freedom df_{res}. Residual sum of squares ss_{res} with formula ss_{res}=\sum_{i=1}^{n}\hat{e}_{i}^{2}\;(4) is estimated as the sum of squared regression residuals \hat{e}_{i}. Regression residuals \hat{e}_{i} with formula \hat{e}_{i}=y_{i}-\hat{y}_{i}\;(5) are estimated as differences between actual y_{i} and fitted \hat{y}_{i} values. Residual degrees of freedom df_{res} with formula df_{res}=n-p-1\;(6) are the number of observations n minus number of independent variables p minus constant term.

Matrix (x'x)^{-1} with dimension (p+1) x (p+1) is the inverse of the matrix product between the transpose of matrix x and matrix x. Matrix x with dimension n x (p+1) is the independent variables matrix including constant term column of ones.

Next, we can estimate coefficient k t-statistic with formula t_{\hat{\beta}_{k}}=\frac{\hat{\beta}_{k}}{se_{\hat{\beta}_{k}}}\;(7) and do t-test with individual null hypothesis that independent variable x_{k} coefficient is equal to zero with formula H_{0}:\hat{\beta}_{k}=0\;(8). If individual null hypothesis is rejected, then adding independent variable x_{k} improved linear regression model.

Below, we find an example of coefficients analysis from multiple linear regression of house price explained by its lot size and number of bedrooms [1].

Table 1. Microsoft Excel® coefficients analysis from multiple linear regression of house price explained by its lot size and number of bedrooms.


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[1] Data Description: Sales prices of houses sold in the city of Windsor, Canada, during July, August and September, 1987.

Original Source: Anglin, P., and Gencay, R. (1996). Semiparametric Estimation of a Hedonic Price Function. Journal of Applied Econometrics, 11, 633–648.

Source: AER R Package HousePrices Object. Christian Kleiber and Achim Zeileis. (2008). Applied Econometrics with R. Springer-Verlag, New York.

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