12.4 . You have fit a multiple linear regression model and the (𝐗^' 𝐗)^-1 matrix is: (𝐗^' 𝐗)^-1=

step 1 First we have to find how many regressors are there in this model. So the given matrix is a symm

12.4 . You have fit a multiple linear regression model and...

$12.4 .$ You have fit a multiple linear regression model and the $\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}$ matrix is: $$ \left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}=\left[\begin{array}{rrr} 0.893758 & -0.0282448 & -0.0175641 \\ -0.028245 & 0.0013329 & 0.0001547 \\ -0.017564 & 0.0001547 & 0.0009108 \end{array}\right] $$ (a) How many regressor variables are in this model? (b) If the error sum of squares is 307 and there are 15 observations, what is the estimate of $\sigma^{2} ?$ (c) What is the standard error of the regression coefficient $\hat{\beta}_{1} ?$

$12.4 .$ You have fit a multiple linear regression model and the $\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}$ matrix is:
$$ \left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}=\left[\begin{array}{rrr} 0.893758 & -0.0282448 & -0.0175641 \\ -0.028245 & 0.0013329 & 0.0001547 \\ -0.017564 & 0.0001547 & 0.0009108 \end{array}\right] $$
(a) How many regressor variables are in this model?
(b) If the error sum of squares is 307 and there are 15 observations, what is the estimate of $\sigma^{2} ?$
(c) What is the standard error of the regression coefficient $\hat{\beta}_{1} ?$

Key Concepts

Standard Error of Regression Coefficients

The standard error of a regression coefficient measures the accuracy of the coefficient’s estimated value. It is obtained by multiplying the estimate of the error variance by the corresponding diagonal element in (X'X)^(-1) and then taking the square root. This standard error is used to assess the precision of the estimated coefficients and to perform hypothesis tests.

Error Variance Estimation

Error variance, denoted by sigma squared (?²), estimates the variability of the errors in the regression model. It is typically estimated by dividing the error sum of squares (SSE) by its degrees of freedom, which is equal to the number of observations minus the number of parameters estimated in the model. This estimate is foundational for constructing confidence intervals and significance tests for the regression coefficients.

Multiple Linear Regression

Multiple linear regression is a statistical technique for modeling the linear relationship between a response variable and two or more predictor variables. The method involves estimating the coefficients that best predict the response based on the values of the predictors, and it assumes that the underlying relationship between the predictors and the response is linear.

Design Matrix and Its Inversion

The design matrix (often denoted X) collects the values of the predictor variables for all observations. Its transpose multiplied by itself (X'X) is central to the normal equations used in finding the least squares estimates. The inverse of this matrix, (X'X)^(-1), is crucial because its diagonal elements are used to compute the variances and consequently the standard errors of the estimated regression coefficients.

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