A stockbroker is interested in the factors influencing the...

A stockbroker is interested in the factors influencing the rate of return on the common stock of banks. For a sample of 30 banks, the following regression was estimated by least squares: $$ \begin{aligned} & \hat{y}=2.37+\underset{(0.39)}{0.84 x_1}+\underset{(0.12)}{0.15 x_2}-\underset{(0.09)}{0.13 x_3} \\ & +1.67 x_4 \quad R^2=0.317 \\ & \end{aligned} $$ where $y=$ percentage rate of return on common stock of bank $x_1=$ percentage rate of growth of bank's earnings $x_2=$ percentage rate of growth of bank's assets $x_3=$ loan losses as percentage of bank's assets $x_4=1$ if bank head office is in New York City and 0 otherwise The numbers below the coefficients are the coefficient standard errors. a. Interpret the estimated coefficient on $x_4$. b. Interpret the coefficient of determination, and use it to test the null hypothesis that, taken as a group, the four independent variables do not linearly influence the dependent variable. c. Let $e_i$ denote the residuals from the fitted regression and $\hat{y}_i$ the in-sample predicted values of the dependent variable. The least squares regression of $e_i^2$ on $\hat{y}_i$ yielded coefficient of determination 0.082 . What can be concluded from this finding?

A stockbroker is interested in the factors influencing the rate of return on the common stock of banks. For a sample of 30 banks, the following regression was estimated by least squares:
$$
\begin{aligned}
& \hat{y}=2.37+\underset{(0.39)}{0.84 x_1}+\underset{(0.12)}{0.15 x_2}-\underset{(0.09)}{0.13 x_3} \\
& +1.67 x_4 \quad R^2=0.317 \\
&
\end{aligned}
$$
where
$y=$ percentage rate of return on common stock of bank
$x_1=$ percentage rate of growth of bank's earnings
$x_2=$ percentage rate of growth of bank's assets
$x_3=$ loan losses as percentage of bank's assets
$x_4=1$ if bank head office is in New York City and 0 otherwise
The numbers below the coefficients are the coefficient standard errors.
a. Interpret the estimated coefficient on $x_4$.
b. Interpret the coefficient of determination, and use it to test the null hypothesis that, taken as a group, the four independent variables do not linearly influence the dependent variable.
c. Let $e_i$ denote the residuals from the fitted regression and $\hat{y}_i$ the in-sample predicted values of the dependent variable. The least squares regression of $e_i^2$ on $\hat{y}_i$ yielded coefficient of determination 0.082 . What can be concluded from this finding?

Key Concepts

Detection of Heteroscedasticity through Regression of Squared Residuals

Heteroscedasticity refers to the circumstance where the variance of the errors is not constant across observations, which can affect the validity of standard statistical tests. One diagnostic approach involves regressing the squared residuals from the fitted model on the predicted values to see if there's a systematic pattern. A low R-squared in this auxiliary regression suggests that there is little evidence of heteroscedasticity in the error term variance.

Standard Errors and Hypothesis Testing of Regression Coefficients

Standard errors in regression provide an estimate of the sampling variability of the coefficient estimates. They are used to construct confidence intervals and conduct hypothesis tests (using t-statistics) to determine whether individual predictors have a statistically significant relationship with the dependent variable.

Coefficient of Determination (R-squared) and Overall Significance Testing

The coefficient of determination, R-squared, quantifies the proportion of variability in the dependent variable that is explained by the independent variables in the model. It is a measure of overall model fit. To test whether the predictors, taken together, have a statistically significant effect on the dependent variable, an F-test is commonly used under the null hypothesis that all regression coefficients (except the intercept) are zero.

Dummy Variable in Regression Analysis

In regression analysis, dummy variables are used to represent categorical data by coding categories as 0 and 1. The coefficient on a dummy variable measures the expected difference in the dependent variable between the group coded as 1 and the baseline group (coded as 0), while holding all other variables constant.

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