A local public utility would like to be able to predict a...

A local public utility would like to be able to predict a dwelling unit's average monthly electricity bill. The company statistician estimated by least squares the following regression model: $$ y_t=\beta_0+\beta_1 x_{1 t}+\beta_2 x_{2 t}+\varepsilon_t $$ where $$ \begin{aligned} y_t= & \text { average monthly electricity bill, in dollars } \\ x_{1 t}= & \text { average bimonthly automobile gasoline bill, } \\ & \text { in dollars } \\ x_{2 t}= & \text { number of rooms in dwelling unit } \end{aligned} $$ From a sample of 25 dwelling units, the statistician obtained the following output from the SAS program: $$ \begin{array}{crcc} \hline \text { Parameter } & \text { Estimate } & \begin{array}{c} \text { Student's } t \text { for } H_0: \\ \text { parameter }=0 \end{array} & \begin{array}{c} \text { Std. error } \\ \text { of estimate } \end{array} \\ \hline \text { Intercept } & -10.8030 & & \\ x_1 & -0.0247 & -0.956 & 0.0259 \\ x_2 & 10.9409 & 18.517 & 0.5909 \\ \hline \end{array} $$ a. Interpret, in the context of the problem, the least squares estimate of $\beta_2$. b. Test, against a two-sided alternative, the null hypothesis $$ H_0: \beta_1=0 $$ c. The statistician is concerned about the possibility of multicollinearity. What information is needed to assess the potential severity of this problem? d. It is suggested that household income is an important determinant of size of electricity bill. If this is so, what can you say about the regression estimated by the statistician? e. Given the fitted model, the statistician obtains the predicted electricity bills, $\hat{y}_t$, and the residuals, $e_t$. He then regresses $\hat{e}_t^2$ on $\hat{y}_t$, finding that the regression has a coefficient of determination of 0.0470 . Interpret this finding.

A local public utility would like to be able to predict a dwelling unit's average monthly electricity bill. The company statistician estimated by least squares the following regression model:
$$
y_t=\beta_0+\beta_1 x_{1 t}+\beta_2 x_{2 t}+\varepsilon_t
$$
where
$$
\begin{aligned}
y_t= & \text { average monthly electricity bill, in dollars } \\
x_{1 t}= & \text { average bimonthly automobile gasoline bill, } \\
& \text { in dollars } \\
x_{2 t}= & \text { number of rooms in dwelling unit }
\end{aligned}
$$
From a sample of 25 dwelling units, the statistician obtained the following output from the SAS program:
$$
\begin{array}{crcc}
\hline \text { Parameter } & \text { Estimate } & \begin{array}{c}
\text { Student's } t \text { for } H_0: \\
\text { parameter }=0
\end{array} & \begin{array}{c}
\text { Std. error } \\
\text { of estimate }
\end{array} \\
\hline \text { Intercept } & -10.8030 & & \\
x_1 & -0.0247 & -0.956 & 0.0259 \\
x_2 & 10.9409 & 18.517 & 0.5909 \\
\hline
\end{array}
$$
a. Interpret, in the context of the problem, the least squares estimate of $\beta_2$.
b. Test, against a two-sided alternative, the null hypothesis
$$
H_0: \beta_1=0
$$
c. The statistician is concerned about the possibility of multicollinearity. What information is needed to assess the potential severity of this problem?
d. It is suggested that household income is an important determinant of size of electricity bill. If this is so, what can you say about the regression estimated by the statistician?
e. Given the fitted model, the statistician obtains the predicted electricity bills, $\hat{y}_t$, and the residuals, $e_t$. He then regresses $\hat{e}_t^2$ on $\hat{y}_t$, finding that the regression has a coefficient of determination of 0.0470 . Interpret this finding.

Key Concepts

Regression Coefficient Interpretation

This concept involves understanding the meaning of the estimated parameters in a linear regression model. The coefficient represents the expected change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other variables constant. It is crucial for inferring the strength and direction of the relationship between variables.

Hypothesis Testing for Regression Coefficients

This involves using statistical tests, such as the t-test, to determine whether an independent variable has a statistically significant relationship with the dependent variable. The null hypothesis typically states that the coefficient is zero (indicating no effect), and a t-test compares the estimated coefficient to its standard error to assess significance.

Multicollinearity

Multicollinearity refers to a situation in regression analysis where two or more independent variables are highly correlated. This high correlation can make it difficult to isolate the individual effect of each independent variable on the dependent variable, potentially leading to unstable and unreliable coefficient estimates. Common measures to assess multicollinearity include examining correlation matrices or variance inflation factors (VIF).

Omitted Variable Bias

This concept occurs when a relevant variable that influences the dependent variable is left out of the regression model. If the omitted variable is correlated with one or more of the included independent variables, the estimated coefficients of the included variables may be biased and inconsistent, potentially leading to misleading inference.

Heteroskedasticity Testing

Heteroskedasticity refers to the condition where the variance of the error terms varies across observations. Testing for heteroskedasticity, such as regressing squared residuals on predicted values, helps detect whether this non-constant variance exists in the data. The coefficient of determination (R-squared) from such a regression can indicate the extent to which the predicted values explain the variability of the error variance, with a low value suggesting weak evidence of heteroskedasticity.

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