91Ó°ÊÓ

Added-variable plots This problem uses the United Nations example in Section 3.1 to demonstrate many of the properties of added-variable plots. This problem is based on the mean function $$\mathrm{E}\left(\log (\text {Fertility}) | \log (P P g d p)=x_{1}, \text {Purban}=x_{2}\right)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$$ There is nothing special about the two-predictor regression mean function, but we are using this case for simplicity. A. Show that the estimated coefficient for \(\log (P P g d p)\) is the same as the estimated slope in the added-variable plot for \(\log (P P g d p)\) after \(P\)urban. This correctly suggests that all the estimates in a multiple linear regression model are adjusted for all the other terms in the mean function. Also, show that the residuals in the added-variable plot are identical to the residuals from the mean function with both predictors. B. Show that the \(t\) -test for the coefficient for \(\log (P P g d p)\) is not quite the same from the added-variable plot and from the regression with both terms, and explain why they are slightly different.

Step by step solution

01

Establish the Multiple regression model

We start with the multiple regression model, where the expected value of the logarithm of fertility is functionally dependent on two variables: \(\log(Ppgdp)\) and \(P_{\text{urban}}\). The model is given by \[ \mathrm{E}(\log (\text {Fertility}) | \log (P P g d p) = x_1, P_{\text{urban}} = x_2) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 \]

02

Compare the regression coefficient with the added-variable plot slope

Compare the estimated coefficient of \(\log (P P g d p)\) in the multiple regression model with the estimated slope of the added-variable plot. The added-variable plot is generated by first regressing \(\log (P P g d p)\) on \(P_{\text{urban}}\) and then on the response variable \(\log (\text {Fertility})\). If our model is correct, the slope in the added-variable plot should be the same as \(\beta_1\). This confirms the interpretation of \(\beta_1\) as the effect of \(\log (P P g d p)\) on \(\log (\text {Fertility})\), adjusting for the effect of \(P_{\text{urban}}\).

03

Compare the residuals

Check if the residuals from the mean function considering both predictors and the residuals from the added-variable plot are identical. This would support the interpretation of residuals as the portions of the response variable (\(\log (\text {Fertility})\)) that cannot be explained by predictors \(\log (P P g d p)\) and \(P_{\text{urban}}\).

04

Perform the t-test

Perform the t-test for the coefficient of \(\log (P P g d p)\) in the multiple regression model. Compare the result with the t-test from the added-variable plot.

05

Explain the difference in t-tests

The t-statistics will not be identical due to the added-variable plot considering \(\log (P P g d p)\) in isolation, when adjusted for \(P_{\text{urban}}\), resulting in slightly different standard errors and hence differing t-statistics. The difference is due to the interplay of predictors in the multiple regression model, which is not exactly mirrored in the added-variable plot.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiple Regression Model

Multiple regression models allow us to understand the relationship between one dependent variable and two or more independent variables. By predicting the dependent variable based on the values of the independent ones, analysts can determine how significant each independent variable is in contributing to the dependent variable.

In the provided exercise, the model predicts \(\log(\text{Fertility})\) based on \(\log(Ppgdp)\) and \(P_{\text{urban}}\). It's structured as \(\mathrm{E}(\log (\text {Fertility}) | \log (P P g d p) = x_1, P_{\text{urban}} = x_2) = \beta_0 + \beta_1 x_1 + \beta_2 x_2\). The coefficients \(\beta_1\) and \(\beta_2\) correspond to the changes in the dependent variable for a one-unit change in their respective independent variables, holding other variables constant.

Regression Coefficients

Regression coefficients in a multiple regression model represent the individual contribution of each independent variable to the prediction of the dependent variable. The example illustrates this with coefficients \(\beta_1\) and \(\beta_2\), which measure the change in \(\log (\text{Fertility})\) for a unit change in \(\log(Ppgdp)\) and \(P_{\text{urban}}\), respectively.

Understanding these coefficients is crucial as they can indicate the strength and direction of the association between the dependent and independent variables. Higher absolute values suggest a stronger relationship, while the sign indicates if the relationship is positive or negative.

Residuals Analysis

Residuals analysis in regression is key to validating a model's accuracy. Residuals are the differences between the observed values of the dependent variable and the values predicted by the regression model.

In our case, we check whether the residuals from the model that includes both predictors match the residuals depicted in the added-variable plot. Matched residuals would imply that the model accounts for the variability in \(\log (\text{Fertility})\) properly and that the effects of the independent variables have been correctly adjusted. Residuals analysis can uncover patterns that suggest potential issues with the model, such as non-linearity or heteroscedasticity.

T-test

The t-test in regression analysis assesses whether the regression coefficients are significantly different from zero, indicating a meaningful contribution of the independent variables to the model. The t-statistic uses the standard error to determine if a coefficient is likely to be due to chance. In the exercise, comparing the t-tests from the multiple regression and the added-variable plot reveals slight differences due to the unique way they account for the variables.

These differences highlight the importance of context when interpreting t-tests. By examining the t-statistics within the framework of the entire model, one can understand the influence and significance of each predictor on the dependent variable.