Based on data from 63 counties, the following model was...

Based on data from 63 counties, the following model was estimated by least squares: $$ \hat{y}=0.58-\underset{(0.019)}{0.052 x_1}-\underset{(0.042)}{0.005 x_2} \quad R^2=0.17 $$ where $\hat{y}=$ growth rate in real gross domestic product $x_1=$ real income per capita $x_2=$ average tax rate, as a proportion of gross national product The numbers below the coefficients are the coefficient standard errors. After the independent variable $X_1$, real income per capita, was dropped from the model, the regression of growth rate in real gross domestic product on $X_2$, average tax rate, was estimated. This yielded the following fitted model: $$ \hat{y}=0.060-\underset{(0.034)}{0.074 x_2} \quad R^2=0.072 $$ Comment on this result.

Based on data from 63 counties, the following model was estimated by least squares:
$$
\hat{y}=0.58-\underset{(0.019)}{0.052 x_1}-\underset{(0.042)}{0.005 x_2} \quad R^2=0.17
$$
where
$\hat{y}=$ growth rate in real gross domestic product
$x_1=$ real income per capita
$x_2=$ average tax rate, as a proportion of gross national product
The numbers below the coefficients are the coefficient standard errors. After the independent variable $X_1$, real income per capita, was dropped from the model, the regression of growth rate in real gross domestic product on $X_2$, average tax rate, was estimated. This yielded the following fitted model:
$$
\hat{y}=0.060-\underset{(0.034)}{0.074 x_2} \quad R^2=0.072
$$
Comment on this result.

Key Concepts

Omitted Variable Bias

This concept refers to the distortion in estimated relationships that occurs when a relevant variable—one that is correlated with both the independent variable of interest and the dependent variable—is left out of the model. In the given scenario, dropping the variable 'real income per capita' appears to change the estimated coefficient on average tax rate, suggesting that its omission may be biasing the coefficient estimates of the remaining variable.

Model Specification

Model specification involves choosing the appropriate set of explanatory variables to include in a regression analysis. The change in the estimated coefficient and R² when ‘real income per capita’ is removed illustrates how excluding an important variable can alter both the magnitude and the significance of other coefficients as well as the overall explanatory power of the model.

Goodness-of-Fit (R²)

The R² statistic measures how well the model explains the variation in the dependent variable. In this case, a noticeable drop in R² when moving from the full model to the reduced model indicates that the omitted variable 'real income per capita' plays a significant role in explaining the variability of the growth rate in GDP, thereby reducing the model's overall fit when it is excluded.

Confounding

Confounding occurs when the effect of an independent variable on the outcome is mixed with the effect of an omitted variable that is related to both the independent variable and the outcome. Here, the change in the estimated effect of average tax rate when 'real income per capita' is dropped suggests that the relationship between tax rate and GDP growth is confounded by income, underscoring the importance of controlling for all relevant variables.

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