Consider a regression model that uses 48 observations. Let...

Consider a regression model that uses 48 observations. Let $e_i$ denote the residuals from the fitted regression and $\hat{y}_i$ be the in-sample predicted values of the dependent variable. The least squares regression of $e_i^2$ on $\hat{y}_i$ has coefficient of determination 0.032 . What can you conclude from this finding?

Key Concepts

Residual Analysis

Residual analysis involves studying the errors (differences between observed and predicted values) from a regression model. It helps identify patterns that indicate potential problems with the model such as non-linearity, autocorrelation, or heteroscedasticity, and is a key step in verifying model assumptions.

Heteroscedasticity

Heteroscedasticity refers to the situation where the variance of residuals is not constant across different levels of the independent variables. This violation of the constant variance assumption can lead to inefficient estimates and biased standard errors, which in turn affect hypothesis testing and confidence intervals.

Coefficient of Determination

The coefficient of determination, often denoted as R², measures the proportion of variability in the dependent variable that is explained by the independent variable(s) in a regression model. In the context of a diagnostic test, a low R² value when regressing squared residuals on predicted values indicates that the model does not explain much of the variability in the squared residuals, suggesting an absence of systematic heteroscedasticity.

Regression Diagnostic Testing

Regression diagnostic testing includes various methods to assess whether the assumptions underlying a regression model are met. For detecting heteroscedasticity, one common approach is to regress the squared residuals on the fitted values. The resulting statistic, such as the R² from this regression, helps evaluate whether the variance of the errors changes with the level of the predicted outcome.

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