Society of Actuaries (SOA) PA Practice Exam

In regression analysis, what do residuals represent?

1. The predicted values of the response variable

2. The difference between observed and predicted values

3. The overall model fit statistics

4. The correlation between variables

The correct answer is: The difference between observed and predicted values

Residuals in regression analysis represent the difference between the observed values of the response variable and the values predicted by the regression model. They serve as a crucial component in assessing the accuracy of the model. When a regression model is fitted to a dataset, it attempts to estimate the relationship between the variables, predicting what the response variable should be based on the values of the explanatory variables. However, no model is perfect, and the actual observed values often differ from these predictions. This discrepancy is quantified as the residual. The sum of all residuals is typically zero, as positive and negative differences balance each other out, but the examination of these residuals can provide insights into the model's accuracy and whether it is appropriate for the data. In contrast, the other options refer to different aspects of regression analysis. The predicted values are not what residuals represent, but rather the output of the model given the input data. Overall model fit statistics relate to how well the model captures the data as a whole, incorporating measures such as R-squared, while correlation assesses the strength and direction of a linear relationship between variables, none of which encompass the definition of residuals.