The OLS residuals ( ûi) in the multiple regression model

The OLS residuals ( ûi) in the multiple regression model

A are typically the same as the population regression function errors.

B cannot be calculated because there is more than one explanatory variable.

C are zero because the predicted values are another name for forecasted values.

D can be calculated by subtracting the fitted (predicted) values from the actual values of the dependent variable.

The Correct Answer and Explanation is :

The correct answer is:

D) can be calculated by subtracting the fitted (predicted) values from the actual values of the dependent variable.

Explanation:

In multiple regression, the Ordinary Least Squares (OLS) residuals are the differences between the observed (actual) values of the dependent variable and the values predicted by the model. These residuals are critical for diagnostics and model validation.

To understand this, let’s break it down:

1. Multiple Regression Model: This model involves predicting a dependent variable ( Y ) based on multiple independent variables ( X_1, X_2, …, X_k ). The model can be written as: [
   Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + … + \beta_kX_k + \epsilon
   ] where:

• ( Y ) is the dependent variable,
• ( \beta_0, \beta_1, …, \beta_k ) are the coefficients of the independent variables,
• ( X_1, X_2, …, X_k ) are the explanatory variables (independent variables),
• ( \epsilon ) represents the error term (or the residuals).

1. Fitted (Predicted) Values: These are the values of ( Y ) predicted by the model based on the observed values of the independent variables. These predicted values are denoted as ( \hat{Y} ). [
   \hat{Y} = \beta_0 + \hat{\beta}_1X_1 + \hat{\beta}_2X_2 + … + \hat{\beta}_kX_k
   ]
2. Residuals: The residuals (denoted as ( \hat{u}_i )) are the differences between the actual observed values of ( Y ) and the predicted values ( \hat{Y} ). [
   \hat{u}_i = Y_i – \hat{Y}_i
   ]
3. Calculation of Residuals: Residuals are calculated as the difference between the actual observed values of ( Y ) and the predicted values ( \hat{Y} ). This helps in assessing how well the model fits the data. A good model should have residuals that are randomly distributed, showing no pattern. Large residuals may indicate a poor fit, suggesting that the model needs refinement or reconsideration of the variables.

Therefore, option D is correct because residuals are calculated by subtracting the fitted (predicted) values from the actual observed values of the dependent variable.