In the least-squares regression model, yi = β1xi + β0 + εi, εi is a random error term with mean ______ and standard deviation σεi = ______.

Verified step by step guidance

1. Recall that in the least-squares regression model, the error term \( \varepsilon_i \) represents the deviation of the observed value \( y_i \) from the predicted value based on the regression line.
2. By definition, the error term \( \varepsilon_i \) is assumed to have an expected value (mean) of zero, which means \( E(\varepsilon_i) = 0 \). This assumption ensures that the regression line is an unbiased estimator of the true relationship.
3. The standard deviation of the error term \( \varepsilon_i \) is denoted by \( \sigma_{\varepsilon_i} \), which measures the typical size of the deviations from the regression line.
4. In the classical linear regression model, the standard deviation of the error term is assumed to be constant for all observations, often written as \( \sigma_{\varepsilon_i} = \sigma \), where \( \sigma \) is an unknown positive constant.
5. Summarizing, the error term \( \varepsilon_i \) has mean \( 0 \) and constant standard deviation \( \sigma \), which are key assumptions for the validity of least-squares regression inference.

Key Concepts

Random Error Term (εi)

In a regression model, the random error term εi represents the difference between the observed value and the value predicted by the model. It accounts for variability not explained by the predictors and is assumed to have a mean of zero, indicating no systematic bias.

Mean of the Error Term

The mean of the error term εi is assumed to be zero in least-squares regression. This assumption ensures that the regression line is an unbiased estimator of the true relationship between the dependent and independent variables.

Standard Deviation of the Error Term (σεi)

The standard deviation σεi of the error term measures the spread or variability of the errors around their mean (zero). It is constant across observations (homoscedasticity) in classical regression, reflecting consistent variability in the errors.