virhevarianssista

Virhevarianssi refers to the variance of the error term in statistical models, particularly in regression analysis. It quantifies the degree of scatter or dispersion of the observed data points around the predicted regression line. A smaller error variance suggests that the model fits the data well, with most of the variation in the dependent variable explained by the independent variables. Conversely, a larger error variance indicates that the model does not capture the underlying relationships effectively, and a significant portion of the variation remains unexplained.

Estimating the error variance is crucial for several reasons. It is a key component in calculating standard errors of regression coefficients, which are used to construct confidence intervals and perform hypothesis tests. For example, the t-statistic used to test the significance of a coefficient is calculated by dividing the coefficient estimate by its standard error, which directly depends on the error variance.

In the context of ordinary least squares (OLS) regression, the error variance is typically estimated by the residual mean square (MSE). The residuals are the differences between the observed values of the dependent variable and the values predicted by the regression model. The MSE is calculated by summing the squared residuals and dividing by the degrees of freedom, which is the number of observations minus the number of estimated parameters. This estimated error variance, often denoted as $\hat{\sigma}^2$, provides a measure of the model's overall fit and the precision of its predictions.