<Dl> <Dd> β ^ = (X ⊤ X) − 1 X ⊤ Y . (\ displaystyle \ mathbf ((\ hat (\ boldsymbol (\ beta))) = () (X ^ (\ top) X) ^ (- 1) X ^ (\ top) Y). \,) </Dd> </Dl> <Dd> β ^ = (X ⊤ X) − 1 X ⊤ Y . (\ displaystyle \ mathbf ((\ hat (\ boldsymbol (\ beta))) = () (X ^ (\ top) X) ^ (- 1) X ^ (\ top) Y). \,) </Dd> <P> Once a regression model has been constructed, it may be important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters . Commonly used checks of goodness of fit include the R - squared, analyses of the pattern of residuals and hypothesis testing . Statistical significance can be checked by an F - test of the overall fit, followed by t - tests of individual parameters . </P> <P> Interpretations of these diagnostic tests rest heavily on the model assumptions . Although examination of the residuals can be used to invalidate a model, the results of a t - test or F - test are sometimes more difficult to interpret if the model's assumptions are violated . For example, if the error term does not have a normal distribution, in small samples the estimated parameters will not follow normal distributions and complicate inference . With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations . </P>

Regression analysis as a tool to forecast assumes that​