Some issues in multivariate regression 1. Increasing the number of right-hand side variables will always increase the R-squared for the regression. 2. If the right-hand side variables are highly correlated with each other then the standard errors of the OLS coefficients will become large. 3. If we omit relevant variables then the OLS estimates will be biased. 4. If we include irrelevant variables the OLS estimates will be unbiased but inefficient.

Adjusted R-squared Adding extra regressors to the RHS of an equation will always increase R-squared. Therefore it does not make sense to use R-squared as a basis for choosing a set of regressors. An alternative is the adjusted R-squared or R-bar squared. This penalises the loss of degrees of freedom when we add extra regressors.

The regression model in mean deviation form Suppose we have: We can redefine the variables as: This will make the algebra considerably easier in our analysis of the multivariate model.

Multicollinearity Consider the regression model The variance covariance matrix of the regression coefficients can be derived as: where

Now consider the D term. This can be written: where is the correlation coefficient for the two X variables. It follows that that closer the correlation between the X variables, the closer D will be to zero and the larger the variance-covariance matrix for the OLS coefficients.

If the x variables in a regression equation are highly correlated then this can result in estimates with a high variance. This is is known as the problem of multicollinearity. The usual signs of multicollinearity are: For individual variables the standard errors are large and the t-ratios are low. For the regression as a whole the F-statistic is highly significant. The R2 will tend to be high. Some degree of multicollinearity is almost always present in econometric models.

Omitted Variables Estimated Model True Model OLS is therefore biased

Inclusion of Irrelevant Variables Estimated Model True Model In this case OLS is unbiased but inefficient