1. 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.

2. 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.

3. 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.

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

5. 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.

6. 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.

7. Omitted Variables
Estimated Model
True Model
OLS is therefore biased

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