Variances and Covariances of Least Squares
The variances and covariances of the least squares estimator give us information about how precise our knowledge of the parameters is from estimating them. Smaller standard errors mean that our knowledge is more precise.
The precision of least squares (LS) depends on a number of factors.
1. Smaller variation in the dependent variable about its mean, a2, makes LS more precise.
2. Larger samples, N, improve LS precision.
3. More variation in the independent variables about their respective means makes LS more precise.
4. Less correlation between the least squares estimates, corr(b2, b3), also improves LS precision.
The precision of least squares (and other estimators) is summarized by the variance-covariance matrix, which includes a measurement of the variance of the intercept, each slope, and covariance between each pair. The variances of the least squares estimator fall on the diagonal of this square matrix and the covariances in the off-diagonal elements.
All of these have to be estimated from the data, and generally depends on your estimate of the overall variance of the model, a2 and correlations among the independent variables. To print an estimate of the variance-covariance matrix following a regression use the —vcv option with your regression in gretl :
The result is
Coefficient covariance matrix
const price advert
40.343 -6.7951 -0.74842 const
1.2012 -0.01974 price
0. 46676 advert
For instance, the estimated variance of bi-the intercept-is 40.343 and the estimated covariance between the LS estimated slopes b2 and b3 is -0.01974.
A (estimated) standard error of a coefficient is the square root of its (estimated) variance, se(b2) = д/var(b2). These are printed in the output table along with the least squares estimates, t-ratios, and their p-values.