A dependent variable Y (20 × 1) was regressed onto 3 independent variables plus an intercept (so that X was of dimension 20 × 4). The following matrices were computed.
(a) Compute
and write the regression equation.
(b) Compute the analysis of variance of Y. Partition the sum of squares due to the model into a part due to the mean and a part due to regression on the Xs after adjustment for the mean. Summarize the results, including degrees of freedom and mean squares, in an analysis of variance table.
(c) Compute the estimate of σ2
and the standard error for each regression coefficient. Compute the covariance between
1
and
2, Cov(
1,
2). Compute the covariance between
1
and
3, Cov(
1,
3).
(d) Drop X3
from the model. Reconstruct X’X and X’Y for this model without X3
and repeat Questions (a) and (b). Put X3
back in the model but drop X2
and repeat (a) and (b).
(i) Which of the two independent variables X2
or X3
made the greater contribution to Y in the presence of the remaining Xs; that is, compare R(β2|β0, β1, β3) and R(β3|β0, β1, β2).
(ii) Explain why
1
changed in value when X2
was dropped but not when X3
was dropped.
(iii) Explain the differences in meaning of β1
in the three models.
(e) From inspection of X’X how can you tell that X1, X2, and X3
were expressed as deviations from their respective means? Would (X’X)−1
have been easier or harder to obtain if the original Xs (without subtraction of their means) had been used?