Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 7

Suppose that the linear model, written in matrix notation,

satisfies Assumptions E.1, E.2, and E.3. Partition the model as

where X₁ is n × (k₁ 1 1) and X₂ is n × k₂.

(i) Consider the following proposal for estimating ?₂. First, regress y on X₁ and obtain the residuals, say, ÿ. Then, regress ÿ on X₂ to get . Show that  is generally biased and show what the bias is. [You should find  in terms of ?₂, X₂, and the residual-making matrix M₁.]

(ii) As a special case, write

where X_k is an n × 1 vector on the variable x_tk. Show that

where SSR_k is the sum of squared residuals from regressing x_tk on 1, x_t1, x_t2, …, x_{t, k–1}. How come the factor multiplying ?_k is never greater than one?

(iii) Suppose you know ?₁. Show that the regression y – X₁?₁ on X₂ produces an unbiased estimator of ?₂ (conditional on X).

Step 1 of 4

Consider the linear model as,

That is partitioned as,

Here,