Consider a data set consisting of n observations, n_c complete and n_m incomplete, for which the dependent variable, y_i, is missing. Data on the independent variables, x_i, are complete for all n observations, X_c and X_m. We wish to use the data to estimate the parameters of the linear regression model y = Xβ + ε. Consider the following the imputation strategy: Step 1: Linearly regress y_c on X_c and compute b_c. Step 2: Use X_m to predict the missing y_m with X_mb_c. Then regress the full sample of observations, (y_c,X_mb_c), on the full sample of regressors, (X_c,X_m).

a. Show that the first and second step least squares coefficient vectors are identical.

b. Is the second step coefficient estimator unbiased?

c. Show that the sum of squared residuals is the same at both steps.

d. Show that the second step estimator of σ² is biased downward.