18.6 Consider the regression model in matrix form, Y = XB + WG + U, where X is an n * k1 matrix of regressors and W is an n * k2 matrix of regressors. Then, as shown in Exercise 18.17, the OLS estimator B n can be expressed B

n

= (XMWX )-1(XMWY ).

Now let b nBV 1 be the “binary variable” fixed effects estimator computed by estimating Equation (10.11) by OLS and let b nDM 1 be the “de-meaning”

fixed effects estimator computed by estimating Equation (10.14) by OLS, in which the entity-specific sample means have been subtracted from X and Y. Use the expression for B n given above to prove that b nBV 1 = b nDM 1 .

[Hint: Write Equation (10.11) using a full set of fixed effects, D1i, D2i, . . . , Dni and no constant term. Include all of the fixed effects in W. Write out the matrix MWX.]