tailieunhanh - Class Notes in Statistics and Econometrics Part 31

CHAPTER 61 Random Coefficients. The random coefficient model first developed in [HH68] cannot be written in the form y = Xβ + ε because each observation has a different β. Therefore we have to write it observation by observation: y t = xt β t (no separate disturbance term), | CHAPTER 61 Random Coefficients The random coefficient model first developed in HH68 cannot be written in the form y Xj3 e because each observation has a different ft. Therefore we have to write it observation by observation yt xt ftt no separate disturbance term where 3t 3 vt with vt o t2S . For s t vs and vt are uncorrelated. By re-grouping terms one gets yt xt 1 3- xtvt xt 1 3- t where t xtvt hence var t t2xtTSxt and for s t s and t are uncorrelated. 1303 1304 61. RANDOM COEFFICIENTS In tiles this model is Estimation under the assumption S is known To estimate 3 one can use the het-eroskedastic model with error variances t2xtTSxt call the resulting estimate 3- The formula for the best linear unbiased predictor of 3t itself can be derived heuristically as follows Assume for a moment that 3 is known then the model can be written as yt xtT3 xtTvt. Then we can use the formula for the Best Linear Predictor equation applied to the situation r Xt 1 vt 0 2 xtT xt xtTS Vt o T Sxt s where xtTvt is observed its value is yt xtT3 but vt is not. Note that here we predict a whole vector on the basis of one linear combination of its elements only. 61. RANDOM COEFFICIENTS 1305 This predictor is v Sxt xtTSxt -1 yt - xj 3 If one adds 3 to both sides one obtains 3 3 Sxi xiTSxi -1 yi - x 3 If one now replaces 3 by 3 one obtains the formula for the predictor given in JHG 88 p. 438 3 3 Sxt xtTSxt -1 yt - xj 3 - Usually of course S is unknown. But if the number of observations is large enough one can estimate the elements of the covariance matrix t2S. This is the fact which gives relevance to this model. Write t2xtTSxt t2 tr xtTSxt t2 tr xtxtTS zja where zt is the vector containing the unique elements of the symmetric matrix xtxtT with those elements not located on the diagonal multiplied by the factor 2 since they occur twice in the matrix and a contains the corresponding unique elements of t2S but no factors 2 here . For instance if there are three .