Estimated dependent variable regression
In a recent Political Analysis special issue (Volume 13 Number 1, Winter 2005) there is the suggestion of estimating “two level models” for cross-country survey data in two steps. Where the first step is a within country regression and the second steps regresses the estimated coefficients of the first step (say, the intercepts) on country level covariates.
Jeff Lewis kindly provided me his ancient (2000) stata code and I corrected a very small bug and included an option for Borjas weights.
Jeff Lewis and Linzer article:
Lewis, J.B. & Linzer, D.A. Estimating Regression Models in Which the Dependent Variable Is Based on Estimates Political Analysis, 2005
and the (very similar) Borjas weights used in my article (with John Huber and Georgia Kernell) in the same issue:
Huber, J.D.; Kernell, G. & Leoni, E.L. Institutional Context, Cognitive Resources and Party Attachments Across Democracies Political Analysis, 2005, 13, 365-386
See also
Borjas, G.J. & Sueyoshi, G.T. A two-stage estimator for probit models with structural group effects Journal of Econometrics, 1994, 64, 165-182
Hanushek, E.A. Efficient Estimators for Regressing Regression Coefficients American Statistician, 1974, 28, 66-67
[edit 3/27/2007: fixed links]
stata example using simulated data
Hanushek estimates sigma as:
[tex]\hat\sigma^2 =\frac{\sum\hat v_i^2-\sum\omega_i^2+tr((X’X)^{-1}X’GX)}
{N-k}[/tex]
Where [tex]\sum\hat v_i^2[/tex] is the sum of the squared residuals of an unweighted linear regression using the first stage estimates as dependent variables, [tex]\omega_i [/tex]are the first stage standard errors, G is diag([tex]\omega_i[/tex]) N is the number of observations in the second stage and k is the number of regressoes (including a constant) in the second stage.
Borjas (1994) argues that the independence assumption across errors in different levels imply that sigma can be estimated more simply by:
[tex]\hat\sigma^2 =\frac{\sum\hat v_i^2-\sum\omega_i^2}
{N-k}[/tex]