Thesis

to be estimated. We consider a first-order panel VAR model, meaning that we consider one lag in the model.9 vi and uit are (3 x 1) vectors of the panel-specific fixed effects, modeling systematic crosssectional heterogeneity, and the idiosyncratic error, respectively. In the estimation of this model, the former, namely the panel-specific fixed effects, or student fixed effects, causes biased estimates if the meandifferencing procedure is used to eliminate fixed effects, as the fixed effects are related to the lags of the dependent variable. Therefore, we use the Helmert transformation. This transformation eliminates the forward mean, and so preserves the orthogonality between the transformed variables and the lagged regressors. To avoid the Nickell bias (Nickell, 1981), which is especially important in the case of a short time horizon, we use the lagged regressors as instruments and estimate the coefficients by Generalized Method of Moments (GMM). We consider a just-identified model with one lag.10 In just-identified models, GMM is numerically comparable to equation-by-equation 2SLS. One consideration when using skill scores is that these scores may contain measurement error, which can affect their reliability. Measurement error occurs when observed test scores deviate from true skill levels due to random noise, grading inconsistencies, or systematic bias. By estimating the panel VAR with GMM, we correct partly for measurement error using lagged test scores as instruments, under the assumption that the error is random and not correlated over time. If true skills are persistent, past test scores contain useful information about current scores. 9We select the lag order based on the coefficient of determination. We estimate a panel VAR(2) model as robustness check in Table C.2 in the appendix. 10A just-identified model has less risk of overfitting and avoids introducing unnecessary complexity. 91

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