Consider the Monte Carlo setup given in Arellano and Bond (1991) (p. 283) for a simple autoregressive equation with one regressor with \(N=100\) and \(T=7\).

(a) Compute the bias and mean squared error based on 100 replications of the following estimators: OLS, Within, one-step and two-step Arellano and Bond GMM estimators, two Anderson and Hsiao type estimators that use \(\Delta y_{i, t-2}\) and \(y_{i, t-2}\) as an instrument for \(\Delta y_{i, t-1}\), respectively. Compare with Table 8.1, p. 284 of Arellano and Bond (1991).

Table 8.1:

(b) Compute Sargan's test of over-identifying restrictions given below (8.11) and count the number of rejections out of 100 replications. Compare with Table 8.2 of Arellano and Bond (1991).

Table 8.2:

\[\begin{align*}
& E\left[\left(y_{i, 1}\left(u_{i, 3}-u_{i, 2}\right)\right]=0\right.  \tag{8.11a}\\
& E\left[\left(y_{i, 1}\left(u_{i, 4}-u_{i, 3}\right)\right]=0\right.  \tag{8.11b}\\
& E\left[\left(y_{i, 2}\left(u_{i, 4}-u_{i, 3}\right)\right]=0\right. \tag{8.11c}
\end{align*}\]

Transcribed Image Text:

InCi.-1 In Pit In Pnit In Yit OLS 0.97 -0.090 0.024 -0.03 (157.7) (6.2) (1.8) (5.1) Within 0.83 -0.299 0.034 0.10 (66.3) (12.7) (1.2) (4.2) 2SLS 0.85 -0.205 0.052 -0.02 (25.3) (5.8) (3.1) (2.2) 2SLS-KR 0.71 -0.311 0.071 -0.02 (22.7) (13.9) (3.7) (1.5) Within-2SLS 0.60 -0.496 -0.106 0.19 (17.0) (13.0) (0.5) (6.4) FD-2SLS 0.51 -0.348 0.112 0.10 (9.5) (12.3) (3.5) (2.9) FD-2SLS-KR 0.40 -0.341 0.083 0.21 (13.3) (18.4) (4.2) (10.0) GMM-two-step 0.70 -0.396 -0.105 0.13 (10.2) (6.0) (1.3) (3.5) System GMM 0.70 -0.415 -0.003 0.09 (8.8) (4.3) (1.0) (3.4)