16.6 Dynamic panels
Many economic relations are dynamic: \(y_{it} = \rho y_{i,t-1} + \alpha _i + \varepsilon _{it}\), with \(|\rho | < 1\). Here strict exogeneity fails by construction, because \(y_{i,t-1}\) depends on \(\varepsilon _{i,t-1}\), and \(\varepsilon _{i,t-1}\) enters the unit mean \(\bar \varepsilon _i\) subtracted by the within transformation.
Theorem 16.6.1 (Nickell). In the model \(y_{it} = \rho y_{i,t-1} + \alpha _i + \varepsilon _{it}\), with \(\varepsilon _{it}\) independent with variance \(\sigma ^2\), stationary initial conditions and \(T\) periods in the within regression, the fixed effects estimator satisfies, as \(N \to \infty \), \[ \plim (\hat \rho _{\text {FE}} - \rho ) = -\frac {1 + \rho }{T - 1}\,A\Bigl [1 - \frac {2\rho A}{(1 - \rho )(T - 1)}\Bigr ]^{-1}, \qquad A = 1 - \frac {1 - \rho ^T}{T(1 - \rho )} . \] For large \(T\) the bias is approximately \(-(1 + \rho )/(T - 1)\); it does not vanish as \(N \to \infty \), only as \(T \to \infty \).
Sketch of proof. The numerator of \(\hat \rho _{\text {FE}} - \rho \) is \(\frac 1N\sum _i\sum _t(y_{i,t-1} - \bar y_{i,-1})(\varepsilon _{it} - \bar \varepsilon _i)\). The terms \(y_{i,t-1}\varepsilon _{it}\) have mean zero, but \(y_{i,t-1}\bar \varepsilon _i\) does not, because \(y_{i,t-1}\) contains \(\varepsilon _{i,t-1}, \varepsilon _{i,t-2}, \dots \), each of which appears in \(\bar \varepsilon _i\) with weight \(1/T\). Writing \(y_{i,t-1} - \alpha _i/(1 - \rho ) = \sum _j\rho ^j\varepsilon _{i,t-1-j}\) and summing the resulting geometric series over \(t\) gives a negative limit for the numerator, proportional to \(\sigma ^2/T\); the denominator’s limit is computed in the same way, and Nickell (1981) gives the details. □
With \(T = 5\) and \(\rho = 0.5\) the bias is \(-0.33\): the estimate is about a third of the truth. Short panels, the commonest kind, are the worst affected. The remedy, due to Anderson and Hsiao (1981), is to first-difference, \(\Delta y_{it} = \rho \Delta y_{i,t-1} + \Delta \varepsilon _{it}\), which removes \(\alpha _i\) but makes \(\Delta y_{i,t-1}\) correlated with \(\Delta \varepsilon _{it}\) through \(\varepsilon _{i,t-1}\), and then to instrument \(\Delta y_{i,t-1}\) by \(y_{i,t-2}\). The level \(y_{i,t-2}\) is correlated with \(\Delta y_{i,t-1} = y_{i,t-1} - y_{i,t-2}\), and uncorrelated with \(\Delta \varepsilon _{it} = \varepsilon _{it} - \varepsilon _{i,t-1}\) if the \(\varepsilon _{it}\) are serially uncorrelated. Arellano and Bond (1991) used all the available lags \(y_{i,t-2}, y_{i,t-3}, \dots , y_{i1}\) as instruments in each period, which gives an overidentified GMM estimator in the sense of Section 15, with a \(J\) test of the overidentifying restrictions and a test for second-order serial correlation in \(\Delta \varepsilon _{it}\), which would invalidate the instruments. When \(\rho \) is close to one the lagged levels are weak instruments for the differences, and the system GMM estimator of Blundell and Bond (1998), which adds equations in levels instrumented by lagged differences, is preferred.
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