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How much useful information is brought to bear on estimation of the parameters is uncertain, as it depends on the correlation of the instruments with the included exogenous variables in the equation. The farther apart in time these sets of variables become the less information is likely to be present. (The literature on this subject contains reference to "strong" versus "weak" instrumental variables.29) In order to proceed, as noted, we can include the lagged dependent variable in x2i. This set of instrumental variables can be used to construct the estimator, actually whether the lagged variable is present or not. We note, at this point, that on this basis, Hausman and Taylor's estimator did not

29See West (2001).

actually use all the information available in the sample. We now have the elements of the Arellano et al. estimator in hand; what remains is essentially the (unfortunately, fairly involved) algebra, which we now develop. Let

yiT.

Note that W, is assumed to be, a T x (1 + K1 + K2 + L1 + L2) matrix. Since there is a lagged dependent variable in the model, it must be assumed that there are actually T +1 observations available on yit. To avoid a cumbersome, cluttered notation, we will leave this distinction embedded in the notation for the moment. Later, when necessary, we will make it explicit. It will reappear in the formulation of the instrumental variables. A total of T observations will be available for constructing the IV estimators. We now form a matrix of instrumental variables. Different approaches to this have been considered by Hausman and Taylor (1981), Arellano et al. (1991, 1995, 1999), Ahn and Schmidt (1995) and Amemiya and MaCurdy (1986), among others. We will form a matrix V, consisting of T - 1 rows constructed the same way for T - 1 observations and a final row that will be different, as discussed below. [This is to exploit a useful algebraic result discussed by Arellano and Bover (1995).] The matrix will be of the form

The instrumental variable sets contained in Vit which have been suggested might include the following from within the model:

xit and xi t-1 (i.e., current and one lag of all the time varying variables) xi1,..., xiT (i.e., all current, past and future values of all the time varying variables) xi1, ...,xit (i.e., all current and past values of all the time varying variables)

The time invariant variables that are uncorrelated with u, that is z1i, are appended at the end of the nonzero part of each of the first T - 1 rows. It may seem that including x2 in the instruments would be invalid. However, we will be converting the disturbances to deviations from group means which are free of the latent effects—that is, this set of moment conditions will ultimately be converted to what appears in (13-38). While the variables are correlated with u by construction, they are not correlated with sit - si. The final row of V is important to the construction. Two possibilities have been suggested:

a- = [z1 x^] (produces the Hausman and Taylor estimator)

ai = [z1 x! 1, x!2,..., x1iT] (produces Amemiya and MaCurdy's estimator).

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