Advanced Econometrics Takeshi Amemiya
Balestra-Nerlove Model
As we mentioned earlier, this is a generalization of 2ECM in the sense that a lagged endogenous variable yu_, is included among the regressors. Balestra and Nerlove (1966) used this model to analyze the demand for natural gas in 36 states in the period 1950-1962.
All the asymptotic results stated earlier for 2ECM hold also for the Balestra- Nerlove model provided that both N and T go to <», as shown by Amemiya (1967). However, there are certain additional statistical problems caused by the presence of a lagged endogenous variable; we shall delineate these problems in the following discussion.
First, the LS estimator ofobtained from (6.6.18) is always unbiased and is consistent if N goes to oo. However, if xft contain y/f/_!, LS is inconsistent even when both N and T go to °o. To see this, consider the simplest case
У и = Mr-i + V-і +
where we assume |/?| < 1 for stationarity and ую = 0 for simplicity. Solving the difference equation (6.6.33) and omitting i, we obtain
(6.6.34)
Therefore, putting back the subscript i,
(6.6.35)
Therefore
(6.6.36)
which implies the inconsistency of LS.
Second, we show that the transformation estimator fiQl is consistent if and only if T goes to °°.9 We have
where Q is given by (6.6.22). We need to consider only the column of X,, which corresponds to the lagged endogenous variable, denoted y_,, and only the T~xA part of Q. Thus the consistency (as Г—* °°) of jJQ1 follows from
(6.6.38)
Third, if ую (the value of at time t = 1) is assumed to be an unknown parameter for each i, the MLE offi is inconsistent unless T goes to °° (Anderson and Hsiao, 1981,1982). The problem of how to specify the initial values is important in a panel data study, where typically N is large but T is not.
Fourth, the possibility of a negative MLE of pis enhanced by the presence of a lagged endogenous variable, as shown analytically by Maddala (1971) and confirmed by a Monte Carlo study of Nerlove (1971). In his Monte Carlo study, Nerlove compared various estimators of fi and concluded that the FGLS described at the end of Section 6.6.2 performs best. He found that the transformation estimator of the coefficient on yu-i has a tendency of downward bias.
