## Finite Distributed Lag Models

An unrestricted finite distributed lag model would be specified as p yt = a + E&xt-i + et ■ (19-11)

We assume that xt satisfies the conditions discussed in Section 5.2. The assumption that there are no other regressors is just a convenience. We also assume that et is distributed with mean zero and variance of. If the lag length p is known, then (19-11) is a classical regression model. Aside from questions about the properties of the

3For further discussion and some alternative measures, see Geweke and Meese (1981), Amemiya (1985, pp. 146-147), Diebold (1998a, pp. 85-91), and Judge et al. (1985, pp. 353-355).

4See Pagano and Hartley (1981) and Trivedi and Pagan (1979).

independent variables, the usual estimation results apply.5 But the appropriate length of the lag is rarely, if ever, known, so one must undertake a specification search, with all its pitfalls. Worse yet, least squares may prove to be rather ineffective because (1) time series are sometimes fairly short, so (19-11) will consume an excessive number of degrees of freedom;6 (2) et will usually be serially correlated; and (3) multicollinearity is likely to be quite severe.

Restricted lag models which parameterize the lag coefficients as functions of a few underlying parameters are a practical approach to the problem of fitting a model with long lags in a relatively short time series. An example is the polynomial distributed lag (PDL) [or Almon (1965) lag in reference to S. Almon, who first proposed the method in econometrics]. The polynomial model assumes that the true distribution of lag coefficients can be well approximated by a low-order polynomial,

Pi = a0 + a1i + a2i2 +-----+ a piq, i = 0,1,..., p > q. (19-12)

After substituting (19-12) in (19-11) and collecting terms, we obtain yt = Y + a0 Y i0 xt "0 + a1 12 i1 xt-M +-----+ a J Y^Xt-i) + £t

Each Zjt is a linear combination of the current and p lagged values of xt. With the assumption of strict exogeneity of xt, y and (a0, a1,..., aq) can be estimated by ordinary or generalized least squares. The parameters of the regression model, Pi and asymptotic standard errors for the estimators can then be obtained using the delta method (see Section D.2.7).

The polynomial lag model and other tightly structured finite lag models are only infrequently used in contemporary applications. They have the virtue of simplicity, although modern software has made this quality a modest virtue. The major drawback is that they impose strong restrictions on the functional form of the model and thereby often induce autocorrelation that is essentially an artifact of the missing variables and restrictive functional form in the equation. They remain useful tools in some forecasting settings and analysis of markets, as in Example 19.3, but in recent work in macroeco-nomic and financial modeling, where most of this sort of analysis takes place, the availability of ample data has made restrictive specifications such as the PDL less attractive than other tools.

19.3.2 AN INFINITE LAG MODEL: THE GEOMETRIC LAG MODEL

There are cases in which the distributed lag models the accumulation of information. The formation of expectations is an example. In these instances, intuition suggests that

5The question of whether the regressors are well behaved or not becomes particularly pertinent in this setting, especially if one or more of them happen to be lagged values of the dependent variable. In what follows, we shall assume that the Grenander conditions discussed in Section 5.2.1 are met. We thus assume that the usual asymptotic results for the classical or generalized regression model will hold.

6Even when the time series is long, the model may be problematic—in this instance, the assumption that the same model can be used, without structural change through the entire time span becomes increasingly suspect the longer the time series is. See Sections 7.4 and 7.7 for analysis of this issue.

the most recent past will receive the greatest weight and that the influence of past observations will fade uniformly with the passage of time. The geometric lag model is often used for these settings. The general form of the model is yt = a + pYd - X)Xtxt-i + St, 0 <X< 1, i=i (19-14)

where

0 0