Lag Structure

TIME-SERIES models typically assume that the independent variables affect the dependent variable both instantaneously and over time. This relationship is modeled with a lag structure whose parameters are then estimated by some appropriate method. Focusing on a single independent variable, x, and omitting the error term, the most general lag structure is

It is impossible to estimate the parameters of such a model because there are an infinite number of them.

A finite distributed lag structure assumes that after some given lag—say, K—x no longer affects y. Here, the lag structure has only a finite number of parameters, that is, βK+1 = βK+2 = … = 0. Although this can be estimated in principle, MULTICOLLINEARITY makes estimation of each parameter very difficult unless it is assumed that only one or two lags of x affect current y (and even here multicollinearity may still be a problem). Thus, analysts usually impose some parametric structure on the βs. The most common assumption is that they follow a polynomial structure, such as one that is quadratic in time, as in βk = λ0 + λ1k + λ2k2. This restriction means that analysts need only estimate three parameters.

Many analysts prefer to allow for an infinite (or at least unlimited) number of lags of x to affect y, feeling that it is odd to allow for impacts from up to some prespecified lag and then zero impacts thereafter. The most common parameterization for this infinite distributed lag is the geometric lag structure, where the impact of x on y is assumed to decline geometrically, that is, βk = βρk. This can be algebraically transformed into a model with a lagged dependent variable. The assumption that the impact of x on y declines geometrically with lag length is often plausible. If it is too constraining, analysts can combine the finite and infinite lag structures by allowing the (very) few first lags to be freely estimated, as in the finite lag structure model, with the remaining lags assumed to decline geometrically. This may provide a nice balance between the flexibility of the finite lag model and the simplicity of the infinite geometric lag model without forcing the analyst to estimate too many parameters or falling afoul of multicollinearity problems.

With several independent variables, analysts must choose between the simplicity of forcing them all to follow the same lag structure and the flexibility of allowing for each variable to have its own lag structure. The price of the latter flexibility is that analysts have to estimate many more parameters. Analysts can also use lag structures to model the error process. All the various tools available to time-series analysts, such as correlograms, can be usefully combined with theoretical insight to allow analysts to choose a good lag structure.