Friday, August 2, 2013

Allocation Models With Autocorrelated Errors

Not too long ago, I had a couple of posts about "allocation models" (here and here). These models are systems of regression equations in which there is a constraint on the data for the dependent variables for the equations. Specifically, at every point in the sample, these variables sum exactly to the value of a linear combination of the regressors. In practice, this linear combination usually is very simple - it's just one of the regressors.

So, for example, suppose that the dependent variables measure the shares of Canada's exports that go to different countries. These shares must add up to one in value. If we have an intercept (a series of "ones") in each equation, then we have an allocation model.

In one of the comments on the earlier posts, I was asked about the possibility of autocorrelated errors in the empirical example that I provided. In my response, I noted that if autocorrelation is present, and is allowed for in the estimation of the model, then special care is needed. In particular, any modification to the model, to allow for a specific form of autocorrelation, must satisfy the "adding up" constraints that are fundamental to the allocation model.

Let's see what this involves, in practice.

One characteristic of an allocation model of this type is that the (contemporaneous) covariance matrix for the errors of the equations is singular. The nice thing is that if the errors are serially independent, we can deal with the singularity of this matrix by dropping any one of the equations from the system, and estimating the remaining system by MLE.

The results that we obtain are well-known to be invariant to the choice we make as to which equation to drop (e.g., Powell, 1969). Moreover, the parameters associated with the discarded equation can be "recovered" from the restrictions on the coefficients that are implied by the "adding up" characteristic of the allocation model.

If this doesn't sound familiar, then take a look at the original post of mine on this topic.

Now, what if the errors aren't serially independent?

Berndt and Savin (1975) showed that if the errors of the system follow an AR(1) or MA(1) process, then the invariance property of the MLE, mentioned above, will hold only if the autoregressive (moving average) parameters are restricted to be the same for each equation's error term.

(Actually, the restriction is usually expressed a bit more generally than this, for a vector AR(1) or MA(1) process, but it collapses to this restriction if there is no cross-equation autocorrelation.)

It's easy to show that the Berndt-Savin result holds for any simple AR(p) or MA(p) error process. That is, one where the only lag that appears in the process is the p'th, and lower-order lags are restricted to have zero coefficients.

In another old paper (Giles, 1988), I showed that the Berndt-Savin result also holds under some other types of serial dependence. Specifically, if the system's errors are spatially autocorrelated, or if they have serially correlated error components, then we have to restrict the autocorrelation parameters to be the same for each error term if we want the MLE results to be the same, regardless of which equation we drop from the allocation model to deal with the singular covariance matrix.

It's also worth pointing out that although serially correlated error components can be written as a particular restricted ARMA(1,1) process, the results of Berndt and Savin (1975) and Giles (1988) don't apply in general to arbitrary unrestricted ARMA error processes.

If you want the technical details behind these results, I've put a reprint of my '88 paper here.

So, the take-away message is a simple one. If you're estimating allocation models, and you want to allow for autocorrelation in the equations' error terms, some care is needed. You can't simply estimate a different autocorrelation process for each error term - not if you want to retain the essential "adding-up" characteristics of the model, and the invariance of MLE to the choice of equation that you "drop". In particular, this means that you can't allow for autocorrelation in some of the error terms, but not in others!

References

Berndt, E. R. and N. E. Savin, 1975. Estimation and hypothesis testing in singular equation systems with autoregressive disturbances. Econometrica, 43, 937-957.

Giles, D. E. A.,  1988. The estimation of allocation models with autocorrelated disturbances. Economics Letters, 28, 147-150.

Powell, A.  A., 1969Aitken estimators as a tool in allocating predetermined aggregates. Journal of the American Statistical Association, 64, 913-922.

1. Dear Dave,

when I remember correctly there also is a (particularly difficult to read) 1994 paper of PJ Dhrymes, Econometric Theory, 10, 254−285, dealing with general autocorrelation structure in singular regression systems .