Moving Average Errors

"God made X (the data), man made all the rest (especially ε, the error term)."

Emanuel Parzen

A while back I was asked if I could provide some examples of situations where the errors of a regression model would be expected to follow a moving average process. 

Introductory courses in econometrics always discuss the situation where the errors in a model are correlated, implying that the associated covariance matrix is non-scalar. Specifically, at least some of the off-diagonal elements of this matrix are non-zero. Examples that are usually mentioned include: (a) the errors follow a stationary first-order autoregressive (i.e., AR(1)) process; and (b) the errors follow a first-order moving average (i.e., MA(1)) process. Typically, the discussion then deals with tests for independence against a specific alternative process; and estimators that take account of the non-scalar covariance matrix - e.g., the GLS (Aitken) estimator.

It's often easier to motivate AR errors than to think of reasons why MA errors may arise in a regression model in practice. For example, if we're using economic time-series data and if the error term reflects omitted effects, then the latter are likely to be trended and/or cyclical. In each case, this gives rise to an autoregressive process. The omission of a seasonal variable will general imply errors that follow an AR(4) process; and so on.

However, let's think of some situations where the MA regression errors might be expected to arise.

Nicholls et al.  (1975) provide a really good survey of the estimation issues associated with MA and ARMA models. Despite its date, this paper remains very important, and it also gives some good  examples of why MA errors may be expected in regression models estimated from economic data. (H.T. to Des., Adrian and Deane for the Parzen quote.) I'll draw from their survey, and then add some more recent examples.

First, there's a class of models that you used to find discussed frequently in introductory econometrics textbooks. You don't see them mentioned as often these days Basically, they involve replacing an unobservable regressor with a weighted sum of lagged values of an observable variable. The classic examples used to relate to "price expectations" and "permanent income", but there are others too. Here's how it goes.

Suppose that the model of interest is of the form

Y_t = β₀ + β₁ X^*_t + ε_t                                                  (1)

where X^*_t is not observable, but we believe it can represented as a distributed lag of  an observable variable, X_t. If this distributed lag is "rational", it can be expressed as the ratio of two polynomials in the lag operator "L", where L(X_t) = X_t-1; L^p(X_t) = X_t-p  ; etc. That is:

X^*_t = [A(L) / B(L)] X_t ;                              (2)

where A(L) and B(L) are finite-order polynomials in L.

Substituting (2) into (1), we get:

B(L)Y_t = α₀ + β₁ A(L) X_t + B(L) ε_t  .            (3)

For instance, we may have

A(L) = a₀ + a₁L + a₂L²   ;   B(L) = b₀ + b₁L  .

In this case, after some re-arrangement, (3) can be written as:

Y_t = α'₀ + b'₁Y_t-1 + γ₁ X_t-1 + γ₂ X_t-2 + (ε_t + b''₁ ε_t-1)  ,

say. (I've just re-labelled some of the coefficients to allow for dividing both sides of the equation by  b₀ .)

We now have a (dynamic) model in which all of the variables are observable, but the error term follows an MA(1) process. 

(Of course, the presence of the lagged dependent variable as a regressor, together with the MA errors means that OLS will be both biased and inconsistent, and an alternative estimator, such as Instrumental variables, will be needed to get consistent estimates of the parameters.)

Practical examples of such models include ones where Y, X and X* are inventories, actual sales and anticipated sales respectively; or where Y, X and X* are measured consumption and income, and "permanent" income. See Sims (1974) for further discussion of models of this general type.

As a second example, consider the following situation that arises in practice quite frequently, especially when modeling financial data. Suppose that daily data are available, but these are converted to monthly "returns" (log-differences) for modelling purposes. So, one resulting monthly "observation" uses data from July 1st to August 1st (say); the next uses data from July 2nd to August 2nd; etc. The data are "overlapping" in the sense that lots of the daily observations are re-used in the calculation of successive monthly values.

A common example of this with macroeconomic data arises when we see CPI data being measured monthly and then converted and reported in the form of "annualized" inflation rates.1

Rowley and Wilton (1973) and  Hansen and Hodrick (1980)  recognized that working with "overlapping" data will induce a moving average process in the error term of a regression model. Gilbert (1986) shows how invalid inferences can be drawn if this is not recognized and taken into account. More recently, Harri and Brorsen  (2009) have provided a useful discussion of some of the other econometric consequences of modeling with such data.

As a final example of how MA errors can arise in a regression model, let's consider the situation where the underlying economic model is expressed in continuous time. Of course, in practice economic data are observed only discretely, so the estimation of the econometric model involves a type of approximation.

There's a rich literature on "continuous time econometrics", dating back at least to work by Koopmans (1950). Many of the principal contributors to this literature were associated with the "Auckland School" of econometricians, including the late Rex (A. R.) Bergstrom, Cliff. (C. R.) Wymer, and Peter (P. C. B.) Phillips. Peter's Master's thesis (supervised at Auckland by Rex Bergstrom) was in this field, resulting in his first Econometrica paper. So was his Ph.D., supervised by Denis (J. D.) Sargan at the L.S.E.

It's also interesting to note that Bill (A. W.) Phillips - the New Zealander who gave us the Phillips Curve - also made seminal, and very early contributions to "continuous time econometrics". Examples of his contributions to this particular field include Phillips (1956, 1966).

Now, how does this all relate to the issue of errors that follow an MA process? Well, in a nutshell, if the model is written in continuous time, but includes "flow" data that have to be measured discretely, then the errors of the model will follow an MA(1) process. You can find a really good discussion of this in Phillips (1978). Interestingly, estimators that use this discrete approximation are biased, and the bias doesn't disappear as the sampling interval goes to zero - but that's another story!

So, there we have some examples of how MA errors might arise in regression models estimated with economic data. I'm not suggesting that this list is comprehensive, but hopefully it will serve to illustrate that  such errors can arise for quite a diverse range of reasons. It's important to keep this in mind, and to test for this type of model mis-specification.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Gilbert, C. L. (1986). Testing the efficient market hypothesis on averaged data. Applied Economics 18, 1149-1166.

Hansen, L. P. and R. J. Hodrick (1980). Forward exchange rates as optimal predictors of future spot rates: An econometric analysis. Journal of Political Economy, 88, 829-853.

Harri, A. and B. W. Brorsen  (2009). The overlapping data problem. Quantitative and Qualitiative Analysis in Social Sciences, 3 (3), 78-115.

Koopmans, T. C. (1950). Models involving continuous time variable. In T. C. Koopmans, ed., Statistical Inference in Dynamic Economic Models, New York, Wiley.

McCrorie, J. R. and M. J. Chambers (2006). Granger causality and the sampling of economic processes. Journal of Econometrics, 132, 311-336.

Nicholls, D. F., A. R. Pagan and R. D. Terrell (1975). The estimation and use of models with moving average disturbance terms: A survey. International Economic Review 16, 113-134.

Phillips, A. W. (1956). Some notes on the estimation of time-forms in reactions in interdependent dynamic systems. Economica, 23, 99-113.

Phillips, A. W. (1966). Estimation of systems of difference equations with moving average disturbances. Paper presented at the Econometric Society Meeting, San Francisco. Reprinted as Chapter 11 in A. R. Bergstrom, A. J. L. Catt and M. Preston, eds., Stability and Inflation: A Volume of Essays to Honour the Memory of A.W.H. Phillips, New York, Wiley.

Phillips, P. C. B.  (1972). The structural estimation of a stochastic differential equation system. Econometrica, 40, 1021-1041.

Phillips, P. C. B. (1978). The treatment of flow data in the estimation of continuous time systems, in A. R. Bergstrom, A. J. L. Catt and M. Preston, eds., Stability and Inflation: A Volume of Essays to Honour the Memory of A.W.H. Phillips, New York, Wiley, 257–274.

Rowley, J. C. R.  and D. A. Wilton (1973). Quarterly models of wage determination: Some new efficient estimates. American Economic Review 63, 380-389.

Sims, C. A. (1974). Distributed lags. In: M. D. Intriligator and D. A. Kendrick, eds., Frontiers of Quantitative Economics, Vol. 2. North-Holland, Amsterdam, 289-338.

© 2011, David E. Giles