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A PITFALL IN ESTIMATION O F M O D E L S WITH R A T I O N A L EXPECTATIONS Robert P. F L O O D and Peter M. GARBER* Unirersity ~?I" Virginia. Charlottesrille, VA 22901, USA Recent applied studies, which assume that agents form expectations ra4ionally lAbel et ai. {1978j, McCallum {1977)], have employed errors-invariables modeis to estimate equations containing proxies for agents' expectations. For example, Abel et ai. use an actual inflation rate as a proxy for the expected inflation rate in estimating the parameters of a money demand function. Under the hypothesis of rational expectations, the actual inflation rate is the expected rate plus an unforcastable error term with zero mean; therefore, an errors-in-variables model can be employed and estimated by an appropriate two-stage procedure. In practice, however, these researchers, apparently finding that the residuals in equations estimated by two-stage least squares exhibit autocorrelation, have used Fair's (1970) technique, which was designed to produce relatively efficient estimators. Fair's technique is useful in situations in which current and lagged endogenous variables appear as regressors in an equation whose error term follows an AR{I) process. The purpose of this note is to argue that the use of Fair's technique in the context of a rational expectations, errors-in-variables model yields parameter estimates which are not consistent.' To illustrate this result we will use a model similar to the model estimated by Abel et al., z a Cagan-type monetary model of inflation in which real money demand is a function of expected inflation and the lagged forward premium on foreign exchange. 3 Real money demand (which equals real *We would like to thank B.T. McCallum for helpful comments. 'While this is a straightforward result, it is of importance for applied work particularly because the popular computer softwear package TSP uses Fair's technique for estimation of two.stage least squares regressions with serially correlated errors. See Hall and Hall (1976, p. 82). 2We use Abel et al. {1978) purely as an illustration of the econometric problem that can arise by combining errors in variables with Fair's technique. We have no criticism of their substantive point that alternate rates of return may enter the money demand function. "~Abel et al. {1978) use the lagged forward premium because the current premium did not produce reasonable results. Also, they use the actual inflation rate between time t - I and t rather than thai between time t and t + I as argued in their theory.

R.P. Flood and P.M. Garber, A pitfall in estimation of models

434

money supply) is

(1)

m, -- p, ~ Yt = T + f l f - 1 + OcX~'+ 8,,

where mr, p, and y, are the logarithms of nominal money balances, the price level and real balances, respectively, ft-1 is the lagged forward premium, and x~ is the expected rate of inflation between times t and t + 1. e-, is a r a n d o m variable which follows an AR(1) process, i.e., e,, = re,,_ t + ut, where u, is white noise.

If x~ is formed rationally, i.e., such that agents are not systematically in error, then X, --- X,~ "k- V,,

(2)

where x, is the actual rate of inflation between t and t + 1 and vt is a disturbance term with mean zero and with Eviv~=0, i 6 j . t,t is uncorrelated with elements in the information set used by agents at time t. We assume that this information set contains realizations of all relevant variables through t as well as the structure of the money market. Hence, t,, is composed of the innovation to the money supply and the realization of u at time t + 1. Also note that x~' is correlated with st [see, e.g., Sargent {1977)], therefore, x, is correlated with e,. The assumption that ezpectations are formed rationally implies that when is proxied by x, the following standard errors-in-variables model results: (3)

.v,=y + flf,- i +=x, + ( e , - ~ v , ) .

Transforming (3), as in the Fair technique, we have y , - p y , _ ~=~,(1 - p ) + fl(f,_ ~ - o f , - 2) + ~ ( x , - px,_ ~)

+ [(r-ply,_ ~+u,-~(v,-pv,_ ~)],

(4)

where p is a parameter to be estimated and the term in brackets is the disturbance term in the transformed equation. The first stage of Fair's procedure used to estimate (4) consists of finding f,-1 and .~,, the fitted value off,_ ~ and x,, by regressing f,_ ~ and x r on a set of predetermined variables which includes at least 3',-~, f - 2 and x,_ ~. If we define ht =f,- 1 - f , - ~ and w, -- x, - ~,, then the equation whose parameters are estimated in the second stage of the procedure is

y,-py,_, =~(1-p)+/~,_,-pf,_

2)+ ~(.,~,-ox,-, )

+ [(r-p)e,_ l +u,-~(v,-pv,_ l -w,--h,)].

(5)

R.P.

Flood

and

P.M.

Garher,

A pitfall in cst imcrtion qf models

435

We can compute estimates of p, y, B and 3 by employing an iterative procedure to minimize the sum of squared residuals. Computation of these estimates completes the application of Fair’s technique to the rational expectations errors-in-variables model. However, the structural parameter estimates, B, & and 5, produced by this technique are not consistent estimates of /I, z and Y respectively. The difficulties with the use of Fair’s technique, in the present circumstance, center on the presence of pi__ l in the error term in eq. (5). According to eq. (2), X,_ l and t’,_ 1 are correlated. Thus the+error term in (5) is correlated with i, - ps, _ l, and the fi, 2 computed as outlined above are not a consistent estimators of p, x. Notice further that if ply,_ 1 were absent from the error in (5) then the b resulting from minimization of the suns of squared residuals would be a consistent estimator of I=.Moreover, only when fi is a consistent estimator of I’does the correlation of E,_ 1 with _y! ! and J,_ 1 not effect the consistency of the B, z estimators. But, with pr,- 1 present in the error term, the i, which minimizes the sum of squared residuals is not a consistent estimator of r. Thus, the correlation of E,_ ,, with x,_ 1 and &, provides an additional reason for the previously described B, 2 to be an inconsistent estimators of /7, 0~. We have chosen the problem of estimating the hyperinflationary money demand function as an example. However, the inconsistency result will hold models are combined with Fair’s generally when errors-in-variables estimation technique. The problem also arises in the context of rational expectations models with errors-in-variables when such techniques as the Cochrane -0rcutt method for correcting for autoregressive disturbances are employed. Since the increased use of rational expectations is making these combinations more common, we present this note as a caveat against the approach. References Ahcl, A., R. Dornbusch, J. Huizingn, and A. Marcus. 1978. Money demand during !lypcrinflrrtion, Journal of Monetary Economics. forthcoming. Fnia, B.C.. 1970, The estimation of simultaneous equations models with lagged endogenous v;hhlcs and first order serially corrclatcd errors. Econometrica 38, no. 3, May, 507 516. I!irII. R,f:l. and B.H. HillI. 1976. Time scrims processor users manu;tl (Stanford Research Institute. Menlo Park, CA). McC’i~llum. R.T., 1977. The role of spcculi~tic~nin the C’anadian forward exchange market : Some L\stimiWs assuming cational expectations, The Rcvicw of Economics and Statistics LIX, no. 2. May. 145 151. Si~rgcnt, T.J., 1977. The demand for money during hyperinflations under rational expectations. tntcrnational Economic Review IS, Feb., 59 82.