are irrational expectations rational? David Demery Nigel W. Duck1 ...

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Are Irrational Expectations Rational? 7where ρ S (L) is a polynomial in the lag operator with ρ S 0 =1,andwhereσ 2 ηS , the variance of η S j ,andtheelementsofρS (L) are functions of the αs,the γs, and the two variances σ 2 ε,andσ 2 uS .From equations (3), (5) and (6) it follows that, for thisnagent, o the informationsetcanbeseenasconsistingofthehistoryofη S j,t and then o n ohistories of u j,t − u S j,t ,and u js,t − u S j,t for s =1, 2,..S− 1. Thehistoryn oof η S j,t is observed from the joint observation of S series; and the historyn o n oof u j,t − u S j,t ,and u js,t − u S j,t are obtained from separate observation ofx j,t and x js,t respectively.We can write the following expression for x j,t in terms of these twoobserved seriesTXTX´x j,t = ρ S i η S j,t−i + γ i³u jt−i − u S j,t−i(7)i=0i=0This expression captures the two previous examples as special cases: if S =1u jt−i − u S j,t−i is zero for all i and equation (7) simplifies to equation (4). IfS = N, u S j,t−i =0, ηS j,t−i = ε t−i, andρ S,i = α i for all i, and so equation (7)collapses to equation (3).The problem of selecting an optimal information set can now be representedas the problem of selecting the optimal value of S. AvalueofS =1corresponds to the selection of an information set consisting of a single series.The selection of S = N corresponds to the full information set andhence to rational expectations.For an agent who has optimally chosen a value of S
Are Irrational Expectations Rational? 8η S j,t−T , the agent’s forecast of x t will be 7i=TEj,t−1x ORE Xt = λ i η S j,t−ii=1where the values of λ i are given from the OLS formula as,λ i = covar(x t, η S j,t−i )var(η S j )In this case the agent’s expectational error will beAR OREj,ti=T X≡ x t −i=1λ i η S j,t−iWhilst this forecast error will be independent of any element of the agent’sinformation set - a feature guaranteed by the application of the OLS formula- it will not in general be white-noise. The reason for this is that, under ourassumptions, the agent never observes x t and so never observes this forecasterror. Since it is not part of his information set he cannot react to it andthereby remove any pattern in it.3 Simulating the inaccuracy of alternative forecasting methodsClearly the four methods of forming expectations outlined above will deliverforecasts of varying accuracy. But whilst the RE forecasts will deliver forecaststhat are in general more accurate than the other forecasts, it is quitepossible that the extra accuracy that RE provides is not worth the extracost of acquiring the information that RE requires. In this section we providethe results of a series of simulation exercises designed to help assess thelikely inaccuracy caused using one of the other three forecasting techniques.3.1 Pure simulation for simple processesThe first exercise is based on pure simulation. We consider the error-varianceof different forecasting methods when the underlying aggregate and idiosyncraticprocesses are both lst or 2nd order MA processes. In tables 1-4 weshow, for a variety of parameter sets in such processes, the ratio (to thevariance of the RE forecast error) of the variance of the forecast error whenexpectations are (a) ORE with a selected number of individual variablesassumed to be observed; (b) adaptive with a range of values for µ includingits optimal value µ ∗ ;andfinally, (c) naïve.7 The history of u jt − u S j,t contains no information of use for forecasting the value ofthe aggregate variable x t , nor do lagged values of η S j,t−i beyond ηS j,t−T .
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