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Rational Expectations Expectations have been a central issue in macroeconomics from the very foundation of the subject; 'we must remember that the price of capital goods varies not only by reason of past changes but also by reason of expected changes either in gross income or in rates of depreciation and insurance' Walras (1954)p310. Much early empirical work on expectations centred around attempts to provide direct measures of agents expectations, eg Katona(1951, 1958), Tobin(1959) or Eisner(1965), and the thrust of much of this research was towards a psychological understanding of individual expectations formation. The adaptive expectations hypothesis by Cagan(1956) and Nerlove(1958), was an important departure because it allowed the treatment of expectations to be made explicit. Background To Expectations Mechanisms The hypothesis of adaptive expectations, first proposed by Cagan (1956), ( t -1 x - x ) = ( xt -1 -t - 2 x ) e t t -2 e t -1 e t -1 0 < <1 By simply rearranging this we can get, ( t -1 x ) = xt -1 + (1 - )t - 2 x e t e t -1 and ( t -1 x ) = xt -1 + (1 - ) xt - 2 + (1 - ) xt -3 . . . e 2 t and so we may model the unobservable expectation purely in terms of past observations of x. However this model may not always give a sensible result expectation In this case it does But if the data is trended??? expectation The expectation always goes to the mean Muth (1961) introduced the notion of a rational expectation to be 'Essentially the same as the prediction of the relevant economic theory'. This would rule out the adaptive expectations model in many circumstances as being a sensible way to form expectations A rational agent can not make consistent and predictable mistakes The Lucas critique (Lucas 1976) essentially emphasised the idea that policy regimes and particular policy rules will affect the reduced form solution for all the endogenous variables in a model. And hence the expectations of a rational agent Suppose a government controls an instrument G and that agents want to form expectations about a variable X, which is simply the sum of G and Z, X t = Gt + Z t And Z t = X t + t Now under one regime where G is simply held fixed a reasonable expectations rule to form expectations about the future value of X would simply be, e )= Gt+1 (t X 1- t+1 The rule for G is known so agents know Gt+1. An equally good way to forecast X would simply be based on; e (t X t +1 )= X t Suppose the government changed its policy rule and decided that from now on G would grow at 10% per period. The first equation would still be a valid approach, but the second is no longer appropriate as the growth in G would now imply that e (t X t +1 ) = 1.1X t So the model which uses the structure of the system remains unchanged when the policy rule changes. But the time series (adaptive) model is structurally unstable The main disadvantage of the RE approach is the extreme assumption which is required about the information available to the economic agent. An alternative approach to making this extreme assumption would be to assume that agents' expectations are on average correct but not make any specific assumption about how agents arrive at these expectations. We must therefore move to a class of models which, while not containing full information, are able to adapt to regime changes and in effect to 'learn' about the economic environment. The question of learning is also important in the context of the RE assumption where, in particular, the question of how agents come to know the true model is simply not addressed. The most extreme assumption, underlying much of the earlier theoretical literature, gives rise to the rational learning models, Friedman(1975), Townsend (1978, 1983), Bray(1983), Bray and Kreps (1984) or Frydman (1982). A slightly weaker assumption gives rise to the boundedly rational learning models. Recently however we have come to appreciate that the behaviour of the parameters in the learning rule gives an important insight into the form of equilibria which may emerge from the system. Marcet and Sargent(1988) So suppose an agent has a rule which is a linear function of a set of parameters D and the learning process (assumed to be some form of least squares learning) is represented by a mapping S, such that Dt+1 = S(Dt). A fixed point of the mapping is represented by convergence of this sequence to some fixed value, this point is sometimes referred to as an expectations equilibrium or an E- equilibrium. Marcet and Sargent(1989a) demonstrate that this fixed point is also a full rational expectations equilibrium. Econometrics and RE The basic axioms of RE If agents are rational they act on all information in the most efficient way. Forecasts are unbiased, uncorrelated and efficient Then if x x t 1 t e t 1 t +1 RE implies E( t+1 | t ) = E( t+1 | t ) = 0 E( | ) = 2 t +1 2 E( t+1 t+1- j ) = 0 j > 0 However the k step ahead RE forecast is correlated with the k-1 step ahead forecast, eg if xt+1 = xt + et so xt+k = xt + et+1 + ...+ et+k k and so e xt+1 -t x = et+1 t+1 e xt+2 -t xt+2 = et+1 + et+2 The one step ahead forecast is independent white noise but further ahead forecasts are correlated, although all are independent of the information set. The change in expectations between one period and the next depends only on the arrival of new information. e ( t+1 x - x t+ j t e t+ j ) = f( t+1 - t ) Direct tests of RE If some measure of expectations exists then a direct test generally takes the form xt+1 = 0 + 1( t xte+1 ) + 2 t + t where H 0 : 0 = 0 = 2 , 1 = 1 If the expectation is more than one period ahead then the error term will be MA(k-1) x t +k = 0 + 1 ( t x ) + 2 t + t e t +k where H 0 : 0 = 0 = 2 , 1 = 1 but t = 0 et + 1 et -1 + ... k et - k This presents problems of inference and much attention has been devoted to constructing correct tests for this case. Much financial data often contains this problem especially eg monthly data on three month rates of return. Multi period expectations Many models also generate theoretical reasons for expecting multi period expectations. eg the Sargent(1979) adjustment cost model Suppose agents have a desired target Y* and they chose Y but changing Y rapidly is costly. We can represent this as, Min C = E[ D ( a0 ( Y t+i - Y i * 2 2 t+i ) + a1( Y t+i - Y t+i -1 ) )] i=0 the FOC for this are 2( a0 ( Y t+i - Y *+i ) + a1 ( Y t+i - Y t+i -1 ) - a1 ( Y t+i+1 - Y t+i )) = 0 t 1 * Y t+i ( a0 + a1 (-L+ L )) = a0 Y t+i and the general solution to this is Y t+i = 1Y t+i -1 + (1 - 1 )(1 - 1 D) ( 1 D ) Y k -i * t+i+k k =i An illustrative example suppose we have a model Y t = 1( t X )+ 2 ( t X e e t +1 t +2 ) + ut under RE we have, xt+i =t x + t+i e t+i and so Y t = 1( X t+1 ) + 2 ( X t+2 ) + ut - 1 t+1 - 2 t+2 Now the Xs are correlated with the error term and there is an MA(2) error process. This is the general problem created by RE estimation. We could do FIML estimation if we had a model for X, but this is expensive and often not robust. a much more common way of dealing with these problems is the Errors in Variable (EVM) approach. The Error in Variable approach This is a form of instrumental variable estimation which allows for the measurement error of using actual to replace expected variables. IV and 2 step estimators suppose we have a model Y t = 1( t X ) + 2 ( X t,2 ) + ut e t +1,1 we proceed by replacing the expectation of X1 with its actual value, we can then perform instrumental variable estimation using X2 and any other suitable instruments. or we can do a 2-step estimator. X t+1,1 = ( x2 ,other instruments) ˆ and then Y t = 1( X t+1,1 ) + 2 ( X t,2 ) + ut ˆ This is all an instrumental variable estimator is except that the IV estimator does the process in one go. The two will give identical estimates for the parameters but different standard errors and `t' statistics as the IV estimator uses ˆ ˆ ut = Y t - 1( X t+1,1 ) - 2 ( X t,2 ) ˆ To calculate the SE while the 2 step procedure uses the instrumenting variable Extrapolative predictors Often when we only have expectations of exogenous variables we construct expectations series from lagged data (eg a subsidiary VAR system). ( t X te+1 ) - t+1 = X t+1 = (L) xt + t These are extrapolative predictors. Given the Chain Rule of Forecasting we may use this equation to make multi period ahead forecasts. ( t X t+2 ) = (L)[(L) xt ] ˆe note if X2 Granger causes X1 then as we have left X2 out of the extrapolative model the parameters will be biased. Serially Correlated errors RE naturally gives rise to MA errors in overlapping data models, many approaches have been developed to cope with this. Hansens GMM method This uses GMM (or equivalently IV) to estimate the structural equation, suppose we have a model Y t = 1( t X )+ 2 ( t X e e t +1 t +2 ) + ut under RE we have, xt+i =t x + t+i e t+i and so Y t = 1( X t+1 ) + 2 ( X t+2 ) + ut - 1 t+1 - 2 t+2 IV or GMM will yield consistent estimates of the parameters (this is just an application of Quasi Maximum Likelihood) but the standard errors are biased so we can not undertake conventional tests. Hansen and Hoderick proposed a correction to the standard errors, define ˆ ˆ et = Y t - 1( X t+1 ) + 2 ( X t+2 ) Now the corrected covariance in the presence of an MA(2) error process will be 1 1 0 0 ... 0 1 1 1 0 ... 0 E(ee) = 2 0 1 1 1 ... 0 = 2 . . . 0 0 ...0 1 1 where T = ei2 /T 2 i=1 T 1=[ ei ei -1 ]/T 2 i= 2 and VAR( ˆ ) = 2 (X X )-1 (X X)(X X )-1 Newey West The Hansen Hoderick robust estimators are asymptotically correct but in a small sample they may give rise to an estimate of the covariance matrix which is not positive semi- definite, ie it can not be inverted. Newey and West suggested a correction to the HH estimate which ensures positive semi definiteness even in small samples. The Newey West correction is T = ei2 /T 2 i=1 T j=[ w(j,m) e e 2 i i- j ]/T j=1...m i= j+1 w(j, m) = 1 - [j/(m + 1)] Hayashi-Sims A more efficient method would be to actually estimate the moving average parameters rather than to simply estimate using OLS or simple IV. The GLS transformation is usually to lag all the variables so that the error term is transformed to white noise. Consider Y t = 1 ( X t+1 ) ut 1 ut -1 we would normally transform this so that Y t (1 - 1 L ) = (1 - 1 L ) [ 1 ( X t+1 ) + 2 ( X t+2 )] + u t -1 -1 The problem here is that this would transform the model so that the lagged variables were no longer orthogonal to the error term. Hayashi and Sims suggest a forward filtering process which maintains orthogonality and removes the serial correlation. Yt (1 + 1 L ) = (1 + 1 L ) [ 1 ( X t+1 ) + 2 ( X t+2 )] + u* -1 -1 -1 -1 t The Cumbey (1983) 2-step, 2-Stage LS A more efficient estimator can be achieved by performing a generalised form of IV which takes account both of the simultaneity and the MA error. If y = x + q E(q q) = 2 plim( T -1 (xq))NE0 then the 2S2SLS estimator is ˆ 2 = [xV(V V )-1V x ] -1 [xV(V V )-1V y] ˆ 2 ) = 2 [xV(V V )-1V x )-1 var( Cross Equation Restrictions Our model generally consist of a structural equation and a subsidiary model to generate the expectations terms (an AR model, a VAR or an implicit model in the Instruments). Often there may exist cross equation restrictions between the marginal model and the structural model. Yt = 1 ( t X te+1 )+ ut Suppose X t+1 = (L)Xt 1 + vt then we could write this system as Y t = 1 ( (L) X t -1 ) + u t + 1 vt X t+1 = (L) X t -1 + vt Testing this restriction then entails a joint test of the model and the RE assumption. The Importance of Expectations In empirical work expectations have proved enormously useful Perhaps the most striking example of such an area of positive achievement is the exchange rate. The emerging consensus rests on the use of the uncovered arbitrage relationship. et = 1(L) et+1 + 2 (L)( r t - r tf ) + 3 (L) zt Estimation of this form of relationship may be found in Hall(1987a, 1987b), Currie and Hall(1989), Gurney, Henry and Pesaran(1989), Fisher et al(1990), Hall(1992) and Hall and Garratt(1992). Time Inconsistency A closely related problem to the informational problem outlined above is the problem of time inconsistency These issues may be formalised by following the simple illustration of Kydland and Prescott, state a general welfare function for two periods as, S( x1 , x2 , 1 , 2 ) x1 = X 1( 1 , 2 ) x2 = X 2 ( x1 , 1 , 2 ) Now if we derive the first order condition for policy in period 2 from the standpoint of a policy maker in period 1 we get, S x 2 S S x 1 . + + . =0 x 2 2 2 x1 2 Now if we consider the first order conditions in period 2, assuming everything occurred in period 1 as originally planned, the conditions become. S x 2 S . + =0 x2 2 2 Which is clearly different Non-Linearities and Expectations The usual hypothesis which underlies both the RE assumption and the analysis of linear models is that agents take the mathematical expectation of the relevant model as their measure of expectations. A further complication which arises in non-linear models is that generally the deterministic solution to the model will not be the mathematical expectation of the probability distribution of the stochastic model. Ef( yt ) f(E( yt )) So the expected value of the sterling dollar exchange rate does not equal 1/ the expected value of the dollar sterling rate. E (£ / $) 1 / E ($ / £) Hall(1988) argues that many apparent contradictions which arise because of non-linearity’s may be reconciled by using the median as the relevant measure of expectations along with suitable distributional assumptions. • If there is a direst measure of the expected exchange rate (S) such as the forward rate (F) then • S-F should be unforecastable white noise • But overlapping data (weekly observations on three month forward rates) creates a moving average error. • They propose estimating by a consistent technique (OLS sometimes or GMM) • Then correcting the standard errors with the HH correction This table tests using only two lagged dependent variables One significant result This table Test using lagged cross rates from all markets More significant results But now shorten the period to exclude the 1973 oil price crises Less significant number of results Testing using more cross rates Some increase in significance Now consider data from the 1920’s Lots of significant results

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