Docstoc

IDENTIFICATION

Document Sample
IDENTIFICATION Powered By Docstoc
					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(ee) =  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 (xq))NE0
then the 2S2SLS estimator is

     ˆ 2 = [xV(V V )-1V x ] -1 [xV(V V )-1V y]
     
               ˆ 2 ) =  2 [xV(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

				
DOCUMENT INFO