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0 = axe
2 2
;

From (20.101) and (20.104) and noting that xF+l is uncorrelated in the limit w

-w:
plim(xr+l™qr)/T = -B plim(o,+l™w+l)/T =
Substituting these expressions in (20.105):




Thus the OLS estimator for #? is inconsistent and is biased downwards. The bia
the smaller is the variance of the ˜noise™ element 0: in forming expectations.

Instrumental Variables: 2SLS
OLS is inconsistent because of the correlation between the RHS variable x
error term qf which ˜contains™ the RE forecast error or+l. solution to th
The
is to use instrumental variables, IV, on (20.103). However, to illustrate some
nuances when applying IV consider the following model:
+ Bx2t + + ˜r
= Qes
= (YX;,+˜
yr ut


B)™
Q˜ = Ix;r+i, ˜ 2 1 ) 6 = (a,
where x:t+l, x2, are asymptotically uncorrelated with ur. Direct application
(20.110) would require an instrument for xlf+l from a subset of the infor
The researcher is now faced with two options. Direct application of IV would
instrument matrix
w = h + l ™ X2t)
1
Where xa acts as its own instrument, giving




This is also the 2SLS estimator since in the first stage xlr+l is regressed
predetermined (or exogenous variables) in (20.1 10) and the additional instrum
An alternative is to replace in (20.110) by i l , + l and apply OLS to:




This yields a ˜two step estimator™ but as long as xlf+l is regressed on all the pre
variables, then OLS on (20.114) is numerically equivalent to the 2SLS estimat
therefore consistent. However, there is a problem with this approach. The OL
from (20.117) are:
A


e = yr - $ i l t + l - 19x2
but the correct (IV/2SLS) residuals use Xlr+l and not and are:
ilr+l
h



el = y - hXlr+l - BX2r
Hence the variance-covariance matrix of parameters from OLS on (20.117) a
since s2 = e™e/T is an incorrect (inconsistent) measure of cr2 (Pagan, 1984). Th
straightforward, however; one merely amends the OLS programme to produce
residuals el in the second stage.

Extrapolative Predictors
Extrapolative predictors are those where the information set utilised by the econ
is restricted to be lagged values of the variable itself, that is an AR ( p ) mode




The maximum value of p is usually chosen so that er is white noise. OLS
(20.121) yields one-step ahead predictions
x:t+l or to replace x;r+l in (20.110). Using iTr+l an instrument for x;r+l an
as
˜ 2 in the instrument matrix W1 gives consistent estimates. Now consider the
t
method. Having obtain i;t+l the ˜first stage™, the second-stage regression
in
OLS on:
+ Bx2r + 4:
Yr = a?i:r+I
+ a(Xfr+I - x l r + l ) - a?(?lt+l - x1r+1)
4r = ur
Compared with the EVM/IV approach (see equations (20.103) and (20.104))
additional term ( i l t + l -xlt+1) in the error term of our second-stage regressio
The term (xlt+1- is the residual from the first-stage regression (20.12
The variable xzt is part of the agent™s information set, at time t , and may t
used by the agent in predicting x1r+1. If so, then (Xlr+l - and the ˜omitt
from the first-stage regression, namely X Z ˜ are correlated. Thus in (20.124) the
,
between the RHS variable x2t and a component of the error term q: implies t
(20.124) yields inconsistent estimates of (a?, B) (Nelson, 1975). This is usuall
in the literature as follows: if ˜ 2 rGranger causes Xlt+l then the two-step
inconsistent. This illustrates the danger in using extrapolative predictors an
xf+l in the second-stage OLS regression, rather than using 2Tr+l as an inst
applying the IV formula. Viewed from the perspective of 2SLS, the inconsis
second stage (20.124) arises because in the first-stage regression, the research
use all the predetermined variables in the model, he erroneously excludes ˜ 2
paradoxically then, even if ˜2˜ is not used by agents in forecasting Xlt+l it must
in the first-stage regression if the two-step procedure is used: otherwise (Xlr
may be correlated with x;?t. Of course, if the two-step procedure is used and
B)
estimates (&, are obtained, the correct residuals calculated using Xlr+l and n
in equation (20.120)) must be used in the calculation of standard errors.

20.4.3 Serially Correlated Errors and Expectations Variables
Up to this point in our discussion of appropriate estimators we have assumed
errors in the regression equation. We now relax this assumption. Serially corre
may arise because of multiperiod expectations or because of serially correlate
errors. In either case, we see below that two broad solutions to the problem a
The first method uses the Generalised Method of Moments (GMM) approach
(1982) and ˜corrects™ the covariance matrix to take account of serially correl
The second method is a form of Generalised Least Squares estimator under
known as the Two-Step Two-Stage Least Squares estimator (2s-2SLS) (Cu
1983). These two solutions to the problem are by no means exhaustive but
widely used in the literature.

The GMM Approach
This approach is demonstrated by first considering serial correlation that arises i
with multiperiod expectations and then moving on to consider serial correla
structural error.
+ ur
+
Yr = B1x,4tl B2xt4t2

= E(xr+jIQt) ( j = 1,2)
-$+j

RE implies:
+ qt+j
= -$+j ( j = 192)
xt+j

and substituting (20.128) in (20.126) we have our estimating equation:
+ + qr
Yr = B l X r + l B2Xr+2

qr = ur - Bl%+l - B2%+2
2SLS on (20.129) with instrument set A, will yield consistent estimates
However, the usual formula for the variance of the IV estimator is inco
presence of serial correlation and qr is MA(1). Hansen and Hodrick (1980
˜correction™ to the formula for the variance of the usual 2SLS estimator. Putti
in matrix notation:
Y=XP+q
The 2SLS estimator for is equivalent to OLS on
+q
y = Xb*
2 = (%+lc %+2)

= 1,2) on A r
and i t + j are the predictions from the regression of Xr+j(j
estimator is:
b* = ( X X ) - ™ ( X y )
with residuals:
e*=y-m*

Note that in the calculation of e* we use X and not X. To calculate the corre
of /?in the presence of an MA(1) error, note that the variance-covariance m
.. 0
1 p1 0 .......
P1 1 P 1 0 -.
0 P 1 1 . P1
0
-1 P1
**

0 ...........
0 1
p1

where p1 is the correlation coefficient between the error terms. Since ef ar
the consistent estimator b*, then consistent estimators of oi,0; and p are g
following ˜sample moments:
Knowing I: we can calculate the correct formula for var(b*) as follows. Sub
(20.131) in (20.134):
+
b* = B (XX)-™Xq

Since plim(T-™)(Xq) = 0, then b* is consistent and the asymptotic varian
given by:

[
var(b,) = T-™ plim P 1lX˜ [qq™] X [X™XI-™]
X-

PkX]
var(b,) = 0; [a™X]-™ [XX]-™

Above, we assume that the population moments are consistently estimated by t
equivalents. Note that var(b*), the Hansen-Hodrick correction to the covarianc
b*, reduces to the usual 2SLS formula for the variance when there is no seria
(i.e. I: = 0˜1). The Hansen-Hodrick correction is easily generalised to the cas
have an MA(k) error, we merely have to calculate $s(s = 1,2, . . . k) and sub
estimates in I:.
The Hansen-Hodrick correction to the standard errors can also be applied
mation of /?in (20.131) can proceed using OLS. In this case the Hansen-Hodri
for var(b) is given by (20.141)but with X replacing X and the elements of C ar
using the consistent OLS residuals.
In the above derivation we have assumed that the error term is homoscedastic
if the error term is heteroscedastic, as is usually the case with financial data,
also be recomputed to take account of this problem.

A Two-Step Two-Stage Least Squares (2s-2SLS) Estimator
So far we have been able to obtain a consistent estimator of the structural para
(20.126) under RE by utilising IV/2SLS or the EVM. We have then ˜correcte
formula for the variance of the estimator using the Hansen-Hodrick formula
the Hansen-Hodrick correction yields a consistent estimator of the variance it
to obtain an asymptotically more efficient estimator which is also consistent. C
(1983) provide such an estimator which is a specific form of the class of
instrumental variables estimators. The formulae for this estimator look rather
If our structural expectations equation after replacing any expectations variab
outturn values is:
Y=XB+S
with
E(qq™) = a21:and plim[T-™(X™q)] # 0
Then the 2s-2SLS estimator is:
™ ™ ™
= [X™A (A™ A )- A™XI- [X™A (A™ A )- A ™y ]
I: I:
bg2
the error term. We have already discussed above how to choose an appropriate
set and how a ˜consistent™ set of residuals can be used to form 2. This ˜first-stag
of can then be substituted in the above formulae, to complete the ˜second s
estimation procedure (see Cuthbertson (1990)).
In small or moderate size samples it is not possible to say whether the Hanse
correction is ˜better than™ the 2S-2SLS procedure since both rely on asympt
Hence, at present, in practical terms either method may be used. The one clear
emerges, however, is that the normal 2SLS estimator for var(B) is incorrect an
be taken in utilising Cochrane -0rcutt-type transformations to eliminate AR e
this may result in an inconsistent estimator for B.

Summary
There are two basic problems involved in estimating structural (single) equation
expectations terms (such as equation (20.126)) by the EVM. First, correlatio
the ex-post variables xl+j and the error term means that IV (or 2SLS) estimati
used to obtain consistent estimates of the parameters. Second, the error term is
serially correlated which means that the usual IV/2SLS formulae for the varia
parameters are incorrect. l b o avenues are then open. Either one can use the I
to form the (non-scalar) covariance matrix (a2X) and apply the ˜correct™ IV
var(b*) (see equation (20.141)). Alternatively, one can take the estimate of a2
a variant of Generalised Least Squares under IV, for example the 2S-2SLS es
var(&2) in equation (20.145).


FURTHER READING
There are a vast number of texts dealing with ˜standard econometrics™ and a
clear presentation and exposition is given in Greene (1990). More advanced
provided in Harvey (1981)™ Taylor (1986) and Hamilton (1994) and in the latt
larly noteworthy are the chapters on GMM estimation, unit roots and changes
Cuthbertson et a1 (1992) give numerous applied examples of time series tec
does Mills (1993)™ albeit somewhat tersely. A useful basic introduction to
˜general to specific™ methodology is to be found in Charemza and Deadman (
a more advanced and detailed account in Hendry (1995). ARCH and GARCH
in a series of articles in Engle (1995).
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