ńňđ. 49 |

2 2

;

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

-w:

plim(xr+lâ€™qr)/T = -B plim(o,+lâ€™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 â€˜noiseâ€™ element 0: in forming expectations.

Instrumental Variables: 2SLS

OLS is inconsistent because of the correlation between the RHS variable x

error term qf which â€˜containsâ€™ 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)â€™

Qâ€˜ = 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 â€™ 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 â€˜two step estimatorâ€™ 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 = eâ€™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 â€˜first stageâ€™, 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 agentâ€™s information set, at time t , and may t

used by the agent in predicting x1r+1. If so, then (Xlr+l - and the â€˜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 â€˜correctsâ€™ 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

â€˜correctionâ€™ 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 ) - â€™ ( 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 â€˜sample moments:

Knowing I: we can calculate the correct formula for var(b*) as follows. Sub

(20.131) in (20.134):

+

b* = B (XX)-â€™Xq

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

given by:

[

var(b,) = T-â€™ plim P 1lXâ€˜ [qqâ€™] X [Xâ€™XI-â€™]

X-

PkX]

var(b,) = 0; [aâ€™X]-â€™ [XX]-â€™

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 â€˜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(qqâ€™) = a21:and plim[T-â€™(Xâ€™q)] # 0

Then the 2s-2SLS estimator is:

â€™ â€™ â€™

= [Xâ€™A (Aâ€™ A )- Aâ€™XI- [Xâ€™A (Aâ€™ A )- A â€™y ]

I: I:

bg2

the error term. We have already discussed above how to choose an appropriate

set and how a â€˜consistentâ€™ set of residuals can be used to form 2. This â€˜first-stag

of can then be substituted in the above formulae, to complete the â€˜second s

estimation procedure (see Cuthbertson (1990)).

In small or moderate size samples it is not possible to say whether the Hanse

correction is â€˜better thanâ€™ 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 â€˜correctâ€™ 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 â€˜standard econometricsâ€™ and a

clear presentation and exposition is given in Greene (1990). More advanced

provided in Harvey (1981)â€™ 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)â€™ albeit somewhat tersely. A useful basic introduction to

â€˜general to specificâ€™ 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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