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the VAR approach can be applied to any theoretical model involving multiperio
and which is linear in the variables. Not surprisingly, therefore, it can also be
efficiency in the forward and spot markets using the uncovered interest parity
forward rate unbiasedness (FRU) conditions outlined in Chapter 12. This cha
outlines how the VAR methodology can be applied in these cases and
illustrates the VAR methodology applied to FRU and UIP

discusses some practical matters in choosing the appropriate VAR

presents some recent empirical tests of FRU and UIP and provides an

example of tests of a forward-looking version of the flex-price monetary m
spot rate, using the VAR methodology

The Forward Market
Chapter 12 analysed tests of forward market unbiasedness, Efsf+l= fr, base
equation tests for any one pair of currencies (or a set of single equations
increase the statistical efficiency of the tests using a ZSURE estimator). Our te
will now be extended, in a statistical sense at least, by considering a VAR syst
to forecast over multiperiod horizons. The first equation is:

+ a12fpt + Wlr+1
= all As,

where wlr+l is white noise and independent of the RHS variables and f p r
the forward premium. In the one period case when the forward rate refers to
time t 1, then the test of FRU is very simple. Under the null of risk neutra
we expect
Ho :˜ 1 = 0, a12 =1

The above equation then reduces to the unbiasedness condition for the forwa

= fr
+ (a12 - Mf +
- EtASt+l = all As, - s)t
b+l Wlf+l

is independent of the limited information set At = ( A s t ,f p t ) . In the one step
we require only equation (15.1) to test FRU. However, we now consider a two
prediction (which is easily generalised to the case of rn step ahead prediction

Two-Period Case
ft for six mo
Suppose we have quarterly data but are considering forward rates
hence FRU is:
EtSt+2 - st = EfA2St+2 = f P r
To forecast two periods ahead we use the identity

Leading (15.1) one period forward, to forecast E1Asr+2 require a forecast of
Hence we require an equation to determine f p which can be taken to be
+ a22fPt + W2t+l
f =a 2 1 b

Equations (15.1) and (15.8) are a simple bivariate vector autoregression (VAR
are I(1) variables but ( s t , ft) have a cointegrating parameter (1, -1) then f p
is I ( 0 ) and all the variables in the VAR are stationary. Such stationary variab
represented by a unique infinite moving average (vector) process which may
to yield an autoregressive process (see Hannan (1970) and Chapter 20). A
system consisting of equations (15.1) and (15.8) can be used as an illustration
It can be shown that the FRU hypothesis (15.4) implies a set of non-li
equation restrictions among the parameters a,j and these non-linear restricti
that the two-period forecast error implicit in (15.6) is independent of inform
( A s t ,f p t ) . For illustrative purposes these restrictions are derived by ˜substi
then shown how they appear in matrix form and how we can easily incorpo
of high order and use it to forecast over any horizon. Using (15.7), (15.1) and
easy to see that
+ EtASt+l
EtA2sr+2 = E&t+2
+ a12fpt) + alz(a21Asr + a22fpt)
= all(a1lAst
+ (all ASt + a12fpO
Collecting terms and equating the resulting expression for EtA2st+2 with f p t
EtA2St+2 = 81 As1 + 82EfPr = f P t
+ a12a21 4- ail
01 =
02 = a m 2 -k ˜ a12
A2st+2 - Er A2sr+2

which given (15.6) and the VAR (15.10) is given by:

+ 62fpr - fPr + qt+l
where qt+l depends on Wir+l ( i = 12). If 61 and 62 are unrestricted then th
value of the forecast error will in general depend on (Asf, fpt), that is inf
time t. It is only if 81 = 0 and 62 = 1 that the orthogonality property of RE h
previous chapter we noted that these restrictions can be tested using a likelihoo
of the restricted versus unrestricted VAR. From (15.11) we see that the rest
be rearranged to give

+ ad/a12
a21 = - a d 1
a22 = [1 - a12(1 + a11)1/a12

Hence the restricted VAR is

-+ a12fpf
= all Asf

The individual coefficients (all, al2) in equation (15.17) are constrained to
same value in (15.16). The log-likelihood value from the restricted system
(15.17) can be compared with that from the unrestricted system (15.1)
noted in the previous chapter if the difference in log-likelihoods is large (˜s
the restrictions are rejected (not rejected).
The problem with the above is that the restrictions have to be worked out an
programmed into the VAR equations which are estimated subject to non-linea
techniques. Both these tasks can become ˜complex™ when the VAR has a la
of lags or the forecast horizon in the FRU equation is large. As we have alr
a computationally simpler procedure is to estimate the unrestricted (linear in
VAR and undertake a Wald test directly on the unrestricted aij coefficient e
can now be quickly demonstrated how the above problem (with lag length
be represented in matrix form and how it can be generalised. The matrix
unrestricted VAR is:
zr+l = & wr+l

It follows that

Hence the FRU hypothesis (15.6) implies:
el˜(A A2)zt = e2™q
e2™ - el™(A + A2) = 0

It is easy to see that (15.23) are the same restrictions as were worked out
substitution™. A Wald test of (15.23) is easily constructed. The procedure can
alised to include any lag length VAR since a pth order VAR can always be w
the A matrix in companion form, as a first order system. A forward predictio
for any horizon rn is given by

and hence FRU for an m-period forward rate fpj“™ is
C el˜A™z, = f p j m )
EtAmst+m =
i= 1

The FRU restrictions for an rn-period horizon are:
f (A) = e2™ - e l ™ x A i = 0
i= 1

which can be tested using the Wald procedure. We can also use the VAR pre
yield a series for the theoretical forward premium for rn periods,
f pjm” =
i= 1

(see equations (15.10) and (15.26)) which can then be compared to the actu
premium (using graphs, variance ratios and correlation coefficients).

The Spot Market and Forward Market
The uncovered interest parity (UIP) condition can be applied over a multiperi
and the VAR approach used in exactly the same way as described above. The m
should equal the expected change in the exchange rate over the subsequent f
(i.e. rn = 4). The above UIP equation is similar to the FRU equation (15.26
have (rt - r,?)on the RHS and not fpjm˜. However, it should be obvious that
for a VAR in As, and (r, - r:) goes through in exactly the same fashion
If the cross-equation restrictions on the parameters of the VAR are not rejec
do not reject the multiperiod UIP condition. What about testing FME and U
This is easy too. Consider the trivariate VAR where z:+˜= (As, fp, r - r*
obtained the unrestricted estimates A (and put them in companion form) w
two sets of restrictions of the form

2 el™Ai - e2˜ = 0 FRU

2 el˜A˜ - e3˜ = 0 UIP
i= 1

where the vectors eJ have unity in the Jfh element and zeros elsewhere (
We can test each restriction separately, thus testing FRU or UIP only, or tes
together (i.e. a joint test of FRU UIP). Of course, if we accept that cove
parity holds then rejection of either one of FRU or UIP should imply reje

How Big Can a VAR Get?
We noted in Chapter 14 that if we have unbiased estimates of the true param
VAR then adding additional variables will not, in principle, rescue tests of
they have failed the Wald test on a limited information set. The key assumptio
is unbiased parameter estimates. We noted that with a finite sample, ˜bias™ is
hence additional variables may make a difference to the Wald test of the VAR
It is convenient at this point to elaborate on these arguments surrounding the
˜size™ of the VAR since it is a key element in evaluating empirical work.
The reader will have noticed that FRU can be tested in a VAR that co
variables (As, fp. rt - r:) or two variables (As, fp). Which is better? In fa
greatly expand the number of variables in the information set (e.g. add st
long-term interest rates, etc.) even when testing just the FRU.
The first thing to note is that a VAR of rn variables can always be reduc
q variables, where q < rn, and in fact we can even reduce the system to q
an autoregressive equation. To illustrate this consider the 3 x 3 VAR system
above, with lag length p = 1 for ease of exposition
tuting (15.34) for d,-, in the RHS of the first two equations (15.31) and (
As, and f p , as dependent variables we obtain two equations for As, and f p
only on their own lagged values. Similarly we could now repeat the proced
obtaining f p , = g ( A S , - j ) and substitute out for this in the equation with
dependent variable, giving an ARMA model for As, (see Chapter 20).
Statistically, the choice between a univariate model or a multivariate VAR d
trade-off between the statistical requirements of lack of serial correlation and he
ticity in the errors, the overall ˜fit™ of the equations and parsimony (i.e.
explanation with the smallest number of parameters). Test statistics are availab
among these various criteria (e.g. Akaike and Schwartz criteria) but judgemen
required even when using purely statistical tests. This is because the statist
are likely to conflict. For example, maximising a parsimony criterion like
information criterion might result in serially correlated errors.
To the above statistical criteria an economist might have prior views (base
and gut instinct) about what are the key variables to include in a VAR. H
a VAR is a reduced form of the structural equations of the whole econom
always likely to be a rather disparate set of alternative variables one might i
VAR. Hence empirical results based on any particular VAR are always ope
on the grounds that the VAR may not be ˜the best™ possible representation of
particular, the stability of the parameters of the VAR is crucial in interpreting
or otherwise of the Wald test of the cross-equation restrictions (see Hendry
Cuthbertson (1991)).

One can see that the VAR methodology is conducive to testing several variants
basic idea, namely RE cross-equation restrictions. Recent studies recognise the
of the cointegration literature when formulating the variables to include in the V
not only include difference variables in the VAR but also the ˜levels™ or cointe
variables such as f p r and ( r - r*), in the above examples). Recent studies a
unanimous in finding rejection of the VAR restrictions when testing the FRU
and the UIP hypothesis - under the maintained hypothesis of a time invarian
risk premium. The rejection of FRU and UIP is found to hold at several ho
3, 6, 9 and 12 months), over a quite wide variety of alternative information
different currencies and over several time spans of data (see, for example, Hak
Baillie and McMahon (1989), Levy and Nobay (1986), MacDonald and Ta
1988) and Taylor (1989˜)).

Term Structure of Forward Premia
Some studies have combined tests of covered interest parity with the EH o
structure of interest rates applied to both domestic and foreign interest rates
methodology is again useful for multiperiod forecasts. By way of illustratio
where dj" = r;'' - t-,?i) (i = 3,6). If the pure expectations hypothesis of in
holds in both the domestic and foreign country then

+ Efd!:\)/2
dj6' = (dj3'
where the subscript t 3 applies because monthly data is used. The above equa
a term structure of forward premia which involves forecasts of the forward p

Equation (15.38) is conceptually the same as that for the term structure of sp
zero coupon bonds, discussed in Chapter 14. Clearly, given any VAR involvin
fpj3' (and any other relevant variables one wishes to include in the VAR) t
will imply the, by now, familiar set of cross-equation restrictions.
There have been a number of VAR studies applied to (15.38) and they usua
ingly reject the pure expectations hypothesis of the term premia in forward
MacDonald and Taylor (1990)). Since there is strong independent evidence t
interest parity holds for most time periods and most maturities, then rejec
restrictions implicit in the VAR parameters applied to (15.38) is most like
failure of the expectations hypothesis of the term structure of interest rates to
domestic and foreign countries.

The Spot Rate, Fundamentals and the VAR Approach
In our analysis of the flex-price monetary model FPMM in Chapter 13 we s
spot rate may be represented as a forward convolution of fundamental variab

Subtracting xt from both sides of (15.39)

i= 1

where 8, = p/(1 B), EtAxt+l = Erxr+l - xt and ErAxt+j = Er(xt+i - X r + i - 1
Equation (15.41) is now in a similar algebraic form to that outlined for the EH
structure in Chapter 14. We may somewhat loosely refer to q = sr - xr as th
rate spread'. Hence, we can apply the same test procedures to (15.41) as in
@e2™Aiz,= e2™8A(I - 8A)-™zt
el˜z, =
i= 1

Hence the Wald restrictions implied by this version of the FPMM are:

el™ - e2™8A(I - 8A)-™ = 0

el™(1- @A) e2˜0A = 0
Equation (15.44) is a set of linear restrictions which imply that the RE foreca
the spot rate is independent of any information available at time t other than
by the variables in x,. We can also define the ˜theoretical exchange rate sprea

qi = e2™8A(I - ˜A)-™z,

and compare this with the actual spread 4,. To implement the VAR methodolo
to have estimates of y, y* to form the variable qr = st - x, and estimates of /
8. An implication of the FPMM is that

is a cointegrating relationship (given that all the variables are Z(1)). Es
( y , y * , P I , /3*) may be obtained by a (multivariate) estimation procedure kn
Johansen procedure (see Chapter 20). The variables qr = sf - xr and Axr c
constructed and used in the VAR.
MacDonald and Taylor (1993) provide a good illustration of the implem
this procedure. Note that by using (15.41) we take explicit account of the po
stationarity in the data. Hence one avoids possible spurious regression probl
in earlier tests of the FPMM which used levels of trended variables. We also
specification error found in earlier studies that used only first differences of th
and ignored any cointegrating relationships among the I ( 1) variables. MacD
Taylor use monthly data January 1976-December 1990 on the DM/$ rate
Johansen procedure they find they do not reject the null that the coefficients
money supplies and relative income are unity for the ˜home™ (Germany) an
(USA) variables and that the interest rate coefficients are nearly equal and op

They take /3 = P* = 0.05 and the variable denoted x, above is taken to be
m*), ( y , - y;). MacDonald and Taylor find that the Wald tests in (15.43) a
decisively reject the RE cross-equation restrictions and there is excess volat
spot rate. The variance of the actual spread qr exceeds the variance of the
spread qi given by (15.45) by a factor greater than 100 (when /3 = 0.05) (a
standard errors are given).
O.XAsf-2 - 0.42A2Amr - 0.79Ayr - 0.008A2r: - 0.
As, = 0.005
(0.003) (0.07) (0.23) (0.34) (0.003) (0.

where R2 = 0.14, SE = 3.2 percent, ( - ) = standard error. Equation (15.48) pa
usual diagnostic tests, although note that the R2 of 14 percent indicates that
variation in Asf is not explained by the equation. They then perform a ˜rolling
and use (15.48) to forecast over different horizons as the estimated parameters a
They find that the RMSE for the ECM are slightly less than those from the ra
˜no change™ forecasts for horizons of 1, 2, 3, 6, 9 and 12 months. This is
to some of the earlier results in Meese and Roghoff (1983). However, note t
the statistical explanation in (15.48) appears to be due to the ad-hoc dynami
little of the statistical explanation seems due to the long-run error correction
forecasts are likely to be dominated by the difference terms, which probably a
a random walk themselves, hence the reason why the reported RMSEs are app
the same as those for the random walk model˜? The MacDonald and Tay
a valiant attempt to test correctly a sophisticated version of the FPMM but
is barely an improvement on the random walk in predicting monthly movem
DM/$ spot rate.

Tests of risk neutrality and RE based on FRU and UIP over multiperiod foreca
are easily accomplished using the VAR methodology. The VAR methodolog
be applied to forward-looking models of the spot rate based on economic fu
such as relative money supply growth. The main results using this methodo
FOREX market suggest
the FOREX market is not efficient under the maintained hypothesis of ris

and RE since both FRU and UIP fail the VAR tests
the FPMM applied to the $/DMrate indicates that this model does not

particularly well, the VAR restrictions do not hold, the spot rate is excessiv
(relative to its ˜theoretical™ counterpart) and the model performs little b
random walk. However, as noted in Chapter 13 ˜monetary fundamentals™ pr
predictive content for long horizon changes in the exchange rate

These results imply that the FOREX market is inefficient under risk neutral
This may be because of a failure of RE or because there exists a sizeable time
premium. The latter is examined in Chapter 18. Alternatively, there could b
of irrational behaviour in the market caused by ˜fads™ or by the mechanics of
process itself. However, relatively sophisticated tests described in this chapt
rescued the abysmal performance of ˜formal™ fundamentals models of the sp
efficiency in the FRU and UIP relationships is rejected.
indicative that sr may not be cointegrated with the fundamentals (see Ba
L-- 2
Stock Price Volatility

It was seen in Chapter 6 that a definitive interpretation of the results from se
of variance bounds test on stock prices is dogged by the stationarity issue,
appropriate method of ˜detrending™ the series for the actual price Pr and
foresight price P:. However, in the previous two chapters we have noted th
procedure tackles this problem head on by explicitly testing for stationarity in t
and it also allows several alternative metrics for assessing the validity of the

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