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AR Forecasting Scheme
+
Using (14.2) a test of the PEH RE is possible if we assume a weakly ratio
tations generating equation for ArF+l which depends only on its own past
example:
+ +
Art+i = a1 Art a2Art-1 wr+l
(where we exclude a constant term to simplify the algebra). Agents are assu
the limited information set AI = (Art, Art-1) to forecast future changes in in
= a1 Art -k a2Art-l

and the forecast error wt+l = Art+i - Arf+l is independent of At under RE. W
that under the null that the PEH t RE is true then equations (14.2) and (14
cross-equation restriction. Substituting (14.4) in (14.2)



If the PEH is true then (14.3) and (14.5) are true and it can be seen that the
in these two equations are not independent. ˜By eye™ one can see, for ex
the ratio of the coefficients on Art and Arr-1 in each equation are ai/(ai/
i = 1,2). Consider the joint estimation of (14.3) and (14.5) without any rest
the parameters:
+ n2Art-1 +
St = nlArt Vr+1




+ RE is true then (14.3) and (14.5) hold and hence we expect
If the PEH


=a l p , = a2/2
111 7t2

A error term has been added to the unrestricted equation (14.6) for the spread.
n
either because St might be measured with error or given our limited informatio
a,), error term picks up the difference between forecasts by the econometr
the
on Af = (r,, r f - l ) and the true forecasts which use the complete information
The econometrician obtains estimates of four coefficients n to 7t4 but the
l
two underlying parameters, al, a2, in the model. Hence there are two implicit
in the model and from (14.8) these are easily seen to be:
7t3 7t4
--=2=-
n1 7t2

These restrictions can be tested by comparing the log-likelihood values from
of the unrestricted equations (14.6) and (14.7) in which the coefficients 7ti
while the form of equation (14.6) is unchanged:
+ n2Arf-l +
Sf = nlArf
The likelihood ratio test compares the ˜fit™ of the unrestricted two-equation s
that of the restricted system. The variance-covariance matrix of the unrestric
(assuming each error term is white noise but they may be a contemporaneous
i.e. a # 0) is:
,
[ 1
2
C=
, OW
4
awv OWV

The variances and covariances of the error terms are calculated from the res
each equation (e.g. a = CG:/n, a = Ci&Gf/n).The restricted system has
: ,
matrix Cr of the same form as (14.12) but the variances and covariances are
using the residuals from the restricted regressions (14.10) and (14.11)™ that is
respectively .
The likelihood ratio test is computed as
LR = n ln[(defC,)/(det C , ) ]
+




where n = number of observations and ˜det™ indicates the determinant of
ance matrix
det C = a a - (awv)2
$:
If the restrictions hold in the data then we do not expect much change in th
and hence det[Cr] FZ det[C,] so that LR % 0. Conversely, if the data do not c
the restrictions we expect the ˜fit™ to be worse and for the restricted residuals
(on average) than their equivalent unrestricted counterparts. Hence [] > [,
a,
: a:
]
det[C,] > det[C,]
and LR will be large. It may be shown that LR is distributed with a (c
squared distribution (x2) under the null, with q degrees of freedom (where q =
parameter restrictions). Thus we reject the null if LR > xz(q) where xc is the cri
(For a formal derivation of the likelihood ratio test see Harvey (1981) and C
et a1 (1992)).

Interpretation of the RE Restrictions
The two estimated equations (14.10) and (14.11) with the restrictions im
easily seen to be consistent with the zero abnormal profit condition. Unde
the (abnormal) profit AP from investing long rather than short is:


For the expected abnormal profit to be zero we require:
= 2E(SflAf) - E(Arr+lJAr)
the expected profit, conditional on A, is zero. Using the restricted equation
is also easy to see that S , = Ar:+1/2 regardless of the particular values of t
another way, the restrictions on the Xis are such that the PEH always holds. Of
we used the unrestricted equation (14.6) to forecast Ar,+l then since the latt
on 713 and n4, the above relationships would not hold.
The above cross-equation restrictions may be given further intuitive appea
that they also imply that the (RE) forecast error is independent of the limited i
set assumed, that is they enforce the error orthogonality property of RE. T
error is:
+ +
Arr+l - E(Arr+llar) = (7r3Arf n4Arr-1 o r + l > - 2Sf
where we have assumed the PEH hypothesis (14.2) holds. Substituting from (1

+ - 2n2)Arr-l + (o,+l - 2vt+1)
- 2nl)Ar,
- Arf+l = (773 (n4

Hence the expected value of the forecast error will not be independent of inf
time t or earlier unless:
n - 2x1 = 7r4 - 2x2 = 0
3

But the above are just a simple rearrangement of the cross-equation restricti
In fact this example is so simple algebraically that the restrictions in (14.9)
equation™ but they are not non-linear.

Cross-Equation Restrictions: Addition of the Spread
A dispassionate commentator might remark that the AR forecasting schem
is very simple and that investors might use many more variables than this
future interest rates. The Campbell-Shiller VAR methodology recognises this
to the most obvious additional variable that investors might use. Given that in
supposed to believe in the PEH, then from (14.2) a variable that should b
predicting Art+l should be the spread S , itself. Hence, we now augment (14
and obtain a new expectations generating equation:


where 7r3 = a1, n4 = a2, 7r5 = bl. Proceeding as before and substituting the f
Arf+l in the PEH equation (14.2), we obtain:



Wald Test
Campbell and Shiller note a very straightforward way of estimating the restrictio
in the model. Since S , appears on both sides of (14.22), then the equation can
(for all values of the variables) if
1 = b1/2, 0 = a1/2, 0 = a2/2
(14.21), namely bl = 2, a1 = a2 = 0 (or 775 = 2, n = n4 = 0). In this simp
3
we have linear restrictions so the Wald test is the same as a t test on a se
cients. The benefit in using the Wald test is that one doesn™t have to run an
regression on the restricted equation system as we did when we performed the
ratio test(3).
Essentially the Wald test implies that only S, is useful in forecasting Ar
coefficient in (14.21) should equal 2. This is because the PEH (14.2) impli
optimal forecast of Arf+,. Hence any additional variables in the expectations
equation (14.21), namely At-,, Art-l, should be zero as suggested by the Wal
It is also straightforward to show that the restrictions imply that the fo
Art+l - Arf+l is independent of all information at time t or earlier (use equa
and (14.21)) and that expected (abnormal) profits E,(AP) = E(Ar,+l) - 2
(equation (14.21) says it all).

Summary: Cross-Equation Restrictions
( 9 By assuming an expectations generating equation for Ar,+1 one can te
+
hypothesis of the PEH RE by running both the unrestricted and (t
restricted equations and applying a likelihood ratio test. The disadvan
procedure (in more complex cases) is that the restricted equation has to
lated algebraically and then estimated: sometimes this can be rather
implement.
(ii) The Campbell-Shiller methodology uses a Wald test which only requires
mate the unrestricted parameter estimates in the expectations generatio
(iii) Whichever test is used the (non-linear) restrictions on the n ensure that (
i
(abnormal) profits are zero, (b) the PEH holds, that is S, = 2Arf+, at al
(c) that the forecast error for Art+l is independent of the variables in the
set at time t or earlier.
(iv) In the above tests one has to posit an explicit expectations generating e
+
Arf,,, and if the latter is incorrectly specified then tests of the PEH
fail not because the PEH is incorrect, but because agents use a different
scheme for Arf+l, resulting in biased parameter estimates.

It is interesting to compare the above tests involving cross-equation restri
a direct test of the PEH which we discussed in Chapter 10 which only i
unbiasedness and orthogonality properties of RE

+ qr+l
rr+l = E,rr+l
Substituting in the term structure relationship (14.2) and adding 52, gives

+ cS2, +
S = a + bS,
: qf+l

where S = Art+1/2 is the ˜perfect foresight spread™. Under the null of PEH
T
expect a = c = 0 and b = 1. The interesting contrast between these two type
priate form for the equation for Arr+1 a direct test based on (14.25) may
a positive feature in testing the PEH. However, using only a single equatio
result in some loss of ˜statistical efficiency™ compared with estimating a tw
system and using the cross-equation tests or the Wald test approaches. Thes
explored further in Section 14.3 when we discuss conflicting results from emp
in this area.


14.2 THE VAR APPROACH
In the previous example we only had to make a one-period ahead forecast of
When multiperiod forecasts are required we need an equation to forecast futu
the spread S,. The latter can be done by using the Campbell-Shiller vector aut
(VAR) approach which involves matrix notation. As before, this approach i
using a simple example. The PEH applied to a three-period horizon gives:

+ r;+1 + r;+,>
R, = i(rt
which may be reparameterised to give:
+
St =
where S, = RI - r, is the long-short spread, = (rf+l - r,) and Arf+
rf+l). Now assume that both S, and Ar, may be represented as a biva
autoregression of order one (for simplicity):




or in vector notation:
+
=
Zf+l Or+]

where z,+l = (Sr+l,Ar,+l)™, A is the (2 x 2) matrix of coefficients aij,
From (14.30) the optimal prediction of future z™s using the
(wl,+l, wr+l)™.
for forecasting is:




Now let el™ = ( l , O ) , and e2˜ = (0, 1) be 2 x 1 selection vectors. It follows t
S, = el™z,
E, Ar,+1 = e2˜z:+, = e2˜Az,
E,Ar,+2 = e2l ze, + ˜ e2™A2zr
=

Substituting the above in the PEH equation (14.27):
f ( a ) = ell - e2™ ($A + $4™) = 0
where the f(a) has been defined as the set of restrictions. Hence a test o
plus the forecasting scheme represented by the VAR simply requires one to e
unrestricted VAR equations and apply a Wald test based on the restrictions i

Wald Test
It is worth giving a brief account of the form of the Wald test at this point. Afte
our 2 x 2 VAR we have an estimate of the variance-covariance matrix of the
VAR system which as previously we denote




The variance-covariance matrix of the non-linear function f ( a ) in (14.37) is



where fa(a) is the first derivative of the restrictions with respect to the aij
The Wald statistic is:


There is little intuitive insight one can obtain from the general form of the
(but see Buse 1982). However, the larger is the variance of f(a) the smaller
of W .Hence the more imprecise the estimates of the A matrix the smalle
the more likely one is ˜to pass™ the Wald test (i.e. not reject the null). In
the restrictions hold exactly then f ( a ) % 0 and W % 0. It may be shown tha
standard conditions for the error terms (i.e. no serial correlation or heteros
etc.) then W is distributed as central x2 under the null with r degrees of freed
r = number of restrictions. If W is less than the critical value xz then we do no
null f ( a ) = 0. The VAR-Wald test procedure is very general. It can be appli
complex term structure relationships and can be implemented with high order
VAR. Campbell and Shiller show that under the PEH, in general, the sprea
n-period and m-period bond yields (n > m) denoted St(n™m) may be represente




where A”r, = r, - rr-m and k = n/m (an integer). For example, for n =
st(47” - r:™) and:
= Rt(4™
+ a22Arf] + [a23Sr-1 + a24Arr-11
Arf+l = [a2lSf
+ [a25Sf-2 + a26&-2]+ . - + wt+l *



However, having obtained estimates of the aij in the usual way, we can re
above 'high lag' system into a first order system. For example, suppose we h
of order p = 2, then in matrix notation this is equivalent to:
a11 a12 a13 a14

0
0.
Art-1
Equation (14.50) is known as the companion form of the VAR and may be
written:
+
Zt = AZr-1 mr+l
where Zfr+l= [&+I, Arf+l, S,, Art]. Given the ( 2 p x 1) selection vec
[l,O , O , 01, e2 = [0, l , O , 01 we have:




= e2'AJZt
E, At-::;
where in our example n = 4, m = 1, p = 2. If (14.44) are substituted into the g
equation (14.40), Campbell and Shiller demonstrate that the VAR non-linear
are given by:

f(a) = e l f - e2'A[I - (rn/n)(I - A")(I - A")-'](I - A)-' = 0

which for our example gives

f*(a) = el' - e2'A[I - (1/4)(I - A4)(I - A)-'](I - A)-' = 0

and this restriction can be tested using the Wald statistic outlined above.

Interpretation
Let us return to the three-period horizon PEH to see if we can gain some
how the non-linear restrictions in (14.36) arise. We proceed as before and de
optimal forecasts of Ar,+l and Arr+2 from the VAR. We have from (14.29) a
and (14.28):
Using (14.29) and (14.49) in the PEH equation (14.27):
s = 3(a21Arr + a22Sr) + :[(a;, + a22all)Arr + a22(a21 + a12)SrI
r2
s = [fa21 + +(a;1 + a22a11)] Arr + [$a22+ f a 2 2 h + a1211 s
r
r
s = fl(a)Arr + f2(a)Sr
r
Equating coefficients on both sides of (14.50) the non-linear restrictions are:
+ ;(.;I + a22a11)
0 = f 1 (a) = +21
= p 2 2 + $22@21 + a12)
1 =f 2W 2

It has been rather tedious to derive these conditions by the long-hand method
tion and it is far easier to do so in matrix form as derived earlier since we al
general form for the restrictions (suitable for programming for any values of n
The matrix restrictions in (14.37) must be equivalent to those in (14.51). Clea
linear element comes from the A2 term while the left-hand side of (14.51) c
to the vector el. It is left as a simple exercise for the reader to show that for
a12
all
A= [a21 U221

the restrictions in (14.51) are equivalent to those in (14.37). As before, the
cross-equation restrictions (14.51) ensure that

(i) Expected (abnormal) profits based on information 2,in the VAR are ze
(ii) The PEH equation (14.27) holds for all values of the variables in the V
(iii) The error in forecasting Arr+l,Arr+2 using the VAR is independent of
at time t or earlier (i.e. of Sr-j, Arr-j for j 3 0). The latter is the or
property of RE.

It is now straightforward to demonstrate how difficult it can become to formula
mate the restricted model and hence perform a likelihood ratio test of the restric
+
for the three-period horizon case. The unrestricted VAR consists of (14.28)
obtain the restricted VAR we have to use (14.51) to obtain a21 and a 2 2 in term
other aijs and then substitute these (two) expressions in (14.29) which togeth
(unchanged) equation (14.28) constitute the restricted model. The algebraic ma
required become horrendous as either the horizon in the PEH or the lag le
VAR is increased.

The Advantages of the VAR Approach
+
(i) To test the PEH RE restrictions we need only estimate the unrestricted
in the VAR. The Wald test on the parameters of the VAR can be formula
general case of any n and rn (for which k = n/rn is an integer).
The! Ciisadvantages are:

( 0 An explicit forecasting scheme for ( S i , Art) is required which may be m
and hence statistical results are biased.
(ii) The Wald test may have poor small sample properties and it is not inva
precise way the non-linear restrictions are formed (e.g. Gregory and Ve
+
Hence the Wald test may reject the null hypothesis of the PEH RE beca
˜slight deviations™ in the data from the null hypothesis. For example, if f 2
in (14.51) but the standard error on f2(a) was 0.003 one would rej
(on a t test) but an economist would still say that the data largely su
PEH. Campbell and Shiller (1992) recognise that the Wald test restricti
+
rejected and yet the PEH RE may provide a ˜reasonable model™ of th
of interest rates.

Further Testable Implications of the PEH Using the VAR Methodology - Th
Theoretical Spread Si
Campbell and Shiller (1992) suggest some additional ˜metrics™ for measuring th
success of the PEH and these are outlined for the three-period horizon model
(i.e. n = 3, rn = 1):
+
S , = $ArP,, iAr;+,
We have seen that the RHS of (14.52) may be represented as a linear pred
the estimated VAR. If we denote the RHS as the theoretical spread S: then:

+ $A2)Zi = f(A)z, = fl(a)Ar, + fz(a)S,
Si = e2™ ( $ A

The theoretical spread is the econometricians™ ˜best shot™ at what the true (RE

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