ests 0 xogeneity for and Uncovered Interest Rate Parity in the United Kingdom John Hunter, Bane1 University In this article we try to determine whether the purchasing power parity and uncovered interest rate arbitrage conditions are satisfied by data on the UK exchange rate. This study compares the notions of cointegrating exogeneity and weak exogeneity. and then tests these hypotheses. 1. INTRODUCTION In this article we test for cointtgration and exogeneity in an empirical model of the exchange rate. The two cointegrating relationships dealt with are purchasing power parity (PPP) and uncovered interest rate parity (UIP). We hav _: used the data analyzed by Fisher et al. ( 1990) and Johansen and Juselius ( 199 1) to determine the role of cointegration. Johansen and Juselius find partial confirmation of the PPP and UIP hypotheses for a system that includes UK and foreign prices (p, and pz), a UK effective exchange rate (e,*, the value of the basket in terms of pounds sterling), the UK Treasury bill rate (i,), and the eurodollar rate (i2) as endogenous variables, and the world price of oil (p,) as an exogenous variable. We complement their results by testing for co- integration and exogeneity without assuming the exogeneity of oil prices. In Section 2 we briefly discuss the economic theory. In Section 3, we use the notion of cointegrating exogeneity, due to Hunter (1990), to partition the cointegrating variables into vectors of endogenous and Address correspondence to John Hunter, Department of Economics, Brunei University. Uxbridge, Middlesex, VB8 3PH, UK. I would like to thank Neil Ericsson, David Hendry, Denis Sargan, and two anonymous referees for their comments. 1 have also benefited from detailed discussion with Rocco Mosconi and Paul Fisher. I would also like to thank Rocco for the use of his program and Paul for a complete data set. Received April 1991; final draft accepted October 1991. Journal of Policy Modeling 14(4):453-463 ( 1992) 0 Society for Policy Modeling, 1992 453 0 I6 I -8938/92/$5 .OO
Transcript
ests 0 xogeneity for and Uncovered Interest Rate Parity in the United Kingdom
John Hunter, Bane1 University
In this article we try to determine whether the purchasing power parity and uncovered interest rate arbitrage conditions are satisfied by data on the UK exchange rate. This study compares the notions of cointegrating exogeneity and weak exogeneity. and then tests these hypotheses.
1. INTRODUCTION
In this article we test for cointtgration and exogeneity in an empirical model of the exchange rate. The two cointegrating relationships dealt with are purchasing power parity (PPP) and uncovered interest rate parity (UIP). We hav _: used the data analyzed by Fisher et al. ( 1990) and Johansen and Juselius ( 199 1) to determine the role of cointegration. Johansen and Juselius find partial confirmation of the PPP and UIP hypotheses for a system that includes UK and foreign prices (p, and pz), a UK effective exchange rate (e,*, the value of the basket in terms of pounds sterling), the UK Treasury bill rate (i,), and the eurodollar rate (i2) as endogenous variables, and the world price of oil (p,) as an exogenous variable. We complement their results by testing for co- integration and exogeneity without assuming the exogeneity of oil prices.
In Section 2 we briefly discuss the economic theory. In Section 3, we use the notion of cointegrating exogeneity, due to Hunter (1990), to partition the cointegrating variables into vectors of endogenous and
Address correspondence to John Hunter, Department of Economics, Brunei University.
Uxbridge, Middlesex, VB8 3PH, UK.
I would like to thank Neil Ericsson, David Hendry, Denis Sargan, and two anonymous referees for their comments. 1 have also benefited from detailed discussion with Rocco Mosconi and Paul Fisher. I would also like to thank Rocco for the use of his program and Paul for a complete data set.
Received April 1991; final draft accepted October 1991.
Journal of Policy Modeling 14(4):453-463 ( 1992)
0 Society for Policy Modeling, 1992
453 0 I6 I -8938/92/$5 .OO
454 J. Hunter
(potentially) exogenous cointegrated variables. Then we describe the restrictions required for both cointegrating exogeneity and weak ex- ogeneity. In Section 4, we estimate an exchange rate system and test the long-run parameter matrix for the restrictions associated with co- integration and exogeneity . Conclusions are presented in Section 5.
TICAL SPECIFICATIONS OF THE EXCHANGE RATE
Johansen and J uselius purchasing power parity:
(1991) consider two long-run hypotheses,
h - P? = 4,: (1)
and uncovered interest rate parity:
I, - I, * = Ae12. (2)
PPP is the logical conclusion of the law of one price applied to inter- national trade relationships. The UIP model has been formulated in the context of rational expectations by Ballie et al. ( 198 1 ), and it has an older Keynesian tradition associated with the opening up of the ISLM model.
3. CONDITIONAL MODELS AND P’ESTING FOR COINTEGRATION AND EXOGENEITY
In this section we formulate a vector autoregression (VAR) and relate it to an error correction model. The conditions for cointegration are specified in terms of the levels parameters in the error correction model. Cointegration imposes a restriction on the matrix of long-run parameters, implying that questions about the nature of exogeneity need to be discussed in this context. At the end of this section, we look at cointegrating exogeneity and the restrictions on the long-run parameters associated with cointegrating and weak exogeneity.
We take the g-variable, Qh-order VAR in levels with Gaussian errors:
A(/!& = p + ‘-I’D, + E, c, - NIID(O,~) t = l,...,n (3)
where A(L) = I - A,L - A,L’. . . - AkLk, D, are centered dummies, and x,’ = [Par Plr P2r e12r il, ~l,l* ’ Economic series that are I( 1) indi- vidually, form I(0) relationships under cointegration (Granger, 1983;
‘In contrast to is convenient for
Johansen testing.
and Juselius ( 1991). we add p,, to p, and p2 in the VAR, because it
TESTS OF COINTEGRATING EXOGENEITY 455
Engle and Granger. 1987). In that case. (3) has an error correction form in terms of stationary variables:
r(L)Au, = I h, , + ~11 + 'PO, + E,. w
where I(L) = I - I?& - I$. . . - I; _ ,I? ‘, and Ilr, , is a set of nonzero stationary linear combinations of A-, , . The hypothesis of Y cointegrating vectors is
H,(r): II = ap’.
where rank (II) = rank (ar) = rank (g) = r and 0 5 r 5 g. Johansen (1988) and Johansen and Juselius ( 1990) present tests of the rank of II to determine Y. We can test further restrictions on II to determine whether the variables are cointegrating exogenous and/or weakly exogenous.
Engle et al. ( 1983) distinguish between a number of concepts of exogeneity: strict, strong, weak, and super. The cointegration literature has mainly dealt with the weak exogeneity of a variable 2, for l3 (Johansen, 1992). Weak exogeneity is defined in terms of specific parameters of interest and formulated in terms of the distribution of observable variables. The joint density of _Y, in (3) can be partitioned into a conditional density of _v,, given z,, and a marginal density of Z, (Engle et al., 1983):
DWX, ,.$I = D,(yIz,.X, ,&)D.(~,IX, ,.&,.
where .r,’ = I_-& z,‘] and X, = (X0, _Y,, A-?, . . . A-,). Weak exogeneity requires that ;he parameters of interest depend on only the parameters of the conditional density of _Y, (@,I and that there is a sequential cut of the parameter spaces for $, and & (Florens and Mouchart, 1980). If so, the marginal density for Z, can be ignored without loss of in- formation when conducting statistical inference about the parameters of interest. Strong exogeneity combines weak exogeneity with Granger noncausality, so that the marginal density for Z, becomes D,(z, I 2, _ I,
&). Super exogeneity requires weak exogeneity and that the parameters of the conditional process for ,r(, be invariant to changes in the process for 2,.
Cointegv”tir; exogeneity (Hunter, 1990) is a long-run notion of exogeneity that implies that the long-run relations are block triangular. We partition I1 to separate the cointegrating vccto: , into those related to the exogenous variables A, 7 alone and those related to both p1, and 2,:
456 J. Hunter
where CY, is g, X r, ox2 is gZ X r, pr is Y X g, and ps is r X g2. Cointegrating exogeneity is defined by the condition that II,, = o$,’ = 0. It follows from a result by Giannini and Mosconi (1990) that under this hypothesis the model can be parameterized as follows:
such that
where Al; = 1% %I, Pi = lpi, lLl* a,, (gi x c;), and p,> (r, x gi) - Here, we have assumed that Giannini and Mosconi’s r* (= r - r, - rJ equals zero.
Under the assumption that rank (II) = rank (p) = rank (a) = r, the necessary and sufficient condition for Z, to be cointegrating ex- ogenous for [pl, pi,] is
n,, = 0, (8
where the normalization above on 01 and p is adopted. Then the fol- lowing vectors, q ,, and q?,, define r, and rz vectors of stationary variables:
A variable Z, is weakly exogenous for the full cointegrating vector
13 1 w pen
a,, = 0 and a,, = 0 (ii)
(Johansen, 1992). Such weak exogeneity implies that all the cointe- grating vectors enter only the conditional model. The variable z, is both cointegrating exogenous and weakly exogenous for the first r, cointegrating vectors [pl, pS,] when two restrictions hold: (i) above, and
a,, = 0. (iii)
Cointegrating exogeneity means no long-term feedback of y onto z (because II,, = 0), and so implies a ‘-weak” form of Granger non- causality. Thus, cointegrating exogeneity parallels strong exogeneity. The latter permits valid forecasts of y from the conditional model, given forecasts of z from the marginal model. Cointegrating exogeneity permits valid long-run forecasts of _v, given the forecasts of z.
TESTS OF CGINTEGRATING EXOGENEITY 457
Unlike strong exogeneity, cointegrating exogeneity does not require a sequential cut between the parameters of the conditional model and those of the marginal model, and so cointegrating exogeneity does not require weak exogeneity. That has two implications: for statistical inference and for policy analysis. Crs: ) jht estimation of the process for y and z may be necessac to obtr i;;l :.hc parameters in the conditional model of y (used for forecasting ;I). Specifically, joint estimation may be required to obtain the cointegratin g vectors entering the conditional model. Second, under cointegrating erogeneity, the parameters of the conditional model nets ._ot be structurally invariant to changes in the parameters of the marginal process, so cointegrating exogeneity by itself does not sustain valid counterfactual analysis. Rather, the focus of cointegrating exogeneity is long-run forecasting, given knowledge of the cointegrating vectors.
4. A VAR MODEL OF THE K EXCHANGE RATE WITH TESTS OF COINTEGRATION AND EXOGENEITY
We reestimate the model of Johansen and Juselius ( I99 1) and com- pare it with a model that does not impose two restrictions: that the oil price is weakly e?rogenous for the cointegrating vectors p, and that the oil price does not enter p. We then use this “extended” model to test the hypotheses of weak and cointegrating exogeneity.
The model of Johansen and Juselius (1991) is equivalent to a six- variable VAR(2) in (3) that sets to zero both the long-run coefficients of the price of oil and the long-run coefficients in the oil equation, that is,
(5)
The matrix ll,, is estimated over the period 1972:3- 1987:2 and it appears in Table 1. Estimating without the zero restrictions in (5), we obtain Table 2. We refer to the VAR without (5) imposed as the extended VAR. ’
Johansen and Juselius test for and find two cointegrating vectors:
‘Both Johansen and Juselius’s VAR and the extended VAR show some misspecification in
terms of nonnormality and nonconstancy. Additionally. some variables (e.g.. prices) may be
l(2). affecting inference (Johansen. 1991). Noiletheless we will proceed. Ior illustrative purposes.
458 J. Hunter
Table 1: The n,, Matrix for Johansen and JuseliuF’s Model
The first cointegrating vector is essentially (pi - p2 - e,2), augmented by two interest rates. The second is close to (i, - i2), the UIP hy- pothesis. Johansen and Juselius undertake a number of additional tests that confirm that imposing the unit coefficients of the PPP and UIP hypotheses does not affect the results, though PPP on its own does not define a valid cointegrating vector.
We now look at the results of the Johansen procedure for the ex- tended VAR. First we obtain tests of cointegration, as shown in Table 3.’ The trace and max tests imply that r = I or possibly r = 2; to follow Johansen and Juselius ( 199 I) we select r = 2. The first two vectors in l3’ are approximately
Table 2: The II Matrix for the VAR(2) with Oil Prices
PO PI Pr el2
. . 11 12
PO - .224 - s47 .599 .925 1.292 4.787 ____~__-_______________I________________~-~~~~~~~~~~~~~~~~~~~~~
PI -.002 : - .09 .087 .078 .253 .I55 PZ -.Oll i - .046 .020 .071 .I27 .393
‘The computer program PCFIML (Hendry, 1989) was used and the results were then confirmed by Johansen and Juselius’s own procedure, written by Hendrik (Juselius, 1991).
TESTS OF COINTEGRATING EXOGENEITY 459
TabIe 3: Cointegration Tests for the Extended VAR(2)
The first row vector gives the PPP hypothesis, augmented by interest rates and the oil price, while the second row is the UIP condition. The results seem to agree with those of both Fisher et al. ( 1990) and Johansen and Juselius (1991), for whom the PPP relationship requires the interest rates for stationarity.
We consider the tests of restrictions on (x and l3 presented in Johansen and Juselius ( 1990) and ( 1991) and in Giannini and Mosconi ( 1990).4
H4p : P = K,a H4p (R X s), q (s x r) where r I s (r g.
HW : P = (Hwcp,~ICIz), H,p (R )( s), (p, (S x r,), & (g x I-:) and r 5 s 5 g, r = I-, -t- rJ.
H,, : P = W,J-b,cpA H,, (I: x ), S cp~ (.y X r), $, (g X r,) and I- 5 s I K, I- = r, -1 r,.
H4,:a = H&,6, H, (R X s), 6 (s X r) where r I s 5 g.
H ha : a = (h,dh,~d, Hti (K X s). 6, (S X I',), K? (g X r2) and I- 5 s 5 R, r = I-, + r,.
&a : a = hHd2), Ku (R X s), 6, (s X JT~), K: Cg X r,) and r L s I g, r = r, + r2.
4Roc~~ Mosconi has written ;i program that tests the hypotheses on the subspaces of 01 and p
(Giannini and Mosconi, i 990).
J. Hunter
Johansen and Juselius ( 1990) show that these hypotheses can be tested by a likelihood ratio test that is asymptotically distributed x2 under the null. The distributional assumptions are valid for the correct choice of r, which is why we restate the A,,,,.,. test at the start of Table 4. Then we test (a) whether p,, can be excluded from the VAR to validate the Johansen and Juselius model:
6 and p = H, letp = 00000
a = H, ,<,tl = [ 1
___~____ I ‘9, (a) 5
where cp is defined above for H,,, 3 is an equivalent term for Hda, j in H,.j, indexes the particular hypothesis tested, and the same applies for Hi,jp. See Table 4.
The second block of Table 4 contains tests for (b) cointegrating exogeneity of i, and iZ, (c) PPP and UIP, (d) cointegrating exogeneity of e,2, (e) block diagonality of o, (f) weak exogeneity of i, and e,2 for p, and (g) weak exogeneity of i2 for p. The tests are undertaken sequentially and Table 4 provides exact definitions of the null hy- potheses. For example, the test for cointegrating exogeneity of i, and i2 uses the following hypotheses to specify the necessary and sufficient condition in (i):
where q2 is defined above for HTp and ql is an equivalent term for Hhu. We formulate similar hypotheses to carry out (c)-(g) and the other tests in Table 4. Having partitioned (x and p into two subspaces, we impose the unit coefficient restriction for PPP on p.I and that for UIP on 13Z (13i is defined by Table 3). The second restriction is of particular interest as Johansen and Juselius find that UIP does not hold for the whole space of p.
Cointegrating exogeneity for i, and i, having been imposed, co- integrating exogeneity of e, 2 requires only one restriction on CX~~, di- agonal QL requires condition (ii), and weak exogeneity involves additional zero restrictions on 0~~~. At the end of Table 4, we repeat Johansen and Juselius’s tests of PPP and UIP for l3; we test for weak exogeneity (WE) of (e,2, i,, i2), (i,, i2), and (e,2, i,); and we test for diagonal (x.
Tab
le
4:
Tes
ts
of
Coi
nteg
ratio
n an
d bx
ogen
eity
Hyp
othe
sis:
“.’
r=2
(a)
K
lo +
H.,
,,Jr
=
2 (0
Nul
l:’
Sta
tist
ic
(95%
lev
el):
r5
I
-nT
ln(I
-
A,)
=
68.8
6 (6
9.98
)
a,,
= O
.p,,
= 0
.j r=
I.2
x’
(4)
=
23.8
3**
(9.4
9)
(i) W
If,
Ip
+ H
, Jr
=
2
(c)
&
18 +
H,
z,J(
b)
(d)
I& z
,!(c)
(i
i) (
e) H
, ,J
d)
(b)
+
(c)
+ (
d)
+ (e
) (f
) H
, A
(d)
(g)
K
3,1(
d)
(PPP
) H
4.&
=
2 (U
IP)
ffdj
plr
= 2
(iii)
H
4201
r =
2 (i
ii)
H33
ulr
= 2
(iii)
H
,,lr
= 2
H ,
lo +
K&
=
2
(CE
)
(CE
) (D
iago
nal
(r)
WE
) W
E)
WE
) W
E)
(WE
) (D
iago
nal
la)
- 0.
/3,
? =
0. j
=
I,.
, 4
(Y 4, -
-0
a,,
= 0.
j =
1. 2
. 3
(joi
nt
null
for
the
test
s ab
ove)
a.
%? =
0.
LyS
l =
0
ah2
= 0
P:,
- PQ
-
&,
= 0,
j
= 1,
2
P5,
- &
=
0. j
=
I, 2
a I,
= 0.
&,
= 0.
as,
=
0. j
=
1. 2
%
=O
.a,=
O,j=
I,
2 ff
*, =
0,
lyz
, =
0, ,
j =
I, 2
a,?
= 0.
j
= I,
2.
3, (
Y,,
= 0,
i
= 4.
5.
6
x46)
=
7.82
(1
2.59
) x?
(3)
= 2.
01
(7.8
2)
x>(I
) =
0.44
(3
.84)
x“
(3)
= 4.
05
(7.8
2)
~‘(1
3)
= 14
.32
(22.
36)
x2(2
) =
3.07
(5
.99)
x’(1
) =
8.52
**
(3.8
4)
x’(6
) =
2.69
(1
2.59
)
x2(4
) =
26.0
0**
(9.4
9)
x46)
=
13.6
5*
(12.
59)
x?(4
) =
10.5
2*
(9.4
9)
x2(4
) =
4.04
(9
.49)
~$
6)
= 3.
96
(12.
59)
Nor
mal
ize
,d P
and
a
mat
rice
s un
der
(b).
(c
).
(d).
an
d ( e
)
P=
_ 0.
12
0 I
0
-I
0 1 and
(r
= -I
0
3.47
1
3.40
-I
I
‘-0.
996
0.0
’ -0
.062
0.
0 -0
.097
0.
0 0.
0 -0
.216
0.
0 -0
.134
0.0
0.28
7 .
.
“C
= co
inte
grat
ion;
C
E
= co
inte
grat
ing
exog
enei
ty;
WE
=
wea
k ex
ogen
eity
. bC
ondi
tions
(i
)-(i
ii)
are
disc
usse
d in
Sec
tion
3.
‘a,,
and
/3,,
defi
ne
sing
le
elem
ents
of
LY
and
p.
*Sig
nifi
cant
at
the
5 p
erce
nt
leve
l. **
Sign
ific
ant
at t
he
I pe
rcen
t le
vel.
462 J. Hunter
From Table 4, hypotheses (b)-(f) appear satisfied. However, (a) is rejected, meaning that Johansen and Juselius’s restriction on II in (5) appears invalid. Even so, this rejection is tentative because the test used depends upon the extended model having Gaussian errors; and that assumption appears invalid (see footnote 2). Hypothesis (g) is also rejected, implying that iZ is not weakly exogenous. Satisfaction of (b) and (d) means that the conditions for cointegrating exogeneity of i,, L, and e,? for p.i are satisfied. The test of weak exogeneity of e12 and i, for p cannot be rejected, conditional upon (b)-(e). However, the joint tests of weak exogeneity for (c,~, i, , L) and (i, , L) are significant while that for (i,, e,?) is not. The PPP restriction and block diagonal cx are met with or without cointegrating exogeneity being imposed, but UIP holds only for p 1. The model at the end of Table 4 meets all ._ of the restrictions associated with (b)-(e), but it does not impose weak exogeneity of c,? and i:. Given the variables used, it seems odd that the exchange rate and the UK interest rate are weakly exogenous, especially as the CY coefficients for e,? and i, are of a very similar magnitude.
5. CONCLUSION
Our evidence supports a UIP hypothesis and a PPP relationship augmented by both interest rates and the price of oil. Further, the interest rates and the exchange rate appear to be cointegrating exog- enous for the augmented PPP relation. The Treasury bill rate and the exchange rate appear to be weakly exogenous for both the UIP and the augmented PPP relations, implying that no variables in levels enter the marginal equations for the Treasury bill rate and the exchange rate. Thus, these two variables are described by generalized random walks, and the UIP condition (for example) does not affect them in the long run. These results should be treated with some care, as Johansen (1991) has provided evidence that the price series are I(2); and specification tests suggest that the extended VAR may be misspecified.
Appendix A: The Data
P,, is a measure of the real oil price provided by HM Treasury. P, is the UK output price for manufacturing excluding food, drink,
and tobacco (OECD Main Economic Indicators, 1988). P2 is output prices for the six leading industrial countries (OECD). &I! is the inverse of the sterling exchange rate index, rebased to
I, is the UK treasury bill rate average for tire quarter (Bank of England Quarterly Bulletin, various editionc;i.
I, is the 3 month eurodollar rate average for the quarter (Bank of England Quarterly Bulletin, various editions).
The series used are all based on 1980 = 100. All data are quarterly and are in logarithms, which are indicated by lower case.
REFERENCES
Baillie. R.T., Lippens, R.E.. and MacMahon. P.C. C 1981) Testing Rational Expectations and Efficiency in the Foreign Exchange Market. Discussion paper. Series A. University of Birmingham.
Engle, R.F., and &anger. C.W.J. t 1987) Co-Integration and Error Correction: Representation. Estimation and Testing. EconometrEcu. 55: 25 1-276.
Engle, R.F.. Hendry, D.F.. and Richard. J.-F. (1983) Exogeneity. Econometrica 51: 277-304. Florens. J.-P., and Mouchart. M. ( 1980) Initial and Sequential Reduction of Bayesian Exper-
iments. CORE discussion paper 8015. Universitk Catholique de Louvain. Louvain la Neuve, Belgium.
Fisher, P-G., Tanna, SK., Turner. D-S.. Wallis. K-F.. and Whitley. J.D. ( 1990) Ecor,ometric Evaluation of the Exchange Rate in Models of the UK Economy. Economic- lournal 100: 1024-1056.
Giannini, C.. and Mosconi, R. ( 1990) Non-Causality and Neutrality in Camtegrated Systems: Representation Estimation and Testing. Discussion paper No >O-002. Economics De-
partment, Politecnico di Milano. Granger, C. W .J. ( 1983) Cointegrated Variables and Error-Correcting Models. Discussion paper
83-13, University of California, San Diego. Hendry, D.F. (1989) PCGIVE: An Interactive Econometric Modelling System. Oxford: Institute
of Economics and Statistics and Nuffeld College. Oxford Unkersity.
Hunter, J. ( 1990) Cointegrating Exogeneity . Et owntic Letters 34: 33-35.
Johansen. S. ( 1988) Statistical Analysis of L’ointegrating vectors. Journcrl of Economics Dyumics
und Control 12: 23 Y -254. Johansen, S. (1991) A Statistical Analysis of Cointegration for I(2) Vanables. Report. No. 77.
University of Helsinki, Department of Statistics. Forthcoming in Econometric Theov.
Johansen, S. (1992) Cointegration in Partial Systems and the Efficientzy of Single-Equation Analysis. Journal of Econometrics 52: 389-402.
Johansen, S., and Juselius, K. ( 1990) Maximum Likelihood Estimation and Inference on Co-
integration-With Applications to the Demand for Money. Oxford Bulletirl of Ewnomics
and Statistics 52: 169-2 IO. Johansen, S., and Juselius, K. ( 1991) Some Structural Hypotheses in a Multivariate Cointegration
Analysis of the Purchasing Power Parity and the Uncovered Interest Parity for UK. Jortrnul
of Econometrics. fonhcoming . Juselius, K. ( 1991) Cointegration Analysis of Time Series. Th RAT3 Mund. Doan Associates.
