The Economic and Social Review, Vol. 20, No. 3, April, 1989, pp. 281-285
Efficiency in the Forward Exchange Market:
A n Application of Co-Integration: A Comment
A N T H O N Y L E D D I N
National Institute for Higher Education, Limerick
I
n a recent e d i t i o n o f the Review, Lucey (1988) finds evidence o f foreign exchange m a r k e t inefficiency using the co-integration technique. His results relate t o sterling and the dollar and are based o n daily spot and one m o n t h f o r w a r d exchange rates. One h u n d r e d observations were used starting i n December 1987.I t is p o i n t e d o u t b y Lucey that, i n relation t o sterling, his results are con t r a r y t o the findings presentedin L e d d i n ( 1 9 8 8 ) where the three m o n t h sterling f o r w a r d exchange rate was f o u n d t o be an unbiased predictor o f the future spot exchange rate (no other currency was considered). This f i n d i n g was based o n an O L S estimate o f E q u a t i o n ( 1 ) .
ST + 1 = a + / 3 Ft+ et (1)
where S{ + 1 and F are the future spot and f o r w a r d exchange rates respectively
and e{ is the error t e r m . Using non-overlapping quarterly data for the p e r i o d
1979q2 t o 1987q4, an F test c o u l d n o t reject the j o i n t n u l l hypothesis (a, j3) = ( 0 , 1 ) . A d d i t i o n a l tests based on the forecast errors supported this result.
I n w h a t follows, quarterly data relating t o sterling, the deutsche mark ( D M ) and the dollar, extended t o cover the p e r i o d 1979q2 t o 1988q4, is used t o establish i f the three m o n t h f o r w a r d exchange rate and the future spot exchange rate are co-integrated.
V e r y b r i e f l y , t w o variables are said t o be co-integrated (see Granger and Weiss ( 1 9 8 3 ) , Granger ( 1 9 8 6 ) and Engle and Granger (1987) for a precise d e f i n i t i o n ) i f :
(a) Each variable is 1(1).
A variable is said t o be 1(0) i f i t is integrated o f order zero. I n this case the variable is " s t a t i o n a r y " . I t has a f i n i t e variance and deviations over t i m e w i l l t e n d t o converge t o the mean. I f , o n the other hand, a variable is integrated of order one, w r i t t e n 1(1), i t is non-stationary (first differences are however stationary). Deviations f r o m the mean increase over t i m e . Variables w h i c h f o l l o w a r a n d o m w a l k are 1(1). Co-integration means that a long-run stable e q u i l i b r i u m exists between the relevant variables.
The first part o f the co-integration test is to determine i f all o f the relevant variables are 1(1). The test used here is the A u g m e n t e d D i c k e y Fuller ( A D F ) test proposed b y D i c k e y and Fuller ( 1 9 8 1 ) . This test is p e r f o r m e d b y estimat ing E q u a t i o n ( 2 ) .
where T is a t i m e t r e n d . The lagged first differences i n X can be increased a r b i t r a r i l y t o ensure et is w h i t e noise.
The t-statistic u n d e r l y i n g 02 tests the n u l l hypothesis H o : X ~ 1(1). I f
the t-statistic is greater than the critical value, H o is rejected and the alter native hypothesis that X is stationary is accepted. I n the Lucey paper this p a r t o f the co-integration test is n o t reported.
Table 1 presents the results o f the A D F test.
T a b l e 1: Random Walk
t-statistic
v2
Spot Exchange Rates
S t e r l i n g - 3 . 0 7 3.72
D M - 2 . 7 8 4.0
D o l l a r - 1 . 1 4 2.3
3 Month Forward Rates
S t e r l i n g - 3 . 0 3.8
D M - 2 . 7 5 3.8
D o l l a r - 1 . 1 2.37
Note: T h e equations w e r e e s t i m a t e d using logs a n d non-overlapping data over the p e r i o d
$2 is an F-statistic w h i c h tests i f j30 = 0 j = 02 = 0. T h a t is i f the particular
variable follows a r a n d o m w a l k w i t h o u t d r i f t .
The results suggest that the spot and f o r w a r d exchange rates f o l l o w a ran d o m w a l k w i t h o u t d r i f t at the 5 per cent level. A Ljung-Box test ( 1 0 t h order) indicated that the error terms i n each o f the estimated equations were free o f autocorrelation.
The second part o f the co-integration test is t o regress the future spot rate on the f o r w a r d rate and calculate the residuals. The residuals are then used to estimate E q u a t i o n ( 3 ) .
ARes, = a „ R e st_ j + a j S A R e st_ j (3)
where Rest is the residual i n t i m e t. I f the t-statistic on the <xQ coefficient is
greater than the critical value, the n u l l hypothesis of n o n co-integration is rejected i n favour o f the alternative hypothesis o f co-integration.
The results are presented i n Table 2.
T a b l e 2: Co-Integration
t-statistic
Sterling 3.35 D M 3 . 7 4 D o l l a r 2 . 3 8
Note: L o g e s t i m a t i o n . T h e critical values are 3.73 a n d 3.17 at the 1 per cent a n d 5 per
cent levels respectively ( E n g l e a n d Granger, 1 9 8 7 ) .
The results suggest that the sterling and the D M future spot and f o r w a r d exchange rates are co-integrated at the 5 per cent level. However, dollar spot and f o r w a r d rates do n o t appear t o be co-integrated. Again the Ljung-Box statistic i n d i c a t e d that the residuals were w h i t e noise.
F i n a l l y , the A D F test was applied t o the forecast errors. The relevant t-statistics are given i n Table 3.
T a b l e 3: Forecast Errors
t-statistic
Sterling 3.6 D M 3.6 D o l l a r 2.66
The n u l l hypothesis t h a t the forecast errors are non-stationary is rejected at the 5 per cent level for sterling and the D M b u t accepted i n the case o f the dollar. Hence the sterling and D M forecast errors appear t o be stationary. This is consistent w i t h the earlier findings. The results relating t o sterling are also consistent w i t h the findings i n L e d d i n (1988) b u t n o t Lucey ( 1 9 8 8 ) .
The differences i n the L e d d i n and Lucey results c o u l d be a t t r i b u t e d t o the different estimation periods. However, the t w o papers differ i n another poten tially i m p o r t a n t respect. I n the L u c e y paper, the data overlap (the sampling p e r i o d is smaller than the forecast period) and this can result i n residual auto c o r r e l a t i o n . Referring back t o E q u a t i o n ( 1 ) , the basic p r o b l e m is that the error t e r m is being generated as new i n f o r m a t i o n becomes available between t i m e t and t i m e t + 1 (the forecast p e r i o d ) . Consequently, the residuals based on overlapping data m u s t be autocorrelated.
Hansen and H o d r i c k (1980) andHsieh (1982) have shown that this p r o b l e m can be overcome i f a Generalised Least Squares estimation procedure is used. I t is desirable to use overlapping data where possible t o overcome the "data shortage" p r o b l e m . Sometimes referred t o as the "peso p r o b l e m " . The d i f f i c u l t y is that financial/economic data m a y require large samples t o allow the sample m o m e n t s t o converge t o the true p o p u l a t i o n moments. However, the results here are i n t e n d e d t o act as a comparison t o the earlier findings i n L e d d i n ( 1 9 8 8 ) . Consequently, the data frequency have n o t been changed.
A second d i f f i c u l t y is that the number o f observations used i n the co-integration tests (39) does n o t concur w i t h the number o f observations (100) used b y Engle and Granger (1987) t o derive the c r i t i c a l values. This m a y or may n o t prove a p r o b l e m . U n f o r t u n a t e l y , c r i t i c a l values for less t h a n 100 observations are n o t as y e t available.
I t is interesting that future spot and f o r w a r d exchange rates appear t o be co-integrated i n the case o f sterling and the D M b u t n o t the dollar. There is no obvious reason w h y this should be the case. This m a y be a f r u i t f u l avenue t o explore i n order t o determine the existence or otherwise o f a risk p r e m i u m .
REFERENCES
D I C K E Y , D . A . , a n d W . A . F U L L E R , 1 9 8 1 . " L i k e l i h o o d R a t i o Statistics for Autoregressive T i m e Series w i t h a U n i t R o o t " , Econometrica, 4 9 , p p . 1 0 5 7 - 1 0 7 2 .
E N G L E , R . , a n d C . W . Granger, 1 9 8 7 . " C o - I n t e g r a t i o n a n d E r r o r C o r r e c t i o n : R e p r e s e n t a t i o n , E s t i m a t i o n a n d T e s t i n g " , Econometrica, 5 5 , p p . 2 5 1 - 2 7 6 .
F U L L E R , W . A . , 1 9 7 6 . Introduction to Statistical Time Series, N e w Y o r k : J o h n W i l e y and S o n s .
G R A N G E R , C . W . G . , a n d A . A . W E I S S , 1 9 8 3 . " T i m e Series A n a l y s i s of E r r o r C o r r e c t i n g M o d e l s " , i n Studies in Econometrics, Time Series and Multivariate Statistics, N e w Y o r k : A c a d e m i c Press, p p . 2 5 5 - 2 7 8 .
H A N S E N , L . P . , a n d R J . H O D R I C K , 1 9 8 0 . " F o r w a r d E x c h a n g e R a t e s as O p t i m a l Pre d i c t o r s of F u t u r e S p o t R a t e s : A n E c o n o m e t r i c Analysis" Journal of Political Economy, 8 8 , p p . 8 2 9 - 8 5 3 .
H S I E H , D . A . , 1 9 8 2 . " T e s t s of R a t i o n a l E x p e c t a t i o n s a n d N o R i s k P r e m i u m in F o r w a r d E x c h a n g e M a r k e t s " , NBER Working Paper No. 843.
L E D D I N , A . , 1 9 8 8 . " P r i c e s , Interest R a t e s a n d F o r e i g n E x c h a n g e M a r k e t E f f i c i e n c y " , P a p e r p r e s e n t e d at the C e n t r a l B a n k of I r e l a n d , J u n e .
