The Econ'omic and Social Review, Vol. 20, No. 1, October 1988, pp. 25-36
Efficiency in the Forward Exchange Market:
An Application of Cointegration
B R I A N M . L U C E Y *
Central Bank of Ireland, Dublin
Abstract: T h i s p a p e r defines a c o i n t e g r a t e d s y s t e m a n d discusses the i m p l i c a t i o n s o f c o i n t e g r a t i o n for
e f f i c i e n c y i n the f o r w a r d e x c h a n g e m a r k e t u s i n g I r i s h d a i l y d a t a . T e s t s are d i s c u s s e d a n d c a r r i e d o u t a n d the results are c o m p a r e d a n d c o n t r a s t e d w i t h those o f a n e a r l i e r p a p e r b y L e d d i n ( 1 9 8 8 ) w h o e x a m i n e d e f f i c i e n c y i n the I r i s h f o r w a r d m a r k e t u s i n g different t e c h n i q u e s . I n c o n t r a s t w i t h the r e s u l t s w h i c h w e r e o b t a i n e d b y H a k k i o a n d R u s h ( 1 9 8 7 ) , w h o u s e d the c o i n t e g r a t i o n a p p r o a c h to s t u d y e f f i c i e n c y in the U S a n d G e r m a n f o r w a r d e x c h a n g e m a r k e t s , the s t u d y f i n d s s u b s t a n t i a l e v i d e n c e of m a r k e t i n e f f i c i e n c y .
I I N T R O D U C T I O N
E
c o n o m i c t h e o r y i n m a n y cases postulates t h a t the t i m e series represen tations o f economic series should t r e n d together. A n example w o u l d be c o n s u m p t i o n t h e o r y , w h i c h w o u l d tend t o suggest that c o n s u m p t i o n and income w o u l d move i n b r o a d l y similar directions. M o r e f o r m a l l y , the q u a n t i t y theory o f m o n e y , w h i c h states t h a t M V = P Y , w o u l d say t h a t the " w e d g e " between n o m i n a l income and m o n e y (whatever that measurement is) w o u l d be constant i f v e l o c i t y were constant, i.e., i f the monetarist m o d e l holds. I n that case, we should expect t o see m o n e y and n o m i n a l income t r e n d i n g very closely together. F i n a l l y , i f the m a r k e t for foreign exchange is an efficient one, i n the sense o f Fama ( 1 9 7 0 ) , then the f o r w a r d rate and the spot rate should t r e n d together, w i t h the f o r w a r d rate being an efficient and unbiased p r e d i c t o r o f the future spot rate.S t r u c t u r a l economic models using standard econometric techniques a t t e m p t to m o d e l the relationships between series and t o estimate the parameters w h i c h u n d e r l y the relationship. F r o m these parameter estimates various inferences about the u n d e r l y i n g t h e o r y can be made, and the process o f falsification can proceed. A n o t h e r w a y i n w h i c h t r e n d i n g variables m i g h t arise is f r o m the assumption o f e q u i l i b r i u m relationships. I f i t is assumed that e q u i l i b r i u m holds, then i t can be seen that the difference between the e q u i l i b r i u m and the actual should n o t be "large". M o r e generally, however, i t is the case t h a t econometric techniques and/or data l i m i t a t i o n s m a y n o t allow f o r adequate inference t o be d r a w n . Recent advances i n the t h e o r y o f w h a t are called co-integrating vectors a l l o w f o r the possibility o f m o r e general and easier inference to be d r a w n concerning economic t h e o r y . This paper defines cointegration i n the c o n t e x t o f a simple 2-variable system. T h e n , the relationship between c o i n t e g r a t i o n and m a r k e t efficiency is e x p l o r e d . Tests are then carried o u t on I r i s h f o r w a r d and spot exchange rates t o ascertain i f there is evidence t h a t they f o r m a cointegrated system. F i n a l l y , the i m p l i c a t i o n s o f these tests f o r m a r k e t efficiency are discussed.
I I D E F I N I N G C O I N T E G R A T I O N
The W o l d d e c o m p o s i t i o n theorem states that a stationary t i m e series process w i t h no deterministic c o m p o n e n t has an i n f i n i t e m o v i n g average representa t i o n . I t is easy t o show ( B o x and Jenkins, 1970) t h a t this can be represented a p p r o x i m a t e l y b y a f i n i t e autoregressive m o v i n g average ( A R M A ) process. F r o m this comes the d e f i n i t i o n o f i n t e g r a t i o n .
D e f i n i t i o n : A series w i t h no deterministic process w h i c h has a stationary invertible A R M A process after differencing d times is said to be integrated o f order d. [Represented as 1(d)]
For purposes where d=0 then the level w i l l be stationary, and where d = l then the change is stationary. The m a i n differences between processes w h i c h are integrated w i t h d=0 and d = l , are w e l l k n o w n , and are discussed i n Engle and Granger ( 1 9 8 7 ) . F o r cases where d = l , as is c o m m o n i n economic series, i.e., where the first differences o f a series are stationary, then as t-+ °°, v ( x()
->• °°where t is t i m e , x is the variable integrated o f order 1 ( xt ~ 1(1)) and
v ( xt) is the variance o f xt. Put simply, i t may appear (and i f we w a i t l o n g enough
1(d), t h e n i n general, f o r a * 0, the linear c o m b i n a t i o n zt = xt - a yt w i l l be
also 1(d). Engle a n d Granger ( 1 9 8 7 ) , adapting f r o m Granger ( 1 9 8 1 ) a n d Granger and Weiss ( 1 9 8 3 ) define c o i n t e g r a t i o n as b e l o w .
D e f i n i t i o n : The components o f a vector are xt are cointegrated o f order
( d , b ) , represented b y x{- C I ( d , b ) i f :
(I) x{ is 1(d), i.e. a l l elements o f the vector xt are each 1(d)] [ ( I I ) a
non-zero vector exists such t h a t z{ = a1 X{~ I ( d - b ) , b > 0 where a
is called the cointegrating vector.
Put m o r e s i m p l y , i f xt ~ C I ( 1 , 1 ) , t h e n i t means t h a t xt achieves s t a t i o n a r i t y
after differencing once, a n d at least one linear c o m b i n a t i o n o f its elements is stationary w i t h o u t differencing, i.e., a r e d u c t i o n o f 1 i n the n u m b e r o f degrees o f differencing r e q u i r e d t o achieve s t a t i o n a r i t y . Engle a n d Granger ( 1 9 8 7 ) state the f o l l o w i n g , f o r the C I ( 1 , 1 ) case, i t can be said t h a t " . . . equi l i b r i u m w i l l occasionally occur, at least t o a close a p p r o x i m a t i o n " .
C o i n t e g r a t i o n o f order (1,1) therefore implies t h a t t w o series do n o t wander too far apart; the difference between t h e m w i l l be stationary, and therefore, we m a y expect t o see close " t r a c k i n g " o f one series b y the other; i f t h e y were n o t cointegrated, the value o f the difference between t h e m w o u l d diverge a r b i t r a r i l y w i t h p r o b a b i l i t y 1, being itself a 1(1) series. N o t e t h a t as y e t , n o t h i n g has been said a b o u t causal d i r e c t i o n a l i t y : a least one causal d i r e c t i o n is i m p l i e d , b u t c o i n t e g r a t i o n does n o t i m p l y any e x p l i c i t causal relationship.
The close l i n k between C o i n t e g r a t i o n and E r r o r C o r r e c t i o n has been estab lished b y Granger ( 1 9 8 1 , 1 9 8 3 ) . I f t w o variables X a n d Y are cointegrated w i t h a cointegrating factor or cointegrating vector a, then they can be w r i t t e n as the E C M :
i t - x H = . [ i t . | - « Y t . I ] + b [ Y t - Y t . I ] + e ,
This standard E C M is similar t o t h a t o f Davidson, H e n d r y , Srba and Y e o ( 1 9 7 8 ) . I t relates the change i n the variables t o lagged changes and t o a lagged linear c o m b i n a t i o n , w h i c h is stationary i n the levels and may be seen as the e q u i l i b r i u m error. As p o i n t e d o u t b y a referee, we m a y view this E C M as a description o f h o w the e c o n o m y eliminates the e q u i l i b r i u m error. This there fore implies t h a t t h e o r y describes the l o n g r u n , and t h a t r a n d o m shocks i n the short r u n can o n l y be s l o w l y adjusted t o . While this view o f e q u i l i b r i u m sees changes i n the variables as arising f r o m the e q u i l i b r i u m error, another _yiew sees the reverse: Campbell a n d Shiller ( 1 9 8 8 ) see the e q u i l i b r i u m error
as resulting f r o m the i n c o r r e c t forecasts b y agents o f the variables. T h e y show h o w an E C M can be w r i t t e n as a V A R ( V e c t o r Autoregressive) system w h i c h can t h e n be tested f o r restrictions. Engle and Granger (1987) infer f r o m the a b i l i t y o f an E C M t o be given a V A R representation, that, given t h a t standard V A R models force c o i n t e g r a t i o n via system restrictions, V A R ' s estimated w i t h cointegrated data tend t o be mis-specified w h e n differences are used, and o m i t valid restrictions (the E C M ) i f levels are used. Given the increasing p o p u l a r i t y o f highly aggregated V A R systems f o r the forecasting o f major macroeconomic elements, and given t h a t the efficiency o f the V A R lies i n the a b i l i t y t o use O L S efficiently, this causes p o t e n t i a l l y major p r o blems f o r atheoretic forecasting, w h i c h w i l l use biased estimates.
A m o r e recent development, b y H y l l e b e r g et al. ( 1 9 8 8 ) is the concept o f seasonal c o i n t e g r a t i o n . Put s i m p l y , this states t h a t although t w o series m a y appear n o t t o be cointegrated at, say, the annual frequency, they m a y be so at other frequencies. I f t h a t is the case, t h e n the standard tests discussed b e l o w are rendered i n v a l i d . The existence o f seasonal u n i t roots has similar con sequences f o r shocks as for the case where the r o o t is present at the annual frequency.
I l l T E S T I N G F O R C O I N T E G R A T I O N : E N G L E - G R A N G E R TESTS
Testing f o r c o i n t e g r a t i o n is closely tied t o the p r o b l e m o f testing for u n i t roots. I n t u i t i v e l y , one can see a simple test o f cointegration as being a test o f the s t a t i o n a r i t y o f the difference between t w o variables: i f there is a u n i t r o o t present i n the series ( Xt - Y ) then the variance w i l l be b o u n d e d b y
i n f i n i t y (explosive variance) and the variable w i l l be non-stationary. Camp bell and Shiller (1987) l o o k at the p r o b l e m this w a y . L e t X( be the vector
o f variables t o be tested for cointegration. The n u l l hypothesis is that a1Xt
is stationary. I f a1, the proposed cointegrating vector, is k n o w n (or pre
d i c t e d b y economic t h e o r y ) then m o d i f i e d D i c k e y - F u l l e r ( 1 9 8 1 ) tests can be used, w i t h t and F statistics corrected for residual a u t o c o r r e l a t i o n .1 Engle
and Granger p r o v i d e a n u m b e r (7) o f tests f o r testing. However, i t must be stated that the Engle-Granger tests are somewhat l i m i t e d i n a p p l i c a b i l i t y , as
(a) T h e y assume C I (1,1) t o be the true case. (b) T h e y are derived for t w o variables.
(c) T h e y were derived f r o m M o n t e Carlo simulations o f 100 observations.
Recently, Engle and Y o o (1987) have p r o d u c e d expanded critical values for t w o o f the Engle and Granger's test statistics. These are calculated for 50, 100 and 200 observations and f o r systems o f up t o five variables.
performance and select one preferred test. M o n t e Carlo simulations w i t h 10,000 replications were used t o o b t a i n the c r i t i c a l values.
(a) The Cointegrating Regression Durbin-Watson Tests ( C R D W )
The cointegrating regression (sometimes called the E q u i l i b r i u m Regression ( H a k k i o and Rush ( 1 9 8 7 ) ) o f Xi t o n X .{, i ¥= j is estimated.
The residuals are then tested for stationarity. I f the residuals are non-stationary, then the Durbin-Watson statistic w i l l t e n d t o zero. Thus, i f the D W is " t o o large", the variables can be seen as cointegrated, i.e., a stationary difference is present between t h e m .
(b) D i c k e y - F u l l e r Tests ( D F )
The residuals f r o m the cointegrating regression are tested b y means o f an a u x i l i a r y regression. This regresses the first difference o f the residuals o n a lagged residual and the coefficient o f the lagged value is t h e n tested f o r significance via a t test. This assumes t h a t any serial c o r r e l a t i o n present is first order.
(c) A u g m e n t e d D i c k e y - F u l l e r Test ( A D F )
This is the same as that f o r the D F test, except t h a t a d d i t i o n a l lags (of the residual) are i n c l u d e d i n the a u x i l i a r y regression. As a result, i t is over paramaterised i f any a u t o c o r r e l a t i o n present is first order, b u t is correct i n higher order cases.
(d) -(g) V e c t o r Autoregression Tests ( V A R )
These tests estimate various V A R representations o f the i m p l i e d ECM.
C o m m e n t i n g o n the v a l i d i t y , p o w e r and usefulness o f these tests, Engle and Granger r e c o m m e n d test (c), the A D F . However, they feel that the C R D W test is a very useful first test, due t o its s i m p l i c i t y . I t is, however, somewhat l i m i t e d . I n e m p i r i c a l tests i n the paper, Engle and Granger m a i n l y use the C R D W , D F and A D F tests t o examine t i m e series data f o r the presence o f cointegration.
I V C O I N T E G R A T I O N A N D M A R K E T E F F I C I E N C Y
a b o u t w h i c h the rate fluctuates, the first differences should always be serially uncorrelated. I f t w o asset prices are cointegrated, then an E C M can be f o u n d t o express t h e m . B u t , this w o u l d mean that part o f the change i n price c o u l d be p r e d i c t e d f r o m the E C M . L e t X{ and Y be t w o asset prices. C o i n t e g r a t i o n
implies:
(Xt " Xt - 1 ) = atXt - 1 " dYt - 1 1 + b [Yt " Yt - 1 1 + e
where X{ and Y are cointegrated w i t h a cointegrating factor o f d. However,
b o t h Xt_ j and Yt_ j are available i n the t - 1 i n f o r m a t i o n set. So, t w o asset
prices f r o m efficient markets cannot be cointegrated.
I n the m a r k e t for foreign exchange, there are further i m p l i c a t i o n s f o r efficiency. I f the f o r w a r d rate is t o be an efficient and unbiased estimate o f the future spot rate, the f o r w a r d forecast error, F - S j , m u s t be a serially uncorrelated r a n d o m variable. Efficiency tests w h i c h rely o n regressing one o n the other w o u l d w o r k o n l y i n the case where b o t h were stationary. N o n -s t a t i o n a r i t y w o u l d lead t o explo-sive -standard error-s. C o i n t e g r a t i o n ha-s -several i m p l i c a t i o n s for efficiency. First, Ft and S j must be cointegrated. Were they
n o t t o be, t h e n the f o r w a r d forecast error w o u l d be non-stationary and w o u l d lead t o Ft and S j diverging a r b i t r a r i l y far apart. Secondly, i t is necessary
for the cointegrating factor t o equal 1. This is demonstrated somewhat more f o r m a l l y i n H a k k i o and Rush ( 1 9 8 7 ) . F i n a l l y , the E C M m u s t have a certain shape ( H a k k i o and Rush).
T h e non-stationarity o f the series means t h a t i t is d i f f i c u l t t o f o r m a l l y test for a c o i n t e g r a t i o n factor = 1, b u t this factor itself can be observed. Cointe gration itself is relatively easily tested. Cointegration also means i n the for w a r d / s p o t m a r k e t t h a t the risk p r e m i u m , i f present, is stationary, necessarily. I f there is a " r i s k " p r e m i u m , then
St = R P + Ft_1 +et
where RP is risk (or t i m e varying) p r e m i u m , e is w h i t e noise. I f spot and for w a r d rates are cointegrated, the f o r w a r d forecast error is stationary. I n the above case, the f o r w a r d forecast error is RP + e{. B u t , e is w h i t e noise, so
V T E S T I N G E X C H A N G E M A R K E T E F F I C I E N C Y I N I R E L A N D
D a i l y data o n I r i s h P o u n d spot and f o r w a r d rates were analysed, starting i n December 1 9 8 7 ; 100 observations were used. F o r w a r d exchange rates were the one m o n t h f o r w a r d rates, w h i c h were m a t c h e d t o the appropriate spot exchange rate. This was done b y allocating t h e m t o the nearest business day t o 30 days ahead. A l l data was i n n o m i n a l terms. Thus, i f a f o r w a r d rate was t o be m a t c h e d t o a Saturday, i t w o u l d actually be matched t o M o n d a y and so o n . I t is evident, f r o m l o o k i n g at the relationship between the f o r w a r d and spot rates, f o r b o t h the dollar and the p o u n d sterling t h a t there is a consistent failure t o track b y the f o r w a r d rate. This w o u l d i m m e d i a t e l y raise doubts as t o the efficiency o f the f o r w a r d rate i n p r e d i c t i n g the future spot rate.
As c o i n t e g r a t i o n t h e o r y does n o t i m p l y a n y t h i n g a b o u t causality i t m a y be felt t h a t i t is necessary t o test for c o i n t e g r a t i o n o f X and Y b y l o o k i n g at X o n Y and Y o n X . However, i f X is cointegrated w i t h Y , d e f i n i t i o n a l l y Y is cointegrated w i t h X . T h e y are t w o elements o f a cointegrated system.
The f o l l o w i n g relationships were tested:
(a) The spot exchange rates against the dollar and against sterling are p a r t o f a cointegrated system;
(b) The f o r w a r d rates are themselves p a r t o f a cointegrated system; (c) The spot and f o r w a r d rates, for each currency, are p a r t o f a cointe
grated system.
As stated, i n all cases 100 observations were used, t o p r o v i d e strict c o m p a r a b i l i t y o f the estimated test statistics t o those given i n Engle and Granger ( 1 9 8 7 ) .
Table 1 provides details o f the tests carried o u t o n the relationships h y p o thesised above. Details o f the e q u i l i b r i u m and a u x i l i a r y regressions m a y be f o u n d i n A p p e n d i x 1. £ = IR£/£ spot rate, £ 3 0 = I R £ / £ 30 day rate. Analo gously for the dollar.
T a b l e 1: Testing for Cointegration
Test Critical Value Values of Test Statistic for System Being Analysed
1% 5% £-$ £ - £ 3 0 $ - $ 3 0 $30-£30
C R D W . 5 1 1 . 3 8 6 . 1 4 6 . 1 5 3 . 0 8 9 . 2 6 7 9 D F 4 . 0 7 3 . 3 7 0 . 3 7 4 1 0 . 5 6 0 3 - 1 . 5 0 - 2 . 6 4 A D F 3 . 7 7 3 . 1 7 - 0 . 2 9 7 9 - 0 . 4 9 6 1 - 1 . 3 4 - 2 . 1 3
H o : N o c o i n t e g r a t i o n . R e j e c t f o r l a r g e v a l u e s .
V I I N T E R P R E T I N G T H E E V I D E N C E
Spot Exchange Rates
T h e evidence against c o i n t e g r a t i o n is strong. A p r e l i m i n a r y conclusion i n favour o f m a r k e t efficiency, as evidenced b y the seeming absence o f cointe gration, can be registered. N o n e o f the tests reject the n u l l hypothesis, o f a non-cointegrated system, a l l being w e l l b e l o w the c r i t i c a l values.
Forward Exchange Rates
A g a i n , there is n o evidence o n the basis o f the test carried o u t here that the f o r w a r d dollar and p o u n d rates f o r m a cointegration system. This w o u l d again be evidence f o r m a r k e t efficiency.
Spot and Forward Rates
As stated, i t is necessary, b u t n o t sufficient, for m a r k e t efficiency that the spot and f o r w a r d rates for a currency be cointegrated.
There is n o evidence o n the basis o f the tests carried o u t that the spot and f o r w a r d rates are cointegrated.
I t can be seen ( A p p e n d i x 1) t h a t i n no case is the coefficient o n the for w a r d rate at all close ( i n terms o f standard errors) t o 1. A c c o r d i n g l y , the testing t h a t - a = b = 1 i n an estimated E C M representation, is n o t done, as the assump t i o n d = 1 w o u l d n o t seem t o be j u s t i f i e d , and one o f the necessary c o n d i t i o n s is v i o l a t e d . So, I conclude against m a r k e t efficiency i n this case.
V I I C O N C L U S I O N
REFERENCES
B O X , G . , a n d G . J E N K I N S , 1 9 7 0 . Time Series Analysis, Forecasting and Control, S a n F r a n c i s c o : H o l d e n D a y .
C A M P B E L L , J . , a n d R . S H I L L E R , 1 9 8 7 . " C o i n t e g r a t i o n a n d t e s t s o f p r e s e n t v a l u e M o d e l s " ,
Journal of Political Economy, V o l . 9 5 , N o . 5 , O c t o b e r , p p . 1 0 6 2 - 1 0 8 8 .
C A M P B E L L , J . , a n d R . S H I L L E R , 1 9 8 8 . " I n t e r p r e t i n g C o i n t e g r a t e d S e r i e s " , Journal of
Economic Dynamics and Control, V o l . 1 2 , p p . 2 - 3 .
D A V I D S O N , J . , D . H E N D R Y , F . S R B A , a n d B . S . Y E O , 1 9 7 8 . " E c o n o m e t r i c M o d e l l i n g o f t h e A g g r e g a t e T i m e S e r i e s R e l a t i o n s h i p b e t w e e n C o n s u m e r ' s E x p e n d i t u r e a n d I n c o m e i n T h e U n i t e d K i n g d o m " , Economic Journal, V o l . 8 8 , p p . 6 6 1 - 6 9 2 .
D I C K E Y , D . , a n d W . F U L L E R , 1 9 8 1 . " T h e L i k e l i h o o d R a t i o S t a t i s t i c s f o r A u t o r e g r e s s i v e T i m e S e r i e s w i t h a U n i t R o o t " , Econometrica, V o l . 4 9 , N o . 4 , J u l y , p p . 1 0 5 7 - 1 0 7 2 . E N G L E , R . , a n d C . G R A N G E R , 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,'Vo\. 5 5 , N o . 2 , M a r c h , p p . 2 5 1 - 2 7 6 . E N G L E , R . , a n d B . S . Y O O , 1 9 8 7 . " F o r e c a s t i n g a n d T e s t i n g i n C o i n t e g r a t e d S y s t e m s " ,
Journal of Econometrics, N o . 3 5 , 1 9 8 7 .
F A M A , 1 9 7 0 . " E f f i c i e n t C a p i t a l M a r k e t s : A R e v i e w o f T h e o r y a n d E m p i r i c a l W o r k " ,
Journal of Finance, N o . 2 5 , M a y , p p . 3 8 3 - 4 1 7 .
G R A N G E R , C , 1 9 8 1 . " S o m e P r o p e r t i e s o f T i m e S e r i e s D a t a a n d t h e i r U s e i n E c o n o m e t r i c M o d e l S p e c i f i c a t i o n " , Journal of Econometrics, N o . 1 6 , p p . 1 2 1 - 1 3 0 .
G R A N G E R , C , 1 9 8 3 . " C o i n t e g r a t i n g V a r i a b l e s a n d E r r o r C o r r e c t i n g M o d e l s " , U n i v e r s i t y o f C a l i f o r n i a , S a n D i e g o D i s c u s s i o n P a p e r , p p . 8 - 1 3 .
G R A N G E R , C , 1 9 8 6 . " D e v e l o p m e n t s i n t h e S t u d y o f C o i n t e g r a t e d E c o n o m i c V a r i a b l e s " ,
Oxford Bulletin of Economics and Statistics, V o l . 4 8 , N o . 3 , A u g u s t , p p . 2 1 3 - 2 2 8 .
G R A N G E R , C , a n d A . W E I S S , 1 9 8 3 . " T i m e S e r i e s A n a l y s i s o f 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 P r e s s .
H A K K I O , C . , a n d M . R U S H , 1 9 8 7 . " M a r k e t E f f i c i e n c y a n d C o i n t e g r a t i o n " , R e s e a r c h W o r k i n g P a p e r R W P 8 7 - 0 5 , F e d e r a l R e s e r v e B a n k o f K a n s a s C i t y , R e s e a r c h D i v i s i o n , F o r t h c o m i n g , Journal of International Money and Finance.
H Y L L E B E R G , S . , R . F . E N G L E , C . G R A N G E R , a n d B . S . Y O O , 1 9 8 8 . " S e a s o n a l I n t e g r a t i o n a n d C o i n t e g r a t i o n " , D i s c u s s i o n P a p e r 8 8 - 3 2 , D e p a r t m e n t o f E c o n o m i c s , U n i v e r s i t y o f S a n D i e g o .
L E D D I N , A . , 1 9 8 8 . " P r i c e s , I n t e r e s t 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 : T h e I r i s h E x p e r i e n c e i n t h e E u r o p e a n M o n e t a r y S y s t e m " , P r e s e n t e d t o C e n t r a l B a n k , J u n e . L E V I C H , R . , 1 9 8 5 . " E m p i r i c a l S t u d i e s o f E x c h a n g e R a t e s : P r i c e B e h a v i o u r , R a t e D e t e r m i n a t i o n a n d M a r k e t E f f i c i e n c y " , i n R . J o n e s a n d P . K e n e n ( e d s . ) , Handbook of Inter
T H E E C O N O M I C A N D S O C I A L R E V I E W
A P P E N D I X 1
Equilibrium and Auxiliary Regression
(t statistics i n brackets)
1. Equilibrium Regression
£
=
1.115608 + .004369 $(36.018) (.225)
DW
=
.146R2
=
.0005£
=
1.126961 - .003779 £ 3 0(192.390) ( - . 7 4 9 )
DW
=
.153R2
=
.0057$
= 1.628826 - .017457 $30 ( 2 4 . 2 5 2 ) (- .446)DW
=
.089R2 .002
$ 3 0 - 1.766255 - .044798 £ 3 0
(22.415) ( - . 6 5 7 )
DW
=
.267R2
=
.0042. First Auxiliary Regression: Dickey-Fuller Regressions
Resij = Residuals o f e q u i l i b r i u m regression o f currency i o n currency j . Difresij = First difference o f Resij.
Difrespd = 0.001076 Respd DW = 1.61
(.73) R2 = - . 0 0 4
Difrespp = 0.024702 Respp DW = 1.62
(.56) R2 = - . 0 0 5
Difresdp = - 0 . 1 3 3 8 5 7 Resdp DW = 2.17
( - 2 . 6 4 ) R2 = .067
Difresdd = - 0 . 0 4 4 6 5 9 Resdd DW = 1.59
3. Second Auxiliary Regressions: Augmented Dickey-Fuller Regression
Dependent Variable: Difresdp
Independent Variable Lag Coefficient t Statistic
Resdp 1 - . 1 2 6 0 - 2 . 1 3
Difresdp 1 - . 1 2 0 9 - 1 . 1 2
Difresdp 2 - . 0 2 1 5 - 0 . 1 9
Difresdp 3 . 0 7 5 1 0.69
Difresdp 4 .0984 0.92
Difresdp 5 - . 1 2 5 9 - 1 . 1 9
R2 = .124
D W = 2.00
Dependent Variable: Difresdd
Independent Variable Lag Coefficient t Statistic
Resdd 1 - . 0 4 2 8 - 1 . 3 4
Difresdd 1 .2282 2.18
Difresdd 2 - . 1 0 1 7 - 0 . 9 5
Difresdd 3 .1476 1.38
Difresdd 4 . 0 8 4 1 - 0 . 7 7
Difresdd 5 - . 1 4 5 3 - 1 . 3 6
R2 = .122
Dependent Variable: Difrespd
Independent Variable Lag Coefficient t Statistic
Respd 1 - 0 . 2 5 9 .340
Difrespd 1 - . 2 5 1 6 1.94
Difrespd 2 - . 0 0 8 2 .06
Difrespd 3 .0147 0.11
Difrespd 4 .0254 0.15
Difrespd 5 - . 0 9 4 9 - 0 . 5 5
R2 = .04
D W = 1.98
Dependent Variable: Difrespp
Independent Variable Lag Coefficient t Statistic
Respp 1 - . 0 3 8 6 - . 4 9
Difrespp 1 .2612 2.00
Difrespp 2 - 0 . 0 1 9 3 - . 1 5
Difrespp 3 .0741 .56
Difrespp 4 - 0 . 0 3 1 5 - . 1 9
Difrespp 5 - 0 . 0 7 0 8 - . 4 2
Naming Conventions
Respp/Resdd are the residuals o f the e q u i l i b r i u m regressions o f the spot and f o r w a r d rates f o r the p o u n d and the dollar. Diffrespp is the first difference o f the above.
