COINTEGRATION TESTING using the RANGES

COINTEGRATION TESTING using the RANGES Felipe M. Aparicio and Alvaro Escribano*

U niversidad Carlos III de Madrid Department of Statistics and Econometrics C /Madrid, 126-128 28903 Getafe (Madrid) Spain Fax: +34 1 624 98 49 Phone: +34 1 624 98 71 E-mail: [email protected] September 23, 1999

*This research, initiated at the Department of Economics (Aparicio and Granger, 1995 [2]) and the Institute for Nonlinear Science of the University of California at San Diego, was partially supported by the Spanish DGICYT under Grant PB95-0298. We are grateful to Ignacio Pena and to Z. Ding for kindly providing the foreign exchange-rate and the stock return series, respectively.

2

1 Abstract In this paper we propose a method for testing the hypothesis of cointegration in pairs of univariate time series. One of our method's main advantages lies in that it does not impose any restriction on the time series models. Another is that cointegration can be tested regardless of the form of the relationship. Essentially, our test rests on a definition of cointegration which requires the sinchronicity up to a constant delay of the relevant informational events for the series. Thus cointegration can bp tested independently on what form of relationship holds between the variables. We propose three alternative test statistics and obtain, under some assumptions,' their asymptotic null distribution. We also propose some graphical techniques consisting in plotting functions of the range sequences for the pairs of series. These plots could help in detecting nonlinearities as well as nonstationarities in the cointegrating relationship. Also we show how nonlinearity and/or nonstationadty' ill. the relationship can be detected by analyzing the cross-difference of ranges. We :Qnally report some

,

experiments on financial and monetary time series that compare the performances of our test statistics with more standard ones.

KEY WORDS; Linear and nonlinear cointegration, Dickey-Fuller test, integrated

time series, order statistics, ranks, range, comovement, marked point processes.

1

3

Introduction

Processes which exhibit common trends or similar long waves in their sample paths are often called cointegrated. The concept of cointegration was inherently linear and originated in macroeconomics and finance (c.f. Granger, 1981[16]; Granger and Engle, 1987[11]), where the theory suggests the presence of economic or institutional forces preventing two or more series to drift too far apart from each other. Take for example, those series as income and expenditure, the prices of a particular good in different markets, the interest rates in different countries, the velocity of circulation of money and short-run interest rates, etc. Cointegration relatiollships may also appear in engineering applications. For instance, between the outputs signals from different sensing or processing devices having a limited storage capacity or memory, and driven by a common persistent input flow (c.f. Aparicio, 1995l1]). i

'

Underlying the idea of cointegration is that of a stochastic equilibrium , relationship (i.e. one which, apart from deterministic elements, holds on the average) between two cointegrated variables, Yt, :rt· A strict equilibrium exists when for some

()!

i:

0, one has Yt =

()!

Xt. This

unrealistic situation is replaced, in practice, by that of (linear) cointegration, in which the equilibrium error Zt = Xt -

()!

Xt is different from zero but fluctuates around the mean much

more frequently than the individual series.

So far, attempts to extend the concept of cointegration beyond the assumption of linearity in the relationship have met with little success. This is essentially due to the fact that a general null hypothesis of cointegration encompassing nonlinear relationships is too wide to be tested. real, and

Notwithstanding, the possibility of nonlinear cointegrating relationships is

th(~refore

it has prompted some interesting definitions and an ongoinJ.'; research on

the subject. The first of these attempts was due to Hallman (1990)[20] and to Granger and Hallman (1991) [17]. Following this, for a pair of series Yt,

1:1.

to have a cointegrating nonlinear

attractoT, there must be nonlinear measurable functions f(·), .9(.) such that .f(ilt) and g(Xt) are both I(d), d> 0, and Wt = f(Yt) - g(Xt) is Assuming that

J

rv

I(d w ), with dw < d.

and 9 can be expanded as Taylor series up to some order

]J

2: 2 around

the ongm, we may write Wt

=

Co

+ CIZt + HOT(Yt, Xt),

where Zt

=

4 Yt - a:r:t, and with

HOT(., .) denoting higher-order terms. It follows that the linear approximation, Zt, to the

true cointegration residuals differs from the latter by some higher-order terms which express that the strengh of attraction onto the cointegration line Yt = aXt varies with the levels of both series, ]It and Xt. As with linear cointegration, the case where dx = dy = 1, dz = 0 and the cointegration residuals have finite variance is most important in practice, since it allows a straighforward interpretation in terms of equilibrium concepts. Figure 1 illustrates the case of nonlinear cointegratioll, with simulated nonlinear transformations of random walks.

-10 '-----1-'-OO---1-:------:-L---'-:c--::-'-------L:---L---1--L ----"1000 g0O 0

Figure 1: Two simulated nonlinearly cointegrated series. The upper series was obtained as Xt

=

Wt

+ e;r,t,

while the lower one corresponds to Yt

= g(Wt) + ey,t, where g(.) repre-

sents a third-order polynomial of its argument random walk variable Wt, and Ex,t, Ey,t are independenti. i.d. sequences.

Further Ellgle and Granger (1987[11]) proposed a method for testing the hypothesis of noncointegration against that of linear cointegration. These tests can be decomposed as follows: • A test for long-memory in the variables, say !it

r-v

I(d,J, :X:t

r-v

I(d x ), and estimation

of the long-memory parameters d x , d y . Then a test of significance for the stochastic

difference dx

-

5 dy. If it is too large as compared to both d x and dy, then the variables

cannot be cointegrated. Otherwise, we assume dx

= dy =

d and go to next step .

• A test for long-memory in the cointegrating residuals Zt = Yt - aXt, and estimation of its long-memory parameter, d z . Then a test of significance for the difference d - d z . Large positive values of this difference as compared to cl, can be taken as evidence of the existence of a (maybe fractional) cointegrating relationship between :tit and Xt. A most investigated case corresponds to when the long-memory features in the variables are only due to unit roots (i.e. dx = d y = 1). This simplifies the procedure since no ,estimation of long-memory parameters is required in this case. However, a test for unit roots is needed to confirm this hypothesis. The second part of the test consists in checking for a unit root in the cointegration residuals ~t = :tt -

(\ q . "J J J l ' o Letting 6i'IJ) be an i.i.d. sequence with P(6~v) = 1) = p(6i ) = -1) = 1/2, we have -see where

J'

Gourieroux (1997[15])- that: E(u(V) t

I u(v) ) t-l

var(u(V)

I u(v) ) t-l

t

where

o

(49)

(50)

j

6i v ) 6ri v ) and hi v ) follows the GARCH(p, q) model above.

n)v) =

Now, if all the coefficients in this representation are non-negative, and if L{=1 a~/J) + Lj=1

b)v)


0, under linear cointeq'f"ation.

2. P[lim/l-+ oo (n - j) -1 L:~=J+1 ZtZt-j 2:: 0] < 1, V j > 0, under nonlincaT and nonco'lntegmtion.

PROOF: Suppose that there exists a nonzero real number a such that Et = Yt-aXt both Xt a.nd Yt will dominate over and r~Y) -

T?r)

= (a - 1) r~x)

Et

rv

1(0). Therefore

after a transient t > to. Thus dY~ = ar~2) ..

.

+ o(r~x), r~Y»).

+ o(r~x), r~Y»)

The latter is equal to (a =, 1) r~x), after t

,

> to for

some finite to, and the result in PR1(1) follows. Under non linear cointegration and non-cointegration one could write r~Y) ~ CtT~:r;), where defines

et

Ct

sequence possibly dependent on the original series. Then Zt = (Ct - 1) r~x) and n

P[ n-+oo lim (n - .j)-1

L

n

ZtZt-j 2:: 0]

t=j+1

P[ n-+oo lim (n - .7)-1

L

(Ct - C) (Ct-j - c) ~~ -(c - 1)2]

t=.i+1 n

< P[ lim (n - j)-1 n-+oo

L

(Cl - C)2 2:: -(c - 1)2]

=

1 (52)

I=j+l

and the reslllt in PR1(2) follows. 0

REMARKS: AS3 could be replaced by the weaker assumption that peUt > u) > P( Ut,j > u), Vu and Vj

> 0, Wh(Te Ut

= (Ct - 1)2 and Ut,j = (Ct - l)(ct-i - 1). With this conditioll, the sample

averages in AS3 need not converge at all.

32 11

EXRPY/EXRPM I EXRPY/EXRPD

11-1.196

I

EXRPM/EXRPD

I

1- 4 .889

1-5.249

Table 8: Values taken by the Dickey-Fuller test statistic

STR1/STR2

\-13.486 Trlj(;r;,

1

1

y) = N(jj - 1) on the two

pairs of foreign exchange rate series and the pair of stock return series. Here jj is the OLS estimator of the parameter in the regression of Vt on 11

EXRPY/EXRPM

11-4.649

1

EXRPY/EXRPD

1-7.005

1

:X:t.

EXRPM/EXRPD

STR1/STR2

I

1-7.460

1-34.522

1

1

= N(jj - 1) on the two

Table 9: Values taken by the Dickey-Fuller test statistic T(1r(1:, y)

pairs of foreign exchange rate series and the pair of stock return series. Here jj is the OLS estimator of the parameter in the regression of the series of ranks for for

4

j

Yt

on the series of ranks

,

Xt.

Experiment on monetary and financial' data

Some of the statistics proposed in the previous sections for testing cointegration were evalua ted on two pairs of exchange rate series (figure 11), and on a pair of stock return series (STR1,8TR2) from a Japanese food company (figure 10). The former group of series were the rates of exchange of the US Dollar (EXRPD), the Deutsch .\1ark (EXRPM) and the Japanese Yen (EXRPY) (in units of 100 yens) against the Spanish Peseta. We took the first n = 1000 daily observations from series starting at January the 1st. 1987. For the exchange-rate data, EXRPD was taken as the reference series.

First of all, we run an Augmented Dickey-Fuller (AD F) test (the conventional DF test was augmented with one lag in the first differences of the series) on the regression residuals of the three pairs of data sets considered above. If we denote by jj the OL8 estimator of the parameter in the regression of Yt on

Xt,

then the ADF test statistic is

T

= N(jj-1) and its val-

ues are shown in the tables 8 and 9, for the levels and for the ranks of the series, respectively.

33 Using the critical values given by Mackinnon (1990) [23] (-2.57, -1.94 and -1.62 at the 1%, 5% and 10% levels, respectively), the hypothesis of (linear) cointegration (i.e. that

T

takes values smaller than the tabulated critical values) is accepted in all cases except for the pair (EXRPY,EXRPM) when the test statistic is computed on the levels of the series, and in all cases when it is computed on their ranks.

Plots of the .iump series (first differences ofthe range series) are shown in figures 12,15,14 and 13 for the pairs (STR1,STR2), (EXRPY,EXRPM), (EXRPY,EXRPD) and (EXRPM,EXRPD) respectively.

"

As we pointed in a previous section, the relative way in which jumps Cluster along the horizontal axis in these plots is related to the likelihood of the cointegration hypothesis. Accordingly, the evidence of co integration is comparatively weak for the pair (EXRPY,EXRPM), since no translational relation between the two sets of arrival times transpires from the figures (the two jump series have almost no overlapping support). For the pairs (EXRPY,EXRPD) and (EXRPM,EXRPD), most of the jumps of one series are synchronous or close to synchronous with those of the other. However, at some time spells, .iumps appear for one series and not for the other, thus suggesting that there may be cointegration in a nonstationary or in a nonlinear way. Finally, for the pair (STR1,STR2) the jump arrival times for the series are more aligned, thus supporting the evidence of linear cointegration.

The values obtained with the range statistic

pt2 for n =

lOOO are given in table 10 below.

It shows that only for the pair (EXRPY,EXRPM) the hypothesis of cointcgration could

be easily rejected, while linear cointegration seems to be the most likely outcome for the remaining pairs of series, especially for the pair (STR1,STR2).

Finally, table 11 shows the values taken by our test statistic: R~~~ on the four pairs of series.

34 11

EXRPYjEXRPM

I

EXRPYjEXRPD

I

EXRPMjEXRPD

I

STRljSTR2

1

11

0.037

1

0.152

1

0.231

1

0.589

1

Table 10: Values taken by

pt2 on the four pairs of financial time series, for n =

1000.

All these values are consistent with the hypothesis of cointegration at the 10% significance level.

11

EXRPYjEXRPM

1

EXRPYjEXRPD

1

EXRPMjEXRPD

1

STRljSTR2

1

11

(J.()8009

1

0.07427

1

0.07427

1

0.09429

1

Table 11: Values taken by R~~2 on the four pairs of financial time series, for n = 1000. j

,

Thus the results obtained with the standard and the proposed testing procedures point to the same conclusion.

5

Conclusion

In this paper we have proposed using first differences of ranges for testing the hypothesis of cointegration in the bivariate time series case. The method is fully non parametric and postulates no model at all for the individual series. The plots of the sequences of first differences of ranges suggest a new definition of cointegration where the relevant feature is the simultaneity of the arrival times of significant informational events. That is, one could say that a pair of series are non-cointegrated when the sequences of first differences of ranges have orthogonal supports (i.e. they do not overlap). Comparison of the behaviour of the jump sequences obtained for each series led

lIS

to propose

two complementary testing procedures that when used in combination allow to discriminate between the alternatives of cointegration, independence, and comovements. The first testing

35 stage, based on a correlation measure for the jump series, is unable to find whether the series are integrated or not, but rejects the null hypothesis of independent random walks when there is a linear relationship between the series. The second testing stage, based on a ratio of counts, can solve most of the ambiguities of the former stage, and has a role similar to a unit-root testing device, with the advantage of not being bound to any particular model (in this case, a unit-root time series model). We have shown that the proposed statistics behave similarly to standard measures such as the ADF and RADF test statistics.

References j

,

[1] F.M. Aparicio. Nonlinear modelling and analysis under long-range dependence with an application to positive time series. PhD thesis, Swiss Federal 111stitute of Technology

(EPFL) -Signal Processing Laboratory-, Lausanne, Switzerland, 1995. [2] F.M. Aparicio and C.W.J. Granger. Nonlinear cointegration and some new tests for comovements. Working Paper (Dept. of Economics). University of California at San Diego, March 1995. [3] T. Bollerslev.

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[4] J. Brei tung and C. Gourieroux. Rank tests for unit roots. .Joumal of econometrics, 81:727. 1997. [5] P.K. Clark. A subordinated stochastic process model with finite variance for speculative prices. Econometrica, 41:135-156, 1973. [6] D.J. Daley and D. Vere-Jones. An introduction to the theory of point processes. SpringerVerlag, N.Y., 1988. [7] H.A. David. Order statistics. John Wiley, N.Y., 1981.

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(Dept. of Statistics and Econometrics). Universidad Carlos HI d~·Madrid, 1997.

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c.c.

Heyde. Martingale limit theory and

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38

4 3.5 3 2.5 2 0

100

200

300

400

500

600

700

800

900

600

700

800

900

1000

(a)

4.5 4 3.5 3 2.5

0

100

200

300

400

500 (b)

j

,

1000

• Figure 10: Two stock return series from the Japanese food comp~ny Ajinomoto.

,

140,-----------,

100,----------,

90L----~----~

o

500 EXRPD

1000

500 EXRPY

1000

72,-----------,

500 EXRPM

1000

Figure 11: Daily foreign exchange rate series from January 1987: EXRPD (Peseta/US Dollar), EXRPY (Peseta/lOO Yens), EXRPM (Peseta/Deutsch Mark).

39

0.2 0.15 0.1 0.05 0

0

100

200

300

500

400

600

700

800

900

1000

600

700

800

900

1000

(a)

0.15

0.1

0.05

0

0

100

200

300

400

500 (b)

• Figure 12: Jump series for the pair (STR1,STR2):

,

0.5

O~---L~~~---L--~

o

100

200

300

____

400

~~~

500

____

600

~

700

__

~

____WL__

800

900

~

1000

(a) 2,----,----,----,---,----,----,----,----,----,----.

1.5

0.5 O~~~

o

__I L U U l l L - L_ _

100

200

300

~

_ _ _ _~_ _~_ _ _ _~_ _~_ _~~~-D

400

500

600

700

800

900

1000

(b)

Figure 13: Jump series for the pair (EXRPM,EXRPD).

40

1.5 ,----,-----,.--,----,------,--,-----,-----r--..,------,

0.5

O~~LL_~L__~-ruLWL-~

o

100

200

300

400

__

500 (a)

~

_

600

_ L_ _

700

800

900

1000

2,--,---,--,------,-----r--..,------,---,--,---, 1.5

0.5 OW-~LL~~~~~_-L_~

o

100

200

300

400

_ _ _ ~_

500 (b)

600

_L_~L_~~~-U

700

800

900

1000

Figure 14: Jump series for the pair (EXRPY,EXRPD).

,

1.5,--,---,--,--,---,--,---,---,--,---,

0.5

O~~~-~~_L_~~~~

o

100

200

300

400

__

500 (a)

L __

600

_L~~

700

800

900

1000

1.5 ,------,---,---,----,------,--,-------,-----r--..,-------,

0.5

O___

o

_

~-W~~_L_

100

200

300

_

_L_~~_~

400

500

600

_L_~

700

____

800

~

900

__

~

1000

(b)

Figure 15: Jump series for the pair (EXRPY,EXRPM).