Testing for Cointegration in Markov Switching ErrorCorrection Models

Liang Hu�and Yongcheol Shiny

Leeds University Business School

Abstract

This paper proposes an e¢ cient test designed to have power against alternativeswhere the cointegrating regression error follows stationary MS regime switching dy-namics. We model an equilibrium process where its error correction adjustmentsare di¤erent in di¤erent regimes characterized by the hidden state Markov chainprocess. Using a general nonlinear MS ECM framework and following a pragmaticresidual-based procedure, we propose an optimal test for the null of no cointegrationagainst an alternative of a globally stationary MS cointegration. The Monte Carlostudies demonstrate that our proposed tests have good size and superior powerproperties compared to the linear tests. In an application to price-dividend rela-tionships, our test is able to �nd cointegration whereas linear based tests fail to doso.

JEL Classi�cation: C12, C13, C32.Key Words: Markov Switching Error Correction Models, Optimal Tests, Monte CarloSimulations, Price and dividend.

�Email: lh@lubs.leeds.ac.ukyEmail: ys@lubs.leeds.ac.uk

1

1 Introduction

Estimation of nonlinear error correction models subject to regime-switching dynamics hasrecently assumed great signi�cance. In the literature, most attention has fallen almostexclusively on the three types. Balke and Fomby (1997) popularise the three-regimethreshold error correction model (ECM), the case where an error correction process mayfollow a unit root in a middle regime whilst at the same time being globally geometricallyergodic in outer regimes. Another popular nonlinear schemes being applied are basedon the smooth transition regression ECM as in Michael, Nobay and Peel (1997), whichmakes the threshold ECM as a special case by allowing the transition from one regime toanother as a smooth function. Psaradakis, Sola and Spagnolo (2004) consider Markov-switching ECM in which deviations from the long-run equilibrium follow a process whichis nonstationary in one regime and mean-reverting in the other and the nature of theregime is governed by the hidden state Markov Chain. See also Krolzig (1997). All thesestudies demonstrate that the assumption of linear adjustment is likely to be too limitedin various economic situations particularly where transaction costs, policy interventionsand so on are present.However, most studies applying regime-switching ECMs have routinely adopted the

two-step testing approach popularised by Balke and Fomby (1997). The �rst step esti-mates the cointegrating parameters using the linear models whilst the second step testsfor the presence of a particular form of nonlinear asymmetry and estimates the associatednonlinear ECMs once cointegration has been established. Whilst such tests based on lin-ear models will have power against nonlinear alternatives, it seems far more sensible touse a test that is designed to have power against the alternative of interest. Two maindi¢ culties arise. First of all, the nuisance parameters characterising the nonlinearity arenot identi�ed under the null hypothesis, which renders the testing problem nonstandard.Second of all, the data generating process under the null hypothesis is nonstationary.There is now a signi�cant number of studies proposing a more e¢ cient procedure for test-ing for cointegration in nonlinear alternative frameworks. Hansen and Seo (2002) proposeto test linear VECM against a two-regime threshold VECM using supLM test to takecare of the unidenti�ed threshold parameters. They derive the asymptotic distributionfor the test statistics and suggest using bootstrap for critical values. Testing for smoothtransition Error Correction utilises the Taylor Expansion of the transition function andtransforms the nonstandard testing into t or F standard tests in auxiliary regression, seeKapetanios, Shin and Snell (2006).However, studies that directly address the similar issue in MS-ECMs are almost nonex-

istent even though these models have been popularly applied in the literature. For examplePsaradakis, Sola and Spagnolo (2004) conduct cointegration analysis with MS alternativein mind, they all adopt linear cointegration tests to establish the existence of cointegra-tion and only allow nonlinearity to enter the analysis at the estimation stage. See alsoSarno and Valente (2000), Krolzig et al. (2002) and Chaudhuri and Kumary (2006). Thissurprising absence is due to more complex nature of hidden Markov chain. Unlike theother two models, estimation of a MS-ECM model would normally be very complicated

[1]

as the sample path grows exponentially with time and one needs to estimate the regimeat each time.In this paper we propose to model an equilibrium process where its error correction

adjustments are di¤erent in di¤erent regimes characterized by the hidden state Markovchain process. Therefore, we focus on the case in which the error correction term followsa globally stationary process under the MS ECM alternative. In this regard we arealso able to allow for the empirically plausible case where the deviations from the long-run equilibrium may be allowed to be nonstationary locally in the state in which thecointegrating residuals are too small or irrelevant for the mean reversion behavior to takeplace. In particular, our approach is motivated by the implication of economic theoryapplied to asset arbitrage under noise trading and transaction costs, e.g. Campbell andKyle (1993).Using a general nonlinear MS ECM framework and following a pragmatic residual-

based procedure in the style of Engle and Granger (1987), we propose that a null hypoth-esis of no cointegration against an alternative of a globally stationary MS cointegrationbe tested directly by adopting an optimal testing procedure for the parameter constancyin a class of MS models proposed by Carrasco, Hu and Ploberger (2009, CHP there-after). This requires the model estimation only under the null and therefore facilitatesthe computation of the test statistics.The small sample performance of the suggested tests is compared to that of the linear

EG and Johansen (1995) tests via Monte Carlo experiments. We �nd that our proposednonlinear tests have good size and superior power properties compared to the linear tests.In particular, our proposed tests are superior to both linear or nonlinear EG tests whenthe regressors are weakly exogenous in a cointegrating regression. This supports similar�ndings made in linear models by Kremers, Ericsson and Dolado (1992), Hansen (1995)and Arranz and Escribano (2000).We provide an application to investigating the presence of cointegration of asset prices

and dividends for eleven stock portfolios (Germany, Belgium, Canada, Denmark, France,Ireland, Italy, Japan, Netherlands, United Kingdom, and United States) allowing fornonlinear adjustment to equilibrium driven by hidden markov chain. Interestingly, ournew test is able to reject the null of no cointegration in majority of cases, whereas thelinear EG test rejects only once. Given the strength of evidence in favor of MS ECMwe also estimate adjustment parameters under the alternative, and we �nd that theseestimates are well de�ned in all cases.The plan of the paper is as follows: Section 2 derives the nonlinear MS error correction

models. Section 3 discusses the stationarity issues. Section 4 develops the proposed teststatistics and derives their asymptotic distributions. Section 5 focuses on the optimal-ity issue and extension of models our test has power against. Section 6 evaluates thesmall sample performance of the proposed tests. Section 7 presents an empirical appli-cation to price and dividend relationships. Section 8 contains some concluding remarks.Mathematical proofs are collected in the appendix.

[2]

2 Markov Switching Error Correction Models

We start with the following linear vector error correction model (ECM) for the m � 1vector of I(1) stochastic processes, zt:

�zt = ��0zt�1 +

pXi=1

�i�zt�i + "t; t = 1; 2; :::; T; (2.1)

where �(m� r); � (m� 1) and �i (m�m) are parameter matrices with � and � of fullcolumn rank.In this paper we aim to analyse at most one conditional long-run cointegrating rela-

tionship between yt and xt, and focus on the conditional modelling of the scalar variableyt given the k-vector xt (k = m�1) and the past values of zt and Z0, where we decomposezt = (yt;x

0t)0. So we obtain the following conditional error correction model for �yt and

the marginal VAR model for �xt:1

�yt = �ut�1 + !0�xt +

pXi=1

0i�zt�i + et; (2.2)

�xt =

pXi=1

�xi�zt�i + "xt; (2.3)

(2.2) and (2.3) are the standard linear ECM, where the adjustment towards the long-run equilibrium is linear. But it is well documented in the literature that the adjustmenttowards long-run steady states could be asymmetric and nonlinear. A typical example isthe stock prices and dividends. It is noted that during some periods, the deviation of thestock prices from the fundamentals cannot be explained by the standard linear ECM (e.g.Shillier (1989) and Fama and French (2002)). Namely, the stock prices and dividendsare cointegrated I(1) processes, but at di¤erent time periods the speed of adjustmentcan be di¤erent. During some periods, the cointegration can even fail, that is, thereis no short-run dynamic adjustment towards the long-run relationship suggested by thefundamentals. Nonlinear ECM is proposed to solve this issue. Following Saikkonen (2004),a general nonlinear ECM takes the following form:

�yt = �ut�1 + g(ut�1) + !0�xt +

pXi=1

0i�zt�i + et; (2.4)

�xt =

pXi=1

�xi�zt�i + "xt

1In practice di¤erent lag orders for �yt and �xt can be selected in a data dependent way without lossof generality or change in the asymptotic analysis, e.g. Ng and Perron (1995). Once the MS cointegrationis detected, then it is also possible to allow for regime switching changes for other dynamic parameters.More discussion follows in the next section.

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We follow Saikkonen (2005) and assume:Assumption 1. (i) The errors (et; "xt)0 in (2.4) are iid(0;�), with � being an m �mpositive de�nite matrix, and E j"xtj` < 1 for some ` > 4 and et is independent to "xt.(ii) The distribution of (et; "xt)0 is absolutely continuous with respect to the Lebesguemeasure and has a density which is bounded away from zero on compact subsets of Rm.(iii) The initial observations Z0 � (z�p; :::; z0) are given. (iv) Let A (z) be given by(1� z) In � ��0z �

Ppi=1 �i (1� z) zi. If detA (z) = 0, then jzj > 1 or z = 1, where the

number of unit roots is equal to m� r. (v) g(�) is asymptotically no greater than a linearfunction of xt:The representation (2.4) makes economic sense in that many economic models predict

that the underlying system tends to display a dampened behavior towards an attractorwhen it is (su¢ ciently far) away from it, but shows some instability within the locality ofthat attractor.Under Assumption 1, Saikkonen (2004, Theorem 2) proves that there exists a choice

of initial values z�p; :::; z0 such that �zt and �0zt�1 are strictly stationary.

Examples of nonlinear ECM include Threshold ECM (as in Balk and Fomby (1997))and Smooth Transition ECM (as in Kapatanios et al (2006)). As noted in Psaradakis etal (2004), Markov Switching ECM can best capture sudden shocks to the economy (e.g.policy changes or �nancial crisis), we propose to use the MS ECM with one cointegrationrelationship in this paper. We make the following assumption:Assumption 2. (i) We assume the conditional VAR for �yt is of the following form:

�yt = �stut�1 + !0�xt +

pXi=1

0i�zt�i + et (2.5)

where st is a scalar geometric ergodic Markov chain with an n-dimensional state space.st has transition probabilities, pij = Pr(st = j j st�1 = i) for i; j = 1; :::; n. (ii) There isno cointegration among the k-vector of I(1) variables, xt.Assumption 1 and 2 imply that the process xt are weakly exogenous and certain

stationarity properties are achieved. We will discuss more about stationarity in the nextsection.Assume that st follow the two-state latent random variables de�ned on f0; 1g, then

we may write �st as�st = �0 + (�1 � �0) st; �0 = 0; �1 < 0;

which is the special case considered by Psaradakis et al. (2004). The importance ofdeveloping the direct testing procedure for the null hypothesis of no cointegration againstthe alternative of cointegration with MS ECM adjustment was clearly acknowledged andhighlighted, but no attempt has been made to develop such tests. Instead the (ine¢ cient)two-step testing and estimation approach has been adopted, which may be due to a fewtechnical complexities involved in the estimation of the alternative model.In this paper we explicitly adopt the optimal testing procedure by CHP and set the

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null and the alternative hypotheses respectively as2

H0 : �st = 0 vs H1 : �st = �t; (2.6)

where �t is a random variable of parameter changes under the alternative,

�t = cst;

where c is a scalar specifying the amplitude of the change and c2 = V ar (�t). Moreover,using that Corr (st; ss) = �jt�sj, where � denotes the autocorrelation coe¢ cient with�1 < � � � � �� < 1, we have that Cov (�t; �s) = c2�jt�sj. Actually our test statistic onlydepends on the covariance structure of the Markov Chain. This is one of the advantagesof the proposed test. It does not require fully speci�ed structure of the Markov Chain yetthe serial correlation is fully taken account of. Moreover, robustness to misspeci�cation isalso possible. Therefore, we de�ne the nuisance parameter vector, � = (c;�)0. Note that� is not identi�ed under the null hypothesis.

3 Global and local stationarity

Francq and Zakoïan (2001) point out that stationarity within each regime is neithersu¢ cient nor necessary for the strict stationarity or second-order stationarity of MS au-toregressive processes. As a consequence, imposing stationarity constraints within eachregime does not imply global stationarity.

3.1 Strict stationarity

The strict stationarity of MS ECM models can be proved along Bougerol and Picard(1992). As shown in Francq and Zakoïan (2001), it involves rewriting the process as anVector Autoregressive process with order 1. Then the strictly stationarity follows if thetop Lyapunov exponent associated with the AR coe¢ cients is strictly negative.To show the strict stationarity of our process (2.2), we can rewrite the model as follows:�

ytxt

�=

� �1 + �st

����st

0 Ik

���yt�1xt�1

�+

pXi=1

�0i�zt�i +

�et + !

0"t"t

�

= �st ��yt�1xt�1

�+

pXi=1

�0i�zt�i +

�et + !

0"t"t

�The top Lyapunov exponent is then

= inf E1

tlog �st ��st�1 ::: ��s0

= inf E

1

tlog���1 + �st� � �1 + �st�1� ::: � �1 + �s1���

= E log��1 + �st��

2See also a companion paper by Hu (2008) for testing for stochastic unit roots in the MS regressionframework.

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Therefore, a su¢ cient condition for strict stationarity is that

nXi=1

�i log j1 + �ij < 0: (3.7)

where �i is the ergodic probability for regime i:We could see that the existence of explosive regime (j�ij > 1) does not preclude strict

stationarity.

3.2 Second-order stationarity

Second-order stationarity for Markov Switching Autoressive models have been addressedin Yao and Attali (1998) and Francq and Zakoïan (2001) among others. Following Francqand Zakoïan (2001), we �nd that the (global) stationarity condition for (2.2) is given by

�(M) < 1; (3.8)

where

M =

26664p11(1 + �1)

2 p21(1 + �1)2 ::: pn1(1 + �1)

2

p12(1 + �2)2 p22(1 + �2)

2 ::: pn2(1 + �2)2

......

...p1n(1 + �n)

2 p2n(1 + �n)2 ::: pnn(1 + �n)

2

37775 ;and �(M) refers to spectral radius associated with M .3

We then assume:Assumption 3. (i) � belongs to a compact set, �. (ii) The conditions (3.7) and (3.8)holds under the alternative.

4 Test Statistics

In this section we will develop the testing procedure for the null of no cointegration againstthe alternative of MS cointegration. First of all we need to address the important technicalissue that both the cointegrating parameters, �x and the regime switching parameter, stare not identi�ed under the null of no cointegration. To this end we follow Engle andGranger (1987) and Kapetanios et al. (2006) and take a pragmatic residual-based twostep approach. In the �rst stage, we obtain the residuals, ut = yt � �

0xxt from (??) with

�x being the OLS estimate of �x. Then in the second stage, in order to overcome theDavies problem that st is not identi�ed under the null, we follow the testing procedureadvanced by CHP. This procedure only needs to estimate the model under the null andrequires to evaluate the summary statistic over a compact subset of �. Combining these

3The spectral radius of the matrix M is de�ned as �(M) = max�2E

j�j, where E is the set of eigenvalues

of M .

[6]

two steps we will propose a number of operational versions of the cointegration test underthe nonlinear MS ECM framework given by (2.2).Since the parameters of interest are the error correction coe¢ cients, �st, we directly

work with the concentrated log-likelihood function of (2.2) under the null given by

`T (�) _ �T

2ln�2��2e

�� 1

2�2e(�y � �u�1)

0MT (�y � �u�1) ; (4.1)

where �y = (�y1; :::;�yT )0, u�1 = (u0; :::; uT�1)

0, MT = IT �W(W0W)�1W0, W =(w1; :::;wT ), wt = (�xt;�zt�1; :::;�zt�p) and � = (�; �2e)

0 with �st = � = 0 under thenull. Then, given �, our proposed test bstatistic can be derived as

ST (�) = ST

��; �

�=

1

2pT 3

TXt=1

�t

��; �

�� 1

8T 2

TXt=1

�2t

��; �

�; (4.2)

where

�t

��; �

�=�`t(2) + `2t(1)

�V ar (�t) + 2

t�1Xs=1

`t(1)`s(1)Cov (�t; �s) ; (4.3)

where `t(1) and `t(2) are �rst and second derivative of `t (�) with respect to �, and � is themaximum likelihood estimator of � obtained under H0.In the general case where nuisance parameters, � = (c;�)0, are unknown, the test

procedure will su¤er from the Davies (1987) problem since ��s are not identi�ed underthe null. Most solutions are achieved by constructing the summary statistics over a gridset of �. Following Andrews and Ploberger (1994) and Hansen (1996), we consider thesupremum and the exponential average of the statistic de�ned respectively by

Sexp =

Z��

exp (ST (�)) dJ (�) and Ssup = sup�2��

ST (�) ; (4.4)

where J (�) is some prior distribution for � with support �� on a compact subset of� = fc2; � : c2 > 0; � < � < ��g with �1 < � < �� < 1.4

Theorem 1 Let Assumptions 1-3 hold. Then, Ssup and Sexp statistics de�ned by (4.4)are admissible.

Remark: Testing for Markov Switching is di¤erent to testing for other regime switch-ing models like structural change or Threshold. The crucial problem is that the scorefunction is identically zero, therefore information matrix is singular under H0; namely,P (stjy1; :::; yT ) is a constant. See Hamilton (1989). As a result, general theory breaksdown here.

4To avoid the case where the test statistic grows without bound, we need to rule out the case that �is on the boundary of the parameter space.

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We now describe the proposed testing procedure using the models, (2.2) and (2.3) indetails. To this end we consider the following concentrated regression of (2.2):

�~yt = �st~ut�1 + ~et; (4.5)

where �~yt and ~ut�1 are the least squares residuals obtained from the regression of �yt andut�1 on (�xt;�zt�1; :::;�zt�p), respectively. Then, the �rst and the second derivatives of(concentrated) log likelihood function with respect to �st = � under the null are derivedas follows:

`t(1) =1

~�2e~ut�1~et; `t(2) = �

1

~�2e~u2t�1;

where ~�2e is the ML estimator of �2e obtained under the null. Then, we have �t =�

c2=~�2e�~�t, where

~�t =1

~�2e

�~e2t � ~�2e

�~u2t�1 + 2

t�1Xs=1

�t�s~et~�e

~es~�e~ut�1~us�1:

For given � = (c; �), the test statistic, ST (�) can be written as

ST (�) =c2=~�2e

2pT 3

TXt=1

~�t �c4=~�4e8T 2

TXt=1

~�2t :

For the Ssup statistic we can �nd an analytic solution form when maximizing ST (�) withrespect to c2, which is given by

Ssup = supf�:�<�<��g

1

2

0@max0@ PT

t=1 ~�tqPTt=1 ~�

2t

; 0

1A1A2

: (4.6)

The Ssup statistic can now be easily calculated since we need a grid search only over theset,

�� : � < � < ��

.

Theorem 2 Let Assumptions 1-3 hold. Then, under the null hypothesis of no cointegra-tion, the Ssup statistic, de�ned by (4.6), has the following asymptotic distribution:

Ssup ) supf�:�<�<��g

1

2

24max0@R 10 B (a)2 dW (a)qR 1

0B (a)4 da

; 0

1A352 ; (4.7)

where �)�denotes a weak convergence and W (a) is a standard Brownian motion de�nedon a 2 [0; 1], B (a) =W (a)�Wx (a)

0�R 1

0Wx (a)Wx (a)

0 da��1 R 1

0Wx (a)W (a) da, and

Wx (a) are k-vector standard Brownian motions independent of W (a).

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Remark: The limiting distribution is pivotal for some simple models. For more generalmodels as represented by (2.2), the limiting distribution is model dependent. Nevertheless,we could tabulate the critical values through Monte Carlo simulation. Refer to e.g. Hansen(1996).To accommodate deterministic components in the cointegrating regression, we extend

to consider the regression with an intercept

yt = a0 + �0xt + ut; (4.8)

and the regression with an intercept and a linear deterministic time trend,

yt = a0 + a1t+ �0xt + ut: (4.9)

The respective test statistics are then obtained as follows: First, the appropriate residualsare obtained from (4.8) or (4.9), and then the corresponding MS ECM regressions areconstructed by

�y�t = �stu�t�1 + !

0�x�t +

p�1Xi=1

0i�z�t�i + error; (4.10)

�y+t = �stu+t�1 + !

0�x+t +

p�1Xi=1

0i�z+t�i + error; (4.11)

where u�t = y�t � �0x�t , and u+t = y+t � �0x+t and superscripts ���and �+� indicate thedemeaned data and the demeaned and detrended data, respectively. The appropriateSsup statistics are then obtained from (4.10) or (4.11), respectively following exactly thesame procedure described above.5

5 Optimality and generalised models

The sequence of local alternatives is given by �st =1

4pT 3�t: That is, the local alternative

is of order T�3=4: Notice that since nuisance parameters are not identi�ed under H0; wedo not have point optimal test in this context. The admissibility implies that the test willbecome optimal in the sense that it is asymptotically equivalent to the likelihood-ratiotest statistic for the same sequence of alternatives. See CHP and Hu (2008).Notice that by far we only introduce markov switching on the speed of adjustment to

the long-run equilibrium, namely �st : There is reason to believe that at di¤erent regimethe process can have di¤erent behaviours in terms of other short-run dynamics or evenvolatility. Especially in �nance, it is very important to model volatility associated witheach regime since it is the measurement of risk at di¤erent state.

5Asymptotic (pivotal) critical values of the Ssup statistics for the above three cases can be easilyevaluated following Theorem 2 too.

[9]

Therefore we should allow for the generalised model as:

�yt = �stut�1 + !0st�xt +

pXi=1

0i;st�zt�i + �stet: (5.12)

That is, we allow the alternative model to include possible Markov Switching on allparameters while the null hypothesis remains the same as before.We �nd that this does not alter the optimality results asymptotically. Under H1; the

ML estimator of the long-run parameter � is T -consistent while all the other estimatedparameters are

pT -consistent. Therefore, the T�3=4 neighborhood for our test remains

unaltered asympotically when the model under H1 is (5.12).Remark: In CHP, it is established that for stationary processes with parameter constancyunder the null hypothesis, the right local alternative for Markov Switching is of orderT�1=4: That is, the likelihood ratio of an alternative in the T�1=4 neighborhood will givethe nondegenerate distribution of the likelihood ratio. However, in the ECM setting, theorder of the local alternative is T�3=4, which is di¤erent to the Threshold ECM, where theorder is of T�1:

6 Monte Carlo Studies

We undertake a small-scale Monte Carlo investigation of the �nite sample size and powerperformance of our proposed sup type test statistics in conjunction with the linear coin-tegration tests of Engle and Granger, denoted as Ssup and teg respectively. To this end weconsider experiments based on a bivariate ECM similar to that adopted by Arranz andEscribano (2000) and Kapatenios et al. (2006), and generate the data as follows:

�yt = �stut�1 + ��xt + "t;

�xt = vt;

ut = yt � �xxt;�"tvt

�� iidN

�0;

��21 00 �22

��:

Here we �x �x = 1 and �21 = 1, �22 = 4. In order to investigate the impact of thecommon factor (COMFAC) restriction, � = 1, we consider di¤erent parameter values for� = f0:5; 1g. Under the null, �st = � = 0.To compute the Ssup test statistics over the grid set, f� : � < � < ��g as described

in Section 3, we use 40 draws for � from an equi-spaced grid over the interval��; ���=

(0; 0:98)6.In this simple case, the limiting distribution of our test statistic is actually pivotal.

Asymptotic critical values is tabulated in Table 1 via stochastic simulation with T = 1; 000

6We �nd thatan increase in the number of draws does not a¤ect the results signi�cantly.

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and 50; 000 replications. For demeaned and detrended cases, we replace in the limitingdistribution the demeaned and detrended Brownian Motions.

Table 1 about here

Now we set the sample size to 100. We use 1000 replications to calculate the empiricalcritical values and then evaluate the size-corrected powers with 1000 iterations. We alsoconsider the results for the case with no intercept and no trend, the demeaned case andthe detrended case, respectively.We consider the two experiments under the alternative. Experiment 1a considers the

case with two regimes in which we let the Markov chain, st, take binary values withtransition probabilities given by pii = Pr (st = ijst�1 = i), i = 1; 2. For conveniencewe let st = 0 correspond to the null regime where �0 = 0, and st = 1 correspondto the other regime where �1 < 0. We consider the di¤erent values for (�0; �1) =f(0;�0:1) ; (0;�0:2) ; (0;�0:4)g, and in each case we consider the di¤erent values for(p00; p11) = f(0:98; 0:98) ; (0:9; 0:9) ; (0:9; 0:98) ; (0:98; 0:9)g. Expecting that our proposedtest statistic will be more powerful against the alternative of two stationary MS regimeswe also consider (�0; �1) = (�0:1;�0:3).Table 2 summarises the results for Experiment 1a. A close look tells that the powers

of the tests depend on error correction parameters as well as the transition probability.The power increases monotonically not only with the distance from the null, measured bythe magnitude of �1, but also with p11 that measures the time when the process spendsin the regime di¤erent from the null. Moreover, when the regimes are less absorbing,implying that the transition from one regime to another is more frequent, the powerincreases. It is also interesting to notice that the EG test performs reasonably well onlywhen the common factor restriction holds exactly (see Panel A), a �nding consistent withthe linear literature, e.g. Kremers, Ericsson and Dolado (1992). In a more general casewhere such restrictions are invalid, however, our proposed test performs much better thanthe EG test (see Panel B). Power performance for both demeaned and detrended dataare somewhat worse than the previous case with no deterministic component as is alsoconsistent with the �ndings in the linear models. But we have qualitatively similar �ndingthat our proposed tests signi�cantly out-perform the EG test unless the common factorrestriction holds exactly, a condition which is most likely to be violated in practice.

Table 2 about here

In Experiment 1b, we consider the case where all the parameters are subject to MarkovSwitching mechanism and generate the data as follows:

�yt = �stut�1 + �st�xt + "st ; (6.1)

�xt = vt; (6.2)

ut = yt � �xxt; (6.3)�"tvt

�� iidN

�0;

��2st 00 �2v

��: (6.4)

[11]

Here we �x �x = 1 and �2v = 4. Under the null, �st = � = 0, �st = � = 0:5; �2st = �2 = 1:

Again, the Markov chain st take binary values with transition probabilities givenby pii = Pr (st = ijst�1 = i), i = 1; 2. But now st = 0 correspond to the null regimewhere �0 = 0, �0 = 0:5; �20 = 4; and st = 1 correspond to the other regime where�1 < 0; �1 = 0:1; �

21 = 1. Namely, regime 0 corresponds to no cointegration with relatively

big variance, while regime 1 corresponds to cointegration with smaller variance.Again we consider the same sets for (�0; �1) with di¤erent combination of transition

probabilities.The results are listed in Table 3.

Table 3 about here.

Compare Panel A of Table 2 with Table 3, we could see that if the data are actually gen-erated with switching on all parameters, our test has better small-sample performances,which is a favorable property.Next we consider Experiment 2 with three regimes, st = f0; 1; 2g and with transi-

tion probabilities given by pii = Pr (st = ijst�1 = i), i = 0; 1; 2. We let st = 0 cor-respond to the null regime where �0 = 0 whilst st = 1 and 2 corresponds to theother regimes where �i < 0, i = 1; 2. We consider the di¤erent parameter values for(�0; �1; �2) = f(0;�0:1;�0:2) ; (0;�0:1;�0:4)g and consider two di¤erent transition prob-ability matrices for each case. The �rst transition probabilities are given by

P1 =

24 0:9 0:05 0:450:05 0:9 0:450:05 0:05 0:1

35 ;where the associated stationary probabilities are [ 0:474 0:474 0:052 ], implying thatthe simulated process will stay in the �rst two regimes with equal probability of 47.4% andit will stay in the third regime with probability of 5.2%. The other transition probabilityis:

P2 =

24 0:253 0:023 0:1460:232 0:973 0:7350:515 0:004 0:119

35 ;and the associated stationary is [ 0:034 0:942 0:024 ].

Table 4 about here

Table 4 provides the simulation results for Experiment 2 and con�rms that the current�ndings are qualitatively similar to the results obtained in Table 2. Overall we mayconclude that the power performance of our proposed test is quite satisfactory.

[12]

7 Empirical Application to Asset Pricing

Campbell and Shiller(1987) investigate the existence of linear cointegration between aggre-gate US stock prices and US dividends implied by a simple equilibrium model of constantexpected asset returns. Null hypothesis of no cointegration was marginally rejected intheir data, but the implied estimate of long-run asset returns was implausible. Impos-ing a more credible long-run return caused non-rejection of the null of no cointegration.Mixed results were found in the literature that follows, e.g. Campbell and Shiller(1988),Froot and Obstfeld (1991), and Cuthbertson, Hayes, and Nitzsche (1997) among others.In this section we apply our proposed tests for cointegration to asset prices and divi-

dends for 11 stock portfolios allowing for nonlinear adjustment to equilibrium in the formof MS ECM. Nonlinear adjustment is well acknowledged in the literature in analysingasset prices and dividends and di¤erent models have been proposed, e.g. Theshold ECmodel by Balke and Fomby (1993) and Smooth Transition EC models by Kapetanios et al(2006). But MS ECM is best suited to situations where changes in regime are triggered bya sudden shock to the economy, which might not be well captured by smooth transition orthreshold models. The deviation of stock prices from the underlying fundamentals couldbe explained by time varying discount factor or intrinsic bubbles. Psaradakis et al (2004)�nd a two state MS ECM can capture very well, in US stock prices and dividends, theadjustments towards long-run equilibrium occuring all the time but at di¤erent rate, oreven taking place in one state of the nature. In all of the studies, they use the standardtwo-step testing approach. Now we apply our tests �rst and then estimate the MS ECMspeci�cation if it�s supported by the test results.

7.1 Data Description

We collect monthly data from January 1974 to December 2006. Data are start periodnominal prices and dividends. Eleven countries are considered: Belgium, Canada, Den-mark, France, Germany, Ireland, Italy, Japan, Netherlands, United Kingdom and UnitedStates. Plots of the data and price-dividend ratio are available. For most of the series,it�s clear that prices and dividends are upward trending and in some time periods thebehavior of prices does not re�ect the behavior of dividends. The price-dividend ratiotakes values well above its sample average value for a long period of time. There are someexceptions. We�ll get back to it again later.

7.2 Testing

The error correction term is obtained from

pt = �+ �dt + ut (7.1)

And then a MS-ECM speci�cation is:

�pt = �stut�1 + !�dt + "t

[13]

We want to compare our Ssup test with Engle-Granger test. To compute the Ssup test,we rely on bootstrap to tabulate the empirical critical values with 1000 iterations. Asdescribed in Section 3, we use 40 draws for � from an equi-spaced grid over the interval[0; 0:98] when computing the test statistic and the critical values. tEG is compared withempirical critical values with 1000 iterations too. The test results are as follows:

Table 5 about here

As we can, Engle-Granger cannot reject the null of no cointegration in majority caseswith the only exception of United Kingdom where the rejection is at 5% level.Our Ssup test rejects the null of no cointegration in four out of eleven countries at

1% level and three more countries at 5% level, which strongly suggests that ignoringthe nonlinear adjustment to equilibrium may lead to false conclusion of nonexistence ofcointegration between price and dividends.

7.3 Estimation

Now we consider a two-step estimation procedure for those countries that we reject thenull of no cointegration and favor the alternative of MS ECM.In step 1 we run the linear cointegration regression (7.1), and save the residual ut:

Estimation results for the linear ECM model as follows are also given:

�pt = a+ �ut�1 + !�dt + "t

In step 2, we run the following MS ECM regression for both 2 regimes and 3 regimes:

�pt = ast + �stut�1 + !st�dt + "st

where "st � N(0; �2st):For 2 regime models, 10 parameters have to be determined (including the transition

probabilities p11 and p22):For 3 regime models, 18 parameters have to determined (including 6 parameters for

transition probabilities).To determine the speci�cation, we suggest to use either AIC or BIC.The estimation results of four most representative countries are as follows:

Table 6 about here

Analysis of Estimation resultsUnited StatesFor US data, t�s clear that prices and dividends are upward trending most of the time.

But in some time periods (speci�cally from Millennium till early 2002) the behavior ofprices does not quite re�ect the behavior of dividends. The price-dividend ratio takesvalues well above its sample average value for a long period of time. Especially from early

[14]

1995 till September 2000, the price-dividends ratios has this clear upward trends accom-panied by higher volatility. From September 2000 the ratio is decreasing dramatically tillit hits the lowest record at April 2003. High volatility is observed during this period too.Both AIC and SBC give overwhelming support for a three-regime MS ECM model.

The estimation results show that around 60% of the data stay in regime 1, and it�scharacterized by a relatively small variance and a slow adjustment rate. In fact the ErrorCorrection coe¢ cient is insigni�cant from 0. As we could see, the dotcom bubble periodis classi�ed into this regime. It basically shows that the two markets are less bounded andhave di¤erent behaviors. Regime 2 captures large variance and slow adjustment rates. Infact Regime 2 well identi�es the 1987 crash. Regime 3 is the stationary regime whichcorresponds to very small variances. 36% of the data lie in regime 3.

United KingdomEngle-Granger test rejects the null of no cointegration only for UK. As we look at

the price and dividend, they are bounded together with upward trends most of the time.During the dotcom bubble period, we observe big decreases in prices accompanied byrelatively �at dividends.Again AIC and SBC favors a three regime model. In all the regimes, the EC coe¢ cient

is signi�cant. Namely, the process is stationary in all three regimes. Regime 1 correspondsto smallest variance and fair adjustment rate. 28% of data lie in Regime 1. Regime 2corresponds to a big adjustment and a very large variance. 1987 big crash is identi�ed inthis regime. 64% of data lie in Regime 3, where the adjustment is modest as well as thevariance.

JapanJapan is di¤erent in pattern in the sense that dividends seem to vary around a �xed

level up to the end of 2003 accompanied by huge volatilities in prices. Moreover, pricesand dividends are less bounded together compared with other countries.We use a three-regime model following AIC and SBC. Regime 1 is dominating with

64% of data. It has a negative (insigni�cant) adjustment rate with a modest variance.Regime 2 corresponds to a slightly explosive process with very small variance. Yet theadjustment rate is still insigni�cant. Regime 3 corresponds to a very negative adjustmentspeed with highest variance. This regime could identify those big crashes in the history.Only 5.8% of data lie in this regime.

ItalyFirst regime corresponds to a slightly explosive process but with a small variance. 17%

of data lie in this regime. Second regime corresponds to a very negative adjustment rateand a large variance. This regime coincides with big crashes. Majority of data (76%) liein the third regime. No cointegration is found in this regime and it�s accompanied by amodest variance.

[15]

7.4 Dynamic causal e¤ect

It is interesting to look at the policy implications from our MS ECM model. Most ofthe literatures focus on the impulse response analysis. See Ehrmann, Ellison and Valla(2003), Krolzig (2006) and Kim (2006) among others. We propose to look at this issuefrom a more straightforward respective, namely, the dynamic causal e¤ects of dividendson the price.we could rewrite it as the following form for pt :

pt = a�st + (1 + �st)pt�1 + !stdt � (�st � + !st)dt�1 + �st"t

Using lag operators, we obtain�1� (1 + �st)L

�pt = a�st + !st

�1�

��st!st

� + 1

�L

�dt + �st"t

We denote

�(L) = 1� (1 + �st)L

�(L) = 1���st!st

� + 1

�L

We can see that the lag polynomials are state dependent.So we have

�(L)pt = a�st + !st�(L)dt + �st"t:

Premultiplying by �(L)�1 to both hand sides, we get

pt =a�st�(1)

+ !st�(L)

�(L)dt +

�st�(L)

"t:

Denote (L) = �(L)�(L)

; we have the dynamic multiplier e¤ects of dt on pt to be

m(h) =hXj=1

@pt+j@dt

=hXj=1

!st+j�(L)

�(L):

We report the price and dividends, dynamic causal e¤ect and smoothes probabilityfor each regime plots for US and UK data in this paper.We can see that for US data, in regime 1, dividends has persisitent positive e¤ects on

the level of price and it keeps increasing. Regime 2 is assciated with high volatility andwe could see that the dynamic causal e¤ect plummets in the �rst three periods. Then itstarts to incease with a fast speed. In the mean-reverting regime with least volatility, thedynamic causal e¤ect is slightly negative in the early periods but soon grows with time.In about 80 periods, the dynamic causal e¤ects in all three regimes approaches the longrun equilibrium.We observe similar patterns in the UK dynamic causal e¤ect plot but they converge

to the long run equilibrium much faster than the US data.In summary, the dynamic causal e¤ect is very di¤erent in di¤erent regimes.

[16]

8 Concluding Remarks

Empirical analysis of cointegration and the associated error correction model has been anintegral part of time series econometrics. However, the emphasis of the earlier literaturewas on the examination of the linear model, implicitly disregarding any possible dynamicnonlinearities and/or asymmetries under consideration. This paper complements otherrecent studies (e.g. Hansen and Seo 2002, Kapetanios et al., 2006) in trying to �ll thisvacuum.Its main contribution has been to develop a new cointegration test statistic designed

to be more powerful against a stationary MS ECM processes than the linear Engle andGranger (1987) and Johansen (1995) tests, which has been routinely applied in the �rststage of the so-called two step approach. As acknowledged in the literature the develop-ment of such a testing procedure was deemed to be complicated due to the estimationcomplexities under the alternative of MS models. We overcome this complexity and pro-vide a simple operational test by adopting the optimal testing procedure recently advancedby CHP, which only requires the model estimation under the null.We show that our proposed test is asymptotically equivalent to a likelihood ratio test,

and derive its limiting distribution. Monte Carlo simulations and an empirical applicationdemonstrate that the suggested testing procedure may be quite useful in practice. As isalways the case when working with nonlinear models there may be several generalisations.More importantly, it could be extended to establish the existence of possibly multiplecointegrating equilibrium relationships in the nonlinear system vector error correctionframework.

[17]

9 Appendix: Mathematical Proofs

9.1 Proof of Theorem 1

The null hypothesis is given byH0 : �t = 0;

and the sequence of local alternatives is given by

H1T : �t =1

4pT 3�t:

Notice the alternative is of order T�3=4; which is di¤erent from the usual T�1 order usedin TAR framework. This is due to the fact that the information matrix is singular underthe null hypothesis and we have to estimate the parameters at each regime. So the rateof convergence is slower in the Markov Switching case.Consider

�T =1

2pT 3

Xt

�t

��; �

�;

where

�t

��; ��=�`t(2) + `2t(1)

�V ar (�t) + 2

t�1Xs=1

`t(1)`s(1)Cov (�t; �s) :

Notice that �t��; ��is a martingale di¤erence sequence and thus Mt =

tPs=1

�s

��; ��is a

martingale.We now derive some preliminary results. First, we de�ne

QT =Xt

E

��t

��; �

�2j Ft�1

�where Ft�1 is the �-algebra up to time t � 1. Notice that QT is the quadratic variationassociated withMt de�ned below, e.g. Hall and Heyde (1980, p.54). Moreover, the follow-ing proposition explicitly provides an important property for martingales, see Karatzasand Shreve (1991, p.32):

Proposition 3 Mt is a continuous and square-integrable martingale. For a sequence ofrandom partitions � of [0; T ] with 0 = t0 � t1 � ::: � tm = T , de�ne the second variationof M over the partition � to be

V 2t (�) =

mXk=1

��Mtk �Mtk�1

��2 :then we have

plimk�k!0

V 2t (�) = hMit

where hMit is the quadratic variation of the process Mt.

[18]

Proposition 3 simply says that summing up squared di¤erences forMt at every possiblespot time then provides a consistent estimator for hMit7.Let Q�T denote the joint distribution of (�1; :::; �T ), indexed by the unknown parameter

�. Let P�;� be the probability measure on y1; y2; :::; yT corresponding to H1T , and P� bethe probability measure on y1; y2; :::; yT corresponding to H0: The ratio of the densitiesunder H0 and H1T is given by

`T �dP�;�dP�

=

Z TYt=1

ft��t=T

3=4�dQ�T=

TYt=1

ft (�0) :

where ft (�) are likelihood functions.By the Neyman-Pearson Lemma, a test based on `T is a best test of a given signi�cance

level for testing the simple null hypothesis thatQTt=1 ft (�0) is the true density versus the

simple alternative thatR QT

t=1 ft��t=T

3=4�dQ�T is true. In addition, a test base on `T has

the best weighted average power for weight function Q�T of all tests of a given signi�cancelevel for testing the simple null hypothesis that

QTt=1 ft (�0) is the true density versus the

alternative thatQTt=1 ft

��t=T

3=4�is true for some � 2 ��: We de�ne the likelihood ratio

of simple nullQTt=1 ft (�0) and simple alternative

QTt=1 ft

��t=T

3=4�for some given � 2 ��

L�T =

QTt=1 ft

��t=T

3=4�QT

t=1 ft (�0)= exp

TXt=1

l�t

!where

l�t = lt��t=T

3=4�� lt (�0) ;

is the log likelihood function de�ned as before.Now we establish the optimality of our tests. To this end we need to show our test is

asymptotically equivalent to the Neyman-Pearson test, or we could express it in terms ofexpectations as follows:

E�

"exp

TXt=1

l�t � ST (�)

!#= 1

where E� denotes the conditional expectation with respect to (�1; :::; �T ) and ST (�) isour test statistic process as de�ned in (4.2).We have the following theorem from Hu (2008).

Theorem 4 we have under H0,

L�T= exp

1

2T 3=2

TXt=1

�t �1

8T 2E��2t j Ft�1

�! P! 1: (9.1)

7For example, if we assume that the asset price is free of arbitrage, then the quadratic variationassociated with the return measures the realized sample-path variation of squared return processes.Proposition (3) suggests that we may approximate the quadratic variation by cumulating products of(high-frequency) returns.

[19]

where the convergence in probability is uniform over �. Moreover, we have P�;� is con-tiguous with respect to P�:

This gives the asymptotic equivalence between the likelihood ratio and the proposedtest statistic in (9.1). Then by Neyman-Pearson Lemma, the test above statistic is op-timal. Notice that our test statistic in (4.2) is obtained by plugging in a consistentestimator of E (�2t j Ft�1), which does not a¤ect the asymptotic admissibility of the test.This completes the proof of Theorem 1.

9.2 Proof of Theorem 2

To �nd the limiting distribution of th Ssup statistic, we need to �nd a limiting behavior of~�t which is a Martingale di¤erence sequence indexed by nuisance parameter � with valuesin the space of continuous functions de�ned on a compact subset. We now consider

1pT 3

TXt=1

~�t =1pT

TXt=1

1

~�2e

"1

~�2e

�~e2t � ~�2e

�� ~ut�1pT

�2+ 2

Xs<t

�t�s~et~�e

~es~�e

~ut�1pT

~us�1pT

#: (9.2)

Under H0, ~�2e = Op(pT ) and then it is easily seen from functional CLT that that as

T !1,

T�1=2[Ta]Xt=1

1

~�2e

�~e2t � ~�2e

�)p2W (a) ; a 2 [0; 1] ;

whereW (a) is the scalar standard Brownian motion. Moreover, it is also well-established(see Kapetanios et al., 2006) that

T�1=2~ut�1 ) �uB (a) ; a 2 [0; 1]

where �u = �e=�(1), B (a) =W (a)�Wx (a)0�R 1

0Wx (a)Wx (a)

0 da��1 R 1

0Wx (a)W (a) da

andWx (a) are k-vector standard Brownian motions independent of W (a). Since ~et and~ut�1 are asymptotically independent, we now have under H0, as T !1,

1pT

TXt=1

"1

~�2e

�~e2t � ~�2e

�� ~ut�1pT

�2#) �2u

Z 1

0

B(a)2dW (a) ; a 2 [0; 1] : (9.3)

Next, we examine the second part of (9.2). First, under H0, as T !1, it is straight-forward to show (see Theorem 1 of Andrews and Ploberger, 1996):

1pT

TXt=1

~et~�e

Xk<t

�t�k~ek~�e)

1Xi=0

�i� i =1p1� �2

W (1) ;

where � i � iidN (0; 1) and W (1) � N (0; 1). Hence, under H0, as T !1,

1pT

[Ta]Xt=1

~et~�e

Xk<t

�t�k~ek~�e) 1p

1� �2W (a) ; a 2 [0; 1] :

[20]

Secondly, noting that

T�1~ut�1~us�1 = T�1~u2t�1 + op (1) for s < t;

and using the fact that ~et and ~ut�1 are asymptotically independent, we obtain:

1pT

TXt=1

"Xs<t

�t�s~et~�e

~es~�e

~ut�1pT

~us�1pT

#) �2up

1� �2

Z 1

0

B (a)2 dW (a) ; a 2 [0; 1] : (9.4)

Hence, combining (9.3) and (9.4),

1pT 3

TXt=1

~�t ) G (�) � 1 +

2p1� �2

!�2u

Z 1

0

B (a)2 dW (a) ; a 2 [0; 1] ; (9.5)

where G (�) depends on the correlation coe¢ cient of the Markov chain, � and is de�nedon a compact set.Next, using the quadratic variation de�nition, e.g. Mikosch (1998) we can establish

that1

T 2

TXt=1

~�2t ) 1 +

2p1� �2

!2�4u

Z 1

0

B(a)4da; a 2 [0; 1] ; (9.6)

Combining (9.5) and (9.6) and using Continuous Mapping Theorem, we get the desiredresult, (4.7).

[21]

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[24]

Table 1: Asymptotic critical values for the simplest modelModel 90% 95% 99%no interept/time trend 4:68 5:02 5:19demeaned 4:36 4:89 4:98detrended 4:15 4:37 4:65

Table 2: Power of Alternative Tests for Experiment 1a with T = 100Panel A: The case with no intercept and no trend

� = 1(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 17:1 15:2 15:8 14:9 8:8 7:6 31:1 28:2(0;�0:2) 30:9 30 31 29:5 9:7 8:9 63:1 65:7(0;�0:4) 40:5 39:6 51:1 49:9 11:5 10:7 86:2 85:2

(�0:1;�0:5) 78:9 79:6 88:2 90:7 57:4 60:3 98:7 98:4

� = 0:5(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 27:3 12:1 28:4 11:7 10:1 6:7 51:4 24:2(0;�0:2) 39:8 25:7 49:6 25:8 12:8 8:3 82:4 62:3(0;�0:4) 43:6 35:7 64:4 48:8 11:1 8:9 90:9 85

(�0:1;�0:5) 91:1 79:8 97 91:8 81:9 56:8 99:6 98:8

Panel B: The demeaned case

� = 1(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)

(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 8:5 8:9 7:6 7:9 3:6 5:1 14:5 16:4(0;�0:2) 18:7 20:7 15:3 19:7 4:6 5:7 39:3 46:9(0;�0:4) 30:1 33:5 32:1 38:5 5:8 8 72:6 79:5

(�0:1;�0:5) 63:3 68:8 68:9 79:7 35:7 42:4 91:8 96

� = 0:5(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 12:2 7:7 9:6 6:4 4:1 4:2 22:4 11:8(0;�0:2) 25:4 14:7 23:5 11:5 4:6 5:2 58:8 35:6(0;�0:4) 33:4 29:2 40:5 31:8 4:8 6:7 82:2 75:8

(�0:1;�0:5) 78:6 62:5 86:3 74:9 54:7 30:4 97:8 95

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Panel C: The detrended case

� = 1(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 8 8 7:3 8 5:1 5:4 12:2 13:3(0;�0:2) 14:8 17:9 12:4 12:3 5:6 5:7 32:5 36:8(0;�0:4) 29:3 32:6 26:2 31:4 6:4 7:3 65:4 76:4

(�0:1;�0:5) 56:6 62:1 59:2 70:7 29:2 32:3 86:8 94

� = 0:5(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 7:8 4:4 6:6 4:3 4 4:3 14:3 7:6(0;�0:2) 20:2 10:6 13:6 7:2 4:7 5 45:6 23:1(0;�0:4) 32:1 26:4 29:3 20:8 5:2 6:6 71:4 66:6

(�0:1;�0:5) 68:9 53:4 75 59:6 39:8 20:3 94:8 90:9

Table 3: Power of Alternative Tests for Experiment 1b with T = 100

All parameters switch(p; q) (0:98; 0:98) (0:9; 0:9) (0:98; 0:9) (0:9; 0:98)(�0; �1) Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1) 21:3 10:2 24:2 11 13:2 9:9 30:5 29:1(0;�0:2) 36:3 14 41:1 19:3 20 15:3 70:2 65:7(0;�0:4) 69:4 29:1 75:8 34:7 27:4 20:1 92:4 88:2

(�0:1;�0:5) 87:3 47:3 88:9 59:7 64:3 49:9 98:6 90:3

Table 4: Power of Alternative Tests for Experiment 2 with T = 100

Panel A: The case with no intercept and no trend

� = 0:5 � = 1(�0; �1; �2) P1 P2 P1 P2

Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1;�0:2) 30:1 13:3 69:2 35 17:4 16:1 38:7 40:9(0;�0:1;�0:4) 33:2 19:1 71:3 38:4 20:7 20:9 41:2 44:3(0;�0:2;�0:4) 52:1 32:5 98:2 91:8 36:2 36:4 86:8 92:1

(�0:1;�0:2;�0:4) 91:7 71:7 98:5 92:5 68:8 74:5 88:0 92:4

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Panel B: The demeaned case

� = 0:5 � = 1(�0; �1; �2) P1 P2 P1 P2

Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1;�0:2) 11:5 7:2 35:4 15:6 8:1 9:8 19:7 22:7(0;�0:1;�0:4) 12:7 8:4 36:5 17:9 9:4 12:7 21:4 25:5(0;�0:2;�0:4) 26:1 16:8 85:2 60:8 17:9 23:2 59:0 73:6

(�0:1;�0:2;�0:4) 66:2 39:6 86:3 62:8 39:9 49:6 61:0 74:5

Panel C: The detrended case

� = 0:5 � = 1(�0; �1; �2) P1 P2 P1 P2

Ssup tEG Ssup tEG Ssup tEG Ssup tEG(0;�0:1;�0:2) 6:4 5:2 21:1 9:2 7:4 8:2 15:9 16:7(0;�0:1;�0:4) 7:1 6:2 22:5 9:9 8:2 9:6 16:9 18:4(0;�0:2;�0:4) 16:7 9:7 69:9 38:2 14:8 17:6 47:6 57:7

(�0:1;�0:2;�0:4) 47:6 22:6 71:4 39:1 32:5 37:2 48:2 59:0

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Table 5: Cointegration tests for price and dividends

Country tEG SsupBelgium 0:937 0:101Canada 0:864 0:005Denmark 0:948 0:190France 0:774 0:167Germany 0:989 0:012Ireland 0:534 0:030Italy 0:988 0:012Japan 0:389 0:002Netherlands 0:976 0:170United Kingdom 0:017 0:004United States 0:110 0:001

Notes: p-value of two tests are provided. The p-values for Ssup test are obtained usingthe bootstrapped empirical critical vhalues with 1000 replications.

[28]

Table 6: MS ECM estimation for price and dividends

Parameters US UK Italy Japana1 :007

(:0032)

� :012(:0024)

�� :042(:0052)

�� :0042(:0035)

ast a2 �:019(:044)

�:0092(:031)

:34(:082)

�� :0106(:0029)

��

a3 :009��(:0027)

:011(:0035)

�� :00079(:0041)

:00071(:029)

�1 �:016(:0089)

�:066(:024)

�� :077(:014)

�� �0:0168(:0099)

�st �2 �:042(:081)

�:238(:106)

� �:55(:159)

�� :0065(:0086)

�3 �:028(:0094)

�� �:032(:0158)

� �:017(:013)

�:159(:081)

�

!1 :226(:306)

:159(:140)

�:265(:087)

�� �:26(:24)

!st !2 �1:192(2:379)

�4:680(3:334)

�:262(:000?)

�� :096(:19)

!3 �:180(:200)

�:169(0:206)

:046(:081)

4:85(3:51)

�1 :002(:00019)

�� :00041(7:5e�5)

�� :00093(:00024)

:0028(:000266)

��

�st �2 :011(:0052)

� :0146(:0042)

�� :0084(:0031)

:00075(:000125)

��

�3 :00045(:000075)

�� :0022(:00025)

�� :0043(:00038)

:0078(:00256)

��

P1 0:601 :283 :174 :646Pst P2 0:031 :073 :058 :296

P3 0:368 :643 :768 :058

Standard errors are given in the parentheses under coe¢ cients. The coe¢ cient is signi�-cant at �5% or ��1% level.

[29]

US price and dividends

0

1

2

3

4

5

6

7Ja

n­74

Jan­

77

Jan­

80

Jan­

83

Jan­

86

Jan­

89

Jan­

92

Jan­

95

Jan­

98

Jan­

01

Jan­

04

pricedividends

Figure 1: US price and dividends

­1.5

­1

­0.5

0

0.5

1

1.5

2

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81

r1r2r3

Figure 2: US Dynamic Causal E¤ect

[30]

Figure 3: US: smoothed probabilities

[31]

UK price and dividends

0

1

2

3

4

5

6

7Ja

n­74

Jan­

77

Jan­

80

Jan­

83

Jan­

86

Jan­

89

Jan­

92

Jan­

95

Jan­

98

Jan­

01

Jan­

04

pricedividends

Figure 4: UK price and dividends

­6

­5

­4

­3

­2

­1

0

1

2

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81r1r2r3

Figure 5: UK dynamic causal e¤ect

[32]

Figure 6: UK: smoothed probabilites

[33]