^{1}

^{2}

This paper proposes and estimates a statistical model of nonlinear cointegration, with applications to the stock markets of Japan and the United States. We define nonlinear cointegration as a long-run stable relationship between two time series variables even in the presence of temporary nonlinear divergence from this long-run relationship. More concretely, extending the bubble model of Asako and Liu (2013) [1] to stock price ratio variables, both upward and downward divergent bubble processes are estimated at a time. We conclude that, although two stock price indexes are not linearly cointegrated, they are considered to be cointegrated nonlinearly.

In this paper we propose and develop the recursive estimation method of a nonlinear statistical model of speculative bubbles and utilize this model in establishing an idea of nonlinear cointegration. We then apply this idea to the stock market indexes of Japan and the United States, and we detect how these indexes commove in the long run although they deviate from the long-run relationship nonlinearly in the short run.

So far, whether stock markets of different countries commove together has mainly been tested by utilizing the linear cointegration relationship à la Engle and Granger (1987) [

As empirical investigation of comovement of stock markets, there have been a number of research and the results vary depending on the countries and sample periods. Asako, Zhang and Liu (2014) [

The construction of the present paper is as follows. In Section 2, we propose a time series model of the boom and bust and develop its recursive estimation method. Section 3 modifies this basic model to apply for a ratio variable, which has more restrictive feature within the model of booms and busts. In Section 4, we apply the modified model to detect the nonlinear cointegration relationship between the stock price indexes of Japan and the United States. Section 5 conclude the paper.

In this section, we develop a model of nonlinear cointegration and explain how to estimate the relevant parameters.

As an extended model to Asako and Liu (2013) [

where

Like

Let us consider briefly the implication of this model. Our basic model consists of two regimes or models (A) and (B). At period t, x_{t} is expressed by a divergent time series model when a speculative bubble continues. We describe this phenomenon by the autoregressive model (A) with parameter _{t}.

More concretely, we assume that the probability of bubble continuation

where α and γ are positive unknown parameters. This formulation implies that π_{t} decreases as the deviation between x_{t} and θ_{t} becomes grater in its absolute value. To put it another way, the probability of a bubble crash, 1-

In principle, we can generalize our formulation by considering a broader class of stochastic models for u_{t} such as ARMA process or by introducing the fundamental values into the functional form of the transition probability (3). However, we have tried to keep our model as simple as possible because this paper is only meant to be a first step in this research direction. The specification, (3), of the probability turns out to be one of the few analytically tractable formulations in the following analyses.

When the probability structure of crashes is taken into consideration, we see that the bubble cannot continue forever. As it grows, the probability of a crash approaches unity and x_{t} will sooner or later be pulled back to the fundamental value θ_{t}. In this way, the time series of x_{t} never diverges, but exhibits more or less stable behavior in the longer run.

Note that letting θ_{t} = 0 and assuming away the constraint x_{t} > 0 leads us to the models of Asako, Kanoh and Sano (1990) [_{t} is not a ratio variable but is a stock price bubble measured as deviations from their fundamental values. The model of nonlinear cointegration, which is developed in Section 4, adds to this basic model the property that ratio bubbles are symmetric between upwards and downwards.

In Liu, Asako and Kanoh (2011) [

One notable difference between the present model (1)-(4) and the earlier ones is that Liu, Asako and Kanoh (2011) [_{t} = 0. Once we allow for θ_{t} > 0, whether θ_{t} is known or unknown causes a big difference in the Bayesian recursive estimation. If it is unknown and to be estimated in the same way as the other parameters of the model, the estimation process becomes too complicated for us to manipulate the model explicitly. On the other hand, if θ_{t} is known and treated as a predetermined parameter even though we have to somehow “estimate” it eventually, this estimation can be separated from the estimation of the entire model and its recursive estimation process remains, in terms of hardness, almost at the same level as Asako and Liu (2013) [_{t} be known and propose its two candidates in Section 3.

In this section, we describe a Bayesian recursive technic to estimate the parameters of our model. Before proceeding to this task, we put _{s} (s = 1; 2; : : : ; t) is either 1, 2, or 3.

With these new notations, we write down the joint density for

where ^{1} over time conditioned on X^{t} and ^{t-1} terms at stage t. Then, in view of (2), the joint prior density function for

Now our main task is to calculate the updated posterior density (6) by utilizing the Bayes’ theorem:

Introducing a new parameter

for the sake of later convenience in notation, from (1) and the normality of u_{t}, we have

Therefore, in view of (7), the multiplication of (6) and (9) yields the updated formula of (6) for period t if and only if we have, to begin with

where the first and second terms within the large brackets represent, respectively, the probability density function of exponentially and mutually independently distributed ^{2}. The integer function

is introduced to simplify the mathematical expression.

Moreover, for the unspecified coefficient functions, we must have

and

Also for means and variances of the normal distributions, it must be

and

Finally, it must be recalled, that by making use of the relationship that applies for conditional density functions

and knowing that

which appears in the denominators of (7) and (11). This establishes all requirement that enable Bayesian recursive estimation to update consistently.

The estimates of

and

We also obtain the probability estimate of bubble continuation from period t-1 to t as

or we can directly obtain the conditional expectation as

Finally, the estimate of the variance of

In carrying out the recursive procedure explained above, two variance parameters are to be specified. These are the dispersions of the random terms in (1) and (2), i.e.,

Let us put

On the other hand, since

and

we have, like (16)

Therefore, the log likelihood function of

and the resulting set of variances

So far is the complete and mathematically rigorous description of the Bayesian recursive estimation and we can estimate parameters for any length of sample periods. However, the number of terms we need to compute in equations from (11) to (14) and others increases at a rate of 3^{t} to exceed a standard capacity of computer as the number of time series data increases. For this reason and to reduce the computational burden, we introduce the so-called condensation procedure first proposed by Harrison and Stevens (1981) [

What we have to do in practice is to approximate the posterior density (5) at period t or the left hand side of (7) by a joint density of the following form

where we utilize the fact that

so that the joint prior density at period t + 1 can be written as

where

whereas

The basic bubble model (1)-(4) formulates the feature that a ratio variable returns to its fundamental value in the long run as the probability that a bubble crashes reaches 100% insofar as the divergent bubble continues. In other words, although short-run bubbles generate explosive discrepancies between _{t}, divergent booms would bust eventually and in this sense there is a stable relationship in the long run. This phenomenon is what we call the nonlinear cointegration.

Unlike the definition of linear cointegration, the definition of nonlinear relationship is model-specific. There may be other models of nonlinear cointegration and our nonlinear cointegration should more restrictively be named speculative bubble nonlinear cointegration or boom and bust nonlinear cointegration.

Such being the case, there is no established method to test the nonlinear cointegration relationship. Instead, we are obliged to accept the existence of the nonlinear relationship only passively. We especially put emphasis on the bubble process in (2) and thereby we detect whether

In the empirical analysis in Section 4, we compute the pseudo-t statistics:

in order to sense the “significance“ regarding the validity of β_{t} > 1. Since the present estimation technic is Baye- sian in the sense that we utilize prior information besides the information extracted from the data, statistics like (33) may not obey Student’s t-distribution. Nonetheless, we would presume that t = 1.65, which is one sided 5% significant for a standard t test, is a critical level to rely on.

In detecting the validity of the nonlinear cointegration, we may as well examine into the probability of bubble continuation

The basic model we developed in Section 2 is applicable to any series of x_{t}. In this section, we modify the basic model to deal with a ratio variable x_{t} > 0．A ratio variable may exhibit both upwards and downwards bubble processes with θ_{t} > 0, which necessitates certain nontrivial revision in recursive estimation.

We alter the basic model into a double regime switching model. One regime switching is that the basic model is of the boom-and-bust type. The other regime switching is that a ratio variable has both upwards (or positive) and downwards (or negative) bubble processes. On the other hand, we maintain (2) or the transition equation of

Then, we can naturally regard it a bubble by β_{t} > 1 once _{t} < 1 for certain periods of time. In such a case, we may misunderstand what is really happening because β_{t} < 1 is usually a case for a stationary autoregressive process. This is quite embarrassing and we may as well be advised to treat the upwards and downwards bubbles asymmetrically. For this aim, we take the reciprocal of the original ratio when the ratio itself is smaller than θ_{t} as in (3), thus resulting in a drastic regime switch for negative downwards bubbles.

Let _{t} by

With this new x_{t}_{,}, we assume that every aspect of the basic model (1)-(4) is valid, i.e.,

except that

replaces (8).

Note that integrating artificially two regimes most likely causes heteroscedasticity in innovation term u_{t} in (1) or (35). In fact, we will introduce proportional variance of u_{t} to

Lastly, we need to revise the probability of bubble continuation. That is, in (3), we have

or

that replaces (4). In (38) or (39), the greater deviation is _{t} − 1/θ_{t} for the negative downwards bubble.

As we have already noted, the fundamental stock prices ratio θ_{t} is assumed known and given to us exogenously at period t. There may be several candidates for θ_{t}. Here we propose two alternative ones^{3}.

The first candidate is the simple arithmetic average of all the past data:

Although we put equal weight on each data, the informational role of the current data decreases over time as (40) by definition is rewritten as θ_{t} = {(t-1) θ_{t}_{-1}^{+}y_{t}}/t, which in turn is rewritten as

Equation (41) implies that θ_{t} follows a random-walk type sticky movement except that the drift term is not stochastic but is given deterministically. As t increases, the contribution of the second term on the right hand side of (41) decreases over time.

The second candidate approximates the fundamental value by the fixed period (say 12 months) moving average up to the current one. Thus in place of (40) we have

And thereby in place of (41), we have

for t > 12. As for the first 12 months, we use the simple average (40).

At period t, we compute θ_{t} once we get a new data y_{t} and we determine which regime we are in, i.e., whether a positive bubble (y_{t} ≥ θ_{t}) or a negative bubble (y_{t} < θ_{t}). If we are rigorously interested in whether the stock price ratio is in positive upwards phase or in negative downwards phase, we may watch where we have been in the past. For example, we would recognize regime shifts only if the opposite new regime continues at least a few consecutive periods. This will exclude a fake regime shift that occurs unsystematically. The idea of this rule of thumb stems from the Bry-Boschan method in the judgment of the business cycle phase.

Once θ_{t} and thereby the data x_{t} of (34) is obtained, we are ready to utilize the recursive estimation technic developed in Section 2. We estimate the basic model as applied to the stock market prices of Japan and the United States.

Asako, Zhang and Liu (2014) attempted to apply the nonlinear cointegration to the stock markets of Japan, the United States and China. They first checked whether there is a linear cointegration relationship between these countries and concluded negatively for any pair of countries. Then they estimated the basic model of (1)-(4) and of three ways of the known fundamental stock prices ratio including (40) and (42). Among these, in what follow, we develop the most representative case of the nonlinear cointegration; namely the one between the stock price indexes of Japan and the United State.

The monthly time series data we have chosen are the Nikkei225 index (hereinafter Nikkei225) for Japan and the Dow-Jones Industrial Average Stock Price Index (hereinafter DJ) for the United States.

_{t}.

Next, we construct from the time series y_{t} that of the artificial variable x_{t} by (34). Referring to the realized y_{t} and two fundamental stock prices ratio θ_{t}, the time series of x_{t} consists of negative bubble (y_{t} < θ_{t}) up to the mid 1990s and thereby, by definition, x_{t} equals the reciprocal of y_{t}. On the contrary, during the latter half of the sample period, x_{t} consists of positive bubble (y_{t} > θ_{t}) and x_{t} is y_{t} itself. In the case of _{t} > θ_{t} and y_{t} < θ_{t} interchange with small intervals, as does x_{t}.

We need to obtain the maximum likelihood estimates for the variances of

Judging on the log likelihood, between the two fundamental stock prices ratio, ^{4}.

With the above preparation, _{t}. The percentage of samples that satisfies the necessary requirement for bubbles β_{t} > 1 amounts to 82.9% for _{t} > 1 exceed more than 80%. These observations may as well support the view that the model (1)-(4) with reasonable modification fits the data and the stock markets of Japan and the United States are cointegrated nonlinearly in the long run. But how reliable is this result?

To answer to this question, we checked the pseudo t t-statistic (33) and found, as summarized in _{t} > 1 is one sided 5% “pseudo-significant” is nil for _{t} < 1 is not significant). These suggest that the standard deviation of β_{t} is relatively large, and the reliability of the estimates is limited. Note, on the contrary, that β_{t} > 1 is 93.3% pseudo-significant for

A clue to this is that the maximum likelihood variance estimate _{t} is theoretically regarded constant in (2). But, like the parameters _{t} does not have to stay unchanged over time. Even if the variance of

β_{t} = β_{t}_{-1} + constant,

and β_{t} can be different from β_{t}_{-1}. Moreover, even if the constant term is 0 and β_{t} = β_{t}_{-1}, in theory, because β_{t} is estimated as the expected value of the posterior distribution à la Bayesian, it can differ from β_{t}_{-1} once the data increases information in the posterior distribution in (18).

In _{t}. As π_{t}, the conditional expectation (21), rather than the point estimate (20), is chosen^{5}. With

Asako, Zhang and Liu (2014) [

1970 -72 | 1973 -76 | 1977 -80 | 1981 -84 | 1985 -88 | 1989 -92 | 1993 -96 | 1997 -2000 | 2001 -04 | 2005 -08 | 2009 -12 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

β_{t} ≥ 1 | 35 | 48 | 48 | 48 | 48 | 21 | 0 | 40 | 48 | 44 | 48 | ||

significant at 5% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

β_{t} < 1 | 0 | 0 | 0 | 0 | 0 | 27 | 48 | 8 | 0 | 4 | 0 | ||

significant at 5% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

β_{t} ≥ 1 | 35 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | ||

significant at 5% | 9 | 48 | 47 | 40 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | ||

β_{t} < 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

significant at 5% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

case of Var (_{t} > 1 fits the data reasonably well, thus establishing the latent nonlinear boom and bust relationship between relevant stock prices.

In this paper we proposed and developed the recursive estimation method of the nonlinear cointegration. The purpose of this attempt has been to show the usefulness of introducing the idea of nonlinear cointegration. By applying this idea to the stock market indexes of Japan and the United States, we have seen that these indexes commove in the long run although they deviate from this relationship in the short run.

Kazumi Asako,Zhentao Liu, (2016) Comovement of Stock Markets—An Analysis by Nonlinear Cointegration*. Open Journal of Social Sciences,04,64-75. doi: 10.4236/jss.2016.45010