Modelling of Cointegration with Student’s T-errors

Two or more non-stationary time series are said to be co-integrated if a certain linear combination of them becomes stationary. Identification of co-integrating relationships among the relevant time series help s the researchers to develop efficient forecasting methods. The classical approach of analyzing such series is to express the co-integrating time series in the form of error correction models with Gaussian errors. However, the modeling and analysis of cointegration in the presence of non-normal errors needs to be developed as most of the real time series in the field of finance and economics deviates from the assumption of normality. This paper focuses on modeling of a bivariate cointegration with a student’s-t distributed error. The co-integrating vector obtained from the error correction equation is estimated using the method of maximum likelihood. A unit root test of first order non stationary process with student’s t-errors is also defined. The resulting estimators are used to construct test procedures for testing the unit root and cointegration associated with two time series. The likelihood equations are all solved using numerical approaches because the estimating equations do not have an explicit solution. A simulation study is carried out to illustrate the finite sample properties of the model. The simulation experiments show that the estimates perform reasonably well. The applicability of the model is illustrated by analyzing the data on time series of Bombay stock exchange indices and crude oil prices and found that the proposed model is a good fit for the data sets.


Introduction
According to Wold's theorem, a single stationary time series {X t } with no deterministic components can be expressed as a linear combination of shocks at previous time points, t − 1, t − 2, ..., which could be approximated by a linear Autoregressive Moving Average (ARMA) model of appropriate order [1]. Box-Jenkins method of analysis assumes that a time series becomes stationary after a finite number (d) of differences and introduced a class of models known as autoregressive integrated moving average (ARIMA) models. If the underlying time series becomes stationary after d differences then the original series is said to be integrated of order d, denoted as I(d). In practice, all naturally occurring time series are non-stationary and some of them may be of I(d). If the time series is stationary then it is said to be of I(0). The integrated series are special class of time series whose non-stationarity is due to the presence of one or more unit roots in the underlying characteristic polynomial. The unit root non-stationarity also indicates the presence of stochastic trend in the time series. Several time series from related domains may wander extensively throughout a research period. However the underlying theory will propose forces which tend to keep them together in a state of equilibrium. That is, there may exist two or more non-stationary time series, and yet a certain linear combination of those variables could be stationary. If that is the case, the original time series are said to be co-integrated.
The commonly used test procedures for cointegration in the literature are Engle and Granger two step estimator [2] and Johansen likelihood ratio test [3]. Johansen [3] developed the maximum likelihood estimator of the cointegration parameters and the likelihood ratio test for testing the presence of cointegration. Both the test are based on the assumption that the possibly cointegrated vector autoregressive(VAR) or error correction model (ECM) has normally distributed errors.The assumption of normality need not hold good if the data under study are generated by some heavy-tailed distributions. Such situations often arise in the study of financial time series. So in the present work, we focus on a bivariate cointegration model with student's-t errors.
The following is a definition of bivariate cointegration.
Let {X 1t } and {X 2t } be two I(1) series. These series are said to be cointegrated, if there exist a vector α = (α 1 , α 2 ) with both elements non zero such that, The studies on cointegration and unit root when the error densities are non-normal is not extensively seen in the literature.
[4] discussed the effects of fiscal policy on the exportation of agricultural products in nigeria using a Vector Error Correction Model (VECM) approach, when the error density follows a normally distribution. [5] [9] discusses an autoregressive models in time series with non normal errors represented by a member of a wide family of symmetric Student's t-distributions. Creal et al. [10] introduced the multivariate Student's t generalised autoregressive score model for volatilities and correlations, where the multivariate normal distribution is a special case. Here we propose the modelling of two cointegrating time series with the errors generated from a bivariate Student's t-distribution.
The rest of the paper is organised as follows. The next section briefly discusses the cointegration model with bivariate Student's t-errors. We discuss the estimation procedure in Section 3. Section 4 deals with the unit root test and cointegration test associated with the autoregressive model of order 1. To evaluate the accuracy of the estimators and test statistic, we carried out a simulation study in section 5. Finally, an application of the proposed model is illustrated in section 6.

Model Description
Let {X 1t } and {X 2t } be two cointegrating time series and both are of I(1). For further analysis, it is customary to express the series in the form of ECM Following [2], let us write (1) It can be shown that the reduced form for the process in (1) and (2) will make the variables X 1t and X 2t as linear combinations of u 1t and u 2t and therefore both the series will be nonstationary (Integrated of order 1). That is, and From (3) and (4), it is clear that {X 1t } and {X 2t } are non stationary as they are linear combinations of a stationary and a non stationary series. Since {X 1t } and {X 2t } are integrated series, equation (2) describes a stationary linear combination of the nonstationary variables. Thus the variables X 1t and X 2t are cointegrated and hence we can say that they have a long run relationship in equilibrium. But if φ → 1, then the series are uncorrelated random walks and hence they are no longer cointegrated. The model given above has been studied by [2] in detail with possibly correlated white noise.
To analyse further,it is convenient to reparameterise the model in (1) and (2) by subtracting the lagged values from both sides. Let ∆ be a difference operator defined by ∆X t = X t − X t−1 . Applying ∆ on X 1t and X 2t of both sides of equations (1) and (2) and after some algebra we will get, 236

Modelling of Cointegration with Student's T-errors
where Z t−1 = X 1,t−1 + αX 2,t−1 is the stationary cointegrating relationship and δ = 1−φ α−β . This model was studied by [2] for obtaining the stationary cointegrating relationship between the time series.
Assume that the error variable η t = (η 1t , η 2t ) in (5) and (6) follows a bivariate Student's t-distribution with the probability density function (pdf) given by, It should be noted that both the marginal distributions are student's t-distribution with the same degrees of freedom ν ( see [11]) with the pdf The above bivariate distribution can be obtained by the transformation, where Z is N (0, Σ), Σ is variance covariance matrix and s ∼ χ 2 r . Here s and Z are independent. The case with ν=1 reduces to a bivariate Cauchy distribution.

Conditional MLE for ECM with bivariate student's t-errors
In this section, we focus on the problem of maximum likelihood estimation of cointegrated model with bivariate Student's t-distributed errors. When there exist an explicit density function for the error random variables, one can use the conditional maximum likelihood estimation for estimating the model parameters. Let the parameter vector to be estimated are the elements of θ = (ρ, α, β, δ, ν) . The explicit form of the conditional log likelihood function for θ = (ρ, α, β, δ, ν) , given (x 10 , x 20 ) = (0, 0) based on sample (x 1t , x 2t ), t = 1, 2, · · · n of size n is given by , where (x 1t , x 2t ) are obtained from the equations (5) and (6). The form of the log likelihood function suggests that we have to maximise it by using some numerical methods. Here we obtain the parameter estimates by using a two step estimation procedure. First we obtain the estimates of ρ, α, β, δ by solving the respective score equations. Then we use the profile likelihood technique for estimating the parameter ν. That is, the estimate of ν is obtained by maximizing the log likelihood function usingρ,α,β,δ. The discussion on profile likelihood and its asymptotic properties can be seen in [12].
On differentiating the log-likelihood function with respect to the parameters ρ, α, β, δ, we get the four equations given by, where Since there do not exist analytically closed form expressions for the estimators, we have to maximize the score functions by using some numerical methods. We have solved these equations by using Newton Raphson method for obtaining the parameter estimates. Let the estimates of ρ, α ,β ,δ beρ,α,β,δ respectively.
Once the model parameters are estimated, the profile likelihood function L ν,α,β,δ,ρ is maximized over ν to getν. The profile log likelihood function is given by, where The properties of first order autoregressive models with student's t-distributed errors are needed to develop a test procedure for cointegration model in (5) and (6), which we discuss in section 4.

Unit root test for Auto Regressive model of order one with Student's t-errors
A sequence that contains one or more roots of its characteristic polynomial that are equal to one is called a unit root process.
The simplest model that may contain a unit root is the Auto Regressive model of order one (AR(1)), where a t denotes a serially uncorrelated white noise with 0 mean and constant variance.
If φ= 1, equation (12) becomes a random walk without drift model, that is, a non-stationary process. If, |φ| < 1, then the series {X t } is stationary. Dickey et al [13] developed a test procedure to determine whether a variable has a unit root, or equivalently, the variable follows a random walk model. Here, our interest is to analyse the first order autoregressive equation in the presence of Student's t-errors. We consider the first order autoregressive process {X t } with the errors {a t } is a sequence of Student's t-random variables.
Our interest is to find the maximum likelihood estimator of φ, sayφ and the test procedure for unit root under the hypothesis H 0 : φ = 1 against H 1 : |φ| < 1. Suppose n observations are available for the analysis and we obtain the likelihood function based on n observations generated by the model (12). The corresponding joint density function of (a 1 , a 2 · · · · · · , a n ) is given by,

Modelling of Cointegration with Student's T-errors
And the conditional likelihood of φ given x 0 for the model (12) is, The corresponding log-likelihood function is, The critical points of the above log likelihood function can be obtained by solving the score functions given by, We obtain the estimator of φ numerically by using the Newton Raphson method. Once the estimate of φ is obtained, then the profile log likelihood function is maximized over ν to getν. We maximize the above profile log likelihood function by using numerical methods for obtaining the estimate of ν. Now let us consider the hypothesis H 0 : φ = 1 against the alternative H 1 : |φ| < 1, that is the time series {X t } was generated by a stationary model. Under H 0 , the maximum value of the likelihood function is and under the alternative, the maximum value of likelihood function is, .
Hence, forφ ∈ H 1 , the likelihood ratio test statistic is We reject H 0 if −2logλ is either too small or too large.

Test for cointegration
We know that, if φ → 1 in (12) , then the time series contains a unit root. Here one can extend the idea of testing the presence of unit root for testing the presence of cointegration using the approach discussed in [2]. First, we have to confirm all the variables are integrated of same order and are non stationary in nature. Once the series are confirmed to be non stationary with same order of integration, we can test for the presence of cointegration by using the residuals from the fitted error correction model. Here the hypothesis of interest for testing the cointegration is, H 0 : δ = 0 against δ = 0 in the model (5 and 6).
Once we obtain the parameter estimates from the data, we tests the residuals from the error correction model using the test procedure described below. We denote the residuals from the error correction model (5) and (6) by,η 1t = ∆x 1t −βδẑ t−1 andη 2t = ∆x 2t +δẑ t−1 , whereẑ t−1 = x 1,t−1 +αx 2,t−1 ,α andβ,δ are the parameter estimates of α, β and δ . And, if the residuals obtained from the ECM are stationary, (ie; the null hypothesis of no cointegration is rejected) then we can conclude that the variables will be cointegrated. Now we will obtain the likelihood ratio test statistic for testing the presence of cointegration in an error correction model.
Under the null hypothesis, the maximum value of the likelihood function is, , and under the alternative hypothesis, the maximum value of the likelihood function is, The likelihood ratio test statistic obtained is given by, Here we reject the null hypothesis of no cointegration if log (L 1 /L 0 ) is large in absolute.
To evaluate the performance of the estimators and test statistic, a simulation study is carried out, which is illustrated in section 5.

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Modelling of Cointegration with Student's T-errors

Simulation study for t distributed errors
Let us begin with a simulation study for the first order autoregressive model with Student's t-distributed errors. For the study, we generate the error random variable from a Student's t-distribution. Then for specified values of the model parameter, we simulated the sequence {x t }, t=1,2,...,n using the relation described in (12). Based on this sample, we obtain the maximum likelihood estimates by solving the equation (14) and (15). For the given values of the model parameter, we repeated the experiment 100 times for computing the estimates and then averaged them over the repetitions. Using the estimate of φ, we obtained the maximum likelihood estimate of ν by maximizing the profile log likelihood function. Next we compute the likelihood ratio test statistic in equation (16) -0.5 500 500 500 500 500 500 500 500 500 500 500 500  φ=-0.2 500 500 500 500 500 500 500 500 500 500 500 500  φ=0.5 500 500 500 500 500 500 500 500 500 500 500 500  φ=0.8 356 417 453 489 500 500 500 -0.5 500 500 500 500 500 500 500 500 500 500 500 500  φ=-0.2 500 500 500 500 500 500 500 500 500 500 500 500  φ=0.5 500 500 500 500 500 500 500 500 500 500 500 500  φ=0.8 337 395 417 458 472 477 498 500 500 500 (5) and (6). Finally we obtained the MLE of the parameters by solving the likelihood equations in (7) to (10). Using the estimated parameter values of ρ, α, β, δ, we maximized the profile log likelihood function in (11) and obtained the estimate of ν. We then repeated the experiment 100 times for computing the estimates and then averaged them over the repetitions. Once the parameters of ECM being estimated, we test for cointegration using the residuals from the error correction model. We use the test statistic given in (17) to compute the number of rejections of the null hypothesis under the various alternatives. Table 5 and 6 corresponds to the parameter estimates of the error correction model associated with t distribution with ν = 3 and ν = 4. Table 7 and 8 gives the number of rejections of the null hypothesis under the various alternatives.        Using the parameter estimates of the ECM, we tested whether the marginals of the residuals follow a Student's t-distribution using Kolmogorov-Smirnov test. The p values obtained for the series are 0.638 and 0.562 respectively, which indicates that Student's t-distribution is suitable for the residuals.
PP-plot and histogram for the marginals given in Figure 2 and 3 also confirms that the chosen distribution is suitable for the   Since we have assumed a bivariate student's t-distribution for the errors, it is also important to check whether the joint distribution of the residual series is following this assumption. We have examined the assumption of bivariate student's tdistribution for the residual series using a bivariate Kolmogorov Smirnov test [14]. Under the null hypothesis, where the residuals follows a bivariate student's t-distribution, the p value obtained for the test is 0.99, which indicates that bivariate student's t-distribution is suitable for modelling the joint residual series. We have also obtained the bivariate skewness and kurtosis measures to confirm whether the fitted distribution has heavy tails [15]. These measures are found to be 0.56 and 8

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Modelling of Cointegration with Student's T-errors respectively, which confirms the presence of a heavy tailed distribution for the joint residual series.

Conclusions
We developed a bivariate cointegration model generated from a bivariate student's-t distributed errors. The estimation method and testing procedure for the presence of unit root and cointegration are developed. The model parameters are estimated using the method of maximum likelihood. Simulation experiments and data analysis are conducted to illustrate the applications of the proposed model.