A note on the specification of conditional heteroscedasticity using a TAR model

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1 A note on the specification of conditional heteroscedasticity using a TAR model Fabio H. Nieto Universidad Nacional de Colombia Edna C. Moreno Universidad Santo Tomás, Bogotá, Colombia Reporte Interno de Investigación No. 21 Departamento de Estadística Facultad de Ciencias Universidad Nacional de Colombia Bogotá, COLOMBIA

2 A note on the specification of conditional heteroscedasticity using a TAR model Fabio H. Nieto Universidad Nacional de Colombia Edna C. Moreno Universidad Santo Tomás, Bogotá, Colombia Abstract Clusters of large values are observed in sample paths of threshold autoregressive (TAR) stochastic processes. In order to characterize the stochastic mechanism that generates this empirical TAR-model stylized fact, three types of marginal conditional distributions of the underlying stochastic process are found in this paper. One of them permits to find the conditional variance function that explains the aforementioned stylized fact and that can be compared, for example, with that of the familiar GARCH processes, in the context of financial time series. As a byproduct, a sufficient condition for having weak stationarity in a TAR stochastic process is derived. Corresponding author: fhnietos@unal.edu.co The authors acknowledge the financial support given by DIB, the investigation division of Universidad Nacional de Colombia at Bogotá 1

3 Key words and phrases: Conditional heteroscedasticity, Stationary nonlinear stochastic process, TAR model. 1 Introduction Sample paths of a threshold autoregressive (TAR) stochastic process can exhibit clusters of large values, as happens in financial time series. In this kind of time series, that stylized fact has been characterized very well by means of Engel s (1982) ARCH model and, after that, by means of GARCH processes (Bollerslev, 1986) and almost all of its extensions. The key element to do that task has been the conditional variance function of the observable stochastic process, from which the time series becomes. The literature on this topic is abundant. Following this route, it would be important to find a conditional variance function of a TAR process, in order to determine if the large-value clusters are explained by some kind of conditional mechanism. TAR models have been used for explaining the nonlinear behavior of a variable of interest via a so-called threshold variable. Tong (1990) proposed this model to specify the dynamic of an open-loop system and, among many others, Nieto (2005, 2008) and Nieto et al. (2013) proposed a Bayesian methodology to fit a particular class of these models to an observed time series. The model class studied by Nieto (2005) is termed TARSO models (Tong, 1990, pp. 101). Nevertheless, the issue of obtaining the univariate conditional distributions of this type of TAR processes or even their univariate marginal distributions have not been studied up to now, in the knowledge of the present authors. Another important problem in this context is under 2

4 which conditions a TAR stochastic process is weakly stationary, since solving this problem one could try to do comparisons with ARCH/GARCH models in order to establish if other stylized facts of financial time series are accomplished by sample paths of Nieto s (2005) TAR processes. Obviously, the issue of weak stationarity has been considered in particular cases of the similar class of self-exciting threshold autoregressive (SETAR) models. Among others, Petruccelli and Woolford (1984) and Chen and Tsay (1991) have worked on this topic, considering very particular cases. The scope of this paper is to obtain univariate marginal conditional distributions of Nieto s (2005) TAR process, in order to find a conditional variance function that can be able to characterize the mechanism that produces largesample clusters, in sample paths of these processes. The paper is organized as follows. In Section 2 we present the basic specification of the entertained TAR process. Section 3 includes the new results about conditional distributions that we have found. In Section 4 we present two examples to illustrate the properties we have found and some conclusions are exposed in Section 5. 2 The TAR model We specify here a particular case of Tong s (1990) TARSO model. Let {X t : t Z} and {Z t : t Z} be stochastic processes related through the equation (TAR model) X t = a (j) 0 + k j i=1 a(j) i X t i + h (j) ε t if r j 1 < Z t r j, (2.1) where j = 1,..., l, with l indicating the number of the so-called regimes in the sample space of variable Z t, which are determined by the real numbers 3

5 (threshold values) r 0 < r 1 <... < r l 1 < r l, where r 0 = and r l =. Here, a (j) i and h (j), i = 0, 1,..., k j, j = 1,..., l, are real numbers and they are called nonstructural parameters. The nonnegative number k j is the autoregressive order in regime j, j = 1,..., l, and l,r 1,...,r l 1, k 1,...,k l 1 and k l are called the structural parameters of the model. Additionally, {ε t } is a Gaussian zero-mean white noise process with variance 1 such that E(X s ε t ) = 0 for s < t, {Z t } is exogenous for {X t }, in the sense that there is no feedback from {X t } towards it, and {Z t } and {ε t } are mutually independent. To describe the dynamic stochastic behavior of {Z t }, we additionally assume that {Z t } is a homogeneous pth order Markov chain, p 1, with invariant or stationary distribution. For future reference, we set R j = (r j 1, r j ], j = 1,..., l, with the convention (r l 1, ] = (r l 1, ). Then, the set of regimes R 1,...,R l is a partition of the real line R. We shall use the symbol TAR(l; k 1,..., k l ) to denote this model and we will say that {X t } is a TAR process, with {Z t } as its threshold process. The class of TAR models was introduced by Tong (1978) and Tong and Lim (1980), in the case where the threshold variable is the lagged variable X t d, where d is some positive integer. In this case, the model is known as the self-exciting TAR (SETAR) model. Tong (1990) extended this notion of nonlinear time series models to the class of TARSO models, under the philosophy that there exists an open-loop system that relates the output process {X t : t Z} to the input {Z t : t Z}. Model (2.1) is a particular case of the general TARSO model and Nieto (2005, 2008) and Nieto et al. (2013) have developed a Bayesian approach for fitting and forecasting TAR models and the procedure has been applied in the fields of the hydrology and 4

6 meteorology (Nieto; 2005, 2013), the economy (Hoyos, 2006), and finance (Moreno, 2011). In a multivariate setting, Tsay (1998) also has analyzed TARSO models, among many other authors. 3 New characteristics of a TAR process 3.1 Weak stationarity In order to simplify a conditional distribution to be presented below, we study sufficient conditions under which the proposed TAR process is weakly stationary. It is easy to see that for all t Z, the marginal cumulative distribution function (cdf) of X t is given by F t (x) = p j F t,j (x), (3.1) j=1 where F t,j (x) = P Xt ((, x] R j ) for any x R and p j = P Z (R j ), j = 1,..., l, with P Xt and P Z denoting, respectively, the induced probability measures of X t and Z t onto the Borelian measurable space (R, B). Notice that l j=1 p j = 1 and that for computing p j, j = 1,..., l, the invariant distribution P Z of the Markov chain {Z t } is used. In this way, for each t Z, the marginal distribution of X t is a mixture of conditional distributions, where the conditioning events are the regimes. Moments for X t can be obtained from this marginal cdf. Indeed, if we denote µ 1,j,t = E(X t R j ) = xdf t,j (x), j = 1,..., l, then µ t = E(X t ) = l j=1 p jµ 1,j,t (a weighted average of the regime means at time t). A similar expression holds for the second moment around zero. Indeed, if we denote µ 2,j,t = E(X 2 t R j ) = x 2 df t,j (x), we get 5

7 that µ 2,t = E(X 2 t ) = l j=1 p jµ 2,j,t. Obviously, Var(X t ) = µ 2,t µ 2 t for all t Z. Proposition 1. Let C be the complex number set. If for each j = 1,..., l, the roots of the polynomial ϕ j (z) = 1 k j unit circle, then where ψ j (z) = 1 ϕ j (z) = standard normal cdf. F t,j (x) = Φ 0,1 ( i=0 ψ(j) i Proof. See the Appendix. i=1 a(j) i z i, z C, are outside of the ) x ψ j (1)a (j) 0 h (j) σ, (3.2) j z i, σ 2 j = i=0 (ψ(j) i ) 2, and Φ 0,1 ( ) denotes the It is important to remark here that this conditional cdf does not depend on t and that, as is noted in the proof, the Gaussian assumption on the process {ε t } is basic for this result. Also, we can state that the distribution of X t conditional on the regime R j is N(ψ j (1)a (j) 0, [h (j) σ j ] 2 ), for all j = 1,..., l. Consequently, for all t Z, the marginal distribution of X t is the same and it is a mixture of conditional normal distributions, where the conditioning sets are the regimes. Moreover, for all t, µ 1,j,t = µ 1,j = ψ j (1) a (j) 0, j = 1,..., l, and the mean function µ t does not depend on t. Let µ = µ t for all t. Likewise, the regime-based conditional variances are constant for t and then, we set σ 2 j = (h (j) σ j ) 2, j = 1,..., l. Also, µ 2,j,t = µ 2,j and µ 2,t = µ 2 = l j=1 p jµ 2,j for all t Z and Var(X t ) = µ 2 µ 2 for all t Z. Since the marginal distribution of X t is a mixture, this can be multimodal. Of course, if µ j = µ, a constant, for all j = 1,..., l, the distribution is unimodal. Proposition 2. Under the conditions in Proposition 1, the autocovari- 6

8 ance function (ACVF) of {X t } is given by Cov(X t, X t n ) = p jk q jk (n) µ 2, (3.3) j,k=1 for all integer numbers t and n, where p jk = P(Z t R j, Z t k R k ) and for j, k = 1,..., l. Proof. See the Appendix. q jk (n) = µ 1,j µ 1,k + h (j) h (k) m=0 ψ (k) m ψ (j) n+m, It is worth noticing here that expression (3.3) is an extension of the ACVF of a linear stochastic process. Indeed, if l = 1 (the only regime is the real line R), k 1 = k, h (1) = h, then Cov(X t, X t n ) = h 2 m=0 ψ m ψ n+m. This expression corresponds to the general form of the ACVF of a linear stochastic process {X t = i=0 ψ iε t }, where the real-number sequence {ψ i } is absolutely summable and {ε t } is a zero-mean white noise process with variance h 2 (see Brockwell and Davis, 1991). Obviously, in our case, X t = k i=1 a ix t i + hε t ; that is, {X t } is an AR(k) linear process. For future reference, we set γ(h) = Cov(X t, X t h ), h Z. Corollary. stationary stochastic process. Proof. Under the conditions in Proposition 1, {X t } is a weakly Because of Propositions 1 and 2, it only remains to show that E(X 2 t ) < for all t Z, to fulfill Brockwell and Davis (1991) s definition of weak stationarity. But this fact is obvious because of the finiteness of the normal distribution second moment. 7

9 3.2 Univariate conditional distributions In the previous subsection, a first type of conditioning emerged; namely, on the regimes R j, j = 1,..., l. As was seen there, under the conditions in Proposition 1, the conditional distribution of X t given the regime R j is normal with mean µ 1,j and variance σ 2 j, for each j = 1,..., l, and it does not depend on t. We remark here that, by no means, we are saying that conditional on a regime the process is linear and autoregressive. Now, for each t > max{k j j = 1,..., l}, we study the conditional distribution of X t given the information set x t 1 = {x t 1,, x 1 } and a regime. It is easy to see that the conditional distribution of X t given x t 1 and the ( regime R j is N a (j) 0 + k j i=1 a(j) i x t i, [h (j) ] ), 2 for each j = 1,..., l. Thus, the parameters h (j), j = 1,..., l, are a kind of conditional standard deviation. A third type of conditioning occurs when the conditioning set is only x t 1 (it is something like a conditional marginalization through the regimes). In this case, for each t > max{k j j = 1,..., l}, the cdf of X t x t 1 is given by F t (x x t 1 ) = p j F t 1,j (x R j, x t 1 ), (3.4) j=1 where F t 1,j (x R j, x t 1 ) = P Xt ((, x] R j, x t 1 ), j = 1,..., l. This is the ( conditional cdf of the N a (j) 0 + k j i=1 a(j) i x t i, [h (j) ] ), 2 as we noted in the last paragraph. Consequently, the distribution of X t x t 1 is a mixture of conditional normal distributions. Setting µ t 1,j = a (j) 0 + k j i=1 a(j) i x t i, j = 1,..., l, we can observe that Var(X t x t 1 ) = p j [h (j) ] 2 + p j µ 2 t 1,j ( p j µ t 1,j ) 2, (3.5) j=1 j=1 j=1 8

10 for each t > max{k j j = 1,..., l}. Note that independent of the TAR-model parameter values, we always have Var(X t x t 1 ) 0. It is worth noticing that via the TAR model proposed in this paper, expression (3.5) gives a way for computing this conditional variance function. Without taking into account the threshold process {Z t }, this conditional variance function can be computed using the GARCH model family. Consequently, these two forms of computing the function that explains conditional heteroscedasticity in a stochastic process can be compared. It is interesting to note that the summand l j=1 p j [h (j) ] 2 in expression (3.5) does not depend on t and thus, it can be interpreted as a communality term in this type of conditional variance. All in all, we have analyzed three exhaustive types of conditioning information sets: (i) only a regime R j, (ii) x t 1 and a regime R j, and (iii) only x t 1. We shall call Type I, Type II, and Type III, respectively, these three types of conditioning. 4 Some examples 1. A simulated model. We consider the TAR(2;1,1) model given by X t 1 + ε t if Z t 0, X t = X t ε t if Z t > 0, where Z t = 0.5Z t 1 +a t and {a t } is a zero-mean Gaussian white noise process with variance 1, which is independent of {ε t }. The length of the simulated time series is 200. In Figure 1 (top) we observe the graph of the simulated time series {x t } and we can note clusters of large values. Since the roots of the polynomials ϕ 1 (z) = z and ϕ 2 (z) = z, 9

11 Time series Conditional variance function Figure 1: Time series and Type-III conditional variance function for Example 1 z C, are outside the unit circle, the process {X t } is weakly stationary. Then µ 1,1 = 0.31 (rounding to two decimal digits) and µ 1,2 = Clearly, p 1 = p 2 = 0.5, thus µ = E(X t ) = Because of ψ 1 (z) = i=0 ( 0.6)i z i and ψ 2 (z) = i=0 ( 0.7)i z i, then σ 2 1 = 1.56 and σ 2 2 = In this way, the regime-based conditional variances are σ 2 1 = 1.56 and σ 2 2 = and the marginal variance is Var(X t ) = 99. In order to get the ACVF of the process {X t }, we need to compute the quantities p jk and q jk (h) for j, k = 1, 2 and h Z. Initially, we observe that q 12 (h) = ( 0.7) h, q 21 (h) = ( 0.6) h, q 11 (h) = ( 0.6) h, and q 22 (h) = ( 0.7) h. To calculate p jk = P (Z t R j, Z t k R k ), we remark that the process {Z t } is Gaussian; hence, all of its bivariate distributions are multinormal and p jk is a double integral of 10

12 the joint probability density function (pdf) on the bidimensional set R j R k. Thus, γ(h) = 2 j,k=1 p jkq jk (h) 0.01 for all h Z. From the above results, the pdf of X t, for all t, is given by f(x) = 0.5[f 1 (x) + f 2 (x)], x R, where the pdf f j corresponds to a normal distribution with mean µ 1,j and variance σ 2 j, j = 1, 2. Now, the distribution of X t conditional on past data up to t 1 and a regime is normal, with mean x t 1 and variance 1 in the first regime and mean x t 1 and variance 100 in the second. Conditioning only on x t 1, the distribution of X t is a mixture of these conditional normal distributions and (rounding to two decimal digits) Var(X t x t 1 ) = x t 1 + 0, 75x 2 t 1. Here, the communality value is 50.5 and, as we can see in Figure 1, the bottom line for the Type-III conditional variance function is around that value. Furthermore, we see there that this conditional heteroscedasticity is large (peaks), in the time periods where large-value clusters of the simulated time series have occurred. In Figure 2 we plot the pdf s for each regime (the left vertical scale on the top panel is for the pdf of regime 1) and the mixture distribution pdf of variable X. 2. Analysis of BOVESPA index returns. In order to illustrate with real data how well a TAR model might explain clusters of large values in a financial time series, we consider the daily Dow Jones index (DJ) as the threshold variable and the daily BOVESPA index (Sao Paulo s stock exchange) as the output variable. More exactly, we set X t = ln(bi t ) ln(bi t 1 ) and Z t = ln(dj t ) ln(dj t 1 ), where BI denotes BOVESPA index. 11

13 REGIME1 REGIME MIXTURE Figure 2: Density functions in the simulated example: for each regime (top) and for the marginal mixture distribution (bottom) The sample period to be considered is December 8, 2000-June 2, 2010 (2474 daily data) and in Figure 3 we plot the BOVESPA and Dow Jones returns. Morettin (2008) have studied the stylized facts of BOVESPA time series. Beginning with Nieto s (2005) fitting approach and then using Tong s (1990) NAIC (information criterium), we obtained the following TAR(3;2,0,4) model for BOVESPA returns: X t X t ε t, if Z t , ε t, if < Z t , X t = X t X t X t X t ε t, if < Z t. Here, the thresholds are the percentiles 25 and 75 of the Dow Jones returns (which is a stationary process); hence, p 1 = p 3 = 0.25 and p 2 = 0.5. Using the Corollary on Section 3.1, {X t } is a weakly stationary process, as 12

14 Figure 3: DOW JONES (top) and BOVESPA (bottom) returns expected. To obtain its marginal first two moments, we analyze the regimebased conditional distributions. On the first regime, the expected return is µ 1,1 = 1.32% with a s.d. of 2%; on the second, the expected return for BOVESPA is µ 1,2 = 0.07% with a s.d. of 1.41%; and on the third, BOVESPA has an expected return of µ 1,3 = 0.97% with a s.d. of 2%. Then, the marginal expected return is 0.05% (empirical is 0.058%) and its marginal standard deviation is 1.7% (empirical is 2%). Now, the expected return at time t, conditional on a regime and the past observed data x t 1 for X, is 0.068% (constant) on the second regime, on the first it depends on the past two data, and on the third, on the past four observations. The conditional variances based on the regimes and the past data for X ([h (j) ] 2, j = 1, 2, 3) indicate that there is more variability in the 13

15 first and third regime (almost the same) than in the second regime. Conditioning only on the information set x t 1, we obtained that the conditional variance function is given by Var(X t x t 1 ) = µ 2 t 1, µ 2 t 1, µ 2 t 1,3 (0.25µ t 1, µ t 1, µ t 1,3 ) 2, for each t > 4, where µ t 1,1 = x t x t 2, µ t 1,2 = (rounding to four decimal digits), and µ t 1,3 = x t x t x t x t 4. In Figure 4 we see the graph of the marginal density implied by the model along with the estimated density (using an Epanechnikov kernel) and the normal density with mean and variance (sample values). We recall that the model-based marginal distribution is a mixture of regime-based conditional normal distributions and, consequently, it is not appropriate to interpret kurtosis. Nevertheless, we see that the maximum value of the TAR-model pdf is between that of the estimated pdf and that of the normal pdf. Remark. Moreno (2011) fitted the following GARCH model to BOVESPA returns: X t = X t 3 + a t, a t = ϵ t σ t, σt 2 = a 2 t σt 1 2, where {ϵ t } is a zero-mean Gaussian white noise process with variance 1. We computed the GARCH-model conditional variance function and that derived from the TAR model using equation (3.5) and plot them in Figure 5. As we 14

16 30 25 DATA MIXTURE STNORMAL Figure 4: Densities for BOVESPA data: estimated density, marginal density, and normal density can see, the two functions signal the large-value clusters by means of their peak values, although the GARCH-model function is smoother than that of the TAR model. Additionally, we can see that they have almost the same bottom line, which is around the communality value Before concluding this section, we remark that some stylized facts of financial time series, other than large-value clusters, were not accomplished by the simulated time series in Example 1 and by the BOVESPA returns. Specifically, in Example 1, the sample autocorrelation function of the original data is stronger than that of the squared data, and in the empirical application using BOVESPA, the squared residuals of the TAR model exhibit autocorrelation while those of the GARCH model do not. These results 15

17 TAR MODEL GARCH MODEL Figure 5: Conditional variance functions for BOVESPA returns can be provided upon request. On the other hand, although is feasible to speak about the univariate marginal distribution of the weakly stationary TAR process, it does not have sense to speak about kurtosis because, in general, this distribution is a mixture of distributions and, consequently, it can be multimodal. 5 Conclusions In this paper, we have found that a certain kind of TAR stochastic processes, which are not linear, can explain large-value clusters in a time series. In order to find the model characteristic that explains that empirical fact, we found the marginal conditional distributions of the the entertained univariate TAR 16

18 stochastic process. Three conditioning types emerge; namely, the Type I conditioning based only on a regime; the Type II conditioning based on the past data for the variable of interest and a regime, and conditioning of Type III that is based only on the past data of the interest variable. In the first case, the conditional distributions are normal with means and variances that do not depend on time if the process is weakly stationary. In the second, they are normal with means given by the autoregressive part of the model in the corresponding regime and variances equal to the squared weights of the white noise, in correspondence with the given regime. In the last case, the conditional distribution is a mixture of the conditional distributions of the second case. Under the Type-III conditional scenario, we derived an expression for computing the conditional variance function of the TAR stochastic process and found how this function can explain the presence of large-value clusters, as happens with the analogous function in the GARCH-model family. By no means, we are claiming that the stochastic mechanism that generates the clusters of large values, in sample paths of TAR stochastic processes, is the same than that explained by GARCH models. As a byproduct of this research, we found sufficient conditions for having weakly stationary TAR processes, in the spirit of Nieto s (2005) TARSO model. In this case, the marginal distribution of the process is time-invariant and it is a mixture of the Type-I conditional distributions. APPENDIX Proof of Proposition 1. Because for each j = 1,..., l, the roots of the polynomial ϕ j (z) = 1 k j i=1 a(j) i z i are outside the unit circle, the inverse 17

19 operator of ϕ j (B) exists. Let ψ j (B) = ϕ j (B) 1 = Then, for all x R, i=0 ψ(j) i B i, with ψ (j) 0 = 1. F t,j (x) = P (X t x R j ) = P (a (j) 0 ψ j (1) + h (j) ψ j (B)ε t x). Now, the random variable ψ j (B)ε t is well defined for all t and it has a normal distribution with mean 0 and variance This ends the proof. i=0 [ψ(j) i F t,j (x) = P (ψ j (B)ε t x a(j) 0 ψ j (1) h (j) ) = P ( ψ j(b)ε t σ j ] 2 = σ 2 j. Consequently, x a(j) 0 ψ j (1) h (j) σ j ). Proof of Proposition 2. Initially, we note that from expression (2) in Section 2.1 and Proposition 1, the mean function of {X t } is E(X t ) = l j=1 p j µ 1,j for all t Z, where µ 1,j = a (j) 0 ψ j (1). Now, in order to obtain the autocovariance function of the stochastic process {X t }, we obtain firstly the bivariate cdf of variables X t and X t h for any integer numbers h and t, which we denote F t,t h. Let (x t, x t h ) R 2, then F t,t h (x t, x t h ) = P (X t x t, X t h x t h ) { ( ) ( l = P Xt,X t h (, x t ] (, x t h ] (R j R k )) }, j,k=1 where P Xt,X t h is the joint probability induced by the random vector (X t, X t h ). Now, it is easy to show that {R j R k : j, k = 1,..., l} constitutes a partition of R 2 because the regimes R j, j = 1,, l, constitute a partition of R. In this 18

20 sense, we can say that {R j R k : j, k = 1,..., l} is the set of bidimensional regimes in R 2. Hence, F t,t h (x t, x t h ) = Now, F t,t h (x t, x t h ) = j,k=1 ) )} P Xt,Xt h {((, x t ] (, x t h ] (R j R k P Xt,X t h Z t R j,z t h R k ((, x t ] (, x t h ] R j R k ) j,k=1 P Zt,Z t h (R j R k ), where P Zt,Z t h is the joint probability induced by the random vector (Z t, Z t h ) and P Xt,X t h Z t R j,z t h R k is the joint probability of (X t, X t h ) conditional on the event (Z t R j ) (Z t R j ) (or the regimes j and k). Let P Zt,Z t h (R j R k ) = p jk ; j, k = 1., l. Now, let F t,t h;j,k (x t, x t h ) = P Xt,X t h Z t R j,z t h R k ((, x t ] (, x t h ] R j R k ), then, we obtain that F t,t h (x t, x t h ) = p jk F t,t h;j,k (x t, x t h ), j,k=1 where l j,k=1 p jk = 1. This means that the joint cdf of X t and X t h is a mixture of conditional bivariate cdf s. Thus, E(X t X t h ) = p jk E(X t X t h Z t R j, Z t h R k ). j,k=1 Following the proof of Proposition 1, we have that X t = µ 1,j +. s=0 ψ(j) s ε t s if Z t R j (ψ (j) 0 = 1), and X t h = µ 1,k + m=0 ψ(k) m ε t h m if Z t h R k. Then, E(X t X t h Z t R j, Z t h R k ) = µ 1,j µ 1,k + h (j) h (k) 19 m=0 ψ (k) m ψ (j) h+m,

21 where, without loss of generality, we have assumed that h > 0. Consequently, ( ) E(X t X t h ) = p jk µ 1,j µ 1,k + h (j) h (k) ψ m (k) ψ (j) h+m, and Cov(X t, X t h ) = j,k=1 ( p jk µ 1,j µ 1,k + h (j) h (k) j,k=1 m=0 m=0 ψ (k) m ψ (j) h+m ) ( ) 2 p j µ 1,j. As we can see, this covariance depends only on the lag h. This completes the proof. j=1 References Brockwell, P.J. and Davis, R.A.(1991) Time Series: Theory and Methods. New York: Springer-Verlag. Chen, R. and Tsay, R.S. (1984) On the ergodicity of TAR(1) processes. The Annals of Applied Probability, 1, Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation. Econometrica, 50, Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31,

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