Univariate Conditional Distributions of an Open-Loop TAR Stochastic Process

Clusters of large values are observed in sample paths of certain open-loop threshold autoregressive (TAR) stochastic processes. In order to characterize the stochastic mechanism that generates this empirical stylized fact, three types of marginal conditional distributions of the underlying stochastic process are analyzed in this paper. One allows us to find the conditional variance function that explains the aforementioned stylized fact. As a by-product, we are able to derive a sufficient condition to have asymptotic weak stationarity in an open-loop TAR stochastic process.


Introduction
In the context of nonlinear stochastic processes, Tong (1990) proposed the open-loop threshold autoregressive (TAR) process and, among many others, Nieto (2005Nieto ( , 2008) ) and Nieto, Zhang & Li (2013) developed a Bayesian methodology to analyze particular cases, namely-when the threshold variable is not a covariable.Sample paths in this kind of TAR process can exhibit clusters of (either positive or negative) large values, which is an empirical fact that is also observed, for example, in financial and meteorological/hydrological time series.In terms of financial data, that stylized fact has been very well characterized by Engle's (1982) ARCH model and, then by the GARCH process (Bollerslev 1986) and almost all of its extensions.The key element in these models has been the conditional variance function of the underlying stochastic process.Along these lines, it is important to find a conditional variance function of an open-loop TAR process in order to determine if large-value clusters are explained by some type of conditional probabilistic mechanism.
The issue of characterizing univariate conditional distributions for this type of TAR processes or, even, their univariate marginal distributions, has not in the authors knowledge been studied until to now (see also Tong 2011).Thus, the main goal of this paper is to obtain univariate marginal conditional distributions for the open-loop TAR process in order to explain the presence of large-sample clusters in a sample path.As a by-product, we provide a sufficient condition to have asymptotic weak stationarity of the TAR process which is also considered in this paper.The issues of weak and strict stationarity and ergodicity have been contemplated in other cases related to the general family of TAR processes.For example, Petruccelli & Woolford (1984) and Chen & Tsay (1991) have worked on the so-called SETAR model class.Tong (1978), Wong & Li (2000), and Li, Ling & Tong (2012) have undertaken research in which their threshold moving average (TMA) equation has a stationary solution.Tong's (2011) paper is an excellent and comprehensive review of the literature of these issues.
The paper is organized as follows: In Section 2 we present the basic specification of the considered TAR process.Section 3 includes the main results of our research, and in Section 4 we present three illustrative examples (two are real-data examples).Section 5 concludes.

The Proposed Open-Loop TAR Model
Let Z be the set of integer numbers.Following Nieto (2005) , let {X t : t ∈ Z} and {Z t : t ∈ Z} be stochastic processes, such that for all t ∈ Z, for some j = 1, . . ., l, where l denotes the number of the so-called regimes in the sample space of variable Z t .These regimes are determined by the extended real numbers (threshold values) Revista Colombiana de Estadística 39 (2016) 149-165 Let 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.
Additionally, {ε t } is a Gaussian zero-mean white a noise process with a variance 1 such that E(X s ε t ) = 0 for s < t, Z s and the set {X t , X t−1 , . ..} are mutually independent for all s > t, and {Z t } and {ε t } are mutually independent.Equation (1) describes a dynamical system without feedback with input {Z t } and output {X t }.To describe the stochastic behavior of {Z t }, we additionally assume that {Z t } is a homogeneous mth order Markov chain, m ≥ 1, and, as a sequence of random variables, it converges weakly to the distribution F .
We shall use the symbol TAR(l; k 1 , . . ., k l ) to denote this model and we will say that {X t } is an open-loop TAR process with {Z t } as its threshold process.Notice that if we define, for each t ∈ Z, the random variable J t as J t = j, if and only if Z t ∈ R j , then {J t } is an indicator process with sample space {1, • • • , l}.Hence, the processes {X t } and {J t } conform a particular case of the general TAR process, in Tong's (1990Tong's ( , 2011) ) sense.
This TAR model has been applied in the fields of the hydrology and meteorology (Nieto 2005, Nieto 2008, Nieto et al. 2013), the economics (Hoyos 2006), and finance (Moreno 2011).In a multivariate setting, Tsay (1998) among many other authors, has also analyzed open-loop models, where variable Z t is included as a covariable.In what follows, we refer to the above open-loop TAR process simply as a TAR process (or model).

New Characteristics of the TAR Process
In this section, our goal is to obtain the distribution of X t for all t, conditional on (i) a regime, R j say, (ii) a regime R j and the past data xt−1 = (x t−1 , . . ., x 1 ) with t > max{k j | j = 1, . . ., l}, and (iii) xt−1 .That is to say, we will consider three types of conditioning sets.Before finding these univariate conditional distributions, we shall derive a sufficient condition to characterize the univariate marginal distributions of the process {X t }.Nieto (2008) presented the first steps in characterizing some of these conditional distributions, as well as the univariate marginal distributions, but the results were shown in an intuitive way without formal analytical proofs.

Univariate Marginal Distributions and Asymptotic Weak Stationarity
We assume that all the random elements considered in this paper are defined on the same probability space (Ω, F, P ).Initially, we note that for all t ∈ Z, the marginal cumulative distribution function (cdf) of X t is given by for any x ∈ R, where F t,j (x) = P (X t ≤ x | Z t ∈ R j ) and p t,j = P (Z t ∈ R j ), j = 1, . . ., l (assuming this last probability is positive).Because l j=1 p t,j = 1, 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 = xdF t,j (x) for all j = 1, . . ., l and all t, then µ t = E(X t ) = l j=1 p t,j µ 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 = x 2 dF t,j (x) for all j = 1, . . ., l and all t, we obtain µ 2,t = E(X 2 t ) = l j=1 p t,j µ 2,j,t .Obviously, Var(X t ) = µ 2,t − µ 2 t for all t ∈ Z. Now, since {Z t } converges weakly to F , p t,j → p j = F (r j ) − F (r j−1 ) as t → ∞ for all j = 1, . . ., l.Of course, if {Z t } has identical univariate marginal distributions, as in the case of a strictly stationary process, then the cdf of Z t is F for all t and, thus, p t,j = p j for all t and for all j = 1, . . ., l.
Proposition 1.Let C be the complex number set.If for each j = 1, . . ., l, the roots of the polynomial φ j where , and Φ(•) denotes the standard normal cdf.
Proof .See the Appendix.
This result was quoted by Moreno (2011) but without a formal proof, and in the particular case where the distribution of Z t is the same for all t.It is important to remark here that, under the assumptions of Proposition 1, the conditional cdf F t,j does not depend on t and that, as is noted in the proof, the Gaussian assumption on the process {ε t } is the basis for this result.Thus, let F j = F t,j , µ 1,j,t = µ 1j , and µ 2,j,t = µ 2j for all t and for all j = 1, . . ., l.Also, we can state that the distribution of X t conditional on the regime R j is N (ψ j (1)a 2 ) for all t and for all j = 1, . . ., l.We note that µ 1j = ψ j (1)a (j) 0 , and for future reference we set σ 2 j = [h (j) σj ] 2 .Furthermore, the sequence {X t } converges weakly to a random variable with cdf F X = l j=1 p j F j .Based on this fact, the mean and the variance function of this process converge, respectively, to a constant value, as t → ∞.As noted above, if the univariate marginal distributions of {Z t } are identical, the cdf of X t is F X for all t.Notice that the limit distribution is a mixture of conditional normal distributions where the conditioning sets are the regimes, and hence, this distribution is potentially multimodal.Of course, if µ 1j = µ 1 for all j = 1, . . ., l, the distribution is unimodal.
Proof .See the Appendix.Moreno (2011) proposed an expression for this ACVF and provided a proof for it.However, in the proof, some conditional probability measures are not well established, there is a mistake in a probability statement, and only the case in which the distribution of Z t is the same for all t was considered.
Using the Chapman-Kolmogorov equations, we can show that the sequence of two-dimensional random vectors {(Z t , Z t−n )} converges weakly to a two-dimensional random vector with cdf has identical univariate marginal distributions, the ACVF of {X t } is given directly by this limit form.We additionally note that E(X 2 t ) < ∞ for all t ∈ Z because of the finiteness of the normal distribution second moment.The results obtained in the above propositions provide us with evidence to state the following definition.
Definition 1.The TAR stochastic process {X t } is said to be an asymptotic weak stationary process if it satisfies the conditions given in Proposition 1.

Remarks.
(1) If {Z t } has identical marginal univariate distributions, {X t } is weakly stationary.(2) We must be careful in trying to use this function for model identification purposes, as in the case of linear ARMA processes.The reason is that the usual ACVF only captures linear dependence and the process {X t } is not linear.(3) The usefulness of the computation of the ACVF in Proposition 2 is in that it leads to the definition of the asymptotic weak stationarity of the process {X t }. (4) In future research, the analytical properties of this ACVF will be investigated, specifically for describing its mathematical properties and, in particular, the rate of convergence towards zero (under ergodicity conditions).
It is worth noticing that the limit form of expression ( 4) is an extension of the ACVF of a linear stochastic process.Indeed, if l = 1, the only regime is the real line R and setting h (1) = h, we obtain the limit form of the autocovariance in (4), which is This expression is analogous to the general form of the ACVF of a linear stochastic process {X t = ∞ i=0 ψ i ε 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 & Davis 1991).Obviously, in this particular case, X t = k i=1 a i X t−i + hε t , with k = k 1 , and the roots of the polynomial φ are outside of the unit circle; that is, {X t } is an AR(k) linear process.For future reference, we put γ(n) = lim t→∞ Cov(X t , X t−n ), for all n ∈ Z.

Univariate Conditional Distributions
In the previous subsection, the first type of univariate conditional distribution emerged; namely when the conditioning set is the event Z t ∈ R j (in the σ-algebra F) for some j = 1, . . ., l.We have referred to this fact by stating that the conditioning is "on the regime"R j .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 1j and variance σ 2 j for all t and for all j = 1, . . ., l.Also notice that, under this scenario, the conditional variance function of the process {X t } is given by V ar(X t | R j ) = σ 2 j if at time t the observation z t ∈ R j .This is a step function that essentially depends on the variable-Z values.
The second type of conditional distribution for the variable X t occurs when the conditioning set is xt−1 and the regime R j , for some j = 1, . . ., l. Trivially, this distribution is N µ t−1,j , [h (j) ] 2 , for all j = 1, . . ., l and for all t > max{k j | j = 1, . . ., l}, where µ t−1,j = a Here, the conditional variance function for the process ] 2 ) for t > max{k j | j = 1, . . ., l}.Clearly, R j occurs at time t if and only if z t ∈ R j .As in the previous case, it is a step function that intrinsically depends on the values of variable Z. Interestingly, under this conditional distribution, we can interpret the parameter h (j) as a conditional standard deviation, j = 1, . . ., l.Now, we only consider xt−1 as the conditioning information set.In order to formally establish the main results, we set the following notation: let N be the set of natural numbers, X t = (X t , . . ., X 1 ), P = P(N), the power set of N, F t = σ(B t × P), the σ-algebra generated by the cartesian product between the t-dimensional Borelian σ-algebra and P, λ t the Lebesgue measure on the measurable space (R t , B t ), µ the counting measure on the measurable space (N, P), and P (Xt,Jt) the probability induced by (X t , J t ) on the product measure space (R t × N, F t , λ t × µ).Proposition 3. Suppose that P (Xt,Jt) has a Radon-Nikodym derivative with respect to the product measure λ t × µ, then, (i) for all t > max{k j | j = 1, . . ., l}, the conditional distribution of X t given xt−1 is a mixture of conditional normal distributions and (ii) the conditional variance function of the process {X t } is given by Var Revista Colombiana de Estadística 39 (2016) 149-165 for t > max{k j | j = 1, . . ., l}.
Proof .See the appendix.
As in Propositions 1 and 2, Moreno (2011) and Moreno & Nieto (2014) proposed and used in their applications the results shown in Proposition 3.However, no formal analytic proof, such as the one presented here that is based on measure theory, was made to support those important results.Because of this, some notwell defined expressions were used in those papers.
We note that independent of the TAR-model parameter values, we have that the right-hand side of equality ( 6) is always nonnegative.Because p t,j ≈ p j for a t that is large enough when {X t } is asymptotically weak stationary, the approximate summand l j=1 p j [h (j) ] 2 in expression ( 6) can be interpreted as a communality term in this type of conditional variance.It is very interesting to note that in the context of Mixture Autoregressive Models (MAR), expression ( 6) is very similar to Wong & Li's (2000) conditional variance function and that our communality term is analogous to their base-line conditional variance.It is worth noting that, additionally on xt−1 , the function given by expression ( 6) depends on all the regimes via the parameters p t,j and [h (j) ] 2 , t > max{k j | j = 1, . . ., l} and j = 1, . . ., l.Hence, this function involves more information about the conditional probabilistic mechanisms of the TAR process than the previous two.
In the following section, we shall call the above three conditional distributions Type I, Type II, and Type III, respectively.

Some Examples
In order to illustrate the three types of univariate conditional distributions of a TAR process and how the Type-III conditional distribution can explain the presence of large-value clusters in its sample paths, we present the following examples.

A Simulated Model
We consider the TAR(2;1,1) model given by 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 }.Notice that {Z t } is a first-order Markov chain and a strict stationary stochastic process.The length of the simulated time series is 200. Figure 1 (top) shows the graph of the simulated time series {x t } and clusters of large values can be seen.
Since the roots of the polynomials φ 1 (z) = 1 + 0.6z and φ 2 (z) = 1 + 0.7z, z ∈ C, are outside of the unit circle, the process {X t } is weakly stationary.Now, µ 11 = −0.31(rounding to two decimal digits) and µ 12 = 0.53.Clearly, 1 = 1.56 and σ2 2 = 1.96.In this way, the Type-I conditional variances are σ 2 1 = 1.56 and σ 2 2 = 196.08,and the marginal variance is Var(X t ) = 99 for all t.As was signaled in the above remarks ( 2), (3), and ( 4) we have to be careful with the interpretation and use of the ACVF of the process {X t }.Here, and only for theoretical illustration purposes, we describe the way in which this function can be obtained.In order to do this, we need to compute the quantities p n,jk and q jk (n) of expression (5), for j, k = 1, 2. Initially, we observe that q 12 (n) = −0.16+17.24(−0.7)n , q 21 (n) = −0.16+17.24(−0.6)n , q 11 (n) = 0.10+1.56(−0.6)n , and q 22 (n) = 0.28 + 196.08(−0.7)n .To calculate p n,jk = P (Z t ∈ R j , Z t−n ∈ R k ), we remark that the process {Z t } is Gaussian; hence, all of its bivariate distributions are multinormal and p n,jk is a double integral of the joint pdf on the bidimensional set R j × R k .In this paper we do not compute these values.Thus, γ(n) = 2 j,k=1 p n,jk q jk (n) − 0.01 for all n ∈ 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 1j and variance σ 2 j , j = 1, 2. The distribution of X t , conditional on past data up to t − 1 and a regime, is normal with mean −0.5 − 0.6x t−1 and variance 1 in the first regime and mean 0.9 − 0.7x t−1 and variance 100 in the second.Now, the Type-III conditional variance function is given by (rounding to two decimal digits) Here, the communality value is 50.50 and, as we can see in Figure 1, the base line for the Type-III conditional variance function is around that value.Furthermore, we can see there that this function has local extreme values in the time periods where large-value clusters of the simulated time series have occurred.

Analysis of BOVESPA Index Returns
In order to illustrate with financial data how well our proposed TAR model might explain clusters of large values in a 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 the BOVESPA index.The sample period is December 8, 2000-June 2, 2010 (2473 daily data), and in Figure 2 we plot the BOVESPA and Dow Jones returns.Morettin (2008) has undertaken extensive statistical analysis of the BOVESPA time series.Using Nieto's (2005) fitting approach, Moreno (2011) found the following TAR(3;2,0,4) model for BOVESPA returns: Here, the thresholds are the percentiles 25 and 75 of the Dow Jones returns (which is a strict stationary process); hence, p 1 = p 3 = 0.25 and p 2 = 0.5.Since the conditions in Proposition 1 are fulfilled, {X t } is a weakly stationary process, as expected.To obtain its marginal first two moments, we analyze the Type-I conditional distributions.For the first regime, the expected return is µ 11 = −1.32% with s.d. of 2%; for the second, the expected return is µ 12 = 0.07% with s.d. of 1.41%; and for the third, BOVESPA has an expected return of µ 13 = 0.97% with s.d. of 2%.Then, the marginal expected return is −0.05% (the empirical is 0.058%) and its marginal standard deviation is 1.7% (empirical is 2%).
Using the Type-II conditional distributions, we found that the conditional expected return at time t is 0.068% (constant) for the second regime, for the first it depends on the past two data, and for the third, on the past four observations.The conditional variances in this case are 0.0198 2 , 0.0137 2 , and 0.0191 2 , indicating that there is more Type-II conditional variability in the first and third regime (almost the same) than in the second regime.
Conditioning by only using the information set xt−1 , we found that the Type-III conditional variance function is given by for each t > 4, where µ t−1,1 = −0.0127+ 0.1113x t−1 − 0.0685x t−2 , µ t−1,2 = 0.0007 (rounding to four decimal digits), and Note.Moreno (2011) fitted the following GARCH model to BOVESPA returns: where { t } is a zero-mean Gaussian white noise process with variance 1.We computed the GARCH-model based conditional variance function and plotted it jointly with the Type-III conditional variance function in Figure 3. Interestingly, the two functions locally signal the large-value clusters in the BOVESPA returns by means of their extreme values, although the GARCH-model function is smoother than that of the TAR model.Additionally, we can observe that they have almost the same base line.It is important to remark here that, by no means, are we trying to compare the TAR and GARCH models.Our main motivation was only to signal that the two conditional variance functions (obtained from two conceptually different models) have, approximately, the same behavior.2013) fitted a TAR model to the daily rainfall (in mm.), as the input or threshold variable, and the daily flow of the Bedon river (in m 3 /sec), as the output or target variable, in a geographical region on southern Colombia.Specifically, the data were collected at a meteorological station with coordinates 2.23 north (latitude) and 76.23 west (longitude) and at a hydrological station with coordinates 2.19 north and 76.15 west.These stations are located in the San Rafael's lagoon neighborhood.The sample period is January 1, 1992-November 30, 2000 (3256 data), and the data was assembled by IDEAM, the official Colombian agency for hydrological and meteorological studies.In Figure 4, one can see the two time series, where the relationship between the two variables is clear as it is the presence of large-value clusters in the river flow time series.Let P t and Xt , respectively, be the rainfall and river flow at day t.Because of the universal convention for measuring these two variables, we need to make Z t = P t−1 .As was suggested by Nieto (2005), the river flow variable needs two transformations, namely-square root of the data and an adjustment for conditional heteroscedasticity via an ARCH(1) model.We denote the adjusted river flow variable as X t .The two time series had missing data but they were estimated using Nieto's (2005) procedure.From now on, we work with the complete (or interpolated) time series.The fitted model was a TAR(4;3,2,2,2) that is given by The stochastic dynamic of Z t was described by means of a first-order Markov chain with identical univariate marginal distribution, which is given by f n (z) = ph n (z) + (1 − p)g(z), where p = P(Z = 0), g(•) denotes a truncated normal density at z = 0 with mean 3.24 and variance 7.76 2 , and with n a positive integer number (see Nieto's (2005) paper for its interpretation).Nieto et al. (2013) found that p 1 = 0.60, p 2 = 0.20, p 3 = 0.12, and p 4 = 0.08 for all t.
Using expression (6), we computed the conditional variance function of the river flow process, which is plotted in Figure 5.As can be see there, and from a local point of view, the time periods at which extreme values of both the conditional variance and the time series occur are similar.The communality value is 2.93.

Conclusions
In this paper, we have characterized three types of univariate conditional distributions for a particular case of the open-loop TAR stochastic processes.We have called them Type I, Type II, and Type III conditional distributions.In the first type, the conditioning information set is a regime; in the second, the conditioning set is constituted by past information of the output variable and a regime; and in the third, the distribution is only conditional on the past information of the output variable.As a by-product, we have found that there is a sufficient condition to have asymptotic weak stationarity in the process and, under that condition and the fact that the model white noise is Gaussian, the Type-I conditional distribution is normal.The Type-III conditional distribution is a mixture of normal distributions.Using some examples, we have illustrated the fact that the Type-III conditional variance function can explain the presence of large-value clusters in time series that obey the TAR model of this paper.An interesting Revista Colombiana de Estadística 39 (2016) 149-165 future investigation could be the ACVF characterization of these kinds of TAR processes.

Figure 1 :
Figure 1: Time series and Type-III conditional variance function for Example 1.

Figure 5 :
Figure 5: Conditional variance function for the river flow.
Conditional variance functions for BOVESPA returns.