Bayesian inference for threshold moving average models - La Sapienza

METRON - International Journal of Statistics 2003, vol. LXI, n. 1, pp. 119-132

MOHAMED A. ISMAIL – HUSNI A. CHARIF

Bayesian inference for threshold moving average models Summary - Bayesian inference for Self Exciting Threshold Moving Average (SETMA) models is introduced. The prior distribution of the model parameters is specified, using informative as well as noninformative priors, and the posterior density is approximated by a multivariate t distribution. Simulation results show that the Bayesian methodology gives sound inferences for the model parameters. Finally, a parsimonious SETMA model is fitted to the U.S. unemployment rates series, and the results compare favorably when paralleled to previous literature on this data set. Key Words - SETMA; Posterior distribution; Multivariate t; Monte Carlo simulation; Quasi conjugate analysis.

1. Introduction In the last three decades there has been a growing interest in finite parameter nonlinear time series modeling. The reason is that the linearity assumption does not always hold, and there are situations where real data sets exhibit features such as limit cycle, amplitude dependent frequencies, and jump phenomena which cannot be described by linear models. Most of the suggested nonlinear models include some type of nonlinearity in the autoregressive part. Among these models, the self exciting threshold autoregressive (SETAR) model is perhaps the most popular one. The SETAR model was introduced by Tong and Lim (1980) and studied steadily in Tong (1983,1990) and Tasy (1989) among others. However, its dual counterpart, the self exciting threshold moving average (SETMA) model, has received less attention. The SETMA model was considered by De Gooijer and Kumar (1992) and studied in detail by De Gooijer (1998). In particular, he discussed the identification, estimation, and testing problems. Invertibility conditions for some SETMA models are given in De Gooijer and Brannas (1995). Received September 2001 and revised November 2002.

120

M. A. ISMAIL – H. A. CHARIF

Unfortunately most of the reported work on threshold models in the literature focused on the non Bayesian analysis. Kheradmandnia (1991) introduced the Bayesian analysis of threshold autoregressive models. Unaware of Kheradmandnia’s work, Broemeling and Cook (1992) and Geweki and Terui (1993) proposed Bayesian analysis of threshold autoregressive models. Chen and Lee (1995) employed a Gibbs sampling approach to develop a Bayesian analysis for SETAR models, and Chen (1998) extended it to a generalized SETAR model where exogeneous variables are added and the threshold is allowed to be a function in an exogeneous variable. On the other hand, Bayesian analysis of SETMA models is not known. Bayesian analysis of moving average models is difficult even for linear models since the likelihood function is nonlinear in the parameters, which causes problems in the prior specification and the posterior analysis. Thus, the integrations involved in Bayesian analysis must be done numerically. Obviously Bayesian analysis for nonlinear time series models is even harder. A possible solution to the problem is the Quasi Conjugate Analysis (QCA), where any constraints on the model coefficients are ignored and moving average terms are replaced by sensible estimates. The advantage of using this approach is that it leads to nice standard distribution results for the posterior densities which facilitates the calculations required for inference. Several authors used the QCA approach including Broemeling and Shaarawy (1988), Chen (1992) and Ismail (1994) among others and it is going to be employed in this paper. The objective of this paper is to develop an approximate non numerical Bayesian inferential technique for SETMA models. The approximate posterior densities for the model coefficients and precision in each regime are derived. Informative as well as noninformative priors are used. The adequacy of the proposed technique is checked using Monte Carlo simulations and a real data set. This paper is organized as follows. Section 2 briefly describes the SETMA model. The Posterior analysis is developed in Section 3. Section 4 addresses the adequacy of the Bayesian approach using simulations, and the new methodology is illustrated on a real data set. Section 5 concludes.

2. Threshold moving average models A time series yt is said to follow a SETMA of order (l;q1 ,q2 ,· · ·,ql ), if it satisfies q

yt = µ( j) + εt( j) +

j

( j) ( j)

θi εt−i , r j−1 ≤ yt−d ≤ r j

(1)

i=1

where j = 1, · · · , l, and d is a positive integer commonly referred to as the delay (or threshold lag) of the model, µ( j) , θ ( j) are constants. The l sequences

Bayesian inference for threshold moving average models

121

( j)

ε ( j) satisfy εt = σ j εt , where εt ∼ NID (0, 1) and where 0 < σ j < ∞. The real numbers r j (called thresholds) satisfy −∞ = r0 < r1 < · · · < rl = ∞ and form a partition of the space of yt−d and the partition r j−1 ≤ yt−d ≤ r j forms the j th regime of the SETMA model. Model (1) is said to be self exciting because changes in the model parameters are generated by past values of yt itself. In this paper, we shall take up the case where l = 2 in detail with µ(1) = µ(2) = 0. This model may be expressed as

yt =

q1  (1) (1)  ε + θi(1) εt−i ,    t i=1 q2

 (2) (1)    ε (2) + θi εt−i , t

if yt−d ≤ r1 (2) if yt−d > r1

i=1 ( j)

where Var(εt ) = τ j−1 . The upper and lower equations in (2) are sometimes called the lower regime and upper regime, respectively. De Gooijer and Brannas (1995) found empirically that the condition for this model to be invertible is that the product |θ (1) ||θ (1) + θ (2) | be less than one. The identification, estimation, and testing of this model were addressed in De Gooijer (1998).

3. Posterior analysis 3.1. The Conditional Likelihood Function Suppose Sn = (y1 , y2 , · · · , yn ) is a realization of a SETMA (2; q1 , q2 ) process. In this study we employ the concept of arranged autoregression where the cases (observations) are rearranged according to the threshold variable yt−d . The autoregression approach was used by Tsay (1989), Chen and Lee (1995) for SETAR model, and by Chen (1998) for generalized SETAR model. Employing the arranged autoregression approach, the SETMA(2; q1 , q2 ) given in equation (2) can be rewritten as

yπi +d =

q1    θ j(1) επ(1)i +d− j + επ(1)i +d    j=1 q2     θ j(2) επ(2)i +d− j + επ(2)i +d 

if i ≤ s otherwise,

j=1

) and γ (2) = where s satisfies yπs ≤ r < yπs +1 , γ (1) = (θ1(1) , θ2(1) , · · · , θq(1) 1 (θ1(2) , θ2(2) , · · · , θq(2) ). The initial errors ε1−q , · · · , ε−1 , ε0 (where q = max(q1 , q2 )) 2 are assumed zeros. The index of the i-th smallest observation of {y1 , y2 , · · ·, yn−d }

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is denoted by πi and define likelihood function η = (d, r ). The conditional of γ (1) , γ (2) , τ1 , τ2 , η | Sn is denoted by l = L γ (1) , γ (2) , τ1 , τ2 , η | Sn . It is obvious that s 2

n−s−d 2

s 2

n−s−d 2

l ∝ τ1 τ2

= τ 1 τ2

s n−d

2

2 τ1 τ2 × exp − εt(1) − εt(2) 2 i=1 2 i=s+1

  2  q1 s  τ 1  yπ +d − × exp − θ j(1) επ(1)i +d− j  i  2  i=1

−

τ2 2

j=1



n−d

 yπ

i +d

i=s+1

(3)

2    (2) (2)  − θ j επi +d− j   j=1 q2

It is clear that the likelihood is a complicated nonlinear function in the parameters θ j(i) (i = 1, 2; j = 1, 2, · · · , qi ). Suppose the errors εti (i = 1, 2) are estimated by nonlinear least squares as follows.

eπ(1)t +d

= yπt +d −

q1

θˆj(1) eπ(1)t +d− j +

t = 1, 2, · · · sˆ ,

j=1 ( j)

( j)

( j)

e1−q = · · · = e−1 = e0 = 0; ( j = 1, 2) eπ(2)t +d = yπt +d −

q2

(4)

θˆj(2) eπ(2)t +d− j + t = sˆ + 1, sˆ + 2, · · · n − d − sˆ

j=1

ˆ rˆ are nonlinear least squares estimates found where yπsˆ ≤ rˆ < yπsˆ +1 and θˆj , d, by minimizing

S(γ

(1)

, γ

(2)

, d, r ) =

s t=1

(εt(1) )2

+

n−d−s

(εt(2) )2

i=s+1

with respect to γ (1) , γ (2) , d, r. Substituting the estimated errors et(i) , i = 1, 2, in the conditional likelihood function l = L(γ (1) , γ (2), τ1, τ2 , η|Sn ) results in an approximate conditional likelihood function l which may be written in the

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form: s 2

n−s−d 2

l ∝ τ1 τ2 s

n−s−d 2

= τ12 τ2

s n−d τ1 τ2 exp − (et(1) )2 − (e(2) )2 2 t=1 2 t=s+1 t

  2  q1 s  τ 1  yπ × exp − − θ j(1) eπ(1)i +d− j  i+d  2  i=1

− s 2

n−s−d 2

= τ1 τ2

τ2 2

j=1



n−d

2    (2) (2)  − θ j eπi +d− j   j=1

(5)

q2

 yπ

i +d

i=s+1

τi (i)T (i) (i)T (i) (i)T (i) (i) × exp − y y − 2γ B +γ A γ 2 i=1 2

where y (1) = yπ1 +d , yπ2 +d , · · · , yπs +d

T

is a vector containing the observa-

tions of the first regime, y (2) = yπs+1 +d , yπs+2 +d , · · · , yπn−d +d

T

is a vector

containing the observations of the second regime, X(1) = X 1(1) , · · · , X s(1)

where X t(1) = eπ(1)t +d−1 , eπ(1)t+d −2 , · · · eπ(1)t +d−q1

T

T

T

T

,

(2) (2) , X(2) = X s+1 ,· · ·, X n−d , where

X t(2) = eπ(2)t +d−1 , eπ(2)t +d−2 , · · · eπ(2)t +d−q2 , A(i) = X(i) X(i) , and B (i) = X(i) y (i) , i = 1, 2. Note that X(i) , A(i) , and B (i) , i = 1, 2 are functions in η = (d, r ); however, the arguments are suppressed for simplicity. T

T

3.2. Prior information Prior specification is an important step in developing any Bayesian analysis. Both proper and improper prior distributions are used to represent prior information about the parameters. In both cases η = (d, r ) is treated as independent of the coefficients γ (i) (i = 1, 2) and precision parameters τi , i = 1, 2. Thus, a suitable choice for the proper prior distribution for the model parameters γ (i) and τi is a normal gamma distribution. That is −1

ζ (γ (i) | τi ) ∼ N (µ(i) , τi−1 V(i) ), ζ (τi ) ∼ (ai , bi ) The approximate conditional likelihood function for each regime, as a function in the parameters γ (i) , τi (given d, r) is a normal gamma density.

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Hence the joint prior of γ (i) , τi may be written as:

q +2a τi ( i 2 i )−1 (i) (i) (i) T (i) (i) (i) ζ γ , τi ∝ τ exp − 2bi +(γ − µ ) V (γ − µ ) (6) 2 where V(i) is a square positive definite matrix of order qi , ai > 0, bi > 0. The marginal prior for η, namely ζ (η) could be any distribution. So, the joint prior of the model parameters, ζ γ (1) , γ (2) , τ1 , τ2 , η may be written as:

ζ γ (1), γ (2), τ1 , τ2 , η = ζ (η)

2

ζ γ (i) | τi ζ (τi )

i=1 2

q

τi

∝ ζ (η)

i +2ai −1 2

exp −

i=1

τi 2bi +(γ (i) −µ(i) )T 2 (i)

V (γ

(i)

(i)

−µ )

(7)

This class of priors is flexible enough to be used in many applications. It also simplifies the mathematical calculations. It is, at least conditionally, the conjugate prior for the approximate conditional likelihood since the approximate likelihood given by (5) conditioning on η = (d, r ), is a normal gamma function, which has the same form as ζ γ (i) , τi given by (6). However, the use of the proposed prior distribution (6) is not a necessity and any other distribution could be used provided we are prepared to use numerical integration. If one has little information about the hyperparameters, or is unwilling to determine them, one may use the improper Jeffreys’ prior, namely

1 (8) ζ γ (1) , γ (2) , τ1 , τ2 ∝ τ1 τ2 Jeffreys’ prior distribution is a special case of the normal gamma class when bi = 0, V(i) = 0, ai = − q2i . 3.3. Posterior analysis Multiplying the approximate conditional likelihood function l by the prior distribution ζ (γ (1) , γ (2) , τ1 , τ2 , η) given by (7) results in the approximate joint posterior distribution which may be written as:

ζ (γ

(1)

,γ

(2)

, τ1 , τ2 , η | Sn ) ∝ ζ (η)

2

n

τi

i +qi +2ai −1 2

exp −

i=1

− 2γ

(i)T

(i)

B (γ

(i)

τi T 2bi + Y (i) Y (i) 2

(i) T

(i)

− µ ) V (γ

where n i is the number of observations in the i th regime.

(i)

(i)

−µ )

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In principle, the marginal posterior distribution of γ (i) for example can be found from ζ

γ

(1)

| Sn =

γ (2)

τ1

τ2

η

ζ γ (1), γ (2), τ1 , τ2 , η | Sn ) dγ (2) dτ1 dτ2 dη.

It is not difficult to show that the marginal posteriors of γ (1) , γ (2) , τ1 , τ2 and η are all non-standard. So, all exact Bayesian inferences must be done numerically. However, the following two theorems show that the approximate posterior distributions of γ (i) and τi given η are standard. Theorem 3.1. Given n observations Sn = (y1 ,y2 , · · · yn ) from the SETMA (2; q1 ,q2 ) model given by (2), a conditional likelihood function approximated by (5) and a prior distribution given by (7), then the approximate conditional marginal posterior distribution of γ (i) given η isa Tm distribution with n+ 2a degrees of freedom, location vector E γ (i) | Sn , η and precision matrix T γ (i) | Sn , η , where

E γ (i) | Sn , η = A(i) + V(i)

T γ

(i)

| Sn , η =

n i + 2ai Ci

−1

B (i) + V(i) µ(i) ,

A(i) + V(i) ,

T

T

Ci = 2bi + y (i) y (i) + µ(i) V (i) µ(i) − B (i) + V(i) µ(i)

× A(i) + V(i)

−1

B (i) + V(i) µ(i)

T

and X(i) , A(i) , B (i) are defined in connection to (5). Theorem 3.2. Under the same conditions of Theorem 3.1, the approximate coni ditional posterior of τi given η is a gamma distribution with parameters ni +2a 2 Ci and 2 . The approximate conditional posterior distribution of τi , ζ (τi | η, Sn ), can be used to make inferences about τi . Unlike γ (i) which has a symmetric T distribution, τi has a skewed distribution. Quoting the mean and the variance could be misleading. Instead, an approximate credible interval is constructed as follows. It is not difficult to show that Ci τi ∼ χn2i +2ai . Therefore, a (1 − α) equal tailed credible interval for τi is given by : χn2 +2a , i

i

α 2

Ci

≤ τi ≤

χn2 +2a , i

i

1− α2

Ci

where χn2i ,α is defined so that p χn2i > χn2i ,α = α.

,

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The next two corollaries report the approximate posterior distributions of γ (i) and τi given η when little information about the parameters are known a priori. Corollary 3.1. If the parameters γ (i) , τi and η are assumed independent a priori and have Jeffreys’ prior distribution (8) and the other conditions of theorem 3.1 are not changed, the approximate posterior distribution of γ (i) given η is a T distribution −1 with (n i − qi ) degrees of freedom, location vector A(i) B (i) and precision matrix n i −qi A(i) . (i)T (i) (i)T (i)− 1 (i) y

y

−B

A

B

Corollary 3.2. Under the same set-up as Corollary 3.1, the approximate posterior i ) and distribution of τ given η is gamma with parameters ( ni −q 2

T

B (i) A(i) 2

−1 (i) B

The marginal posterior function of η may be obtained by integrating out γ (i) , τi from the approximate joint posterior function. ζ (η | Sn ) ∝ ζ (η)

2

i=1

n i +ai 2

| A(i) + V(i) |

n

(Ci )−

i +2ai 2

.

where, Ci and A(i) are defined as before. If Jeffreys’ prior for γi and τi is employed, the marginal posterior of η takes the form:

2 n i −qi

− ni +2ai 2 2 (i) T (i) (i) T (i) −1 (i) y y −B A B ζ (η | Sn ) ∝ ζ (η) i=1

| A(i) |

It’s clear that the marginal posterior of η is non-standard. Thus all inferences about η must be done numerically. 3.4. Posterior analysis of a SETMA (2; 1, 1) Model The approach described previously is illustrated in this section for the simple SETMA (2; 1, 1) model given by: yπi +d =

(1) θ (1) ε (1) 1 πi +d−1 + επi +d

θ1(2) επ(2)i +d− j + επ(2)i +d

i ≤s otherwise.

Assume θ1(1) , θ1(2) , τ1 , τ2 and η are independent a priori, and have a vague prior,

1 . ζ θ (1) , θ (2) , τ1 , τ2 , η ∝ τ1 τ2

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Using Corollaries (3.1) and (3.2) we get

B (i) (n i − 1) A(i) ζ (θ (i) | Sn , η) ∼ T1 n i − 1, (i) , A Ci ! " n i − 1 Ci ζ (τi | Sn , η) ∼

, 2 2

where,

A

(1)

=

n 1

2 eπ(1)i +d−1

,

B

(1)

=

n 1

i=1



A(2) = 

eπ(1)i +d−1

,

i=1

n 1 +n 2

i=n 1 +1

2





eπ(2)i +d−1  ,

B (2) = 

T

C1 = y (1) y (1) − B (1) A(1)

−1



n 1 +n 2

i=n 1 +1

y (1) = yπ1 +d , yπ2 +d ,· · ·, yπn1 +d T

yπ(1) i +d

T

,

B (1),

y (2) = yπn

, yπ(2) e(2) i +d πi +d−1

T

+d , yπn +2 +d ,· · ·, yπn 1 +n 2 +d 1 +1 1 T

T

C2 = y (2) y (2) − B (2) A(2)

−1

,

B (2) .

Therefore, for i = 1, 2, E(θ

(i)

(i) −1

| Sn , η) = A

E(τi | Sn , η) =

(i)

B ,

ni − 1 , Ci

Var (θ

(i)

| Sn , η) =

Var (τi | Sn , η)

=

Ci −1 A(i) , ni − 3

2(n i − 1) . Ci2

The marginal posterior distribution of η is given by

2 n i −1

− ni −1 T T −1 2 2 ζ (η | Sn ) ∝ y (i) y (i) − B (i) A(i) B (i) . i=1

| A(i) |

4. Illustrative examples In this section, the proposed methodology is illustrated with a simulation study and a real data set. In both cases, a non informative prior is used.

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4.1. Simulation This study deals with generating 100 sets of n = 200 observations from the following SETMA(2;1,1) model yt =

−0.5ε (1) + ε (1) t t−1

yt−1 ≤ 0.4

(2) 0.5εt−1 + εt(2)

otherwise,

(9)

where the delay parameter = 0.4, # and where

d = 1, the threshold parameter # r$ $ (1) (2) (1) −1 −1 and εt(2) are εt ∼ N 0, τ1 = 0.5 , εt ∼ N 0, τ2 = 1 , and εt independent. The above SETMA (2;1,1) model is a dual model to the SETAR used in the simulation study of Chen and Lee (1995). The initial values of the errors are set to zeros and using the equation (9), 400 observations are generated. The first 200 observations are deleted to remove the initialization effect. A lower limit of 1 and upper limit of 5 are used for d. Thus, the candidate domain for d is 1, 2, · · · , 5. The candidate domain for the threshold is chosen to be P10 , P20 , · · · P90 where Pi is the i th percentile of the data. For each generated series, the least squares estimates are calculated and the residuals are computed. Then, exact and posterior means of θ1(1) , θ1(2) , τ1 and τ2 are calculated. The mode of the marginal posterior distribution is chosen as a Bayesian estimator for both of d and r . The simulation results are shown in Table 1. Columns 3 through 9 contain the mean, standard deviation, minimum (Min), first quartile (Q 1 ), median, third quartile (Q 3 ), and maximum (Max) for 100 posterior modes of d and r and for 100 posterior means of the other parameters. Each of the parameters θ1(1) , θ1(2) , τ1 and τ2 has a block of two rows in Table 1. The upper row shows descriptive statistics for approximate posterior estimates obtained by the proposed approximation while the second row presents descriptive statistics for exact posterior estimates obtained by numerical integration. The last row of Table 1 presents the previous descriptive statistics for nonlinear least squares estimates of r . The frequency histogram of posterior modes of r for the 100 generated data sets is displayed in figure 1. It seems that the distribution is nearly symmetric. Inspection of the results in Table 1 shows that the proposed Bayesian methodology gives sound inferences for the parameters. The delay parameter d is correctly estimated with d = 1 in all 100 data sets. The estimates of θ1(1) , θ1(2) have negative bias while those of other parameters are nearly unbiased. The comparison of elements of each pair of rows for each coefficient and precision parameter shows clearly how close are the approximate posterior estimates to those obtained by numerical integration. Moreover, the investigation of the last two rows of Table 1 indicates that the Bayesian estimates for r are closer to the true value than the classical nonlinear least squares ones.

129

Bayesian inference for threshold moving average models Table 1: Simulation Results for 100 simulated data sets from model (9). Parameters

True Value

Mean

Standard Deviation

Min

Q1

Median

Q3

Max

−0.5

−0.603 −0.602

0.111 0.099

−0.959 −0.8

−0.677 −0.77

−0.597 −0.6

−0.529 −0.5

−0.384

0.5

0.398 0.405

0.091 0.085

0.142 0.1

0.345 0.4

0.41 0.4

0.457 0.5

0.629 0.6

τ1

2

2.055 2.032

0.34 0.344

1.416 1.417

1.826 1.824

2.016 2.001

2.266 2.193

3.121 3.158

τ2

1

1.04 1.103

0.16 0.176

0.704 0.77

0.924 0.968

1.025 1.115

1.159 1.217

1.442 1.57

d

1

1

0

1

1

1

1

1

r

0.4

0.363 0.28

0.259 0.32

0.201 0.103

0.382 0.358

0.502 0.491

1.292 0.999

(1)

θ1

(2)

θ1

−0.658 −0.91

45

40

35

30

25

20

15

10

5

0

-1

-0.5

0

0.5

1

1.5

Figure 1. Histogram of posterior modes for r .

4.2. Unemployment Rates The series of US unemployment rates was analyzed by McCulloch and Tsay (1993). They identified a SETAR (2;2,4) model for the changes of the series with d = 1 and r = 0.3. Chen and Lee (1995) developed a Bayesian analysis

130


of the series Employing a Gibbs sampling approach to calculate the posterior densities of the parameters. A SETAR (2;2,4) model with d = 1 and r = 0.299 was used. 2

1.5

1

0.5

0

-0.5

-1

0

20

40

60

80

100

120

140

160

180

Figure 2. Time Plot for Changes of Us Unemployment Rates.

Assuming a non informative prior, the proposed Bayesian methodology is applied to the changes of the US unemployment rates series from the second quarter of 1948 to the first quarter of 1993 using a SETMA (2;1,1) model. The posterior mode of d is 1 which is identical to that of McCulloch and Tsay (1993) and Chen and Lee (1995). The posterior mode of r is 0.267 which is close to its estimates found by McCulloch and Tsay (1993) and Chen and Lee (1995). Conditioning on d = 1 and r = 0.267, the Bayesian estimates of the coefficients and precision parameter in each regime are reported in Table 2. Table 2: Bayesian Estimates of Changes in US. Unemployment Rates with d = 1 and r = 0.267. (1)

θ1 Mean Standard Error

0.569 0.093 (2)

Mean Standard Error

θ0 0.141 0.061

(2)

θ1 0.754 0.104

τ1

n1

11.779 1.394

144

τ2 4.649 2.374

n2 34


131

It should be noted that our methodology like Chen and Lee (1995), does not need a subjective determination of the threshold parameter r via scatter plots. In addition, our estimated model is more parsimonious than SETAR models which are used in McCulloch and Tsay (1993) and Chen and Lee (1995).

5. Conclusion and open problems We have shown that a complete Bayesian inference for SETMA model is possible and is no more difficult than doing switching regression. It is shown that, conditioning on the delay and threshold parameters, the conditional posteriors of the model coefficients and precision are approximated by a multivariate t and gamma densities. Therefore, the need for numerical integration is avoided. The proposed technique is demonstrated by working out details for a SETMA (2;1,1) model. Moreover, a Monte Carlo study is conducted and the results indicate the success of the Bayesian approach in analyzing SETMA time series data. We assumed that the model is identified, i.e. the model orders are known. However, the identification problem may be investigated by employing Bayes Factors and posterior odds to identify a model from the different models corresponding to all possible values of the orders, see for example Barbieri and Conigliani (2000). In addition, the forecasting problem may be studied by deriving the predictive density of future observations. Other analytic approximations such as Lindley (1980) and Tierney and Kadane (1986) can be tried to implement a Bayesian analysis for a general nonlinear ARMA model. Moreover, the relationship between Broemeling and Shaarawy’s approximation and the above mentioned approximations need to be investigated. In addition, Monte Carlo based methods such as Gibbs Sampling needs to be compared with the analytic approximations.

Acknowledgments The authors wish to express thanks to the referee for the helpful comments which have improved the exposition of the paper.

REFERENCES Barbieri, M. M. and Conigliani, C. (2000) “Bayesian analysis of threshold autoregressive time series with change points”, Technical Report, Universit`a di Roma “La Sapienza”, Roma, n.19. Broemeling, L. D. and Cook, P. (1992) “Bayesian analysis of threshold autoregressions”, Communications in Statistics: Theory and Methods, 21, 2459-2482.

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MOHAMED A. ISMAIL Department of Statistics UAE University P.O. Box 17555 Al-Ain (United Arab Emirates) [email protected]

HUSNI A. CHARIF UAE University P.O. Box 17555 Al-Aim (United Arab Emirates)