GIBBS SAMPLING FOR SARMA MODELS

GIBBS SAMPLING FOR SARMA MODELS

Mohamed A. Ismail* and Ayman A. Amin**

Working Paper No. 7

2010

*Faculty of Economics and Political Science, Statistics Department, Cairo University, Egypt, and

consultant, Economic Issues Program (EIP), Information and Decision Support Center (IDSC) of the

Egyptian Cabinet, Cairo, Egypt (email: [email protected]). **Faculty of Commerce, Statistics Department, Menoufia University, Menoufia, Egypt (email:

[email protected]).

This study was presented at the Inauguration Conference for Launching Information and Decision

Support Center-Working Paper Series (IDSC-WPS), held on Sunday 28 March 2010, in Cairo, Egypt.

The contents of the study and opinions expressed are the sole responsibility of the author(s) and do

not necessarily reflect the views of the IDSC.

ABSTRACT

This paper introduces a fast, easy and accurate Gibbs sampling algorithm to develop a

Bayesian inference for a multiplicative seasonal autoregressive moving average (SARMA)

model. The proposed algorithm uses values generated from normal and inverse gamma

distributions and does not involve any Metropolis-Hastings generation. Simulated examples

and a real data set are used to illustrate the proposed algorithm.

صلخم

ة دم درا ا ا ة وارز ة )Gibbs( ا ر ة ا هو ة وا د ر وا دالل طو زي اال وذج ا

دار ذا اال ط -ا و رك ا و ا ر ا وارزم دم). SARMA( ا رح ا وز ا ا ا

ط وز ا ا، وب و دة ا وال جا دام و وب ا د. )Metropolis-Hastings( أ و

دام ة ا ث ات اآاة أ ا وارزم و ة و رح ا .ا

1

1. INTRODUCTION

Seasonal Autoregressive Moving Average (SARMA) modeling of time series has been successfully

applied in a great number of fields including economic forecasting. Bayesian analysis of

Autoregressive Moving Average (ARMA) type models is difficult even for non-seasonal models,

since the likelihood function is analytically intractable, which causes problems in prior specification

and posterior analysis. Different solutions, including Markov Chain Monte Carlo (MCMC) methods,

have been suggested in the literature for the Bayesian time series analysis. Several authors have

considered Bayesian analysis of ARMA models (Newbold 1973; Monahan 1983; Broemeling, and

Shaarawy 1984; Shaarawy, and Ismail 1987; Marriott, and Smith 1992).

Bayesian time series analysis has been advanced by the emergence of MCMC methods,

especially the Gibbs sampling method. Assuming a prior distribution on the initial observations and

initial errors, Chib, and Greenberg (1994); Marriott et al. (1996) developed Bayesian analysis for

ARMA models using the MCMC technique. Barnett, Kohn, and Sheather (1996, 1997) used the

MCMC methodology to estimate the multiplicative and ARMA model. Their algorithm was based on

sampling functions of the partial autocorrelations. A merit of their approach is that one for one draws

of each partial autocorrelation can be obtained but at the cost of a more complicated algorithm.

Recently, Ismail (2003a, 2003b) used Gibbs sampling algorithm to analyze multiplicative

seasonal autoregressive and seasonal moving average models. His algorithm was based on

approximating the likelihood function via estimating the unobserved errors. Then, the approximate

likelihood is used to derive the conditional distributions required for implementing Gibbs sampler.

Rather than restricting the parameters space to satisfy stationarity and invertibility conditions as in

Barnett, Kohn, and Sheather (1997); Marriott et al. (1996) among others, the process could be made

stationary and invertible by choosing the hyperparameters, which ensure that the prior for the model

coefficients lies in the stationarity and invertibility region. The latter approach was used by

Broemeling (1985); McCulloch, and Tsay (1994); Ismail (2003a, 2003b) among others and is going to

be used in this paper.

The objective of this paper is to extend Ismail's (2003a, 2003b) algorithm to multiplicative

seasonal ARMA models. The proposed algorithm does not involve any Metropolis-Hastings iteration,

which is an advantage over other algorithms in the literature. In addition, our analysis is not

conditional on the initial values; that is, we assume that the series starts at time t = 1 with unknown

initial observations and errors. Moreover, various features of the SARMA models, which may be

complicated to check in the classical framework, may be routinely tested in the sampling based

Bayesian framework. As an example, there is often interest in testing the significance of interaction

parameters, which are the product of the nonseasonal and seasonal coefficients in the model. The

proposed algorithm can easily construct confidence intervals for interaction parameters and therefore

test their significance.

The paper is organized as follows. Section 2 briefly describes the multiplicative SARMA

model. Section 3 is devoted to summarizing posterior analysis and the full conditional posterior

distributions of the parameters. The implementation details of the proposed algorithm including

convergence monitoring are given in Section 4. The proposed methodology is illustrated in Section 5

using simulated examples and Egyptian imports of durable consumption goods series. Finally, the

conclusion is given in Section 6.

2

2. MULTIPLICATIVE SARMA MODELS

A time series { is said to be generated by a multiplicative SARMA model of orders p, q, P and Q,

denoted by SARMA (p, q)(P, Q)s, if it satisfies (1)

where ; I is a sequence of independent normal variates with zero mean and variance ,

and I is the set of integers. The backshift operator B is defined such that , s is the

number of seasons in the year. The nonseasonal autoregressive polynomial is … with order p, the nonseasonal moving average polynomial is … with order q, the seasonal autoregressive polynomial is … with order P, and … is the seasonal moving average polynomial with order Q.

The nonseasonal and seasonal autoregressive coefficients are … and … , and the nonseasonal and seasonal moving average coefficients are … and … . Each of the nonseasonal and seasonal orders p, q is always

less than or equal to the number of seasons in the year s. The time series ; I is assumed to start

at time t = 1 with unknown initial values … and unknown initial errors … .

The model (1) can be written as: ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ (2)

where, , … , ; ; , , … , ; ; … ; , , … , , , … , ; ; , , … , ; ; … ; , , … , , , … , ; ; , , … , ; ; … ; , , … , , … , ; ; , , … , ; ; … ; , , … , , (3)

and and are (s – p – 1) and (s – q – 1) row vectors of zeros, respectively. Model (2) shows that

the multiplicative SARMA model can be written as an ARMA model of order p + Ps and q + Qs with

some zero coefficients and some coefficients that are products of nonseasonal and seasonal

coefficients. Thus, the model is nonlinear in , , and , which complicates the Bayesian analysis.

However, the following sections explain how Gibbs sampling technique can facilitate the analysis.

The SARMA model (2) is stationary if the roots of the polynomials and lie outside

the unit circle, and when the roots of the polynomials and lie outside the unit circle,

the process is invertible. For more details about the properties of SARMA models. (Box and Jenkins

1976).

3

3. POSTERIOR ANALYSIS

3.1 Likelihood Function

Suppose that … is a realization from the multiplicative SARMA model (2),

assuming that the random errors ~ , and employing a straightforward random variable

transformation from to , the likelihood function , , , , , , | = is given by:

∑ (4)

where, ∑ ∑ ∑ ∑ — ∑ ∑ ∑ ∑ — . (5)

The likelihood function (4) is a complicated function in the parameters , , , , and . Suppose

the errors are estimated recursively as:

∑ ∑ ∑ ∑ ∑ ∑∑ ∑ , (6)

where , , , are sensible estimates. Several estimation

methods, such as the Innovations Substitution (IS) method proposed by Koreisha and Pukkila (1990),

give consistent estimates for , , and . The idea of the IS method is to fit a long autoregressive

model to the series and obtain the residuals. Then appropriate lagged residuals are substituted into

SARMA model (2). Finally, the parameters are estimated using the ordinary least squares method.

Substituting the residuals in the likelihood function (4) results in an approximate likelihood

function: ∑ (7)

where, ∑ ∑ ∑ ∑ — ∑∑ ∑ ∑ — . (8)

, are defined in (3), is a matrix with tth row , … , ; ; , , … , ; ; … ; , , … , , where is a (s – q – 1) row vector of zeros.

3.2 Prior Specification

For multiplicative SARMA models, suppose that, given the error variance parameter , the

parameters , , , , and are independent a priori, i.e.,

4

, , , , , , | | | | || , ∑ , ∑ , ∑ , ∑, ∑ , ∑ , (9)

where, μ, ∆ is the r-variate normal distribution with mean µ and variance ∆ and IG(α, β) is the

inverse gamma distribution with parameters α and β. Such prior distribution is a normal inverse

gamma distribution and can then be written as: , , , , , , ∑ ∑ ∑ ∑∑ ∑ (10)

The prior distribution (10) is chosen for several reasons. It is flexible enough to be used in

numerous applications; it also facilitates the mathematical calculations and it is, at least conditionally,

a conjugate prior.

Multiplying the joint prior distribution (10) by the approximate likelihood function (7) results

in the joint posterior , , , , , , | , which may be written as:

, , , , , , | exp ∑∑∑ ∑ ∑ ∑ ∑ ∑ ∑ (11)

The conditional posterior distribution for each of the unknown parameters is obtained from the

joint posterior distribution (11) by grouping together terms in the joint posterior that depend on this

parameter, and finding the appropriate normalizing constant to form a proper density. In this study, all

conditional posteriors are normal and inverse gamma distributions for which sampling techniques

exist. The full conditional distributions of the SARMA parameters are derived in the Appendix.

4. THE PROPOSED GIBBS SAMPLER

The proposed Gibbs sampling algorithm for the multiplicative SARMA model given in equation (2)

can be conducted as follows:

Step 1: Specify starting values , , , , , and and set . A set of initial

estimates of the model parameters can be obtained using the IS technique of Koreisha, and

Pukkila (1990).

Step 2: Calculate the residuals recursively using (6).

Step 3: Obtain the full conditional posterior distributions of the parameters.

5

Step 4: Simulate

• ~ | , , , , , , ,

• ~ | , , , , , , ,

• ~ | , , , , , , ,

• ~ | , , , , , , ,

• ~ | , , , , , , ,

• ~ | , , , , , , , and

• ~ | , , , , , , .

Step 5: Set and go to step (4).

This algorithm gives the next value of the Markov chain

{ , , , , , , by simulating each of the full conditional

posteriors where the conditioning elements are revised during a cycle.

This iterative process is repeated for a large number of iterations and convergence is

monitored. After the chain converges, after n iterations, the simulated values { , ,, , , , , , are used as a sample from the joint posterior. Posterior

estimates of the parameters are computed directly via sample averages of the simulation outputs.

A large and growing literature deals with techniques for monitoring convergence of Gibbs

sampling sequences. In what follows, we shall summarize the diagnostics that will be used in the case

of a multiplicative SARMA model:

1. Autocorrelation estimates, which indicate how much independence exists in the sequence of each

parameter draws. A high degree of autocorrelation indicates that more draws are needed to get

accurate posterior estimates.

2. Raftery, and Lewis (1992, 1995) proposed a set of diagnostics which includes:

• a thinning ratio (Thin) which is a function of the autocorre1ation in the draws;

• the number of draws (Burn) to use for initial draws or 'burn-in' before starting to sample the

draws for the purpose of posterior inference;

• the total number of draws (Total) needed to achieve the desired level of accuracy;

• the number of draws (Nmin) that would be needed if the draws represented an iid chain.

• (I-stat), which is the ratio of the (Total) to (Nmin). Raftery and Lewis suggested that a

convergence problem may be indicated when values of (I- stat) exceed 5.

3. Geweke (1992) proposed two groups of diagnostics:

a) The first group includes the numerical standard errors (NSE) and relative numerical efficiency

(RNE). The NSE estimates reflect the variation that can be expected, if the simulation were to be

repeated. The RNE estimates indicate the required number of draws to produce the same

numerical accuracy when iid sample is drawn directly from the posterior distribution. The

estimates of NSE and RNE are based on spectral analysis of time series, where two sets of these

estimates are obtained. The first set is based on the assumption that the draws come from iid

process. The second set is based on different tapering or truncating of the periodgram window.

When there are large differences between the two sets, the second set of estimates would be

chosen because it would take into consideration autocorrelations in the draws.

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b) The second group of diagnostics includes a test of whether the sampler has attained an

equilibrium state. This is done by carrying out Z-test for the equality of the two means of the first

and last parts of draws and the Chi squared marginal probability is obtained. Usually, the first

and last parts are the first 20 percent and the last 50 percent of the draws.

LeSage (1999) implemented calculations of the above convergence measures using the Matlab

package. These diagnostics will be used in Section 5 to monitor the convergence of the proposed

algorithm.

5. ILLUSTRATIVE EXAMPLES

5.1 Simulated Examples

In this subsection, we present two examples with simulated data to evaluate the efficiency of the

proposed methodology. The two examples deal with generating 250 observations from

SARMA(1,1)(1,1)4 and SARMA(1,1)(1,1)12 models, respectively. The two simulated examples are as

follows:

1) . . . . . .

2) . . . . . .

The analysis was implemented using Matlab and running on Pentium PC 2.53 GHZ took

several seconds (90 seconds on average) to complete. The error variance was chosen to be 0.5 in

the first example and 1 in the second example. A non informative prior was assumed for , , , , , and via setting ∑ ∑ ∑ ∑ (all these matrices are

scalars), and . A normal prior with zero mean and variance was used for the

initial observations vector , and with zero mean and variance was used for the initial errors

vector . The starting values for the parameters , , and were obtained using the IS method.

The starting values for and were assumed to be zeros.

Now, the implementation of the proposed Gibbs sampler is straightforward. For each data set,

the Gibbs sampler was run 11,000 iterations, where the first 1,000 draws are ignored and every tenth

value in the sequence of the last 10,000 draws is recorded to have an approximately independent

sample. All posterior estimates are computed directly as sample averages of the simulated outputs.

Table 1 presents the true values and Bayesian estimates of the parameters for example 1.

Moreover, a 95 percent confidence interval using the 0.025 and 0.975 percentiles of the simulated

draws is constructed for every parameter. From table 1, it is clear that Bayesian estimates are close to

the true values and the 95 percent confidence interval includes the true value for every parameter.

Sequential plots of the parameters generated sequences together with marginal densities are displayed

in figure 1. The marginal densities are computed using non parametric technique with Gaussian

kernel. Figure 1 shows that the proposed algorithm is stable and fluctuates in the neighborhood of true

values. In addition, the marginal densities show that the true value of each parameter (which is

indicated by the vertical line) falls in the constructed 95 percent confidence interval.

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TABLE 1. Bayesian Results for Example 1

Parameter True

valuesMean

Std.

Dev.

Lower

95% limitMedian

Upper

95% limit

0.2 0.118 0.095 -0.062 0.117 0.306

0.8 0.805 0.023 0.760 0.805 0.849

0.3 0.385 0.107 0.184 0.390 0.592

0.9 0.802 0.065 0.675 0.801 0.927

0.5 0.536 0.046 0.453 0.533 0.634

Figure 1. Sequential Plots and Marginal Posterior Distributions of Example 1

The convergence of the proposed algorithm is monitored using the diagnostic measures

explained in Section 4. The autocorrelations and Raftery, and Lewis diagnostics are displayed in table

2, whereas table 3 presents the diagnostic measures of Geweke (1992). Table 2 shows that the draws

for each of the parameter have small autocorrelations at lags 1, 5, 10 and 50, which indicates no

convergence problem. This conclusion was confirmed by the diagnostic measures of Raftery and

Lewis where the thinning estimate (Thin) is 1, the reported (Nmin) is 937, which is close to the 1000

draws we used and I-stat value is 0.953, which is less than 5. Scanning the entries of table 3 confirms

the convergence of the proposed algorithm, where Chi squared probabilities show that the equal

means hypothesis cannot be rejected and no dramatic differences between the NSE estimates are

found. In addition, the RNE estimates are close to 1, which indicates the iid nature of the output

sample.

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TABLE 2. Autocorrelations and Raftery - Lewis Diagnostics for Example 1

Par. Autocorrelations Raftery - Lewis Diagnostics

Lag 1 Lag 5 Lag 10 Lag 50 Thin Burn Total(N) (Nmin) I-stat

-0.041 0.025 0.006 -0.094 1 2 893 937 0.953

0.001 0.084 0.084 -0.001 1 2 893 937 0.953

-0.005 0.032 0.032 -0.037 1 2 893 937 0.953

0.006 -0.011 -0.011 -0.025 1 2 893 937 0.953

0.007 -0.035 -0.035 -0.010 1 2 893 937 0.953

TABLE 3. Geweke Diagnostics for Example 1

Par. NSE

iid

RNE

iid

NSE

4%

RNE

4%

NSE

8%

RNE

8%

NSE

15%

RNE

15%

0.00196 1 0.00188 1.078 0.0016 1.498 0.00121 2.60 0.369

0.00123 1 0.00116 1.118 0.0012 0.996 0.00110 1.25 0.899

0.00250 1 0.00242 1.062 0.0022 1.255 0.00175 2.04 0.515

0.00222 1 0.00255 0.758 0.0028 0.647 0.00260 0.73 0.019

0.00139 1 0.00151 0.844 0.0014 1.020 0.00146 0.90 0.996

A procedure similar to that used for example 1 is repeated for example 2 and the true values

and Bayesian results are shown in table 4. Similar conclusions to those of example 1 are obtained. The

convergence diagnostics for example 2 are displayed in tables 5 and 6. Our Gibbs sampler is applied

to several simulated data from other SARMA(1,1)(1,1)s models, which do not appear here. The results

for these data sets are similar to results of examples 1 and 2 and therefore are not included.

TABLE 4. Bayesian Results for Example 2

Parameter True

Values Mean

Std.

Dev.

Lower

95% limit Median

Upper

95% limit

0.5 0.517 0.050 0.423 0.517 0.612

0.8 0.817 0.024 0.774 0.816 0.867

0.4 0.407 0.075 0.266 0.407 0.555

0.7 0.713 0.064 0.591 0.713 0.839

1.0 0.950 0.085 0.797 0.943 1.138

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TABLE 5. Autocorrelations and Raftery - Lewis Diagnostics for Example 2

Par. Autocorrelations Raftery - Lewis Diagnostics

Lag 1 Lag 5 Lag 10 Lag 50 Thin Burn Total(N) (Nmin) I-stat

-0.014 0.046 0.046 -0.052 1 2 969 937 1.034

0.006 -0.026 -0.041 0.010 1 2 969 937 1.034

-0.030 0.064 0.012 -0.029 1 2 969 937 1.034 Θ -0.027 -0.010 -0.012 -0.047 1 2 969 937 1.034

0.034 0.008 0.015 -0.004 1 2 969 937 1.034

TABLE 6. Geweke Diagnostics for Example 2

Par. NSE

iid

RNE

iid

NSE

4%

RNE

4%

NSE

8%

RNE

8%

NSE

15%

RNE

15%

0.00157 1 0.00140 1.266 0.0011 2.143 0.00090 3.07 0.763

0.00075 1 0.00075 1.005 0.0007 1.000 0.00067 1.26 0.819

0.00238 1 0.00209 1.288 0.0017 1.958 0.00176 1.82 0.962

0.00203 1 0.00169 1.444 0.0014 2.213 0.00118 2.97 0.593

0.00269 1 0.00271 0.983 0.0027 0.975 0.00294 0.84 0.033

5.2 Egyptian Imports of Durable Consumption Goods

The Egyptian imports of durable consumption goods series (y ) consists of 228 monthly values from

January 1980 to December 1998. Using the Box-Jenkins methodology, the following

SARIMA(0,1,1)(1,1,1)12 model may be identified for y : (12)

where,

In this section, the proposed Bayesian analysis is applied to the differenced Egyptian imports of

durable consumption goods series . The hyperparameters and starting values are chosen as in the

simulated examples. Table 5 summarizes the Bayesian results for the differenced Egyptian imports of

durable consumption goods series. Sequential plots and marginal densities of the differenced Egyptian

imports of durable consumption goods series are displayed in figure 2.

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TABLE 7. Bayesian Results for the Differenced Egyptian Imports of Durable Consumption Goods Series Using

SARMA(0,1)(1,1)12

Parameter Mean Std.

Dev.

Lower

95% limit Median

Upper

95% limit

0.091 0.069 -0.047 0.089 0.219

-0.644 0.071 -0.780 -0.644 -0.509

-0.646 0.087 -0.816 -0.647 -0.476

0.118 0.011 0.097 0.117 0.141

Figure 2. Sequential Plots and Marginal Posterior Distributions of the Differenced Egyptian Imports of Durable

Consumption Goods Series

It is worthwhile to test the significance of the interaction parameter in the above

SARIMA(0,1,1)(1,1,1)12 model. Although the testing procedure of the significance of is

complicated or even impossible in the classical approach framework, it is straightforward in the

suggested Bayesian framework. Using the proposed Gibbs sampling algorithm, the marginal posterior

distribution of is obtained and displayed in figure 3. Moreover, a 95 percent credible interval for

is . . , which supports the significance of the interaction coefficient in the model.

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Figure 3. Marginal Posterior Distribution of the Interaction Parameter η

6. CONCLUSION

In this paper, we developed a simple and fast Gibbs sampling algorithm for estimating the parameters

of the multiplicative SARMA model. The empirical results of the simulated examples and real data

set showed the accuracy of the proposed methodology. An extensive check of convergence, using

several diagnostics, showed that the convergence, of the proposed algorithm was achieved.

Although the employed prior distribution in Section 3 is informative, a noninformative prior is

used for the parameters , , , and in the illustrative examples for the sake of simplicity.

However, if one needs to use an informative prior, the hyperparameters of the prior distribution must

be elicited. One way to elicit the hyperparameters is the training sample approach where the data is

divided into two parts; the first part constitutes the training sample and is used to provide proper

priors. Then, posterior distributions are obtained by combining these priors with the likelihood based

on the second part of the data (non-training sample). The training sample approach is used by

Lempers (1971); Spiegelhalter, and Smith (1982), among others.

Future work may investigate model identification using stochastic search variable selection,

outliers detection, and extension to multivariate seasonal models.

APPENDIX

This appendix displays the full conditional posterior distributions of the SARMA model, which are

used in developing the Gibbs sampling algorithm explained in Section 4.

1. The Conditional Posterior of

The conditional posterior of is ~ | , , , , , , , ),

where, , , is an matrix with ∑ and is an matrix with

.

12


The conditional posterior of is ~ | , , , , , , , ),

where, , , is an matrix with ∑ and is an matrix with

.


The conditional posterior of is ~ | , , , , , , , ), where, , ,

is an matrix with ∑ , and is an matrix with

.


The conditional posterior of is ~ | , , , , , , , ), where, , ,

is an matrix with ∑ and is an matrix with

.


The conditional posterior of is ~ | , , , , , , , ,

where, and ∑∑ ∑ ∑∑ ∑ .

Thus, the parameter can be sampled from Chi-square distribution using the transformation

13

~ .

6. The Conditional Posterior of y

Using model (2), the equations for the elements of and errors ε can be written as

Where,

,

,

,

, , … , , , … , and , , … , , which has the normal distribution with zero mean and variance

( where is the unit matrix of order . Using linear regression results and

standard Bayesian techniques, the conditional posterior of y is ~ | , , , , , , , ), where, , , and .

14

7. The Conditional Posterior of ε

The conditional posterior of ε is ~ | , , , , , , , ), where, , ,

and .

REFERENCES

Barnett, G., Robert J. Kohn, and S. Sheather. 1996. Bayesian estimation of autoregressive model

using Markov Chain Monte Carlo. Journal of Econometrics 74 (2): 237-254.

———. 1997. Robust Bayesian estimation of autoregressive moving-average models. Journal of

Time Series Analysis 18 (1): 11-28.

Box, G.E.P., and G. M. Jenkins. 1976. Time series analysis, forecasting and control. San Francisco:

Holden-Day Inc.

Broemeling, L. D. 1985. Bayesian analysis of linear models. New York: Marcel Dekker Inc.

Broemeling, L. D., and S. Shaarawy. 1984. Bayesian inferences and forecasts with moving average

processes. Communications in Statistics: Theory and Methods 13 (15): 1871-1888.

Chib, S., and E. Greenberg. 1994. Bayes inference in regression models with ARMA(p,q) errors.

Journal of Econometrics 64 (1-2): 183-206.

Geweke, J. 1992. Evaluating the accuracy of sampling-based approaches to the calculations of

posterior moments. In J. M. Bernardo, J. O. Berger, and A. P. Dawid, ed., Bayesian Statistics 4.

USA: Oxford University Press.

Ismail, M. A. 2003a. Bayesian analysis of seasonal autoregressive models. Journal of Applied

Statistical Science 12 (2): 123-136.

———. 2003b. Bayesian analysis of seasonal moving average model: A Gibbs sampling approach.

Japanese Journal of Applied Statistics 32 (2): 61-75.

Koreisha, S., and T. Pukkila. 1990. Linear methods for estimating ARMA and regression model with

serial correlation. Communications in Statistics: Simulation and Computation 19 (1): 71-102.

Lempers, F. B. 1971. Posterior probabilities of alternative linear models. The Netherlands:

Rotterdam University press.

LeSage, J. P. 1999. Applied econometrics using Matlab. Dept. of Economics, University of Toledo,

http://www.econ.utoledo.edu.

15

Marriott, J., and A. F. M. Smith. 1992. Reparameterization aspects of numerical Bayesian

methodology for autoregressive moving-average models. Journal of Time Series Analysis 13

(4): 327-343.

Marriott, J., N. Ravishanker, A. Gelfand, and J. Pai. 1996. Bayesian analysis of ARMA processes:

Complete sampling based inference under full likelihoods. In D. Berry, K. Chaloner, and J.

Geweke, ed., Bayesian statistics and econometrics: Essays in honor of Arnold Zellner. New

York: Wiley-Interscience.

McCulloch, R. E., and R. S. Tsay. 1994. Bayesian analysis of autoregressive time series via the Gibbs

sampler. Journal of Time Series Analysis 15 (2): 235-250.

Monahan, J. F. 1983. Fully Bayesian analysis of ARMA time series models. Journal of Econometrics

21 (3): 307-331.

Newbold, P. 1973. Bayesian estimation of Box Jenkins transfer function noise models. Journal of the

Royal Statistical Society: Series B (Methodological) (JRSSB) 35 (2): 323-336.

Raftery, A. E., and S. Lewis. 1992. One long run with diagnostics: Implementation strategies for

Markov Chain Monte Carlo. Statistical Science 7 (4): 493-497.

———. 1995. The number of iterations, convergence diagnostics and generic Metropolis algorithms.

In W. R. Gilks, D. J. Spiegelhalter, and S. Richardson, ed., Practical Markov Chain Monte

Carlo. London: Chapman and Hall.

Shaarawy, S., and M. A. Ismail. 1987. Bayesian inference for seasonal ARMA models. The Egyptian

Statistical Journal 31 (1): 323- 336.

Spiegelhalter. D. I., and A.F.M. Smith. 1982. Bayes factors for linear and log-linear models with

vague prior information. JRSSB 44: 377-387.

GIBBS SAMPLING FOR SARMA MODELS

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