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ISSN 1744-6783

THE STRUCTURE OF THE AUSTRALIAN GROWTH PROCESS: A BAYESIAN MODEL SELECTION VIEW OF MARKOV SWITCHING

A. TAYLOR, D. SHEPHERD, S. DUNCAN

Tanaka Business School Discussion Papers: TBS/DP04/22 London: Tanaka Business School, 2004

The Structure of the Australian Growth Process: A Bayesian Model Selection View of Markov Switching By Andrew Taylor*, David Shepherd+ and Stephen Duncan*

* Department of Engineering Sciences, University of Oxford + Imperial College, University of London

Abstract In this paper we examine the structure of the GDP growth process in Australia. Our objective is to determine whether the growth process exhibits well-defined business cycle features and whether any significant regime shifts have occurred in the mean and variance of the growth process, as well as its autoregressive structure. The analysis is based on a general MarkovSwitching Autoregressive Model, implemented from a Bayesian perspective using Markov Chain Monte Carlo numerical methods. Our results suggest that there are two regimes in GDP growth and that the dominant statistical feature is a switch from higher to lower volatility in growth. In contrast to previous studies for Australia, our results suggest that there are no distinct business cycle features present in the growth process.

JEL Classification Numbers: C1, C4, E3

Corresponding Author: Dr David Shepherd Tanaka Business School Imperial College London South Kensington Campus London SW7 2AZ United Kingdom Email: [email protected]

The Structure of the Australian Growth Process: A Bayesian Model Selection View of Markov Switching

1. Introduction Time series models of real GDP growth provide important insights about the structure of the underlying growth process, which complement those derived from formal structural models based on fully specified production function relationships. The time series models are particularly useful in determining whether the growth process exhibits a cyclical structure (represented by an autoregressive process) and whether non-stationary elements (typically represented by regime shifts) are significant. There is in fact now a large body of evidence to suggest that non-stationary elements, in the form of regime shifts, are potentially important in many economic processes, including the growth process. Examples of the diverse range of work in this area include Hamilton (1989), Perron (1990, 1997), Engel (1994), Acemoglu and Scott (1997), Hall, Psaradakis and Sola (1997), Raymond and Rich (1997), Kim and Nelson (1999b). The message from these and other studies is that failure to allow for the presence of regime shifts (structural changes) in the stochastic process can lead to invalid inferences concerning the nature and strength of the autoregressive process. This is an important issue, since the structure of the autoregressive process effectively describes the degree of persistence in response to shocks affecting the system and whether the responses to positive and negative shocks are symmetric. Recent studies of the growth process in Australia suggest that regime shifts are a potentially significant feature of Australian GDP growth and that simple linear models do not provide an adequate explanation of the historical pattern of growth. For example, based on an examination of non-linear and Markov-Switching models, Bodman and Crosby (2002) find evidence in support of the view that GDP growth follows a cyclical pattern, but with distinct regime shifts in the mean growth rate. Their results also suggest that there is no asymmetry in the response to shocks of the kind identified for the US economy by Beaudry and Koop (1993) and Pesaran and Potter (1997). Henry and Summers (2000) consider output growth in the context of a non-linear threshold adjustment model, which is equivalent to a model with regime changes, and find evidence that Australian growth follows a cyclical path, but with an asymmetric response to shocks originating in the United States. These and other studies (Layton, 1997; Bodman, 1998) suggest that the full complexity of the Australian growth process is unlikely to be captured by a purely linear autoregressive model. In this paper we examine the structure of the growth process in Australia with the aid of an autoregressive model of real GDP growth that allows for a range of potential regime shifts together with their non-switching counterparts. The analysis is based on a MarkovSwitching model of the form suggested by Hamilton (1989), but we adopt a more general

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structure that allows for state switches in both the autoregressive parameters and the variance, as well as up to three different levels in the mean, and the case in which the mean and variance of the time series arise through independent Markov chains. The model thus allows us to consider a broader range of possible regime shifts than have previously been considered for Australia and elsewhere. For example, an important implication of Hamilton's original work, that recessionary phases in the US have a permanent negative impact on output, depends on a model that allows for only two mean states and is at odds with other evidence (Beaudry and Koop, 1993; Sichel, 1993) that recessions (when output falls quickly) are typically followed by a period of unusually higher growth, until the economy reaches its normal growth path. By allowing for three possible mean states, we are able to investigate the relevance of this issue for Australia. Similarly, while Bodman and Crosby (2002) estimate a switching model with three mean states, their analysis does not allow for possible switches in the variance and the autoregressive parameters and hence there is no way of knowing whether their results are conditioned by these restrictions. Apart from its greater generality, a further distinctive feature of our contribution is that we use Markov Chain Monte Carlo (MCMC) numerical methods (Robert and Casella, 1999) in a fully Bayesian estimation and inference procedure that allows us to address the model specification problem in an efficient manner. In particular the alternative model structures are evaluated on an equal basis using Bayes factors (Kass and Raftery, 1995). This provides new understanding of the relative validity of each structure considered, given the data. Although the likelihood approach applied by Hamilton (1989, 1994) can be extended to incorporate model selection through penalty terms such as AIC (Akaike, 1978) and BIC (Schwarz, 1978), these terms are asymptotic approximations to the bias in the likelihood function and require large sample sizes. The procedure used here is similar to the one described by Albert and Chib (1993) and Kim and Nelson (1999a) where posterior distributions are examined in some detail, giving improved insight into the significance of the regime changes which are identified in the time series. The plan of the paper is as follows. In section 2 we outline the structure of the general Markov Switching Model used to explain GDP growth. Section 3 discusses the estimation and model evaluation procedures that are used in this study. Section 4 presents the results of the analysis. The final section provides a brief summary of the analysis and what the main results of the paper imply about the structure of Australian growth process. 2. General Markov Switching Models In time series analysis, Markov Switching Models, also known as Hidden Markov Models, are a method suited to dealing with the problem of non-stationarity, where a single probability density cannot be applied to all points in a time series. These models can be

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regarded as a natural extension of Mixture Modelling (McLachlan and Peel, 2000) where the latent structure of an observed sequence, usually a time series, is assumed to be determined by a Markov chain, S , whose “hidden” values may not be observed. In the Markov Switching Models considered in this paper, the unobserved state variable s , is assumed to evolve according to a discrete Markov chain and when a state transition occurs in this Markov chain, a change also takes place in the parameters of the model that describes the observed data. This means that the hidden state, st depends on it's preceding value, st −1 according to the state transition matrix, P with entries pij such that k

p( st = j | st −1 = i ) = pij

∑p

and

j =1

ij

=1

(1)

For example, in a Markov Switching Model let the observations be Gaussian AR(1) conditional on the state of the Markov chain. In this model, the parameter vector φ j is the mean, the autoregressive parameter and the variance for Markov chain state j and an observation at time t , yt , depends on both st and yt −1 . The distribution of yt given st is

f ( yt | st = j , yt −1 ) = f j ( yt | φ j , yt −1 )

(2)

In this paper, we use a more general version of the model introduced by Hamilton (1989, 1994). Our general Markov Switching Model covers a range of individual models, where parameters can be fixed or Markov switching. The linear equation for each model in

yt is shown in equation (3), with µ, θ and σ

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representing respectively the mean,

autoregressive parameters and variance of the stochastic process m

yt = µ t + ∑θ k ( yt −k − µ t −k ) + et

(3)

k =1

where et ~ N(0,σ t ) . The model parameters, φ = ( µ ,θ * ,σ 2 ) , are determined by the hidden 2

Markov chain state, st , given the level of Markov switching that is specified in the model design. For each parameter this means

θ k ,t = θ k , j when st = j for AR parameter switching models = θ ∀st for non -AR parameter switching models

µ t = µ j when st = j for mean switching models = µ ∀st for non - mean switching models

σ t2 = σ j when st = j for variance switching models = σ 2 ∀s t for non − variance switching models such that the parameters adopt state dependent values when switching applies and a single value when switching does not apply.

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In the context of the growth analysis in this paper, yt represents the growth rate of real GDP and the parameters, µ and σ 2 describe the mean and variance (or volatility) of the growth rate in any given state. The autoregressive parameters θ , can be interpreted as determining the cyclical component of the time series in any given state, characterising persistence or local dependencies (Engle and Kozicki, 1993; Vahid and Engle, 1997). A cyclical feature in the series may can also be indicated by a regular switching pattern in the mean, with switches between higher and lower growth in a two-state model or switches between below-average and above-average growth in a three-state model1. The principle of applying an m state hidden Markov model to GDP growth is that there are m recurring and aperiodic regimes which dominate the data. In some cases, it is possible that the underlying mechanism that brings about the non-stationarity may be a series of independent changepoints and the hidden Markov model would then represent an approximation to such a process. The state at time t , st , is assumed to follow a discrete Markov chain with m states. In this paper we consider m ∈ (1,2,3) such that the non-switching case is compared with 2 and 3 state switching models. The Markov chain that determines the state of the system evolves according to a m × m transition matrix P with entries pij such that

pij = P( st = j | st −1 = i ) This means that the future state of the depends only on the current state of the Markov chain and the values in the transition matrix. By equation (3), the observed system behaviour depends on the values of pij in P . For example, a tendency to reside for long periods in

st = i would be caused by a high value of pii (the probability of remaining in state i ) whereas a very low value of pii indicates a transient state perhaps due to one off events.

3. Estimation and Model Evaluation Procedures Markov Switching Model parameters can be estimated through a number of techniques. In this paper we employ Bayesian methods, using Gibbs sampling to resolve the problem of intractable integrals in a Bayesian procedure (Robert and Casella, 1999; 1

Given that we are examining variations in the growth rate of GDP, our analysis implicitly associates the business cycle as equivalent to a ‘growth cycle’. Some authors, such as Harding and Pagan (2003a) have argued that the business cycle should be viewed in terms of the so-called ‘classical cycle’, which emphasises the identification of recession and recovery phases in the level of GDP. According to this view, the cycle is best identified by examining the time pattern of the sign of the growth rate, with recession and recovery identified according to some given rule such as two or more negative quarters of growth followed by two or more positive quarters of growth. The relative merits of these different approaches are discussed in the interchange between Hamilton (2003) and Harding and Pagan (2003b). We discuss the relevance of the issue for Australia in the results section.

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Gamerman, 1997; O'Ruanaidh and Fitzgerald, 1996). A special form of the EM algorithm known as the forwards backwards recursion (Dempster, Laird and Rubin, 1977) can also be used to obtain maximum likelihood and posterior mode parameter estimates (Rabiner, 1989; Bengio,1999; Ghahramani, 2001). There are two inferential benefits of this Bayesian procedure. The first benefit is that the this method allows us to use Bayes factors to compare each of the candidate GDP models on an equal footing before selecting the most probable model given the data. Non-switching models are included in the set of possible models, so that stationarity is tested alongside the non-stationary models. The second benefit, for inferential purposes, is that the Bayesian approach estimates the full posterior distribution of each parameter, as opposed to the point estimates and asymptotic standard errors that are often used, and which are subject to assumptions. In particular, this can be seen in our later Markov state analysis, where the state membership probabilities, P ( st = j | .) , are plotted for each observation in contrast to plotting the state sequence that maximises the likelihood function. Further examination of posterior distributions for Markov switching models can be found for a variety of econometric datasets in Albert and Chib (1993) and Kim and Nelson (1999a), where in many cases the posterior distribution of parameter vector φ is found to be non-Gaussian and in some cases the distribution is found to have multiple modes. The use of these methods, with repeat MCMC runs and careful analysis of the MCMC diagnostics, helps this paper avoid the model fit and specification issues which appear to be found in some Markov switching studies (Bruenig and Pagan, 2001). Bayesian Statistics and Gibbs Sampling Our approach is to use Bayesian methods in order to determine the most likely model (given the data) and the times that the state of the underlying growth process changes. A key result of this is that in addition to estimating all model parameters, a common measure is used to select the most suitable model from the available alternatives. This Bayesian measure is known as the ‘model evidence’ and one of its features is that penalises more complex models in order to prevent over-fitting the data. In Bayesian analysis (Box and Tiao, 1974), as opposed to maximum likelihood or least squares inference, the results analysed are usually a full marginal posterior distribution for each parameter in question. This paper uses numerical methods to evaluate the posterior distribution of the model and its parameters through a technique known as Gibbs sampling, because closed form expressions are not readily available for these posterior distributions. Gibbs sampling is a form of Markov Chain Monte Carlo (MCMC) analysis (Robert and Casella, 1999; Gamerman, 1997; O'Ruanaidh and Fitzgerald, 1996), which is a name for Bayesian simulation methods in posterior distribution

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estimation, where a target density is estimated by constructing and then sampling a Markov chain whose steady state distribution is that density. Model Estimation Procedure In evaluating the general switching model, given by equation (3), the Markov states,

s are treated as an unknown parameter of the model in the same manner as are µ and θ . The aim of the analysis is to examine the posterior mode of the system state, arg max st P( st | y ) , in order to understand the nature of each state and to determine the evidence in favour of each model against a reference model. Using the recursive property of Bayes theorem, the model likelihood, given complete knowledge of the Markov chain S and φ = ( µ t ,θ t ,* ,σ t ) is 2

T

P(y | φ , s) = ∏ (2πσ t ) exp( t =1

2

−1 2 2σ t

m

∑ k =1

( yt − µ t − θ k ,t ( yt −k − µ t −k )) 2 )

(4)

Again using Bayes theorem, assuming a prior distribution for the model parameters, including the unknown states, P(φ | M ) P( P | M ) P(s | P ) , the posterior distribution for the parameters of model M is

P(φ , P, s | y ) ∝ P(φ | M ) P(P | M ) P(y | φ , s)

(5)

where a proportionality symbol replaces the equality due to the omission of a normalising coefficient. The normalising coefficient is P( M | y ) , which is also the model evidence statistic that is discussed later. Equation (5) is the joint posterior distribution of all model parameters and, being unwieldy, is of limited practical use. A more useful expression is the marginal distribution of each parameter, which can be obtained by integrating the posterior over all values of all other parameters. For such a complex model (see later for the actual priors used) this kind of integration is not practical and, as mentioned above, a numerical technique known as Gibbs sampling is used instead. The Gibbs sampler factorises the posterior probability density of all unknown parameters, φ , s and P , into full conditional distributions of each individual parameter, which are conditional on the value of all other parameters. For example, P( µ1 | .) is the full conditional distribution of µ1 given the values of the other parameters. It can be shown (Robert and Casella, 1999) that a Markov chain can be simulated whose stationary

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distribution is φ , s, P through drawing random samples in turn from the full conditional distribution of each parameter. One cycle of the Gibbs sampler is as follows: Initialise. Sample at random from the prior distributions in φ , s, P . Step 1. Sample at random from P( µ1 | .) and repeat for all µ j depending on the model specification. Step 2. Sample at random from P(σ 1

2

| .) and repeat for all σ j . 2

Step 3. Sample at random from P( s1 | .) , repeat for all t . Step 4. Sample at random from P( p 00 , p 01 , p10 , p11 | .) , i.e. the first row of the transition matrix, and repeat for all m rows. Step 5. Sample at random from P (θ 1,1 | .) and repeat for all P (θ i , j | .) This process is repeated, storing each new sampled value for later analysis. Once the system has been run for a reasonable number of cycles (10,000 cycles have been used in our analysis) it reaches a steady state and the stored values may be shown to be genuine random samples from the distribution of each parameter given only the data (Robert and Casella, 1999). This means that the results may be treated as marginal posterior distributions. Before analysing these stored values in order to study the posterior distribution of the model, which is the objective of our MCMC simulation, it is necessary check the MCMC Markov chain for convergence and in order to ensure the presence of reasonable steady state variation. This is to guard against invalid results, such as trapping Markov states We perform these checks visually through time series plots of parameter values against MCMC iteration number to confirm that we have a random series of steady state samples from a Markov chain, not subject to any single long term trend2.

Further to this, to avoid potential convergence

problems in the application of Markov switching models (Breunig and Pagan, 2001), our results are based on three independent MCMC simulations. The Prior Distributions Bayesian methods require the specification of a set of prior distributions. Our analysis is based on the following priors, chosen to cover a reasonably feasible range of parameters. These priors reflect the fact that we use a standardised measure of GDP growth (zero mean and unit variance) in the MCMC calculations. Under the Gaussian assumption used in our model (4), the results are invariant to scale and location changes. 2

More extensive checks and diagnostics are discussed in Robert and Casella (1999).

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State Means, µ j : The prior chosen for ( µ1 , µ 2 , µ 3 ) is a hierarchical Gaussian form where the values of µ 2 and µ 3 are taken to depend on the value of µ1 plus or minus a constant, δ which itself is given a Gaussian prior such that

µ1 ~ N(0,1) µ 2 ~ N( µ1 + δ ,10)

µ 3 ~ N( µ1 − δ ,10) δ ~ N(0.5,3) where N(a, b) refers to a Gaussian distribution with mean a and inverse variance b . This form favours posterior distributions where means are different between states, a feature employed to ensure the MCMC simulation converges and to prevent the problem of label switching where the results become scrambled (Richardson and Green, 1998). Inverse variance, τ : The inverse variance is used for convenience as this can be used with a conjugate Gamma prior to give a tractable posterior distribution (Box and Tiao, 1974). This is given a Gamma prior with hyperparameters a1 and a 2 such that.

a1 ~ G amma(10,1) a 2 ~ Gamma(1,1)

τ ~ Gamma(a1 , a2 ) This prior enables estimation to proceed without assuming specific values for these parameters (Migon and Gamerman, 1999; Carlin and Louis, 2000). AR parameters, θ k : The autoregressive parameters, where specified, are given a Gaussian prior distribution with zero mean and precision, τ k , which is assigned a Gamma prior with hyper-parameters, b1,k ~ Gamma(10,1) and b2,k ~ Gamma(1,1) . The range of values that this prior generates encompasses anything likely to be encountered in a study such as ours. Transition matrix, P : The transition matrix prior is assumed to be row-wise Dirichlet (Devroye, 1986). This is the natural conjugate prior for a set of probabilities that sum to one (Box and Tiao, 1974). The functional form of each of these priors can be found in (Robert and Casella, 1998).

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Model Selection Procedure In a Bayesian context, the model evidence (E) statistic can be used to compare different models (O'Ruanaidh and Fitzgerald, 1996). The evidence is the normalising divisor in the posterior probability equation for model parameters φ , s, P , given model M i and prior information on parameters H

P(φ , s, P | y , M i , H ) =

P(φ , s, P | H ) P( y | φ , s, P, M i , H )

∫ P(φ , s, P | H ) P(y | φ , s, P, M i , H )dφ , s, P

(6)

Using Bayes theorem, the model evidence, Ei is given by

Ei = ∫ P(φ , s, P | H ) P(y | φ , s, P, M i , H )dφ , s, P = P( M i | y , H )

(7)

The model evidence Ei is a combination of prior information and the likelihood, given the data, integrated over model parameter space. It is therefore a measure of the probability of the model given the data. Given the evidence over a range of models, the Bayes Factor Bi , for model Mi , is the ratio Ei / E0 where M 0 is a reference model. The reference must be a fixed point in the model space, which might for example be the non-switching non-AR model. The model with the highest Bayes factor is then regarded as the most probable model given the data. The model choice can therefore be based on either Ei or Bi directly3. We consider both

Ei and Bi to determine the preferred model, although they point unambiguously in the same direction. In this analysis, the integral for E in equation (7) is not analytically tractable and is estimated by accumulating values of the likelihood function in the course of the Gibbs sampler iterations described above (Carlin and Louis, 2000). 4. Model Application and Results In this section we use the general Markov-Switching Model to examine the nature of the growth process in Australia. The analysis is based on a consideration of seasonally adjusted quarterly data for the volume index of real GDP, covering the period 1962Q1 to 2001Q1, obtained from the OECD Main Economic Indicators database. Our measure of GDP growth is the first difference of the logarithm of the quarterly GDP series. A plot of the series is shown in figure 1. 3

A table of suggested thresholds for the interpretation of Bayes factors is provided in Kass and Rafferty (1995).

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Figure 1. Time Series Plot of Australian GDP Growth A u s t ra lia n R e a l G D P

G ro w th

0 .0 6

0 .0 4

GDP Growth Rate

0 .0 2

0

-0 .0 2

-0 .0 4

-0 .0 6 1 9 6 5

1 9 7 0

1 9 7 5

1 9 8 0

1 9 8 5

1 9 9 0

1 9 9 5

2 0 0 0

A visual inspection of the series suggests that there may have been some change in the volatility of the growth rate over the period, perhaps occurring somewhere in the middle 1980s. Beyond that, it is difficult to say much about the other properties of the series from a purely visual inspection of the plot and a more formal statistical analysis is required. The application of standard least squares methods suggests a mild autoregressive structure in the time series process, but at this stage we have no we have no way of knowing whether such a result is conditional on the assumption that there are no state switches in the process. Our formal analysis of the series is based on an examination of the general MarkovSwitching model, as described by equation (3), which is repeated for convenience: m

yt = µ t + ∑θ k ( yt −k − µ t −k ) + et

(3)

k =1

The y variable is the standardised growth rate of real GDP, measured as the first difference of the logarithm of the quarterly GDP series corrected to zero mean and unit variance. Our objective is to determine the most likely structure of the growth process in a set of models that incorporate the possibility of regime switches in the mean, variance and autoregressive parameters. In this model, switches in the mean state imply regime shifts in the average growth rate of the economy, switches in the variance imply regime shifts in the volatility of the growth process, and switches in the autoregressive parameters imply regime shifts in the underlying cyclical structure of the process in any given state. As noted earlier, the presence of a cyclical feature in the growth process is indicated by an autoregressive model structure and/or a repeating pattern of switches in the mean growth rate. We consider a set of models that allow for the possibility that there are no regime switches in any of the parameters and up to two switches in the variance and autoregessive

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parameters, and three switches in the mean parameter. The individual models examined can be summarised as follows:

Non-Switching Models: AR(0) to AR(8). In this case, the relatively higher order is chosen to ensure that any rejection of the non-switching model is not the result of a limited AR order. Two-State Switching Models: AR(0) to AR(4) mean switching, variance switching, mean and variance switching and AR parameter switching. Three-State Switching Models: AR(0) to AR(6) models with mean switching. In this case, the higher order is tested to ensure that the evidence value does not increase for higher orders in the last set of model structures. Two-State Switching Models with two state vectors: In this case, the concept of two independent state variables is applied to the data in order to permit the current mean and variance of the time series to be determined through independent random processes. This model is a straightforward extension of equation (3).

A total of 33 models are therefore considered, comprising 9 non-switching models, 15 twostate models, 7 three-state models and one model in which two independent Markov chains determine the parameter vector. The evaluation of the alternative models is summarised in table 1, which shows both the log mean model Evidence log( E i ) and the Bayes factor Bi for

i = 1,..,30 . These are calculated as the average values from 3 independent MCMC simulations, each with 10,000 replications. The results are summarised in table 1. The results suggest that the most probable GDP growth model for Australia is the two-state AR(0) model with Markov switching mean and variance. However, the Bayes factors for the various alternatives suggest that three other models offer feasible results for this data, and these need also to be considered. The next most probable model is the variance only switching model with white noise error and constant mean. The Bayes factor statistics that are given in this table provide the estimated probability of each given model being the one that generated the data expressed relative to the model with the highest model evidence. As a result, with a Bayes factor of 93.6%, this second most probable model is almost as likely to have been responsible for generating the data as the model that has 100% Bayes factor. The remaining two models with high Bayes factors are the stationary AR(1) process with MS mean and variance, and the stationary white noise process with independent MS mean and variance.

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Table 1 - Model Evidence and Bayes Factors Summary Model description Stationary white noise Stationary AR(1) process Stationary AR(2) process Stationary AR(3) process Stationary AR(4) process Stationary AR(5) process Stationary AR(6) process Stationary AR(7) process Stationary AR(8) process Stationary white noise with MS mean Stationary AR(1) process with MS mean Stationary AR(2) process with MS mean Stationary AR(3) process with MS mean Stationary AR(4) process with MS mean MS AR(1) process with MS mean MS AR(1) process with MS mean MS AR(1) process with MS mean MS AR(1) process with MS mean Stationary white noise with MS mean Stationary AR(1) process with three state MS mean Stationary AR(2) process with three state MS mean Stationary AR(3) process with three state MS mean Stationary AR(4) process with three state MS mean Stationary AR(5) process with three state MS mean Stationary AR(6) process with three state MS mean Stationary AR(7) process with three state MS mean Stationary white noise with MS mean and variance Stationary AR(1) process with MS mean and variance Stationary AR(2) process with MS mean and variance Stationary AR(3) process with MS mean and variance Stationary AR(4) process with MS mean and variance Stationary white noise with independent MS mean and variance Stationary white noise with MS variance

Log Evidence -210.3 -210.9 -207.5 -206.5 -205.4 -205.4 -202.9 -203.4 -200.8 -212.4 -212.9 -210.7 -211.1 -211.9 -213.0 -214.1 -215.1 -216.0 -203.4 -203.3 -205.4 -205.2 -205.3 -206.2 -197.7 -199.9 -191.9 -193.9 -194.0 -194.1 -195.5 -192.9 -192.0

Bayes Factor 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 100.0 26.4 10.7 2.7 0.0 35.6 93.6

Although each of these four most probable models could reasonably be selected to describe the growth rate data, in the sense that they have significant Bayes factors, closer examination of the results shows that the dominant statistical feature for this group of models is the same main state change in the variance of the growth process and that other apparent differences are relatively insignificant. This can be seen from an inspection of table 2, which reports the parameter estimates for the four most probable models. The table shows the posterior centile statistics, reported in the form of the posterior mean and a 95% Bayesian confidence region for each parameter (Migon and Gamerman, 1999). The parameters of these

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models include the overall mean, or the two Markov state means, µ1 and µ2 , the associated inverse variances, τ1 and τ2 , and the autoregressive parameter, θ. It is important to note that all of these statistics are derived from the MCMC, which has been applied to the standardised GDP data. Table 2 – GDP Growth Switching Model Parameters

µ

Stationary white noise with MS Stationary white noise with MS mean and variance variance 0.1011 97.50% -0.0217 mean

µ2

τ1 τ2

θ

Stationary AR(1) process with MS mean and variance

-0.1433

2.50%

µ1

Stationary white noise with independent MS mean and variance

97.50%

0.3265

0.2030

0.3306

mean

0.0195

-0.4665

0.0572

2.50%

-0.2578

-1.1440

-0.2013

97.50%

0.1181

0.2770

0.1024

mean

-0.0321

0.0536

-0.0480

2.50%

-0.1747

-0.2712

-0.1941

97.50%

0.8846

0.8903

0.9211

0.8750

mean

0.6496

0.6573

0.6769

0.6399

2.50%

0.4330

0.4446

0.4569

0.4273

97.50%

5.1910

5.2810

6.4840

5.0440

mean

3.3530

3.4400

4.1450

3.3420

2.50%

2.0460

2.1850

2.4610

2.1470

97.50%

0.1645

mean

0.0099

2.50%

-0.1353

Bayes Factor

100.00

93.6

35.6

26.4

The parameter estimates of the most probable model are shown in the first column of table 2. These values suggest that output growth in Australia is best characterised as a white noise process with two distinct states of higher and lower volatility, which each are associated with a different mean growth rate (higher and lower output growth are associated respectively with higher and lower growth volatility). However, the differences in the two mean growth rates are relatively small and the 95% posterior confidence regions for these values are clearly overlapping, which suggests that the means are not significantly different in statistical terms.

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The next most probable model (shown in the second column of table 2) suggests a similar variance switch, but with a constant mean. The implication is that the two most probable models are telling essentially the same story, which is that the most pronounced difference between the states is in terms of their volatility levels, with the variance of the higher growth state roughly five times that of the lower growth state. Equally important these two models suggest that there are no significant autoregressive features in the growth process. This conclusion is effectively confirmed by the results for the third most likely model in the group of four (shown in column 3 of table 2). In this case, the mean and variance are allowed to evolve according to independent Markov chains and it is more likely that we would be able to identify any significant differences in the mean growth rate between states. Apart from the fact that the Bayes factor for this model is lower than for the two previous models, the results suggest that the mean growth rates of the two states are not significantly different, that there are no autoregressive features, and that the dominant feature is again a switch from higher to lower growth volatility. Finally, turning to the fourth (and least likely) model in the selected group, it can be seen from the parameter estimates in table 2 that the mean differences are again small and insignificant. In addition, although this model does at first sight exhibit an AR(1) structure, it has a weak AR parameter value and the 95% posterior distribution spans zero, suggesting that θ is not significantly different to zero according to usual inference procedures. Taken together, the results for the four models considered point to essentially the same conclusion, which is that there are no significant autoregressive or mean switching features in the Australian growth process and the dominant feature is a major shift in the volatility of the process. An additional fact is that the non-switching models are not significant. Consequently, our results suggest that the identification of a cyclical (autoregressive) structure in the growth process is dependent on the traditional assumption that there are no state switches in the mean and variance of the process. The model evidence statistics point to the fact that this is assumption is unlikely to be valid for the case of Australia. In so far as the cyclicality of the series can be identified as an autoregressive feature, this carries the implication that output growth in Australia does not exhibit any distinct business cycle features of the kind that have been identified for countries such as the United States. In other words, the variability in the Australian growth process appears to be confined to state variability in the variance of the process rather than the traditionally-defined business cycle variability. The final piece of information we need is the timing in the switches of the Markov states, from higher growth/higher volatility states to lower growth/lower volatility states. Figure 2 shows time series plots of the estimated posterior probabilities of Markov state membership with higher GDP variance (state 1) for each of the models. In the case of the two 15

Markov chain model, two plots are given, one each for the mean and variance. Each point on these charts represents the relative number of occasions that Gibbs sampler visited state 1 after it had been allowed to settle. The shaded areas of the plot represent periods in which the economy is most likely to have been in the higher growth/higher volatility states (state 1). The non-shaded areas represent the periods of lower growth/lower volatility (state 2).In the two Markov chain model, the shaded area represents state 1, the lower of the two state means. Figure 2 - Markov State Probabilities for the four most likely Growth models.

Focussing on the white noise process with switching mean and variance, the probability results suggest that the economy was in the higher volatility state from the beginning of the period until 1984Q2, when it entered a period of low volatility. An almost identical switch can be seen in the most likely state of the three other models studied in figure 2. Apart from this predominant feature, the posterior probability values provide some evidence of low volatility in the period around 1979Q4. It should be noted, however, that the evidence in support of this first brief switch from higher to lower volatility is marginal (somewhere in the region of 50% in favour of either the low volatility state, or the higher volatility state). The strong result of the analysis is that economy moved to the lower volatility state in 1984Q2 and remained in that state for the remainder of the period. As can be seen from figure 2, this second switch is associated with a very high probability.

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While the statistical analysis of the series does not of course provide any direct information about the causes of the above Markov state switches, it should be possible to identify likely economic causes by considering the events that were happening at the time in the domestic and/or international economy. With respect to the first state switch, in the late 1970s, we have already said that the result is marginal and it may be that there was no switch at this time. Assuming that it did happen, there is no obvious and direct explanation for the brief switch to the lower volatility phase, although this is clearly a period in which many economies were still adjusting to the impact of the oil shocks of the early 1970s and the subsequent rise in inflation that occurred. The second and most dominant switch occurs in 1984, from higher to lower volatility. A similar reduction in growth volatility, also dated to the mid 1980s, has been identified for the United States by several authors, including Kim and Nelson (1999b) and Stock and Watson (2002). Stock and Watson suggest that at least some of the reduction in the volatility of US output growth may have occurred in response to improvements in the operation of monetary policy and a more favourable set of productivity and commodity price shocks, although a large proportion (up to 60%) is unexplained. In the case of Australia, the reduction in growth volatility is most likely to be connected with important events that occurred in the domestic economy around that time (Dixon, Shepherd and Thomson, 2001). For example, the election of the Hawke Labour government in 1983 was followed by a series of important policy events, which included: considerable financial deregulation, floating of the currency, removal of price support schemes for agricultural products, a marked reduction in tariffs on imports, and changes in wage-setting arrangements. These are all policy changes that would be expected to influence the growth environment, particularly in a relatively open economy such as Australia. One could also point to the fact that this was a period during which significant changes in the industrial structure of the economy were occurring. In contrast, work by Simon (2001) suggests that the decline in volatility in Australia may be due to a reduction in the volatility of productivity shocks rather than an increase in the structural stability of the economy. The final matter we need to consider in relation to the timing of the state switches is what they imply, if anything, about the business cycle. In the Markov switching context, the presence of any cyclicality would most obviously be indicated by a series of repeated switches in the mean growth rate or its volatility, with alternation between high and low phases. Our results suggest that no such cyclical feature is present in the series and that process is dominated by a one-off state switch in the mid-1980s. In the absence of any alternating state switches, and the absence of any autoregressive features, our conclusion is that there are no business cyclical features present in the Australian growth process and that it is best characterised as a white noise process with a major volatility shift in the mid-1980s. Having said this, it should be emphasised that our failure to identify a business cycle feature

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does not of course mean that the Australian economy has never experienced periods of recession and recovery4. What it suggests, rather, is that any such recession and recovery phases are best regarded as random events contained within the noise component of the model and not part of any identified regularity in the series5.

5. Summary and Conclusions In this paper we have examined the structure of the GDP growth process in Australia using a general Markov Switching Model that allows for regime changes in all of the model parameters, including the variance and autoregressive parameters as well as the mean. The model was analysed with Bayesian estimation and inference methods, utilising a Monte Carlo Markov Chain simulation procedure. Our objective was to determine whether any significant regime shifts can be identified in the mean and variance of the growth process, as well as its autoregressive structure, and whether the growth process exhibits well-defined business cycle features. Apart form its greater generality, a distinctive feature of our contribution is that it provides an efficient means of assessing the relative validity of the different model structures that could plausibly be used to explain the data. The Bayes factor analysis shows that a model with an autoregressive structure is a plausible candidate to explain the behaviour of GDP growth, but the estimation results indicate that any such autoregressive element is weak, and probably insignificant, and that non-autoregressive model structures are more likely. From the various models considered, the results strongly suggest that Australian GDP growth is best characterised as a nonautoregressive process with switches between phases of higher and lower growth volatility. The results suggest that there are two possible points at which the economy has experienced a switch from higher to lower growth volatility phases, in 1977 and in 1984. Of these two switch points, by far the most significant is the switch to more a stable growth path that occurred in 1984 and which extends to the end of the sample period. We suggested that this state switch could be explained by important domestic events affecting the economy around that time. The fact that we are not able to identify a significant autoregressive structure in the stochastic growth process, or any regular switching in the mean growth rate, has an important implication for the analysis of the business cycle, at least as far as Australia is concerned. 4

The significant downturns in activity in the early 1980s and the beginning of the 1990s are the obvious examples. 5 We use the term random in the context of our single equation model. It may be that some of the variation in activity contained within the noise component could be explained by say international events, such as a downturn in activity in foreign markets. A multivariate model would be needed to examine such possibilities. This would take us beyond the scope of the present paper.

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Previous studies for Australia have identified an autoregressive structure in GDP growth models and have taken this to be an indicator of the presence of a cyclical feature. In contrast, our results suggest that the main variability in GDP growth is explained by a one-off regime shift in the volatility of the process rather than a traditional business cycle feature. This does not of course imply that there are never any ups and downs in GDP, or that it isn’t possible to identify peaks and troughs in the level of GDP, as advocated by Harding and Pagan (2003a, 2003b). What it does mean, however, is that some care is needed in defining exactly what is meant by a business cycle and whether the cycles should be thought of as one-off or recurring events. Our analysis deals explicitly with the growth process and regards cyclicality as being indicated by an autoregressive process or regular switches in the mean growth rate. Viewed from this perspective, our results suggest that, unlike many other countries, the cycles in the Australian growth rate do not exhibit a degree of regularity or repetition of the kind that one would normally associate with the business cycle. Finally, we have considered only a subset of the possible models that could be estimated from the general switching model family and, strictly speaking, our results are conditional on that limitation. In principle it would be possible to consider a wider and more complex set of models than have been considered in this paper. This is a line of investigation that the authors are currently pursuing, in the context of a comparative study of the growth process in several industrial economies. The MCMC procedure is computationally intensive and the major limitation on estimating models with more complex switching possibilities would appear to be the computational time involved.

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