Title

Bayesian Estimates of Potential Output and NAIRU for Taiwan

Shin-Hui Chen （陳馨蕙）Department of Economics,

National Dong Hwa University

Jin-Lung Lin (林金龍 )Department of Finance

National Dong Hwa University

Motivation and Framework

Section2

Section3

Section4e

Section5le

This paper aims to develop the corresponding Bayesian sampling algorithms for Watson’s decomposition method and Apel and Jansson’s systems approach.

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SectionSection

1 Introduction

Econometric Models2

3Sampling Algorithms forBayesian State Space Models

The Monte Carlo Simulation4

5Application to Taiwan’s SeasonallyUnadjusted Data

2. Econometric Models

2.1 A Summary of SSM and Kalman Filter

2.2 Watson’s Decomposition

2.3 Apel and Jansson’s decomposition

A Brief Summary of SSM and Kalman Filter• A general state space model can be written as:

where θt, Yt, Vt, Wt are the state variables, observed variables measurement

error terms, and disturbance terms, respectively. Ft and Gt are known

matrices and could be time-varying or time-invariant.

The state variables t is assumed to follow Gaussian distribution and 0 has initial prior

System Equation

Observation Equation

Dt−1 denote the information provided by set of past observations Y1, · · · , Yt−14

A Summary of SSM and Kalman Filter (Cont.)Our interest is to compute the conditional densities p(θs|Dt).

When s = t, the Kalman Filter recursion is applied to compute the conditional densities, p(θs|Dt).

If θt−1|Dt−1~N(mt−1,Ct−1), then the Kalman Filter for model (1) is

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A Summary of SSM and Kalman Filter (Cont.)

When s < t, the concept of smoothing is applied. If

then the smoothing recursion for model (1) is

Excellent exposition of DLM can be found in West and Harrison (1997), Koopman and Ooms (2006) and Petris, Petrone and Campagnoli (2009). 6

Watson’s DecompositionWatson (1986) decomposed observed GDP as the sum

of potential GDP and output gap and the model is listed below

For seasonally unadjusted series, we frequently observe seasonal unit root rather than regular unit root

We replace the random walk equation with seasonal unit root equation and keep the specification of output gap unchanged

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Watson’s Decomposition in Sate Space Form

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Apel and Jansson’s DecompositionApel and Jansson (1999) added inflation rates and unemployment to the model. Output and employment is linked via Okun’s law while the relationship between output and inflation is governed by Phillips curve.

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2.3 Apel and Jansson’s Decomposition

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3. Sampling Algorithms for Bayesian State Space Models

3.1 Sampling the States from p(θ0:T |ψ, y1:T ): FFBS Algorithm

3.2 Sampling the Unknown Parameters from p(ψ|θ0:T , y1:T )

3.3 Algorithm for Watson’s Model

3.4 Algorithm for Apel and Jansson’s Model

),|( :1:0 TT yp

MCMC and Gibbs Sampling• Consider a general DLM model as model (1), our primary

interest is the joint conditional distribution of the state vectors and the unknown parameters given the data

where y1:T and θ0:T denote the data (y1, · · · , yT ) and (θ0, · · · , θT ), respectively.

To achieve greater efficiency, Markov Chain Monte Carlo (MCMC) and in particular Gibbs sampling algorithm are used for approximating the joint posterior p(θ0:T , ψ|y1:T ).

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Algorithm 1 Gibbs Sampling Algorithm

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FFBSConjugated Priors

M-H Algorithm

Forward Filtering Backward Sampling (FFBS)

14Note that the last factor in the product, p(θT |DT ), is exactly the Kalman filter.

The Forward Filtering Backward Sampling (FFBS) algorithm (Carter and Kohn,1994; Fruhwirth-Schnatter,1994) provides a more efficient way to sample the full set θ0:T from the complicated and high-dimensional full posterior.

Algorithm 2: FFBS Algorithm

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FFBSConjugated Priors

M-H Algorithm

FFBS is essentially a simulation version of the smoothing recursions.

Excellent summaries of FFBS can be found in Doucet, Logothetis and Krishnamurhty(2000), Petris, Petrone and Campagnoli (2009) and Cargnoni, Muller and West (2010).

Conjugate Priors—Normal Gamma Priors

Unknown Parameters Conjugate Priors

The disturbance (σ2) If we assume that σ2 ~ IG(c/2, s/2) then

The unknown coefficients (β).

The full posterior distributions will be

where β| · · · ~ (mn, Cn) and V = (σ2)I .16

Typically, we can decompose the unknown parameters into two components, the unknown coefficients (β) and the disturbance (σ2).

When were known (drawn by FFBS algorithm), the state space model would reduce to a normal linear regression model and the unknown parameters are conditionally conjugated (see e.g., Durbin and Koopman, 2002; Koop, 2003 §4).

)(:0iT

]2/)(,2/)[(~...)|(1

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n

iiusncIGp

The Metropolis-Hastings Algorithm

However, in practice the unknown coefficients (β) can be multidimensional and their posterior distribution depends heavily on the model specification.

In Stock and Watson’s model, for example, the potential GDP follows a random walk while the output gap is an AR(2) process.

In this case, p(1|2) does not have a standard form and is difficult to simulate from.

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Algorithm 3 Metropolis-Hastings Algorithm

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This is not the first paper that samples an intractable posterior distribution arises in a stationary AR(2) process by Metropolis-Hastings algorithm. For example, see Chib and Greenberg (1994).

3.3 Algorithm for Watson’s Model

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3.4 Algorithm for Apel and Jansson’s ModelEmpirical analysis indicates that inflation and output

depend negatively on lagged cyclical unemployment by the parameters (η1, η2) and (1, 2), respectively (see e.g., Apel, 1999; Schumacher, 2008).

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4. The Monte Carlo Simulation

4.1 A Simulation Study for Watson’s Model

4.2 A Simulation Study for for Apel and Jansson’s Model

4.1 A Simulation Study for Watson’s Model

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Fig 1: Estimated results of Watson’s model with simulated data

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Simulated Estimated

Figure 2: Diagnostic plots for simulatedWatson’s model

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Table 2: Model Calibration and the Estimating Results of Apel’s model

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Figure 3: Estimated Results of Simulated Apel and Jansson’s model

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Simulatedoutput gap

Simulatedunemployment gap

EstimatedEstimated

The evolution of estimated the output gap and the unemployment gap are almost identical to their respective simulated series.

Table 2: Model calibration and the estimating results of Apel’s model

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First note that the posterior means of δ’s, η’s and ’s are much closer to their true values than the ML estimates.

Second, the ML estimates of the standard deviations, σ’s, tend to be smaller than their corresponding posterior means.

This example indeed shows that our Bayesian sampling algorithms are practical and flexible and do not merely duplicate the ML estimates.

Figure 4: Prior and posterior distribution of parameters

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These summaries demonstrate that the initial prior brief has onlya modest effect on the posterior shrinkage.

The bivariate scatterplots of (1, 2), (δ1, δ2) and (η1, η2), together with their corresponding marginal histograms show that there is a strong dependence between (1, 2), (δ1, δ2) and (η1, η2). This confirms that drawing these pairs of parameters simultaneously is essential in improving the mixing of the chain.

5 Application to Taiwan’s Seasonally Unadjusted Data

5.1 Results for Watson’s Model

5.2 Results for Apel and Jansson’s Model

Lin and Chen (2010) document that there appears to be a rising trend in the Taiwan’s unemployment gap, possibly as a result of structural changes or the periodicity of the cycle become longer.

Distinct classes of NAIRU specifications are implemented to mitigate the concern about implausible estimates and misspecification.

Lin and Chen (2010) documented that specifying the unemployment gap as an integrated AR(1) process leads a dramatic decline in the sum of δ’s and both the unemployment gap and output gap became more stable.

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Previous Literature on Taiwan’s Potential Output and NAIRU

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Estimating potential output and NAIRU for Taiwan with conventional, methods is problematic !!!

The Application to TaiwanWith appropriate priors, prior information about

the structure of the economy based on theory and country-specific circumstance can be embedded in models (Waliszewski, 2010)

We take Lin and Chen’s (2010) estimates as a reference for the prior means but leave a considerable amount of uncertainty around

the prior variances.

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Results of Watson’s model with real data—Table3 and Figure 5

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Comparison of the posterior means of σz and σy show that the transitory component plays a major role in the output fluctuations.

First, when seasonal unit root is explicitly considered, the estimates of the states exhibit no seasonal fluctuation.

Figure 6: Estimated Results of Apel and Jansson’s Model with Real Data

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First, when seasonal unit root is explicitly considered, the estimates of the states exhibit no seasonal fluctuation.

The recession in the 2000s appears to be very severe compared to Watson’sunivariate model.

The financial crisis of 2008 further induced sharp increase in the NARIU.

Table 4: Empirical results of Apel’s model

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Table 4 shows that the 95% posterior interval traps the ML estimates for mostunknown coefficients (δ’s, η’s, ’s and α).

The posterior standard deviations of unknown parameters and their corresponding standard errors generally exceed the standard errors reported by the ML

Table 4: Empirical results of Apel’s model

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1. Traditional Kalman filter (ML) puts too little weight on the variance of the permanent component.2. Bayesian estimates also allow for

more stochastic variation in the cyclical component, the unemployment gap and output gap.

Figure 7: Comparison between the Bayesian State Estimates and the ML State Estimates

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Red line : Bayesian(AR2), the Bayesian estimates with AR(2) cyclical unemployment.Black line : ML(AR2) denotes the maximum likelihood estimates with AR(2) cyclical unemployment. Blue line: ML(DAR1) the maximum likelihood estimates with an integrated AR(1) process.

Figure 7: Comparison between the Bayesian State Estimates and the ML State Estimates

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The similarity between the Bayesian output gap and the ML(DAR1) output gap demonstrates that the Bayesian framework is rich enough to cope with model misspecification and identification problems.

What is particularly striking is that without any particular model specification, the path of the Bayesian output gap is almost the same as the path of ML(DAR1) output gap (the blue line).

Compared to the ML(AR2) output gap (black line), the Bayesian(AR2) output gap is slowly trending downwards and contains a more pronounced cyclical pattern.

Comparison between the Bayesian State Estimates and the ML State Estimates

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Although the posterior means of δ’s are almost the same as the ML estimates, we find that the posterior distribution indeed allows for more parameter uncertainty

Figure 7: Comparison between the Bayesian State Estimates and the ML State Estimates

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All the unemployment gap estimates show that the economy has experienced a significant increase in the unemployment gap during 2000s.

Even with same model specification, the Bayesian(AR2) output gap is slowly trending upwards and presents a significantly morenegative unemployment gap during 1987 to 1999.

SummaryThis paper develops the corresponding Bayesian

sampling algorithms for Watson’s decomposition method and Apel and Jansson’s system approach.

Simulation and empirical analyses show that our Bayesian sampling algorithms are flexible and do not merely duplicate the maximum likelihood estimates.

We find that the maximum likelihood generally understates the parameter variability and puts too little weight on the variance.

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SummaryWhile a Bayesian approach allows for more stochastic

variation in the permanent and cyclical component

Our results demonstrate that the posterior distribution facilitates assessment of the parameter uncertainty.

A Bayesian approach is rich enough to cope with model

specification issues and provides more relevant information for conducting monetary and fiscal policies .

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Thank You for Your

Listening!

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