Estimation of Regional Business Cycle in Japanusing Bayesian Panel Spatial Autoregressive

Probit Model∗

Kazuhiko Kakamu† Hajime Wago‡ Hisashi Tanizaki§

Abstract

This paper considers the panel spatial autoregressive probit modelfrom Bayesian point of view. The probit model is useful for qualitative(specifically, binary) data analyses. As it is well known, it is not easy toestimate the panel probit model by the maximum likelihood method.Therefore, in this paper, we consider the Markov chain Monte Carlo(MCMC) method to estimate the parameters as an alternative estima-tion method. Moreover, including the spatial autoregressive effect intothe panel probit model we consider estimating the panel spatial autore-gressive probit model. Our model (i.e., the panel spatial autoregressiveprobit model) is illustrated with simulated data set and it is comparedwith the panel probit model. Furthermore, taking into account the

∗An earlier version of this paper was presented at the 2005 Japanese Joint StatisticalMeeting at Hiroshima, the 2005 Fall meeting of Japanese Economic Association at ChuoUniversity, MODSIM05 in Melbourne and ISBA08 in Hamilton Island. We would like tothank seminar/conference participants, especially Hideo Kozumi, Koji Miyawaki, YasuhiroOmori, Mike K.P. So and Cathy W.S. Chen for their helpful comments and suggestions.However, responsibility for any errors remains entirely with us. This research was partiallysupported by the Ministry of Education, Science, Sports and Culture (Grant-in-Aid forYoung Scientists (B) #20730144) for K. Kakamu and Japan Society for the Promotion ofScience (Grant-in-Aid for Scientific Research (C) #18530158) for H. Tanizaki.†Faculty of Law and Economics, Chiba University, Inageku, Chiba 263-8522, Japan.

Email: kakamu@le.chiba-u.ac.jp.‡Department of Economics, Kyoto Sangyo University, Kitaku, Kyoto 603-8555, Japan.

Email: wago@ism.ac.jp§Graduate School of Economics, Kobe University, Nadaku, Kobe 657-8501, Japan.

Email: tanizaki@kobe-u.ac.jp

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spatial interaction, we explore the regional business cycle across 47 pre-fecture data in Japan, where the estimation period is from 1979 to 2003.Using Japanese regional data, we conclude from the empirical resultsthat the spatial interaction plays an important role in the regional busi-ness cycle.

Key Words: Business cycle; Markov chain Monte Carlo (MCMC);Panel spatial autoregressive probit model.

JEL Classification: C11, C15, C23, R11.

1 Introduction

The probit model has been widely used in qualitative data regression, espe-cially in microeconometrics or business cycle analysis, and it has been extendedto the panel analysis. Maddala (1987) gives excellent surveys of the panel pro-bit model. Although the probit model is useful for qualitative data analysis,it is difficult to estimate the panel probit model by the maximum likelihoodmethod, as pointed out by Bertschek and Lechner (1998). Therefore, alterna-tive estimation methods such as the generalized method of moments (GMM)are proposed (see Bertschek and Lechner, 1998; Greene, 2004 and so on) forestimation of the panel probit model.

Since the seminal work by Anselin (1988), the spatial dependency becomesthe concern of economic activity. Therefore, the probit model with the spatialdependency (i.e., the spatial autoregressive probit model) has been examinedboth analytically and empirically. For example, Holloway et al. (2002) ap-ply the spatial probit model to high-yielding variety (HYV) adoption amongBangladeshi rice producers. Smith and LeSage (2004) also propose the spatialdependency with individuals in each location, and apply to the 1996 presi-dential election results for 3110 US counties. As it is difficult to estimate thespatial autoregressive probit model by the maximum likelihood method, allthe past studies are estimated by the Markov chain Monte Carlo (MCMC)method (e.g., LeSage, 2000).

In the business cycle analysis, it is important to capture the turning point.Several models are proposed, i.e., probit and logit model (Maddala, 1992);sequential probability recursion model (Neftci, 1982); Markov switching model(Hamilton, 1989); dynamic Markov-switching factor model (Watanabe, 2003).However, they do not take into account the regional business cycle with the

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spatial dependency. In this paper, incorporating the spatial dependency intothe panel probit model, we consider the panel spatial autoregressive probitmodel as a regional business cycle model.

From Bayesian point of view, Kakamu and Wago (2008) examine the panelspatial autoregressive model. This paper considers the panel spatial autore-gressive probit model in Bayesian framework, by extending LeSage (2000) andKakamu and Wago (2008). Our model (i.e., the panel spatial autoregressiveprobit model) is illustrated with simulated data set and it is compared withthe panel probit model, where the spatial autoregressive effect is ignored. Wefind from the simulation results that the bias appears in the parameter esti-mates when the estimated model is the panel probit model and the true modelis the panel spatial autoregressive model. In addition, the bias increases as thespatial autoregressive effect is large, which implies that the spatial interactionplays an important role.

Furthermore, taking into account the spatial interaction, we also explorethe regional business cycle in Japan, where 47 prefecture data during theperiods from 1979 to 2003 are utilized. From the empirical results, (i) weconfirm that the occurrence and collapse of the bubble economy is a commonphenomena in Japan, i.e., the probability that the growth rate of regional IIP(Index of Industrial Production) is positive falls drastically in 1992-1993 andrises after 1993 (note that the bubble economy in Japan burst in 1991), (ii)the business cycle in Japan is very similar to the regional business cycle ineach prefecture, and (iii) the posterior distributions of the spatial interactiondo not include zero in the 95% credible intervals, when the business cyclephase (expansion or contraction) in most of the prefectures is similar to eachother (i.e., 1979, 1984, 1988-1993, 1998 and 2001). Thus, we conclude fromJapanese regional data that the spatial interaction plays an important role inthe business cycle.

The rest of this paper is organized as follows. In the next section, wesummarize the panel spatial autoregressive probit model. Section 3 discussescomputational strategy of the MCMC methods. In Section 4 our approach isillustrated with simulated data set. Section 5 presents the empirical resultsbased on the business index records in Japan, where 47 prefectures data from1979 to 2003 are examined. Section 6 summarizes the results with concludingremarks.

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2 Panel Spatial Autoregressive Probit Model

Let yit be a binary data and xit be a 1 × k vector of exogenous variables,where i indicates the region (i = 1, 2, · · · , n) and t represents time period(t = 1, 2, · · · , T ). Let wij denote the spatial weight of the jth region in the ithregion, which is given by: (i) wij = 0 for all i = j and (ii) wij = 1/mi whenthe jth region is contiguous with the ith region and wij = 0 otherwise, wheremi denotes the number of regions which is contiguous with the ith region.Note that we have

∑nj=1wij = 1 for all i. In the panel model, we consider

that the unobservable component which is specific to the ith region affectsthe dependent variable (e.g., Kakamu and Wago, 2008). αi is denoted by theunobservable component. Then, the panel spatial autoregressive probit modelwith parameters ρt, αi and β is written as follows:

yit =

{1, if zit ≥ 0,0, if zit < 0,

zit = ρtn∑

j=1

wijzjt + αi + xitβ + εit, εit ∼ N (0, 1), (1)

where zit is a latent variable (see Tanner and Wang, 1987). In (1), ρt representsthe spatial interaction at time t. We may assume ρt = ρ. In the model above,however, it might be more plausible to consider that the spatial interactionvaries over time. Let us define:

ρ = (ρ1, ρ2, · · · , ρT )′, α = (α1, α2, · · · , αn)′,

zt = (z1t, z2t, · · · , znt)′, z = (z′1,z′2, · · · ,z′T )′,

yt = (y1t, y2t, · · · , ynt)′, y = (y′1,y′2, · · · ,y′T )′,

X t = (x′1t,x′2t, · · · ,x′nt)′, X = (X ′1,X

′2, · · · ,X ′T )′.

W is called the spatial weight matrix (see, e.g., Anselin, 1988), where theelement of W in row i and column j is denoted by wij, defined above. Then,the likelihood function of the model (1) is given by:

L(y|ρ,α,β,z,X,W ) =T∏

t=1

f(yt|ρt,α,β,zt,X t,W ), (2)

where

f(yt|ρt,α,β,zt,X t,W ) = (2π)−n2 |In − ρtW | exp

(−e

′tet2

)

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×n∏

i=1

T∏

t=1

{yit1[0,∞)(zit) + (1− yit)1(−∞,0](zit)

}.

In indicates the n×n unit matrix, et is given by et = zt−ρtWzt−X tβ−α,and 1(a,b)(x) denotes the indicator function, which takes 1 when x lies on theinterval between a and b.

3 Posterior Analysis

3.1 Joint Posterior Distribution

Since we utilize Bayesian method for estimation, we adopt the following hier-archical prior:

π(ρ,α,β, µ, ξ2) =

{T∏

t=1

π(ρt)

}{n∏

i=1

π(αi|µ, ξ2)

}π(β)π(µ)π(ξ2),

which is used in Kakamu and Wago (2008), where µ and ξ2 indicate the meanand variance of αi, respectively.

Given the prior density π(ρ,α,β, µ, ξ2) and the likelihood function (2), thejoint posterior distribution can be expressed as:

π(ρ,α,β, µ, ξ2|z,y,X,W )

∝ π(ρ,α,β, µ, ξ2)T∏

t=1

f(yt|ρt,α,β,zt,X t,W ). (3)

We assume the prior distributions, i.e., π(ρt), π(αi|µ, ξ2), π(β), π(µ) and π(ξ2),as follows:

ρt ∼ U(−1, 1), αi|µ, ξ2 ∼ N (µ, ξ2), β ∼ N (β0,Σ0),

µ ∼ N (µ0, ξ20), ξ2 ∼ IG(

ν0

2,λ0

2),

where IG(a, b) denotes the inverse gamma distribution with scale parametera and shape parameter b.

3.2 Posterior Simulation

From the joint posterior distribution (3), now we can implement the MCMCmethod. The Markov chain sampling scheme can be constructed from the full

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conditional distributions of {zt}Tt=1, {ρt}Tt=1, {αi}ni=1, β, µ and ξ2, which areshown in Sections 3.2.1 – 3.2.3. Using the Gibbs sampler (e.g., see Gelfandand Smith, 1990), the random draws from the posterior distribution of {zt}Tt=1,{ρt}Tt=1, {αi}ni=1, β, µ and ξ2 are generated.

3.2.1 Sampling {zt}Tt=1

In the case of the probit model, it is required to generate the latent vari-ables {zt}Tt=1. Let us define z−t = {z1, · · · ,zt−1, zt+1, · · · , zT}, where zt isexcluded from {zt}Tt=1. Tanner and Wang (1987) proposed the data augmen-tation method to generate the latent variables. We utilize Tanner and Wang(1987) in this paper. The full conditional distribution of zt follows:

zt|ρ,α,β, µ, ξ2,z−t,y,X,W ∼MT Nat≤zt≤bt(zt, V t), (4)

with zt = V t{In − ρtW }′(α +X tβ), V−1

t = (In − ρtW )′(In − ρtW ), at =(a1t, a2t, · · · , ant)′ and bt = (b1t, b2t, · · · , bnt)′. MTNa<z<b(µ,V ) denotes themultivariate truncated normal distribution with mean µ and scale matrix V ,where z is distributed between a and b. In this paper, the truncation is setto be (ait, bit) = (0,∞) when yit = 1 and (ait, bit) = (−∞, 0) when yit =0. The multivariate truncated normal random variable zt is sampled by theMetropolis-Hastings algorithm, which is proposed by Miyawaki (2008).

3.2.2 Sampling {ρt}Tt=1

Define ρ−t = {ρ1, · · · , ρt−1, ρt+1, · · · , ρT}, where ρt is excluded from ρ. From(3), the full conditional distribution of ρt is written as:

p(ρt|ρ−t,α,β, µ, ξ2, z,y,X,W ) ∝ |In − ρtW | exp

(−e

′tet2

), (5)

with et = (In−ρtW )zt−X tβ−α. Using the Metropolis algorithm (e.g., seeTierney, 1994), a random draw of ρt is sampled from the conditional distribu-tion p(ρt|ρ−t,α,β, µ, ξ2,z,y,X,W ).

The following Metropolis step is used: (i) sample ρnewt from:

ρnewt = ρoldt + ctηt, ηt ∼ N (0, 1), (6)

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where ct is called the tuning parameter and ρoldt denotes the random drawpreviously sampled, (ii) evaluate the acceptance probability:

ω(ρoldt , ρnewt ) = min

(p(ρnewt |ρ−t,α,β, µ, ξ2,z,y,X,W )

p(ρoldt |ρ−t,α,β, µ, ξ2,z,y,X,W ), 1

),

and (iii) set ρt = ρnewt with probability ω(ρoldt , ρnewt ) and ρt = ρoldt otherwise.The proposal random draw of ρt generated from (6) takes the real value withinthe interval (−∞,∞), although the prior distribution of ρt lies on the intervalbetween −1 and 1 (see Section 3.1). If the candidate of ρt does not lie on theinterval (−1, 1), the conditional posterior should be zero and accordingly theproposal value is rejected with probability one (see Chib and Greenberg, 1998).We need the tuning parameter ct in sampling ρt. In the numerical examplediscussed below, we choose the tuning parameter such that the acceptancerate becomes between 0.4 and 0.6 (see Holloway et al., 2002).

3.2.3 Sampling the Other Parameters

The full conditional distribution of β is:

β|ρ,α, µ, ξ2,z,y,X,W ∼ N (β, Σ), (7)

with β = Σ{X ′(z − (Ψ ⊗W )z −∆α) + Σ−10 β0}, ∆ = (1T ⊗ In), 1T is a

T × 1 unit vector, and Σ = (X ′X + Σ−10 )−1. The element of Ψ in row t and

column s is given by ρt for t = s and zero for t 6= s, i.e., Ψ = diag(ρ).Given z, ρ and β, (1) is written as:

zit −n∑

j=1

ρtwijzjt − xitβ = εit, εit ∼ N (αi, 1).

Let us define α−i = {α1, · · · , αi−1, αi+1, · · · , αn}, where αi is excluded from α.The full conditional distribution of αi follows:

αi|ρ,α−i,β, µ, ξ2,z,y,X,W ∼ N (αi, ξ2), (8)

with αi = ξ2{∑Tt=1(zit −∑n

j=1 ρtwijzjt − xitβ) + ξ−2µ} and ξ2 = (T + ξ−2)−1.Full conditional distributions of µ and ξ2 are given by:

µ|ρ,α,β, ξ2, z,y,X,W ∼ N (µ, σ2), (9)

ξ2|ρ,α,β, µ, z,y,X,W ∼ IG(ν

2,λ

2), (10)

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with µ = σ2(ξ−2∑ni=1 αi + ξ−2

0 µ0), σ2 = (ξ−2n + ξ−20 )−1, ν = n + ν0 and

λ = (α− µ)′(α− µ) + λ0.Thus, from (4), (5), (7) – (10), the random draws of (z,ρ,α,β, µ, ξ2) are

easily sampled from Gibbs sampler (e.g., Gelfand and Smith, 1990).

4 Numerical Example by Simulated Data

To illustrate the panel spatial autoregressive probit model using Bayesian es-timation, yit and zit (i = 1, 2, · · · , n and t = 1, 2, · · · , T ) are generated from:

yit =

{1, if zit ≥ 0,0, if zit < 0,

zit = ρn∑

j=1

wijzjt + αi + 1.0x1it + 1.0x2it + εit, εit ∼ N (0, 1),

where αi is sampled from the normal distribution N (0, 2), and xit = (x1it, x2it)is generated as a vector of two standard normal variates, i.e., k = 2. In thisnumerical example, we construct the spatial weight matrix W as: (i) generatew∗ij for i > j from Bernoulli distribution with probability of success 0.2, (ii) setw∗ji = w∗ij for i 6= j and w∗ij = 0 for i = j, and (iii) compute

∑nj=1w

∗ij = mi and

wij = w∗ij/mi for all i, j. To see the effect of the spatial autocorrelation, weconsider the cases of ρ = 0.3, 0.6, 0.9. In this simulation study, ρ is assumed tobe constant over time. For the number of regions, n = 50 is taken. To see theeffect of T , we consider the cases of T = 5, 10, 15. For the prior distributions,we set the hyper-parameters as follows:

β0 = 0, Σ0 = 100× Ik, µ0 = 0, ξ20 = 100, ν0 = 2, λ0 = 0.01. (11)

Since it is interesting to see the cases where the spatial autocorrelationis ignored, we estimate the panel spatial autoregressive probit model for twocases: ρ = 0 and ρ 6= 0. Using the simulated data, we implement the MCMCalgorithm, where 300, 000 iterations are performed and the first 100, 000 itera-tions are discarded as the initial burn-in period. Taking every ten draws out ofthe remaining iterations (i.e., 20, 000 random draws are chosen), we obtain theposterior statistics for the parameters. Based on the convergence diagnostictest proposed by Geweke (1992), we make sure whether the 100, 000 burn-initerations are practically large enough. In the burn-in iterations, the tuning

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Table 1: Simulated Data: Posterior Means and Standard Deviations

(a) Panel Spatial Autoregressive Probit Model with No Restriction (ρ 6= 0)T True Value ρ = 0.3 ρ = 0.6 ρ = 0.95 µ 0.0 −0.131 (0.208) −0.027 (0.263) 0.064 (0.194)

β1 1.0 0.773 (0.143) 1.290 (0.245) 1.136 (0.264)β2 1.0 1.063 (0.171) 1.368 (0.251) 1.254 (0.272)ξ2 2.0 1.674 (0.727) 2.908 (1.461) 1.584 (1.079)ρ 0.290 (0.204) 0.342 (0.169) 0.900 (0.031)

10 µ 0.0 0.030 (0.177) −0.067 (0.214) −0.047 (0.214)β1 1.0 0.847 (0.103) 1.132 (0.132) 1.140 (0.144)β2 1.0 0.870 (0.101) 1.151 (0.130) 0.985 (0.143)ξ2 2.0 1.372 (0.402) 2.163 (0.679) 2.121 (0.697)ρ 0.329 (0.137) 0.561 (0.086) 0.887 (0.023)

15 µ 0.0 −0.031 (0.203) −0.018 (0.212) −0.042 (0.194)β1 1.0 0.921 (0.092) 0.983 (0.095) 1.025 (0.114)β2 1.0 0.917 (0.087) 0.942 (0.094) 1.074 (0.114)ξ2 2.0 1.841 (0.555) 2.133 (0.613) 1.784 (0.545)ρ 0.169 (0.117) 0.584 (0.077) 0.898 (0.018)

(b) Panel Spatial Autoregressive Probit Model with Restriction (ρ = 0)T True Value ρ = 0.3 ρ = 0.6 ρ = 0.95 µ 0.0 −0.167 (0.213) −0.051 (0.277) 0.160 (0.090)

β1 1.0 0.754 (0.139) 1.237 (0.241) 0.451 (0.084)β2 1.0 1.036 (0.165) 1.350 (0.259) 0.560 (0.096)ξ2 2.0 1.523 (0.666) 2.887 (1.504) 0.025 (0.045)

10 µ 0.0 0.057 (0.182) −0.077 (0.211) 0.038 (0.113)β1 1.0 0.846 (0.104) 1.066 (0.124) 0.620 (0.079)β2 1.0 0.863 (0.100) 1.035 (0.117) 0.476 (0.073)ξ2 2.0 1.335 (0.389) 1.871 (0.581) 0.411 (0.144)

15 µ 0.0 −0.058 (0.202) −0.105 (0.196) −0.364 (0.102)β1 1.0 0.913 (0.092) 0.905 (0.089) 0.526 (0.062)β2 1.0 0.899 (0.085) 0.831 (0.082) 0.558 (0.060)ξ2 2.0 1.781 (0.525) 1.688 (0.468) 0.365 (0.117)

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0:20:10:0 0:0 0:5 1:0� = 0:3, T = 5Skew = �0:246 0:20:10:0 0:0 0:5 1:0� = 0:6, T = 5Skew = �0:288 0:20:10:0 0:0 0:5 1:0� = 0:9, T = 5Skew = �0:7040:20:10:0 0:0 0:5 1:0� = 0:3, T = 10Skew = �0:280 0:20:10:0 0:0 0:5 1:0� = 0:6, T = 10Skew = �0:344 0:20:10:0 0:0 0:5 1:0� = 0:9, T = 10Skew = �0:3120:20:10:0 0:0 0:5 1:0� = 0:3, T = 15Skew = �0:153 0:20:10:0 0:0 0:5 1:0� = 0:6, T = 15Skew = �0:428 0:20:10:0 0:0 0:5 1:0� = 0:9, T = 15Skew = �0:307Figure 1: Simulated Data: Posterior Distributions of ρ

parameter ct is chosen between 0.024 and 0.146. Ox version 4.1 (see Doornik,2006) is used to obtain all the results in this paper.

Table 1 shows the posterior estimates of the parameters. (a) in Table 1represents the panel spatial autoregressive probit model with no restriction(i.e., ρ 6= 0), while (b) in the table indicates the panel spatial autoregressiveprobit model with restriction (i.e., ρ = 0), which is equivalent to the panelprobit model. The values in the parentheses give us the standard deviationsbased on the random draws generated from the posterior distributions. It isshown from Table 1 that the posterior estimates of (a) are not too differentfrom those of (b) in the case of ρ = 0.3. However, as ρ increases, distinctdifferences between (a) and (b) appear for all the estimates of µ, β and ξ2.As it is easily expected, we can observe serious biases in the restricted model(b) as ρ is large. In the case of the panel spatial autoregressive model (a),the mean estimates of β, ξ2 and ρ are not too different from the true values.That is, under the condition that there exists the spatial dependency in thetrue model, the biases of the estimated posterior means become serious if weestimate the panel probit model in which the spatial autocorrelation is ignored.In addition, we can see that all the posterior means go to the true values as Tis large (i.e., see the case of T = 15).

The posterior distributions of ρ are displayed in Figure 1. We cannot findthe features of the empirical distributions at a glance. However, it is shown

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that the posterior distributions are skewed to the left. Note that in each figureSkew indicates skewness of the empirical distribution based on the MCMCdraws. As ρ is large, the skewness to the left becomes large in absolute value.

5 Regional Business Cycle in Japan

As an empirical application, we consider the regional business cycle across 47prefectures in Japan (i.e., n = 47) during the periods from 1979 to 2003 (i.e.,T = 25). Let IIPit be Index of Industrial Production by Region, which is takenfrom Ministry of Economy, Trade and Industry. The dependent variable yittakes one if the growth rate of IIPit is positive and zero otherwise. The busi-ness cycle is not observed in general. Therefore, when the growth rate of IIPit

is positive, we consider that the ith region at time t is in the expansion periodand then yit = 1 is set as a proxy variable of the business index. Contrarily,when the growth rate of IIPit is negative, the ith region at time t is in the con-traction period and accordingly yit = 0 is set. As explanatory variables xit, wetake Scheduled Cash Earnings for Male Workers (WAGE) and Job Offers-to-Applicants Rate (JOB) from Ministry of Health Labor and Welfare, Loans andDiscounts Outstanding of Domestically Licensed Banks by Prefecture (BANK)from Bank of Japan, and Index of Financial Potential (FINANCE) from Min-istry of Internal Affairs and Communications. Note that FINANCE is givenby the average of the revenue-expenditure ratio data in the past three yearsfor each prefecture. FINANCE shows a large value when a prefecture is fi-nancially generous. All the data are also available from Statistical Informa-tion Institute for Consulting and Analysis (called Sinfornica, Japanese web-site: http://www.sinfonica.or.jp/datalist/index.html). Furthermore,all the data except for FINANCE are transformed into per capita data, andall the data are converted into the percent change data. As discussed in Section2, the weight matrix W consists of contiguity dummy variables, proposed byKakamu et al. (2008),1 and the average number of dummy variables is about

1Except for Okinawa prefecture, all Japanese prefectures are located in the four majorislands, i.e., Hokkaido, Honshu, Shikoku and Kyushu. We consider that these four islandsare connected with each other by trains and roads, despite the fact that the four islandsare geographically separated. For example, we consider that Hokkaido is contiguous withHonshu through the Seikan railway tunnel. Honshu and Shikoku are contiguous with eachother through the Awaji and Seto Bridges, and Kyushu is also contiguous with Honshuthrough the Kanmon Tunnel and Bridge. Okinawa is the only one prefecture which isindependent of all the other prefectures. For Japanese prefectures, note as follows. Shikoku

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Table 2: Empirical Results

Mean SD 2.5% 97.5% CD

µ 0.156 0.073 0.015 0.301 0.770WAGE −0.170 1.303 −2.780 2.340 0.304JOB 3.142 0.440 2.308 4.034 0.196BANK 3.516 1.340 0.930 6.154 0.182FINANCE −0.362 1.629 −3.546 2.827 0.353ξ2 0.026 0.025 0.002 0.089 0.115

4. Using the same hyper-parameters as in Section 4, the MCMC algorithm isimplemented, using 300, 000 iterations and discarding the first 100, 000 itera-tions as the initial burn-in period. Out of the remaining draws (i.e., 200, 000draws), we take every ten draws to obtain the posterior statistics for the pa-rameters. The tuning parameters (i.e., ct, t = 1, 2, · · · , T ) are chosen between0.027 and 0.424 in the burn-in period.

Table 2 shows the estimation results, where Mean, SD and CD representthe posterior mean, the standard deviation and Geweke’s convergence diag-nostics (Geweke, 1992), respectively. CD represents the p value based on thetest statistic on difference between two sample means (i.e., dividing all thegenerated random draws into three parts, we compute two sample means fromthe first 20% and last 50% of the random draws), where the test statistic isasymptotically distributed as a standard normal random variable. We judgethat the random draws generated by MCMC do not converge to the randomdraws generated from the target distribution when CD is less than 0.05 orgreater than 0.95. See Geweke (1992) for detail discussion on CD. From theestimation results in Table 2, we can conclude that the random draws gen-erated by MCMC converge to the random draws generated from the targetdistribution, because CD is between 0.05 and 0.95.

It might be expected in Table 2 that all of JOB, BANK and FINANCEare positively correlated with the business cycle indicator yit. That is, we

consists of four prefectures, i.e., Tokushima, Kagawa, Ehime and Kochi prefectures (seeFigure 9). Kyusyu includes seven prefectures, i.e., Fukuoka, Saga, Nagasaki, Kumamoto,Oita, Miyazaki and Kagoshima prefectures (see Figure 10). Honshu consists of 34 prefectures(i.e., all the prefectures except for Hokkaido in Figure 3 and all the prefectures in Figures 4– 8).

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can consider that a prefecture is in the expansion period when JOB, BANKor FINANCE is large. A large JOB indicates that there are a lot of joboffers relative to applicants, i.e., JOB represents mobility in labor market. Alarge BANK represents that a bank can lend a lot of money to residents, i.e.,BANK indicates liquidity in money market. A large FINANCE indicates thata prefecture is financially generous. As for WAGE, there are two aspects:(i) an increase in WAGE results in tight labor market and accordingly thebusiness cycle phase switches from expansion to recession due to an decreasein profits, and (ii) the expansion in the business cycle continues because of anincrease in WAGE. Therefore, WAGE has both positive and negative effectsto the business cycle indicator yit.

The estimation results in Table 2 are as follows. JOB and BANK affectthe business cycle index (i.e., yit) positively, because the posterior means ofJOB and BANK are estimated positively and the 95% credible intervals do notinclude zero. The mobility in labor market and the liquidity in money marketlead to expansion in the business cycle. Thus, from the results in Table 2, wecan conclude that the business cycle index (i.e., yit) positively depend on JOBand BANK. However, WAGE and FINANCE do not affect the business index,because the 95% credible intervals of WAGE and FINANCE include zero. Asmentioned above, WAGE includes both positive and negative effects to thebusiness cycle, and therefore we obtain the estimation results that WAGE ispositive but insignificant. As for FINANCE, we expect that there is a posi-tive effect to the business cycle, but we have the result that FINANCE doesnot affect the business cycle. For one reason, as mentioned above, the FI-NANCE data are constructed by the average of the revenue-expenditure ratiodata in the past three years for each prefecture, and accordingly they show thesmoothed movements, compared with the business cycle fluctuations. There-fore, FINANCE does not necessarily reflect the business cycle fluctuations.

Figure 2 describes the movements of ρt (i.e., regional interdependencyat time t), where the solid line represents the posterior means of ρt, t =1, 2, · · · , T , and the two dotted lines are given by 95% credible intervals. Ac-cording to Cabinet Office, Government of Japan, the trough-to-peak (i.e.,expansion) periods in Japan are given by October 1977 to February 1980,February 1983 to January 1985, November 1986 to February 1991, October1993 to May 1997, and January 1999 to November 2000, which correspondto the shaded areas in Figure 2. We can find that the spatial dependenciesin Japanese business cycle vary over time, which are positively estimated in1979-1980, 1984, 1988-1993, 1997-1998 and 2001, because the 95% credible in-

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1:00:50:0�0:5�1:01980 1985 1990 1995 2000

Figure 2: Posterior Mean (solid line) and 95% Credible Intervals (dotted lines)of ρt

tervals of ρt do not include zero for these periods. According to Figure 2, thesignificantly positive estimates of ρt in 1979-1980, 1984, 1988-1991 and 1997are observed in the expansion periods (i.e., the shaded areas), while those in1992-1993, 1998 and 2001 are in the contraction periods. It is known that inJapan the bubble burst in 1991. Except for 1997, ρt is significantly positive inthe expansion periods before the bubble burst, but in the contraction periodsafter the bubble burst. The relationship between the spatial autoregressiveparameter estimates (Figure 2) and the regional business cycles (Figures 3 –10) will be discussed later.

Figures 3 – 10 show the probability that the business cycle in each prefec-ture is in the expansion periods, i.e., the probability that zit > 0 occurs givenprefecture i and time t. There, the number of the random draws of zit greaterthan zero is divided by the number of all the generated random draws of zit.Note that zit is generated from (4). We divide Japan into eight regions, follow-ing Kakamu and Fukushige (2005), and discuss their features for each region.The shaded areas in Figures 3 – 10 represent the trough-to-peak periods inJapan, which are shaded in the same areas as Figure 2. We can observe fromFigures 3 – 10 that the business cycle in Japan is very similar to the regionalbusiness cycle in each prefecture, because a lot of the prefectures are in theshaded areas when the expansion probability is large.

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1:00:50:0 1980 1985 1990 1995 2000Hokkaidor r r r r r r r r r r r r

r r r r r r r r rr r r

r r Aomorib b b b b b b b b b b b b

b b b b b bb b b

b b bb b Iwate

� � � � � � � � � � � � �� � � � � �

� � ��

� �� � Miyagi

} } } } } } } } } } } } }} } } } } }

} } }} } }} } Akita

� � � � � � � � � � � � �� � � � � �

� � �� � �

� � Yamagata FukushimaFigure 3: Hokkaido and Tohoku Region1:00:50:0 1980 1985 1990 1995 2000Ibaragi

r r r r r r r r r r r r rr r r r r r

r r r r r rr r To higi

b b b b b b b b b b b b bb b b b b b

b b b b b bb b Gunma� � � � � � � � � � � � � � � � � � �

� � �� � �

� � Saitama} } } } } } } } } } } } } } } } } } } } } }

} } }} } Chiba

� � � � � � � � � � � � � � � � � � �� � �

� � �� � Tokyo KanagawaFigure 4: Kanto Region1:00:50:0 1980 1985 1990 1995 2000Nigata

r r r r r r r r r r r r rr r r r r r

r r rr r r

r r Toyamab b b b b b b b b b b b b

b b b b b b b b bb b b

b b Ishikawa� � � � � � � � � � � � �

� � � � � � � � �� � �

� � Fukui} } } } } } } } } } } } }

} } } } } }} } }

} } }} } Yamanashi

� � � � � � � � � � � � �� � � � � �

� � �� � �

� � NaganoFigure 5: Hokuriku and Koshinetsu Region1:00:50:0 1980 1985 1990 1995 2000Gifu

r r r r r r r r r r r r r r r r r r rr r r

r r rr r Shizuoka

b b b b b b b b b b b b b b b b b b bb b b b b b

b b Ai hi� � � � � � � � � � � � � � � � � � �

� � � � � �� � Mie

Figure 6: Tokai Region

15

1:00:50:0 1980 1985 1990 1995 2000Shigar r r r r r r r r r r r r

r r r r r rr r r r r r

r r Kyotob b b b b b b b b b b b b b b b b b b

b b b b b bb b Osaka� � � � � � � � � � � � �

� � � � � �� � �

� � �� � Hyogo

} } } } } } } } } } } } } } } } } } }} } } } } }

} } Nara� � � � � � � � � � � � � � � � � � �

� � � � � �� � Wakayama

Figure 7: Kinki Region1:00:50:0 1980 1985 1990 1995 2000Tottorir r r r r r r r r r r r r r r r r r r

r r rr r rr r Shimane

b b b b b b b b b b b b bb b b b b b b b b

b b bb b Okayama� � � � � � � � � � � � � � � � � � �

� � �� � �

� � Hiroshima} } } } } } } } } } } } } } } } } } }

} } } } } }} } Yamagu hi

Figure 8: Chugoku Region1:00:50:0 1980 1985 1990 1995 2000Tokushimar r r r r r r r r r r r r

r r r r r r r r r r r rr r Kagawab b b b b b b b b b b b b

b b b b b b b b b b b bb b Ehime

� � � � � � � � � � � � �� � � � � � � � �

� � �� � Ko hi

Figure 9: Shikoku Region1:00:50:0 1980 1985 1990 1995 2000Fukuokar r r r r r r r r r r r r r r r r r r

rr r r r r

r r Sagab b b b b b b b b b b b b b b b b b b

b b b b b bb b Nagasaki

� � � � � � � � � � � � �� � � � � � � � �

� � �� � Kumamoto

} } } } } } } } } } } } }} } } } } }

} } }} } }

} } Oita� � � � � � � � � � � � �

� � � � � � � � �� � �

� � Miyazaki Kagoshima4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 Okinawa

Figure 10: Kyushu and Okinawa Region

16

Hokkaido and Tohoku Region: Each line in Figure 3 shows the move-ments of the probability that the business cycle is in the expansion period.Hokkaido and Tohoku region is located in the most north of Japan. The busi-ness cycle in this region shows the following features. In 1979, 1984, 1988-1992,1998 and 2000-2001, the expansion probabilities take almost the same valuesfor all the prefectures in Hokkaido and Tohoku region. Note that the periodsfrom 1988 to 1992 (around the 3rd shaded area in the figure) correspond tothe bubble economy in Japan. On the other hand, the expansion probabilitiesare different for each prefecture in the other periods. Especially, the expansionprobabilities between 1992 and 1997 are quite different for each prefectures.

Kanto Region: In Figure 4, the movements of the expansion probabilitiesin Kanto region are displayed. Note that Tokyo belongs to this region. Fromthe figure, each prefecture in Kanto region takes almost the same expansionprobabilities in 1979, 1984, 1988-1989, 1992-1993, 1995 and 1998. However,we obtain the results that Kanto region is different from Hokkaido and Tohokuregion in 1990-1991 and 2000. The prefectures in Kanto region take differentvalues in 1990-1991 and 2000, but all the prefectures in Hokkaido and Tohokuregion (Figure 3) take similar values.

Hokuriku and Koshinetsu Region: Figure 5 displays the movements ofthe expansion probabilities in Hokuriku and Koshinetsu region. In 1979, 1983-1985, 1988-1993, 1998 and 2000-2001, we have similar expansion probabilitiesfor all the prefectures in Hokuriku and Koshinetsu region.

Tokai Region: In Figure 6, the movements of the expansion probabilitiesin Tokai region are displayed. In 1979-1980, 1983-1985, 1987-1989, 1992-1993,1995-1998, 2000 and 2003, all the expansion probabilities are similar for allthe prefectures.

Kinki Region: Figure 7 shows the expansion probabilities in Kinki region.In 1979-1980, 1984, 1988-1990, 1992, 1997-1998 and 2001, all the expansionprobabilities in Kinki region take similar values. The remarkable point is themovement of the expansion probabilities in Osaka. Osaka economy is thesecond largest in Japan, but the expansion probabilities after 1998 are lessthan 0.5. The recession in Osaka economy is serious after 1998.

17

Table 3: Business Cycle Pattern in Each Region

Region\Year 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03HT (Fig.3) ◦ – – – – ◦ – – – ◦ ◦ ◦ ◦ × – – – – – × – ◦ × – –Ka (Fig.4) ◦ – – – – ◦ – – – ◦ ◦ – – × × – ◦ – – × – – – – –HK (Fig.5) ◦ – – – ◦ ◦ ◦ – – ◦ ◦ ◦ ◦ × × – – – – × – ◦ × – –To (Fig.6) ◦ ◦ – – ◦ ◦ ◦ – ◦ ◦ ◦ – – × × – ◦ ◦ ◦ × – ◦ – – ◦Ki (Fig.7) ◦ ◦ – – – ◦ – – – ◦ ◦ ◦ – × – – – – ◦ × – – × – –Ch (Fig.8) ◦ – – – – ◦ – – – ◦ ◦ – – – × – – – – × – ◦ × – –Sh (Fig.9) ◦ – – – – ◦ – – – ◦ ◦ ◦ ◦ × – – ◦ – – × – – × – –KO (Fig.10)∗ – – – – – ◦ – – – – – ◦ ◦ – – – – – – × – – – – ◦

(i) KT, Ka, HK, To, Ki, Ch, Sh and KO indicate Hokkaido and Tohoku, Kanto, Hokurikuand Koshinetsu, Tokai, Kinki, Chugoku, Shikoku, and Kyushu and Okinawa regions,respectively.

(ii) See Figures 3 – 10 for the prefectures included in each region.(iii) ∗ represents the results in which Okinawa prefecture is excluded.

Chugoku Region: Figure 8 displays the expansion probabilities in Chugokuregion. In 1979, 1984, 1988-1989, 1993, 1998 and 2000-2001, all the expansionprobabilities are similar. However, during the rest of the periods, the expansionprobabilities are quite different for each prefecture.

Shikoku Region: Figure 9 indicates the expansion probabilities in Shikokuregion. In 1979, 1984, 1988-1992, 1995, 1998 and 2001, we can observe thatthe expansion probabilities are similar for all the prefectures in Shikoku region.

Kyushu and Okinawa Region: Figure 10 displays the expansion proba-bilities in Kyushu and Okinawa region. Except for Okinawa, the expansionprobabilities in 1984, 1990-1991, 1998 and 2003 are similar. For the rest of theperiods, the expansion probabilities take very different values for all the pre-fectures in Kyushu and Okinawa region. In addition, Okinawa is very differentfrom the other prefectures, because the expansion probabilities between 1991and 2001 are less than 0.5 and the cases where the expansion probability isgreater than 0.5 are only 9 out of 25 years. It might be plausible to concludethat the recession in Okinawa is very serious during the periods from 1979 to2003.

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Based on Figures 3 – 10, Table 3 summarizes the business cycle patternwithin each region. ◦ indicates that all the prefectures within each region takessimilar values in the expansion period, where all the expansion probabilities aregreater than 0.5. × represents that all the prefectures within each region takessimilar values in the recession period, where all the expansion probabilities areless than 0.5.

Compared with Figure 2 and Table 3, when most of the regions are inexpansion periods (in Table, 3 see the columns where the number of ◦ is atleast 5), the spatial autocorrelation estimates are significantly greater thanzero (see Figure 2). That is, in Table 3 we have at least five ◦’s in 1979, 1984and 1988 – 1990, and in Figure 2 the spatial autocorrelation estimates are sig-nificantly positive in the periods, judging from the 95% credible interval band.Contrarily, when most of the regions are in contraction periods (in Table 3,see the columns where the number of × is at least 5), the spatial autocorrela-tion estimates are significantly positive (see Figure 2). That is, in Table 3 wehave at least five ×’s in 1992, 1998 and 2001, and in Figure 2 the spatial au-tocorrelation estimates are significantly positive in the periods. Thus, beforethe bubble burst the interdependency among 47 prefectures increases in theexpansion periods, but after the bubble burst the interdependency increases inthe contraction periods. Therefore, we can consider that a structural changeoccurred when the bubble burst in 1991. Kakamu and Fukushige (2005) foundthat (i) in the 1980’s the interregional income inequality increases and simi-larly the individual income inequality also increases and (ii) in the 1990’s theinterregional income inequality decreases although the individual income in-equality increases. In this paper, we can observe the structural change on theinterdependency when the bubble burst in 1991. This result is consistent withKakamu and Fukushige (2005).

6 Concluding Remarks

This paper has examined the panel spatial autoregressive probit model fromBayesian point of view. We have expressed the joint posterior distribution (3),and considered the MCMC methods to estimate the parameters of the modelin Section 3.2. We have illustrated our approach using both simulated dataand actual data. In Section 4, we find from the simulation results that (i)when there exists the spatial dependency, the biases of the estimated posteriormeans become serious if we ignore the spatial autocorrelation, and (ii) all the

19

posterior means go to the true values as T is large. Thus, the simulationresults imply that the spatial interaction plays an important role in regionaleconomic models.

Moreover, in Section 5 we have considered the regional business cycle inJapan. We have the following findings: (i) we confirm that the occurrenceand collapse of bubble economy is a common phenomena in Japan, i.e., theprobability that the growth rate of IIP is positive falls drastically from 1992-1993 and rises after 1993 (note that the bubble economy burst in 1991), (ii)the business cycle in Japan is very similar to the regional business cycle ineach prefecture, and (iii) the posterior distributions of the spatial interactionρt do not include zero in the 95% credible intervals when the business cyclephase (i.e., expansion or contraction) in most of the prefectures is similar toeach other (see 1979, 1984, 1988-1990, 1992, 1998 and 2001 in Table 3), i.e.,the spatial interaction effect is significantly positive in the expansion periodsbefore the bubble burst and in the contraction periods after the bubble burst.Thus, we have observed the structural change on the regional interdependencywhen the bubble burst in 1991. We conclude from Japanese regional data thatthe spatial interaction plays an important role in the business cycle.

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