Spatial dependence in regional business cycles: evidence from Mexican states

Abstract

This study investigates how regional business cycles are spatially dependent in Mexico by developing a Markov switching model with a spatial autoregressive process. The Markov switching model with two regimes distinguishes business cycles between expansion and recession phases (i.e., high- and low-growth rate regimes). The objective of this study is twofold. First, this study aims to identify which states transitioned from expansion to recession during the Great Recession in 2008–2009. Second, it numerically examines the extent to which states that experienced this transition caused a deterioration in neighboring states’ economies. Employing Bayesian inference for the Markov switching model with quarterly data of state economic activity during the period 2003:Q1–2015:Q4, this study finds that Mexican states with higher manufacturing sector shares tended to be in recession during the Great Recession. Although some states experienced economic downturns in this period, they were not in a recessionary regime. This study also finds that business cycles across states were spatially dependent during the Great Recession. The numerical simulations of spatial spillover effects suggest that states that regime-switched from expansion to recession during the Great Recession caused a reduction in the quarterly growth rate of their neighboring economies by an average of 0.39 percentage points.

Appendices

Appendix A Drawing ${\varvec{\rho}}$ by the metropolis–hastings algorithm

We use a truncated normal distribution as a proposal distribution. When the random variable $x$ has the truncated normal distribution ${\mathrm{TN}}_{(a,b)}(\mu ,{\sigma }^{2})$, the probability density function (p.d.f.) is as follows:

$$q(x)=\left\{\begin{array}{ll}\frac{(1/\sigma )\phi ((x-\mu )/\sigma )}{\Phi ((b-\mu )/\sigma )-\Phi ((a-\mu )/\sigma )},& \mathrm{if}\,a<x<b,\\ 0,& \mathrm{otherwise},\end{array}\right.$$

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where $\phi (\cdot )$ and $\Phi (\cdot )$ are the p.d.f. and the cumulative distribution function (c.d.f.) of the standard normal distribution, respectively.

To avoid high autocorrelation and poor mixing, generating $\rho$ from the posterior distributions is repeated $H$ times within the $g$ th iteration. Superscript $(h)$ refers to the sample from the posterior distributions obtained in the $h$ th iteration within the $g$ th iteration as ${\rho }^{(g-1,h)}$. Note that the index $h$ is reset in each $g$ th iteration.

We use the probability integral transformation method for sampling from the truncated normal distribution. We set ${\mu }^{(g-1,h)}={\rho }^{(g-1,h)}$, ${\sigma }^{2}=1$, $a=1/{\omega }_{\mathrm{min}}$, and $b=1$. Following Holloway et al. (2002), we introduce a tuning parameter $c$ into the variance term, so that the acceptance rate might fall within the interval [$0.3, 0.7$].²⁴

For convenience of explanation, we omit the superscript $(g-1)$ as ${\rho }^{(h)}={\rho }^{(g-1, h)}$. Note that ${\rho }^{(g-1)}={\rho }^{(g-\mathrm{1,0})}$ if $h=0$. When $u$ is distributed as a uniform distribution $\mathrm{U}(\mathrm{0,1})$, we can draw ${\rho }^{\prime}$ from ${\mathrm{TN}}_{(1/{\omega }_{\mathrm{min}},1)}({\rho }^{(h-1)},1)$ as follows:

$${\rho }^{{\prime}}={\rho }^{(h-1)}+c{\Phi }^{-1}\left(\Phi (1/{\omega }_{\mathrm{min}}-{\rho }^{(h-1)})+u\left[\Phi (1-{\rho }^{(h-1)})-\Phi (1/{\omega }_{\mathrm{min}}-{\rho }^{(h-1)})\right]\right).$$

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The acceptance probability $\alpha ({\rho }^{(h-1)},{\rho }^{{\prime}})$ is calculated by:

$$\alpha \left({\rho }^{\left(h-1\right)},{\rho }^{\mathrm{{\prime}}}\right)=\mathrm{min}\left[\frac{\pi ({\rho }^{\mathrm{{\prime}}}|{\varvec{Y}},{{\varvec{S}}}^{(h)},{\boldsymbol{\Omega }}^{(h)},{{\varvec{\mu}}}^{(h)},{\boldsymbol{\Phi }}^{(h)})\left(\Phi (1-{\rho }^{(h-1)})-\Phi (1/{\omega }_{\mathrm{min}}-{\rho }^{(h-1)})\right)}{\pi ({\rho }^{(h-1)}|{\varvec{Y}},{{\varvec{S}}}^{(h)},{\boldsymbol{\Omega }}^{(h)},{{\varvec{\mu}}}^{(h)},{\boldsymbol{\Phi }}^{(h)})\left(\Phi (1-{\rho }^{\mathrm{{\prime}}})-\Phi (1/{\omega }_{\mathrm{min}}-{\rho }^{\mathrm{{\prime}}})\right)}, 1\right].$$

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where $\pi (\rho |{\varvec{Y}},{{\varvec{S}}}^{(h)},{\boldsymbol{\Omega }}^{(h)},{{\varvec{\mu}}}^{(h)},{\boldsymbol{\Phi }}^{(h)})$ is calculated from Eq. (10). Because a standard normal distribution is symmetric, $\phi ({\rho }^{{\prime}},{\rho }^{(h-1)})$ and $\phi ({\rho }^{(h-1)},{\rho }^{{\prime}})$ are offset. We repeat this step $H=10$ times in each $g$ th iteration to avoid high autocorrelation and poor mixing. Following step 8(c) in the algorithm, we judge whether ${\rho }^{{\prime}}$ is accepted or not after the $H$ iteration as follows:

$${\rho }^{(g)}=\left\{\begin{array}{ll}{\rho }^{{\prime}},& \mathrm{if}\,u<\alpha ({\rho }^{(g-1,H)},{\rho }^{{\prime}}),\\ {\rho }^{(g-1)},& \mathrm{otherwise}.\end{array}\right.$$

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Appendix B Multi-move Gibbs sampling for ${{\varvec{s}}}_{{\varvec{n}}}$

Kim and Nelson (1998, 1999a, b) were the first to apply multi-move Gibbs sampling to a Markov switching model. Our explanation here is based on Kim and Nelson (1999b). For convenience of explanation, we define vectors ${\tilde{{\varvec{s}}}}_{n}^{t}$ and ${{\varvec{s}}}_{n}^{t}$, and a matrix ${\tilde{{\varvec{Y}}}}^{t}$ using the following notation:

$${\tilde{{\varvec{s}}}}_{n}^{t}=\left(\begin{array}{c}{s}_{1,n}\\ {s}_{2,n}\\ \vdots \\ {s}_{t,n}\end{array}\right),\,{{\varvec{s}}}_{n}^{t}=\left(\begin{array}{c}{s}_{t,n}\\ {s}_{t+1,n}\\ \vdots \\ {s}_{T,n}\end{array}\right),\,{\tilde{{\varvec{Y}}}}^{t}=\left(\begin{array}{cccc}{y}_{1,n}& {y}_{\mathrm{1,2}}& \cdots & {y}_{1,N}\\ {y}_{2,n}& {y}_{\mathrm{2,2}}& \cdots & {y}_{2,N}\\ \vdots & \vdots & \ddots & \vdots \\ {y}_{t,n}& {y}_{t,2}& \cdots & {y}_{t,N}\end{array}\right).$$

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The aim here is to obtain $p({\tilde{{\varvec{s}}}}_{n}^{T}|{\tilde{{\varvec{Y}}}}^{T},{\varvec{\theta}})$. This can be rewritten as follows:

$$\begin{aligned} p(\tilde{\varvec{s}}_{n}^{T} |\tilde{\varvec{Y}}^{T} ,{\varvec{\theta}}) & = p(s_{T,n} |\tilde{\varvec{Y}}^{T} ,{\varvec{\theta}})p(\tilde{\varvec{s}}_{n}^{T - 1} |s_{T,n} ,\tilde{\varvec{Y}}^{T} ,{\varvec{\theta}}) \\ & = p(s_{T,n} |\tilde{\varvec{Y}}^{T} ,{\varvec{\theta}})\mathop \prod \limits_{t = 1}^{T - 1} p(s_{t,n} |{\varvec{s}}_{n}^{t + 1} ,\tilde{\varvec{Y}}^{T} ,{\varvec{\theta}}). \\ \end{aligned}$$

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Furthermore, the second term can be expressed as follows:

$$p({s}_{t,n}|{{\varvec{s}}}_{n}^{t+1},{\tilde{{\varvec{Y}}}}^{T},{\varvec{\theta}})\propto p({s}_{t+1,n}|{s}_{t,n},{\varvec{\theta}})p({s}_{t,n}|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}}),$$

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where the first term on the RHS represents the transition probability. Incorporating the normalizing constant, we have the following probability mass function:

$$p({s}_{t,n}=i|{{\varvec{s}}}_{n}^{t+1},{\tilde{{\varvec{Y}}}}^{T},{\varvec{\theta}})=\frac{p({s}_{t+1,n}|{s}_{t,n}=i,{\varvec{\theta}})p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})}{{\sum }_{j=0}^{1}p({s}_{t+1,n}|{s}_{t,n}=j,{\varvec{\theta}})p({s}_{t,n}=j|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})},$$

(30)

where $p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})$ is calculated using the Hamilton filter (see "Appendix C" for details). The calculation step for Eq. (28) can be summarized as follows: First, we draw ${s}_{T,n}$ conditional on ${\tilde{{\varvec{Y}}}}^{T}$ and ${\varvec{\theta}}$; second, given ${s}_{T,n}$, the sampling ${s}_{t,n}$ for $t=T-1,\dots ,1$ is implemented by backward recursion based on Eq. (30).

Appendix C Hamilton filter with spatial lag

Hamilton’s (1989) filter is applied to calculate the conditional probabilities $p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})$ for region $n$ at date $t$. Based on Chib (1996, 2001), we explain how the Hamilton filter is applied in this study. Using scalar notation, model (2) can be rewritten as follows:

$${y}_{t,n}=\rho \sum_{m=1}^{N}{w}_{nm}{y}_{t,m}+{\phi }_{n}{y}_{t-1,n}+{\mu }_{n,0}(1-{s}_{t,n})+{\mu }_{n,1}{s}_{t,n}+{\varepsilon }_{t,n},\,{\varepsilon }_{t,n}\sim \mathrm{i}.\mathrm{i}.\mathrm{d}.\,\mathrm{N}(0,{\sigma }_{n}^{2}).$$

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For the conditional p.d.f. $f({y}_{t,n}|{s}_{t,n},{{\varvec{y}}}_{t,-n},{\varvec{\theta}})$, the expected value and variance become $\mathrm{E}({y}_{t,n}|{s}_{t,n},{{\varvec{y}}}_{t,-n},{\varvec{\theta}}) =\rho {\sum }_{m=1}^{N}{w}_{nm}{y}_{t,m}+{\phi }_{n}{y}_{t-1,n}+{\mu }_{n,0}(1-{s}_{t,n})+{\mu }_{n,1}{s}_{t,n}$ and $\mathrm{Var}({y}_{t,n}|{s}_{t,n},{{\varvec{y}}}_{t,-n},{\varvec{\theta}})={\sigma }_{n}^{2}$, where the subscript $-n$ of ${{\varvec{y}}}_{t,-n}$ indicates that the $n$ th element is excluded from the vector, and for simplicity we assumed that for each region $n$, the spatial lag $\sum_{m=1}^{N}{w}_{nm}{y}_{t,m}$ is exogenously given. Therefore, the conditional p.d.f., which is used in the iteration process of the Hamilton filter, is given by the following:

$$f({y}_{t,n}|{s}_{t,n},{{\varvec{y}}}_{t,-n},{\varvec{\theta}})=\frac{1}{\sqrt{2\pi {\sigma }_{n}^{2}}}\mathrm{exp }\left(-\frac{({y}_{t,n}-\rho {\sum }_{m=1}^{N}{w}_{nm}{y}_{t,m}-{\phi }_{n}{y}_{t-1,n}-{\mu }_{n,0}(1-{s}_{t,n})-{\mu }_{n,1}{s}_{t,n}{)}^{2}}{2{\sigma }_{n}^{2}}\right).$$

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The algorithm of the Hamilton filter consists of two steps: prediction and update. The conditional p.d.f. $p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})$ is obtained by forward recursion $t=1, 2, \dots , T$.

• 1. Prediction Step: Calculate the probability

  $$p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t-1},{\varvec{\theta}})=\sum_{j=0}^{1}p({s}_{t,n}=i|{s}_{t-1,n}=j,{\varvec{\theta}})p({s}_{t-1,n}=j|{\tilde{{\varvec{Y}}}}^{t-1},{\varvec{\theta}}),$$

  (33)

  where, when $t=1$, $p({s}_{0,n}=i|{\tilde{{\varvec{Y}}}}^{0},{\varvec{\theta}})$ is replaced by the steady-state probabilities as follows:

  $${\pi }_{n,0}=\frac{1-{p}_{n,11}}{2-{p}_{n,00}-{p}_{n,11}}\,\mathrm{and}\,{\pi }_{n,1}=\frac{1-{p}_{n,00}}{2-{p}_{n,00}-{p}_{n,11}}.$$

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• 2. Update Step: Calculate the probability

  $$p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})=\frac{f({y}_{t,n}|{s}_{t,n}=i,{{\varvec{y}}}_{t,-n},{\varvec{\theta}})p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t-1},{\varvec{\theta}})}{{\sum }_{j=0}^{1}f({y}_{t,n}|{s}_{t,n}=j,{{\varvec{y}}}_{t,-n},{\varvec{\theta}})p({s}_{t,n}=j|{\tilde{{\varvec{Y}}}}^{t-1},{\varvec{\theta}})}.$$

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The probabilities $p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t},{\varvec{\theta}})$ are used in the multi-move Gibbs sampling. The probabilities $p({s}_{t,n}=i|{\tilde{{\varvec{Y}}}}^{t-1},{\varvec{\theta}})$ are also used for calculating the likelihood function in the model selection.

Appendix D Robustness check by spatial econometrics

4.1 D. 1. Spatial autocorrelation

To investigate the time-varying spatial dependence in regional business cycles, this study calculates Moran’s $I$ statistics across states at time $t$ as follows:

$${I}_{t}=\frac{{{\varvec{y}}}_{t}^{\mathrm{\top }}{\varvec{W}}{{\varvec{y}}}_{t}}{{{\varvec{y}}}_{t}^{\mathrm{\top }}{{\varvec{y}}}_{t}},$$

(36)

where the SWM is based on Eq. (19), the economic and route distance between states with the distance decay parameter $\eta =4$.

Figure 6 shows the calculation results of Moran’s $I$ in the study period. Panel (a) shows the results in each quarterly period. The red marker indicates statistical significance at the 10% level. Importantly, spatial autocorrelation is not always significant throughout the entire period. However, spatial autocorrelation occurred during the Great Recession of 2008–2009. To mitigate the fluctuations, the centered moving average of order 3 is calculated in panel (b). The degree of spatial autocorrelation increased gradually during the Great Recession of 2008–2009 and fell after the Great Recession.

Fig. 6

Time-series of Moran's $I$. The variable used for Moran's $I$ is the quarterly growth rate of the Quarterly Indicator of State Economic Activity (Indicador Trimestral de la Actividad Económica Estatal, ITAEE). The red marker indicates statistical significance at the 10% level. The spatial weight matrix is based on the gross regional products and the route distance across states with the distance decay parameter $\eta = 4$. Centered moving average of order 3 is calculated in panel (b)

Figure 7 shows Moran’s scatterplot using annually aggregated data for 2008 and 2009. In other words, the quarterly data are pooled on a yearly basis. As discussed above, the positive spatial autocorrelation is confirmed visually during the Great Recession. Although spatial autocorrelation across regional business cycles is not obvious from the data, it is confirmed as being significant during the Great Recession.

Fig. 7

Moran scatter plot. The variable is the quarterly growth rate of ITAEE. The spatial weight matrix is based on the gross regional products and the route distance across states with the distance decay parameter $\eta = 4$. The quarterly data are pooled on a yearly basis

4.2 D. 2. Spatial panel econometrics

The spatial autoregressive process of dependent variable ${\varvec{W}}{{\varvec{y}}}_{t}$ considers contemporaneous interdependence across regions. One may consider another possibility of a spatial autoregressive process, that is, ${\varvec{W}}{{\varvec{y}}}_{t-1}$ instead of ${\varvec{W}}{{\varvec{y}}}_{t}$. Consider a simpler version of model (2) without a temporal autoregressive process as follows:

$${{\varvec{y}}}_{t}=\rho {\varvec{W}}{{\varvec{y}}}_{t-1}+{\varvec{\mu}}+{{\varvec{\varepsilon}}}_{t},$$

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where ${\varvec{\mu}}=({\mu }_{1},{\mu }_{2},\dots ,{\mu }_{N})$ is the fixed effect of state $n$. By successive iteration, we can show that ${{\varvec{y}}}_{t}=({\varvec{I}}-\rho {\varvec{W}}{)}^{-1}{\varvec{\mu}}+({\varvec{I}}-\rho {\varvec{W}}{)}^{-1}{{\varvec{\varepsilon}}}_{t}$, which is equivalent to ${{\varvec{y}}}_{t}=\rho {\varvec{W}}{{\varvec{y}}}_{t}+{\varvec{\mu}}+{{\varvec{\varepsilon}}}_{t}$. Therefore, note that simultaneous spatial autoregressive processes result from dynamic spatial dependence. See LeSage and Pace (2009) for a discussion on time dependence in spatial econometric models.

To control for common external shocks across the Mexican states, this study estimated a spatial panel econometric model with fixed effects (Lee and Yu, 2010):

$${{\varvec{y}}}_{t}=\rho {\varvec{W}}{{\varvec{y}}}_{t}+{\varvec{\mu}}+{\psi }_{t}+{{\varvec{\varepsilon}}}_{t},$$

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where ${\varvec{\mu}}=({\mu }_{1},{\mu }_{2},\dots ,{\mu }_{N}{)}^{\top }$ is the fixed effect of state $n$, ${\psi }_{t}$ is the fixed effect of time $t$, and SWM is based on Eq. (19), the economic and route distance across states with the distance decay parameter $\eta =4$. Time fixed effects aim to control for common external shocks across the county. Year and quarter and year $\times$ quarter fixed effects are included. The parameter of interest is $\rho$, which measures the spatial dependence in economic activities.

Table 7 shows the estimation results obtained by maximum likelihood estimation. In column (1), in which time fixed effects are not controlled for, the estimate of $\rho$ is 0.230 and significantly positive throughout the entire period. When the year and quarter fixed effects are controlled for in column (2), the magnitude of spatial dependence becomes 0.199 but remains statistically significant at the 1% level. When the year $\times$ quarter fixed effects are controlled for in column (3), the magnitude of spatial dependence becomes 0.028 and is statistically insignificant.

Table 7 Maximum likelihood estimation results of spatial panel econometric models

To estimate spatial dependence in regional business cycles under control for common external shocks, the entire period is divided into three subperiods. In columns (4) and (6), the coefficient estimates of the spatial lag are insignificant and close to zero in the pre and post periods of the Great Recession. In column (5), the parameter estimate of spatial dependence is 0.236 and significantly positive at the 10% level only during the Great Recession of 2008:Q2–2009:Q2, suggesting that significant spatial dependence in a subperiod results in statistical significance in the entire period. Note that the split of the study period is exogenously determined within this regression, although the Markov switching model endogenously estimates expansion and recession phases by state.

Summing up, after controlling for common external shocks across the Mexican states, we confirmed significant spatial dependence in regional business cycles only during the Great Recession. Although time-invariant spatial dependence in regional business cycles was assumed in the model, time-varying spatial dependence in regional business cycles will be more precise. Therefore, the quantitative magnitude of spatial spillover effects on neighboring economies might have a wider range than that estimated, whereas the qualitative discussion about the spatial spillover effects does not change.

Appendix E Model selection

We use the log marginal likelihood to compare different econometric models. Chib (1995) proposed a procedure for calculating marginal likelihood under Gibbs sampling. However, in this study, a parameter measuring spatial dependence $\rho$ is drawn by the MH algorithm, and thus we employ a method proposed by Chib and Jeliazkov (2001).

The calculation of the marginal likelihood is based on the following equation:

$$m({\varvec{Y}})=\frac{L({\varvec{Y}}|{\varvec{\theta}})\pi ({\varvec{\theta}})}{\pi ({\varvec{\theta}}|{\varvec{Y}})},$$

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which is termed the basic marginal likelihood identity (BMI). The BMI consists of the likelihood function, prior distribution, and posterior distribution. This identity holds at any ${\varvec{\theta}}$. In this study, the mean of the posterior distribution ${{\varvec{\theta}}}^{*}$ is used. Thus, by taking the logarithms of the BMI and evaluating them at ${{\varvec{\theta}}}^{*}$, we can calculate the log marginal likelihood estimate as follows:

$$\mathrm{log }\, \widehat{m}({\varvec{Y}})=\mathrm{log }\, L({\varvec{Y}}|{{\varvec{\theta}}}^{*})+\mathrm{log }\, \pi ({{\varvec{\theta}}}^{*})-\mathrm{log }\, \widehat{\pi }({{\varvec{\theta}}}^{*}|{\varvec{Y}}).$$

(40)

Based on Eq. (40), we calculate the following three terms: the likelihood function, the prior distribution, and the posterior distribution, all evaluated at ${{\varvec{\theta}}}^{*}$.

The first term on the RHS of Eq. (40) is the log likelihood function. Note that the Markov switching model includes hidden variables $\{{{\varvec{s}}}_{t}{\}}_{t=1}^{T}$. The likelihood function thus takes the following form:

$$L({\varvec{Y}}|{{\varvec{\theta}}}^{*})=\prod_{t=1}^{T}\left[\sum_{j=0}^{1}f({{\varvec{y}}}_{t}|{{\varvec{s}}}_{t}=j,{{\varvec{\theta}}}^{*})p({{\varvec{s}}}_{t}=j|{\tilde{{\varvec{Y}}}}^{t-1},{{\varvec{\theta}}}^{*})\right].$$

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The second term in the brackets must be calculated in advance. This term can be obtained from the prediction step in the Hamilton filter.

The second term on the RHS of Eq. (40) is the logarithm of the joint prior distribution. As we assumed an independent prior distribution across parameters and regions, the prior distribution can be obtained as follows:

$$\pi ({{\varvec{\theta}}}^{*})=\pi ({\rho }^{*})\left[\prod_{n=1}^{N}\pi ({\sigma }_{n}^{2*})\pi ({{\varvec{\mu}}}_{n}^{*})\pi ({\phi }_{n})\pi ({p}_{n,11}^{*})\pi ({p}_{n,00}^{*})\right].$$

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The third term on the RHS of Eq. (40) is the logarithm of the joint posterior distribution, which can be rewritten as follows:

$$\begin{aligned} \hat{\pi }({\varvec{\theta}}^{*} |{\varvec{Y}}) & = \hat{\pi }\left( {\rho^{*} {|}{\varvec{Y}}} \right) \left[ {\mathop \prod \limits_{n = 1}^{N}} \hat{\pi }(\sigma_{n}^{{2{*}}} |\rho^{*} ,{\varvec{Y}})\hat{\pi }({\varvec{\mu}}_{n}^{*} |\rho^{*} ,\sigma_{n}^{{2{*}}} ,{\varvec{Y}})\hat{\pi }(\phi_{n}^{*} |\rho^{*} ,\sigma_{n}^{{2{*}}} ,{\varvec{\mu}}_{n}^{*} ,{\varvec{Y}}) \right. \\ & \quad \times \left. \hat{\pi }(p_{n,11}^{*} |\rho^{*} ,\sigma_{n}^{{2{*}}} ,{\varvec{\mu}}_{n}^{*} ,\phi_{n}^{*} ,{\varvec{Y}})\hat{\pi }(p_{n,00}^{*} |\rho^{*} ,\sigma_{n}^{{2{*}}} ,{\varvec{\mu}}_{n}^{*} ,\phi_{n}^{*} ,p_{n,11}^{*} ,{\varvec{Y}}) \right], \end{aligned}$$

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where

$$\widehat{\pi }({\rho }^{*}|{\varvec{Y}})=\frac{{G}^{-1}{\sum }_{g=1}^{G}\alpha ({\rho }^{(g)},{\rho }^{*}|{\varvec{Y}},{{\varvec{S}}}^{(g)},{\boldsymbol{\Omega }}^{(g)},{{\varvec{\mu}}}^{(g)},{\boldsymbol{\Phi }}^{(g)},{{\varvec{p}}}_{11}^{(g)},{{\varvec{p}}}_{00}^{(g)})q({\rho }^{(g)},{\rho }^{*})}{{J}^{-1}{\sum }_{k=1}^{J}\alpha ({\rho }^{*},{\rho }^{(k)}|{\varvec{Y}},{{\varvec{S}}}^{(k)},{\boldsymbol{\Omega }}^{(k)},{{\varvec{\mu}}}^{(k)},{\boldsymbol{\Phi }}^{(k)},{{\varvec{p}}}_{11}^{(k)},{{\varvec{p}}}_{,00}^{(k)})},$$

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$$\widehat{\pi }({\sigma }_{n}^{2*}|{\rho }^{*},{\varvec{Y}})=\frac{1}{J}\sum_{k=1}^{J}\pi ({\sigma }_{n}^{2*}|{\rho }^{*},{{\varvec{\mu}}}_{n}^{(k)},{\phi }_{n}^{(k)},{p}_{n,11}^{(k)},{p}_{n,00}^{(k)},{{\varvec{s}}}_{n}^{(k)},{\varvec{Y}}),$$

(45)

$$\widehat{\pi }({{\varvec{\mu}}}_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{\varvec{Y}})=\frac{1}{J}\sum_{k=1}^{J}\pi ({{\varvec{\mu}}}_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{\phi }_{n}^{(k)},{p}_{n,11}^{(k)},{p}_{n,00}^{(k)},{{\varvec{s}}}_{n}^{(k)},{\varvec{Y}}),$$

(46)

$$\widehat{\pi }({\phi }_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\varvec{Y}})=\frac{1}{J}\sum_{k=1}^{J}\pi ({{\varvec{\mu}}}_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{p}_{n,11}^{(k)},{p}_{n,00}^{(k)},{{\varvec{s}}}_{n}^{(k)},{\varvec{Y}}),$$

(47)

$$\widehat{\pi }({p}_{n,11}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{\varvec{Y}})=\frac{1}{J}\sum_{k=1}^{J}\pi ({p}_{n,11}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{p}_{n,00}^{(k)},{{\varvec{s}}}_{n}^{(k)},{\varvec{Y}}),$$

(48)

$$\widehat{\pi }({p}_{n,00}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{p}_{n,11}^{*},{\varvec{Y}})=\frac{1}{J}\sum_{k=1}^{J}\pi ({p}_{n,00}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{p}_{n,11}^{*},{{\varvec{s}}}_{n}^{(k)},{\varvec{Y}}).$$

(49)

The superscript $(g)$ refers to the sample from the posterior distribution in the $g$ th iteration and $(k)$ refers to the sample from the reduced Gibbs runs obtained in the $k$ th iteration. Note that some of the parameters are given as a mean in the reduced Gibbs runs, and that ${\rho }^{(k)}$ is drawn from a proposal distribution $q({\rho }^{*},{\rho }^{(k)})$. Besides the $G$ iterations, we need to implement an additional $5\times J$ iterations for the reduced Gibbs runs. The first reduced run is for the denominator of $\widehat{\pi }({\rho }^{*}|{\varvec{Y}})$ and $\widehat{\pi }({\sigma }_{n}^{2*}|{\rho }^{*},{\varvec{Y}})$; the second is for $\widehat{\pi }\left({{\varvec{\mu}}}_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{\varvec{Y}}\right);$ the third is for $\widehat{\pi }\left({\phi }_{n}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\varvec{Y}}\right);$ the fourth is for $\widehat{\pi }({p}_{n,11}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{\varvec{Y}})$; and the fifth is for $\widehat{\pi }({p}_{n,00}^{*}|{\rho }^{*},{\sigma }_{n}^{2*},{{\varvec{\mu}}}_{n}^{*},{\phi }_{n}^{*},{p}_{n,11}^{*},{\varvec{Y}})$. We set $J$ to have the same number of iterations as $G$. Moreover, the numerical standard errors of the marginal likelihood estimates are also calculated. For calculation of the numerical standard errors, we need to select a lag at which the autocorrelation is small enough to be neglected. Thus, we set the lag length equal to 40. See Chib and Jeliazkov (2001) for more details.

Table 8 presents the log marginal likelihood estimates with numerical standard errors using the different econometric models. First, it is useful to compare estimates of the log marginal likelihood between the Markov switching model with a spatial autoregressive process (MS-SAR) and the Markov switching (MS) model because MS is a spatial case of MS-SAR when $\rho =0$. Consequently, it is supported to take into account spatial dependence in regional business cycles. Our estimation results also indicate that the Markov switching model with a first-order autoregressive process MS-AR(1) fits the data almost as well as MS-SAR. The Markov switching model with a spatial autoregressive process and a first-order autoregressive process MS-SAR-AR(1) is supported against MS-AR(1) or MS-SAR. See Supplementary Information for the full estimation results of MS, MS-AR(1), MS-SAR, and MS-SAR-AR(1).

Table 8 Log marginal likelihood estimate

Appendix F Map of Mexico

State codes and names appear in Fig. 8. In the empirical analysis, the 32 states are divided into two groups in terms of the manufacturing specialization index. Average specialization index of manufacturing sector between 2003 and 2005 is used. Colored states show the manufacturing specialization index above the median. Figure 9 shows details of the geographical distribution of manufacturing specialization index.

Fig. 8

Map of Mexico. Author's creation. The 32 states are divided into two groups. Colored states show the specialization index of manufacturing sector above the median

Fig. 9

Geographical distribution of manufacturing sector in Mexico. Author's creation. Average specialization index of manufacturing sector between 2003 and 2005 is used