Is Decoupling in Action?

Abstract

As explained in Chap. 2, the decoupling hypothesis essentially refers to changes in the degree of business cycle interdependence between the two groups of economies (EEs and AEs). It implies two main consequences that should be empirically observable: (1) a decreasing comovement of economic cycles between AEs and EEs over time, (2) an increasing resilience of the EEs to adverse scenarios in AEs. These two points were studied in this chapter by using two different tools: the Euclidean Distance Indicator and the Time-Varying Panel VAR Econometric Model.

Appendices

Appendix 1: Set of Countries

See Table 3.5.

Table 3.5 Countries

Appendix 2: Main Descriptive Statistics

See Table 3.6.

Table 3.6 Descriptive statistics of GDP growth rate (%) data for each country

Appendix 3: Euclidean Distance Computed on the Economic “Deviation Cycles”

See Figs. 3.9 and 3.10.

Fig. 3.9

Euclidean distance indicator computed on deviation cycles: EEs plus DEs vs. AEs and EEs vs. AEs. (1) EEs plus DEs vs. AEs, (2) EEs vs. AEs

Fig. 3.10

Euclidean distance indicator computed on deviation cycles: EEs regional groups vs. all AEs. (1) Latin America EEs vs. AEs, (2) MENA EEs vs. AEs, (3) Sub Sahara Africa EEs vs. AEs, (4) Asia EEs vs. AEs, (5) Europe EEs vs. AEs

Appendix 4: Priors, Posteriors Distributions and the Computational Method

The prior distributions proposed in this chapter are chosen according to previous experiences from related literature and because of their intuitiveness and convenience in the applications.

Prior densities are assumed for ${\xi}_0=\left(\;{\Omega}^{-1},B,\;{\theta}_0\right)$, and ${\sigma}^2$ is assumed to be known. The elements of ${\xi}_0$ are assumed to be independent. They are the parameters of the probability density functions of the innovations ${\upsilon}_t$ and ${\eta}_t$ and the initial state of the coefficients in system (3.6)–(3.7); hence, the assumption of independencies is very reasonable and, in fact, is a very common assumption in the related literature; therefore

$$ P\left({\theta}_0,\Omega, B\right)=P\left({\theta}_0\right)P\left(\Omega \right)P(B) $$

(3.8)

The matrix $\Omega$ with dimension $\left(NG\times NG\right)$ is the variance/covariance matrix of the residuals ${E}_t$ and so ${\Omega}^{-1}$ is the precision matrix. In Bayesian statistics, the Wishart distribution,³⁷ namely a probability distribution of symmetric positive-definite random matrix, is often used as the prior for the precision matrix, and also in this work it is assumed that the matrix ${\Omega}^{-1}$ has the W distribution with ${z}_1$ degrees of freedom and scale matrix ${Q}_1$, namely:

$$ {\Omega}^{-1}\sim W\left({z}_1,\;{Q}_1\right) $$

(3.9)

The matrix $B$, whose dimension is $R\times R$ where $R={\sum}_{f=1}^F{dim}_f$, is the variance/covariance matrix of the innovation ${\eta}_t$. To guarantee orthogonality of factors, the matrix $B$ must be block-diagonal, hence, $B= diag\left({B}_1,\dots, {B}_F\right)$. For $f=1,\dots, F$ each block ${B}_f$, whose dimension is ${dim}_f\times {dim}_f$, is assumed to be ${B}_f={b}_f{I}_f$, where ${I}_f$ is the identity matrix with dimension ${dim}_f$ and ${b}_f$ is a scalar³⁸ which is distributed like an inverse gamma with shape parameter $\left(\raisebox{1ex}{$\rho $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ and scale parameter $\left(\raisebox{1ex}{$\varrho $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$:

$$ {b}_f\sim IG\left(\raisebox{1ex}{$\rho $}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\;\raisebox{1ex}{$\varrho $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) $$

(3.10)

The law of motion for the factors (Eq. 3.5) implies that for $t=1,\dots, T\;{\theta}_t\Big|{\theta}_{t-1},B\sim N\left({\theta}_{t-1},B\right)$, and so

$$ P\left({\theta}_1,\dots, {\theta}_T\Big|{\theta}_0,B\right)=\prod_{t=1}^TP\left({\theta}_t\Big|{\theta}_{t-1},B\right)\propto {\left|B\right|}^{-\frac{T}{2}} exp\left\{-\frac{1}{2}{\sum}_{t=1}^T{\left({\theta}_t-{\theta}_{t-1}\right)}^{\prime }{B}^{-1}\left({\theta}_t-{\theta}_{t-1}\right)\right\} $$

(3.11)

where the symbol “$\propto$” stands for “proportional”. The prior for ${\theta}_0$ is assumed to be normal

$$ {\theta}_0\sim N\left({\overline{\theta}}_0,{\overline{R}}_0\right) $$

(3.12)

The values of the vector of hyperparameters $\mu =\left({z}_1,{Q}_1,\alpha, b,{\overline{\theta}}_0,{\overline{R}}_{0,}{\sigma}^2\right)$ are selected either to produce rather loose priors (this is the case for ${z}_1,{Q}_1,\alpha, b,{\overline{R}}_0$), or chosen observing the data (the case for ${\overline{\theta}}_0$), or chosen to maximize the model in-sample fit of data (the case for ${\sigma}^2$).

The hyperparameter ${z}_1$ is set equal to $NG+50$, namely 162 (i.e., dimension of the squared matrix $\Omega$ plus 50) because, for the Wishart distribution to be proper, the degrees of freedom must be³⁹ at least equal to the dimension of the matrix $\Omega$. In some related literature [e.g., Cogley and Sargent (2003), Cogley (2003), Primiceri (2005), or Canova et al. (2007)], the scale matrix ${Q}_1$ is chosen to be the inverse of variance/covariance matrix of the corresponding Ordinary Least Square estimates on a training sample. In this work, owing to available data, there is not a training sample and the scale matrix has been set equal to the identity matrix. This prior assumption means that the prior expected variance covariance matrix of residuals is a diagonal matrix, namely uncorrelated residuals between equations, and all equal elements on the diagonal; however, also in this case the prior probability density function on Ω allows for posterior non diagonal matrix, namely allows for the case of posterior correlated residuals between equations. The hyperparameters $\rho$ and $\varrho$ are set at 2 and 10, respectively. Following the related literature [e.g., Canova and Ciccarelli (2009)] ${\overline{\theta}}_0$ is the Ordinary Least Squares estimate in the time-invariant version of the model, and the matrix ${\overline{R}}_0$ is assumed to be ${\overline{R}}_0={I}_R$ where ${I}_R$ is identity matrix whose dimension is $R\times R$ (remind that $R={\sum}_{f=1}^F{dim}_f$). Since the in-sample fit improves if ${\sigma}^2$ goes to zero, an exact factorization of ${\delta}_t$ is used.

To compute the posterior distribution for $\xi =\left({\Omega}^{-1,},{b}_1,\dots, {b}_F,{\left\{{\theta}_t\right\}}_{t=1}^T\right)$ the prior is combined with the likelihood $\mathcal{L}\left(\xi \Big|{Y}_1,\dots, {Y}_T\right)$ which (conditional to the first $p$ observations before ${Y}_1$) is proportional to:

$$ \mathcal{L}\left(\xi \Big|{Y}_1,\dots, {Y}_T\right)\propto {\left|\Omega \right|}^{-\frac{T}{2}} exp\left\{-\frac{1}{2}{\sum}_{t=1}^T{\left({Y}_t-{W}_t\Xi {\theta}_t\right)}^{\prime }{\Omega}^{-1}\left({Y}_t-{W}_t\Xi {\theta}_t\right)\right\} $$

(3.13)

As explained in Greenberg (2008), using the Bayes rule,⁴⁰ the likelihood of data (3.13) and the prior probability density functions (3.8)–(3.12) the posterior distribution for $\xi$ is proportional to:

$$ \begin{array}{c}P\left({\Omega}^{-1,},{b}_1,\dots, {b}_F,{\left\{{\theta}_t\right\}}_{t=1}^T\Big|{Y}_1,\dots, {Y}_T\right)\propto {\left|\Omega \right|}^{-\frac{T}{2}} exp\left\{-\frac{1}{2}{\sum}_{t=1}^T{\left({Y}_t-{W}_t\Xi {\theta}_t\right)}^{\prime }{\Omega}^{-1}\left({Y}_t-{W}_t\Xi {\theta}_t\right)\right\}\\ {}\times {\left|\Omega \right|}^{-\frac{\left({z}_1-NG-1\right)}{2}} exp\left\{-\frac{1}{2}tr\left({Q}_1{\Omega}^{-1}\right)\right\}\\ {}\begin{array}{c}\times {\prod}_{i=1}^F{b}_i^{-\left(\frac{\rho }{2}+1\right)} exp\left\{-\frac{\varrho }{2{b}_i}\right\}\\ {}\times {\left|B\right|}^{-\frac{T}{2}} exp\left\{-\frac{1}{2}{\sum}_{t=1}^T{\left({\theta}_t-{\theta}_{t-1}\right)}^{\prime }{B}^{-1}\left({\theta}_t-{\theta}_{t-1}\right)\right\}\end{array}\end{array} $$

(3.14)

and to compute the conditional posterior distribution of each parameter we can consider only the terms in the joint posterior (3.14) that contain the parameter of interest. Let be ${\xi}_{-\mathcal{K}}$ the vector $\xi$ excluding the parameter $\mathcal{K}$. After few algebraic passages (see Koop and Korobilis 2010) it is obtained:

$$ {\Omega}^{-1}\Big|{Y}_1,\dots, {Y}_T,\;{\xi}_{-\Omega}\sim W\left({z}_1+T,\;{\left({Q}_1^{-1}+{\sum}_t\left({Y}_t-{W}_t\Xi {\theta}_t\right){\left({Y}_t-{W}_t\Xi {\theta}_t\right)}^{\prime}\right)}^{-1}\;\right) $$

(3.15)

$$ {b}_f\Big|{Y}_1,\dots, {Y}_T,\;{\xi}_{-{b}_f}\sim IG\left(\frac{\rho +T*{dim}_f}{2},\;\frac{\varrho +\sum_t{\left({\theta}_{ft}-{\theta}_{ft-1}\right)}^{\prime}\left({\theta}_{ft}-{\theta}_{ft-1}\right)}{2}\right),\;f=1,\dots, F $$

(3.16)

The Eqs. (3.15) and (3.16) have been used in the Gibbs sampling algorithm⁴¹ to draw samples of the parameters ${\Omega}^{-1}$, ${b}_f$ for $f=1,\dots, F$; but to complete the algorithm it is needed a means of drawing from $P\left({\theta}_t\Big|{Y}_1,\dots, {Y}_T,{\xi}_{-{\theta}_t}\right)$. Canova et al. (2007) (see also Canova 2007, p. 378) show that

$$ {\theta}_t\Big|{Y}_1,\dots, {Y}_T,{\xi}_{-{\theta}_t}\sim N\left({\overline{\theta}}_{t\Big|T},{\overline{V}}_{t\Big|T}\right),\;t\le T $$

(3.17)

where ${\overline{\theta}}_{t\Big|T}$ and ${\overline{V}}_{t\Big|T}$ are the one-period-ahead forecasts of ${\theta}_t$ and the variance/covariance matrix of the forecast error, respectively, calculated with the Kalman smoother as described in Chib and Greenberg (1995).

The posterior density functions (3.15)–(3.17) have been used for sampling in the Gibbs sampling algorithm, as follows:

• 1. Initiate the algorithm by choosing ${b}_f^0=0.01$ for $f=1,\dots, F$

• 2. At the first iteration, draw

  ${\theta}^{(1)}=\left({\theta}_1^1,{\theta}_2^1,\dots, {\theta}_T^1\right)$ with the Kalman smoother as in Chib and Greenberg (1995) knowing ${b}_f^0$ for $f=1,\dots, F$

  ${\Omega}^{(1)}$ from the conditional posterior distribution $P\left(\Omega \Big|{Y}_1,\dots, {Y}_T,\;{\theta}^{(1)}\right)$

  ${b_f}^{(1)}$ from the conditional posterior distribution $P\left({b}_f\Big|{Y}_1,\dots, {Y}_T,\;{\theta}^{(1)}\right)$, $f=1,\dots, F$

• 3. At the gth iteration, draw

  ${\theta}^{(g)}=\left({\theta}_1^g,{\theta}_2^g,\dots, {\theta}_T^g\right)$ with the Kalman smoother as in Chib and Greenberg (1995) knowing ${b}_f^{g-1}$ for $f=1,\dots, F$

  ${\Omega}^{(g)}$ from the conditional posterior distribution $P\left(\Omega \Big|{Y}_1,\dots, {Y}_T,\;{\theta}^{(g)}\right)$

  ${b}_f^g$ from the conditional posterior distribution $P\left({b}_f\Big|{Y}_1,\dots, {Y}_T,\;{\theta}^{(g)}\right)$, $f=1,\dots, F$

until the desired number of iterations is obtained. Once the posterior distribution of ${\theta}_t$ is available it can be constructed the posterior distribution of the indicators for example, the posterior mean of the indicator ${GI}_t$ can be approximated by $\left(1/\mathcal{G}\right){\sum}_{g=}^{\mathcal{G}}{\mathcal{W}}_{1t}{\theta}_{1t}^g$, where $\mathcal{G}$ is the number of draws, and a credible 68 % interval can be obtained by ordering the draws of ${\mathcal{W}}_{1t}{\theta}_{1t}^g$ and taking the 16th and the 84th percentiles of the distribution.

The convergence of the sampler to the posterior distribution has been checked by increasing the length of the chain. The results presented in this chapter are based on 100,000 runs of 200 elements drawn 500 times, and the last observations of the final 450 times is used for inference.

Appendix 5: Historical Decomposition of GDP Fluctuations

The Fig. 3.11 plots the historical fluctuations of GDP, the global component, the regional component and the idiosyncratic component (namely the country specific component and the residual term).

Fig. 3.11

Historical decomposition of GDP fluctuations

Appendix 6: Additional Results

This appendix proposes additional results obtained with CAs which, differently from those presented in Sect. 3.2.3.5.2, have been performed on the model estimated on the entire data sample (1970–2010). Using the full set of data, instead of on the data until the time of the simulated shock, allows to consider a richer set of information in the estimation procedure.

The main advantage of using the model estimated on the full set of information is that the CAs can be performed also for the first years of the sample period because also the parameters estimated at the beginning of the sample period are based on a rich set of information; and so I believe that also this exercise, which is in some way in the same logic of the ex-post forecast econometric exercises often presented in the empirical literature, could confirm and make robust the results discussed in the Sect. 3.2.3.5 of Chap. 3.

Both the immediate impact and the cumulated impact have been computed on the sample period from 1981 to 2010.

Table 3.7 presents the median responses together with the lower and upper values of the credible intervals (16th and 84th percentiles, respectively) for different sub-samples.

Table 3.7 Dynamic impact of the adverse scenario in the AEs on the EEs; additional results

The resilience of EEs (see also Fig. 3.12) was higher during the last 15 years of the sample period than in the preceding 15, both when the evaluation is made in terms of the immediate impact and when it is made in terms of the cumulated impact. During the period 1996–2010, for example, the cumulated impact was about 15 % lower than in the period 1981–1995.

Fig. 3.12

Impact (median, 16th–84th percentiles) on EEs of shocks spreading from the AEs; additional results. (1) Immediate impact, (2) Cumulated impact

Let us break down the whole sample period in smaller sub-samples (5-years). If we observe the last 15 years of the sample period, we see that the resilience of the EEs fell rather than rose. The immediate impact (also shown in Fig. 3.13, panel 1) measured during the 5-year period 2006–2010 was up 12 % from that measured in the 5-year period 2001–2005, and in turn the latter was 40 % higher than that measured during the previous 5-year period 1996–2000. The same conclusions are reached if the results obtained for the cumulated impacts are considered (Fig. 3.13, panel 2). Nevertheless it can be observed that, despite the increase in the intensity of impacts during the last 15 years (1996–2010), the cumulated effect of the shock simulated in the 5-year period 2006–2010 was approximately 9 % lower than that simulated in the 5-year period 1981–1985. Therefore, the EEs’ resilience at the end of the 2000s is greater than it was during the early 1980s.

Fig. 3.13

Impact (median, 16th–84th percentiles) on EEs of shocks spreading from the AEs; additional results. (1) Immediate impact, (2) Cumulated impact

The above results broadly confirm the conclusion exposed in the Sect. 3.2.3.5.2 in this chapter.