Which Global Cycle? A Stochastic Factor Selection Approach for Global Macro-Financial Cycles

3.1.1 Centered Parameterization of the DFM

Observation Equation In order to model the global dynamics of macroeconomic and financial variables we adopt a two-level dynamic factor model. Intuitively this means that we want to include a global common factor that can influence all time series in the model, i.e. the global macro-financial factor (G) on the first level of the hierarchy, and we need sector-specific common factors on the second level, i.e. a global macroeconomic (S ^m ) and a global financial factor (S ^f ) that only load on their respective series. This gives us a factor structure with potentially three global factors.

We have observable variables y i t k , with k ∈ {m, f} denoting either macroeconomic (m) or financial time series (f) for country i = 1, …, N at time t = 1, …, T, assumed to be driven by the global factors and an idiosyncratic component:

with γ i k and ϕ i k as the factor loadings and μ i t k being the idiosyncratic component.

Stacking y i t k over all k and i we get a (N × 2) × 1 vector y t = y 1 t m , … , y N t m , y 1 t f , … , y N t f ′ that allows to re-write the model in matrix form as

with μ t = μ 1 t m , … , μ N t m , μ 1 t f , … , μ N t f ′ and F t = G t , S t m , S t f ′ being J × 1 with J = 3, equaling the number of factors. Λ = (Γ, Φ) is the (N × 2) × J loading matrix with Γ = γ 1 t m , … , γ N t m , γ 1 t f , … , γ N t f ′ , and the sector-factor loading matrix Φ featuring a block structure, i.e. ϕ i m is set to zero if k = f and unrestricted otherwise. Consequently we restrict ϕ i f = 0 for k = m and leave it unrestricted otherwise. To add clarity in regard to this two-level structure, the factor loading matrix for the model is

Processes for Factors and Idiosyncratic Components The dynamics in the unobserved components F _t and μ _t are assumed to follow zero-mean autoregressive processes,

where P(L) and Π(L) are lag-polynomials of order p and q respectively. The specification allows for no spillovers in the state equation between global factors, i.e. P l = diag ρ l G , ρ l m , ρ l f for l = 1, …, p. The same holds for the idiosyncratic components, i.e. we assume Π = diag π i l k and Ω _ν to be a diagonal covariance matrix. The baseline model adopts an AR(1) specification for both the factor and the idiosyncratic component processes. In contrast to much of the literature, Q _ɛ is not normalized to the identity matrix, but only restricted to be diagonal, i.e. diag σ ε G 2 , σ ε m 2 , σ ε f 2 leaving each standard deviation of the factor innovation unconstrained. More on this follows in the identification section below.

3.1.2 Identification

One of the key issues in the factor model literature is to separately identify the loadings Λ and factors F _t in Eq. (2). We require additional restrictions to avoid observational equivalent models (Bai and Ng 2013). For any non-singular J × J matrix H we have

To this end, other studies typically set Q _ɛ to the identity matrix for scale identification and restrict one of the loadings for each factor to be strictly positive for sign identification (Kose, Otrok, and Whiteman 2003). However, this setup is not feasible for the present research design. For one, the stochastic factor selection approach adopted wants to test whether the innovation variance of the factors is equal to zero and not impose it. And second, if a factor drops out of the model it is not possible to restrict one of its loadings to be strictly positive.

Given these constraints we follow the identification approach of Berger, Everaert, and Pozzi (2021). First of, we assume uncorrelated factor innovations, i.e. Q ε = diag σ ε 2 with σ ε 2 = σ ε G 2 , σ ε m 2 , σ ε f 2 . Furthermore, we restrict the average of the relevant (non-zero) factor loadings for each factor to be normalized to one.^[6]

3.1.3 Non-Centered Parameterization of the DFM

To employ the stochastic model specification search of Frühwirth-Schnatter and Wagner (2010) we need to re-write the DFM using its non-centered parameterization, i.e. we note that Q ε = Σ ε Σ ε ′ and divide Eq. (2) by the standard deviation of the state equation innovations. Having Σ _ɛ now show up in the observation equation allows it to be treated as regression coefficients to which the variable selection technique can be applied to.

The observation equation of the state-space model is re-written as

where Σ _ɛ  = diag(σ _ɛ ), with σ ε = σ ε 1 , σ ε 2 , … , σ ε J ′ and f j t = F j t σ ε j in f _t = (f _1t , …, f _Jt ) are rescaled to have innovations with unit variances.

The non-centered parameterization of the state equations follows to be

A crucial caveat to this non-centered parameterization is the fact that the standardized factors f _jt and the corresponding standard deviation σ ε j are only identified to a sign change (Frühwirth-Schnatter and Wagner 2010). We can multiply f _jt and σ ε j by −1 without changing the product. Therefore the likelihood is bimodal with modes σ ε j and − σ ε j if F _jt exists, i.e. σ ε j 2 > 0 . If σ ε j 2 = 0 , then the likelihood is unimodal around zero. Hence, even without the full stochastic factor selection procedure, we can obtain first evidence on the existence of different factors by looking at the posterior distributions of the corresponding standard deviation σ ε j .

3.1.4 Parsimonious DFM

The last step needed for the DFM with stochastic factor selection involves the introduction of binary indicators δ _j multiplied by each rescaled factor f _jt . We now have

where Δ = diag(δ), with δ = (δ ₁, …, δ _J )′ and for

1. δ _j  = 1, f _jt is included in the model with σ ε j as unconstrained parameter to be estimated from data.

2. δ _j  = 0, f _jt is excluded from the model with σ ε j set to zero.

Sampling the binary indicators together with the unknown parameters of the model allows calculating posterior factor inclusion probabilities and by combining them we can ultimately obtain posterior model probabilities. More specifically, we calculate the marginal likelihoods of the models with and without the respective rescaled factor. After obtaining the posterior probability from combining the marginal likelihoods with the prior, we sample the binary indicator from a Bernoulli distribution. The posterior inclusion probability is then the fraction of draws from the posterior sampler in which the global factor was selected to exist.

3.2.1 Prior Distributions

To conduct Bayesian estimation of our model, we need to specify prior distributions for the respective parameters. An overview of the assumed prior distributions can be found in Table 1. All regression coefficients have a Gaussian prior distribution. The elements of the loading matrix Λ have a prior mean of 1 and the autoregressive parameters ρ _jl and π i l k a prior mean of 0.5. The prior variance is set equal to 0.15² for both the loadings and the AR parameters. Also following a Gaussian prior are the factor standard deviations σ ε j ∼ N ( v 0 , V 0 ) with prior mean v ₀ = 0 and an uninformative prior variance V ₀ = 10. Typically in DFMs, the factor innovation variances are modelled using an inverse gamma distribution. However, as shown by Frühwirth-Schnatter and Wagner (2010), I G -distribution with no probability mass at zero pushes the posterior of σ ε j 2 away from zero. Since, we want to test if σ ε j 2 = 0 we use the non-centered parameterization to model σ ε j using a Gaussian prior. The variances of the idiosyncratic innovations σ v i k 2 have an inverse gamma distribution with shape s ₀ T and scale s ₀ b ₀ T, expressing the prior belief b ₀ = 1 and strength of the belief s ₀ = 0.1 as a fraction of the sample size T (see Bauwens, Lubrano, and Richard 2000). Finally, we choose a Bernoulli prior for the binary indicators δ _j . In the baseline case, we set the prior probability of each indicator equaling 1 to 0.5, i.e. p(δ = 1) = p ₀ = 0.5. Conscious of the multiplicity control issue in Bayesian variable selection (see Scott and Berger 2010), we add p ₀ = 0.25 and p ₀ = 0.75 as robustness checks. Although this should in principle only be a problem in settings with a large number of factors where the prior inclusion probability p ₀ = 0.5 is likely to lead the fraction of included factors to approach 0.5.

3.2.2 Overview of the MCMC Algorithm for Posterior Inference

The posterior density is given by p(β, δ, f|y) where we collect the parameters in β = λ , ρ , ϕ , σ ε , σ v i k 2 . Since the joint posterior density does not allow for a closed form solution we employ a Gibbs sampler to sample the binary indicators δ, parameters β, and factors f conditional on the data.

1. Sample the binary indicators δ from p(δ|y, β, f) while marginalizing over the parameters σ _ɛ for which the factor selection is carried out.

2. Sample the parameters β from the conditional distribution p(β|y, δ, f)

   1. σ _ɛ are sampled from p σ ε | y , δ , λ , ρ , π , σ v i k 2 , f using Eq. (9) for f _j if δ _j  = 1. Otherwise if δ _j  = 0, we set the corresponding σ ε j = 0 .

   2. Sample λ and σ v i k 2 jointly using Eq. (9) and impose identifying normalizations.

   3. Sample AR parameters ρ and π from their respective conditionals p ρ | y , δ , λ , σ ε , π , σ v i k 2 , f and p π | y , δ , λ , σ ε , ρ , σ v i k 2 , f . We follow the approach of Chib and Greenberg (1994) by adding a MH-step within the Gibbs sampler.

3. Sample the factors f from p(f|y, δ, β)

   1. We use multi-move sampling (see Carter and Kohn 1994; Kim and Nelson 1998) to sample the included factors.

   2. Introduce random sign switch on σ ε j and f _j to make use of the non-identified signs. Specifically, with probability 0.5, σ ε j and f _j stay the same and with the same probability they are multiplied by −1.

4. Calculate additional quantities, e.g. scaled factors F j ≡ σ ε j f j and variance shares from the variance decomposition.

3.2.3 Variance Decomposition

After establishing evidence for the existence of certain common factors, we are interested in assessing the relevance of these factors. A convenient measure is the share of variance of each country’s variables explained by the global factors. The variance decomposition of Eq. (1) gives us the variance shares explained by the global macro-financial factor G _t as

the sector factors S t k as

and the idiosyncratic components as

After obtaining the individual variance shares we calculate the averages across the macroeconomic and financial sector variables to get the average variance shares of the global and sector factors and of the idiosyncratic components.

4.2.1 Descriptive Look at Evidence for Global Factors

We start our main analysis by reminding of the fact that the standardized factors and their respective standard deviations are only identified to a sign change. To make use of this property, the Gibbs sampler includes a random sign switch. Hence, we should see a bimodal posterior distribution for the standard deviation if the corresponding factor exists and an unimodal posterior distribution centered around zero if the factor does not exist. To take a first look at the evidence for the common global cycles we plot the posteriors of factor standard deviations in Figure 1.

Figure 1:

Posterior distributions of the global factor standard deviations. Top: sample with credit growth data. Bottom: sample with gross capital flow data. Sample of 30,000 MCMC draws. All binary indicators set to 1.

For all potential factors we observe bimodality except for the macroeconomic factor in the sample with gross capital flows which is much more unimodal. Especially strong, i.e. with the least probability mass at zero, is the bimodal posterior for the global financial cycle. Regardless of which financial variable is included, the weakest bimodality can be observed for the global business cycle. The posterior standard deviation of S ^m extracted from the model with credit growth (top panel in Figure 1) does not display as clear an unimodal density as in the model with gross capital flows (bottom panel) but it still features significant probability mass around zero.

4.2.2 Results from Stochastic Factor Selection

For the more rigorous analysis we move beyond merely looking at the posterior distributions of the factor standard deviations. The stochastic factor selection approach tests for the existence of the global factors. The main results of our testing procedure are summarized in Table 2. Given are the posterior factor inclusion probabilities. To control for possible prior sensitivity, we also present the results for the prior inclusion probabilities of p ₀ = 0.25 and p ₀ = 0.75, next to the baseline of p ₀ = 0.5. We observe strong evidence for the global macro-financial and financial factors in the models with credit growth and with capital flows. For all p ₀, the posterior inclusion probabilities are unity.

For the global macroeconomic factor however, we observe posterior inclusion probabilities well below 50 % across prior specifications and models. In regard to the model with credit growth, the baseline result (i.e. for p ₀ = 0.5) shows only a 16 % posterior inclusion probability which rises to 37 % for p ₀ = 0.75. The finding is confirmed in the model with capital flows. Here we find an even lower posterior inclusion probability of 4 % for p ₀ = 0.5 and 15 % for p ₀ = 0.75.^[11]

As laid out in Section 3.1.4 we can use the posterior inclusion probabilities to obtain posterior model probabilities. Table 3 indicates that irrespective of the factor inclusion prior the model with credit growth and the model with capital flows feature only the global macro-financial and the financial factor, i.e. they have the highest posterior probability. The baseline models exclude a separate global business cycle with a probability of 84 % and 96 % for the model with credit and the model with capital flows, respectively. Model specifications that would omit other factors did not receive any posterior probability. Hence, we can draw the first main conclusion from these findings: controlling for a global macro-financial factors eliminates the global macroeconomic factor but not a separate global financial factor.

4.3.1 The Estimated Global Factors

Selecting the most likely model, we plot the corresponding factor estimates in Figure 2. What we obtain are the global macro-financial cycle and the global financial cycles, i.e. a global credit cycle and a global capital flow cycle, estimated from models including real GDP growth and credit growth and gross capital flows, respectively. The two estimates of the macro-financial cycle are essentially identical, while the two global financial cycles feature their own dynamics. All cycles feature the strong impact of the great financial crisis of 2007–2009 and to a lesser extend the reaction to the Dotcom crash of 2001 and the double-dip due to the Euro area debt crisis.

Figure 2:

Estimates of the selected global factors. Plotted are the factor estimates of the model selected by the SFS approach. The solid lines give mean estimates and the shaded area the 90 % credible sets obtained from 30,000 MCMC draws. Top: Estimates of G and S ^f of model with credit growth. Bottom: Estimates of G and S ^f of model with gross capital flows.

4.3.2 Variance Decomposition

Having established evidence on the existence of the global and the financial factor, we can asses their relevance for the underlying macroeconomic and financial data by looking at the variance decomposition in Table 4. The results for the model specification SFS-GMF, i.e. the model preferred by the stochastic factor selection approach, shows that the global factor can explain on average around 11 % of the variance in the macroeconomic time series, regardless of the included financial variable. The financial variables are driven by both the global factor and the financial factor and we find a higher total share of the average explained variance, i.e. 22 % for the model with credit growth and 19 % for the model with capital flows. These results are largely in line with the literature on global cycles.^[12] Regardless of the model, the financial factors S ^f explain higher shares of the variance on average compared to the global factor G. Additionally the GMFCy is found to be more important for capital flows than credit growth. The global factor G explains on average 9 % of the variance in capital flows compared to 6 % for credit growth. Next to these main results of the model Table 4 also includes the variance decomposition for the other possible models, i.e. where we either exclude the global factor or the sector-specific factors. Of interest are especially the results for the macroeconomic data. For the model with only S ^m , it appears to captures the variance share explained by G in the our baseline model and for the model that includes both G and S ^m the variance share seems to be split among the two factors. Hence, the inclusion of one of the factors seems to be sufficient to capture the commonality in global macroeconomic dynamics. This does not hold financial dimension however. The global factor is not able to capture all of the comovement in the financial variables. Furthermore, forcing the model to exclude the global factor, i.e. selecting specification MF in Table 4, attributes explanatory power to the global financial cycle that is in fact due to the global macro-financial cycle as it is also significantly driving dynamics in the financial series.

4.3.3 Interpreting the Global Macro-Financial Cycle

A key finding that lets us interpret the nature of the global macro-financial cycle comes from estimating the global cycles G using different financial variables (see Figure 3). While the two global financial cycles (bottom panel) feature their distinct dynamics, we observe that the GMFCys (top panel) estimated with credit growth (dashed line) and gross capital inflow (red line) data are essentially identical. In fact, they are nearly the same as the global business cycle, if estimated using only macroeconomic data (blue line). These results suggest that what we have labelled the global macro-financial cycle is essentially the global business cycle identified in the literature (see Kose, Otrok, and Whiteman 2003). This one common global factor is sufficient to capture the joint macroeconomic dynamics. The inclusion of the financial variables are not important for estimating the shape of the global macro-financial cycle. Nevertheless, we find that the GMFCy explains relevant shares of the variation in both macroeconomic and financial data. Importantly however, as shown in the variance decomposition, the GMFCy does not explain all the comovement in the financial variables of interest. They are additionally driven by a separate global financial cycle.

Figure 3:

Comparison of the different estimates of GMFCy and the unconditional GBCy. Top: G estimated from data set with gross capital flows (red solid line), credit growth (dashed line) and the unconditional global business factor S ^m (blue solid line) estimated using only the macroeconomic data set. Bottom: S ^f estimated from data set with gross capital flows (red solid line) and credit growth (dashed line).