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References

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I. A Bayesian Approach to Estimating Dynamic Factor Models

The estimation procedure we use for our dynamic factor model is a Bayesian approach that exploits Gibbs sampling techniques. These techniques make it computationally feasible to draw from the exact finite sample distribution of the parameters and factors of interest in our model. The approach that we use builds upon the work of Otrok and Whiteman (1998) and does not rely on asymptotics in either the cross-sectional or the time dimension but at the same time is feasible from a computational standpoint for datasets where there are a large number of factors.

Our estimation procedure is better suited to answering the questions addressed in this paper than many of the alternatives. There are, of course, classical approaches to estimating dynamic factor models. Gregory and others (1997) follow Quah and Sargent (1993) and use Kalman filtering and the EM algorithm to estimate a dynamic factor model for the G-7 capturing the world as well as country specific cycles. Unfortunately, for a dataset of the dimension we are interested in (106 countries, 318 variables, and 110 factors), this approach is computationally infeasible. For very large datasets, such as ours, the approximate factor model approaches of Stock and Watson (1989) and Forni and others (1998) are quite efficient at extracting factors. However, the approximate factor models cannot be applied to situations where we wish to impose zero restrictions on some factor loadings to “identify” some factors as belonging to a particular country or a set of zero restrictions to limit other factors to a particular economy type. For example, we identify a country factor by having an unrestricted non-zero factor loading for all variables in that country. The variables of all other countries are restricted to have a zero factor loading on this country factor. The parametric approach we take can easily handle these restrictions.

Since analytic forms for the joint posterior of the factors and parameters are unobtainable, we employ numerical methods to simulate from the joint posterior distribution of the factors and parameters. We use a “data augmentation” algorithm to generate draws from the joint posterior of interest (see Tanner and Wong, 1987; Otrok and Whiteman, 1998). Data augmentation builds on the following key observation: if the factors were observable, under a conjugate prior, the model (1) to (3) would be a simple set of regressions with Gaussian autoregressive errors. Then, conditional on the regression parameters and the data, one can determine the conditional distribution of the factors. It is straightforward to generate random samples from this conditional distribution, and such samples can be employed as stand-ins for the unobserved factors.

To be more specific, the dynamic factor analysis model in equations (1) to (3) can be thought of as consisting of a specification of a Gaussian probability density for the data {yt} conditional on a set of parameters φ and a set of latent variables {ft}. Call this density function gy(Y|φ,F) where Y denotes the IJT × 1 vector of data on the observables, and F denotes the M × 1 vector of dynamic factors. In addition, there is a specification of a Gaussian probability density gf (F) for F itself. Given a prior distribution for φ, π(φ), the joint posterior distribution for the parameters and the latent variables is given by the product of the likelihood and prior, h(φ,F|Y) = gy(Y|φ,F)gf(F)π(φ). As is shown in Otrok and Whiteman (1998), although the joint posterior h(φ,F|Y) is extremely cumbersome, under a conjugate prior for φ the two conditional densities h(φ|F,Y) and h(F|φ,Y) are quite simple. Moreover, it is possible to use this fact and Markov-Chain Monte Carlo methods (MCMC) to generate an artificial sample {φj,Fj} for j = 1,…,J as follows:

  1. Starting from a value F0 in the support of the posterior distribution for F, generate a random drawing φ1 from the conditional density h(φ|F0,Y).

  2. Now generate a random drawing F1 from the conditional density h(F|φ1,Y).

  3. This process is repeated, generating at each step drawings φj ~ h(φ|Fj-1,Y) and Fj ~h(F|φj-1,Y).

Under regularity conditions satisfied here (see Tanner and Wong, 1987), the sample so produced is a realization of a Markov chain whose invariant distribution is the joint posterior h(φ,F|Y). What makes this process feasible is the simplicity of the two conditional distributions. For example, h(φ|F,Y) is easily constructed from equation (1) when F is known. In particular, equation (1) is just a normal linear regression model for yi given the factors (albeit a regression that has autocorrelated errors). Because the prior for the intercept and factor loadings is Gaussian, the conditional posterior for the parameters (σi, ai and the bi‘s) is also Gaussian. The other conditional density, h(F|φ,Y) is a little more complicated because it is the solution to a Gaussian signal extraction problem. Kalman filter techniques are commonly used to solve such problems, but when the time series is short, as in this application, it is straightforward to solve the problem directly. Solving the problem without using the Kalman filter is especially useful when the number of factors is large, as in the problem we study. (With a large number of factors the state equation in the Kalman filter can become computationally very burdensome.) Otrok and Whiteman (1998) do this as follows: first, they write the joint density for the data and the dynamic factors given the parameters as a product of I*J*M independent Gaussian densities (I*J of them are associated with the observable time series, M with the dynamic factors). Second, from this joint distribution, simple normal distribution theory is used to obtain the conditional distribution for any one of the factors given the rest and the parameters. These normal distributions involve inverses of T × T covariance matrices that can be handled using conventional procedures provided T is not large. In the model analyzed here, T = 40 is not problematic.

The prior on all the factor loading coefficients is N(0,10), which is quite diffuse. For the parameters of autoregressive polynomials, the prior is N(0,∑), where Σ=[1000.5000.25]. Because the data are growth rates, this prior embodies the notion that growth is not serially correlated though the prior is loose enough to allow for significant serial correlation; also, the probability that lags are zero grows with the length of the lag.28 Experimentation with tighter and looser priors for both the factor loadings and the autoregressive parameters did not produce qualitatively important changes in the results reported below. The prior on the innovation variances in the observable equations is Inverted Gamma (6, 0.001), which is also quite diffuse.

Since we are not sampling from the posterior itself (the elements of the Markov chain are converging to drawings from the posterior), it is important to monitor the convergence of the chain. We do so in a number of ways. First, we restarted the chain from a number of different initial values, and the procedure always converged to the same results. Second, we experimented with chains of different lengths ranging from lengths of 2,500 to 10,000. The results remained unchanged. The results we report are based on a chain of length 5,000.

II. Testing for Structural Breaks

There is substantial evidence of structural breaks in many macroeconomic time series. One approach to identifying structural breaks in our model would be to specifically allow for breaks in the factor loadings, innovation terms and autoregressive parameters in the model described by equations (1)-(3). Unfortunately, given the size of our model the econometric implementation of such a procedure is infeasible. Instead, we use a sequence of univariate break tests to shed light on the potential for breaks in the data. The tests we use consider breaks in both persistence and volatility and allow for one break at an unknown date for each variable (Andrews 1991).

The first test we employ estimates an AR(1) model to each time series and tests for a break in both the constant term and autoregressive parameter. The Andrews test requires us to estimate the model over the whole sample and then again allowing for a break at each possible date. That is we estimate:

yt=a+D(s)+ρyt1+D(s)αyt1+εt(1)

where D(s) is a dummy variable that takes the value 0 before the break date and 1 at the break date (s = t) and all dates after the break date. For each possible break date we estimate the model in (4) and calculate the Wald statistic:

k*( (SSRSSU)/k)/(SSU/(T2*k))(2)

where SSR is the sum of squared residuals from the regression with no break (D(s) = 0 for all t) and SSU is the sum of squared residuals for the regression that allows for a break. K is the number of parameters that may break, in our case k = 2. T is the number of time series observations. We calculate the Wald statistic for all possible break dates in the middle 2/3 of the sample, since tests near the endpoints have been known to be unreliable, and use the max of the Wald statistics as our potential break date. The statistical significance of this break can be checked against the critical values provided in Andrews (1991), which depend on S/T.

To test for a break in only the volatility in each series, we regress the demeaned absolute value of each series on a constant for the restricted regression and then on a constant and a dummy variable for the unrestricted regression. That is, for the unrestricted regression we estimate:

Yt*=a+D(s)+εt(3)

where yt* is the demeaned absolute value of the series, for all possible break dates. The Wald statistic is as given above with k = 1 degrees of freedom.

III. List of Countries

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Data sources: Primarily from the World Bank’s World Development Indicators (WDI), supplemented with the International Monetary Fund’s World Economic Outlook (WEO) database.
1

Earlier versions of this paper were presented at the University of Wisconsin and Cornell University. We would like to thank Stijn Claessens, Selim Elekdag, Charles Engel, Mark Watson, and seminar participants for useful comments.

2

See Kose, Otrok, and Whiteman (2008) for a brief survey of the literature studying the extent of business cycles comovement among industrial countries.

3

The NBER focuses on the evolution of five indicators—real GDP, real income, employment, industrial production, and wholesale-retail sales. Others have used variables such as real GDP and unemployment to pin down the sources of business cycles (Blanchard and Quah, 1989). King and others (1991) study joint fluctuations in output, consumption and investment to identify trends and cycles.

4

In addition to analysis of business cycles comovement, factor models have been used in a variety of contexts. See Otrok and Whiteman (1998) for an application to a forecasting exercise, and Bernanke, Boivin, and Eliasz (2005) for an application to the analysis of monetary policy.

5

The sign restriction is simply a normalization that allows us to interpret the factors in an intuitive way. For example, we normalize the factor loading for GDP growth in the U.S. on the global factor to be positive. This implies that the global factor falls in 1974 and 1981, consistent with the fact that most countries had a recession in those years. If, instead, we were to normalize using Venezuela, an oil exporting country that benefited from the 1974 oil price shock, the factor would rise in 1974. The U.S. factor loading on the global factor would then be negative.

6

Other measures include: (1) the concordance statistic (Harding and Pagan, 2002), which measures the synchronization of turning points; and (2) coherences (the equivalent of correlations in the frequency domain, although, unlike static correlations, they could allow for lead-lag relationships between two variables). Similar to simple correlations, these other measures are subject to a variety of limitations.

7

Even though the factors are uncorrelated, samples taken at each pass of the Markov chain will not be, purely because of sampling errors. To ensure adding up, we took a further step for these calculations, and orthogonalized the sampled factors, ordering the global factor first, the regional factor second, and the country factor third. Our simulations suggest that the order of orthogonalization has little impact on the results. In particular, all of the results remain qualitatively similar under alternative orderings, and the quantitative differences are small.

8

On average, EMEs also had higher per capita incomes and experienced higher growth rates than ODCs over the last two decades. Over the period 1960–1984, industrial countries on average accounted for more than 70 percent of world GDP (in PPP terms) while the aggregate share of the EMEs was roughly 25 percent. During the globalization period, the share of EMEs in world GDP has increased to 34 percent while that of industrial economies has fallen to 62 percent. The share of ODCs has registered a slight decline over time.

9

The trade openness ratio for EMEs has risen from 28 percent to 78 percent over the last two decades. Similarly, for INCs, the openness measure has increased from 26 percent to 46 percent during the period of globalization. In contrast, the openness ratio for ODCs has been rather stable, around 65 percent. For a detailed account of changes in trade and financial linkages, see Akin and Kose (2008).

10

Moreover, the beginning of the globalization period marks the start of the Uruguay Round negotiations which speeded up the process of unilateral trade liberalizations in many developing countries.

11

See Blanchard and Simon (2001), McConnell and Perez-Quiros (2000), and Stock and Watson (2005). Explanations for this decline in volatility are many, ranging from “the new economy” driven changes to the more effective use of monetary policy.

12

The correlation between the global factor and the world price of oil, measured by the index of average spot prices (from the IMF’s International Financial Statistics), is -0.04.

13

In small samples, it is possible that the global and group-specific factors will be correlated due to a spurious correlation associated with the short sample. In all the results reported here, we impose orthogonality by regressing the group-specific factors on the global factor and retaining the residual. The correlation between the industrial country factor and EMEs (ODCs) factor in our sample is around 0.6 (0.5). The average correlation between the global factor and group-specific factors is less than 0.1. We calculated 5 percent and 95 percent quantile bands for all of the estimated factors, but leave them out of the plots to reduce clutter. Plots showing the quantile bands are available from the authors.

14

Another possibility is that there is a second global factor that we have not accounted for. We checked whether this is the case by conducting additional simulations but the results suggest this is not the case.

15

The NBER reference business cycle dates for the U.S. are as follows: Troughs: Feb. 1961, November 1970, March 1975, July 1980, November 1982, March 1991, and November 2001. Peaks: April 1960, December 1969, November 1973, January 1980, July 1981, July 1990, and March 2001. All other reference business cycle dates are taken from IMF (2002).

16

We also calculated the median (rather than mean) variance shares attributable to each factor for the full sample and each group of countries. These were generally close to the average shares reported in Tables 16, indicating that there are no obvious outlier countries driving our results. Hence, we only report results using means. The results using medians are available from the authors.

17

We do not report the factor loadings; they are available from us upon request.

18

Detailed variance decompositions for each country in our sample are available from the authors upon request.

19

By estimating the model over two sub-samples we are allowing the model parameters, such as the factor loadings and those that determine the structure of propagation of shocks, to vary across subsamples. This will yield a different variance decomposition. However, the estimate of the factor itself is very similar whether estimated over the full sample or over subsamples. This result is not very surprising as the index of common activity in a period should not be affected by data many periods away. It is also consistent with the recent work of Stock and Watson (2007) who show that latent factors can be estimated consistently despite parametric instability.

20

Recent research shows that the implementation of similar macroeconomic policies can lead to a higher degree of business cycle synchronization (see Darvas, Rose, and Szapáry, 2005).

21

For evidence of a European business cycle, see Artis, Krolzig, and Toro (2004) and references therein. But Canova, Ciccarelli, and Ortega (2007) argue that, since the 1990s, the empirical evidence does not reveal a specific European cycle. Bordo and Helbling (2004) find a trend toward increased synchronization among industrial countries, while Monfort et. al (2003) conclude that the degree of comovement among G-7 economies has been declining. Changes in bilateral output correlations often are not significant, a point emphasized by Doyle and Faust (2005).

22

They document that business cycle synchronization among the G-7 countries increased during the 1970s and early to mid-1980s. The subsequent decline reflects decreased synchronization with Japan and, to a lesser extent, Germany. Stock and Watson (2005) report that the share of output fluctuations in the other five G-7 countries that can be attributed to common factors increased from 1960–83 to 1984– 2002.

23

For instance, Kose, Otrok, and Whiteman (2003) use data from 60 countries, but their sample period is limited to1960-1990. The use of recent data is important since globalization really picked up steam only in the mid-1980s. Moreover, our use of a larger sample (and larger sub-samples within each group) allows us to draw a sharper contrast across country groups in terms of their exposure to the global economy.

24

For the impact of trade linkages, see Baxter and Kouparitsas (2005) and the references therein. For financial linkages, see Imbs (2006) and Kose, Prasad, and Terrones (2003).

25

Some recent theoretical papers produce results consistent with the dynamics of investment we report here In Head’s (2002) model, cross-country correlations of investment are positive because of increasing returns to the worldwide variety of intermediate goods. Also see Heathcote and Perri (2004).

26

Helbling and others (2007) report that U.S. recessions are more worrisome for the rest of the world than a mid-cycle slowdown since U.S. import growth turns sharply negative during recessions, and cross-country asset price correlations increase significantly during financial market downturns.

27

While international policy coordination serves an important role in an integrated world economy in the traditional models based on trade multiplier mechanisms, Obstfeld and Rogoff (2001) argue that increased integration may in fact diminish the need for monetary policy coordination. See Canzoneri, Cumby, and Diba (2005) for a survey.

28

Otrok and Whiteman (1998) discuss the procedure for ensuring stationarity of the lag polynomial. The method involves drawing from a truncated Normal distribution in the Metropolis-Hastings step.

Global Business Cycles: Convergence or Decoupling?
Author: Mr. Ayhan Kose, Mr. Eswar S Prasad, and Mr. Christopher Otrok