What is different this time? Investigating the declines of the U.S. real GDP during the Great Recession

Transcription

1 What is different this time? Investigating the declines of the U.S. real GDP during the Great Recession Yu-Fan Huang Sui Luo International School of Economics and Management Capital University of Economics and Business, Beijing, China Last Revision: January 2014 Abstract In this paper, we investigate to what extent the declines of the U.S. real GDP during recession of 2008 are permanent. We use a trend-cycle decomposition model featuring recession-specific plucking effect on the cyclical component. The empirical results indicate that the recession of 2008 is associated with substantial permanent changes in the real GDP: the mean growth rate of and the level of the trend component both declined substantially. However, the cyclical component does not play an important role as it did in previous recessions. After the recession of 2008, permanent loss is estimated to be -2.6% in terms of annual growth rate. Keywords: Recession, Plucking Effect, Trend-cycle Decomposition, Permanent Loss. JEL codes: E32. yufanh.cueb@gmail.com. suiluocueb@gmail.com. 1

2 1 Introduction Therecessionfollowingtheonsetofthesubprimecrisisin2008hasbeennamedthe Great Recession because of its severe and prolonged effects on the economy. Besides the fact that it is the greatest downturn since the beginning of the celebrated Great Moderation, the Great Recession distinguishes itself from other post-wwii recessions in the behaviors of the real output: there has been no sign of a V-shaped recovry, a period of temporary high-speed growth following the end of a recession (Reinhart and Rogoff (2012)). The absence of the rapid recovery has concerned policy makers and individuals around the world even though the Great Recession was deemed by the NBER to have ended in June The lack of a rapid growth after a recession raises the possibility that the real output is not returning to its potential, or trend, level if the economy was operating near full capacity before the Great Recession. 1 This possibility is illustrated by the potential outputs estimated by the Congressional Budget Office shown in Figure (1). However, given a widely accepted assumption that the output gap, the difference between the real and potential output, is a statioanry process, it is hard to explain such persistent gaps without dramastic changes in the potential output during the Great Recession. There exist justifications of changes in the potential output, e.g., the cyclicality of R&D(Ouyang (2011)) or oil supply shocks (Hamilton (2009)), and it is our goal to empirically assess the extent to which the declines of U.S. real GDP can be considered as changes in the potential output. We use a standard trend-cycle decomposition model, proposed by Clark (1987), in which the trend follows a random walk with a drifting slope. 2 In addition to the usual stationary cyclical component, we also consider the plucking effect, a temporary negative pulling force in the cyclical component during a recession (Friedman (1993) and 1 It is, however, equally plausible that after the burst of the housing bubble, the economy is returing to its bubble-free trend level. We will discuss this possibility in the next section. 2 Perron and Wada (2009) point out that allowing the slope to change can avoid potential misleading results. 2

3 Real GDP CBO Trend Our Estimated Trend Figure 1: Potential Output Based on the CBO and Our Estimates Kim and Nelson (1999a)), to capture the possible asymmetry in the business cycle. The magnitude of the plucking effect is specified to be time varying to cope with the possible heterogeneity among U.S. recessions in the post WWII history. Based on the methodology proposed by Eo and Kim (2013), our model can be written as a linear state space model, thus the conventional maximum likelihood estimation can be implemented and the information criteria be constructed easily. The empirical results suggest that the recessions in the post-wwii U.S. history are not all the same in terms of the downward plucking effect. According to the Bayes factor approximated by the Bayesian information criterion, the model featuring time-varying plucking effect strongly outperforms model without plucking which implies symmetric business cycle. This finding is in line with that of Morley and Piger (2012), which obtain a business cycle measure with an asymmetric shape across NBER expansion and recession phases. Our main findings can be well summarized by Figure (1). During the Great Recession, wefindthatthemeangrowthrateandthetrendcomponentoftherealgdpbothdeclined substantially. Correspondingly, the estimated output gap is not as persistent as that 3

4 implied by the CBO trend. Furthermore, the estimated time-varying plucking effects are significant throughout the sample periods but the magnitude is shrinking over time. Surprisingly, the estimated plucking effect during the Great Recession is the smallest among all recessions under consideration. These findings suggest that the declines of the real GDP during the Great Recession were largely driven by permanent factors, and the U.S. economy suffered from a great amount of permanent loss. To evaluate the permanent loss quantitatively, we construct a counterfactual trend component which indicates how much the trend level would have been if the economy operated as it did before a recession. The difference between the counterfactual and the estimated trend component is our measure of the permanent loss. During the Great Recession, we estimate a 4.5%, or 2.6% in terms of annual growth rate, permanent decline in the level of the real GDP. This number is consistent with that obtained using the Markov-switching model (Hamilton (1989)), which implies a L-shaped recovery after a recession. However, our finding is also, by and large, consistent with Piger et al. (2005) in that the real GDP suffered from much less permanent decline in all other recessions than in the Great Recession. Our analysis suggests that the U.S. economy has recovered in the sense that it has been operating near the potential level, which lowered down subsantially after the Great Recession. Therefore, the short-run stimulus policies, e.g., loose money and expansionary government spending, may not be the optimal tools given that they cannot affect the long-run potential output. It might be an appropriate time for thinking about policies boosting long term economic growth. However, understanding the main factor causing the declined potential output needs a structural model and is beyond the scope of this paper. Further research is needed for making specific policy suggestions. The outline of the remainder of the paper is as follows. Section 2 introduce the trendcycle decomposition model we used in this paper and discuss some estimation issues. In section 3, we report the evidence for the time-varying plucking effect and the empirical assessment of the permanent loss. We conclude and discuss some policy implications in section 4. 4

5 970 Lower Mean Growth Rate 970 Negative Permanent Shocks 970 Negative Transitory Shocks Figure 2: Three Stereotypes 2 A Trend-cycle Decomposition Model with Timevarying Plucking Effect 2.1 The Trend Component and the Real GDP during the Great Recession The set of the trend-cycle decomposition models is too vast to be considered all at once. Therefore, we focus on the periods of the Great Recession and extract the most important features that should be taken into account. Figure (2) shows the real GDP (dotted line) since 2004 and three stereotypes of the trend component. The estimation results certainly can be a mixture of these possibilities. The left panel shows the scenario in which the mean growth rate of the real GDP lowered down during the Great Recession, and there was no significant change in the level of the trend. A necessary consequence of this scenario is a large negative output gap, so it will take a long time for the real GDP to catch up with the trend. The substantial output gap justifies the need for the stimulus policies adopted by governments of many countries suffered from the crisis of 2008, while the lowered mean growth rate suggests economic policies towards long-run economic activity. In the scenario shown in the middle panel, the level of the trend shifted downward dur- 5

6 ing the Great Recession while leaving the mean growth rate intact. Note that the trend component can be interpreted as level of output when all the shocks die out. Therefore, this scenario implies that the economy had been operating around the potential output during the Great Recession, and the output gap was negligible. It is a pessimistic viewpoint of the Great Recession because the output will not be able to return to the level it would have been if there was no recession. In third case, the trend has been growing at a constant rate for all the time, but there was a prolonged business over the past 10 years. The period before the Great Recession enjoyed an economic boom, if not a bubble, possibly due to extraordinarily lose monetary policy, and the economic crash associated with the collapsed housing market was driven by the market adjustment mechanism. To take account for these three scenarios, or any combinations of them, the trend component of the empirical model needs to have the following two features: 1. The mean growth rate is subject to change. Because it has widely accepted that there was a productivity slowdown in 1970s, e.g., Perron and Wada (2009), the model should be able to account for at least two structural breaks in the mean growth rate. 2. The trend component can shift in level. Therefore, we consider a trend-cycle decomposition model proposed by Clark (1987) as follows: y t = τ t +c t, (1) τ t = d t 1 +τ t 1 +σ η η t, (2) d t = d t 1 +σ u u t, (3) c t = Φ 1 c t 1 +Φ 2 c t 2 +σ ǫ ǫ t, (4) whereη t, u t andǫ t arepossiblycorrelatedstandardnormaldistributions. AccordingtoOh and Zivot (2006), the three correlation coefficients among these shocks are not identified, 6

7 and at least two restrictions are needed. Following Morley et al. (2003), we assume that η t and ǫ t are correlated with correlation coefficient ρ and that u t is uncorrelated with other two shocks. The trend component, τ t, is assumed to follow a local linear trend process as in Harvey (1985). Assuming the drift term, d t, as a random walk is an easy way to take account for more than one major change in the mean growth rate to occur, and equation (2) also allows for a sudden level shift in τ t without changing the slope. 3 In fact, this model can be considered as a generalization of models incorporating structural breaks in the mean growth rate. 4 Instead of assuming abrupt changes, equation (3) implies a continuously changing drift term if σ u is not very large, which is usually the case in practice. 2.2 The Asymmetric Business Cycle The cyclical component, c t, is assumed to follow a AR(2) process. To ensure the identification, we impose a restriction that all the roots of (1 Φ 1 L Φ 2 L 2 ) lies within a unit circle. As a benchmark, c t is assumed to have mean zero, implying that the cyclical component is symmetric over business cycle. Even though the symmetry over the business cycle does not prevent any one of the three stereotypes shown in Figure (2), we further consider a class of models featuring asymmetry in the cyclical component because, as pointed out by Morley and Piger (2012), the cyclical measure can be highly sensitive to model specification. Within a trend-cycle decomposition, Kim and Nelson (1999a) first incorporates the asymmetric cyclical component according to the Friedman s plucking model (Friedman (1993)), which hypothesizes that the real output cannot exceed a ceiling level but occasionally is plucked downward by recession. See Sinclair (2009) for latest evidence for the plucking model. Following Kim and Nelson (1999a), we specify the ceiling level as described by equa- 3 An alternative for the random walk specification is the mixture innovation model (Giordani and Kohn (2008)), which assumes σ u is non-zero with a probability. Our results are not sensitive to this alternative. 4 For example, Perron and Wada (2009) assumes d t = D t d 0 + (1 D t )d 1, where D t = 0 for sample periods before the break date. 7

8 tions (2) and (3), and the cyclical component is specified as follows: c t = Φ 1 c t 1 +Φ 2 c t 2 +δ t (5) δ t = µs t +σ ǫ ǫ t, (6) where S t = 0,1. In what follows, S t is set to be 1 in the NBER recession dates and 0 otherwise. 5 Therefore, one can interpret the parameter µ as the plucking effect, which pulls the cyclical component downward during a recession. 6 Imposing known S t according to the NBER recession dates has one crucial drawback: it implies all recessions are different only by the duration because the plucking effect is identical for all recessions. We deal with this drawback by allowing recession-specific plucking effect. Following Koop and Potter (2007), we use a random walk process to approximate the time variation of the plucking effect as follows δ t = µ ss t +σ ǫ ǫ t, (7) µ s = µ s 1 +ω s, ω s N(0,σ 2 ω), where the subscript s indicates the s-th recession. There are two main difficulties of estimating model incorporating equation (7). First, µ s does not have the t subscript, so the model is different from the standard state space model. Second, µ s is defined only during the recession, S t = 1, thus it is difficult, if not impossible, to make inference about µ s during a boom without further assumption. We follow the method proposed by Eo and Kim (2013) and resolve these difficulties using a counterfactural prior, which imposes an assumption on what the value of µ s would be in the periods between the s-th and (s+1)-th recession. In this paper, we adopt the most simple specification of the counterfactural prior and assume that the plucking effect in a boom is identical to the plucking effect during the 5 Alternatively, one can specify unknown S t which follows a Markov-switching process and estimate the model using the Kim s filter (Kim and Nelson (1999c)). However, given that our goal is not to detect the turning point of µ, but rather to decompose each recession, we specify S t according to the NBER recession dates. 6 In the estimation, however, we do not restrict µ to be negative. 8

9 previous recession. As the results, the transition dynamic of µ s can be rewritten using the conventional t subscripts as follows δ t = µ t S t +σ ǫ ǫ t, (8) µ t = µ t 1 +k t σ w w t, (9) where w t N(0,1), and k t = 1 if S t 1 = 0 and S t = Dynamic Volatility Because the real GDP has been recognized to be heteroscedastic, we also consider cases in which the volatility of the cyclical component evolves according to the following processes: δ t = µs t +σ ǫ,t ǫ t (10) σ ǫ,t = (1 R t )σ ǫ,0 +R t σ ǫ,1, (11) where σ ǫ,1 = qσ ǫ,0 in the estimation. 7 R t is a unobserved two-state Markov-switching variable that evolve according to transition probabilities given below: Pr[R t = 1 R t 1 = 1] = p 11 Pr[R t = 0 R t 1 = 1] = 1 p 11 Pr[R t = 0 R t 1 = 0] = p 00 Pr[R t = 1 R t 1 = 0] = 1 p 00. All models under consideration can be written into a state-space form with Markovswitching parameters, if any. Details are shown in Appendix A. It can be estimated by MLE and the likelihood function is evaluated by the Kim s filter (Kim and Nelson (1999c)). 7 Allowing for changes in the trend volatility yields very similar estimation results and likelihood improvement is hardly noticeable. The equality hypothesis cannot be rejected at any usual confidence level. 9

10 Table 1: Model Comparison Feature None MSV TVP TVP-MSV LLK AIC BIC * All models consist of equations (1) to (4) with different features of δ t. ** MSV represents Markov-switching volatility, TVP the time-varying plucking effect. *** LLK is the log-likelihood. AIC and BIC are corresponding Akaike and Bayesian information criterion. 3 Empirical Results 3.1 Evidence of the Time-varying Plucking Effect WeuserealGDPdatafrom1947:3to2013:3obtainedonthewebsiteoftheFederalReserve Bank of St. Louis. Table (1) shows the log-likelihood and corresponding Akaike and Bayesian information criterion (AIC and BIC), which are computed as the log-likelihood minus the offsetting penalty. 8 Both AIC and BIC select the model featuring the timevarying plucking effect and the Markov-switching volatility. Even though the information criteria are not used for hypothesis testing, the BIC reported in table (1) does show a strong evidence for the time-varying plucking effect. In fact, The BIC can be used to construct the Bayes factor according to the following approximation: lnb i,j = BIC i BIC j, where B i,j is the Bayes factor comparing model i and j (see Claeskens and Hjort (2008)). Kass and Raftery (1995) argue that the Bayes factor summarizes the evidence provided by the data in favor of one statistical model as opposed to another, and suggest a set of 8 The penalty is the number of parameters in the model for AIC, while it is the number of parameters times one half of the log of the sample size for BIC. 10

11 criteria as follows : 2 ln B i,j Evidence against M j 0 to 2 Not worth more than a bare mention 2 to 6 Positive 6 to 10 Strong >10 Very strong. According to these criteria, a model is 150 times more likely to be the model generating observed data if 2 ln B i,j is equal to 10. Using the concept of the Bayes factor, the values reported in Table (1) reveal how the time-varying plucking effect is preferred by the data. With or without the Markovswitching volatility, the models with time-varying plucking effect is always very strongly preferred by the data comparing to the models without the plucking effect. Therefore, we will use the model featuring time-varying plucking effect and Markov-switching volatility to obtain the decomposition of the real GDP. 3.2 Decomposing the U.S. Recessions The estimates of the parameters are reported in Table (2), and Figure (3) displays the smoothed estimates of the trend component and 95% confidence interval of the mean growth rate. 9 The estimated standard deviation of the trend shock, η t, is large and significant. That the trend exhibits a large extent of variation supports the stochastic trend hypothesis (Morley et al. (2003)). Despite a wide confidence interval, the estimated mean growth rate indicates the first productivity slowdown started around the 1970s (Perron(1989)) and the second after 2000s(Jorgenson et al.(2008) and Fabina and Wright (2013)). As a matter of fact, the stochastic trend is not a consequence of incorporating the plucking effect but a general finding in all the models under consideration. Therefore, the claim that the stochastic trend is an artifacts created by the neglect of a change in the 9 We use a simulation smoothing to obtained smoothed estimates of unobserved components. See Appendix for details. 11

12 Table 2: Maximum Likelihood Estimation φ q (0.0808) (0.1745) φ σ ω (0.0983) (0.1167) σ η σ u (0.2384) ( ) σ ǫ p (0.2557) (0.0823) ρ p (0.1235) (0.0780) lnl= * Standard errors in () RGDP Trend Figure 3: Smoothed Estimates of the Trend and the Mean Growth Rate 12

13 Cycle µ t /(1 φ(1)) Figure 4: Smoothed Estimates of the Plucking Effect and the Cycle slope of the trend function (Perron and Wada (2009)) is not supported by data including the Great Recession. Concerning the recession-specific plucking effect, the left panel of Figure (4) shows substantial and significant time variations across 11 recessions, and the 95 % confidence interval not covering zero indicates that the plucking effect has been present in all recessions. However, the magnitude of the plucking effect has been declining over time, with the most abrupt change occurring in the recession of The right panel of Figure (4) reports the smoothed estimates of the cyclical component and a measure of the long-run plucking effect represented by µ t /(1 Φ(1)). 10 Generally speaking, the plucking effect underwent a structural change around the mid-1980s, which is also the starting point of the celebrated Great Moderation, a period of decreased macroeconomic volatility (Kim and Nelson (1999b)). The association between the Great Moderation and the plucking effect certainly deserves further research. Based on our estimation results, we find some important features of the real GDP summarized as the following: First, the estimated mean growth rate reaches the lowest 10 This measure is not precise in statistical sense because if a recession lasts for a long time, it will become the trend component. 13

14 Permanent Loss RGDP Trend Figure 5: Smoothed Estimates of Trend point after the Great Recession, providing some evidence for the slope-shifting scenario illustrated in the first panel of Figure (2). Second, the trend component appears to be stochastic, implying that the the level-shifting scenario illustrated in the second panel of Figure (2) is also plausible. Third, the plucking effect played a minor role in the Great Recession in that the size of the plucking effect is much smaller than that in the other recessions. These findings suggest that the Great Recession has long-lasting, if not permanent, effect on the real output. 3.3 Measuring of the Permanent Loss The estimated output gap, the difference between the trend and the real GDP, has been closed around 2011, and it is about the same time that the real GDP has recovered to where it was before the recession of However, if the economy grew as it did before the recession, the real GDP could have grown at a faster rate as the dashed line. The difference between the counterfactual growth path and the estimated trend can be naturally regarded as the permanent loss of this recession. Figure (5) illustrates our measure of the permanent loss. It is straightforward to 14

15 formalize the idea as follows: for a recession which began at time t 0 and ended at time t 1, Ψ = τ t1 T τ t 1, (12) where τ t1 T is the smoothed estimate of the trend component at time t 1, and τt 1 is the counterfactual value of the trend given that the mean growth rate is equal to the level at the beginning of the recession and that there is no permanent shock during the recession. It can be easily computed as τt 1 = (t 1 t 0 )d t0 T +τ t0 T, where d t0 T and τ t0 T are the smoothed estimates of mean growth rate and the trend component at time t 0. Note that even though Ψ is referred to as a loss, nothing prevents it from being positive. 11 Table(3) shows the estimated permanent loss. We obtain the standard deviation using the results from the simulation smoothing. 12 Because the duration of a recession is not identical, we transform Ψ in terms of annual rate of growth, denoted by Ψ a. Specifically, Ψ a = 4 Ψ/(t 1 t 0 ). The estimated permanent loss associated with the Great Recession is significantly lower than zero, and its magnitude is greater than that associated with the recession beginning at 1973q4, when the first oil shock struck the U.S. economy. 4 Conclusion This paper finds that the Great Recession is different from other post-wwii recessions. A large portion of the declines of the real GDP during the Great Recession seems to be permanent according to data up to date, while the transitory deviations of the real GDP from the trend play a minor role. Our empirical results shed some light on appropriate macroeconomic policies five years after the Great Recession. Given that the stimulus 11 A positive Ψ can be rationalized by a series of negative demand shocks and positive productivity shock. 12 For each sample drawn from the smoothed distribution, we compute Ψ for each recession. These samples are used to compute the mean and the standard deviation. 15

16 Table 3: Permanent Loss of Each Recession Recession Ψ Ψ a Ψ a /s.d. 53q3 to 54q q3 to 54q q3 to 58q q2 to 61q q4 to 70q q4 to 75q q1 to 80q q3 to 82q q3 to 91q q1 to 01q q4 to 09q * s.d. represents the standard deviation of simulated Ψ. 16

17 policies, e.g., lose monetary policy and expansionary fiscal policy, affect the real economy only in the short term, the findings in this paper suggest policies concerning the economic growth in the long run. 17

18 References Carter, Christopher K. and Robert J. Kohn, On Gibbs sampling for state space models, Biometrika, 1994, 81 (3), 541. Claeskens, Gerda and Nils Lid Hjort, Model selection and model averaging Cambridge series in statistical and probabilistic mathematics, Cambridge University Press, Clark, Peter K, The Cyclical Component of U.S. Economic Activity, The Quarterly Journal of Economics, November 1987, 102 (4), Eo, Yunjong and Chang-Jin Kim, Markov-Switching Models with Evolving Regime- Specific Parameters: Are Post-War Booms or Recessions All Alike?, Working Paper September Fabina, Jake and Mark Wright, Where has all the productivity growth gone?, Chicago Fed Letter, 2013, Jan (306). Friedman, Milton, The Plucking Model of Business Fluctuations Revisited, Economic Inquiry, 1993, 31 (2), Giordani, Paolo and Robert Kohn, Efficient Bayesian Inference for Multiple Change- Point and Mixture Innovation Models, Journal of Business & Economic Statistics, January 2008, 26 (1), Hamilton, James D, A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle, Econometrica, March 1989, 57 (2), Hamilton, James D., Causes and Consequences of the Oil Shock of , Brookings Papers on Economic Activity, 2009, 40 (1 (Spring)), Harvey, A C, Trends and Cycles in Macroeconomic Time Series, Journal of Business & Economic Statistics, June 1985, 3 (3),

19 Jorgenson, Dale W., Mun S. Ho, and Kevin J. Stiroh, A Retrospective Look at the U.S. Productivity Growth Resurgence, Journal of Economic Perspectives, Winter 2008, 22 (1), Kass, Robert E. and Adrian E. Raftery, Bayes factors, Journal of the american statistical association, 1995, pp Kim, Chang-Jin and Charles R Nelson, Friedman s Plucking Model of Business Fluctuations: Tests and Estimates of Permanent and Transitory Components, Journal of Money, Credit and Banking, August 1999, 31 (3), and Charles R. Nelson, Has The U.S. Economy Become More Stable? A Bayesian Approach Based On A Markov-Switching Model Of The Business Cycle, The Review of Economics and Statistics, November 1999, 81 (4), and, State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, The MIT Press, Koop, Gary and Simon M. Potter, Estimation and Forecasting in Models with Multiple Breaks, Review of Economic Studies, 2007, 74 (3), Morley, James and Jeremy Piger, The Asymmetric Business Cycle, The Review of Economics and Statistics, February 2012, 94 (1), Morley, James C., Charles R. Nelson, and Eric Zivot, Why Are the Beveridge- Nelson and Unobserved-Components Decompositions of GDP So Different?, The Review of Economics and Statistics, May 2003, 85 (2), Oh, Kum Hwa and Eric Zivot, The Clark Model with Correlated Components, Working Papers UWEC , University of Washington, Department of Economics January Ouyang, Min, On the Cyclicality of R&D, Review of Economics and Statistics, 2011, 93 (2),

20 Perron, Pierre, The great crash, the oil price shock, and the unit root hypothesis, Econometrica, 1989, pp and Tatsuma Wada, Let s take a break: Trends and cycles in US real GDP, Journal of Monetary Economics, September 2009, 56 (6), Piger, Jeremy, James Morley, and Chang-Jin Kim, Nonlinearity and the permanent effects of recessions, Journal of Applied Econometrics, 2005, 20 (2), Reinhart, Carmen and Kenneth Rogoff, This Time is Different, Again? The United States Five Years after the Onset of Subprime, MPRA Paper October Sinclair, Tara, Asymmetry in the Business Cycle: Friedman s Plucking Model with Correlated Innovations, Studies in Nonlinear Dynamics & Econometrics, December 2009, 14 (1),

21 A The state space form The model with time-varying plucking effect can be written in a state space form as follows: [ y t = Φ 1 y t 1 +Φ 2 y t Φ 1 Φ 2 S t 0 1 ] β t (13) where β t = [τ t,τ t 1,τ t 1,µ t,d t,ǫ t ]. The transition equation of β t is σ η β t = β t d 01,t σ w σ u σ ǫt (R t ) 0 0 ξ t, (14) where ξ t = [η t,ǫ t,w t,u t ] is a vector of normal distribution with mean zero and covariance matrix as follows: 1 ρ 0 0 ρ Σ =. (15) B Simulation Smoothing The state space model given above can be written in a compact form as follows y t c t ȳ t = Hβ t (16) β t = Fβ t 1 +W(R t )ξ t, (17) wherec t, H, d, F, W arematricescontainingknownparameters. R t isamarkov-switching unobserved variable. 21

22 A simulation smoother is a procedure for drawing samples from the conditional distribution of state given the observations. Let x T denote a vector [x 1,...,x T ]. The objective distribution is f(β T y T ). Note that a standard backward sampling scheme, e.g., Carter and Kohn (1994), cannot be directly be implemented if some of the parameters are Markov-switching. However, we can decompose the distribution as follows: f(β T ȳ T ) = f(β T,R T ȳ T )dr T = f(β T R T,ȳ T )f(r T ȳ T )dr T Therefore, we can generate samples from f(r T ȳ T ) and then generate samples from f(β T R T,ȳ T ). The resulting samples of β T can be used to construct smoothed estimates of β T. Using the Kim s filter, we will obtain a series of filtered probability of R t for all t, and generating a sample from f(r T ȳ T ) can be done. See Kim and Nelson (1999c) for details. Given a set of R t for all t, the FFBS developed by Carter and Kohn (1994) can be implemented with some modifications. First, run the basic filtering as follows: β t t 1 = d+fβ t 1 t 1, P t t 1 = FP t 1 t 1 F +W(R t )ΣW(R t ) η t t 1 = ȳ t Hβ t t 1 f t t 1 = HP t t 1 H β t t = β t t 1 +P t t 1 H f 1 t t 1 η t t 1 P t t = P t t 1 P t t 1 H f 1 t t 1 HP t t 1 Save β t t and P t t, and generate β T from N(β T T,Pt T T ). Then do backward sampling by computing β t t,βt+1 = β t t +P t t F (FP t t F +W(R t )ΣW(R t ) ) 1 (β t+1 Fβ t t d), P t t,βt+1 = P t t P t t F (FP t t F +W(R t )ΣW(R t ) ) 1 FP t t, for t = T 1,T 2,...,1. 22

23 Finally, for each t, generate β t from N(β t t,βt+1,p t t,βt+1 ). However, for most of the sample periods, µ t is a constant. It will change only at the time when a recession begins. We follow Eo and Kim (2013) and generate µ t as follows: If S t = 0 and S t+1 = 1, we generate µ t from N(β t t,βt+1 (4,1),P t t,βt+1 (4,4)), where number in parentheses indicates the position in a matrix. Otherwise, we set µ t = β t t,βt+1 (4,1), which by default is µ t+1. Repeat the above procedure for many times, and we will have many sample from the smoothed distribution of β t. These samples form the basis of inference reported in this paper. 23