Supplementary Appendix to A Bayesian DSGE Model of Stock Market Bubbles and Business Cycles

Transcription

1 Supplementary Appendix to A Bayesian DSGE Model of Stock Market Bubbles and Business Cycles Jianjun Miao, Pengfei Wang, and Zhiwei Xu May 9, 205 Department of Economics, Boston University, 270 Bay State Road, Boston, MA 0225, USA, CEMA, Central University of Finance and Economics, and AFR, Zhejiang University, China. miaoj@bu.edu. Tel: Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. pfwang@ust.hk. Tel: Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China. xuzhiwei09@gmail.com. Tel.:

2 This online material contains six appendices to the paper. Appendix A proves proposition in the paper. Appendix B derives the stationary equilibrium. Appendix C studies the bubbly steady state. Appendix D provides the log-linearized equilibrium system around the bubbly steady state. Appendix E presents a table of business cycle moments. Appendix F presents a robustness analysis. A Appendix: Proof of Proposition in the Paper: We use a conjecture and verification strategy to find the decision rules at the firm level. We first study the optimal investment problem by fixing the capacity utilization rate u j t. Using 4 and 6 in the paper, we can write firm j s dynamic programming problem as v t ε j t Kj t + b t,τ ε j t v Ltε j t Lj t = max I j t,lj t+ u j t R tk j t P ti j t Lj t + Lj t+ R ft +Q t [ δ j t Kj t + εj t Ij t ] + B t,τ Q Lt L j t+, A. subject to the investment constraint: 0 P t I j t uj t R tk j t Lj t + Lj t+ R ft + η t K j t. A.2 For ε j t P t /Q t, I j t = 0. Optimizing over L j t+ yields Q Lt = /R ft. For ε j t+ P t/q t, the optimal investment level must reach the upper bound in the above investment constraint. We can then immediately derive the optimal investment rule in 8 of the paper. In addition, the credit constraint 7 in the paper must bind so that R ft L j t+ = Q tξ t K j t + B t,τ. A.3 Substituting the optimal investment rule and Q Lt = /R ft into A. yields: v t ε j t Kj t + b t,τ ε j t v Ltε j t Lj t = u j t R tk j t + Q t δ j t Kj t + B t,τ L j t + max{q t ε j t /P t, 0} u j t R tk j t + η tk j t Lj t + Lj t+ R ft. A.4 Since u j t is determined before observing εj t, it solves the following problem: max u j t u j t R tk j t + Q t δ j t Kj t + G tu j t R tk j t, A.5 2

3 where G t is defined by 20 in the paper. We then obtain the first order condition R t + G t = Q t δ u j t. A.6 Since δ j t = δuj t is convex, this condition is also sufficient for optimality. From this condition, we can immediately deduce that optimal u j t does not depend on firm identity so that we can remove the superscript j. By defining δ t δu t, A.4 becomes v t ε j t Kj t + b t,τ ε j t v Ltε j t Lj t = u t R t K j t + Q t δ t K j t + B t,τ L j t + max{q t ε j t /P t, 0} u t R t K j t + η tk j t Lj t + Lj t+ R ft, where L j t+ /R ft is given by A.3. Matching coefficients yields: { v t ε j t = ut R t + Q t δ t + Q t ε j t /P t u t R t + η t + ξ t Q t if ε j t Pt Q t u t R t + Q t δ t otherwise, A.7 { b t,τ ε j t = Q t ε j t /P t B t,τ if ε j t Pt Q t B t,τ otherwise, A.8 and { v Lt ε j t = Qt ε j t /P t if ε j t Pt Q t otherwise. Using equation 4 in the paper, we then obtain 2 and 22 and 23 in the paper. Q.E.D. B Appendix: Stationary Equilibrium We define the following transformed variables: where Γ t = Z α α t C t C t Γ t, Ĩ t I t Z t Γ t, Ỹ t Y t Γ t, Kt K t Γ t Z t, P t s P t s, Ba Γ t Ba t, Xt X t, Wt W t, t Γ t Γ t Z t Γ t Q t Q t Z t, Pt = P t Z t, Rt = R t Z t, Λt Λ t Γ t, A t. The other variables are stationary and there is no need to scale them. To be consistent with a balanced growth path, we also assume that K 0t = Γ t Z t K 0, where K 0 is a constant. 3

4 The six shocks in the model are given by. The permanent TFP shock, A p t = Ap t λ at, ln λ at = ρ a ln λ a + ρ a ln λ a,t + ε at. B. 2. The transitory TFP shock, ln A m t = ρ a m ln A m t + ε a m,t. B.2 3. The IST shock, Z t = Z t λ zt, ln λ zt = ρ z ln λ z + ρ z ln λ z,t + ε zt. B.3 4. The sentiment shock, ln θ t = ρ θ θ + ρ θ ln θ t + ε θ,t. B.4 5. The labor shock, ln ψ t = ρ ψ ln ψ + ρψ ln ψ t + ε ψt. B.5 6. The financial shock, ln ζ t = ρ ζ ln ζ + ρζ ln ζ t + ε ζt. Here, all innovations are mutually independent and are independently and identically distributed normal random variables. Denote by g γt Γ t /Γ t the growth rate of Γ t. Denote by g γ the nonstochastic steady-state of g γt, satisfying ln g γ α α ln λ z + ln λ a. B.6 On the nonstochastic balanced growth path, investment and capital grow at the rate of λ I g γ λz ; consumption, output, wages, and bubbles grow at the rate of g γ ; and the rental rate of capital, Tobin s marginal Q, and the relative price of investment goods decrease at the rate λ z. After the transformation described in Section 3, we can derive a system of 5 equations for 5 transformed variables: { C t, Ĩt, Ỹt, N t, Kt, u t, Qt, Xt, Pt, Wt, R t, m t, Ba t, R ft, Λ t }.. Resource constraint: where g zt = Z t /Z t. C t Ω Ĩt g zt g γt 2 λ I Ĩ t = Ĩ Ỹt, t B.7 4

5 2. Aggregate Investment: Ĩ t = αỹt + ζ t Qt Xt + B Φ ε t a t, B.8 P t where ε t = P t / Q t. 3. Aggregate output: Ỹ t = u t Xt α N α t. B.9 4. Labor supply: 5. The law of motion for capital: α Ỹt N t Λt = ψ t. B.0 K t+ = δ t X Σ ε t + Ĩt t Φ ε B. t, where Σ ε t ε>ε t εdφ ε. 6. Capacity utilization: where G t = ε>ε t Ỹt α + G t = Q t δ u t, u t Xt ε/ε t dφ ε = Σ ε t ε t + Φ ε t. B.2 7. Marginal Q: Q t = β δ e E t Λt+ Λ t Qt+ g zt+ g γt+ [ ut+ δ u t+ + δ t+ + ζ t+ G t+ ]. B.3 8. Effective capital stock used in production: X t = δ e g zt g γt Kt + δ e K 0. B.4 9. Euler equation for investment goods producers: P t = + Ω 2 Ĩt Ĩ t g zt g γt λ I 2 + Ω Ĩt Λt+ Ĩt+ βe t Ω g zt+ g γt+ Λ λ I t Ĩ t Ĩ t g zt g γt λ I Ĩt+ Ĩ t Ĩt Ĩ t g zt g γt 2 g zt+g γt+. B.5 5

6 0. The wage rate:. The rental rate of capital: W t = α Ỹt N t. R t = αỹt u t Xt. B.6 B.7 2. Evolution of the number of bubbly firms: m t = m t δ e θ t + δ e ω. B.8 3. Evolution of the total value of the bubble: B a t = βe t Λt+ Λ t Ba t+ + G t+ δ e θ t m t m t+. B.9 4. The risk-free rate: R ft = βe t Λt+ Λ t g γt+ + G t+ δ e. B Marginal utility for consumption: Λ t = C t h C t /g γt βe t h C t+ g γt+ h C t. B.2 C Appendix: Steady State The transformed system presented in Appendix B has a nonstochastic steady state. We eliminate W t and R t and then obtain a system of 5 equations for 5 steady-state values: { C, Ĩ, Ỹ, N, K, u, Q, X, P, W, R, m, Ba, R f, Λ}, where we have removed time subscripts. We assume that the function δ is such that the steady-state capacity utilization rate is equal to. In addition, we set Q = which pins down G.. Resource constraint: C + Ĩ = Ỹ, C. where we have used the fact that λ I = λ z g γ. 2. Aggregate investment: Ĩ = αỹ + ζ Q X + B a Φ ε, C.2 P where Φ ε = ε>ε dφ ε, and ε = P / Q. 3. Aggregate output: Ỹ = X α N α. C.3 6

7 4. Labor supply: 5. End-of-period capital stock: α Ỹ N Λ = ψ. C.4 K = δ X + Ĩ Σ ε Φ ε, C.5 where 6. Capacity utilization: where 7. Marginal Q: G = Σ ε εdφ ε. ε>ε α Ỹ X + G = Qδ, ε>ε ε/ε dφ ε = Σ ε ε + Φ ε. [ = β δ e δ λ + δ + z g ζg ]. γ C.6 C.7 8. Effective capital stock used in production: X = δ e λ z g γ K + δe K 0. C.8 9. Euler equation for investment goods producers: P =. C.9 0. The wage rate:. The rental rate of capital: W = α Ỹ N. R = αỹ X. C.0 C. 2. Evolution of the number of bubbly firms: m = m δ e θ + δ e ω. C.2 3. Evolution of the total value of the bubble: B a = β B a + G δ e θ. C.3 7

8 4. The risk-free rate: R f = β g γ + G δ e. C.4 5. Marginal utility for consumption: Λ = C h C/g γ βh Cg γ h C. C.5 For convenience, define ε t = P t /Q t = P t / Q t as the investment threshold. We use a variable without the time subscript to denote its steady-state value in the transformed stationary system. The following proposition characterizes the bubbly steady state. Proposition: Suppose that ω > 0 and 0 < ε min < β δ e θ < β. Then there exists a unique steady-state threshold ε ε min, ε max satisfying If the parameter values are such that B a ε>ε ε/ε dφ ε =. C.6 β δ e θ Ỹ = [ϕ k δ] ϕ x / [ β δ e θ ] Φ ε α ζϕ x > 0, C.7 where we define ϕ x ϕ k δe K 0 + δ e, C.8 λ z g γ K α λ z g γ θ δ β δe θ ζ [ β δ e θ ], C.9 then there exists a unique bubbly steady-state equilibrium with the bubble-output ratio given in C.7. The steady-state growth rate of the bubble is given by θ = R f /g γ, where R f is the steadystate interest rate. In addition, if δ = α, β δ e θ ϕ x then the capacity utilization rate in this steady state is equal to. C.20 Proof: In the steady state, equation B.5 implies that P =. Hence by definition we have ε = / Q. Then by the evolution equation B.9 of the total bubble, we obtain the steady-state relation: β δ e θ = G = ε/ε dφ ε. ε>ε C.2 The bubbleless steady state can be obtained by setting B a = 0 and m = ω = 0. In this case we can remove equations C.3 and C.2. 8

9 Define the expression on the right-hand side of the last equality as a function of ε, G ε. Then we have Gε min = ε min and Gε max = 0. Given the assumption that ε min < β δ e θ, there is a unique solution ε to equation C.2 by the intermediate value theorem. In addition, by the definition of G, we have G = Σ ε ε [ Φ ε ], where Σ ε = ε>ε εdφ ε. Thus Σ ε can be expressed as Σ ε = [G + Φ ε ] ε. C.22 Suppose that the steady-state capacity utilization rate is equal to. The steady-state version of B.3 gives C.7 and the steady-state version of B.2 gives C.6. Using these two equations, we can derive α Ỹ X = Q [ + G g z g γ δ ζg β δ e Substituting equation C.2 into the above equation yields: ]. C.23 Q X Ỹ = ϕ x, C.24 where ϕ x is given by C.9. In order to support the steady-state u =, we use equation B.2 and C.24 to show that condition C.20 must be satisfied. From B.4, the end-of-period capital stock to the output ratio in the steady state satisfies K Ỹ = ϕ X k Ỹ, C.25 where ϕ k is given by C.8. Then from equation B., we can derive the steady-state relation: Ĩ Ỹ = Φ ε Σ ε = X [ϕ k δ ] Ỹ Φ ε [G + Φ ε ] [ϕ k δ ] = [ Φ ε ] [ϕ k δ ] ϕ x G + Φ ε, C.26 Q X Ỹ where the second line follows from C.22 and ε = / Q and the last line follows from C.24. After substituting C.2 into the above equation, we solve for Φ ε : Φ ε = / [ β δ e θ ] C.27 Ĩ/ Ỹ [ϕk δ ] ϕ x, 9

10 From B.8, the steady-state total value of bubble to GDP ratio is given by B a Ỹ = Ĩ Ỹ Q X Φ ε α ζ Ỹ. Substituting C.2, C.26 and C.24 into the above equation yields C.7. We require B a /Ỹ > 0. By 23 and 34 in the paper, the growth rate of bubbles of the surviving firms in the steady state is given by θ = R f /g γ. Q.E.D. D Appendix: Log-linearized System We eliminate equations for W t and R t. The log-linearized system for 3 variables { C t, Ĩt, Ỹt, N t, K t, u t, Qt, Xt, Pt, m t, Ba t, R ft, Λ t } including two growth rates are summarized as follows:. Resource constraint: 2. Aggregate investment: Ŷ t = C Ĉ t + Ĩ Î t. Ỹ Ỹ D. Î t = α ζϕ α + ζϕ x + B Ŷ t + x ˆζt a /Ỹ α + ζϕ x + B + ˆQ t + ˆX t a /Y B a /Ỹ + α + ζϕ x + B ˆB a t a µˆε t ˆP t, /Ỹ D.2 where 3. Aggregate output: µ = φ ε ε Φ ε, ˆε t = ˆP t ˆQ t. D.3 Ỹ t = α û t + ˆX t + α ˆN t. D.4 4. Labor supply: ˆΛ t + Ŷt ˆN t = ˆψ t. D.5 5. End of period the capital stock: ˆK t+ = δ û t + δ ˆXt + δ Î t µ ˆε t, D.6 ϕ k ϕ k ϕ k ϕ G where ϕ G Φ ε G. D.7 0

11 6. Capacity utilization: Ŷ t ˆX t + [ β δ e θ ] ϕ Gˆε t = ˆQ t + + δ δ û t. D.8 7. Marginal Q: ˆQ t = E t ˆΛt+ ˆΛ t + E t ˆQt+ ĝ zt+ ĝ γt+ + β δ eδ λ z g γ δ δ E tû t+ + ζβ δ e G λ z g γ E t ˆζt+ + ϕ Gˆε t+. D.9 8. Effective capital stock ˆX t = δ e λ z g γ ϕ k ˆKt ĝ zt ĝ γt. D.0 9. Euler equation for investment goods producers: ˆP t = E t [ + β Ωgγ 2 λ 2 zît + Ω λ 2 zgγ 2 ĝ γt + ĝ zt Ω λ 2 zgγît 2 D. βω λ 2 zgγ 2 Ît+ + ĝ zt+ + ĝ γt+ ]. 0. Evolution of the number of bubbly firms: ˆm t = δ e θ ˆm t + δ e θˆθ t. D.2. Evolution of the total value of the bubble: 2. The risk-free rate ˆB a t = E t ˆΛt+ ˆΛ t + ˆB a t+ + δ e θ E t ˆm t+. δ e θ + [ β δ e θ ] ϕ G E tˆε t+ D.3 ˆR ft = E t ˆΛt+ ˆΛ t ĝ γt+ + [ β δ e R f /g γ ] ϕ G E tˆε t+. D.4 3. Marginal utility for consumption: ˆΛ t = [ g γ g γ + h Ĉt ] ĝ γt g γ βh g γ hĉt g γ h βh [ g γ βh E t g γ Ĉt+ + ĝ γt+ + h ]. D.5 g γ h g γ hĉt

12 4. The growth rate of consumption goods ĝ γt = 5. The growth rate of the investment goods price: α α ˆλ zt + ˆλat + Âm t t Âm. D.6 ĝ zt = ˆλ zt. D.7 In the above system G is determined by C.3, G =, β δ e θ D.8 Φ ε is given by C.27, and δ satisfies C.20. The log-linearized shock processes are listed below.. The permanent technology shock: ˆλ at = ρ aˆλat + ε at. D.9 2. The transitory technology shock: Â m t = ρ a mâm t + ε a m,t. D The permanent investment-specific technology shock: ˆλ zt = ρ zˆλzt + ε zt. D.2 4. The labor supply shock: ˆψ t = ρ ψ ˆψt + ε ψt. D The financial shock: ˆζ t = ρ ζˆζt + ε ζt. D The sentiment shock: ˆθt = ρ θˆθt + ε θt. D.24 E Appendix: Business Cycle Moments To evaluate our model performance, we present in Table the baseline model s predictions regarding standard deviations, correlations with output, and serial correlations of output, consumption, 2

13 investment, hours, and stock prices. This table also presents results for four estimated comparison models discussed in the paper. The model moments are computed using the simulated data from the estimated model when all shocks are turned on. We take the posterior modes as parameter values. Both simulated and actual data are in logs and HP filtered. F Appendix: Robustness F. Extended Model with Consumer Sentiment Index Table 2 below reports the prior and posterior distributions of estimated parameters in the extended model of Section 5 in the paper with the consumer sentiment index as one of the observation series. The parameters {a j, b j } 5 j= in the table are coefficients in the equation for the sentiment shock and in the observation equation of the consumer sentiment index. The variable σ err represents the standard deviation of the measurement error. F.2 Priors In our baseline model of the paper, we choose 0% as the prior mean of σ θ because we know that the stock market volatility is very high. To see if our result is robust to a smaller prior mean of σ θ, we set the prior as Inv-Gamma with mean 0.0 and standard deviation infinite. We re-do Bayesian estimation and report estimation results in Table 3. We find that these results are very similar to those in the baseline estimation. F.3 A Hybrid Model Our baseline model has abstracted away from many other potentially important shocks such as news shocks or uncertainty shocks. Thus, it is possible that the sentiment shock is not important at all in explaining stock prices and real variables if other shocks are taken into account. To examine this possibility, we follow the methodology of Ireland 2004 and combine the DSGE model with the VAR model. 2 We then estimate this hybrid model using Bayesian methods. 3 Following Ireland 2004, we now shut down all the shocks in the baseline model except the sentiment shock, and introduce four measurement errors into the measurement equations for the data { P sdata t, Ct Data, It Data, ln N Data}. Specifically, let 2 Ireland, Perter N., 2004, A method for Taking Models to the Data, Journal of Economic Dynamics and Control 28, We thank Tao Zha for suggesting us to conduct this analysis. 3

14 Table. Business Cycles Statistics Y C I N SP P Standard Deviations % U.S. Data Baseline Model No Stock Price No Sentiment No Bubble Extended Standard Deviations Relative to Y U.S. Data Baseline Model No Stock Price No Sentiment No Bubble Extended First Order Autocorrelations U.S. Data Baseline Model No Stock Price No Sentiment No Bubble Extended Correlation with Y U.S. Data Baseline Model No Stock Price No Sentiment No Bubble Extended Note: The model moments are computed using the simulated data 20,000 periods from the estimated model at the posterior mode. All series are logged and detrended with the HP filter. The columns labeled Y, C, I, N, SP, and P refer, respectively, to output, consumption, investment, hours worked, the stock price, and the relative price of investment goods. No Bubble corresponds to the model without bubbles. No Sentiment corresponds to the baseline model without sentiment shocks. No Stock Price corresponds to the baseline model without using the stock price data in estimation. Extended corresponds to the model in Section 5 of the paper. 4

15 Table 2. Priors and posteriors of estimated parameters in the extended model Prior Distribution Posterior Distribution Parameter Distr. Mean St.Dev. Mode Mean 5% 95% h Beta Ω Gamma δ /δ Gamma ζ Beta µ Gamma f Gamma f 2 Gamma f 3 Gamma a Gamma a 2 Gamma a 3 Gamma a 4 Gamma a 5 Gamma b Gamma b 2 Gamma b 3 Gamma b 4 Gamma b 5 Gamma ρ a Beta ρ a m Beta ρ z Beta ρ θ Beta ρ ψ Beta ρ ζ Beta σ a % Inv-Gamma Inf σ a m % Inv-Gamma Inf σ z % Inv-Gamma Inf σ θ % Inv-Gamma Inf σ ψ % Inv-Gamma Inf σ ζ % Inv-Gamma Inf σ err % Inv-Gamma Inf

16 Table 3. Prior and posterior distributions of parameters Prior Distribution Posterior Distribution Parameter Distr. Mean St.Dev. Mode Mean 5% 95% h Beta Ω Gamma δ /δ Gamma ζ Beta µ Gamma f Gamma f 2 Gamma f 3 Gamma ρ a Beta ρ a m Beta ρ z Beta ρ θ Beta ρ ψ Beta ρ ζ Beta σ a % Inv-Gamma 0.0 Inf σ a m % Inv-Gamma 0.0 Inf σ z % Inv-Gamma 0.0 Inf σ θ % Inv-Gamma 0.0 Inf σ ψ % Inv-Gamma 0.0 Inf σ ζ % Inv-Gamma 0.0 Inf

17 P sdata t C Data t I Data t ln N Data = ˆP s t Ĉt Ît ˆN t + ln g γ ln g γ ln g γ ln N + ν t, F. where ν t is the vector contains four measurement errors, g γ is the gross growth rate of output, and N is the average hours in the data. Following Ireland 2004, we assume that the measurement errors ν t follow a VAR process: ν t = A ν t + Bˆε ν,t, F.2 where A is the coefficient matrix and B is assumed to be lower-triangular such that the innovations in ˆε ν,t are orthogonal to each other. The measurement errors in equation F.2 can be considered as a combination of all omitted structural shocks in our baseline model and allow for potential model misspecifications. We allow the measurement errors to be flexible enough so that the data are not necessarily driven by the sentiment shock. The idea is that, if the sentiment shock is not the driving force, then equations F. and F.2 form a first-order Bayesian VAR system and the measurement errors should be important in explaining fluctuations in the data of { P sdata t, C Data t, It Data, ln N Data}. On the other hand, if the baseline model is correctly specified and the sentiment shock is the main source of fluctuations, then the estimated measurement errors will be unimportant. The variance decomposition shows that the sentiment shock remains the single most important factor accounting for the stock price variation although its importance is somewhat reduced. It explains about 82 percent of the variation in the stock prices. It still accounts for significant fractions of fluctuations in investment, consumption and output, explaining about 26, 38, 35 percent, respectively. fluctuation in hours. As in the baseline model, the sentiment shock is not important in explaining the We also find that the estimates of the common parameters in the hybrid model are very similar to those in the baseline model. The smoothed sentiment shock is still highly correlated with the consumer sentiment data, the correlation is about These results suggest that the importance of the sentiment shock is robust to the model variation and specification of different shocks. 7