Estimating Macroeconomic Models: A Likelihood Approach

Transcription

1 Estimating Macroeconomic Models: A Likelihood Approach Jesús Fernández-Villaverde University of Pennsylvania, NBER, and CEPR Juan Rubio-Ramírez Federal Reserve Bank of Atlanta Estimating Dynamic Macroeconomic Models p. 1/45

2 Outline We are interested in performing likelihood-based inference in nonlinear and/or non-normal DSGE models. Estimating Dynamic Macroeconomic Models p. 2/45

3 Outline We are interested in performing likelihood-based inference in nonlinear and/or non-normal DSGE models. We apply particle filtering to evaluate the likelihood of the model. Estimating Dynamic Macroeconomic Models p. 2/45

4 Outline We are interested in performing likelihood-based inference in nonlinear and/or non-normal DSGE models. We apply particle filtering to evaluate the likelihood of the model. We estimate a neoclassical business cycle model with investment-specific technological change and stochastic volatility. Estimating Dynamic Macroeconomic Models p. 2/45

5 What is the Particle Filter? The particle filter is a monte carlo algorithm that tracks the distribution of states conditional on a sequence of observables. Estimating Dynamic Macroeconomic Models p. 3/45

6 What is the Particle Filter? The particle filter is a monte carlo algorithm that tracks the distribution of states conditional on a sequence of observables. Original idea seems to be by Handschin and Mayne (1969). Estimating Dynamic Macroeconomic Models p. 3/45

7 What is the Particle Filter? The particle filter is a monte carlo algorithm that tracks the distribution of states conditional on a sequence of observables. Original idea seems to be by Handschin and Mayne (1969). Applied in financial econometrics by Shephard and coauthors. Estimating Dynamic Macroeconomic Models p. 3/45

8 What is the Particle Filter? The particle filter is a monte carlo algorithm that tracks the distribution of states conditional on a sequence of observables. Original idea seems to be by Handschin and Mayne (1969). Applied in financial econometrics by Shephard and coauthors. Reviewed on Doucet, De Freitas, and Gordon (2001). Estimating Dynamic Macroeconomic Models p. 3/45

9 What is the Particle Filter? The particle filter is a monte carlo algorithm that tracks the distribution of states conditional on a sequence of observables. Original idea seems to be by Handschin and Mayne (1969). Applied in financial econometrics by Shephard and coauthors. Reviewed on Doucet, De Freitas, and Gordon (2001). We modify the filter to be more flexible with shocks. Estimating Dynamic Macroeconomic Models p. 3/45

10 Why Are Nonlinearities Important? Estimating Dynamic Macroeconomic Models p. 4/45

11 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Estimating Dynamic Macroeconomic Models p. 4/45

12 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Estimating Dynamic Macroeconomic Models p. 4/45

13 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Linearization eliminates asymmetries, threshold effects, precautionary behavior, big shocks,... Estimating Dynamic Macroeconomic Models p. 4/45

14 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Linearization eliminates asymmetries, threshold effects, precautionary behavior, big shocks,... Moreover, linearization induces an approximation error. Estimating Dynamic Macroeconomic Models p. 4/45

15 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Linearization eliminates asymmetries, threshold effects, precautionary behavior, big shocks,... Moreover, linearization induces an approximation error. This is worse than you may think: Estimating Dynamic Macroeconomic Models p. 4/45

16 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Linearization eliminates asymmetries, threshold effects, precautionary behavior, big shocks,... Moreover, linearization induces an approximation error. This is worse than you may think: 1. Theoretical arguments. Estimating Dynamic Macroeconomic Models p. 4/45

17 Why Are Nonlinearities Important? Most DSGE models are nonlinear. Common practice: estimate a linearized version. Linearization eliminates asymmetries, threshold effects, precautionary behavior, big shocks,... Moreover, linearization induces an approximation error. This is worse than you may think: 1. Theoretical arguments. 2. Computational evidence. Estimating Dynamic Macroeconomic Models p. 4/45

18 Theoretical Arguments Estimating Dynamic Macroeconomic Models p. 5/45

19 Theoretical Arguments We develop these arguments in "Convergence Properties of the Likelihood of Computed Dynamic Models" (2006) and in some work-in-progress. Estimating Dynamic Macroeconomic Models p. 5/45

20 Theoretical Arguments We develop these arguments in "Convergence Properties of the Likelihood of Computed Dynamic Models" (2006) and in some work-in-progress. 1. Second-order errors in the approximated policy function induce first-order errors in the likelihood function. Estimating Dynamic Macroeconomic Models p. 5/45

21 Theoretical Arguments We develop these arguments in "Convergence Properties of the Likelihood of Computed Dynamic Models" (2006) and in some work-in-progress. 1. Second-order errors in the approximated policy function induce first-order errors in the likelihood function. 2. As the sample size grows, the error in the likelihood function also grows and we may have inconsistent point estimates. Estimating Dynamic Macroeconomic Models p. 5/45

22 Theoretical Arguments We develop these arguments in "Convergence Properties of the Likelihood of Computed Dynamic Models" (2006) and in some work-in-progress. 1. Second-order errors in the approximated policy function induce first-order errors in the likelihood function. 2. As the sample size grows, the error in the likelihood function also grows and we may have inconsistent point estimates. 3. Linearization complicates the identification of parameters. Estimating Dynamic Macroeconomic Models p. 5/45

23 Computational Evidence Estimating Dynamic Macroeconomic Models p. 6/45

24 Computational Evidence We build the computational evidence in several papers. Estimating Dynamic Macroeconomic Models p. 6/45

25 Computational Evidence We build the computational evidence in several papers. Several researchers (Ann, King, Schorfheide, Winschel) have gathered similar evidence after our paper first circulated. Estimating Dynamic Macroeconomic Models p. 6/45

26 Computational Evidence We build the computational evidence in several papers. Several researchers (Ann, King, Schorfheide, Winschel) have gathered similar evidence after our paper first circulated. 1. Big differences in the level of the likelihood. Estimating Dynamic Macroeconomic Models p. 6/45

27 Computational Evidence We build the computational evidence in several papers. Several researchers (Ann, King, Schorfheide, Winschel) have gathered similar evidence after our paper first circulated. 1. Big differences in the level of the likelihood. 2. Often, important differences in point estimates. Estimating Dynamic Macroeconomic Models p. 6/45

28 Computational Evidence We build the computational evidence in several papers. Several researchers (Ann, King, Schorfheide, Winschel) have gathered similar evidence after our paper first circulated. 1. Big differences in the level of the likelihood. 2. Often, important differences in point estimates. 3. Big differences in smoothed shocks and states. Estimating Dynamic Macroeconomic Models p. 6/45

29 Computational Evidence We build the computational evidence in several papers. Several researchers (Ann, King, Schorfheide, Winschel) have gathered similar evidence after our paper first circulated. 1. Big differences in the level of the likelihood. 2. Often, important differences in point estimates. 3. Big differences in smoothed shocks and states. 4. Better identification of parameters. Estimating Dynamic Macroeconomic Models p. 6/45

30 Why Are Non-normalities Important? Estimating Dynamic Macroeconomic Models p. 7/45

31 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Estimating Dynamic Macroeconomic Models p. 7/45

32 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. Estimating Dynamic Macroeconomic Models p. 7/45

33 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Estimating Dynamic Macroeconomic Models p. 7/45

34 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Evidence by Geweke (1993 and 1994). Estimating Dynamic Macroeconomic Models p. 7/45

35 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Evidence by Geweke (1993 and 1994). Recent discussion about the Great Moderation: Estimating Dynamic Macroeconomic Models p. 7/45

36 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Evidence by Geweke (1993 and 1994). Recent discussion about the Great Moderation: 1. Kim and Nelson (1999), McConell and Pérez-Quirós (2000), and Stock and Watson (2002). Estimating Dynamic Macroeconomic Models p. 7/45

37 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Evidence by Geweke (1993 and 1994). Recent discussion about the Great Moderation: 1. Kim and Nelson (1999), McConell and Pérez-Quirós (2000), and Stock and Watson (2002). 2. Good luck? Sims and Zha (2005). Estimating Dynamic Macroeconomic Models p. 7/45

38 Why Are Non-normalities Important? Evidence of time-varying volatility in time series. Fundamental issue in financial econometrics. However, macro has lagged despite Engle (1982). Evidence by Geweke (1993 and 1994). Recent discussion about the Great Moderation: 1. Kim and Nelson (1999), McConell and Pérez-Quirós (2000), and Stock and Watson (2002). 2. Good luck? Sims and Zha (2005). 3. Or good policy? Clarida, Galí, and Gertler (2000). Estimating Dynamic Macroeconomic Models p. 7/45

39 State Space Representation of the Model How do we evaluate the likelihood p ( y T ;γ ) of a DSGE model? Estimating Dynamic Macroeconomic Models p. 8/45

40 State Space Representation of the Model How do we evaluate the likelihood p ( y T ;γ ) of a DSGE model? Transition equation: S t = f (S t 1, W t ;γ) p (S t S t 1 ;γ) Estimating Dynamic Macroeconomic Models p. 8/45

41 State Space Representation of the Model How do we evaluate the likelihood p ( y T ;γ ) of a DSGE model? Transition equation: S t = f (S t 1, W t ;γ) p (S t S t 1 ;γ) Measurement equation: Y t = g (S t, V t ;γ) p (Y t S t ;γ) Estimating Dynamic Macroeconomic Models p. 8/45

42 State Space Representation of the Model How do we evaluate the likelihood p ( y T ;γ ) of a DSGE model? Transition equation: S t = f (S t 1, W t ;γ) p (S t S t 1 ;γ) Measurement equation: Y t = g (S t, V t ;γ) p (Y t S t ;γ) We want to track conditional density p ( S t y t 1 ;γ ). Estimating Dynamic Macroeconomic Models p. 8/45

43 Factorization of the Likelihood Why? p ( y T ;γ ) = = T p ( y t y t 1 ;γ ) t=1 T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t Estimating Dynamic Macroeconomic Models p. 9/45

44 Factorization of the Likelihood Why? p ( y T ;γ ) = = T p ( y t y t 1 ;γ ) t=1 T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t Knowledge of { p ( S t y t 1 ;γ )} T t=1 of the likelihood of the model. allows the evaluation Estimating Dynamic Macroeconomic Models p. 9/45

45 Tracking the Conditional Distribution of States S t Filtering problem: forecast and update. Estimating Dynamic Macroeconomic Models p. 10/45

46 Tracking the Conditional Distribution of States S t Filtering problem: forecast and update. 1. Forecast: Chapman-Kolmogorov equation p ( S t y t 1 ;γ ) = p (S t S t 1 ;γ)p ( S t 1 y t 1 ;γ ) ds t 1 Estimating Dynamic Macroeconomic Models p. 10/45

47 Tracking the Conditional Distribution of States S t Filtering problem: forecast and update. 1. Forecast: Chapman-Kolmogorov equation p ( S t y t 1 ;γ ) = p (S t S t 1 ;γ)p ( S t 1 y t 1 ;γ ) ds t 1 2. Update: Bayes theorem p ( S t y t ;γ ) = p (y t S t ;γ)p ( S t y t 1 ;γ ) p (y t y t 1 ;γ) where: p ( y t y t 1 ;γ ) = p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t Estimating Dynamic Macroeconomic Models p. 10/45

48 Intuition Previous equations involve complicated integrals. Estimating Dynamic Macroeconomic Models p. 11/45

49 Intuition Previous equations involve complicated integrals. In a linear and normal world: mean and variance are sufficient statistics translation and spread. Estimating Dynamic Macroeconomic Models p. 11/45

50 Intuition Previous equations involve complicated integrals. In a linear and normal world: mean and variance are sufficient statistics translation and spread. This can be done by the Kalman filter. Estimating Dynamic Macroeconomic Models p. 11/45

51 Intuition Previous equations involve complicated integrals. In a linear and normal world: mean and variance are sufficient statistics translation and spread. This can be done by the Kalman filter. In a nonlinear and/or non-normal world, we need to carry the distribution deformation. Estimating Dynamic Macroeconomic Models p. 11/45

52 Intuition Previous equations involve complicated integrals. In a linear and normal world: mean and variance are sufficient statistics translation and spread. This can be done by the Kalman filter. In a nonlinear and/or non-normal world, we need to carry the distribution deformation. This can be done by the Particle filter. Estimating Dynamic Macroeconomic Models p. 11/45

53 Intuition Previous equations involve complicated integrals. In a linear and normal world: mean and variance are sufficient statistics translation and spread. This can be done by the Kalman filter. In a nonlinear and/or non-normal world, we need to carry the distribution deformation. This can be done by the Particle filter. Alternatives? Unscented Kalman filter, Grid Filter,... Estimating Dynamic Macroeconomic Models p. 11/45

54 A Law of Large Numbers T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t Estimating Dynamic Macroeconomic Models p. 12/45

55 A Law of Large Numbers T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t { { } } N T Suppose we have s i t t 1 i=1 t=1 { p ( S t y t 1 ;γ )} T t=1. Estimating Dynamic Macroeconomic Models p. 12/45

56 A Law of Large Numbers T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t { { } } N T Suppose we have s i t t 1 i=1 t=1 Then: p ( y T ;γ ) T t=1 1 N { p ( S t y t 1 ;γ )} T t=1. N ) p (y t s i t t 1 ;γ i=1 Estimating Dynamic Macroeconomic Models p. 12/45

57 A Law of Large Numbers T t=1 p (y t S t ;γ)p ( S t y t 1 ;γ ) ds t { { } } N T Suppose we have s i t t 1 i=1 t=1 Then: p ( y T ;γ ) T t=1 1 N { p ( S t y t 1 ;γ )} T t=1. N ) p (y t s i t t 1 ;γ Evaluating the likelihood function Drawing from density: i=1 { p ( St y t 1 ;γ )} T t=1 Estimating Dynamic Macroeconomic Models p. 12/45

58 Introducing Particles { } N s i t 1 t 1 N i.i.d. draws from p ( S t 1 y t 1 ;γ ). i=1 Estimating Dynamic Macroeconomic Models p. 13/45

59 Introducing Particles { } N s i t 1 t 1 N i.i.d. draws from p ( S t 1 y t 1 ;γ ). i=1 { } N Each s i t 1 t 1 is a particle and s i t 1 t 1 particles. i=1 a swarm of Estimating Dynamic Macroeconomic Models p. 13/45

60 Introducing Particles { } N s i t 1 t 1 N i.i.d. draws from p ( S t 1 y t 1 ;γ ). i=1 { } N Each s i t 1 t 1 is a particle and s i t 1 t 1 i=1 particles. { } N s i t t 1 N i.i.d. draws from p ( S t y t 1 ;γ ). i=1 a swarm of Estimating Dynamic Macroeconomic Models p. 13/45

61 Introducing Particles { } N s i t 1 t 1 N i.i.d. draws from p ( S t 1 y t 1 ;γ ). i=1 { } N Each s i t 1 t 1 is a particle and s i t 1 t 1 i=1 particles. { } N s i t t 1 N i.i.d. draws from p ( S t y t 1 ;γ ). i=1 { } N Each s i t t 1 is a proposed particle and s i t t 1 of proposed particles. a swarm of i=1 a swarm Estimating Dynamic Macroeconomic Models p. 13/45

62 Introducing Particles { } N s i t 1 t 1 N i.i.d. draws from p ( S t 1 y t 1 ;γ ). i=1 { } N Each s i t 1 t 1 is a particle and s i t 1 t 1 i=1 particles. { } N s i t t 1 N i.i.d. draws from p ( S t y t 1 ;γ ). i=1 { } N Each s i t t 1 is a proposed particle and s i t t 1 of proposed particles. Weight of each proposed particle: q i t = ) p (y t s i t t 1 ;γ ) N i=1 (y p t s i t t 1 ;γ a swarm of i=1 a swarm Estimating Dynamic Macroeconomic Models p. 13/45

63 A Proposition Let { } N s i t t 1 be a draw from p ( S t y t 1 ;γ ). i=1 Estimating Dynamic Macroeconomic Models p. 14/45

64 A Proposition Let { } N s i t t 1 be a draw from p ( S t y t 1 ;γ ). i=1 Let { s i } { } N N t i=1 be a draw with replacement from s i t t 1 i=1 and probabilities qt. i Estimating Dynamic Macroeconomic Models p. 14/45

65 A Proposition Let { } N s i t t 1 be a draw from p ( S t y t 1 ;γ ). i=1 Let { s i } { } N N t i=1 be a draw with replacement from s i t t 1 i=1 and probabilities qt. i Then, { s i t} N i=1 is a draw from p ( S t y t ;γ ) : { } N s i t t = { s i N t} i=1 i=1 Estimating Dynamic Macroeconomic Models p. 14/45

66 A Proposition Let { } N s i t t 1 be a draw from p ( S t y t 1 ;γ ). i=1 Let { s i } { } N N t i=1 be a draw with replacement from s i t t 1 i=1 and probabilities qt. i Then, { s i t} N i=1 is a draw from p ( S t y t ;γ ) : { } N s i t t = { s i N t} i=1 i=1 Proof: Importance sampling and Bayes theorem. Estimating Dynamic Macroeconomic Models p. 14/45

67 Importance of the Proposition Estimating Dynamic Macroeconomic Models p. 15/45

68 Importance of the Proposition { } N 1. Update: We can use a draw s i t t 1 from p ( S t y t 1 ;γ ) i=1 { } N to get a draw s i t t from p ( S t y t ;γ ). i=1 Estimating Dynamic Macroeconomic Models p. 15/45

69 Importance of the Proposition { } N 1. Update: We can use a draw s i t t 1 from p ( S t y t 1 ;γ ) i=1 { } N to get a draw s i t t from p ( S t y t ;γ ). i=1 { } N 2. Forecast: We can use a draw s i t t from p ( S t y t ;γ ), a i=1 draw from p (W t+1 ;γ), and S t+1 = f (S t, W t+1 ;γ) to get a { } N draw s i t+1 t i=1. Estimating Dynamic Macroeconomic Models p. 15/45

70 Particle Filtering I Estimating Dynamic Macroeconomic Models p. 16/45

71 Particle Filtering I { } N Step 0, Initialization: Sample N values s i 1 0 from i=1 p (S 1 ;γ). Go to step 2. Estimating Dynamic Macroeconomic Models p. 16/45

72 Particle Filtering I { } N Step 0, Initialization: Sample N values s i 1 0 from i=1 p (S 1 ;γ). Go to step 2. Step 1, Forecast: Sample N values p (W t ;γ) and S t = f (S t 1, W t ;γ). { } N s i t t 1 i=1 with Estimating Dynamic Macroeconomic Models p. 16/45

73 Particle Filtering I { } N Step 0, Initialization: Sample N values s i 1 0 from i=1 p (S 1 ;γ). Go to step 2. Step 1, Forecast: Sample N values p (W t ;γ) and S t = f (S t 1, W t ;γ). { } N s i t t 1 i=1 with Step 2, Weighting: Assign to each draw s i t t 1 the weight q i t. Estimating Dynamic Macroeconomic Models p. 16/45

74 Particle Filtering I { } N Step 0, Initialization: Sample N values s i 1 0 from i=1 p (S 1 ;γ). Go to step 2. Step 1, Forecast: Sample N values p (W t ;γ) and S t = f (S t 1, W t ;γ). { } N s i t t 1 i=1 with Step 2, Weighting: Assign to each draw s i t t 1 the weight q i t. { } N Step 3, Update: Draw s i t t with replacement from i=1 { } N s i t t 1 with probabilities { q i } N t. If t < T set i=1 i=1 t t + 1 and go to step 1. Otherwise stop. Estimating Dynamic Macroeconomic Models p. 16/45

75 Particle Filtering II Use { { } } N T s i t t 1 i=1 t=1 to compute: p ( y T ;γ ) T t=1 1 N N ) p (y t s i t t 1 ;γ i=1 We can filter, forecast, and smooth Estimating Dynamic Macroeconomic Models p. 17/45

76 An Application: a Business Cycle Model Estimating Dynamic Macroeconomic Models p. 18/45

77 An Application: a Business Cycle Model I want to show the power of the particle filter. Estimating Dynamic Macroeconomic Models p. 18/45

78 An Application: a Business Cycle Model I want to show the power of the particle filter. And learn something about the evolution of the U.S. aggregate fluctuations over the last decades. Estimating Dynamic Macroeconomic Models p. 18/45

79 An Application: a Business Cycle Model I want to show the power of the particle filter. And learn something about the evolution of the U.S. aggregate fluctuations over the last decades. A business cycle model with: Estimating Dynamic Macroeconomic Models p. 18/45

80 An Application: a Business Cycle Model I want to show the power of the particle filter. And learn something about the evolution of the U.S. aggregate fluctuations over the last decades. A business cycle model with: 1. Investment-specific technological change. Greenwood, Herkowitz, and Krusell (1997 and 2000) Estimating Dynamic Macroeconomic Models p. 18/45

81 0.03 A Look at the Data Per Capita Output Growth Q/Q Per Capita Investment Growth Q/Q Per Capita Log Hours Worked 0.04 Relative Price of Investment Growth Q/Q Estimating Dynamic Macroeconomic Models p. 19/45

82 An Application: a Business Cycle Model I want to show the power of the particle filter. And learn something about the evolution of the U.S. aggregate fluctuations over the last decades. A business cycle model with: 1. Investment-specific technological change. Greenwood, Herkowitz, and Krusell (1997 and 2000) 2. Stochastic volatility. Estimating Dynamic Macroeconomic Models p. 20/45

83 2.5 Another x 10-3 look at the Data: The Variance Output Investment Labor Price Estimating Dynamic Macroeconomic Models p. 21/45

84 Households Representative household with utility function: E 0 t=0 ( ) β t e d t log C t + ψ log (1 L t ) Law of motion of d t, the preference shock: d t = ρd t 1 + σ dt ε dt, ε dt N (0,1) We will explain later the law of motion of σ dt. Estimating Dynamic Macroeconomic Models p. 22/45

85 Technology Final Good: C t + X t = A t K α t L 1 α t Law of motion of capital: K t+1 = (1 δ) K t + V t X t Shocks: log A t = ζ + log A t 1 + σ at ε at, ζ 0 and ε at N (0,1) log V t = υ + log V t 1 + σ vt ε υt, υ 0 and ε υt N (0,1) Estimating Dynamic Macroeconomic Models p. 23/45

86 Stochastic Volatility We follow a standard specification: log σ dt = (1 λ d )log σ d + λ d log σ dt 1 + τ d η dt and η dt N (0,1) log σ at = (1 λ a ) log σ a + λ a log σ at 1 + τ a η at and η at N (0,1) log σ υt = (1 λ υ ) log σ υ + λ υ log σ υt 1 + τ υ η υt and η υt N (0,1) Estimating Dynamic Macroeconomic Models p. 24/45

87 Analysis of the Model It can be shown that along a balanced growth path the following variables are stationary: Y t /Z t, C t /Z t, X t /Z t, K t+1 /(Z t V t ), (Y t /L t )/Z t, and L t where Z t = A 1/(1 α) t V α/(1 α) t. Estimating Dynamic Macroeconomic Models p. 25/45

88 Analysis of the Model It can be shown that along a balanced growth path the following variables are stationary: Y t /Z t, C t /Z t, X t /Z t, K t+1 /(Z t V t ), (Y t /L t )/Z t, and L t where Z t = A 1/(1 α) t V α/(1 α) t. The growth rates for the exogenous shocks are: log A t log A t 1 log V t log V t 1 = ζ + σ at ε at = υ + σ vt ε υt Estimating Dynamic Macroeconomic Models p. 25/45

89 Analysis of the Model It can be shown that along a balanced growth path the following variables are stationary: Y t /Z t, C t /Z t, X t /Z t, K t+1 /(Z t V t ), (Y t /L t )/Z t, and L t where Z t = A 1/(1 α) t V α/(1 α) t. The growth rates for the exogenous shocks are: log A t log A t 1 log V t log V t 1 = ζ + σ at ε at = υ + σ vt ε υt Therefore, Y t, C t, X t, and Y t /L t grow at rate (ζ + αυ)/(1 α), K t grows at rate (ζ + υ)/(1 α), and L t is stationary. Estimating Dynamic Macroeconomic Models p. 25/45

90 Transforming the Model The model is nonstationary because of the presence of two unit roots, one in each technological process. We need to transform the model into a stationary problem. We scale variables as C t = C t Z t, X t = X t Z t, and K t = Then: K t Z t V t 1. E 0 t=0 ( β t e d t log C ) t + ψ log(1 L t ) C t + e γ+αυ+σ at ε at +ασ υt ε υt 1 α Kt+1 = e γ+σ atε at Kα t L 1 α t + (1 δ) e υ σ υtε υt Kt Estimating Dynamic Macroeconomic Models p. 26/45

91 Equilibrium Conditions 1. Euler equation: e d t e γ+αυ+ε at +ασ υt ε υt 1 α = C t βe t e d t+1 C t+1 ( αe γ+σ at+1ε at+1 Kα t+1 L 1 α t+1 + (1 δ) e υ σ υt+1ε υt+1 ) 2. A labor supply condition: 3. The resource constraint: ψ ed t Ct = (1 α) e γ+σ atε at Kα 1 L t L α t t C t +e γ+αυ+σ at ε at +ασ υt ε υt 1 α Kt+1 = e γ+σ atε at Kα t L 1 α t +(1 δ) e υ σ υtε υt Kt Estimating Dynamic Macroeconomic Models p. 27/45

92 The Steady State We can find the steady state of the transformed model. We have a cointegration relation between output and investment in nominal terms: ) (e ζ+αυ 1 α (1 δ) e υ X ss Ỹ ss = α exp( ζ+αυ 1 α ) β (1 δ) exp ( υ) Let log K t+1 = log K t+1 K ss and log X t = log X t X ss Estimating Dynamic Macroeconomic Models p. 28/45

93 Solution We solve the model using 2nd order perturbation. Other non-linear solution methods are possible. Aruoba, Fernández-Villaverde, and Rubio-Ramírez (2005). Structure of a 2nd order approximation: log K t+1 = Ψ k1 s t s tψ k2 s t log X t = Ψ x1 s t s tψ x2 s t log L t = Ψ l1 s t s tψ l2 s t where: s t = (1,log K t, d t, σ at ε at, σ υt ε υt,log σ dt,log σ at,log σ υt ) Estimating Dynamic Macroeconomic Models p. 29/45

94 State Space Representation I: Transition Equation f 1 (S t, W t ) = 1 f 2 (S t, W t ) = Ψ k1 s t s tψ k2 s t f 3 (S t, W t ) = ((1 λ a ) log σ a + λ a log σ at + τ a η at+1 ) ε at+1 f 4 (S t, W t ) = ((1 λ υ ) log σ υ + λ υ log σ υt + τ υ η υt+1 ) ε υt+1 f 5 (S t, W t ) = ρd t + e (1 λ d) log σ d +λ d log σ dt +τ d η dt+1 ε dt+1 f 6 (S t, W t ) = (1 λ a )log σ a + λ a log σ at + τ a η at+1 f 7 (S t, W t ) = (1 λ υ )log σ υ + λ υ log σ υt + τ υ η υt+1 f 8 (S t, W t ) = (1 λ d ) log σ d + λ d log σ dt + τ d η dt+1 f 9 16 (S t, W t ) = s t where S t = (s t, s t 1 ). Estimating Dynamic Macroeconomic Models p. 30/45

95 State Space Representation II: Measurement Equation log P t log Y t log X t log L t = υ γ+αυ 1 α γ+αυ 1 α log L ss + Ψ l3 ε υt + Ψ y1 (s t s t 1 ) + 1 ( 2 s t Ψ y2 s t s t 1 Ψ ) y2s t 1 + σ at 1 ε at 1 +ασ υt 1 ε υt 1 1 α Ψ x1 (s t s t 1 ) + 1 ( 2 s t Ψ x2 s t 1 2 s t 1 Ψ ) x2s t 1 + σ at 1 ε at 1 +ασ υt 1 ε υt 1 1 α Ψ l1 s t s tψ l2 s t + 0 ǫ 1t ǫ 2t ǫ 3t Estimating Dynamic Macroeconomic Models p. 31/45

96 Performing Likelihood-Based Inference Time series: 1. Relative price of capital, output, investment, and hours. 2. Sample: 1955:Q1 to 2000:Q4. Vector of parameters γ is: (ρ, β, ψ, α, δ, υ, ζ, τ d, τ a, τ υ,σ d,σ a,σ υ, λ a, λ υ, λ d, σ ǫ 1, σǫ 2, σǫ 3 ) Use a Random-walk Metropolis-Hastings to explore the likelihood: Classical and Bayesian. Estimating Dynamic Macroeconomic Models p. 32/45

97 MLE Table 5.1: Maximum Likelihood Estimates Parameter PointEstimate StandardError(x10 3 ) E E a d a d a d 1² 2² 3² 7.120E E E E E E E E E E Estimating Dynamic Macroeconomic Models p. 33/45

98 Figure 6.1: Model versus Data 0.03 Fit of the Data 0.02 Output Investment Model Data Hours 0.04 Relative Price of Investment Estimating Dynamic Macroeconomic Models p. 34/45

99 Figure 6.2: Smoothed Capital and Shocks 0.7 Shocks Mean of (+/- std) k t 1.5 x Mean of (+/- std) 10-3 a,t a,t Mean of (+/- std),t,t Mean of (+/- std) d t Estimating Dynamic Macroeconomic Models p. 35/45

100 Figure 6.3: Smoothed Volatilities -7.5 Volatility I Mean of (+/- std) a,t Mean of (+/- std),t Mean of (+/- std) d,t Estimating Dynamic Macroeconomic Models p. 36/45

101 Volatility II 10 x 10-3 Mean of (+/- std) Output 9 Figure 6.4: Instantaneous Standard Deviation Mean of (+/- std) Investment Mean of (+/- std) Hours 10 x 10-3 Mean of (+/- std) Relative Price of Investment Estimating Dynamic Macroeconomic Models p. 37/45

102 Counterfactuals I 10 x 10-3 Output Actual Counterfactual Figure 6.5: Counterfactual Exercise 1 10 x 10-3 Output x 10-3 Output Investment 0.05 Investment 0.05 Investment Hours Hours Hours x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment Estimating Dynamic Macroeconomic Models p. 38/45

103 Counterfactuals II 10 x 10-3 Output Actual Counterfactual Figure 6.6: Counterfactual Exercise 2 10 x 10-3 Output x 10-3 Output Investment 0.05 Investment 0.05 Investment Hours Hours Hours x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment Estimating Dynamic Macroeconomic Models p. 39/45

104 Counterfactuals III 10 x 10-3 Output Actual Counterfactual Figure 6.7: Counterfactual Exercise 3 10 x 10-3 Output x 10-3 Output Investment 0.05 Investment 0.05 Investment Hours Hours Hours x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment 10 x 10-3 Relative Price of Investment Estimating Dynamic Macroeconomic Models p. 40/45

105 Are Nonlinearities and Non-normalities Important? Estimating Dynamic Macroeconomic Models p. 41/45

106 Are Nonlinearities and Non-normalities Important? We estimate four version of the model: Table 7.1: Versions of the Model Solution No Stochastic Volatility Stochastic Volatility Linear Version 1 Version 2 Quadratic Version 3 Benchmark Estimating Dynamic Macroeconomic Models p. 41/45

107 Are Nonlinearities and Non-normalities Important? We estimate four version of the model: Table 7.1: Versions of the Model Solution No Stochastic Volatility Stochastic Volatility Linear Version 1 Version 2 Quadratic Version 3 Benchmark We use Likelihood Ratio tests to compare models, Rivers and Vuong (2002). Estimating Dynamic Macroeconomic Models p. 41/45

108 Are Nonlinearities and Non-normalities Important? We estimate four version of the model: Table 7.1: Versions of the Model Solution No Stochastic Volatility Stochastic Volatility Linear Version 1 Version 2 Quadratic Version 3 Benchmark We use Likelihood Ratio tests to compare models, Rivers and Vuong (2002). Loglike benchmark: , loglike version 2: Estimating Dynamic Macroeconomic Models p. 41/45

109 Figure 7.1: Comparison of Smoothed Capital and Shocks 0.6 Comparison with Linear Model linear I Mean of (+/- std) k t quadratic 1 x Mean of (+/- std) 10-3 a,t a,t Mean of (+/- std),t,t Mean of (+/- std) d t Estimating Dynamic Macroeconomic Models p. 42/45

110 Figure 7.2: Comparison of Smoothed Volatilities -6 Comparison with Linear Model linear II Mean of (+/- std) a,t quadratic Mean of (+/- std),t Mean of (+/- std) d,t Estimating Dynamic Macroeconomic Models p. 43/45

111 What are We Doing Now? We are estimating a richer DSGE model with: 1. Nominal and real rigidities. 2. Monetary and fiscal policy. 3. Stochastic volatility. 4. Parameter drifting. Estimating Dynamic Macroeconomic Models p. 44/45

112 What are We Doing Now? We are estimating a richer DSGE model with: 1. Nominal and real rigidities. 2. Monetary and fiscal policy. 3. Stochastic volatility. 4. Parameter drifting. We are working on a model with micro heterogeneity. Estimating Dynamic Macroeconomic Models p. 44/45

113 What are We Doing Now? We are estimating a richer DSGE model with: 1. Nominal and real rigidities. 2. Monetary and fiscal policy. 3. Stochastic volatility. 4. Parameter drifting. We are working on a model with micro heterogeneity. We are exploring the semi-nonparametric estimation of DSGE models. Estimating Dynamic Macroeconomic Models p. 44/45

114 Conclusions 1. Particle filtering is a general purpose and efficient method to estimate DSGE models. 2. We learned about the importance of stochastic volatility to account for U.S. Business Cycle. 3. Much exciting work to do in the next few years! Estimating Dynamic Macroeconomic Models p. 45/45