State Space Models and Filtering. Jesús Fernández-Villaverde University of Pennsylvania

Transcription

1 State Space Models and Filtering Jesús Fernández-Villaverde University of Pennsylvania

2 State Space Form What is a state space representation? States versus observables. Why is it useful? Relation with filtering. Relation with optimal control. Linear versus nonlinear, Gaussian versus nongaussian. 2

3 State Space Representation Let the following system: Transition equation x t+ = Fx t + Gω t+, ω t+ N (,Q) Measurement equation z t = H x t + υ t, υ t N (,R) wherex t are the states and z t are the observables. Assume we want to write the likelihood function of z T = {z t } T t=. 3

4 The State Space Representation is Not Unique Take the previous state space representation. Let B be a non-singular squared matrix conforming with F. Then, if x t = Bx t, F = BFB, G = BG, andh = H B,we can write a new, equivalent, representation: Transition equation Measurement equation x t+ = F x t + G ω t+, ω t+ N (,Q) z t = H x t + υ t, υ t N (,R) 4

5 Example I Assume the following AR(2) process: z t = ρ z t + ρ 2 z t 2 + υ t, υ t N ³, σ 2 υ Model is not apparently not Markovian. Can we write this model in different state space forms? Yes! 5

6 State Space Representation I Transition equation: x t = " ρ ρ 2 # x t + " # υ t where x t = h y t ρ 2 y t i Measurement equation: z t = h i x t 6

7 State Space Representation II Transition equation: x t = " ρ ρ 2 # x t + " # υ t where x t = h y t y t i Measurement equation: z t = h i x t " # Try B = ρ 2 on the second system to get the first system. 7

8 Example II Assume the following MA() process: z t = υ t + θυ t, υ t N ³, σ 2 υ,andeυt υ s =fors 6= t. Again, we have a more conmplicated structure than a simple Markovian process. However, it will again be straightforward to write a state space representation. 8

9 State Space Representation I Transition equation: x t = " # x t + " θ # υ t where x t = h y t θυ t i Measurement equation: z t = h i x t 9

10 State Space Representation II Transition equation: x t = υ t where x t = [υ t ] Measurement equation: z t = θx t + υ t Again both representations are equivalent!

11 Example III Assume the following random walk plus drift process: z t = z t + β + υ t, υ t N ³, σ 2 υ This is even more interesting. We have a unit root. We have a constant parameter (the drift).

12 State Space Representation Transition equation: x t = " # x t + " # υ t where x t = h y t β i Measurement equation: z t = h i x t 2

13 Some Conditions on the State Space Representation We only consider Stable Systems. A system is stable if for any initial state x, the vector of states, x t, converges to some unique x. A necessary and sufficient condition for the system to be stable is that: λ i (F ) < for all i, whereλ i (F ) stands for eigenvalue of F. 3

14 Introducing the Kalman Filter Developed by Kalman and Bucy. Wide application in science. Basic idea. Prediction, smoothing, and control. Why the name filter? 4

15 Some Definitions Let x t t = E ³ x t z t be the best linear predictor of x t given the history of observables until t, i.e. z t. Let z t t = E ³ z t z t = H x t t be the best linear predictor of z t given the history of observables until t, i.e. z t. Let x t t = E ³ x t z t be the best linear predictor of x t given the history of observables until t, i.e.z t. 5

16 What is the Kalman Filter trying to do? Let assume we have x t t and z t t. We observe a new z t. We need to obtain x t t. Note that x t+ t = Fx t t and z t+ t = H x t+ t,sowecangoback to the first step and wait for z t+. Therefore, the key question is how to obtain x t t from x t t and z t. 6

17 A Minimization Approach to the Kalman Filter I Assume we use the following equation to get x t t from z t and x t t : ³ ³ x t t = x t t + K t zt z t t = xt t + K t zt H x t t This formula will have some probabilistic justification (to follow). What is K t? 7

18 A Minimization Approach to the Kalman Filter II K t is called the Kalman filter gain and it measures how much we update x t t as a function in our error in predicting z t. The question is how to find the optimal K t. The Kalman filter is about how to build K t such that optimally update x t t from x t t and z t. How do we find the optimal K t? 8

19 Some Additional Definitions µ ³xt Let Σ t t E x t t ³ xt x t t z t be the predicting error variance covariance matrix of x t given the history of observables until t, i.e. z t. µ ³zt Let Ω t t E z t t ³ zt z t t z t be the predicting error variance covariance matrix of z t given the history of observables until t, i.e. z t. µ ³xt Let Σ t t E x t t ³ xt x t t z t be the predicting error variance covariance matrix of x t given the history of observables until t, i.e. z t. 9

20 Finding the optimal K t We want K t such that min Σ t t. It can be shown that, if that is the case: K t = Σ t t H ³ H Σ t t H + R with the optimal update of x t t given z t and x t t being: ³ x t t = x t t + K t zt H x t t We will provide some intuition later. 2

21 Example I Assume the following model in State Space form: Transition equation x t = µ + υ t, υ t N ³, σ 2 υ Measurement equation z t = x t + ξ t, ξ t N ³, σ 2 ξ Let σ 2 ξ = qσ2 υ. 2

22 Example II Then, if Σ = σ 2 υ,whatitmeansthatx was drawn from the ergodic distribution of x t. We have: K = σ 2 υ +q +q. Therefore, the bigger σ 2 ξ relative to σ2 υ (the bigger q) thelowerk and the less we trust z. 22

23 The Kalman Filter Algorithm I Given Σ t t, z t,andx t t,wecannowsetthekalmanfilter algorithm. Let Σ t t, then we compute: µ ³zt Ω t t E z t t ³ zt z t t z t = E H ³ x t x t t ³ xt x t t H+ υ t ³ xt x t t H + H ³ x t x t t υ t + υ t υ t zt = H Σ t t H + R 23

24 The Kalman Filter Algorithm II Let Σ t t, then we compute: µ ³zt E z t t ³ xt x t t z t µ E H ³ ³ x t x t t ³ xt x t t + υt xt x t t z t = = H Σ t t Let Σ t t, then we compute: K t = Σ t t H ³ H Σ t t H + R Let Σ t t, x t t, K t,andz t then we compute: ³ x t t = x t t + K t zt H x t t 24

25 The Kalman Filter Algorithm III Let Σ t t, x t t, K t,andz t, then we compute: E Σ t t E µ ³xt x t t ³ xt x t t z t ³ xt x t t ³ xt x t t ³ xt x t t ³ zt H x t t K t ³ K t zt H x t t ³ xt x t t + ³ K t zt H x t t ³ zt H x t t K t z t = = Σ t t K t H Σ t t where, you have to notice that x t x t t = x t x t t K t ³ zt H x t t. 25

26 The Kalman Filter Algorithm IV Let Σ t t, x t t, K t,andz t, then we compute: Σ t+ t = F Σ t t F + GQG Let x t t, then we compute:. x t+ t = Fx t t 2. z t+ t = H x t+ t Therefore, from x t t, Σ t t,andz t we compute x t t and Σ t t. 26

27 The Kalman Filter Algorithm V We also compute z t t and Ω t t. Why? To calculate the likelihood function of z T = {z t } T t= (to follow). 27

28 The Kalman Filter Algorithm: A Review We start with x t t and Σ t t. The, we observe z t and: Ω t t = H Σ t t H + R z t t = H x t t K t = Σ t t H ³ H Σ t t H + R Σ t t = Σ t t K t H Σ t t 28

29 x t t = x t t + K t ³ zt H x t t Σ t+ t = F Σ t t F + GQG x t+ t = Fx t t We finish with x t+ t and Σ t+ t. 29

30 Some Intuition about the optimal K t Remember: K t = Σ t t H ³ H Σ t t H + R Notice that we can rewrite K t in the following way: K t = Σ t t HΩ t t If we did a big mistake forecasting x t t using past information (Σ t t large) we give a lot of weight to the new information (K t large). If the new information is noise (R large) we give a lot of weight to the oldprediction(k t small). 3

31 A Probabilistic Approach to the Kalman Filter Assume: Ã" x # " Σxx Σ xy #! Z w =[X wy w] N y, Σ yx Σ yy then: X y, w N ³ x + Σ xy Σ yy (y y ), Σ xx Σ xy Σ yy Σ yx Also x t t E ³ x t z t and: µ ³xt Σ t t E x t t ³ xt x t t z t 3

32 Some Derivations I If z t z t is the random variable z t (observable) conditional on z t,then: Let z t t E ³ z t z t = E ³ H x t + υ t z t = H x t t Let E µ ³zt Ω t t E z t t ³ zt z t t z t = H ³ x t x t t ³ xt x t t H+ ³ υ t xt x t t H+ H ³ x t x t t υ t + υ t υ t zt = H Σ t t H + R 32

33 Some Derivations II Finally, let E µ ³zt E z t t ³ xt x t t z t = µ H ³ ³ x t x t t ³ xt x t t + υt xt x t t z t = = H Σ t t 33

34 The Kalman Filter First Iteration I Assume we know x and Σ,then Ã! x z N z Ã" x H x # " Σ Σ, H H Σ H Σ H + R #! Remember that: X y, w N ³ x + Σ xy Σ yy (y y ), Σ xx Σ xy Σ yy Σ yx 34

35 The Kalman Filter First Iteration II Then, we can write: x z,z = x z N ³ x, Σ where x = x + Σ H ³ H Σ H + R ³ z H x and Σ = Σ Σ H ³ H Σ H + R H Σ 35

36 Therefore, we have that: z = H x Ω = H Σ H + R x = x + Σ H ³ H Σ H + R ³ z H x Σ = Σ Σ H ³ H Σ H + R H Σ Also, since x 2 = Fx + Gω 2 and z 2 = H x 2 + υ 2 : x 2 = Fx Σ 2 = F Σ F + GQG 36

37 The Kalman Filter th Iteration I Assume we know x t t and Σ t t,then Ã! xt z t N z t Ã" xt t H x t t # " Σt t Σ, t t H H Σ t t H Σ t t H + R #! Remember that: X y, w N ³ x + Σ xy Σ yy (y y ), Σ xx Σ xy Σ yy Σ yx 37

38 The Kalman Filter th Iteration II Then, we can write: x t z t,z t = x t z t N ³ x t t, Σ t t where x t t = x t t + Σ t t H ³ H Σ t t H + R ³ z t H x t t and Σ t t = Σ t t Σ t t H ³ H Σ t t H + R H Σ t t 38

39 The Kalman Filter Algorithm Given x t t, Σ t t and observation z t Ω t t = H Σ t t H + R z t t = H x t t Σ t t = Σ t t Σ t t H ³ H Σ t t H + R H Σ x t t = x t t + Σ t t H ³ H Σ t t H + R ³ z t H x t t 39

40 Σ t+ t = F Σ t t F + GQG t t x t+ t = Fx t t 4

41 Putting the Minimization and the Probabilistic Approaches Together From the Minimization Approach we know that: x t t = x t t + K t ³ zt H x t t From the Probability Approach we know that: x t t = x t t + Σ t t H ³ H Σ t t H + R ³ z t H x t t 4

42 But since: K t = Σ t t H ³ H Σ t t H + R We can also write in the probabilistic approach: x t t = x t t + Σ t t H ³ H Σ t t H + R ³ z t H x t t = = x t t + K t ³ zt H x t t Therefore, both approaches are equivalent. 42

43 Writing the Likelihood Function We want to write the likelihood function of z T = {z t } T t= : log ` ³ z T F, G, H, Q, R = TX t= TX t= log ` ³ z t z t F, G, H, Q, R = N 2 log 2π + 2 log Ωt t + 2 TX t= v tω t t v t v t = z t z t t = z t H x t t Ω t t = H t Σ t t H t + R 43

44 Initial conditions for the Kalman Filter An important step in the Kalman Fitler is to set the initial conditions. Initial conditions:. x 2. Σ Wheredotheycomefrom? 44

45 Since we only consider stable system, the standard approach is to set: x = x Σ = Σ where x solves How do we find Σ? x = Fx Σ = F Σ F + GQG Σ =[I F F ] vec(gqg ) 45

46 Initial conditions for the Kalman Filter II Under the following conditions:. The system is stable, i.e. all eigenvalues of F are strictly less than one in absolute value. 2. GQG and R are p.s.d. symmetric 3. Σ is p.s.d. symmetric Then Σ t+ t Σ. 46

47 Remarks. There are more general theorems than the one just described. 2. Those theorems are based on non-stable systems. 3. Since we are going to work with stable system the former theorem is enough. 4. Last theorem gives us a way to find Σ as Σ t+ t Σ for any Σ we start with. 47

48 The Kalman Filter and DSGE models Basic Real Business Cycle model max E X t= β t {ξ log c t +( ξ)log( l t )} c t + k t+ = kt α (e z t l t ) α +( δ) k z t = ρz t + ε t, ε t N (, σ) Parameters: γ = {α, β, ρ, ξ, η, σ} 48

49 Equilibrium Conditions ( ³ = βe t +αe z t+ k c t c t+ α l α t+ η ) t+ ξ = ξ ( α) e z t kt α lt α l t c t c t + k t+ = e z t k α t l α t +( η) k t z t = ρz t + ε t 49

50 A Special Case We set, unrealistically but rather useful for our point, η =. In this case, the model has two important and useful features:. First, the income and the substitution effect from a productivity shock to labor supply exactly cancel each other. Consequently, l t is constant and equal to: l t = l = ( α) ξ ( α) ξ +( ξ)( αβ) 2. Second, the policy function for capital is k t+ = αβe z tk α t l α. 5

51 A Special Case II The definition of k t+ implies that c t =( αβ) e z tk α t l α. Let us try if the Euler Equation holds: = βe t c t ( αβ) e z tk t α = βe l α t ( c t+ ³ αe z t+ k α ( ( αβ) e z tk α t l α = βe t ) t+ l α t+ ³ αe ( αβ) e zt+ kt+ α z t+ k α l α ( αβ ( αβ) = βα ( αβ) α ( αβ) k t+ ) ) t+ l α t+ 5

52 Let us try if the Intratemporal condition holds ξ l = ξ ( αβ) e z tk α t l α ( α) ez t k α t l α ξ l = ξ ( α) ( αβ) l ( αβ)( ξ) l = ξ ( α)( l) (( αβ)( ξ)+( α) ξ) l =( α) ξ Finally, the budget constraint holds because of the definition of c t. 52

53 Transition Equation Since this policy function is linear in logs, we have the transition equation for the model: log k t+ z t = log αβλl α α ρ ρ log k t z t + ² t. Note constant. Alternative formulations. 53

54 Measurement Equation As observables, we assume log y t and log i t subject to a linearly additive measurement error V t = ³ v,t v 2,t. Let V t N (, Λ), where Λ is a diagonal matrix with σ 2 and σ2 2,as diagonal elements. Why measurement error? Stochastic singularity. Then: Ã log yt log i t! = Ã log αβλl α! log k t+ z t + Ã v,t v 2,t!. 54

55 The Solution to the Model in State Space Form F = x t = log k t z t log αβλl α α ρ ρ H =,z t =,G= Ã log αβλl α Ã log yt log i t!!,q= σ 2,R= Λ 55

56 The Solution to the Model in State Space Form III Now, using z T,F,G,H,Q, and R as defined in the last slide......we can use the Ricatti equations to compute the likelihood function of the model: log ` ³ z T F, G, H, Q, R Croos-equations restrictions implied by equilibrium solution. With the likelihood, we can do inference! 56

57 What do we Do if η 6=? We have two options: First, we could linearize or log-linearize the model and apply the Kalman filter. Second, we could compute the likelihood function of the model using a non-linear filter (particle filter). Advantages and disadvantages. Fernández-Villaverde, Rubio-Ramírez, and Santos (25). 57

58 The Kalman Filter and linearized DSGE Models We linearize (or loglinerize) around the steady state. We assume that we have data on log output (log y t ), log hours (log l t ), and log investment (log c t ) subject to a linearly additive measurement error V t = ³ v,t v 2,t v 3,t. We need to write the model in state space form. Remember that bk t+ = P k b t + Qz t and bl t = Rk b t + Sz t 58

59 Writing the Likelihood Function I The transitions Equation: bk t+ z t+ = P Q ρ bk t z t + ² t. The Measurement Equation requires some care. 59

60 Writing the Likelihood Function II Notice that by t = z t + α b k t +( α) b l t Therefore, using b l t = R b k t + Sz t by t = z t + α b k t +( α)(r b k t + Sz t )= (α +( α)r) b k t +(+( α)s) z t Also since bc t = α 5 b l t + z t + α b k t and using again b l t = R b k t + Sz t bc t = z t + α b k t α 5 (R b k t + Sz t )= (α α 5 R) b k t +( α 5 S) z t 6

61 Writing the Likelihood Function III Therefore the measurement equation is: log y t log l t log c t = log y α +( α)r +( α)s log l R S log c α α 5 R α 5 S + v,t v 2,t v 3,t. bk t z t 6

62 The Likelihood Function of a General Dynamic Equilibrium Economy Transition equation: S t = f (S t,w t ; γ) Measurement equation: Y t = g (S t,v t ; γ) Interpretation. 62

63 Some Assumptions. We can partition {W t } into two independent sequences n W,t o and n W2,t o,s.t. Wt = ³ W,t,W 2,t and dim ³ W2,t +dim (Vt ) dim (Y t ). 2. We can always evaluate the conditional densities p ³ y t W t,yt,s ; γ. Lubick and Schorfheide (23). 3. Themodelassignspositiveprobabilitytothedata. 63

64 Our Goal: Likelihood Function Evaluate the likelihood function of the a sequence of realizations of the observable y T at a particular parameter value γ: p ³ y T ; γ We factorize it as: p ³ y T ; γ = = TY Z Z t= TY t= p ³ y t y t ; γ p ³ y t W t,yt,s ; γ p ³ W t,s y t ; γ dw t ds 64

65 A Law of Large Numbers ( ½ If s t t,i,w t t,i ¾ N then: i= ) T t= N i.i.d. draws from n p ³ W t,s y t ; γ o T t=, p ³ y T ; γ ' TY t= N NX i= p µ y t w t t,i,y t,s t t,i ; γ 65

66 ...thus The problem of evaluating the likelihood is equivalent to the problem of drawing from n p ³ W t,s y t ; γ o T t= 66

67 Introducing Particles n s t,i,w t,i o N i= N i.i.d. draws from p ³ W t,s y t ; γ. Each s t,i particles.,w t,i is a particle and n s t,i,w t,i o N i= a swarm of ½ s t t,i,w t t,i ¾ N i= N i.i.d. draws from p ³ W t,s y t ; γ. Each s t t,i,w t t,i is a proposed particle and a swarm of proposed particles. ½ s t t,i,w t t,i ¾ N i= 67

68 ... and Weights q i t = p µ P Ni= p y t w t t,i,y t,s t t,i ; γ µ y t w t t,i,y t,s t t,i ; γ 68

69 AProposition s t t,i,w t t,i ¾ N i= Let n es i, ew i o ½ N be a draw with replacement from i= and probabilities qt.then i n es i, ew i o N is a draw from i= p ³ W t,s y t ; γ. 69

70 Importance of the Proposition. It shows how a draw canbeusedtodraw n s t,i,wt,i ½ s t t,i,w t t,i ¾ N from p ³ W t,s y t ; γ o i= N i= from p ³ W t,s y t ; γ. 2. With a draw n s t,i to get a draw,wt,i o N i= from p ³ W t,s y t ; γ we can use p ³ W,t+ ; γ ½ s t+ t,i,w t+ t,i ¾ N i= and iterate the procedure. 7

71 Sequential Monte Carlo I: Filtering Step, Initialization: Set t à and initialize p ³ W t,s y t ; γ = p (S ; γ). Step, Prediction: Sample N values density p ³ W t,s y t ; γ = p ³ W,t ; γ ½ s t t,i,w t t,i ¾ N Step 2, Weighting: Assign to each draw s t t,i qt i. Step 3, Sampling: Draw n s t,i o N,wt,i i= i= p ³ W t,s y t ; γ. with rep. from,w t t,i from the the weight ½ s t t,i,w t t,i ¾ N i= with probabilities n qto i N. If t < T set t à t + and go to i= step. Otherwise stop. 7

72 Sequential Monte Carlo II: Likelihood Use ( ½ s t t,i,w t t,i ¾ ) N T i= t= to compute: p ³ y T ; γ ' TY t= N NX i= p µ y t w t t,i,y t,s t t,i ; γ 72

73 A Trivial Application How do we evaluate the likelihood function p ³ y T α, β, σ nonnormal process: of the nonlinear, s t = α + β s t +s t + w t y t = s t + v t where w t N (, σ) andv t t (2) given some observables y T = {y t } T t= and s. 73

74 . Let s,i = s for all i. 2. Generate N i.i.d. draws ½ s,i,w,i ¾ N i= from N (, σ). 3. Evaluate p µ y w,i,y,s,i à = p t(2) y à α + β s,i +s,i + w,i!!. 4. Evaluate the relative weights q i = p t(2) à y P Ni= p t(2) à à y α+β s,i +s,i à α+β s,i +s,i +w,i!! +w,i!!. 74

75 with rela- ½ 5. Resample with replacement N values of s,i ¾ N,w,i i= tive weights q i. Call those sampled values n s,i,w,io N i=. 6. Go to step, and iterate -4 until the end of the sample. 75

76 A Law of Large Numbers A law of the large numbers delivers: and consequently: p ³ y y, α, β, σ ' N NX i= p µ y w,i,y,s,i p ³ y T α, β, σ ' TY t= N NX i= p µ y t w t t,i,y t,s t t,i 76

77 Comparison with Alternative Schemes Deterministic algorithms: Extended Kalman Filter and derivations (Jazwinski, 973), Gaussian Sum approximations (Alspach and Sorenson, 972), grid-based filters (Bucy and Senne, 974), Jacobian of the transform (Miranda and Rui, 997). Tanizaki (996). Simulation algorithms: Kitagawa (987), Gordon, Salmond and Smith (993), Mariano and Tanizaki (995) and Geweke and Tanizaki (999). 77

78 A Real Application: the Stochastic Neoclassical Growth Model Standard model. Isn t the model nearly linear? Yes, but:. Better to begin with something easy. 2. We will learn something nevertheless. 78

79 The Model Representative agent with utility function U = E P t= β t ³c θ t ( l t) θ τ τ. One good produced according to y t = e z tak α t l α t with α (, ). Productivity evolves z t = ρz t + ² t, ρ < and² t N (, σ ² ). Lawofmotionforcapitalk t+ = i t +( δ)k t. Resource constrain c t + i t = y t. 79

80 Solve for c (, ) andl (, ) given initial conditions. Characterized by: U c (t) =βe t h Uc (t +) ³ +αae z t+ k α t+ l(k t+,z t+ ) α δ i θ θ c(k t,z t ) l(k t,z t ) =( α) ez t Ak α t l(k t,z t ) α A system of functional equations with no known analytical solution. 8

81 Solving the Model We need to use a numerical method to solve it. Different nonlinear approximations: value function iteration, perturbation, projection methods. We use a Finite Element Method. Why? Aruoba, Fernández-Villaverde and Rubio-Ramírez (23):. Speed: sparse system. 2. Accuracy: flexible grid generation. 3. Scalable. 8

82 Building the Likelihood Function Time series:. Quarterly real output, hours worked and investment. 2. Main series from the model and keep dimensionality low. Measurement error. Why? γ =(θ, ρ, τ, α, δ, β, σ ², σ, σ 2, σ 3 ) 82

83 State Space Representation k t = f (S t,w t ; γ) =e tanh (λ t ) kt α l ³ k t, tanh (λ t ); γ α ³ ³ θ l kt, tanh (λ t ); γ ( α) θ l ³ k t, tanh +( δ) k t (λ t ); γ λ t = f 2 (S t,w t ; γ) =tanh(ρtanh (λ t )+² t ) gdp t = g (S t,v t ; γ) =e tanh (λ t ) kt α l ³ k t, tanh (λ t ); γ α + V,t hours t = g 2 (S t,v t ; γ) =l ³ k t, tanh (λ t ); γ + V 2,t inv t = g 3 (S t,v t ; γ) =e tanh (λ t ) kt α l ³ k t, tanh (λ t ); γ α ³ ³ θ l kt, tanh (λ t ); γ ( α) θ l ³ k t, tanh + V 3,t (λ t ); γ Likelihood Function 83

84 Since our measurement equation implies that p (y t S t ; γ) =(2π) 3 2 Σ 2 e ω(s t ;γ) 2 where ω(s t ; γ) =(y t x(s t ; γ))) Σ (y t x(s t ; γ)) t, we have (2π) 3T 2 Σ T 2 Z TY Z t= ' (2π) 3T 2 p ³ y T ; γ = e ω(s t ;γ) 2 p ³ S t y t,s ; γ ds t p (S ; γ) ds Σ T 2 TY t= N NX i= e ω(si t ;γ) 2 84

85 Priors for the Parameters Priors for the Parameters of the Model Parameters Distribution Hyperparameters θ ρ τ α δ β σ ² σ σ 2 σ 3 Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform,,,,,.5.75,,.,.,.,. 85

86 Likelihood-Based Inference I: a Bayesian Perspective Define priors over parameters: truncated uniforms. Use a Random-walk Metropolis-Hastings to draw from the posterior. Find the Marginal Likelihood. 86

87 Likelihood-Based Inference II: a Maximum Likelihood Perspective We only need to maximize the likelihood. Difficulties to maximize with Newton type schemes. Common problem in dynamic equilibrium economies. We use a simulated annealing scheme. 87

88 An Exercise with Artificial Data First simulate data with our model and use that data as sample. Pick true parameter values. Benchmark calibration values for the stochastic neoclassical growth model (Cooley and Prescott, 995). Calibrated Parameters Parameter θ ρ τ α δ Value Parameter β σ ² σ σ 2 σ 3 Value * * 4 Sensitivity: τ =5andσ ² =

89 Figure 5.: Likelihood Function Benchmark Calibration Likelihood cut at ρ Likelihood cut at τ Likelihood cut at α Likelihood cut at δ Likelihood cut at σ Likelihood cut at β Likelihood cut at θ Nonlinear Linear Pseudotrue x -3

90 Figure 5.2: Posterior Distribution Benchmark Calibration ρ τ α 4 δ σ 5 β x θ 5 σ x -4 σ 2 σ x x -4

91 Figure 5.3: Likelihood Function Extreme Calibration Likelihood cut ρ Likelihood cut τ Likelihood cut α Likelihood cut δ Likelihood cut σ Likelihood cut β Likelihood cut θ Nonlinear Linear Pseudotrue

92 Figure 5.4: Posterior Distribution Extreme Calibration ρ τ α δ σ β θ σ x -4 σ 2 σ x x -4

93 Figure 5.5: Converge of Posteriors Extreme Calibration ρ 8 τ x x 5 α x 5 δ x 5.4 σ β x x 5.4 θ.5 σ x x 5 σ 2 σ x x 5

94 Figure 5.6: Posterior Distribution Real Data ρ 5 τ x 4 α δ x -3 5 σ 5 β θ σ σ 2 σ

95 Figure 6.: Likelihood Function Transversal cut at α Transversal cut at β Exact Particles Particles Particles Exact Particles Particles Particles Transversal cut at ρ Transversal cut at σ ε Exact Particles Particles Particles Exact Particles Particles Particles x -3

96 Figure 6.2: C.D.F. Benchmark Calibration particles 2 particles particles x 4 4 particles x 4 5 particles x 4 6 particles x x 4

97 Figure 6.3: C.D.F. Extreme Calibration particles 2 particles particles x 4 4 particles x 4 5 particles x 4 6 particles x x 4

98 Figure 6.4: C.D.F. Real Data particles 2 particles particles x 4 4 particles x 4 5 particles x 4 6 particles x x 4

99 Posterior Distributions Benchmark Calibration Parameters Mean s.d. θ ρ τ α δ β σ ² σ σ σ

100 Maximum Likelihood Estimates Benchmark Calibration Parameters MLE s.d. θ ρ τ α δ β σ ².7.4 σ σ 2 σ

101 Posterior Distributions Extreme Calibration Parameters Mean s.d. θ ρ τ α δ β σ ² σ σ σ

102 Maximum Likelihood Estimates Extreme Calibration Parameters MLE s.d. θ ρ τ α δ β σ ².35.5 σ σ 2 σ

103 Convergence on Number of Particles Convergence Real Data N Mean s.d

104 Posterior Distributions Real Data Parameters Mean s.d. θ ρ τ α δ β σ ² σ σ 2 σ

105 Maximum Likelihood Estimates Real Data Parameters MLE s.d. θ ρ τ α δ β σ ² σ σ 2 σ

106 Logmarginal Likelihood Difference: Nonlinear-Linear p Benchmark Calibration Extreme Calibration Real Data

107 output hours inv Nonlinear versus Linear Moments Real Data Real Data Nonlinear (SMC filter) Linear (Kalman filter) Mean s.d Mean s.d Mean s.d

108 A Future Application: Good Luck or Good Policy? U.S. economy has become less volatile over the last 2 years (Stock and Watson, 22). Why?. Good luck: Sims (999), Bernanke and Mihov (998a and 998b) andstockandwatson(22). 2. Good policy: Clarida, Gertler and Gaĺı (2), Cogley and Sargent (2 and 23), De Long (997) and Romer and Romer (22). 3. Long run trend: Blanchard and Simon (2). 98

109 How Has the Literature Addressed this Question? So far: mostly with reduced form models (usually VARs). But:. Results difficult to interpret. 2. How to run counterfactuals? 3. Welfare analysis. 99

110 Why Not a Dynamic Equilibrium Model? New generation equilibrium models: Christiano, Eichebaum and Evans (23) and Smets and Wouters (23). Linear and Normal. But we can do it!!!

111 Environment Discrete time t =,,... Stochastic process s S with history s t =(s,..., s t ) and probability µ ³ s t.

112 The Final Good Producer Perfectly Competitive Final Good Producer that solves µz ³ Z max y i s t θ θ ³ ³ di p i s t y i s t di. y i (s t ) Demand function for each input of the form with price aggregator: y i ³ s t = p ³ s t = p ³ i s t p (s t ) θ y ³ s t, Ã Z ³ p i s t θ θ!θ θ di. 2

113 The Intermediate Good Producer Continuum of intermediate good producers, each of one behaving as monopolistic competitor. The producer of good i has access to the technology: ³ ³ ³ y i s t =max ½e z(st ) k α i s t s t φ, l α i ¾. Productivity z ³ s t = ρz ³ s t + ε z ³ s t. Calvo pricing with indexing. Probability of changing prices (before observing current period shocks) ζ. 3

114 Consumers Problem ³ ³ c s t dc ³ s t σ c E s t X β t t= ε ³ c s t σ c ε l ³ s t l ³ s t σ l σ l + ε m ³ s t m ³ s t σ m σ m p ³ s t ³ c ³ s t + x ³ s t + M ³ s t + q ³ s t+ s t+ p ³ s t ³ w ³ s t l ³ s t + r ³ s t k ³ s t + M ³ s t k ³ s t Z B ³ s t+ B s t =( δ) k ³ s t φ x ³ s t k ³ s t + x ³ s t. B ³ s t+ ds t+ = + B ³ s t + Π ³ s t + T ³ s t 4

115 Government Policy Monetary Policy: Taylor rule i ³ s t = r g π g ³ s t π g ³ s t a ³ s t b ³ s t +a ³ s t ³ π ³ s t ³ π t g s +b ³ s t ³ y ³ s t ³ y t ³ g s + ε i s t ³ ³ = π g s t + ε π s t = a ³ s t ³ + ε a s t = b ³ s t ³ + ε b s t Fiscal Policy. 5

116 Stochastic Volatility I We can stack all shocks in one vector: ε ³ s t = ³ ³ ³ ³ ³ ³ ³ ³ ³ ε z s t, ε c s t, ε l s t, ε m s t, ε i s t, ε π s t, ε a s t, ε b s t Stochastic volatility: ε ³ s t = R ³ s t.5 ³ ϑ s t. The matrix R ³ s t can be decomposed as: R ³ s t = G ³ s t ³ H s t G ³ s t. 6

117 Stochastic Volatility II H ³ s t (instantaneous shocks variances) is diagonal with nonzero elements h i s ³ t that evolve: ³ ³ ³ log h i s t =logh i s t + ς i η i s t. G ³ s t (loading matrix) is lower triangular, with unit entries in the ³ diagonal and entries γ ij s t that evolve: ³ ³ ³ γ ij s t = γ ij s t + ω ij ν ij s t. 7

118 Where Are We Now? Solving the model: problem with 45 state variables: physical capital, the aggregate price level, 7 shocks, 8 elements of matrix H ³ s t, and the 28 elements of the matrix G ³ s t. Perturbation. We are making good progress. 8