Advanced Econometrics III, Lecture 5, Dynamic General Equilibrium Models Example 1: A simple RBC model... 2

Transcription

1 Advanced Econometrics III, Lecture 5, Contents 1 Dynamic General Equilibrium Models Example 1: A simple RBC model Approximation Method Based on Linearization Example 2: RBC model with uncertainty Econometric Methods for the Analysis of Dynamic General Equilibrium Models State-Space-Observer Form

2 Advanced Econometrics III, Lecture 5, Dynamic General Equilibrium Models Define a model solution Motivate the need to somehow approximate model solutions Describe basic idea behind Log Linear Approximations 1.1 Example 1: A simple RBC model Model: subject to : max {C t,k t+1 } t=0 β tc1 σ t 1 σ C t + K t+1 (1 δ)k t = K α t, K 0 is given

3 Advanced Econometrics III, Lecture 5, First order condition: C σ t or after substituting out resource constraint: βct+1 σ [ αk α 1 t+1 + (1 δ)] = 0 v(k t, K t+1, K t+2 ) = 0, t = 0, 1,...with K 0 given Solution: a function K t+1 = g(k t ) such that v(k t, g(k t ), g(g(k t ))) = 0, for all K t Problem: This is an infinite number of equations (one for each possible K t ) in an infinite number of unknowns (a value for g for each possible K t ) With only a few rare exceptions, this is very hard to solve exactly Easy case: 1. σ = 1, δ = 1 = g(k t ) = αβk a t 2. v is linear in K t, K t+1, K t+2 Standard approach: approximate v by a log linear function

4 Advanced Econometrics III, Lecture 5, Approximation Method Based on Linearization Three steps: Compute the steady state Do a log linear expansion around steady state Solve the resulting log linearized system 1. Compute steady state K t = K : C σ βc σ [ αk α 1 + (1 δ) ] = 0 = αk α 1 + (1 δ) = 1 β K satisfies: = K = 1 β α (1 δ) 1 1 α v(k, K, K ) = 0

5 Advanced Econometrics III, Lecture 5, First order Taylor expansion of v around steady state to get a difference equation in K t v1(kt K ) + v2(kt+1 K ) + v3(kt+2 K ) = 0 where v 1 = v(k t, K t+1, K t+2 ) Kt =K K t+1 =K t+2 =K t Conventionally, log-linear approximation is much easier to handle: Write this as : (v 1 K ) ˆK t + (v 2 K ) ˆK t+1 + (v 3 K ) ˆK t+2 = 0 : ˆK t = K t K K : α 2 ˆK t + α 1 ˆK t+1 + α 0 ˆK t+2 = 0 : α 2 = v 1 K, α 1 = v 2 K, α 0 = v 3 K 3. Solve the difference equation in ˆK t using undetermined coeffi cient method Posit the following policy rule ˆK t+1 = A ˆK t

6 Advanced Econometrics III, Lecture 5, where A is to be determined Compute A : or A is the eigenvalue of polynomial. α 2 ˆK t + α 1 A ˆK t + α 0 A 2 ˆK t = 0 α 2 + α 1 A + α 0 A 2 = 0 In general, two eigenvalues Can show: in RBC example, one eigenvalue is explosive, the other is not There exist theorem (Stokey-Lucas chapter 6) that says you should ignore the explosive eigenvalue Saddle point equilibrium If somehow you start at single steady state, stay there If you are away from single steady state, you will eventually go back to the steady state Other examples are possible: Both eigenvalues are explosive

7 Advanced Econometrics III, Lecture 5, All possible equilibria involve leaving that steady state Log linear approximation is not useful since it ceases to be valid outside a neighborhood of steady state could have two-period cycle, and log-linearize about the two-period cycle (that s another story) Maybe all explosive root case is unlikely If Blanchard-Kahn conditions are not satisfied, too many explosive roots. Both eigenvalues are stable Many paths converge into steady state: multiple equilibria How are this happen? 1. strategic complementarities between economic agents 2. inability of agents to coordinate 3. combination can lead to multiple equilibria, coordination failure What is the source of strategic complementarities? nature of technology and preferences nature of relationship between agents and the government Example 1:

8 Advanced Econometrics III, Lecture 5, work hard take it easy work hard (3,3) (0,1) take it easy (1,0) (1,1) Everyone "take it easy" equilbrium is a coordination failure: if everyone could get together, they would all choose to work hard Example 2: every firm in economy has a pet investment project which only seems profitbable if the economy is booming If each firm conjecture all other firms will invest, this implies a booming economy, so it makes sense for each firms to invest If each firm conjecture all other firms will not invest, so economy will stagnate and it makes sense for each firms not to invest Example 3 firm production function: y t = A t K a t h 1 α t A t = Y γ t where Y t is the economy wide average output Resource constraint C t + K t+1 (1 δ)k t = Y t

9 Advanced Econometrics III, Lecture 5, Equilibrium condition: Y t = y t economy wide average output is average of individual firms production Household preference t=0 β t u(c t, h t ) γ large enough leads to two stable eigenvalues, multiple equilibria 1.3 Example 2: RBC model with uncertainty Model: max {C t,k t+1 } E 0 t=0 β tc1 σ t 1 σ

10 Advanced Econometrics III, Lecture 5, subject to : C t + K t+1 (1 δ)k t = A t Kt α, K 0 is given where ε t is a stochastic process with Eε t = ε and ˆε t = ρˆε t 1 + e t.e t N(0, σ 2 e) First order condition: E t [ C σ t or after substituting out resource constraint: βct+1 σ [ αk α 1 t+1 ε t+1 + (1 δ) ]] = 0 E t v(k t, K t+1, K t+2 ; ε t+1, ε t ) = 0, t = 0, 1,...with K 0 given Solution: a function K t+1 = g(k t, ε t ) such that E t v(k t, g(k t, ε t ), g(g(k t, ε t ), ε t+1 ), ε t+1, ε t ) = 0, for all K t, ε t With only a few rare exceptions, this is very hard to solve exactly v is log linear

11 Advanced Econometrics III, Lecture 5, Standard approach: approximate v by a log linear function Compute steady state of K t where ε t is replaced with Eε t Replace v by its Taylor expansion around steady state Solve resulting log linear system If actual stochastic system remains in neighbourhood of steady state, log linear approxiamtion is good. Caveat: strategy not accurate in all conceivable situations Example: suppose that where I live { 40 ε = temperature = o C with 50 percent of time 0 o C, with 50 percent of time On average, temperature is quite nice, Eε = 20 Let K be the capital invested in heating and airconditioning EK is very very large! Economist who predicts investment based on replacing ε by Eε would predict K = 0 In standard model, this is not a big problem, because shocks are not so big, and steady state value of K (the value when ε is replaced by Eε) is approximately EK (the average value of K when ε is stochastic)

12 Advanced Econometrics III, Lecture 5, Step 1: steady state Step 2: Log Linearize K = 1 β αε (1 δ) 1 1 α v(k t+2, K t+1, K t, ε t+1, ε t ) α 0 ˆK t+2 + α 1 ˆK t+1 + α 2 ˆK t + β 0ˆε t+1 + β 1ˆε t Step 3: Solve log linearize system Posit Note: ˆK t+1 = A ˆK t + Bˆε t ˆK t+2 = A 2 ˆK t + ABˆε t + Bρˆε t + Be t+1 Pin down A and B by log-linearized Euler equation: E t [ α0 ˆK t+2 + α 1 ˆK t+1 + α 2 ˆK t + β 0ˆε t+1 + β 1ˆε t ] = 0

13 Advanced Econometrics III, Lecture 5, Substistute the posited policy rule into this equation we have α(a) ˆK t + F ˆε t = 0 where α(a) = α 0 A 2 + α 1 A + α 2 F = α 0 AB + α 0 Bρ + α 1 B + β 0 ρ + β 1 Fine A and B that satisfy: α(a) = 0, F = 0 2 Econometric Methods for the Analysis of Dynamic General Equilibrium Models Multiple equation methods

14 Advanced Econometrics III, Lecture 5, State-space-observables form Limited information estimation: match impulse response function Full information estimate: MLE and Bayesian inference Single Equation methods: GMM 2.1 State-Space-Observer Form Suppose we have a model solution in hand: z t = Az t 1 + Bs t s t = P s t 1 + ε t, Eε t ε t = Σ we can rewrite it as state space represenation state equation: ξ t+1 = F ξ t + Cv t+1 observation equation : y t data = A x t + H ξ t + w t where F, Q, R, A, H depends on the model parameters. θ = β, δ,...

15 Advanced Econometrics III, Lecture 5, Then we can Estimate θ and forecast ξ t and y t data can take into account situations in which data represent a mixture of qtr, monthly, daily observation Supose we have a mixed monthly/quarterly data for t = 0, 1, 2,... ξt+1 = F ξt + Cv t+1 y t data = H ξt + w t where H selects elements of ξt to construct y t data, can easily handle distinction between whether quarterly data represent monthly average (as in flow variables) or point-in-time observations on one month in the quarter (as in stock variables) DYNARE and other softwares are available on web Kalman Filter can be used to forecast and smooth.