Lecture VI. Linearization Methods
|
|
- Toby Baker
- 5 years ago
- Views:
Transcription
1 Lecture VI Linearization Methods Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Perturbation Methods p. 1 /40
2 Local Approximations DSGE models are characterized by a set of nonlinear equations, some of which are intertemporal Euler equations (stochastic difference equations), others are intratemporal FOCs, others feasibility/budget constraints Aim: transform the nonlinear system into a system of linear equations and use techniques from linear RE to solve it To do this transformation, we use local approximations: the idea is to obtain a good approximation of the function in a neighborhood of a benchmark point Benchmark point needs to have two features: 1) easy to compute, 2) visited often by the dynamic system. Usually, the SS These methods are useful to solve economies with distortions, but with small shocks and no occasionally binding constraints. G. Violante, Perturbation Methods p. 2 /40
3 Taylor principle The starting point of this theory is Taylor s theorem. Suppose f : R R is a C function, then for any x in its domain we have f (x) f (x )+f (x )(x x )+ 1 2 f (x )(x x ) n! f(n) (x )(x x ) n +O n+1 so the error is asymptotically smaller than any of its terms. Taylor s theorem says the RHS polynomial approximation is good near x. But how near is near? More in general, write the power series expansion of f around x : f (x) = α k (x x ) k where α k = 1 k! f(k) (x ) k=0 G. Violante, Perturbation Methods p. 3 /40
4 Taylor principle Theorem: Consider the power series k=0 α k(x x ) k. If it converges for x = x r, it converges for x < x r. Then, we can define the radius of convergence of the power series as the quantity r > 0 s.t.: r = sup { x x : } α k (x x ) k < k=0 r provides the maximal radius around x s.t. series converges. Definition. A function f : Ω C is said to be analytic on the open set Ω if, for every x Ω, there exists a sequence {α k } and a radius r > 0 such that f admits representation: f (x) = α k (x x ) k for x x < r k=0 G. Violante, Perturbation Methods p. 4 /40
5 Taylor principle f analytic locally given by a convergent power series If f is analytic at x, then f has continuous derivatives of all orders at x (not too surprising...) A point where f is not analytic is called a singularity of f. That s where the local approximation fails Example: tangent function with power series representation: tan(x) = sinx cosx = k=0 ( 1)k x 2k 2k! k=0 ( 1)k x 2k+1 (2k+1)! has a singularity at x 0 = π/2 for which cos(x) = 0. G. Violante, Perturbation Methods p. 5 /40
6 Taylor principle Then, we can state an important theorem for local approximations. Theorem: Let f be analytic at x. If f or any derivative of f exhibits a singularity at x 0, then the radius of convergence of the Taylor series expansion of f in the neighborhood of x is bounded above by x 0 x. This theorem gives us a guideline to use Taylor series expansions: it tells us that the power series evaluated at x cannot deliver an accurate approximation for f at any point farther away from x than any singular point of f. G. Violante, Perturbation Methods p. 6 /40
7 Taylor principle Example I: log(1 x) with singularity at 1. x log(1 x) Taylor 100 at x = Example II: production function y = k α with α (0,1) and steady-state k = 0.5. The function has a singularity at x 0 = 0 (first derivative), and hence the Taylor expansion has a radius bounded above by 0.5. Bottom line: when you linearize, think about the radius of convergence (calibration-specific) G. Violante, Perturbation Methods p. 7 /40
8 Linearization of FOCs Consider the standard deterministic growth model: k t+1 k α t +c t (1 δ)k t = 0 (1) 1 c t β 1 c t+1 ( αk α 1 t+1 +1 δ) = 0 (2) Let ˆk t k t k and ĉ c t c be level-deviations from SS The feasibility constraint (1) is an equation g(k t+1,k t,c t ) = 0: g(k t+1,k t,c t ) = g +g1ˆk t+1 +g2ˆk t +g3ĉ t = ˆk [ t+1 α(k ) α 1 +(1 δ)]ˆkt +ĉ since g = 0 (SS equilibrium condition) G. Violante, Perturbation Methods p. 8 /40
9 Linearization of FOCs 1 β 1 ( αk α 1 t+1 c t c +1 δ) = 0 (2) t+1 The EE (2) is an equation of the form g(c t,c t+1,k t+1 ) = 0: 0 = g +g 1ĉ t +g 2ĉ t+1 +g 3ˆk t+1 0 = 1 (c ) 2ĉt +β Simplifying: [ ] α(k ) α 1 +1 δ (c ) 2 ĉ t+1 β α(α 1)(k ) c α 2 ˆkt+1 0 = ĉ t +β [ ] ( ) α(k ) α 1 c +1 δ ĉ t+1 βα(α 1) k (k ) α 1ˆkt+1... and we are done G. Violante, Perturbation Methods p. 9 /40
10 Log-linearization of FOCs Another common practice is to take a log-linear approximation to the equilibrium It delivers a natural interpretation of the coefficients in front of the variables: these can be interpreted as elasticities. Define log-deviations from SS: ˆx = log(x) log(x ) and now linearize around log(x ) First-order Taylor exp. of g(x) = g(exp(logx)) around log(x ): g(x) g(exp(logx ))+g x (exp(logx ))exp(logx )ˆx = g +g xx ˆx G. Violante, Perturbation Methods p. 10 /40
11 Log-linearization of feasibility constraint g(k t+1,k t,c t ) = k t+1 k α t +c t (1 δ)k t = 0 Log-linearization: g(k t+1,k t,c t ) = g +g 1 k ˆkt+1 +g 2 k ˆkt +g 3 c ĉ t = k ˆkt+1 α(k ) αˆkt +c ĉ t (1 δ)k ˆkt G. Violante, Perturbation Methods p. 11 /40
12 Alternative approach There is a simple way to do it w/o computing derivatives based on three building blocks: 1. Log approximation: ax = ax exp(logx logx ) = ax exp(ˆx) ax (1+ ˆx) 2. Steady-state relation all additive constants, like ax 1 above, drop out because each log-linearized equation is a SS relationship: g(k,k,c ) = 0 3. Second order terms negligible and set to zero: ˆxŷ = 0 G. Violante, Perturbation Methods p. 12 /40
13 Log-linearization of feasibility constraint k t+1 kt α +c t (1 δ)k t = 0 ) [ )] α ) k exp (ˆkt+1 (k )exp(ˆkt +c exp(ĉ t ) (1 δ)k exp (ˆkt = 0 ) [ )] ) k exp (ˆkt+1 (k ) α exp (αˆk t +c exp(ĉ t ) (1 δ)k exp (ˆkt = 0 k ( 1+ˆk t+1 ) [(k ) α( 1+αˆk t )]+c (1+ĉ t ) (1 δ)k ( ) 1+ˆk t = 0 k ˆkt+1 α(k ) αˆkt +c ĉ (1 δ)k ˆk = 0 where the last line uses the SS restriction: k (k ) α +c (1 δ)k = 0 Same expression as the one on slide 11. G. Violante, Perturbation Methods p. 13 /40
14 Log-linearization of Euler equation 0 = 1 c t β 1 c t+1 ( αk α 1 t+1 +1 δ ) 1 0 = c exp(ĉ t ) β 1 c exp(ĉ t+1 ) ( c exp(ĉ t+1 ) = βc exp(ĉ t ) α(k ) α 1 exp [ exp(ĉ t+1 ) = βexp(ĉ t ) α(k ) α 1 exp ( ) ) α 1 α(k ) α 1 exp (ˆkt+1 +1 δ ((α 1)ˆk t+1 ) ((α 1)ˆk t+1 ) ) +1 δ ] +1 δ ] +1 δ = βexp(ĉ t ) [α(k ) α 1( ) 1+(α 1)ˆk t+1 [ ] = exp(ĉ t ) βα(k ) α 1 +β(α 1)ˆk t+1 α(k ) α 1 +β(1 δ) = exp(ĉ t ) [1+β(α 1)ˆk t+1 α(k ) α 1] where the last line uses the SS version of the EE. Taking logs: ĉ t+1 = ĉ t +β(α 1)ˆk t+1 α(k ) α 1 G. Violante, Perturbation Methods p. 14 /40
15 The method of undetermined coefficients This is Uhlig s toolkit method Alternative classic methods for solving linear RE models: Blanchard-Kahn (ECMA, 1980) Christiano (Comp. Economics, 2002) King-Watson (Comp. Economics, 2002) Klein (JEDC, 2000) Sims (Comp. Econ, 2001) G. Violante, Perturbation Methods p. 15 /40
16 Strategy 1. Derive the nonlinear dynamic system that characterizes the model economy (same number of equations and unknowns) 2. Compute the steady state. 3. (Log) linearize the nonlinear system around the SS. The system is linear in controls and (exogenous and endogenous) state variables. Some eqns are difference eqns. 4. Express the linear system in some matrix representation. This representation is what differs across solution methods. 5. Use a matrix decomposition method to derive: (a) Decision rules, i.e., linear functions from states to controls. (b) Laws of motion for the endogenous state variables, i.e., linear functions from the states at t to the states at t+1. G. Violante, Perturbation Methods p. 16 /40
17 Stochastic growth model with leisure and 2 shocks Household problem: s.t. max {c t,k t+1,h t } E 0 t=0 1 β t logc t ϕ h1+ γ t 1+ 1 γ c t +k t+1 = (1 τ)w t h t +(1+r t δ)k t Firm problem: max {h t,k t } ez t kt α h 1 α t w t h t r t k t Government: ḡe g t = τw t h t, with g t+1 = ρ g g t +σ g η t+1 Laws of motion for capital and productivity: k t+1 = (1 δ)k t +i t z t+1 = ρ z z t +σ z ε t+1 G. Violante, Perturbation Methods p. 17 /40
18 Step 1: Equilibrium conditions w t 1 c t = βe t (1 τ) = ϕh 1 γ t c t [ ] 1+rt+1 δ c t+1 w t = (1 α)e z t kt α h α t r t = αe z t kt α 1 h 1 α t ḡe g t = τw t h t c t +k t+1 = (1 τ)w t h t +(1+r t δ)k t k t+1 = (1 δ)k t +i t z t+1 = ρz t +σ z ε t+1 g t+1 = ρ g g t +σ g η t+1 We can exclude the law of motion for k from the system because, once we know k t+1 we know i t, and i t does not enter anywhere else. And we can substitute the tax revenues out from the household BC. G. Violante, Perturbation Methods p. 18 /40
19 Step 1: Reduced system of equilibrium conditions w t 1 c t = βe t (1 τ) = ϕh 1 γ t c t [ ] 1+rt+1 δ c t+1 w t = (1 α)e z t kt α h α t r t = αe z t kt α 1 h 1 α t c t +k t+1 +ḡe g t = w t h t +(1+r t δ)k t z t+1 = ρz t +σ z ε t+1 g t+1 = ρ g g t +σ g η t+1 At any t, (k t,z t,g t ) are pre-determined 3 choice variables (c t,k t+1,h t ) and 2 prices (w t,r t ) to be determined (by the first 5 equations) plus 2 equations for law of motions of the exogenous states G. Violante, Perturbation Methods p. 19 /40
20 Step 2: SS system Set z = ḡ = 0. Evaluate at SS the first 5 equations: Solve for: ( k, h, c, w, r ) r = 1 β 1+δ ( k 1/β 1+δ h = α w = (1 α) c+ḡ = w h+ r k w c (1 τ) = ϕ h 1 γ ( k h ) α ) 1 α 1 G. Violante, Perturbation Methods p. 20 /40
21 Step 3: Log-linearize around SS Euler equation: 1 c t 1 ceĉt 1 eĉt 1 eĉt = βe t [ ] 1+rt+1 δ c t+1 [ 1+ reˆr t+1 ] δ = βe t ceĉt+1 [ ] 1+ r(1+ ˆrt+1 ) δ = βe t eĉt+1 [ ] 1+ r + rˆrt+1 δ = βe t eĉt+1 e ĉ t = E t [ (1+β rˆrt+1 )e ĉ t+1 ] 1 ĉ t = E t [(1+β rˆr t+1 )(1 ĉ t+1 )] 0 = E t [β rˆr t+1 ĉ t+1 +ĉ t ] G. Violante, Perturbation Methods p. 21 /40
22 Step 3: Log-linearize Intratemporal FOC: w t (1 τ) = ϕh 1 γ t c t eŵt = 1 eĥt γ eĉt ĥ t = γ(ŵ t ĉ t ) Firm s FOC for labor: w t = (1 α)e z t kt α h α t ) ŵ t = z t +α(ˆkt ĥt Firm s FOC for capital: ) ˆr t = z t +(1 α) (ĥt ˆk t G. Violante, Perturbation Methods p. 22 /40
23 Step 3: Log-linearize Budget constraint: c t +ḡe g t +k t+1 = w t h t +(1+r t δ)k t ceĉt +ḡe g t + keˆk t+1 = w heŵt+ĥt + ( reˆr t +1 δ) keˆkt c(1+ĉ t )+ḡ(1+g t )+ k (1+ ˆk ) t+1 ( ) = w h(1+ŵ t ) 1+ĥt + r(1+ ˆr t ) k (1+ ˆk ) ) t +(1 δ) k (1+ˆk t cĉ t +ḡg t + kˆk t+1 = w h ( ) ŵ t +ĥt + r k (ˆr t + ˆk ) t +(1 δ) kˆk t G. Violante, Perturbation Methods p. 23 /40
24 Step 3: Log-linearize Final log-linear system: 0 = E t [β rˆr t+1 ĉ t+1 +ĉ t ] ĥ t = γ(ŵ t ĉ t ) ˆr t = z t +(1 α) (ĥt ˆk ) t ) ŵ t = z t +α(ˆkt ĥt cĉ t +ḡg t + kˆk ( ) t+1 = w h ŵ t +ĥt + r k (ˆr t + ˆk ) t +(1 δ) kˆk t Substitute away factor prices (last eqn. uses CRS: ȳ = r k + w h) [ 0 = E t {β r z t+1 +(1 α) (ĥt+1 ˆk } t+1 )] ĉ t+1 +ĉ t ĥ t [ ] = γ z t +α(ˆkt ĥt ) ĉ t cĉ t +ḡg t + kˆk t+1 [ ] = ȳ z t +αˆk t +(1 α)ĥt +(1 δ) kˆk t G. Violante, Perturbation Methods p. 24 /40
25 Step 3: Log-linearize Now substitute away consumption in the EE using the intratemporal FOC: [ E t {β r z t+1 +(1 α) (ĥt+1 ˆk )] ) t+1 z t+1 α(ˆkt+1 ĥt } ĥ t+1 = γ ) z t +α(ˆkt ĥt 1 γ ĥ t [ ) c z t +α(ˆkt ĥt 1 ĥ t ]+ḡg t + kˆk [ ] t+1 = ȳ z t +αˆk t +(1 α)ĥt +(1 δ) kˆk t γ z t+1 = ρz t +σ z ε t+1 g t+1 = ρ g g t +σ g η t+1 We cannot reduce the dimensionality of the system further. Note: the variance of the shock does not affect any of the first two equilibrium equations. Certainty equivalence holds. G. Violante, Perturbation Methods p. 25 /40
26 Step 4: Solve the system To ease the notation, express the system in recursive form and collect terms: ] 0 = E[ˆk +φ 1ˆk +φ2 ĥ +φ 3 ĥ+φ 4 z +φ 5 z 0 = ˆk +φ 8ˆk +φ9 ĥ+φ 10 z +φ 11 g z = ρ z z +σ z ε g = ρ g g +σ g η We are looking for a linear solution to the system of the form: ˆk = λ 1ˆk +λ2 z +λ 3 g ĥ = λ 4ˆk +λ5 z +λ 6 g We want to express the λ s as a function of the φ s and the ρ s G. Violante, Perturbation Methods p. 26 /40
27 Step 4: Solve the system Substituting decision rules into the first two equations: 0 = (λ 1 +φ 1 +φ 2 λ 4 λ 1 +φ 3 λ 4 )ˆk + (λ 2 +φ 2 λ 4 λ 2 +φ 2 λ 5 ρ z +φ 3 λ 5 +φ 4 ρ z +φ 5 )z + (λ 3 +φ 2 λ 4 λ 3 +φ 2 λ 6 ρ g +φ 3 λ 6 )g 0 = (λ 1 +φ 8 +φ 9 λ 4 )ˆk +(λ 2 +φ 9 λ 5 +φ 10 )z +(λ 3 +φ 9 λ 6 +φ 11 )g Since both equations need to be satisfied for all (ˆk,z,g ) : λ 1 +φ 1 +φ 2 λ 4 λ 1 +φ 3 λ 4 = 0 λ 2 +φ 2 λ 4 λ 2 +φ 2 λ 5 ρ z +φ 3 λ 5 +φ 4 ρ z +φ 5 = 0 λ 3 +φ 2 λ 4 λ 3 +φ 2 λ 6 ρ g +φ 3 λ 6 = 0 λ 1 +φ 8 +φ 9 λ 4 = 0 λ 2 +φ 9 λ 5 +φ 10 = 0 λ 3 +φ 9 λ 6 +φ 11 = 0 Linear system of 6 equations into 6 unknowns (λ s) G. Violante, Perturbation Methods p. 27 /40
28 General case Let x be the m = 1 endogenous states (k), y the n = 1 control (h), z as the k = 2 exogenous states (z,g). Write the system as: 0 = Ax +Bx+Cy +Dz 0 = E[Fx +Gx +Hx+Jy +Ky +Lz +Mz] z = Nz +ε Note that in our model F = 0 and you can substitute out z and collapse the other terms with M so L = 0. Note also that the C is a square matrix, in our case (1,1) so it can be inverted. When it is not square, you have to amend a bit the strategy. G. Violante, Perturbation Methods p. 28 /40
29 General case The decision rules can be expressed as: x = Px+Qz y = Rx+Sz where P is (m m), Q is (m k), R is (n m), and S is (n k) Substitute the decision rules into the system: 0 = (AP +B +CR)x+(AQ+CS +D)z 0 = (GP +H +JRP +KR)x+(GQ+JRQ+JSN +KS +M)z Since these equations must hold for any x and z: 0 = AP +B +CR (1) 0 = AQ+CS +D (2) 0 = GP +H +JRP +KR (3) 0 = GQ+JRQ+JSN +KS +M (4) G. Violante, Perturbation Methods p. 29 /40
30 General case Solving equation (1) for R, we obtain: R = C 1 (AP +B)... and here is where it s useful that C is a square matrix (invertible) Substituting this expression for R into equation (3): JC 1 AP 2 ( JC 1 B +KC 1 A G ) P ( KC 1 B H ) = 0 This is a quadratic matrix equation in P. A number of ways to solve this problem. Uhlig suggests using the generalized eigenvalue problem (also called the QZ decomposition or generalized Schur decomposition) P,R. Now we need Q,S G. Violante, Perturbation Methods p. 30 /40
31 General case Now, solve for S as a function of Q with equation (2): S = C 1 (AQ+D) To solve for Q substitute expression above into equation (4): GQ+JRQ JC 1 (AQ+D)N KC 1 (AQ+D)+M = 0 ( G+JR KC 1 A ) Q JC 1 AQN JC 1 DN KC 1 D +M = 0 Q is the only unknown, but not trivial as Q is sandwiched in some terms. Use vec(axb) = ( B T A ) vec(x), where is the Kronecker product, to pull out the sandwiched X matrix. Apply this vectorization to equation (4) to obtain Q: 0 = I ( G+JR KC 1 A ) vec(q) N T ( JC 1 A ) vec(q) vec(jc 1 DN KC 1 D +M) [ vec(q) = I ( G+JR KC 1 A ) N T ( JC 1 A )] 1 vec(jc 1 DN KC 1 D +M) G. Violante, Perturbation Methods p. 31 /40
32 Dynare It is a collection of routines which solves and simulates non-linear models with forward looking variables. It can be used in either MATLAB or Octave (an open source and free Matlab clone). It is freely downloadable from the Dynare website Dynare can solve forward looking non-linear models under perfect foresight, when the model is deterministic, and rational expectations, when the model is stochastic. In the first case, the solution method preserves all the non-linearities. In the second case, Dynare computes a (first or second order) local approximation around the steady state G. Violante, Perturbation Methods p. 32 /40
33 Equilibrium conditions for the stochastic growth model 1 c t w t = βe t = ϕh 1 γ t c t w t r t [ ] 1+rt+1 δ c t+1 = (1 α)e z t kt α h α t = αe z t kt α 1 h 1 α t c t +k t+1 = w t h t +(1+r t )k t k t+1 = (1 δ)k t +i t y t z t+1 = c t +i t = ρz t +σ z ε t 8 endogenous variables (c t,h t,k t+1,w t,r t,i t,y t,z t+1 ) and 8 equations. One exogenous variable ε t G. Violante, Perturbation Methods p. 33 /40
34 Dynare timing convention Predetermined variables at date t must have time index t 1 Rewrite system with this convention: 1 c t w t = βe t = ϕh 1 γ t c t w t r t [ ] 1+rt+1 δ c t+1 = (1 α)e z t k α t 1h α t = αe z t kt 1 α 1 h1 α t c t +k t k t y t z t = w t h t +(1+r t )k t 1 = (1 δ)k t 1 +i t = c t +i t = ρz t 1 +σ z ε t G. Violante, Perturbation Methods p. 34 /40
35 Dynare program blocks Write a file with extension.mod and run it in Matlab by typing dynare filename. The file must contain 6 blocks: 1. Labeling block: declare endogenous variables, exogenous variables, and parameters. 2. Parameter values block: assigns values to the parameters. 3. Model block: between model and end write your n equil. conditions 4. Initialization block: Dynare solves for the SS. 5. Shock variance block: declare the parameters that is the variance of the shock 6. Solution block: solves model and computes IRFs G. Violante, Perturbation Methods p. 35 /40
36 Blocks 1-2 Labeling block (endo vars, exo vars, parameters) var c h y i w r k z; varexo eps; parameters gamma beta alpha delta psi rho sigma_eps; Parameter values block gamma = 0.5; beta = 0.99; alpha = 0.333; delta = 0.025; psi = 1.5; rho = 0.95; sigma_eps = 0.01; G. Violante, Perturbation Methods p. 36 /40
37 Model Block Dynare s default: linear approximation. If we want log-linear approximations, need to specify a variable as exp(x), so that x is the log of the variable of interest and exp(x) is the level Dynare knows that when there is a (+1) there is an expectation model; 1/exp(c) = beta (1/exp(c(+1)) (1+exp(r(+1)) delta)); exp(w)/exp(c) = psi exp(h)ˆ(1/gamma); exp(w) = (1 alpha) exp(z) exp(k( 1))ˆalpha exp(h)ˆ( alpha); exp(r) = alpha exp(z) exp(k( 1))ˆ(alpha 1) exp(h)ˆ(1 alpha); exp(c)+exp(k) = exp(w) exp(h)+(1+exp(r)) exp(k( 1)); exp(k) = (1 delta) exp(k( 1))+exp(i); exp(y) = exp(c)+exp(i); z = rho z+eps; end; G. Violante, Perturbation Methods p. 37 /40
38 Initialization block for steady state Dynare will solve for the steady state numerically, if you provide an initial guess When giving initial values, remember that Dynare is interpreting all the variables as logs initval; k = log(9); c = log(0.7); h = log(0.3); w = log(2); r = log(0.02); y = log(1); i = log(0.3); z = 0; end; G. Violante, Perturbation Methods p. 38 /40
39 Initialization block for steady state Shock variance block shocks; vareps= sigma_epsˆ2; end; Computation of steady state steady; Solution block stoch_simul(hpfilter=1600,order=1,irf=40); order is the order of approximation of the model around the SS G. Violante, Perturbation Methods p. 39 /40
40 Output of Dynare 1. steady state values of endogenous variables 2. a model summary, which counts variables by type 3. covariance matrix of shocks (which in this example is a scalar) 4. the policy and transition functions (in state space notation) c t = c+a c,k (k t 1 k)+a c,z 1 (z t 1 z)+a c,z ε t that can be rewritten, in more familiar notation as: c t = c+a c,k (k t 1 k)+a c,z (z t z) where a c,z = a c,z 1 ρ 5. first and second moments, correl. matrix, autocov. up to order 5 6. IRFs G. Violante, Perturbation Methods p. 40 /40
Session 2 Working with Dynare
Session 2 Working with Dynare Seminar: Macroeconomics and International Economics Philipp Wegmüller UniBern Spring 2015 Philipp Wegmüller (UniBern) Session 2 Working with Dynare Spring 2015 1 / 20 Dynare
More informationFirst order approximation of stochastic models
First order approximation of stochastic models Shanghai Dynare Workshop Sébastien Villemot CEPREMAP October 27, 2013 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October
More informationStochastic simulations with DYNARE. A practical guide.
Stochastic simulations with DYNARE. A practical guide. Fabrice Collard (GREMAQ, University of Toulouse) Adapted for Dynare 4.1 by Michel Juillard and Sébastien Villemot (CEPREMAP) First draft: February
More informationDYNARE SUMMER SCHOOL
DYNARE SUMMER SCHOOL Introduction to Dynare and local approximation. Michel Juillard June 12, 2017 Summer School website http://www.dynare.org/summerschool/2017 DYNARE 1. computes the solution of deterministic
More informationReal Business Cycle Model (RBC)
Real Business Cycle Model (RBC) Seyed Ali Madanizadeh November 2013 RBC Model Lucas 1980: One of the functions of theoretical economics is to provide fully articulated, artificial economic systems that
More informationLinearization. (Lectures on Solution Methods for Economists V: Appendix) Jesús Fernández-Villaverde 1 and Pablo Guerrón 2 February 21, 2018
Linearization (Lectures on Solution Methods for Economists V: Appendix) Jesús Fernández-Villaverde 1 and Pablo Guerrón 2 February 21, 2018 1 University of Pennsylvania 2 Boston College Basic RBC Benchmark
More informationLinearized Euler Equation Methods
Linearized Euler Equation Methods Quantitative Macroeconomics [Econ 5725] Raül Santaeulàlia-Llopis Washington University in St. Louis Spring 206 Raül Santaeulàlia-Llopis (Wash.U.) Linearized Euler Equation
More informationSolving Monetary (MIU) models with Linearized Euler Equations: Method of Undetermined Coefficients
Int. J. Res. Ind. Eng. Vol. 6, No. 2 (207) 72 83 International Journal of Research in Industrial Engineering www.riejournal.com Solving Monetary (MIU) models with Linearized Euler Equations: Method of
More informationG Recitation 9: Log-linearization. Contents. 1 What is log-linearization? Some basic results 2. 2 Application: the baseline RBC model 4
Contents 1 What is log-linearization? Some basic results 2 2 Application: the baseline RBC model 4 1 1 What is log-linearization? Some basic results Log-linearization is a first-order Taylor expansion,
More informationDynare. Wouter J. Den Haan London School of Economics. c by Wouter J. Den Haan
Dynare Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan Introduction What is the objective of perturbation? Peculiarities of Dynare & some examples Incorporating Dynare in other Matlab
More informationSuggested Solutions to Homework #6 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homework #6 Econ 511b (Part I), Spring 2004 1. (a) Find the planner s optimal decision rule in the stochastic one-sector growth model without valued leisure by linearizing the Euler
More informationGraduate Macroeconomics - Econ 551
Graduate Macroeconomics - Econ 551 Tack Yun Indiana University Seoul National University Spring Semester January 2013 T. Yun (SNU) Macroeconomics 1/07/2013 1 / 32 Business Cycle Models for Emerging-Market
More informationLecture 15. Dynamic Stochastic General Equilibrium Model. Randall Romero Aguilar, PhD I Semestre 2017 Last updated: July 3, 2017
Lecture 15 Dynamic Stochastic General Equilibrium Model Randall Romero Aguilar, PhD I Semestre 2017 Last updated: July 3, 2017 Universidad de Costa Rica EC3201 - Teoría Macroeconómica 2 Table of contents
More informationIntroduction to Coding DSGE Models with Dynare
Introduction to Coding DSGE Models with Macroeconomic Theory (MA1213) Juha Kilponen Bank of Finland Additional Material for Lecture 3, January 23, 2013 is preprocessor and a collection of Matlab (and GNU
More informationDynare Class on Heathcote-Perri JME 2002
Dynare Class on Heathcote-Perri JME 2002 Tim Uy University of Cambridge March 10, 2015 Introduction Solving DSGE models used to be very time consuming due to log-linearization required Dynare is a collection
More informationChristiano 416, Fall 2014 Homework 1, due Wednesday, October Consider an economy in which household preferences have the following form:
Christiano 416, Fall 2014 Homework 1, due Wednesday, October 15. 1. Consider an economy in which household preferences have the following form: β t u(c t, n t ), β =.99, and u(c t, n t ) = log(c t ) +
More informationThe full RBC model. Empirical evaluation
The full RBC model. Empirical evaluation Lecture 13 (updated version), ECON 4310 Tord Krogh October 24, 2012 Tord Krogh () ECON 4310 October 24, 2012 1 / 49 Today s lecture Add labor to the stochastic
More informationDSGE-Models. Calibration and Introduction to Dynare. Institute of Econometrics and Economic Statistics
DSGE-Models Calibration and Introduction to Dynare Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics willi.mutschler@uni-muenster.de Summer 2012 Willi Mutschler
More informationAn Introduction to Perturbation Methods in Macroeconomics. Jesús Fernández-Villaverde University of Pennsylvania
An Introduction to Perturbation Methods in Macroeconomics Jesús Fernández-Villaverde University of Pennsylvania 1 Introduction Numerous problems in macroeconomics involve functional equations of the form:
More information... Solving Dynamic General Equilibrium Models Using Log Linear Approximation
... Solving Dynamic General Equilibrium Models Using Log Linear Approximation 1 Log-linearization strategy Example #1: A Simple RBC Model. Define a Model Solution Motivate the Need to Somehow Approximate
More informationMacroeconomics Theory II
Macroeconomics Theory II Francesco Franco FEUNL February 2016 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 1 / 18 Road Map Research question: we want to understand businesses cycles.
More informationPerturbation Methods I: Basic Results
Perturbation Methods I: Basic Results (Lectures on Solution Methods for Economists V) Jesús Fernández-Villaverde 1 and Pablo Guerrón 2 March 19, 2018 1 University of Pennsylvania 2 Boston College Introduction
More informationA simple macro dynamic model with endogenous saving rate: the representative agent model
A simple macro dynamic model with endogenous saving rate: the representative agent model Virginia Sánchez-Marcos Macroeconomics, MIE-UNICAN Macroeconomics (MIE-UNICAN) A simple macro dynamic model with
More informationIntroduction to Numerical Methods
Introduction to Numerical Methods Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan "D", "S", & "GE" Dynamic Stochastic General Equilibrium What is missing in the abbreviation? DSGE
More informationHansen s basic RBC model
ansen s basic RBC model George McCandless UCEMA Spring ansen s RBC model ansen s RBC model First RBC model was Kydland and Prescott (98) "Time to build and aggregate uctuations," Econometrica Complicated
More informationSolving a Dynamic (Stochastic) General Equilibrium Model under the Discrete Time Framework
Solving a Dynamic (Stochastic) General Equilibrium Model under the Discrete Time Framework Dongpeng Liu Nanjing University Sept 2016 D. Liu (NJU) Solving D(S)GE 09/16 1 / 63 Introduction Targets of the
More informationSolving Linear Rational Expectation Models
Solving Linear Rational Expectation Models Dr. Tai-kuang Ho Associate Professor. Department of Quantitative Finance, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan 30013,
More informationDeterministic Models
Deterministic Models Perfect foreight, nonlinearities and occasionally binding constraints Sébastien Villemot CEPREMAP June 10, 2014 Sébastien Villemot (CEPREMAP) Deterministic Models June 10, 2014 1 /
More informationWhen Inequality Matters for Macro and Macro Matters for Inequality
When Inequality Matters for Macro and Macro Matters for Inequality SeHyoun Ahn Princeton Benjamin Moll Princeton Greg Kaplan Chicago Tom Winberry Chicago Christian Wolf Princeton STLAR Conference, 21 April
More informationMacroeconomics Theory II
Macroeconomics Theory II Francesco Franco FEUNL February 2011 Francesco Franco Macroeconomics Theory II 1/34 The log-linear plain vanilla RBC and ν(σ n )= ĉ t = Y C ẑt +(1 α) Y C ˆn t + K βc ˆk t 1 + K
More informationApproximation around the risky steady state
Approximation around the risky steady state Centre for International Macroeconomic Studies Conference University of Surrey Michel Juillard, Bank of France September 14, 2012 The views expressed herein
More informationSolving Deterministic Models
Solving Deterministic Models Shanghai Dynare Workshop Sébastien Villemot CEPREMAP October 27, 2013 Sébastien Villemot (CEPREMAP) Solving Deterministic Models October 27, 2013 1 / 42 Introduction Deterministic
More informationPerturbation and Projection Methods for Solving DSGE Models
Perturbation and Projection Methods for Solving DSGE Models Lawrence J. Christiano Discussion of projections taken from Christiano Fisher, Algorithms for Solving Dynamic Models with Occasionally Binding
More informationADVANCED MACROECONOMICS I
Name: Students ID: ADVANCED MACROECONOMICS I I. Short Questions (21/2 points each) Mark the following statements as True (T) or False (F) and give a brief explanation of your answer in each case. 1. 2.
More informationSimple New Keynesian Model without Capital. Lawrence J. Christiano
Simple New Keynesian Model without Capital Lawrence J. Christiano Outline Formulate the nonlinear equilibrium conditions of the model. Need actual nonlinear conditions to study Ramsey optimal policy, even
More informationSolution Methods. Jesús Fernández-Villaverde. University of Pennsylvania. March 16, 2016
Solution Methods Jesús Fernández-Villaverde University of Pennsylvania March 16, 2016 Jesús Fernández-Villaverde (PENN) Solution Methods March 16, 2016 1 / 36 Functional equations A large class of problems
More information1. Using the model and notations covered in class, the expected returns are:
Econ 510a second half Yale University Fall 2006 Prof. Tony Smith HOMEWORK #5 This homework assignment is due at 5PM on Friday, December 8 in Marnix Amand s mailbox. Solution 1. a In the Mehra-Prescott
More information(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Government Purchases and Endogenous Growth Consider the following endogenous growth model with government purchases (G) in continuous time. Government purchases enhance production, and the production
More informationExtracting Rational Expectations Model Structural Matrices from Dynare
Extracting Rational Expectations Model Structural Matrices from Dynare Callum Jones New York University In these notes I discuss how to extract the structural matrices of a model from its Dynare implementation.
More informationNews-Shock Subroutine for Prof. Uhlig s Toolkit
News-Shock Subroutine for Prof. Uhlig s Toolkit KENGO NUTAHARA Graduate School of Economics, University of Tokyo, and the JSPS Research Fellow ee67003@mail.ecc.u-tokyo.ac.jp Revised: October 23, 2007 (Fisrt
More informationApplications for solving DSGE models. October 25th, 2011
MATLAB Workshop Applications for solving DSGE models Freddy Rojas Cama Marola Castillo Quinto Preliminary October 25th, 2011 A model to solve The model The model A model is set up in order to draw conclusions
More informationSuggested Solutions to Problem Set 2
Macroeconomic Theory, Fall 03 SEF, HKU Instructor: Dr. Yulei Luo October 03 Suggested Solutions to Problem Set. 0 points] Consider the following Ramsey-Cass-Koopmans model with fiscal policy. First, we
More informationMoney in the utility model
Money in the utility model Monetary Economics Michaª Brzoza-Brzezina Warsaw School of Economics 1 / 59 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6
More information1 The Basic RBC Model
IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of
More informationThe Basic RBC model with Dynare
The Basic RBC model with Dynare Petros Varthalitis December 200 Abstract This paper brie y describes the basic RBC model without steady-state growth.it uses dynare software to simulate the model under
More informationGraduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models
Graduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models Eric Sims University of Notre Dame Spring 2011 This note describes very briefly how to conduct quantitative analysis on a linearized
More informationWhen Inequality Matters for Macro and Macro Matters for Inequality
When Inequality Matters for Macro and Macro Matters for Inequality SeHyoun Ahn Princeton Benjamin Moll Princeton Greg Kaplan Chicago Tom Winberry Chicago Christian Wolf Princeton New York Fed, 29 November
More informationMA Advanced Macroeconomics: 7. The Real Business Cycle Model
MA Advanced Macroeconomics: 7. The Real Business Cycle Model Karl Whelan School of Economics, UCD Spring 2016 Karl Whelan (UCD) Real Business Cycles Spring 2016 1 / 38 Working Through A DSGE Model We have
More informationDecentralised economies I
Decentralised economies I Martin Ellison 1 Motivation In the first two lectures on dynamic programming and the stochastic growth model, we solved the maximisation problem of the representative agent. More
More informationADVANCED MACROECONOMIC TECHNIQUES NOTE 1c
316-406 ADVANCED MACROECONOMIC TECHNIQUES NOTE 1c Chris Edmond hcpedmond@unimelb.edu.aui Linearizing a difference equation We will frequently want to linearize a difference equation of the form x t+1 =
More informationPublic Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model. Burkhard Heer University of Augsburg, Germany
Public Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model Burkhard Heer University of Augsburg, Germany October 3, 2018 Contents I 1 Central Planner 2 3 B. Heer c Public Economics: Chapter
More informationThe welfare cost of energy insecurity
The welfare cost of energy insecurity Baltasar Manzano (Universidade de Vigo) Luis Rey (bc3) IEW 2013 1 INTRODUCTION The 1973-1974 oil crisis revealed the vulnerability of developed economies to oil price
More informationThe economy is populated by a unit mass of infinitely lived households with preferences given by. β t u(c Mt, c Ht ) t=0
Review Questions: Two Sector Models Econ720. Fall 207. Prof. Lutz Hendricks A Planning Problem The economy is populated by a unit mass of infinitely lived households with preferences given by β t uc Mt,
More informationDSGE (3): BASIC NEOCLASSICAL MODELS (2): APPROACHES OF SOLUTIONS
DSGE (3): Stochastic neoclassical growth models (2): how to solve? 27 DSGE (3): BASIC NEOCLASSICAL MODELS (2): APPROACHES OF SOLUTIONS. Recap of basic neoclassical growth models and log-linearisation 2.
More informationSolutions Methods in DSGE (Open) models
Solutions Methods in DSGE (Open) models 1. With few exceptions, there are not exact analytical solutions for DSGE models, either open or closed economy. This is due to a combination of nonlinearity and
More informationDynamic stochastic general equilibrium models. December 4, 2007
Dynamic stochastic general equilibrium models December 4, 2007 Dynamic stochastic general equilibrium models Random shocks to generate trajectories that look like the observed national accounts. Rational
More informationLecture 7: Linear-Quadratic Dynamic Programming Real Business Cycle Models
Lecture 7: Linear-Quadratic Dynamic Programming Real Business Cycle Models Shinichi Nishiyama Graduate School of Economics Kyoto University January 10, 2019 Abstract In this lecture, we solve and simulate
More informationLinear and Loglinear Approximations (Started: July 7, 2005; Revised: February 6, 2009)
Dave Backus / NYU Linear and Loglinear Approximations (Started: July 7, 2005; Revised: February 6, 2009) Campbell s Inspecting the mechanism (JME, 1994) added some useful intuition to the stochastic growth
More informationSmall Open Economy RBC Model Uribe, Chapter 4
Small Open Economy RBC Model Uribe, Chapter 4 1 Basic Model 1.1 Uzawa Utility E 0 t=0 θ t U (c t, h t ) θ 0 = 1 θ t+1 = β (c t, h t ) θ t ; β c < 0; β h > 0. Time-varying discount factor With a constant
More informationAdvanced Econometrics III, Lecture 5, Dynamic General Equilibrium Models Example 1: A simple RBC model... 2
Advanced Econometrics III, Lecture 5, 2017 1 Contents 1 Dynamic General Equilibrium Models 2 1.1 Example 1: A simple RBC model.................... 2 1.2 Approximation Method Based on Linearization.............
More informationComputing first and second order approximations of DSGE models with DYNARE
Computing first and second order approximations of DSGE models with DYNARE Michel Juillard CEPREMAP Computing first and second order approximations of DSGE models with DYNARE p. 1/33 DSGE models E t {f(y
More informationECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2
ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2 The due date for this assignment is Tuesday, October 2. ( Total points = 50). (Two-sector growth model) Consider the
More informationLecture 4 The Centralized Economy: Extensions
Lecture 4 The Centralized Economy: Extensions Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 36 I Motivation This Lecture considers some applications
More informationDynamic Stochastic General Equilibrium Models
Dynamic Stochastic General Equilibrium Models Dr. Andrea Beccarini M.Sc. Willi Mutschler Summer 2014 A. Beccarini () Advanced Macroeconomics DSGE Summer 2014 1 / 33 The log-linearization procedure: One
More informationLecture 2. (1) Aggregation (2) Permanent Income Hypothesis. Erick Sager. September 14, 2015
Lecture 2 (1) Aggregation (2) Permanent Income Hypothesis Erick Sager September 14, 2015 Econ 605: Adv. Topics in Macroeconomics Johns Hopkins University, Fall 2015 Erick Sager Lecture 2 (9/14/15) 1 /
More informationproblem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
More informationProjection Methods. Michal Kejak CERGE CERGE-EI ( ) 1 / 29
Projection Methods Michal Kejak CERGE CERGE-EI ( ) 1 / 29 Introduction numerical methods for dynamic economies nite-di erence methods initial value problems (Euler method) two-point boundary value problems
More informationWhat are we going to do?
RBC Model Analyzes to what extent growth and business cycles can be generated within the same framework Uses stochastic neoclassical growth model (Brock-Mirman model) as a workhorse, which is augmented
More informationMacroeconomics II Money in the Utility function
McCless]Prof. McCless UCEMA Prof. McCless UCEMA Macroeconomics II Money in the Utility function [ October, 00 Money in the Utility function Money in the Utility function Alternative way to add money to
More informationNew Keynesian Macroeconomics
New Keynesian Macroeconomics Chapter 4: The New Keynesian Baseline Model (continued) Prof. Dr. Kai Carstensen Ifo Institute for Economic Research and LMU Munich May 21, 212 Prof. Dr. Kai Carstensen (LMU
More informationPerturbation and Projection Methods for Solving DSGE Models
Perturbation and Projection Methods for Solving DSGE Models Lawrence J. Christiano Discussion of projections taken from Christiano Fisher, Algorithms for Solving Dynamic Models with Occasionally Binding
More informationMacroeconomics I. University of Tokyo. Lecture 13
Macroeconomics I University of Tokyo Lecture 13 The Neo-Classical Growth Model II: Distortionary Taxes LS Chapter 11. Julen Esteban-Pretel National Graduate Institute for Policy Studies Environment! Time
More informationFoundation of (virtually) all DSGE models (e.g., RBC model) is Solow growth model
THE BASELINE RBC MODEL: THEORY AND COMPUTATION FEBRUARY, 202 STYLIZED MACRO FACTS Foundation of (virtually all DSGE models (e.g., RBC model is Solow growth model So want/need/desire business-cycle models
More informationThe Real Business Cycle Model
The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.
More informationEndogenous Growth. Lecture 17 & 18. Topics in Macroeconomics. December 8 & 9, 2008
Review: Solow Model Review: Ramsey Model Endogenous Growth Lecture 17 & 18 Topics in Macroeconomics December 8 & 9, 2008 Lectures 17 & 18 1/29 Topics in Macroeconomics Outline Review: Solow Model Review:
More informationBayesian Estimation of DSGE Models: Lessons from Second-order Approximations
Bayesian Estimation of DSGE Models: Lessons from Second-order Approximations Sungbae An Singapore Management University Bank Indonesia/BIS Workshop: STRUCTURAL DYNAMIC MACROECONOMIC MODELS IN ASIA-PACIFIC
More informationA comparison of numerical methods for the. Solution of continuous-time DSGE models. Juan Carlos Parra Alvarez
A comparison of numerical methods for the solution of continuous-time DSGE models Juan Carlos Parra Alvarez Department of Economics and Business, and CREATES Aarhus University, Denmark November 14, 2012
More informationPerturbation Methods
Perturbation Methods Jesús Fernández-Villaverde University of Pennsylvania May 28, 2015 Jesús Fernández-Villaverde (PENN) Perturbation Methods May 28, 2015 1 / 91 Introduction Introduction Remember that
More informationThe Growth Model in Continuous Time (Ramsey Model)
The Growth Model in Continuous Time (Ramsey Model) Prof. Lutz Hendricks Econ720 September 27, 2017 1 / 32 The Growth Model in Continuous Time We add optimizing households to the Solow model. We first study
More informationResolving the Missing Deflation Puzzle. June 7, 2018
Resolving the Missing Deflation Puzzle Jesper Lindé Sveriges Riksbank Mathias Trabandt Freie Universität Berlin June 7, 218 Motivation Key observations during the Great Recession: Extraordinary contraction
More informationMacroeconomic Theory and Analysis Suggested Solution for Midterm 1
Macroeconomic Theory and Analysis Suggested Solution for Midterm February 25, 2007 Problem : Pareto Optimality The planner solves the following problem: u(c ) + u(c 2 ) + v(l ) + v(l 2 ) () {c,c 2,l,l
More informationPANEL DISCUSSION: THE ROLE OF POTENTIAL OUTPUT IN POLICYMAKING
PANEL DISCUSSION: THE ROLE OF POTENTIAL OUTPUT IN POLICYMAKING James Bullard* Federal Reserve Bank of St. Louis 33rd Annual Economic Policy Conference St. Louis, MO October 17, 2008 Views expressed are
More informationSignaling Effects of Monetary Policy
Signaling Effects of Monetary Policy Leonardo Melosi London Business School 24 May 2012 Motivation Disperse information about aggregate fundamentals Morris and Shin (2003), Sims (2003), and Woodford (2002)
More informationThe Basic New Keynesian Model. Jordi Galí. June 2008
The Basic New Keynesian Model by Jordi Galí June 28 Motivation and Outline Evidence on Money, Output, and Prices: Short Run E ects of Monetary Policy Shocks (i) persistent e ects on real variables (ii)
More information0 β t u(c t ), 0 <β<1,
Part 2 1. Certainty-Equivalence Solution Methods Consider the model we dealt with previously, but now the production function is y t = f(k t,z t ), where z t is a stochastic exogenous variable. For example,
More informationInference. Jesús Fernández-Villaverde University of Pennsylvania
Inference Jesús Fernández-Villaverde University of Pennsylvania 1 A Model with Sticky Price and Sticky Wage Household j [0, 1] maximizes utility function: X E 0 β t t=0 G t ³ C j t 1 1 σ 1 1 σ ³ N j t
More informationu(c t, x t+1 ) = c α t + x α t+1
Review Questions: Overlapping Generations Econ720. Fall 2017. Prof. Lutz Hendricks 1 A Savings Function Consider the standard two-period household problem. The household receives a wage w t when young
More informationProblem Set 4. Graduate Macro II, Spring 2011 The University of Notre Dame Professor Sims
Problem Set 4 Graduate Macro II, Spring 2011 The University of Notre Dame Professor Sims Instructions: You may consult with other members of the class, but please make sure to turn in your own work. Where
More informationRising Wage Inequality and the Effectiveness of Tuition Subsidy Policies:
Rising Wage Inequality and the Effectiveness of Tuition Subsidy Policies: Explorations with a Dynamic General Equilibrium Model of Labor Earnings based on Heckman, Lochner and Taber, Review of Economic
More informationPermanent Income Hypothesis Intro to the Ramsey Model
Consumption and Savings Permanent Income Hypothesis Intro to the Ramsey Model Lecture 10 Topics in Macroeconomics November 6, 2007 Lecture 10 1/18 Topics in Macroeconomics Consumption and Savings Outline
More informationLecture VIII. Income Process: Facts, Estimation, and Discretization
Lecture VIII Income Process: Facts, Estimation, and Discretization Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Income Process p. 1 /26 Estimation of income process Competitive
More informationGraduate Macro Theory II: Notes on Solving Linearized Rational Expectations Models
Graduate Macro Theory II: Notes on Solving Linearized Rational Expectations Models Eric Sims University of Notre Dame Spring 2017 1 Introduction The solution of many discrete time dynamic economic models
More informationFoundations for the New Keynesian Model. Lawrence J. Christiano
Foundations for the New Keynesian Model Lawrence J. Christiano Objective Describe a very simple model economy with no monetary frictions. Describe its properties. markets work well Modify the model to
More informationHigh-dimensional Problems in Finance and Economics. Thomas M. Mertens
High-dimensional Problems in Finance and Economics Thomas M. Mertens NYU Stern Risk Economics Lab April 17, 2012 1 / 78 Motivation Many problems in finance and economics are high dimensional. Dynamic Optimization:
More informationNews Driven Business Cycles in Heterogenous Agents Economies
News Driven Business Cycles in Heterogenous Agents Economies Paul Beaudry and Franck Portier DRAFT February 9 Abstract We present a new propagation mechanism for news shocks in dynamic general equilibrium
More informationDSGE Methods. Estimation of DSGE models: Maximum Likelihood & Bayesian. Willi Mutschler, M.Sc.
DSGE Methods Estimation of DSGE models: Maximum Likelihood & Bayesian Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@uni-muenster.de Summer
More informationAdvanced Macroeconomics II The RBC model with Capital
Advanced Macroeconomics II The RBC model with Capital Lorenza Rossi (Spring 2014) University of Pavia Part of these slides are based on Jordi Galì slides for Macroeconomia Avanzada II. Outline Real business
More informationLecture Notes 7: Real Business Cycle 1
Lecture Notes 7: Real Business Cycle Zhiwei Xu (xuzhiwei@sjtu.edu.cn) In this note, we introduce the Dynamic Stochastic General Equilibrium (DSGE) model, which is most widely used modelling framework in
More informationSOLVING LINEAR RATIONAL EXPECTATIONS MODELS. Three ways to solve a linear model
SOLVING LINEAR RATIONAL EXPECTATIONS MODELS KRISTOFFER P. NIMARK Three ways to solve a linear model Solving a model using full information rational expectations as the equilibrium concept involves integrating
More informationabout this, go to the following web page: michel/dynare/
Estimation, Solution and Analysis of Equilibrium Monetary Models Assignment 3: Solution and Analysis of a Simple Dynamic General Equilibrium Model: a Tutorial The primary purpose of this tutorial is to
More information