Perturbation and Projection Methods for Solving DSGE Models

Transcription

1 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 Constraints, 2000, Journal of Economic Dynamics and Control. Discussion of perturbations primarily taken from Judd s textbook. Also: Wouter J. den Haan and Joris de Wind, How well behaved are higher order perturbation solutions? DNB working paper number 240, December For pruning, see Kim, Jinill, Sunghyun Kim, Ernst Schaumburg, and Christopher A. Sims, 2008, Calculating and using second order accurate solutions of discrete time dynamic equilibrium models, Journal of Economic Dynamics and Control, 32(11), Also, Lombardo, 2011, On approximating DSGE models by series expansions, European Central Bank.

2 Outline A Toy Example to Illustrate the basic ideas. Functional form characterization of model solution. Projections and Perturbations. Neoclassical model. Projection methods Perturbation methods Stochastic Simulations and Impulse Responses Focus on perturbation solutions of order two. The need for pruning.

3 Simple Example Suppose that x is some exogenous variable and that the following equation implicitly defines y: Let the solution be defined by the policy rule, g: satisfying for all x X h x,y 0, for all x X y g x R x;g h x,g x 0 Error function

4 The Need to Approximate Finding the policy rule, g, is a big problem outside special cases Infinite number of unknowns (i.e., one value of g for each possible x) in an infinite number of equations (i.e., one equation for each possible x). Two approaches: projection and perturbation

5 Projection Find a parametric function, ĝ x;, where is a vector of parameters chosen so that it imitates the property of the exact solution, i.e., for all x X, as well as possible. Choose values for so that R x; h x,ĝ x; is close to zero for. x X R x;g 0 The method is defined by the meaning of close to zero and by the parametric function, ĝ x;, that is used.

6 Projection, continued Spectral and finite element approximations Spectral functions: functions, ĝ x;, in which each parameter in influences for all example: n ĝ x; i 0 i H i x, ĝ x; x X 1 n H i x x i ~ordinary polynominal (not computationaly efficient) H i x T i x, T i z : 1,1 1,1, i th order Chebyshev polynomial : X 1,1

7 Zeros: Chebyshev polynomials

8 Projection, continued ĝ x; Finite element approximations: functions,, in which each parameter in influences ĝ x; over only a subinterval of x X ĝ x; X

9 Projection, continued Close to zero : two methods Collocation, for n values of x : x 1,x 2,...,x n X choose n elements of 1 n so that R x i ; h x i,ĝ x i ; 0, i 1,...,n how you choose the grid of x i s matters Weighted Residual, for m>n values of xxxxxxxx choose the n x : x 1,x 2,...,x m X m j 1 w j i h x j,ĝ x j ; 0, i 1,...,n i s

10 Example of Importance of Grid Points Here is an example, taken from a related problem, the problem of interpolation. You get to evaluate a function on a set of grid points that you select, and you must guess the shape of the function between the grid points. Consider the Runge function, f k 1 1 k 2, k 5,5 This is called the Runge phenomenon, discovered by Carl Runge in Next slide shows what happens when you select 11 equallyspaced grid points and interpolate by fitting a 10 th order polynomial. As you increase the number of grid points on a fixed interval grid, oscillations in tails grow more and more violent. Chebyshev approximation theorem: distribute more points in the tails (by selecting zeros of Chebyshev polynomial) and get convergence in sup norm.

11 How You Select the Grid Points Matters Figure from Christiano Fisher, JEDC, 2000

12 Example of Importance of Grid Points Here is an example, taken from a related problem, the problem of interpolation. You get to evaluate a function on a set of grid points that you select, and you must guess the shape of the function between the grid points. Consider the Runge function, f k 1 1 k 2, k 5,5 Next slide shows what happens when you select 11 equallyspaced grid points and interpolate by fitting a 10 th order polynomial. As you increase the number of grid points on a fixed interval grid, oscillations in tails grow more and more violent. Chebyshev approximation theorem: distribute more points in the tails (by selecting zeros of Chebyshev polynomial) and get convergence in sup norm.

13

14 Projection, continued Close to zero : two methods Collocation, for n values of x : x 1,x 2,...,x n X choose n elements of 1 n so that R x i ; h x i,ĝ x i ; 0, i 1,...,n how you choose the grid of x i s matters Weighted Residual, for m>n values of xxxxxxxx choose the n x : x 1,x 2,...,x m X m j 1 w j i h x j,ĝ x j ; 0, i 1,...,n i s

15 Perturbation Projection uses the global behavior of the functional equation to approximate solution. Problem: requires finding zeros of non linear equations. Iterative methods for doing this are a pain. Advantage: can easily adapt to situations the policy rule is not continuous or simply non differentiable (e.g., occasionally binding zero lower bound on interest rate). Perturbation method uses Taylor series expansion (computed using implicit function theorem) to approximate model solution. Advantage: can implement procedure using non iterative methods. Possible disadvantages: Global properties of Taylor series expansion not necessarily very good. Does not work when there are important non differentiabilities (e.g., occasionally binding zero lower bound on interest rate).

16 Taylor Series Expansion Let f : R R be k+1 differentiable on the open interval and continuous on the closed interval between a and x. Then, where f x P k x R k x Taylor series expansion about x a : P k x f a f 1 a x a 1 2! f 2 a x a k! f k a x a k remainder: R k x 1 k 1! f k 1 x a k 1, for some between x and a Question: is the Taylor series expansion a good approximation for f?

17 Taylor Series Expansion It s not as good as you might have thought. The next slide exhibits the accuracy of the Taylor series approximation to the Runge function. In a small neighborhood of the point where the approximation is computed (i.e., 0), higher order Taylor series approximations are increasingly accurate. Outside that small neighborhood, the quality of the approximation deteriorates with higher order approximations.

18 Taylor Series Expansions about 0 of Runge Function Increasing order of approximation leads to improved accuracy in a neighborhood of zero and reduced accuracy further away. 1 5 th order Taylor series expansion th order expansion 10 th order expansion 1 1 x x

19 Taylor Series Expansion Another example: the log function It is often used in economics. Surprisingly, Taylor series expansion does not provide a great global approximation. Approximate log(x) by its k th order Taylor series approximation at the point, x=a: log x log a 1 i 1 1 x a i a i 1 This expression diverges as N for k i x such that x a a 1

20 Taylor Series Expansion of Log Function About x= th order approximation Taylor series approximation deteriorates infinitely for x>2 as order of approximation increases. 5 th order approximation log(x) nd order approximation th order approximation x

21 Taylor Series Expansion In general, cannot expect Taylor series expansion to converge to actual function, globally. There are some exceptions, e.g., Taylor s series expansion of f x e x,cos x,sin x about x=0 converges to f(x) even for x far from 0. Problem: in general it is difficult to say for what values of x the Taylor series expansion gives a good approximation.

22 Taylor Versus Weierstrass Problems with Taylor series expansion does not represent a problem with polynomials per se as approximating functions. Weierstrass approximation theorem: for every continuous function, f(x), defined on [a,b], and for every 0, there exists a finite ordered polynomial, p(x), on [a,b] such that f x p x, for all x a,b Weierstrass polynomials may approximate well, even if sometimes the Taylor series expansion is not very good. Our analysis of the Runge function illustrates the point: The Taylor series expansion of the Runge function diverges over a large portion of the domain, while the sequence of polynomials associated with the zeros of the Chebyshev polynomials converges to the Runge function in the sup norm.

23 Ok, we re done with the digression on the Taylor series expansion. Now, back to the discussion of the perturbation method. It approximates a solution using the Taylor series expansion.

24 Perturbation Method Suppose there is a point, x X, where we know the value taken on by the function, g, that we wish to approximate: g x g,somex Use the implicit function theorem to approximate g in a neighborhood of Note: R x;g 0forallx X x R j x;g dj R x;g 0forallj, allx X. dxj

25 Perturbation, cnt d Differentiate R with respect to x and evaluate the result at x : R 1 x d dx h x,g x x x h 1 x,g h 2 x,g g x 0 g x h 1 x,g h 2 x,g Do it again! R 2 x d2 dx h x,g x 2 x x h 11 x,g 2h 12 x,g g x h 22 x,g g x 2 h 2 x,g g x. Solve this linearly for g x.

26 Perturbation, cnt d Preceding calculations deliver (assuming enough differentiability, appropriate invertibility, a high tolerance for painful notation!), recursively: g x,g x,...,g n x Then, have the following Taylor s series approximation: g x ĝ x ĝ x g g x x x 1 2 g x x x n! g n x x x n

27 Perturbation, cnt d Check. Study the graph of R x;ĝ over x X to verify that it is everywhere close to zero (or, at least in the region of interest).

28 Function: Example: a Circle h x,y x 2 y For each x except x= 2, 2, there are two distinct y that solve h(x,y)=0: y g 1 x 4 x 2, y g 2 x 4 x 2. The perturbation method does not require that the function g that solves h(x,g(x))=0 be unique. When you specify the value of the function, g, at the point of the expansion, you select the function whose Taylor series expansion is delivered by the perturbation method.

29 Example of Implicit Function Theorem y h x, y x 2 y g x g x g x x 2 2 x 2 g x h 1 x,g h 2 x,g x g h 2 had better not be zero!

30 Outline A Toy Example to Illustrate the basic ideas. Functional form characterization of model solution. Projections and Perturbations. Neoclassical model. Projection methods Perturbation methods Done! Stochastic Simulations and Impulse Responses Focus on perturbation solutions of order two. The need for pruning.

31 Neoclassical Growth Model Objective: Constraints: E 0 t 0 t u c t, u c t c 1 t 1 1 c t exp k t 1 f k t,a t, t 0,1, 2,.... a t a t 1 t, t ~ E t 0, E t 2 V f k t,a t exp k t exp a t 1 exp k t

32 Why Log Capital? Might hope to get an accurate solution. Consider the special case, 1 In this case, we know the solution is given by: K t 1 exp a t K t, K t exp k t So, in terms of log capital, the solution is exactly linear: k t 1 log a t k t Solution methods often work with polynomials (the perturbation method always does!), so in case, 1, you would get exactly the right answer.

33 Efficiency Condition E t u c t 1 f k t,a t exp k t 1 u c t 1 f k t 1, a t t 1 exp k t 2 period t 1 marginal product of capital f K k t 1, a t t 1 0. Here, k t,a t ~given numbers t 1 ~random variable time t choice variable, k t 1 Parameter,, indexes a set of models, with the model of interest corresponding to 1

34 A policy rule, With the property: Solution k t 1 g k t,a t,. R k t,a t, ;g E t u c t f k t,a t exp g k t,a t, c t 1 u f k t 1 a t 1 g k t,a t,, a t t 1 exp g k t 1 a t 1 g k t,a t,, a t t 1, f K k t 1 a t 1 g k t,a t,, a t t 1 0, for all a t,k t,.

35 Let Projection Methods ĝ k t,a t, ; be a function with finite parameters (could be either spectral or finite element, as before). Choose parameters,, to make R k t,a t, ;ĝ as close to zero as possible, over a range of values of the state. use weighted residuals or Collocation.

36 Occasionally Binding Constraints Suppose we include a non negativity constraint on investment. Lagrangian approach. Add a multiplier, λ t, to the set of functions to be computed, and add the ( complementary slackness ) equation: 0 t 0 Non negativity constraint on investment exp g k t,a t, 1 exp k t 0 Conceptually straightforward to apply preceding solution method. For details, see Christiano Fisher, Algorithms for Solving Dynamic Models with Occasionally Binding Constraints, 2000, Journal of Economic Dynamics and Control. This paper describes a wide range of strategies, including those based on parameterizing the expectation function, that may be easier, when constraints occasionally bind.

37 Perturbation Conventional application of the perturbation approach, as in the toy example, requires knowing the value taken on by the policy rule at a point. The overwhelming majority of models used in macro do have this property. In these models, can compute non stochastic steady state without any knowledge of the policy rule, g. Non stochastic steady state is such that k k g k, a 0 (nonstochastic steady state in no uncertainty case) 0 (no uncertainty) 0, 0 and k log

38 Error function: Perturbation c t R k t,a t, ;g E t u f k t,a t exp g k t,a t, c t 1 u f g k t,a t,, a t t 1 exp g g k t,a t,, a t t 1, for all values of k t,a t,. f K g k t,a t,, a t t 1 0, So, all order derivatives of R with respect to its arguments are zero (assuming they exist!).

39 Four (Easy to Show) Results About Perturbations Taylor series expansion of policy rule: g k t,a t, linear component of policy rule k g k k t k g a a t g second and higher order terms 1 2 g kk k t k 2 g aa a t 2 g 2 g ka k t k a t g k k t k g a a t... g 0 : to a first order approximation, certainty equivalence All terms found by solving linear equations, except coefficient on past endogenous variable, g k,which requires solving for eigenvalues To second order approximation: slope terms certainty equivalent g k g a 0 Quadratic, higher order terms computed recursively.

40 First Order Perturbation Working out the following derivatives and evaluating at k t k,a t 0 R k k t,a t, ;g R a k t,a t, ;g R k t,a t, ;g 0 Implies: problematic term Source of certainty equivalence In linear approximation R k u f k e g g k u f Kk g k u f k g k e g g k 2 f K 0 R a u f a e g g a u f Kk g a f Ka u f k g a f a e g g k g a g a f K 0 R u e g u f k e g g k f K g 0 Absence of arguments in these functions reflects they are evaluated in k t k,a t 0

41 Technical notes for next slide u f k e g g k u f Kk g k u f k g k e g g k 2 f K f k e g g k u f Kk u g k f k g k e g g k 2 f K 0 1 f k 1 e g u f Kk u f k f K g k e g g k 2 f K 0 f k e g f K 1 fk u u f Kk e g f K u u Simplify this further using: f K ~steady state equation f K K 1 exp a 1, K exp k exp 1 k a 1 f k exp k a 1 exp k f K e g f k e g g k g k 2 0 f Kk e g f K g k g k 2 0 f Kk 1 exp 1 k a f KK 1 K 2 exp a 1 exp 2 k a f Kk e g to obtain polynomial on next slide.

42 First Order, cont d Rewriting R k 0 term: u u f KK f K g k g k 2 0 There are two solutions, 0 g k 1, g k 1 Theory (see Stokey Lucas) tells us to pick the smaller one. In general, could be more than one eigenvalue less than unity: multiple solutions. Conditional on solution to, solved for linearly using R a 0 equation. These results all generalize to multidimensional case g k g a

43 Numerical Example Parameters taken from Prescott (1986): 0.99, 2 20, 0.36, 0. 02, 0.95, V Second order perturbation: 3.88 ĝ k t,a t 1, t, k gkk gk k t k gaa k t k a t ga a t 0 g g gka k t k a t 0 gk k t k 0 ga a t

44 Comparing 1 st and 2 nd Order Approximation In practice, 1 st order approximation is often used. Want to know if results change much, going from 1 st to 2 nd order. They appear to change very little in our example. Following is a graph that compares the policy rules implied by the first and second order perturbation. The graph itself corresponds to the baseline parameterization, and results are reported in parentheses for risk aversion equal to 20.

45 If initial capital is 20 percent away from steady state, then capital choice differs by 0.03 (0.035) percent between the two approximations. If shock is 6 standard deviations away from its mean, then capital choice differs by 0.14 (0.18) percent between the two approximations *( k t+1 (2 nd order) - k t +1 (1 st order) ) = 2 = *( k t+1 (2 nd order) - k t +1 (1 st order) ) *( k t - k * ), percent deviation of initial capital from steady state *a t, percent deviation of initial shock from steady state Number in parentheses at top correspond to = 20.

46 Outline Done! A Toy Example to Illustrate the basic ideas. Functional form characterization of model solution. Projections and Perturbations. Neoclassical model. Projection methods Perturbation methods Done! Next Stochastic Simulations and Impulse Responses Focus on perturbation solutions of order two. The need for pruning.

47 Next Stochastic simulations Mapping from shocks and initial conditions to data. 2 nd order approximation and simulation. Pruning: a way to do 2 nd order approximation. Naïve simulation versus pruning. Spurious steady state. Extended example of simulation and pruning. Impulse response functions Conditional impulse response function (IRF). Unconditional IRF.

48 Simulations Artificial data simulated from a model can be used to compute second moments and other model statistics. These can be compared with analog statistics computed in the data to evaluate model fit. Simulated model data can be compared with actual data, to evaluate model fit. The computation of impulse responses, when the solution is nonlinear, requires simulations.

49 Shock and initial conditions Simulation data Mapping from k t,a t,, t 1 to k t+2 : k t 2 g g k t,a t,, a t t 1, Mapping from k t,a t,, t 1, t 2 to k t+3 : k t 3 g g k t 1 g k t,a t,, a t 1 a t t 1,, a t 2 2 a t t 1 t 2, Similarly obtain mapping from to y t+j+1, for j=1,2,. k t,a t,, t 1,..., t j

50 2 nd order Taylor expansion and simulation We do not have the exact policy rule, g, and so must do some sort of approximation in computing the simulations. Consider the 2 nd order expansion of the mapping from k t,a t, t 1, to k t 2 : g g k t,a t,, a t t 1, After some (painful!) algebra ( y t k t k ): y t 2 g k 2 y t g k g a g a a t g a t g kkg 2 k g k g kk y 2 t g k g 2 a 2g ka g a g k g aa g aa 2 a 2 t g aa 2 2 t 1 g g k 1 2 g k g kk g a g ka 1 y t a t g ka g k y t t 1 g ka g a g aa a t t 1 g a t 1

51 Technical note to previous slide Taylor series expansion of about g g k t,a t,, a t t 1, k t k, a t t 1 0 First order terms: k t : g k g k t, a t,, a t t 1, g k k t,a t, g k 2 a t : g k g k t, a t,, a t t 1, g a k t,a t, g a g k t,a t,, a t t 1, g k g a g a t 1 : g a g k t, a t,, a t t 1, g a : g k g k t, a t,, a t t 1, g k t,a t, g g k t,a t,, a t t 1, g k g g 0 Second order terms: k t 2 : g kk g k 2 g k g kk a t 2 : g k g a 2 g ka g a g k g aa g ak g a g aa 2 2 t 1 : g aa 2 2 : g kk g 2 g k g g k g g k g g g k g g k t a t : g kk g a g k g ka g k g k g ka k t t 1 : g ka g k k t : g kk g g k g k g k g k g k 0 a t t 1 : g ka g a g aa a t : g kk g a g g k g a g k g a g ak g g a 0 t 1 : g ak g g a g a g a

52 Pruning: a way to do 2 nd order approximation Second order approximation, mapping to k t+3 from k t,a t,, t 1, t 2 is even more algebraintensive. Mapping to k t+4, k t+5, worse still. Turns out there is a simple way to compute these mappings, called pruning 1 First, draw t 1, t 2, t 3,... Then, solve linear system: ỹ t j 1 g k ỹ t j g a a t j, a t j a t j 1 t j, j 1,2,3,... Finally, Brute force substitution verifies that pruning delivers second order approximation on previous slide. y t j 1 g k y t j g a a t j 1 2 g 2 2 kkỹ t j g aa a t j g g ka ỹ t j a t j 1 Kim, Kim, Schaumburg and Sims, 2008.

53 Pruning: a way to do 2 nd order approximation Second order approximation, mapping to k t+3 from k t,a t,, t 1, t 2 is even more algebraintensive. Mapping to k t+4, k t+5, worse still. Turns out there is a simple way to compute these mappings, called pruning 1 First, draw t 1, t 2, t 3,... Then, solve linear system: ỹ t j 1 g k ỹ t j g a a t j, a t j a t j 1 t j, j 1,2,3,... Finally, Note that if y t+j were used here, then higher order powers than two would appear, and would not be second order expression. y t j 1 g k y t j g a a t j 1 2 g 2 2 kkỹ t j g aa a t j g g ka ỹ t j a t j 1 Kim, Kim, Schaumburg and Sims, 2008.

54 Pruning: a way to do 2 nd order approximation Second order approximation, mapping to k t+3 from k t,a t,, t 1, t 2 is even more algebraintensive. Mapping to k t+4, k t+5, worse still. Turns out there is a simple way to compute these mappings, called pruning 1 By using ỹ t instead of First, draw t 1, t 2, t 3,... Then, solve linear system: ỹ t j 1 g k ỹ t j g a a t j, a t j a t j 1 t j, j 1,2,3,... Finally, to remove higher order terms, you are in effect doing what a gardener who removes (prunes) diseased or unwanted parts of plants. y t j 1 g k y t j g a a t j 1 2 g 2 2 kkỹ t j g aa a t j g g ka ỹ t j a t j y t 1 Kim, Kim, Schaumburg and Sims, 2008.

55 Pruning: a way to do 2 nd order approximation Second order approximation, mapping to k t+3 from k t,a t,, t 1, t 2 is even more algebraintensive. Mapping to k t+4, k t+5, worse still. Turns out there is a simple way to compute these mappings, called pruning By using ỹ t j instead of First, draw t 1, t 2, t 3,... Then, solve linear system: ỹ t j 1 g k ỹ t j g a a t j, a t j a t j 1 t j, j 1,2,3,... Finally, y t j To remove higher order terms, you guarantee stability of simulated paths. y t j 1 g k y t j g a a t j 1 2 g 2 2 kkỹ t j g aa a t j g g ka ỹ t j a t j

56 Naïve simulation versus pruning Naïve simulation (no pruning): y t j 1 g k y t j g a a t j 1 2 g 2 2 kky t j g aa a t j g g ka y t j a t j, j 1,2,3,... Two problems k t,a t,, t 1,..., t j #1: Implied mapping from to y t+j+1, for j=1,2, contains terms of higher order than 2 and so is not a second order approximation. Those higher order terms do not correspond to the ones in the higher order Taylor series expansion (which makes use of derivatives of g of higher order than 2) and so they do not confer the desirable (local) accuracy properties of a Taylor series.

57 Naïve simulation versus pruning Naïve simulation (no pruning): y t j 1 g k y t j g a a t j 1 2 g 2 2 kky t j g aa a t j g g ka y t j a t j, j 1,2,3,... Two problems #2: Mapping from k t,a t,, t 1,..., t j to y t+j+1, for The j=1,2, second contains order terms approximation of higher to order the than policy 2 and rule so has is not a second a (definitely order approximation. spurious, in the Those higher case order of the terms neoclassical do not correspond growth model) to the steady ones in the state. higher order Taylor series expansion (which makes Spurious use of steady derivatives state marks of g of a higher transition order into than 2) and explosive so they dynamics, do not confer profoundly the desirable different (local) from accuracy properties of a Taylor series. actual g, in the case of the neoclassical model.

58 Spurious Steady State Setting a t =0 and ignoring g σσ (it s small anyway), the 2 nd order approximation to the policy rule is: y t 1 g k y t 1 2 g 2 kky t This has two steady states: y t =0 and y 2 1 g k g kk This corresponds to the following value of the capital stock: y k k log K/K K exp y k because g ignored This is very large, probably not worth worrying about in neoclassical model, but the analog object in medium sized New Keynesian models is closer to actual steady state. exp (after rounding)

59 The Spurious Steady State in 2 nd Order Approximation of Neoclassical Model Because of scale problem, it is hard to see the policy rule when graphed in the natural way, k t+1 against k t. Instead, will graph: k t+1 k t against k t k* (recall, k t = log(k t )).

60 y t gk y t ga a t gkk y t gaa a 2 t g gka y t a t y t k t k, k t ~log, capital stock Second order approximation of policy rule This is slightly higher than non stochastic ss because g σσ >0. Precautionary motive. a t 4 Var a t, capital 996 k t+1 - k t a t 2 Var a t, capital 886 a t 0, actual capital a t = Impact of a t reversed at high k k t - k*

61 Stylized Representation of 2 nd Order Approximation of Policy Rule, a=0 K t The shape of the policy rule to the left of the first positive steady state corresponds to what we know qualitatively. In neoclassical model, second steady state seems far enough to right that (perhaps) we don t have to worry about it This (spurious) steady state marks a transition to unstable dynamics. Simulations would explode if capital stock got large enough. Convexity implies you explode really fast if you get into this region K t

62 Next Stochastic simulations Mapping from shocks and initial conditions to data. 2 nd order approximation and simulation. Pruning: a way to do 2 nd order approximation. Naïve simulation versus pruning. Spurious steady state. Extended example of simulation and pruning. Impulse response functions Conditional impulse response function (IRF). Unconditional IRF. Done Next

63 Extended Example of Simulation and Pruning Simplified version of 2 nd order approximation to neoclassical model solution: 2 y t y t 1 y t 1 t, 0.8, 0.5, E 2 t, t 2 y t t 0 45 degree line t 2 After a few big positive shocks, this process will explode. Explosion big because of convexity y t

64 Simulations, std dev = σ/ y t solid line: y t y t 1 y t 1 t line: y t y t 1 t y t Two lines virtually the same for small shocks.

65 Simulations, std dev = σ y t solid line: y t y t 1 y t 1 t 1.5 line: y t y t 1 t 1 By period 71, this explodes to MATLAB s Inf and remains there Two lines wildly different for large shocks.naive simulations explode! y t

66 Pruning Procedure for simulating artificial data, using second order Taylor series expansion of simulated data as a function of the shocks and initial conditions 1. First, draw a sequence, 1, 2,..., T Next, solve for ỹ 1,ỹ 2,...,ỹ T in the linear component of the process: ỹ t ỹ t 1 t The ( pruned ) solution to the 2 nd order difference equation is y 1,y 2,...,y T in 2 y t y t 1 ỹ t 1 t Note that the y t s cannot explode. 1 For pruning with higher order perturbations, see Lombardo (2011).

67 Simulations, std dev = σ/ y t solid line: y t y t 1 ỹ t 1 t ( pruned ) line: ỹ t ỹ t 1 t The two lines roughly coincide y t

68 Simulations, std dev = σ y t solid line: y t y t 1 ỹ t 1 t ( pruned ) line: ỹ t ỹ t 1 t The two lines now differ a lot y t

69 Next Stochastic simulations Mapping from shocks and initial conditions to data. 2 nd order approximation and pruning. Naïve simulation of 2 nd order approximation, versus pruning. Spurious steady state. Extended example. done Impulse response functions Conditional impulse response function (IRF). Unconditional IRF. done

70 Impulse Response Function (Conditional) impulse response function: Impact of a shock on expectation of future variables. E y t j t 1,shock t 0 E y t j t 1,shock t 0, j 0,1,2,... Impulse responses are useful for building intuition about the economic properties of a model. When you condition on initial information, can compare responses in recessions versus booms. Can also used for model estimation, if you have the empirical analogs from VAR analysis.

71 Impulse Response Function, cnt d Example: Obviously: Also t 1 y t y t 1 t y t 1 t E y t t 1, t 0 y t 1 y t t 1 y t 1 E y t t 1, t 0 t So that: 2 y t 1 t 1 t E y t 1 t 1, t 2 y t 1 t, E y t 1 t 1, t 0 2 y t 1 E y t 1 t 1, t E y t 1 t 1, t 0 t In general: E y t j t 1, t 0 E y t j t 1, t 0 j t

72 Impulse Responses, cnt d Easy in the linear system! Impulse responses not even a function of Different story in our 2 nd order approximation, especially because of the spurious steady state.. Example: 2 y t y t 1 y t 1 t t 1 Same form as our 2 nd order approximation Obviously: t 1 y t 1 y 2 t 1 t E y t t 1, t 2 y t 1 y t 1 E y t t 1, t 0 t Easy

73 IRF, cont d One period out IRF Note: y t y t 2 y t 1 2 y t 1 y t 1 t 2 y t 1 y t 1 t 2 t 1 Then, 2 2 E y t 1 t 1, t 0 y t 1 y t 1 t y t 1 y t t 2 y t 1 y t 1 t 2 2 E y t 1 t 1, t 0 y t 1 y t 1 y t 1 y t 1 2 So, Ouch! Much more complicated is a function of elements of t 1 E t 1 t 1, t E y t 1 t 1, t E y t 1 t 1, t 0 t 2 2 t 2 y t 1 y t 1 t 0

74 IRF s, cnt d Too hard to compute IRF s by analytic formulas, when equations are not linear. What we need: Fix a value for Compute: Subtract: 2 y t y t 1 y t 1 t 2 t 1 y t 1 y t 1 our example E y t j t 1, t, j 1,2,3,...,T, for a given value of t 0 E y t j t 1, t 0, j 1,2,3,...,T. E y t j t 1, t E y t j t 1, t 0, j 1,2,3,...,T

75 Computational strategy IRF s, cnt d From a random number generator, draw: 1 t 1, 1 1 t 2,..., t T Using the stochastic equation, y t 1 y t 1 and the given compute (by pruning) Repeat this, over and over again, R (big) times, to obtain Finally, y 1 t 1,y 1 1 t 2,...,y t T y 1 t 1,y 1 1 t 2,...,y t T... y R t 1,y R R t 2,...,y t T E y t j t 1, t 1 R l 1 R y l t j, j 1,2,...,T 2 t

76 IRF s, cnt d To get E y t j t 1, t 0, j 1,2,3,...,T., just repeat the preceding calculations, except set t 0 To do the previous calculations, need R and T. Dynare will do these calculations. In the stoch_simul command, R is set by including the argument, replic=r. T is set by including irf=t.

77 Next Stochastic simulations Mapping from shocks and initial conditions to data. 2 nd order approximation and pruning. Naïve simulation of 2 nd order approximation, versus pruning. Spurious steady state. Extended example. done done Impulse response functions Conditional impulse response function (IRF). Unconditional IRF. done

78 A Different Type of Impulse Response Function The previous concept of an impulse response function required specifying the information set, t 1. How to specify this is not often discussed in part because with linear solutions it is irrelevant. With nonlinear solutions, makes a difference. How to choose? t 1 t 1 One possibility: nonstochastic steady state. Another possibility: stochastic mean.

79 Unconditional Impulse Response Functions

80 Note that Unconditional IRF E y t j t 1, t E y t j t 1, t 0, j 1,2,3,...,T is a function of t 1 (i.e., it is a random variable) Evaluate the mean of this random variable as follows: Suppose there is date 0, date t and date T, where T>t and t is itself large. Draw R sets of shocks (no need to draw t ) 1 0, 1 1,..., 1 t 1, 1 t 1, 1 1 t 2,..., t T... R 0, R 1,..., R t 1, R t 1, R R t 2,..., t T

81 Unconditional IRF Using t 0, t 0together with 1 0, 1 1,..., 1 t 1, 1 t 1, 1 1 t 2,..., t T... R 0, R 1,..., R t 1, R t 1, R R t 2,..., t T Compute two sets (by pruning) y 1 0,y 1 1,...,y 1 t 1,y 1 t,y 1 t 1,y 1 1 t 2,...,y t T... y R 0,y R 1,...,y R t 1,y R t,y R t 1,y R R t 2,...,y t T The period t+j IRF is computed by averaging across l=1,,r, for given t+j, j=0,1,,t. Then, subtract, as before. In Dynare, t is set with drop=t parameter in stoch_simul command.

82 Conclusion For modest US sized fluctuations and for aggregate quantities, it may be reasonable to work with first order perturbations. This assumption deserves much further testing. Can do this by studying the error function. Also, try fancier approximations and see if it changes your results. One alternative to first order perturbations is higher order perturbations. These must be handled with care, as they are characterized by spurious steady states, which may be the transition point to unstable dynamics. Must do some sort of pruning to compute IRF s, or just to simulate data. An alternative is to apply projection methods. Perhaps these have less problems with spurious steady states. Computation of solutions is more cumbersome in this case. First order perturbation: linearize (or, log linearize) equilibrium conditions around non stochastic steady state and solve the resulting system. This approach assumes certainty equivalence. Ok, as a first order approximation.

83 Solution by Linearization (log) Linearized Equilibrium Conditions: E t 0 z t 1 1 z t 2 z t 1 0 s t 1 1 s t 0 Posit Linear Solution: z t Az t 1 Bs t List of endogenous variables determined at t s t Ps t 1 t 0. To satisfy equil conditions, A and B must: Exogenous shocks 0 A 2 1 A 2 I 0, F 0 0 B P 1 0 A 1 B 0 If there is exactly one A with eigenvalues less than unity in absolute value, that s the solution. Otherwise, multiple solutions. Conditional on A, solve linear system for B.