DYNARE COURSE Amsterdam University. Motivation. Design of the macro-language. Dynare macro language and dynare++ k-order approximation

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1 DYNARE COURSE Amsterdam University Dynare macro language and dynare++ k-order approximation Dynare macro language Michel Juillard October 23, 2008 Motivation Design of the macro-language The Dynare language used in MOD files) is well suited for describing economic models However, it lacks some useful features, such as: a loop mechanism for automatically repeating similar blocks of equations such as in multi-country models) an operator for indexed sums or products inside equations a mechanism for splitting large MOD-files in smaller modular files the possibility of conditionally including some equations or some runtime commands The Dynare Macro-language was specifically designed to address these issues Being flexible and fairly general, it can also be helpful in other situations The Dynare Macro-language provides a new set of macro-commands which can be inserted inside MOD-files Language features include: file inclusion loops conditional inclusion if/then/else structures) expression substitution The macro-processor transforms a MOD file with macro-commands into a MOD file without macro-commands doing text expansions/inclusions) and then feeds it to the Dynare parser The key point to understand is that the macro-processor only does text substitution like the C preprocessor or the PHP language)

2 Macro Directives Inclusion directive Directives begin with an at-sign followed by a pound and occupy exactly one line However, a directive can be continued on next line by adding two anti-slashes \\) at the end of the line to be continued A directive produces no output, but serves to give instructions to the macro processor This directive simply includes the content of another file at the place where it is inserted. "filename" "modelcomponent.mod" Note that it is possible to include a file from an included file nested includes). Variables The macro processor maintains its own list of variables distinct of model variables and of Matlab variables) Variables can be of four types: integer character string declared between double quotes) array of integers array of strings No boolean type: false is represented by integer zero true is any non-null integer Macro-expressions /2) It is possible to construct macro-expressions, using standard operators. Operators on integers arithmetic operators: +,-,*,/ comparison operators: <,>,<=,>=,==,!= logical operators: &&,,! integer ranges: :4 is equivalent to integer array [,2,3,4] Operators on character strings comparison operators: ==,!= concatenation: + extraction of substrings: if s is a string, then one can write s[3] or s[4:6]

3 Macro-expressions 2/2) Define directive Operators on arrays dereferencing: if v is an array, then v[2] is its 2 nd element concatenation: + difference -: returns the first operand from which the elements of the second operand have been removed extraction of sub-arrays: e.g. v[4:6] Macro-expressions can be used at two places: inside macro directives, directly in the body of the MOD-file, between an at-sign and curly braces the macro processor will substitute the expression with its value The value of a macro-variable can be defined with directive. variable_name = expression x = y = v = [, 2, 4 w = [ "foo", "bar" z = 3+v[2] Expression substitution Dummy example Before x = [ "B", "C" i = 2 A After macro-processing A = C; Loop directive variable_name in array_expr Example: before country in [ "home", "foreign" ] GDP_@{country} = K_@{country}^a * Example: after macro-processing GDP_home = K_home^a * L_home^-a); GDP_foreign = K_foreign^a * L_foreign^-a);

4 Conditional inclusion directive integer_expr body included if expr!= Example: alternative monetary policy linear_mon_pol linear_mon_pol i = w*i-) + -w)*i_ss + i = i-)^w * i_ss^-w) * Syntax integer_expr body included if expr!= body included if expr == Echo and error directives The echo directive will simply display a message on standard output The error directive will display the message and make Dynare stop only makes sense inside a conditional inclusion directive) string_expr "Information "Error message!" Saving the macro-expanded MOD file Modularization For debugging or learning purposes, it is possible to save the output of the macro-processor This output is a valid MOD-file, obtained after processing the macro-commands of the original MOD-file Just add the savemacro option on the Dynare command line after the name of your MOD-file) If MOD file is filename.mod, then the macro-expanded version will be saved in filename-macroexp.mod directive can be used to split MOD-files into several modular components Example setup: modeldesc.mod: contains variable declarations, model equations and shocks declarations simul.mod: includes modeldesc.mod, calibrates parameters and runs stochastic simulations estim.mod: includes modeldesc.mod, declares priors on parameters and runs bayesian estimation Dynare can be called on simul.mod and estim.mod but it makes no sense to run it on modeldesc.mod)

5 Indexed sums or products Example: moving average Multi-country models MOD-file skeleton example Before window = 2 var x MA_x; MA_x = i in -window:window ); After macro-processing var x MA_x; MA_x = /5* +x-2) +x-) +x0) +x) +x2) countries = [ "US", "EU", "AS", "JP", "RC" nth_co = co in countries var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ; parameters a_@{co} ; co in countries Y_@{co} = K_@{co}^a_@{co} * if co!= nth_co +i_@{co}) = +i_@{nth_co}) * E_@{co}+) / E_@{co}; // UIP else E_@{co} = Endogeneizing parameters /3) When doing the steady-state calibration of the model, it may be useful to consider a parameter as an endogenous and vice-versa) Example: y = ) α ξ l ξ + α) ξ k ξ ξ ξ lab_rat = wl py During simulation or estimation, the share parameter α is a parameter, and lab_rat is an endogenous variable But for steady-state calibration, we may want to impose an economically relevant value for lab_rat, and deduce the implied value for α during calibration, α is endogenous and lab_rat is a parameter Endogeneizing parameters 2/3) Create modeqs.mod with variable declarations and model equations For declaration of alpha and steady var alpha; parameter parameter alpha; var Create steady.mod: begins steady = then "modeqs.mod" initializes parameters including lab_rat, excluding alpha) computes steady state using hints for endogenous, including alpha) saves values of parameters and endogenous at steady-state to a file

6 Endogeneizing parameters 3/3) Possible future developments Create simul.mod: begins steady = 0 then "modeqs.mod" loads values of parameters and endogenous at steady-state from file computes simulations Note: functions for saving and loading parameters and endogenous are not yet in Dynare distribution they should be soon, ask me if you re interested) Find a nicer syntax for indexed sums/products Implement other control structures: elsif, switch/case, while/until loops Implement macro-functions or templates), with a syntax QUADRATIC_COSTx, x_ss, phi) = phi/2*x/x_ss-) The general problem Stochastic case: E t {fy t+, y t, y t, u t )} = 0 Dynare++ and k-order approximation y : vector of endogenous variables u : vector of exogenous stochastic shocks u t = σǫ t Eǫ t ) = 0 [ ] E[ǫ t ] i [ǫ t ] i2 [ǫ t ] ik ) = Σ k) ǫ i i 2 i k Then, E[u t ] i [u t ] i2 [u t ] ik ) = σ k [ Σ k)] i i 2 i k

7 The solution function k order approximation of the structural model y t = g y t, u t,σ) and F y t, u t, u t+,σ) = f g g y t, u t,σ), u t+,σ), g y t, u t,σ), y t, u t ) or with s t = [ yt u t ] F s t, u t+,σ) = f g g s t,σ), u t+,σ), g s t,σ), s t ) E t {F i s, u,σ )} = = 0 F i s, 0, 0) + E t { p j= j! j k=0 m=k j ] [F is k u m kσj m [ŝ] α [ŝ] α k [ u ] β [ u ] β m k σ j m } α α k β β m k where s = s t, ŝ = s t s t and u = u t+. In tensor notation, the same index used first as subscript and then superscript of two tensors implies summation of the products: [A] αβ [B] α [C] β = A ij B i C j. i j Reduction of conditional expectation Perturbation approach For each j =,,p, find [g s j] E t {F i s t, u t+,σ )} = = 0 F i s, 0, 0) + p j= j! j k=0 m=k j ] [F is k u m kσj m [ŝ] α [ŝ] α k σ m k [Σ ǫ ] β β m k σ j m α α k β β m k such that [F is j ] and find such that j m=k α α j = 0 [g s k σ j k] [ F i s k u m k σ j m ] α α k [Σ ǫ ] β β m k = 0

8 Faà di Bruno formula Most of what follows depends on the kth order partial derivatives of the composite of two functions, given the partial derivatives of each function. If Fr) = fzr)), [F ir j ] α α j = j [f iz ]ββl l l= c M l,j l m= [z r cm ] β m αc m ) where M l,j is the set of all partitions of the set of j indices with l classes,. is the cardinality of a set, c m is m-th class of partition c, and αc m ) is a sequence of α s indexed by c m. Note that M,j = {{,, j}} and M j,j = {{}, {2},, {j}}. Example for the third partial derivatives [ ] [F i r ]αα2α3 = f i 3 z [z r 3] β α β α 2 α 3 + [f i z [z 2 r ] ]ββ2 β α [z r 2] β 2 α 2 α 3 ] [f i z 3 + [z r ] β α 2 [z r 2] β 2 α α 3 + [z r ] β α 3 [z r 2] β 2 α α 2 )+ β β 2 β 3 [z r ] β α [z r ] β 2 α 2 [z r ] β 3 α 3 A linear equation that is hard to solve efficiently Recovering g y j Applying twice the Faá di Bruno formula to F y t, u t, u t+,σ) = f g g y t, u t,σ), u t+,σ), g y t, u t,σ), y t, u t ) and grouping together all terms not involving g y j, one gets [ fy +] β [g y] β γ or, in matrix form [ ] γ gy j + [ [ β j f α α y +] gy j β j] γ γ j m= [g y ] γ m αm + [ ] [ ] β f y 0 gy j + [B] β α α α j α j = 0 f y +g y g y j + f y +g y j g y g y g y ) + f y 0g y j + B = 0 where the derivatives are unrolled along the rows of matrix g y j An asset pricing model AX + BXC C C = D dynare++ uses an efficient algorithm proposed by Ondra Kamenik Urban Jermann 998) Asset pricing in production economies Journal of Monetary Economics, 4, real business cycle model consumption habits investment adjustment costs compares return on several securities log linearizes RBC model + log normal formulas for asset pricing

9 Simple asset pricing formulas In a production economy: Euler equation: Uc t = E t {βuc t+ r t+ } Rate of return on firm s capital: Risk free rate: rf t = Expected risk premium: r t = A t k α t n α t { [E t β Uc }] t+ Uc t E t {r t+ } rf t = E t {r t+ } { [E t β Uc }] t+ Uc t Firms The representative firm maximizes its value: with Y t D t log A t E t t+k = A t Kt α X tn t ) α = Y t W t Nt I t K t = δ)k t + X t = ρ log A t + e t = + g)x t β k µ t+k µ t D t a ξ It K t ) x + a2 ) K t Households Interest rate The representative households maximizes current value of future utility: E t β k C t χc t ) τ τ k=0 subject to the following budget constraint: W t N t + D t = C t and with N t =. Good market equilibrium imposes Risk free interest rate: r f = } E t {βg τ µ t+ µ t where µ t is the utility of a marginal unit of consumption in period t. µ t = c t χc t /g) τ χβ gc t+ χc t ) τ Y t = C t + I t

10 Rate of return jermann98.mod Rate of return of firms { ) git ξ r t = E t a αz t+ g α kt α k t // //. Variable declaration // δ + a ξ + git+ a git+ k t k t ) ξ ) ξ + a 2 gi t+ k t )} var c, d, erp, i, k, m, r, rf, w, y, z, mu; varexo ez; suite suite // // 2. Parameter declaration and calibration // parameters alf, chihab, xi, delt, tau, g, rho, a, a2, betstar, bet; alf = 0.36; // capital share in production function chihab = 0.89; // habit formation parameter xi = /4.3; // capital adjustment cost parameter delt = 0.025; // quarterly deprecition rate g =.005; //quarterly growth rate note zero growth =>g=) tau = 5; // curvature parameter with respect to c rho = 0.95; // AR) parameter for technology shock a a2 betstar bet = g-+delt)^/xi); = g-+delt)-g-+delt)^/xi))/-/xi)))* g-+delt)^-/xi))); = g/.038; = betstar/g^-tau)); // // 3. Model declaration // g*k = -delt)*k-) + a/-/xi))*g*i/k-))^-/xi)+a2)*k-); d = y - w - i; w = -alf)*y; y = z*g^-alf)*k-)^alf; c = w + d; mu = c-chihab*c-)/g)^-tau)-chihab*bet*c+)*g-chihab*c)^-tau); mu = betstar/g)*mu+)*a*g*i/k-))^-/xi))*alf*z+)*g^-alf)* k^alf-))+-delt+a/-/xi))*g*i+)/k)^-/xi)+a2))/ a*g*i+)/k)^-/xi))-g*i+)/k); logz) = rho*logz-)) + ez;

11 suite continued m = betstar/g)*mu+)/mu; rf = /m; r = a*g*i/k-))^-/xi))*alf*z+)*g^-alf)*k^alf-))+ -delt+a/-/xi))*g*i+)/k)^-/xi)+a2)/ a*g*i+)/k)^-/xi))-g*i+)/k); erp = r - rf; initval; m = betstar/g; rf = /m); r = /m); erp = r-rf; z = ; k = g/betstar)--delt))/alf*g^-alf)))^/alf-)); y = g^-alf))*k^alf; w = -alf)*y; i = -/g)*-delt))*k; d = y - w - i; c = w + d; mu = c-chihab*c/g))^-tau))-chihab*bet*c*g-chihab*c)^-tau)); ez = 0; vcov = [0.000]; Running dynare++ Three different concepts dynare++ --order=2 jermann98.mod. deterministic) steady state 2. stochastic steady state 3. unconditional expectation

12 Deterministic steady state Nonlinear decision rule A linearized decision rule cuts the main diagonal at the deterministic steady state ). In general, the decision is shifted at the deterministic steady state: agents don t decide to stay at the deterministic steady state. K t K t B A A K t K t Nonlinear decision rule Stochastic steady state The stochastic steady state, s, describe the point where agents decide to stay in absence of shocks this period, but taking into account the distibution of shocks in the future. K t K t B s C A B A K t s K t

13 Unconditional expectation Further issues Because of Jensen inequality, the unconditional expectation, EK), is somewhere below the quadratic decision rule, but not on it. In absence of shocks, agents don t decide to go to the unconditional expectation. K t s B A C Impulse response functions depend of state at time of shocks and history of future shocks. For large shocks higher order approximation simulation may explode pruning algorithm Sims) truncate normal distribution Judd) s K t State dependent risk premium premium.m Compute expected risk premium at different state points Compute capital stocks between deterministic steady state plus/minus two standard error according to first order approximation) Warning: this generates unlikely state points as previous consumption remains at steady state value! v_k = dyn_vcov8,8); k_steadystate = dyn_steady_states8); kk = k_steadystate-2*sqrtv_k):0.5: k_steadystate+2*sqrtv_k); nk = sizekk,2); for i=:nk k = kki); start = dyn_steady_states; start8) = k; temp = dynare_simul jermann98,0,start); resultsi) = temp2); end figure; plotkk,results);