126 lines
3.8 KiB
Modula-2
126 lines
3.8 KiB
Modula-2
/*
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Check the policy functions obtained by perturbation at a high approximation
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order, using the Burnside (1998, JEDC) model (for which the analytical form of
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the policy function is known).
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As shown by Burnside, the policy function for yₜ is:
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yₜ = βⁱ exp[aᵢ+bᵢ(xₜ−xₛₛ)]
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where:
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θ² ⎛ 2ρ 1−ρ²ⁱ⎞
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— aᵢ = iθxₛₛ + σ² ─────── ⎢i − ────(1−ρⁱ) + ρ² ─────⎥
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2(1−ρ)² ⎝ 1−ρ 1−ρ² ⎠
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θρ
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— bᵢ = ───(1−ρⁱ)
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1−ρ
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— xₛₛ is the steady state of x
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— σ is the standard deviation of e.
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With some algebra, it can be shown that the derivative of yₜ at the deterministic
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steady state is equal to:
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∂ᵐ⁺ⁿ⁺²ᵖ yₜ ∞ (2p)!
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──────────────── = ∑ βⁱ bᵢᵐ⁺ⁿ ρᵐ ───── cᵢᵖ exp(iθxₛₛ)
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∂ᵐxₜ₋₁ ∂ⁿeₜ ∂²ᵖs ⁱ⁼¹ p!
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where:
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— s is the stochastic scale factor
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θ² ⎛ 2ρ 1−ρ²ⁱ⎞
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— cᵢ = ─────── ⎢i − ────(1−ρⁱ) + ρ² ─────⎥
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2(1−ρ)² ⎝ 1−ρ 1−ρ² ⎠
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Note that derivatives with respect to an odd order for s (i.e. ∂²ᵖ⁺¹s) are always
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equal to zero.
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The policy function as returned in the oo_.dr.g_* matrices has the following properties:
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— its elements are pre-multiplied by the Taylor coefficients;
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— derivatives w.r.t. the stochastic scale factor have already been summed up;
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— symmetric elements are folded (and they are not pre-multiplied by the number of repetitions).
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As a consequence, the element gₘₙ corresponding to the m-th derivative w.r.t.
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to xₜ₋₁ and the n-th derivative w.r.t. to eₜ is given by:
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1 ∞ cᵢᵖ
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gₘₙ = ────── ∑ ∑ βⁱ bᵢᵐ⁺ⁿ ρᵐ ──── exp(iθxₛₛ)
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(m+n)! 0≤2p≤k-m-n ⁱ⁼¹ p!
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where k is the order of approximation.
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*/
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@#define k =7
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var y x;
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varexo e;
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parameters beta theta rho xbar;
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xbar = 0.0179;
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rho = -0.139;
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theta = -1.5;
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theta = -10;
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beta = 0.95;
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model;
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y = beta*exp(theta*x(+1))*(1+y(+1));
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x = (1-rho)*xbar + rho*x(-1)+e;
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end;
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shocks;
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var e; stderr 0.0348;
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end;
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initval;
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x = xbar;
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y = beta*exp(theta*xbar)/(1-beta*exp(theta*xbar));
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end;
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steady;
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stoch_simul(order=@{k},k_order_solver,irf=0,drop=0,periods=100);
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% Verify that the policy function coefficients are correct
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sigma2=M_.Sigma_e;
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i = 1:800;
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c = theta^2*sigma2/(2*(1-rho)^2)*(i-2*rho*(1-rho.^i)/(1-rho)+rho^2*(1-rho.^(2*i))/(1-rho^2));
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b = theta*rho*(1-rho.^i)/(1-rho);
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for ord = 0:@{k}
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g = oo_.dr.(['g_' num2str(ord)])(2,:); % Retrieve computed policy function for variable y
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for m = 0:ord % m is the derivation order with respect to x(-1)
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v = 0;
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for p = 0:floor((@{k}-ord)/2) % 2p is the derivation order with respect to s
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if ord+2*p > 0 % Skip the deterministic steady state constant
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v = v + sum(beta.^i.*exp(theta*xbar*i).*b.^ord.*rho^m.*c.^p)/factorial(ord)/factorial(p);
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end
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end
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if abs(v-g(ord+1-m)) > 1e-14
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error(['Error in matrix oo_.dr.g_' num2str(ord)])
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end
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end
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end
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% Verify that the simulated time series is correct
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xss = oo_.steady_state(2);
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xlag = xss;
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for T = 1:size(oo_.endo_simul,2)
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e_ = oo_.exo_simul(T);
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y_ = oo_.steady_state(1);
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for ord = 0:@{k}
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g = oo_.dr.(['g_' num2str(ord)])(2,:); % Retrieve computed policy function for variable y
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for m = 0:ord
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y_ = y_ + g(ord+1-m)*(xlag-xss)^m*e_^(ord-m)*nchoosek(ord,m);
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end
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end
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if abs(y_-oo_.endo_simul(1,T)) > 1e-14
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error(['Error in k_order_simul DLL'])
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end
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xlag = oo_.endo_simul(2,T);
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end
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