Approximation around the risky steady state

Transcription

1 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 are ours and do not necessarily represent the views of Bank of France.

2 Motivation deterministic steady state : a state point where the system rests in absence of shocks (present or future). risky steady state : a state point where the system rests in the absence of shocks this period, but taking into account the likelihood of shocks in the future. Local approximation around the deterministic steady state are easy to compute. The risky steady state may be more central in the ergodic distribution. The risky steady state may exist for problems where the a unique steady state doesn t exist (i.e. portfolio choice). The term stochastic steady state is ambiguous.

3 Risky steady state The risky steady state, K sss, describes 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 sss C B K ss A K ss K sss K t-1

4 Outline 1. General model 2. 2nd order approximation around the deterministic steady state 3. Approximation around the risky steady state 4. Examples 5. Risky steady state and portfolio problems

5 General model E t {f(y t+1, y t, y t 1, u t )} = 0 E(u t ) = 0 E(u t u t ) = Σ u E(u t u τ ) = 0 t τ y : vector of endogenous variables u : vector of exogenous stochastic shocks

6 The stochastic scale variable E t {f(y t+1, y t, y t 1, u t )} = 0 At period t, the only unknown stochastic variable is y t+1, and, implicitly, u t+1. We introduce the stochastic scale variable, σ and the auxiliary random variable, ǫ t, such that u t+1 = σǫ t+1

7 The stochastic scale variable (continued) E(ǫ t ) = 0 E(ǫ t ǫ t) = Σ ǫ E(ǫ t ǫ τ) = 0 t τ and Σ u = σ 2 Σ ǫ

8 Solution function y t = g(y t 1, u t,σ) where σ is the stochastic scale of the model. If σ = 0, the model is deterministic. For σ > 0, the model is stochastic. Under some conditions, the existence of g() function is proven via an implicit function theorem. See H. Jin and K. Judd Solving Dynamic Stochastic Models (

9 Solution function (continued) Then, y t+1 = g(y t, u t+1,σ) = g(g(y t 1, u t,σ), u t+1,σ) F(y t 1, u t, u t+1,σ) = f(g(g(y t 1, u t,σ), u t+1,σ), g(y t 1, u t,σ), y t 1, u t ) E t {F(y t 1, u t,σǫ t+1,σ)} = 0

10 The perturbation approach Obtain a Taylor expansion of the unkown solution function in the neighborhood of a problem that we know how to solve. The problem that we know how to solve is the deterministic steady state. One obtains the Taylor expansion of the solution for the Taylor expansion of the original problem. One consider two different perturbations: 1. points in the neighborhood from the steady sate, 2. from a deterministic model towards a stochastic one (by increasing σ from a zero value).

11 The perturbation approach (continued) The Taylor approximation is taken with respect to y t 1, u t and σ, the arguments of the solution function y t = g(y t 1, u t,σ). At the deterministic steady state, all derivatives are deterministic as well.

12 Second order approximation of the model E t F ( y, u,σε ) { =E t F (y, 0, 0)+F y ŷ + F u u + ( F u ε ) + F σ σ [ F yy (ŷ ŷ)+f uu (u u) =0 + ( F u u ( ε ε ) + F σσ + 2F u σε ) σ 2] + F yu (ŷ u)+f yσ (ŷ σε ) + F uσ ( u σε ) +O(2) }

13 Representing the second order derivatives The second order derivatives of a vector of multivariate functions is a three dimensional object. We use the following notation 2 F x x = 2 F 1 x 1 x 1 2 F 1 x 1 x F 2 x 1 x 1 2 F 2 x 1 x F m x 1 x F m x 1 x 2 2 F 1 x 2 x F 2 x 2 x F m x 2 x 1 2 F 1 x n x n 2 F 2 x n x n... 2 F m x n x n

14 Second order approximation of the model E t F ( y, u,σε ) { =E t F (y, 0, 0)+F y ŷ + F u u + ( F u ε ) + F σ σ [ F yy (ŷ ŷ)+f uu (u u) =0 + ( F u u ( ε ε ) + F σσ + 2F u σε ) σ 2] + F yu (ŷ u)+f yσ (ŷ σε ) + F uσ ( u σε ) +O(2) }

15 Second order approximation of the model (cont d) Distributing the conditional expectation, one gets E t F ( y, u,σε ) [ =F y ŷ + F u u + F σ σ F yy (ŷ ŷ) + F uu (u u)+ (F u u Σε + F σσ )σ 2] =0 + F yu (ŷ u)+o(2) This equation can only be verified for all values of ŷ, u and σ if each partial derivatives of F is equal to 0. These are the conditions that let us recover the partial derivatives of the solution function g().

16 Second order decision function ( y t = ȳ + 0.5g σσ σ 2 + g y ŷ + g u u g yy (ŷ ŷ) ) + g uu (u u) + g yu (ŷ u) We can fix σ = 1.

17 Graphical interpretation K t B K ss A K ss K t-1

18 Approximating the risky steady state with derivatives computed at the deterministic steady state Even on the basis of a second order approximation around the deterministic steady state, it is possible to compute the risky steady state. The risky steady state must solve ỹ = ȳ + 0.5g σσ + g y (ỹ y ) + 0.5gyy ((ỹ y ) (ỹ y ))

19 Approximation around the risky steady state Let s define the risky steady state as the point ỹ such that E t F ( ỹ, 0, u ) = E t { f(g(g(ỹ, 0,σ),σǫ,σ), g(ỹ, 0,σ), ỹ, 0) } = 0, and ỹ = g ( ỹ, 0,σ ), for σ = 1. Remarks: The risky steady state depends upon the decision function But, in the perturbation approach, the determination of g() depends upon the risky steady state... ỹ and the derivatives of g() must be determined simultaneously.

20 Second order approximation of the risky steady state The condition that ỹ = g ( ỹ, 0,σ ) requires that g σ = g σσ = 0. A second order approximation of the model at the risky steady sate, when y t 1 = ỹ, u t = 0 and σ = 1 gives E t { f(ỹ, ỹ, ỹ, 0)+f y+ g u σε (f y + g uu + f y+y + (g u g u ))σ 2( ε ε )} Reducing the conditional expectation, the first order term disappears and we get f (f y + g uu + f y+y + (g u g u ))σ 2 Σε = 0 Remember that Σ u = σ 2 Σ ε.

21 Fixed point algorithm 1. Evaluate the derivatives of the model at an arbitrary guess value for the risky steady state, possibly the deterministic steady state when it exists. 2. Compute the derivatives of the solution function: g y, g u, g yy, g uu, g yu. 3. Compute the residuals f ( ỹ, ỹ, ỹ, 0 ) (f y + g uu + f y+y + (g u g u )) Σ u Use a non linear solver to find ỹ that sets the residuals to 0.

22 Third order approximation of the risky steady state The third order approximation is given by 1 f + 2 (f y + g uu + f y+y + (g u g u )) Σ u (2) + 1 ( f y+ g uuu 6 + 3f y+y + (g uu g u )+f y+y +y + (g u g u g u ) ) Σ (3) u = 0 where Σ (2) and Σ (3) represent the 2nd and, respectively, 3rd moments of the distribution of shocks u. If the shocks are drawn from symmetrical probability distribution, Σ (3) = 0 and this expression is identical to the one obtained with the second order approximation.

23 Fixed point algorithm 1. Evaluate the derivatives of the model at an arbitrary guess value for the risky steady state, possibly the deterministic steady state when it exists. 2. Compute E t F y and E t F u using previous approximation of g y and g u 3. Compute the derivatives of the solution function: g y, g u, g yy, g uu, g yu. 4. Compute the residuals 1 f + 2 (f y + g uu + f y+y + (g u g u )) Σ u (2) + 1 ( 6 f y+ g uuu + 3f y+y + (g uu g u )+f y+y +y + (g u g u g u ) ) Σ (3) u = 0 Use a non linear solver to find ỹ that sets the residuals to 0. Make sure that the derivatives have converged to a fixed point as well.

24 Burnside (1998) model A simple asset pricing model: An homogeneous good is produced by a single tree. The household uses equity shares to transfer wealth from one period to the next. Household period utility is cθ t θ. This model has an analytical solution.

25 Equilibrium conditions y t = βe t {exp(θx t+1 )(1+y t+1 )} x t = (1 ρ) x +ρx t 1 +ε t where y t is the price/dividends ratio, and x t the growth rate of dividends.

26 Exact solution Burnside (1998) provides the following closed form solution: y t = β i exp(a i + b i (x t x)) i=1 where a i = θ xi + θ2 σ 2 2(1 ρ) 2 [i 2ρ( 1 ρ i) 1 ρ + ρ2( 1 ρ 2i) ] 1 ρ 2 and b i = θρ( 1 ρ i) 1 ρ

27 Exact value of the risky steady state As the stochastic process for x t is linear, its risky steady state is identical to its deterministic steady state, x. The exact risky steady state for y t is y t = β i exp(a i ). i=1 For the numerical simulation, we sum over the first 800 terms.

28 Quantitative experiment Calibration: θ = 10 ρ = x = σ ε = Deterministic steady state of y Exact value of risky steady state nd order approximation of risky steady state

29 The different approximations Around the deterministic steady state: y t (x t x)+3.04(x t x) 2 Around the risky steady state: y t (x t x)+3.75(x t x) 2

30 Moments of simulated variables A simulated trajectory of periods for x t Approximation around Exact Deterministic SS Risky SS Solution Mean of y S.D. of y

31 Approximation errors measure A simulated trajectory of periods for x t Comparing the two approximated solutions with the exact values for y t We report the mean relative error (in percent): E 1 = N N y t yt y t t=1 the maximum relative error (in percent): E = 100 max{ y t y t y t }

32 Approximation errors results Approximation around Deterministic SS Risky SS E E

33 Equation errors In absence of exact solution, we define err t = E t f (ĝ(ĝ(y t 1, u t ),u t+1 ),ĝ(y t 1, u t ),y t 1, u t ) The integration necessary to compute the conditional expectation is evaluated numerically with a 7-point Hemite formula. Mean approximation error E 1i = 1 N N err it t=1 Maximum approximation error E i = max{ err it }

34 Equation error results The first equation is normalized: 1 = βe t {exp(θx t+1 )(1+y t+1 )}/y t Approximation around Deterministic SS Risky SS E E Equation error is not necessarily a good measure of accuracy. Remember that the approximation around the deterministic steady state generates too little variance in y.

35 Jermann (1998) model Asset prices in production economies: RBC model consumption habits investment adjustment cost comparing risk free rate with expected rate of return on capital

36 The firm The representative firm maximizes its value: with E t k=0 β kµ t+k µ t D t+k Y t = A t Kt 1 α (X tn t ) 1 α D t = Y t W t N t I t K t = (1 δ)k t 1 +φ log A t = ρ log A t 1 + e t X t = (1+g)X t 1 ( It K t 1 ) K t 1

37 The household The representative household maximizes current value of future utility: E t k=0 β k(c t+k χc t+k 1 ) 1 τ 1 τ subject to the following budget constraint: and with N t = 1. W t N t + D t = C t

38 Risk free rate and rate of return on capital The risk free rate 1 r f t = } E t {β µ t+1 µ t The expected rate of return of on capita { ( ) 1 ( It ξ r t = E t a 1 αa t+1 Kt α 1 K t 1 + ( 1 δ + a 1 It ξ a 1 ( It+1 K t ) 1 ) 1 ξ K t 1 ξ + a 2 I t+1 K t )} The risk premium erp t = r t r f t

39 Deterministic and risky steady state Variable Deterministic Risky erp r f r Ĉ µ D Î K Ŵ Ŷ

40 Moments of simulated variables Mean S.D. Variable Deterministic Risky Deterministic Risky erp r f r Ĉ µ D K Ŷ

41 Approximation errors results Approximation around Deterministic SS Risky SS Equation E 1 E E 1 E Marginal utility Euler equation

42 Portfolio problem: A simple two-assset endowment model Agents in both countries maximize welfare: U t = E t τ=t θ t C 1 ρ 1 ρ where θ t is a time-varying discout factor determined by θ τ = θ τ 1 ωc A η τ where C A represents aggregate consumption.

43 Budget constraint Agents in home country face the following budget constraint: a hτ a fτ = a hτ 1 r ht a fτ 1 r f t + yk hτ + yl hτ c hτ assets are a ht, a f t (in zero net supply) payoffs: r hτ = yk hτ /z hτ 1 r fτ = yk fτ /z fτ 1 Agents in foreign country face the following budget constraint: a hτ + a fτ = a hτ 1 r ht + a fτ 1 r f t + yk fτ + yl fτ c fτ the two budget constraints, for home and for foreign agents, imply equilibrium in the good market: yk hτ + yl hτ + yk fτ + yl fτ = c hτ + c fτ.

44 First order conditions Agents in both countries choose their consumption level and the amount of equity, both home and foreign, that they want to hold. The first order conditions for the optimality of these decisions in the home country are given by { } η ρ ρ c h t = ωe t c h E t { c h ρ t+1 r ht+1 t+1 r ht+1 } { ρ = E t c h t+1 r f t+1 { ρ t = ωe t c f c f η ρ E t { c f ρ t+1 r ht+1 } t+1 r f t+1 } = E t { c f ρ t+1 r f t+1 } }

45 Endowment dynamics Finally, the exogenous dynamics of the endowments is given by ln yk ht = ln yk + ek ht ln yl ht = ln yl + el ht ln yk f t = ln yk + ek f t ln yl f t = ln yl + el f t. The identical mean of the processes in both countries reflects their symmetry. The variances are var (ek ht ) = var (ek f t ) = σ k var (el ht ) = var (el f t ) = σ k cov (ek ht, el h ) = cov (ek f t, el f t ) = σ kl cov (ek ht, ek f ) = cov (el ht, el f t ) = 0.

46 The model Bringing all equations together, the model is r hτ = yk hτ /z hτ 1 (1) r fτ = yk fτ /z fτ 1 (2) { } η ρ ρ c h t = ωe t c h t+1 r ht+1 (3) { } η ρ ρ c f t = ωe t c f t+1 r f t+1 (4) a hτ a fτ = a hτ 1 r ht a fτ 1 r f t + yk hτ + yl hτ c hτ (5) a hτ + a fτ = a hτ 1 r ht + a fτ 1 r f t + yk fτ + yl fτ c fτ (6) { } { } ρ E t c h t+1 r ρ ht+1 = E t c h t+1 r f t+1 (7) { } { } ρ E t c f t+1 r ρ ht+1 = E t c f t+1 r f t+1 (8)

47 The singularity problem The deterministic steady state is indeterminate { ρ ) E t c h t+1( } rht+1 r f t+1 = 0 { ρ ) E t c h t+1( } rht+1 r f t+1 = 0 Even if one chooses a particular portfolio allocation, the Jacobian is rank deficient Devereux and Sutherland (2011) suggest to solve the real model at first order (eq 1-6) and to compute a second order approximation of the porfolio choice equations (eq. 7-8). In the symmetric case, the Jacobian is rank deficient even at the risky steady state.

48 Endogenous state variables and shocks In this model, endogenous state variables are s t 1 = [ ] z ht 1 z f t 1 and shocks are u t = [ ] ek ht el ht ek f t el f t

49 First order solution for real variables We can write a first order approximation of the four variables appearing in equations 7 and 8 as ĉ ht = g c h s ŝt 1 + g c h u u t r ht = g r h s ŝ t 1 + g r h u u t ĉ f t = g c f s ŝt 1 + g c f u u t r f t = g r f s ŝt 1 + g r f u u t where ĉ ht and ŝ t indicate relative deviation from the deterministic steady state and r ht, absolute deviation. Note that the value of g c h s, g c h u,g c f s, g c f u changes with the value of â h and â f, but not with higher order terms of a ht and a f t.

50 Second order approximation of the portfolio equation Then, a second order approximation of equation (7), conditional on a first order approximation of the real variables, is E t {ĉ h ρ r h ρc ρ 1 ( h r h g c h s ŝt + g c ) h ρ u u ( t+1 +ch g r h s ŝ t + g r ) h u u t+1 ρ 1 0.5ρc ( h g c h s ŝt + g c )( h u u t+1 g r h s ŝ t + g r ) } h u u t+1 = E t {ĉ h ρ r f ρc ρ 1 ( h r f g c h s ŝt + g c ) h ρ u u ( t+1 +ch g r f s ŝt + g r ) f u u t+1 ρ 1 0.5ρc ( h g c h s ŝt + g c )( h u u t+1 g r f s ŝt + g r ) } f u u t+1

51 Simplifying Resolving the conditional expectations and simplifying using the symmetry of the solution, we get and ( g c h u g r h u g c h u g r f u) Σu = 0 ( g c f u g r h u g c f u g r f u) Σu = 0. Combined with the first order approximation of equations 1 to 6, as a function of a h and a f, the two equations above can be solved numerically for â h and â f.

52 Second order solution for both real and portfolio variables ĉ ht =g c h s ŝt 1 + g c h u u t ( g c h ss (ŝ t 1 ŝ t1 )+2g c h su(ŝ t 1 u t ) +g c h ss (u t u t )+g c ) h σ 2 r ht = g r h s ŝ t 1 + g r h u u t ( g r h ss (ŝ t 1 ŝ t1 )+2g r h su (ŝ t 1 u t ) +g r h ss (u t u t )+g r h σ 2 ) ĉ f t = g c f s ŝt 1 + g c f u u t ( g c f ss(ŝ t 1 ŝ t1 )+2g c f su(ŝ t 1 u t ) +g c f ss(u t u t )+g c f σ 2 ) r f t = g r f s ŝt 1 + g r f u u t ( g r f ss(ŝ t 1 ŝ t1 )+2g r f su(ŝ t 1 u t ) +g r f ss(u t u t )+g r f σ 2 )

53 Second order approximation of the portfolio equation E t {ĉ h ρ r h ρc h ρ 1 r h ( g c h s ŝt + g c h u u t ( g c h ss (ŝ t ŝ t ) +2g c h su(ŝ t u t+1 )+g c h ss (u t+1 u t+1 )+g c h σ 2 )) +ch ρ ( g r h s ŝ t +g r h u u t ( g r h ss (ŝ t 1 ŝ t1 )+2g r h su (ŝ t u t+1 )+g r h ss (u t+1 u t+1 )+g r h σ 2 )) 0.5ρc h ρ 1 ( g c h s ŝt + g c h u u t+1 )( g r h s ŝ t + g r h u u t+1 ) } = E t {ĉ h ρ r f ρc h ρ 1 r f ( g c h s ŝt + g c h u u t ( g c h ss (ŝ t ŝ t ) +2g c h su(ŝ t u t+1 )+g c h ss (u t+1 u t+1 )+g c )) h ρ ( +ch g r f σ 2 s ŝt+g r f u u t ( g r f ss(ŝ t 1 ŝ t1 )+2g r f su(ŝ t u t+1 )+g r f ss(u t+1 u t+1 )+g r )) f σ 2 0.5ρc h ρ 1 ( g c h s ŝt + g c h u u t+1 )( g r f s ŝt + g r f u u t+1 ) }.

54 Symmetrical case E t {ĉ h ρ r h ρc h ρ 1 r h ( g c h s ŝt + g c h u u t ( g c h ss (ŝ t ŝ t ) +2g c h su(ŝ t u t+1 )+g c h ss (u t+1 u t+1 )+g c h σ 2 )) +ch ρ ( g r h s ŝ t +g r h u u t ( g r h ss (ŝ t 1 ŝ t1 )+2g r h su (ŝ t u t+1 )+g r h ss (u t+1 u t+1 )+g r h σ 2 )) 0.5ρc h ρ 1 ( g c h s ŝt + g c h u u t+1 )( g r h s ŝ t + g r h u u t+1 ) } = E t {ĉ h ρ r f ρc h ρ 1 r f ( g c h s ŝt + g c h u u t ( g c h ss (ŝ t ŝ t ) +2g c h su(ŝ t u t+1 )+g c h ss (u t+1 u t+1 )+g c )) h ρ ( +ch g r f σ 2 s ŝt+g r f u u t ( g r f ss(ŝ t 1 ŝ t1 )+2g r f su(ŝ t u t+1 )+g r f ss(u t+1 u t+1 )+g r )) f σ 2 0.5ρc h ρ 1 ( g c h s ŝt + g c h u u t+1 )( g r f s ŝt + g r f u u t+1 ) }.

55 Comparing with Devereux and Sutherland Using the same order of approximation of the real economy and the portfolio equations. This is only important for economies that are not symmetrical. The solution is then robust to the way of writing the model. It is still necessary to proceed in two steps. One for the real economy, one for the portfolio equations. The risky steady state can only be computed in presence of a unit root.

56 Calibration: the symmetrical case ρ = 1 ψ = 0.7 η = 0.9 g = 0.25 ȳ k = 1 ȳ l = 1 ν = 0.5 ω = 0.75 σ k = 0.02 σ l = 0.01

57 Results Variable DS RSS c h ln ze h r h a h c f ln ze f r f a f Both approaches provide basically the same results.

58 Different volatilities and utility curvatures σ k h = 0.2, σ k f = 0.4, ρ h = 1, ρ f = 10 Variable DS RSS c h ln ze h r h a h c f ln ze f r f a f

59 Conclusion A formal definition of the risky steady state. An algorithm to derive jointly a second order approximation the risky steady state and a second order approximation of the decision rules. An approximation around the risky steady state is not necessarily more accurate. The issue of (spurious) multiplicity of risky steady states remains to be studied. The approach permits to study the feedback of the portfolio choice on real variable and uses the same order of approximation for the real part of the model and the portfolio equations. This is important for asymmetrical models. As in Devereux and Sutherland (2011), we need to proceed in two steps.

60 References Devereux, M. B. and A. Sutherland (2011). Country portfolios in open economy macro models. Journal of the European Economic Association 9,