Example Environments

Transcription

1 Example Environments David N. DeJong University of Pittsburgh Optimality Closing the Cycle Spring 2008, Revised Spring 2010

2 Lucas (1978 Econometrica) One-Tree of Asset Prices Text reference: Ch. 5.3, pp Environment: max U = E 0 c t β = 1 1+ρ, 0 < ρ < 1, subject to t=0 β t u(c t ), (1) Optimality Closing the Cycle c t + p t (s t s t 1 ) = d t s t 1 + q t, (2) s 0 given.

3 Establish Optimality Value function: V (s t 1 ) = max c t E t fu(c t ) + βv (s t )g FONC: = maxe t fu(d t s t 1 + q t p t (s t s t 1 )) + βv (s t )g s t Envelope Condition: E t u 0 (c t ) [ p t ] + βv 0 (s t ) = 0 V 0 (s t ) = u 0 (c t+1 ) [d t+1 + p t+1 ] Optimality Closing the Cycle

4 Optimality, cont. Combining FONC and Env. Cond. Yields Pricing Kernel: u p t = 0 (c t+1 ) βe t u 0 (c t ) (d t+1 + p t+1 ) (3) Optimality Closing the Cycle

5 Since agents are identical, market clearing requires s t = s t 1 8t. Hereafter, we ll normalize: s t = 1 8t. From the budget constraint (2), this implies Optimality Closing the Cycle c t = d t + q t.

6 Closing the At this point, we have two equations and four unknowns: c t = d t + q t, u p t = 0 (c t+1 ) βe t u 0 (c t ) (d t+1 + p t+1 ). Optimality Closing the Cycle To close the model, we specify stochastic processes for the exogenous state variables s t = [d t q t ] 0 : d t = de gt e ω dt, ω dt = ρ d ω dt 1 + ε dt, q t = qe gt e ω qt, ω qt = ρ q ω qt 1 + ε qt, with υ t iidn (0, Σ).

7 Closing the, cont. Exercise: De ning ed t = d t /e gt, show that the assumed SP for d t implies ln ed t = (1 ρ d ) ln d + ρ d ln ed t 1 + ε dt, Optimality Closing the Cycle and thus: I dividends feature a deterministic constant-growth component I logged dividends exhibit AR(1) uctuations about a linear trend.

8 Closing the, cont. In general, when specifying SPs for exogenous forcing variables, it is important that the speci cations correspond with their empirical counterparts. (For an analysis of the importance of this issue: Gorodnichenko and Ng, 2007, U. Mich. WP.) Optimality Closing the Cycle In this case, support for the assumption of trend-stationarity comes from DeJong and Whiteman (1991 AER; 1994 ET ).

9 The Data Logged Deviations from Linear Trend Optimality Closing the Cycle

10 We seek a speci cation of the model written in terms of stationary versions of variables. Stationarity will be induced in the data analogously. In this case, conversion to stationarity requires trend removal. Optimality Closing the Cycle h From the exercise, we ve shown that d et So too is ec t, since c t = d t + q. What of p t? eq t i 0 is stationary.

11 , cont. Guess: ep t is also stationary. To verify, determine whether this guess is consistent with the pricing kernel. In terms of deterministic components of variables, under the guess we have u pe gt = 0 (ce gt+1 ) β u 0 (ce gt (d + p)egt+1, ) or p = β u 0 (ce gt+1 ) u 0 (ce gt (d + p)eg. ) Optimality Closing the Cycle

12 , cont. Assuming we have γ u(c t ) = c1 t 1 γ, p = β e γg (d + p)e g = βe (1 γ)g (d + p), Optimality Closing the Cycle and thus the guess is veri ed. Solving for p, we obtain p = 1 (1 + ρ) e (γ 1)g 1 d.

13 Hereafter, p t is in fact p t e gt, etc. The model is then " ct+1 γ p t = βe (1 γ)g E t (d t+1 + p t+1)# c t (4) c t = d t + q t (5) d t = de ω dt, ω dt = ρ d ω dt 1 + ε dt, (6) q t = qe ω qt, ω qt = ρ q ω qt 1 + ε qt. (7) Optimality Closing the Cycle State: s t = [d t q t ] 0 Shocks: υ t = [ε dt ε qt ] 0 Controls: c t = [c t p t ] 0 Parameters: µ = β γ g d q vec(σ) 0. Note that (6) and (7) constitute s t = f (s t 1, υ t ) ; (4) and (5) will be used to construct an approximation of c t = f (s t ). Our goal will be to transform this system into a h i 0 likelihood function over the observables X t = d et ep t ec t.

14 As speci ed, the model features three observable variables, but only two sources of stochastic behavior. This gives rise to an issue known as stochastic singularity. Whenever there are more observable variables than structural shocks, various xed combinations of observable variables are predicted to be deterministic. (See Ingram, Kocherlakota, and Savin, 1994 JME for details.) Possible remedies: Optimality Closing the Cycle I Augment the model with additional structural shocks. I Introduce measurement error.

15 Cycle (Class-Long Exercise) Text reference: Ch. 5.1, pp subject to Exercise: max U = E 0 c t,l t y t = z t kt α nt 1 α, 1 = n t + l t, y t = c t + i t, β t t=0 k t+1 = i t + (1 δ)k t, c ϕ t l 1 t 1 φ ϕ 1 φ z t = z 0 e gt e ω t, ω t = ρω t 1 + ε t. I Establish associated Value Function I Derive FONCs I Establish Non-Linear System I Derive steady states., Optimality Closing the Cycle