Risks For The Long Run And The Real Exchange Rate

Transcription

1 Riccardo Colacito, Mariano M. Croce

2 Overview International Equity Premium Puzzle Model with long-run risks Calibration Exercises Estimation Attempts & Proposed Extensions Discussion

3 International Equity Premium Puzzle Asset returns imply a highly volatile stochastic discount factor governing returns on domestic and foreign currency denominated assets. Currency exchange rates should adjust to adjust to prevent arbitrage opportunities but variance of depreciation rate is low. Implication is a high correlation between stochastic discount factors, but cross-country correlation of consumption is low. Need highly correlated component of consumption growth...

4 International Equity Premium Puzzle Two countries: home (h) and foreign (f) with m i t = log(m i t): E t [exp(m f t+1)r f t+1] = 1 = E t [exp(m h t+1)r h t+1] If the foreign asset is traded in the home country, with exchange rate e t, E t [exp(mt+1)r f t+1] f = 1 = E t [exp(mt+1) h e t+1 R e t+1]. f t If markets are complete: π t+1 = m f t+1 m h t+1 where π t+1 = log(e t+1 /e t ). (Backus, Foresi and Telmer 1996)

5 International Equity Premium Puzzle Var(π t+1 ) = Var(m f t+1 m h t+1) ρ m h,m f = σ2 m h + σ 2 m f σ 2 π 2σ m hσ m f σ 2 m i >.20 from Hansen-Jagannathan bounds σ 2 π Implies that ρ m h,mf >.96 but covariance in consumption is only 27%. In a one county model, consumption growth does not vary enough to explain the excess return over the risk free rate. In a two country model, consumption growth does not co-vary enough to keep track of the movements in the exchange rate.

6 Model: Preferences Two countries (i {h, f }), each with a representative consumer, each with a single county-specific good. Complete home bias- each country derives utility only from its own endowment. Epstein-Zin preferences: Ut i = {(1 δ)(ct i ) 1 γ θ + δ[e t [(U i t+1) 1 γ ]] 1/θ } θ 1 γ δ is discount rate, γ is CRRA, ψ is IES, θ = 1 γ 1 1/ψ

7 Model: Assets E t [M i t+1r i j,t+1] = 1 m i t+1 = log M i t+1 = θ log δ θ ψ ci t+1 + (θ 1) log R i c,t+1 where R i t+1 is the return on the asset that pays dividend stream {C i t }. R i t+1 = v i c,t v i c,t exp c t+1 where vc,t+1 i is the price-consumption ratio in country i. Complete markets: exchange rate adjusts to equalize returns across countries. In equilibrium, autarky in goods and asset holdings. (Not a model of trade.)

8 Model: Consumption Process Consumption follows an exogenous law of motion: c i t = x i t 1 + ɛ i c,t x i t = ρ x x i t 1 + ɛ i x,t Shocks are i.i.d. with correlations governed by Σ: [ɛ h c,tɛ f c,tɛ h x,tɛ f x,t] N(0, Σ) Σ = σ 2 [ Hc 0 0 φ 2 eh x ] [ 1 ρ hf c H c = ρ hf c 1 ] [ 1 ρ hf x H x = ρ hf x 1 ]

9 Model: First-Order Approximations First-order Taylor approximation for price-consumption ratio ( ) vc,t i = vc i (ψ 1) 1 + ψ(1 ρ x δ) x t i Solutions to first-order approximation of model: mt+1 i = log δ 1 ψ x t i + δ 1 1 ψ 1 ρ hf ɛ i x,t+1 + ɛ i c,t+1 x e t+1 = mt+1 f mt+1 h e t r i c,t+1 = r c + 1 ψ x i t + δ 1 1 ψ 1 ρ x δ ɛi x,t+1 + ɛ i c,t+1 r i f,t+1 = r f + 1 ψ x i t where r i f,t+1 is the log risk-free rate and r j is the average log return on asset j.

10 Model: Two Propositions Proposition 1. For a given choice of parameters, and provided that ρ h,f x > ρc h,f, the lowest cross country correlation of the stochastic discount factor is achieved (uniquely) by (ψ, ρ x ) = ( 1 δ, γ 0) where 1 2ρx δ+δ2 δ = δ 2 (1 ρ 2 x ). Proposition 2. For a given choice of parameters, the lowest volatility of the depreciation rate is achieved for ρ hf x = 1.

11 Calibration: Baseline

12 Calibration: Varying Paramters

13 Calibration: Matching Other Moments Introduce dividend growth process: d i t = µ d + λx t 1 + ɛ i d,t [ɛ h c,tɛ f c,tɛ h x,tɛ f x,tɛ h d,t ɛf d,t ] N(0, Σ) [ ] [ Σ 0 Σ = 0 σ 2 φ 2 d H H d = σ 2 1 ρ hf d d ρ hf d 1 ]

14 Calibration: Matching other Moments

15 Calibration: Matching other Moments Match most moments fairly well including correlation of excess returns. Correlation of risk-free rate is one: depends only on x i t which are perfectly correlated. Add stochastic volatility: Doesn t change results, correlation of risk-free rate drops to.98

16 Estimation of Long Run Risks Attempts to estimate the persistent component Compare frequency spectra with and without predictable component. Can t distinguish two cases. Use Kalman filter to estimate state-space system. Estimates reasonably close to calibrated values but very large standard errors. Estimate state space system with additional data from Germany and Japan. Doesn t tighten confidence intervals. Simulated Method of Moments estimation with additional moments. Get tighter estimates for consumption laws of motion but not for preference parameters or dividend process.

17 Estimation: MLE Estimates

18 Estimation: Additional Countries

19 Estimation: Additional Moments

20 Extensions Additional factors to match yield curve evidence. Yield curves have sharper features at low frequencies but can t be identified in one-factor model.

21 Extensions Economic interpretations of x t. Include wider set of moments. Relaxing home bias assumption to study traded and non-traded goods.

22 Discussion Paper shows how long run risk can be used to solve international equity premium puzzle. Results are very sensitive to high persistence and high cross-country correlation. Attempt to estimate the model but are unable to properly identify parameters or to provide evidence that the persistent component is present. No discussion of approximation errors.

23 Discussion Is there hope of identifying model? Look at country pairs where would expect less correlation in consumption process. Is there more freedom from the data to lower the correlation? Relax home bias assumption. Implications for bi-lateral trade?