Risks for the Long Run and the Real Exchange Rate

Risks for the Long Run and the Real Exchange Rate Riccardo Colacito - NYU and UNC Kenan-Flagler Mariano M. Croce - NYU Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 1/29

Set the stage Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ ] E t M i t+1 Rt+1 i = 1, i {us, uk} Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 2/29

Set the stage Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ ] E t M i t+1 Rt+1 i = 1, i {us, uk} How to evaluate M i t+1? 1. From prices: σ ( M i t+1 ) E[R i t+1 Rf t+1] σ(r i t+1 Rf t+1) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 2/29

Set the stage Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ ] E t M i t+1 Rt+1 i = 1, i {us, uk} How to evaluate M i t+1? 1. From prices: σ ( M i t+1 ) E[R i t+1 Rf t+1] σ(r i t+1 Rf t+1) e.g. if E ˆR i R f 7%, σ `R i R f 17% then σ `M i 40% Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 2/29

Set the stage Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ ] E t M i t+1 Rt+1 i = 1, i {us, uk} How to evaluate M i t+1? 1. From prices: σ ( M i t+1 ) E[R i t+1 Rf t+1] σ(r i t+1 Rf t+1) e.g. if E ˆR i R f 7%, σ `R i R f 17% then σ `M i 40% 2. From quantities: M i t+1 = Ui / C i t+1 U i / C i t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 2/29

Set the stage Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ ] E t M i t+1 Rt+1 i = 1, i {us, uk} How to evaluate M i t+1? 1. From prices: σ ( M i t+1 ) E[R i t+1 Rf t+1] σ(r i t+1 Rf t+1) e.g. if E ˆR i R f 7%, σ `R i R f 17% then σ `M i 40% 2. From quantities: M i t+1 = Ui / C i t+1 U i / C i t e.g. with CRRA preferences M i t+1 = δ C i t+1 C i t «γ Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 2/29

Puzzle By no arbitrage E t [M f t+1 Rf t+1 ] [ = 1 = E t M h t+1 Rt+1 h ] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage E t [M f t+1 Rf t+1 ] = 1 = E t [M h t+1 ] e t+1 R f t+1 e t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage E t [M f t+1 Rf t+1 ] = 1 = E t [M h t+1 ] e t+1 R f t+1 e t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f + σ 2 m h 2ρ mf,m hσ m fσ m h = σ2 π Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% + σ 2 m h }{{} 20% 2 ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% + σ 2 m h }{{} 20% 2 ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: log M i t+1 = γ c i t+1 γ 2 σ 2 c f + γ 2 σ 2 c h 2ρ cf, c hγ2 σ c fσ c h = σ 2 π Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% + σ 2 m h }{{} 20% 2 ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: log M i t+1 = γ c i t+1 γ 2 σ 2 c f }{{} 20% + γ 2 σ 2 c h }{{} 20% 2ρ cf, c h γ2 σ c fσ c h }{{} 20% = σ 2 π Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

Puzzle By no arbitrage log M f t+1 log Mh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% + σ 2 m h }{{} 20% 2 ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: log M i t+1 = γ c i t+1 γ 2 σ 2 c f }{{} 20% + γ 2 σ 2 c h }{{} 20% 2 ρ cf, c h }{{} 0.3 γ 2 σ c fσ c h }{{} 20% = σ 2 π }{{} 28% Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 3/29

The puzzle (cont d) Brandt, Cochrane and Santa-Clara (2005): ρ mf,m h σ2 π Prices high low Quantities low high Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 4/29

The puzzle (cont d) Brandt, Cochrane and Santa-Clara (2005): ρ mf,m h σ2 π Prices high low Quantities low high How does this puzzle look in a cross section of countries? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 4/29

The puzzle in a cross section of countries HJ bound and depreciation rate: high correlation of SDF and low variance of depreciation rate. 1.0 0.9 0.85 Prices UK France Germany Netherlands Sweden Correlations 0.35 0.3 0.15 0 0 2 4 17 18 21 24 27 30 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 5/29

The puzzle in a cross section of countries Consumption data and CRRA preferences: low correlation of SDF and high variance of depreciation rate. 1.0 0.9 0.85 Prices UK France Germany Netherlands Sweden Correlations 0.35 0.3 Quantities 0.15 0 0 2 4 17 18 21 24 27 30 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 5/29

The puzzle in a cross section of countries This paper: high correlation of SDF, low correlation of c and low variance of depreciation rate. 1.0 0.9 0.85 corr(m h,m f ) UK France Germany Netherlands Sweden Correlations 0.35 0.3 corr( c h, c f ) 0.15 0 0 2 4 17 18 21 24 27 30 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 5/29

Rules of the game and outline 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 6/29

Rules of the game and outline 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities low volatility of depreciation rate low volatility of consumption growth low correlation of consumption growths Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 6/29

Rules of the game and outline 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities low volatility of depreciation rate low volatility of consumption growth low correlation of consumption growths 2. Can we match other key features of financial markets? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 6/29

Rules of the game and outline 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities low volatility of depreciation rate low volatility of consumption growth low correlation of consumption growths 2. Can we match other key features of financial markets? 3. Can we estimate this model? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 6/29

Rules of the game and outline 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities low volatility of depreciation rate low volatility of consumption growth low correlation of consumption growths 2. Can we match other key features of financial markets? 3. Can we estimate this model? 4. Extensions and future research... Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 6/29

Setup of the economy Endowment economy. Two country specific goods. Complete home bias in consumption. Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 7/29

Setup of the economy Endowment economy. Two country specific goods. Complete home bias in consumption. Epstein, Zin and Weil preferences: U i t = {(1 δ)(ct) i 1 γ θ + δ [ E t (Ut+1) i 1 γ] 1 } θ 1 γ θ, i {h, f} where θ = 1 γ 1 1/ψ Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 7/29

Setup of the economy Endowment economy. Two country specific goods. Complete home bias in consumption. Epstein, Zin and Weil preferences: U i t = {(1 δ)(ct) i 1 γ θ + δ [ E t (Ut+1) i 1 γ] 1 } θ 1 γ θ, i {h, f} where θ = 1 γ 1 1/ψ What do stochastic discount factors look like? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 7/29

Stochastic discount factors Assume ψ = 1: U i t = (1 δ) log C i t + δ 1 γ log E t The stochastic discount factors are [ exp(1 γ)u i t+1 ] log M i t+1 = log Ui / C i t+1 U i / C i t = log δ + log Ci t C i t+1 + log exp{(1 γ)u i t+1} E t [ exp{(1 γ)u i t+1 } ] Brandt, Cochrane and Santa-Clara use: log M i t+1 = log δ + log Ci t C i t+1 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 8/29

Remainder of the economy Home country Foreign country c h t = µ c + x h t 1 + σε h c,t x h t = ρx h t 1 + σϕ e ε h x,t c f t = µ c + x f t 1 + σεf c,t x f t = ρx f t 1 + σϕ eε f x,t Shocks are i.i.d. within each country Shocks are correlated across countries ρ c = corr(ε h c,t, ε f c,t) ρ x = corr(ε h x,t, ε f x,t) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 9/29

Calibration δ γ ψ θ µ c σ ρ ϕ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θ log δ θ ψ ci t + (θ 1) log R i c,t+1 c i t = µ c + x i t 1 + σε i c,t x i t = ρx i t 1 + σϕ e ε i x,t Preferences: Low risk aversion (γ) IES from Bansal, Gallant and Tauchen (2004) Monthly model: high discounting Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 10/29

Calibration δ γ ψ θ µ c σ ρ ϕ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θ log δ θ ψ ci t + (θ 1) log R i c,t+1 c i t = µ c + x i t 1 + σε i c,t x i t = ρx i t 1 + σϕ e ε i x,t Consumption process: Average consumption growth 2% Standard deviation of consumption growth 2.5% Variance explained by long run risk 7 8% Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 10/29

Calibration δ γ ψ θ µ c σ ρ ϕ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θ log δ θ ψ ci t + (θ 1) log R i c,t+1 c i t = µ c + x i t 1 + σε i c,t x i t = ρx i t 1 + σϕ e ε i x,t Cross correlations of shocks: Correlation of consumption growths 0.3 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 10/29

Three ingredients We can solve the puzzle by appropriately combining three ingredients: Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 11/29

Three ingredients We can solve the puzzle by appropriately combining three ingredients: 1. Use Epstein and Zin preferences: BCSC (2005): m i t+1 = E [ g( c i t+1) I t+1 ] = γ c i t+1 This paper: m i t+1 = E [ g( c i t+1, c i t+2, c i t+3,...) I t+1 ] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 11/29

Three ingredients We can solve the puzzle by appropriately combining three ingredients: 1. Use Epstein and Zin preferences: BCSC (2005): m i t+1 = E [ g( c i t+1) I t+1 ] = γ c i t+1 This paper: m i t+1 = E [ g( c i t+1, c i t+2, c i t+3,...) I t+1 ] Alter the conditional distribution of ( c h, c f) : by assuming 2. High persistence ρ i c i t+1 = µ c + x i t + σε i c,t+1 x i t+1 = ρ i x i t + σϕ e ε i x,t+1 3. High cross country correlation corr ( ) ε h x,t+1, ε f x,t+1 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 11/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Intertemporal Elasticity of substitution (ψ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 ρ=0.987 ρ=0.9 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Stochastic discount factors m i t+1 = θ log δ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σϕ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 ρ x =1 ρ x =0.75 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 12/29

Exchange rate depreciation V ar ( et+1 e t ) { = 2(1 ρ x) 1 ψ 2 1 ρ + 2 [ δ(1 γψ) (1 ρδ) ] } 2 ϕ 2 eσ 2 + 2γ 2 (1 ρ c )σ 2 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 13/29

Exchange rate depreciation V ar ( et+1 e t ) { = 2(1 ρ x) 1 ψ 2 1 ρ + 2 [ δ(1 γψ) (1 ρδ) ] } 2 ϕ 2 eσ 2 + 2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 13/29

Exchange rate depreciation V ar ( et+1 e t ) { = 2(1 ρ x) 1 ψ 2 1 ρ + 2 [ δ(1 γψ) (1 ρδ) ] } 2 ϕ 2 eσ 2 + 2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 ρ x =0.9 ρ x =1 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 13/29

Exchange rate depreciation V ar ( et+1 e t ) { = 2(1 ρ x) 1 ψ 2 1 ρ + 2 [ δ(1 γψ) (1 ρδ) ] } 2 ϕ 2 eσ 2 + 2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 ρ x =0.9, ρ=.98 ρ x =1, ρ=.987 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ) Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 13/29

Every assumption counts Ingredients needed to solve the puzzle: 1. Disentangle elasticity of substitution from risk aversion 2. Highly persistent predictable component 3. Highly correlated predictable components Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 14/29

Every assumption counts Ingredients needed to solve the puzzle: 1. Disentangle elasticity of substitution from risk aversion 2. Highly persistent predictable component 3. Highly correlated predictable components Can we match key moments of international financial markets? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 14/29

Introducing dividends The system becomes c i t = µ c + x i t 1 + σε i c,t d i t = µ d + λx i t 1 + σϕ d ε i d,t x i t = ρx i t 1 + σϕ e ε i x,t sssssssssssssssssssssssssssssssssssss i {h, f} Shocks are i.i.d. within each country Shocks are correlated across countries Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 15/29

Introducing dividends The system becomes c i t = µ c + x i t 1 + σε i c,t d i t =.0007 + 3 x i t 1 + σ 5 ε i d,t x i t = ρx i t 1 + σϕ e ε i x,t sssssssssssssssssssssssssssssssssssss i {h, f} Shocks are i.i.d. within each country Shocks are correlated across countries Calibrate coefficients of dividend growth to match: Average dividend growth 1% Standard deviation of dividend growth 12% Leverage is 5 Small correlation of dividend growths: corr ε h d,t, εf d,t 0 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 15/29

Introducing dividends: results US UK Model ρ ( m h, m f) - - 0.93 σ ( et+1 e t ) 11.21 11.83 E (r d r f ) 7.02 9.17 7.01 σ ( (r d r f ) ) 17.13 22.83 19.60 ρ rd h rh f, rf d rf f 0.60 0.58 E (r f ) 1.47 1.62 1.33 σ ( (r f ) ) 1.53 2.92 1.19 ρ rf h, rf f 0.65 1.00 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 16/29

Introducing stochastic volatility The system becomes i {h, f} c i t = µ c + x i t 1 + σ t ε i c,t d i t = µ d + λx i t 1 + σ t ϕ d ε i c,t x i t = ρx i t 1 + σ t ϕ e ε i x,t ( σ 2 t ) i = σ 2 + ν 1 [ (σ 2 t 1 ) i σ 2 ] + σ w ε i σ,t Shocks are i.i.d. within each country Shocks are correlated across countries Guidelines to calibrate stochastic volatility given by ( ) ( ) V ar t r d t+1 = (1 ν 1 )k 0 + ν 1 V ar t 1 r d t + k1 σ w ε σ,t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 17/29

Introducing stochastic volatility The system becomes i {h, f} c i t = µ c + x i t 1 + σ t ε i c,t d i t = µ d + λx i t 1 + σ t ϕ d ε i c,t x i t = ρx i t 1 + σ t ϕ e ε i x,t ( ) σ 2 i [ (σ t =.0068 2 ) ] 2 i +.96 t 1.0068 2 +.23e 5 ε i σ,t Shocks are i.i.d. within each country Shocks are correlated across countries Guidelines to calibrate stochastic volatility given by ( ) ( ) V ar t r d t+1 = (1 ν 1 )k 0 + ν 1 V ar t 1 r d t + k1 σ w ε σ,t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 17/29

Introducing stochastic volatility The system becomes i {h, f} c i t = µ c + x i t 1 + σ t ε i c,t d i t = µ d + λx i t 1 + σ t ϕ d ε i c,t x i t = ρx i t 1 + σ t ϕ e ε i x,t ( ) σ 2 i [ (σ t =.0068 2 ) ] 2 i +.96 t 1.0068 2 +.23e 5 ε i σ,t Shocks are i.i.d. within each country Shocks are correlated across countries Guidelines to calibrate stochastic volatility given by ( ) ( ) V ar t r d t+1 = (1 ν 1 )k 0 + ν 1 V ar t 1 r d t + k1 σ w ε σ,t Cross correlation of ε σ,t has small impact on results. Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 17/29

Results US UK No stoch vol W/Stoch vol ρ ( m h, m f) - - 0.93 0.92 σ ( et+1 e t ) 11.21 11.83 12.67 E (r d r f ) 7.02 9.17 7.01 7.03 σ ( (r d r f ) ) 17.13 22.83 19.60 19.41 ρ rd h rh f, rf d rf f 0.60 0.58 0.57 E (r f ) 1.47 1.62 1.33 1.33 σ ( (r f ) ) 1.53 2.92 1.19 1.22 ρ rf h, rf f 0.65 1.00 0.98 σ (r c ) - - 4.74 4.75 ρ ( ) rc h, rc f - - 0.85 0.85 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 18/29

Estimating long run risks Can we justify high persistence and high correlation: [ [ c h t c f t x h t x f t ] ] = = [ [ µ h c µ f c ] + ρ h 0 0 ρ f [ ][ x h t 1 x f t 1 ] x h t 1 x f t 1 + ] [ + 1 0 1 ρ 2 c ρ c [ 1 0 ρ x 1 ρx 2 ][ σ h ε h c,t σ f ε f c,t ][ ] σ h ϕ h eε h x,t σ f ϕ f eε f x,t ] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 19/29

Estimating long run risks Can we justify high persistence and high correlation: [ [ c h t c f t x h t x f t ] ] = = [ [ µ h c µ f c ] + ρ h 0 0 ρ f [ ][ x h t 1 x f t 1 ] x h t 1 x f t 1 + ] [ + 1 0 1 ρ 2 c ρ c [ 1 0 ρ x 1 ρx 2 ][ σ h ε h c,t σ f ε f c,t ][ ] σ h ϕ h eε h x,t σ f ϕ f eε f x,t ] Roadmap: 1. Use consumption data only Use Kalman filter to get a recursive representation of the likelihood function Multi-country provide inconclusive evidence Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 19/29

Estimating long run risks Can we justify high persistence and high correlation: [ [ c h t c f t x h t x f t ] ] = = [ [ µ h c µ f c ] + ρ h 0 0 ρ f [ ][ x h t 1 x f t 1 ] x h t 1 x f t 1 + ] [ + 1 0 1 ρ 2 c ρ c [ 1 0 ρ x 1 ρx 2 ][ σ h ε h c,t σ f ε f c,t ][ ] σ h ϕ h eε h x,t σ f ϕ f eε f x,t ] Roadmap: 1. Use consumption data only Use Kalman filter to get a recursive representation of the likelihood function Multi-country provide inconclusive evidence 2. Use consumption and price data Focus on a set of moments of interest Sharply identify departure from i.i.d. Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 19/29

Consumption data only: results Home = US and Foreign = UK Quarterly data from 1970 to 1998. Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 20/29

Consumption data only: results Home = US and Foreign = UK Quarterly data from 1970 to 1998. ρ h ρ f ρ x ρ c Calibrated 0.987 0.987 1.000 0.300 Real Data 0.909 0.940 0.897 0.208 (T=120) [0.547,0.995] [0.308,0.995] [0.696,1.000] [0.004,0.406] Simulations 0.941 0.934 0.844 0.312 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 20/29

Consumption data only: results Home = US and Foreign = UK Quarterly data from 1970 to 1998. ρ h ρ f ρ x ρ c Calibrated 0.987 0.987 1.000 0.300 Real Data 0.909 0.940 0.897 0.208 (T=120) [0.547,0.995] [0.308,0.995] [0.696,1.000] [0.004,0.406] Simulations 0.941 0.934 0.844 0.312 (T=120) [0.731,1.000] [0.630,1.000] [0.467,1.0] [0.066,0.530] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 20/29

Consumption data only: results Home = US and Foreign = UK Quarterly data from 1970 to 1998. ρ h ρ f ρ x ρ c Calibrated 0.987 0.987 1.000 0.300 Real Data 0.909 0.940 0.897 0.208 (T=120) [0.547,0.995] [0.308,0.995] [0.696,1.000] [0.004,0.406] Simulations 0.941 0.934 0.844 0.312 (T=120) [0.731,1.000] [0.630,1.000] [0.467,1.0] [0.066,0.530] Simulations 0.987 0.987 0.986 0.302 (T=10000) [0.983,0.990] [0.983,0.991] [0.953,1.0] [0.285,0.318] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 20/29

Consumption data only: results (cont d)... other parameters ϕ h e ϕ f e σ h σ f Calibrated 0.048 0.048 68 68 Real Data 0.351 0.184 38.1 77.9 (T=120) [0.000,0.424] [0.000,1.092] [28.6,38.1] [72.3,78.0] Simulations 0.142 0.119 66.1 66.1 (T=120) [0.000,0.475] [0.000,1.165] [51.2,71.6] [37.4,68.2] Simulations 0.049 0.049 67.9 68.0 (T=10000) [0.039,0.059] [0.041,0.057] [67.1,69.1] [67.3,68.7] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 21/29

Does adding other countries help? US, UK and Germany US, UK and Japan US, UK, Germany and Japan ρ US 0.911 0.906 0.919 [.530,.990] [.552,.984] [.796,.996] ρ UK 0.928 0.947 0.934 [.049,.986] [.214,.978] [.746,.994] ρ Ger 0.932-0.993 [.274,.985] - [.614,.996] ρ Jpn - 0.989 0.99 - [.215,.989] [.504,.998] ρ US,UK x 0.920 0.973 0.913 [.890,.999] [.861,.999] [.700,.999] ρ US,Ger x 0.899-0.891 [.874, 1.000] - [.700,.997] ρ US,Jpn x - 0.991 0.905 - [.877, 1.000] [.701, 1.000] Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 22/29

Introducing prices Consumption information is not enough: we need price information. Select 22 moments to match: Consumption: V ar ` c h t, cov ` c h t, c h t 1 V ar c f t, cov c f t, cf t 1, cov c f t, cf t 2 cov c h t, cf t 1, cov c h t 1, cf t Dividend: V ar ` d h t, V ar d f t, cov d h t, df t Excess returns: V ar r d,h t r f,h t, cov r d,h t r f,h t V ar r d,f t r f,f t, cov r d,f t r f,f t, r d,f t 1 rf,f t 1, cov r d,h t r f,h t, r d,f t r f,f t Risk free rates: V ar r f,h t, V ar r f,f t Depreciation rate: V ar et+1 e t Use Simulated Method of Moments, cov ` c h t, c h t 2,, cov c h t, cf t,, r d,h t 1 rf,h t 1, Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 23/29

Prices and consumption: results Consumption only Whole Model Point Estimate 95% CI Point Estimate 95% CI ρ h 0.736 [0.349,0.996] 0.997 [0.927,1.000] ρ f 0.904 [0.015,0.997] 0.996 [0.912,0.999] ϕ h e 1.422 [0.190,17.318] 0.024 [0.005,0.213] ϕ f e 0.182 [0.000,3.502] 0.041 [0.003,0.157] σ h 27.629 [4.527,34.799] 32.792 [28.036,36.676] σ f 79.916 [44.407,87.421] 79.569 [69.767,88.879] ρ hf x 0.999 [0.353,1.000] 0.998 [0.853,1.000] ρ hf c 0.222 [-0.988,0.999] 0.349 [0.215,0.492] σ et+1 e t ρ `m h, m f Unconditional moments Unconditional moments - 11.692-0.922 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 24/29

Future Research What s next? Extend the list of moments that can be matched Relax assumption of complete home bias in consumption Where does x t come from? Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 25/29

Extending the basic model Set Ψ = 1 (risk sensitive preferences): U i t = log c i t + δ (1 γ i )(1 δ) log E [ { t exp (1 γ i )(1 δ)ut+1}] i Two common factors ( If λ 1 = 1, λ 2 = 0, ρ ( ) ρ ε c,h t, ε z 1 t = ρ c h t = µ c,h + λ 1 z 1,t 1 + λ 2 z 2,t 1 + ε c,h t c f t = µ c,f + λ 2 z 1,t 1 + λ 1 z 2,t 1 + ε c,f t z 1,t = ρ 1 z 1,t 1 + ε z 1 t z 2,t = ρ 2 z 2,t 1 + ε z 2 t ε c,h t ( ε c,f t, ε z 2 t ), ε c,f t = 0.3, ρ (ε z 1 t, ε z 2 t ) = 1, ) = 0, then we have the basic model. Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 26/29

Two factors One factor: yields are perfectly correlated. corr(r f ) corr(m) share 50 45 var(π) var(r f ) var(m) 1 40 0.8 35 30 0.6 25 20 0.4 15 0.2 10 5 0 0 0 1 2 3 4 5 0 1 2 3 4 5 λ 2 λ 2 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 27/29

Two factors Two factors: low corr(r fh, r ff ) and high corr(m h, m f ). corr(r f ) corr(m) share 50 45 var(π) var(r f ) var(m) 1 40 0.8 35 30 0.6 25 20 0.4 15 0.2 10 5 0 0 0 1 2 3 4 5 0 1 2 3 4 5 λ 2 λ 2 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 27/29

Yields and long run risks In this model: r fi t = k 0 + k 1 z 1,t + k 2 z 2,t Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 28/29

Yields and long run risks In this model: r fi t = k 0 + k 1 z 1,t + k 2 z 2,t 3.5 x 10 4 US 3 2.5 2 1.5 1 0.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 US and UK 0 0 1 2 3 0 0 1 2 3 1.2 x 10 3 UK 1 0.8 0.6 0.4 0.2 0 0 1 2 3 Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 28/29

Concluding remarks Key ingredients Separate elasticity of substitution from risk aversion Highly persistent predictable component Highly correlated predictable components It is possible to explain low volatility of the depreciation of the US dollar high equity premium high persistence of the risk free rate high correlation of int l financial markets correlation of bonds low correlation of consumption growths low persistence of consumption growths Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 29/29