Volatility Risk Pass-Through - PDF Free Download

Volatility Risk Pass-Through Ric Colacito Max Croce Yang Liu Ivan Shaliastovich 1 / 18

Main Question Uncertainty in a one-country setting: Sizeable impact of volatility risks on growth and asset prices Typically, high aggregate volatility is bad": - Lowers output and investment - Lowers asset valuations - Increases risk premia and marginal utility Open question: How are volatility risks shared internationally? - Novel empirical investigation on G17 - Novel theoretical insights on volatility risk-sharing 2 / 18

Main Findings 1. International pass-through of output vol shocks to consumption vol - Trade channel: higher vol lower net exports - Consumption vol more cross-country correlated than output vol 2. Volatility pass-through is significant and size-dependent - Quantification by a Pass-through index - Smaller countries feature a stronger pass-through 3. Volatility disconnect puzzle - corr(σ t ( e t+1 ),σ t ( c t+1 c t+1 ))=.3 - Beyond the Backus & Smith 93 puzzle 3 / 18

Empirical Analysis 4 / 18

Empirical Analysis Quarterly data for 17 major industrialized countries from 1971 to 214 Output is consumption plus net exports - Abstract for now from investment and government expenditure For variable of interest in each country, run a filter: z t = µ(1 ρ) + ρz t 1 + e σ t (z)/2 η t σ t (z) = µ σ (1 ν) + νσ t 1 (z) + σ w w t - z is real output, consumption, net exports, exchange rates σ(z) is our estimate of the short-run volatility 4 / 18

Macroeconomic Volatilities 1. Substantial persistent movements in macro vols 5 / 18

Macroeconomic Volatilities 1. Substantial persistent movements in macro vols 2. Across countries: ρ(σ y t,σy t ) =.3 < ρ(σ c t,σc t ) =.5 5 / 18

Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Macroeconomic Volatilities 1. Substantial persistent movements in macro vols 2. Across countries: ρ(σ y t,σy t ) =.3 < ρ(σ c t,σc t ) =.5 3. Within countries: ρ(σ c t,σy t ) =.7 < 1 international pass-through. 5 / 18

Measuring Relative Impulse Impact Identify impact of relative output vols on quantities In benchmark case, stack country variables, relative to US: σ t ( y i ) σ t ( y US ) y i y US Ỹ i,t = σ t ( c i ) σ t ( c US ), c i c US (NX/Y ) i (NX/Y ) US Estimate a pooled VAR(1) across countries Trace impulse response of relative output vol shocks on consumption, net exports, and consumption volatility 6 / 18

"(NX/Y) "c "y <("y) "(NX/Y) 1 2 3 4 5 6 7 8 9 1 Motivation #1-3 Empirical Analysis Model Risk-Sharing Conclusions Appendix 1-1 Response to Volatility Shocks -2 1 2 3 4 5 6 7 8 9 1 Periods Data Model Take-aways:.3.2.1 1 2 3 4 5 6 7 8 9 1 #1 1-3 High output volatility decreases the growth rate of output -1-2 1 2 3 4 5 6 7 8 9 1 5 #1-4 -5-1 -15 1 2 3 4 5 6 7 8 9 1 #1-3 1-1 -2 1 2 3 4 5 6 7 8 9 1 Periods Data Model 7 / 18

"(NX/Y) "c "c "y "y <("y) "(NX/Y) 1 2 3 4 5 6 7 8 9 1 Motivation #1-3 Empirical Analysis Model Risk-Sharing Conclusions Appendix 1-1 Response to Volatility Shocks -2 1 2 3 4 5 6 7 8 9 1 Periods.3.3.2.2.1.1 Data Model 1 2 3 4 5 6 7 8 9 1 #1-3 1 2 3 4 5 6 7 8 9 1 1 #1-3 1-1 Take-aways: High output volatility decreases the growth rate of output -1-2 1 2 3 4 5 6 7 8 9 1-2 #1-4 1 2 3 4 5 6 7 8 9 1 5 #1-4 5-5 -1-5 -15-1 1 2 3 4 5 6 7 8 9 1-15 #1-3 1 1 2 3 4 5 6 7 8 9 1 #1-3 1-1 -1-2 1 2 3 4 5 6 7 8 9 1-2 Periods 1 2 3 5 4 6 7 8 9 1 DataPeriods Model Data Model However, net imports increase, and consumption falls by less Evidence of international risk-sharing 7 / 18

<("c) <("y) <("c) <("y) Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix.25 Volatility Pass-Through.2.15.1.5 Output Vol 1 2 3 4 5 6 7 8 9 1 Consumption Vol.25.25.2.2.15.15.1.1.5.5 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 1 Periods Data Model.25.2 High Output Vol increases Consumption Vol less than one-to-one.15.1.5 1 2 3 4 5 6 7 8 9 1 Periods Data Model 8 / 18

Volatility Pass-Through Index (VPTI) Details Pass-through Index := 1 (σ t( c i ) σ t ( c US )) (σ t ( y i ) σ t ( y US )) Interpretation of VPTI with one good and CRRA - no risk sharing, i.e., autarky ( c i,t = y i,t ) - 1 perfect risk sharing ( c i,t = c j,t ) In the data: - G7 countries, VPTI = 5% - Bottom-1 G17 countries, VPTI = 6% - Bottom-1 G17 countries, VPTI = 7% w.r.t. shocks originating in small countries 9 / 18

Volatility Disconnect Puzzle By no-arbitrage + CRRA, FX and consumption diff.s connected: e t+1 = γ ( c h,t+1 c f,t+1 ) 1 / 18

Volatility Disconnect Puzzle By no-arbitrage + CRRA, FX and consumption diff.s connected: e t+1 = γ ( c h,t+1 c f,t+1 ) Backus & Smith (1993): empirical disconnect of levels Corr( e t+1, c h,t+1 c f,t+1 ) This paper: empirical disconnect of vols Corr(Var t [ e t+1 ],Var t [ c h,t+1 c f,t+1 ]).2 - Puzzle with CRRA - Puzzle for EZ models that address the Backus-Smith puzzle (among others, Colacito Croce (211,213)) 1 / 18

Model 11 / 18

Model Two countries: home (h) and foreign (f ) Recursive EZ utility over the consumption aggregate C t C h t = ( ) xt h α ( ) y h 1 α t, C f t = ( ) xt f 1 α ( ) y f α t x h, x f, y h, and y f are allocations of each good to each country α > 1/2 captures home bias Endowments are co-integrated, and feature long-run and volatility risks: logx t = µ x + z 1,t 1 τlog(x t 1 /Y t 1 ) + e σ x,t /2 σε x,t logy t = µ y + z 2,t 1 + τlog(x t 1 /Y t 1 ) + e σ y,t /2 σε y,t z j,t = ρz j,t 1 + σ z ε j,t, j {1,2} Focus on short-run volatilities of endowments, as in the data. - Can extend to accommodate long-run volatility risks 11 / 18

Equilibrium Allocations and Relative Size Under complete markets, compute efficient allocations by solving Pareto problem with time-varying weights Optimal allocations depend on ratio of Pareto weights (country size) S t : S t = S t 1 Mh t M f t ( C h t /Ct 1 h C f t /Cf t 1 ), t 1 Evolution of S t depends on pricing kernels M h and M f Under recursive preferences, volatility news are priced, and affect consumption allocations Details 12 / 18

Model Calibration Calibration for level shocks: similar to Colacito Croce (JPE 211, JF 213) - Risk aversion is 7 - Intertemporal elasticity of substitution is 1.5 Calibration for vol shocks: median estimates in our data - Output volatility shocks are persistent - Negatively correlated with endowment shocks (-.12, as in the data) - Weakly correlated across countries (.3) Same successes of Colacito Croce (213) + explains VPTI and vol disconnect 13 / 18

sdf " (NX/Y) " c <("y) sdf 2 4 6 8 1 2 4 6 8 1 Motivation#1 Empirical Analysis Model Risk-Sharing Conclusions Appendix #1-3 2-2 2-2 Risk Sharing 2 4 6 8 1 Periods 2 4 6 8 1 Periods HOME FOREIGN.2.15.1.5 EZ.2.15.1.5 CRRA Home country receives vol shock 2 4 6 8 1 #1-4 2 2 4 6 8 1 #1-4 2 1 1-1 -1-2 2 4 6 8 1 #1-5 5-2 2 4 6 8 1 #1-5 5-5 -5 2 4 6 8 1 #1-3 2 4 6 8 1 #1-3 2 2-2 -2 2 4 6 8 1 Periods 2 4 6 8 1 Periods HOME FOREIGN 14 / 18

sdf " (NX/Y) sdf " c <("y) " (NX/Y) " c sdf <("y).15 2 4 6 8 1.15 2 4 6 8 1 Motivation#1 Empirical Analysis Model Risk-Sharing Conclusions Appendix #1-3.1.1 2.5 2 4 6 8 1 #1-2 -4 2 2.5 2 4 6 8 1 #1-2 -4 2 Risk Sharing 1-1 2 4 6 8 1 Periods HOME 2 4 6 8 1 1 Periods FOREIGN -1-2 2 4 EZ 6 8 1.2 #1-5.155.1.5-2 2 4CRRA6 8 1.2 #1-5.155.1.5 Home country receives vol shock -5 2 4 6 8 1 2 4 6 8 1 #1-4 2 #1-3 1 2-1 -2 2 4 6 8 1 2 4 6 8 1 #1-5 Periods 5-5 2 4 6 8 1 2 4 6 8 1 #1-4 2 #1-3 1 2-1 -2 2 4 6 8 1 2 4 6 8 1 #1-5 Periods 5 Under EZ utility, vol shock is bad news Home SDF HOME FOREIGN -5-5 2 4 6 8 1 2 4 6 8 1 #1-3 #1-3 2 2-2 -2 2 4 6 8 1 Periods 2 4 6 8 1 Periods HOME FOREIGN 14 / 18

" (NX/Y) sdf " c " (NX/Y) <("y) sdf " c <("y) " (NX/Y) " c sdf <("y).15 2 4 6 8 1.15 2 4 6 8 1 Motivation#1 Empirical Analysis Model Risk-Sharing Conclusions Appendix #1-3.1.1 2.5 2 4 6 8 1 #1-2 -4 2 2.5 2 4 6 8 1 #1-2 -4 2 Risk Sharing 1-1 2 4 6 8 1 Periods HOME 2 4 6 8 1 1 Periods FOREIGN -1-2 2 4 EZ 6 8 1.2 #1-5.155.1.5-2 2 4CRRA6 8 1.2 #1-5.155.1.5 Home country receives vol shock -5 2 4 6 8 1 #1-4 2 4 6 8 1 2 #1-3 1 EZ.22.15-1.1-2.5 2 4 6 8 1 #1-5 2 4 6 8 1 2 4 5 Periods6 8 1 #1-4 2 HOME 1-5 -1 2 4 6 8 1-2 #1-3 2 4 6 8 1 2 #1-5 5-2 2 4 6 8 1-5 Periods 2 4 6 8 1 HOME #1-3 2-5 2 4 6 8 1 2 4 6 8 1 #1-4 2 #1-3 CRRA 1.22.15-1.1-2.5 2 4 6 8 1 2 4 6 8 1 #1-5 4Periods6 2 8 1 5 #1 FOREIGN 2-4 1-5 -1 2 4 6 8 1-2 #1-3 2 4 6 8 1 2 #1-5 5-2 2 4 6 8 1 Periods -5 2 4 6 8 1 FOREIGN #1-3 2 Under EZ utility, vol shock is bad news Home SDF Under EZ utility, high vol country receives resources from abroad Home Consumption Home NX 14 / 18

Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption and output growth within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and 15 / 18 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Unconditional co-movements of volatilities Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 st ; 4 th ] mark (σ σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5

Unconditional co-movements of volatilities Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 st ; 4 th ] mark (σ σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Within countries: high correlation of consumption vol and GDP vol Across countries: lower correlation of consumption vols Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption and output growth within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and 15 / 18

Unconditional co-movements of volatilities Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 st ; 4 th ] mark (σ σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Within countries: high correlation of consumption vol and GDP vol Across countries: lower correlation of consumption vols CRRA overshoots with both correlations Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption and output growth within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and 15 / 18

Unconditional co-movements of volatilities Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 st ; 4 th ] mark (σ σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Within countries: high correlation of consumption vol and GDP vol Across countries: lower correlation of consumption vols CRRA overshoots with both correlations Time-varying vol (TVV) brings model with EZ preferences closer to the data Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption and output growth within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and 15 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption and output growth within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping the US as home country. For each country, we compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4. For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in section 2.1. Panel B reports estimated pass-through coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping the US as home country. For each country, we compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4. For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect to both domestic (US) and foreign volatility shocks for both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping the US as home country. For each country, we compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4. For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect benchmark to bothmodel domestic matches (US) and the empirical foreign volatility amount shocks of Vol for pass-through both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping the US as home country. For each country, we compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4. For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix US/Bottom-1 G17 Countries Data [.72.77] [.45.57] [.66.78] Model (EZ).72.39.7 Model (CRRA).72.38.37 Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect benchmark to bothmodel domestic matches (US) and the empirical foreign volatility amount shocks of Vol for pass-through both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping 2 US vs bottom G17 countries the US as home country. For each country, we compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4. For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix US/Bottom-1 G17 Countries Data [.72.77] [.45.57] [.66.78] Model (EZ).72.39.7 Model (CRRA).72.38.37 Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect benchmark to bothmodel domestic matches (US) and the empirical foreign volatility amount shocks of Vol for pass-through both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping 2 US vs bottom G17 countries the US as home country. For each country, we compute the moments of interest over the post BrettonUS Wood hasperiod, a much1971:q1 213:Q4. larger SWC For each moment, we report first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix US/Bottom-1 G17 Countries Data [.72.77] [.45.57] [.66.78] Model (EZ).72.39.7 Model (CRRA).72.38.37 Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect benchmark to bothmodel domestic matches (US) and the empirical foreign volatility amount shocks of Vol for pass-through both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping 2 US vs bottom G17 countries the US as home country. For each country, we compute the moments of interest over the post BrettonUS Wood hasperiod, a much1971:q1 213:Q4. larger SWC For each moment, we report first and fourth cross-countryus quintiles. unloadsthe lessentries vol to from smaller thecountries model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration is reported in table 4. 16 / 18

Panel A: Unconditional comovements Avg. Quintiles Bench- No TVV CRRA [ 1 Pass-through ; 4 ] mark (σ and size σ = ) (γ = 7) corr(σ t ( c t+1 ), σ t ( y t+1 ).65 [.26;.8].88.98 corr(σ t ( c t+1 ), σ t ( c t+1)).45 [.35;.66].35 -.93.5 Panel B: Pass-through and size SWC US vol shock Foreign vol shock US/G7 Countries: Data [.44.51] [.43.54] [.51.63] Model (EZ).5.53.53 Model (CRRA).5.3.3 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix US/Bottom-1 G17 Countries Data [.72.77] [.45.57] [.66.78] Model (EZ).72.39.7 Model (CRRA).72.38.37 Notes: In panel A, we report correlations between the conditional volatility (σ t ) of consumption1 and USoutput vs G7growth countries within and across countries. Conditional volatilities are obtained by estimating equation (2.1) country by country. The data refer to G-17 countries and are described in similar section Shares 2.1. Panel of World B reports Consumption estimated pass-through (SWC) coefficients (see equation (2.4)) with respect benchmark to bothmodel domestic matches (US) and the empirical foreign volatility amount shocks of Vol for pass-through both the G7 and bottom-1 G17 countries. SWC denotes the share of world consumption, S/(1+S), keeping 2 US vs bottom G17 countries the US as home country. For each country, we compute the moments of interest over the post BrettonUS Wood hasperiod, a much1971:q1 213:Q4. larger SWC For each moment, we report first and fourth cross-countryus quintiles. unloadsthe lessentries vol to from smaller thecountries model are obtained from 1 repetitions of small samples. smaller Our benchmark countries unload quarterly a lot calibration of vol risk reported to US in table 4. 16 / 18

FX and Consumption Disconnect in the Model CRRA EZ e t+1 = ( c f t+1 ch t+1) St+1 17 / 18

FX and Consumption Disconnect in the Model EZ CRRA e t+1 = ( c f t+1 ch t+1) St+1 17 / 18

FX and Consumption Disconnect in the Model EZ CRRA e t+1 = ( c f t+1 ch t+1) St+1 G-17 Data Model Avg. Quintiles Bench- No TVV CRRA [1 st ;4 th ] mark ( =) ( =7) Levels Disconnect corr( cd t+1, e t+1 ) -.13 [-.19; -.4] -.25 -.27 1. corr( cdt+4 b, be t+4 ) -.14 [-.29; -.5] -.21 -.24 1. Notes: This table reports key moments for real consumption growth di erentials ( cd = c c ) and exchange rate growth ( e). Foreign variables are marked by ; cumulative good long-run risks and volatility shocks decrease relative consumption growth rates are denoted by b. Conditional log-volatilities are denoted by t. The empirical moments andare size obtained of country by estimating equation (2.1) country by country, as detailed in section 2.2. The data refer to G-17 countries and are described in section 2.1. For each country, we Produces weak negative correlation between the levels of FX and compute the moments of interest over the post Bretton Wood period, 1971:Q1 213:Q4, as detailed consumption in section 2.1. differential, For each moment, as in thewe data report (i) its GDP-weighted average across countries; and (ii) its first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration 17 / 18

FX and Consumption Disconnect in the Model EZ CRRA e t+1 = ( c f t+1 ch t+1) St+1 G-17 Data Model Avg. Quintiles Bench- No TVV CRRA [1 st ;4 th ] mark ( =) ( =7) Levels Disconnect corr( cd t+1, e t+1 ) -.13 [-.19; -.4] -.25 -.27 1. corr( cdt+4 b, be t+4 ) -.14 [-.29; -.5] -.21 -.24 1. Volatility Disconnect corr( t ( cd t+1 ), t( e t+1 )).2 [-.1.42].56 -.75 1. corr( t ( cd b t+4 ), t( eb t+4 )).26 [-.2.52].47 -.75 1. Notes: This table reports key moments for real consumption growth di erentials ( cd = c c ) and exchange rate growth ( e). Foreign variables are marked by ; cumulative CRRA and model with no TVV cannot match this moment growth rates are denoted by b. Conditional log-volatilities are denoted by t. The empirical moments Volatilities are obtained of consumption by estimating equation differential (2.1) and country consumption by country, asshare: detailed in section 2.2. The data refer to G-17 countries and are described in section 2.1. For each country, we compute themove moments in the ofsame interest direction over theinpost Bretton response to volatility Wood period, shocks 1971:Q1 213:Q4, as detailed inmove section in the 2.1. opposite For each moment, directionwe in response report (i) its to long-run GDP-weighted shocks average across countries; and (ii) its first and fourth cross-country quintiles. The entries from the model are obtained from 1 repetitions of small samples. Our benchmark quarterly calibration 17 / 18

18 / 18 Motivation Empirical Analysis Model Risk-Sharing Conclusions Appendix Conclusions 1. Domestic volatility risks are "passed through" internationally 2. Volatility pass-through is significant - Smaller countries better share volatility risks 3. FX-Vol Disconnect Puzzle 4. Resolve these puzzles with recursive risk sharing of vol shocks

Table 1: Data Summary Statistics G7 Avg. G17 Avg. G17 Quintile Simple Simple Weighted 1 st 4 th Consumption growth Mean 1.91 1.63 1.89 1.26 2.2 Std. Dev. 1.75 1.99 1.67 1.34 2.47 AR(1).11.7.17 -.16.31 Output growth Mean 1.94 1.71 1.93 1.43 2. Std. Dev. 2.21 2.97 2.2 2.1 4.43 AR(1). -.9.7 -.26.9 Net Exports over Output: Mean.3.8.4 -.3.34 Std. Dev. 1.6 2.48 1.45 1.79 3.24 AR(1). -.9.7 -.26.9 Within-Country Correlations: Consump. and output growth.67.51.71.35.72 Consump. and output vol.54.47.65.26.8 Across-Country Correlations: Consump. growth.27.24.25.13.33 Output growth.15.14.14.6.2 Consump. vol.51.47.45.35.66 Output vol.32.3.3.18.45 Notes: This table shows summary statistics for consumption growth, output growth, change in net-export-to-output ratio, and consumption and output volatility. G7 Avg. ( G17 Avg. ) refers to simple (both simple and GDP-weighted) averages of key moments for G7 (G17) countries. The rightmost two columns show the first and fourth quintiles of the mo- 1 / 9

Table 2: Volatility Risk Pass-Through Panel A: Contemporaneous adjustments to relative volatility shocks ( y) y ( c) c (NX/Y) Passthrough US/G7 Countries:.21 -.46.1 -.2 -.25.52 [.2.22] [.9.11] [-.44.3] [-.44.3] [-.49 -.2] [.48.56] US/Bottom-1 G17 Countries:.21 -.57.8 -.16 -.39.61 [.21;.22] [-.95; -.19] [.7;.9] [-.41;.9] [-.73; -.6] [.56;.65] Panel B: Pass-through and size Origin of Vol Shock: U.S. Foreign Country US/G7 Countries:.49.57 [.43;.54] [.51;.63] US/Bottom-1 G17 Countries:.51.72 [.45;.57] [.66;.78] Notes: Panel A shows the estimates of the contemporaneous responses ( e 1j) of the VAR(1) specified in equations (2.2) (2.3) with respect to a shock to relative output volatility. Responses of output growth, consumption growth, and net-exports-to-output ratio are annualized, in percentages. Volatility pass-through is defined as in equation (2.4). Panel B reports pass-through measures based on the estimates of the VAR in equations (2.5) (2.6) with respect to volatility shocks a ecting either the US or the remaining countries. We report 95% credible intervals in brackets. Our quarterly data range from 1971:q1 to 213:q4. 2 / 9

Table 3: Volatility Disconnect Puzzle G7 Avg. G17 Avg. G17 Quintile Simple Simple Weighted 1 st 4 th Levels Disconnect corr( cd t+1, e t+1 ) -.14 -.11 -.13 -.19 -.4 corr( cdt+4 b, be t+4 ) -.14 -.17 -.14 -.29 -.5 Volatility Disconnect corr( t ( cd t+1 ), t( e t+1 )).2.21.2 -.1.42 corr( t ( cdt+4 b ), t( be t+4 )).27.25.26 -.2.52 Notes: This table shows correlations between the level and conditional volatility of consumption growth di erentials (cd i t c US t c i t) and exchange rate growth ( e i USD t ), respectively. In both cases, the US is considered the benchmark home country. Cumulative growth rates are denoted by c. G7 Avg. ( G17 Avg. ) refers to simple (both simple and GDP-weighted) averages of key moments for G7 (G17) countries. The rightmost two columns show the first and fourth quintiles of the moments of interest in the G17 crosssection. Consumption is seasonally adjusted, real, and per capita. Volatility estimates are based on the specification reported in equation (2.1). Quarterly observations are from the 1971:Q1 213:Q4 sample. 3 / 9

Table 4: Calibration Description Parameter Value Panel A: Standard Parameters Relative Risk Aversion 7 Intertemporal Elasticity of Substitution 1.5 Subjective Discount Factor 4.98 Degree of Home Bias.96 Mean of Endowment Growth µ 4 2.% Short-Run Risk Volatility p4 1.87% Long-Run Risk Autocorrelation 4.953 Relative Long-Run Risk Volatility z/ 6.9% Cross-Correlation of Short-Run Shocks X.15 Cross-Correlation of Long-Run Shocks z.92 Panel B: Time-Varying Short-Run Risk Persistence of Short-Run Volatility.9 [.89.93] Volatility of Short-Run Volatility sr.15 [.15.16] Cross-Correlation of Short-Run Volatility,.3 [.13.45] Short-Run Volatility Correlation with, y -.12 Short-Run Shocks [-.15 -.5] Notes: All parameters are calibrated at quarterly frequency. In panel B, the entries for the data are from the VAR specified in equations (2.5) (2.6). Numbers in brackets denote the 95% credible intervals. Data are from the OECD dataset and refer to G-17 countries. The 4 / 9

Table A1: Robustness of Pass-Through Results Panel A: Contemporaneous adjustments to relative volatility shocks ( y) y ( c) c (NX/Y) Passthrough Global Benchmark, G17 Countries:.16 -.44.6 -.6 -.37.61 [.15;.16] [-.67; -.21] [.6;.7] [-.2;.9] [-.56; -.18] [.57;.64] US/Pooled G7:.19 -.52.9 -.26 -.26.53 [.19;.2] [-.83; -.23] [.8;.1] [-.5;-.2] [-.49; -.3] [.49;.56] VAR(2) Model:.21 -.41.9 -.11 -.29.59 [.2;.21] [-.71; -.11] [.8;.9] [-.34;.13] [-.53; -.6] [.55;.62] Panel B: Pass-through and size Origin of Vol Shock: US Foreign Country Global Benchmark/G17 Countries:.52.62 [.45;.59] [.58;.66] US/Pooled G7:.47.64 [.43;.52] [.58;.7] VAR(2):.55.63 [.5;.6] [.58;.68] Notes: Panel A shows the estimates of the contemporaneous responses ( e 1j) of the VAR specified in equations (2.2) (2.3) with respect to a shock to relative output volatility. Responses of output growth, consumption growth, and the net-exports-to-output ratio are 5 / 9

Table B1: Standard Unconditional Moments G-17 Data Model Avg. Quintiles Bench- No TVV CRRA [1 st ;4 th ] mark ( =) ( =7) corr( c, c ).25 [.13;.33].38.37.74 ( c)(%) 1.67 [1.34; 2.47] 1.85 1.82 1.64 ( c)/ ( y).88 [.57;.82].93.94.83 ACF 1( c).17 [-.16;.31].6.7.8 (M)/E(M)(%) 47.86 47.85 11.49 ( e)(%) 1.5 [1.2; 11.4] 12.8 12.65 8.31 E(r f )(%) 1.35 [1.44; 2.41] 2.17 2.19 14.91 (r f )(%) 1.79 [1.61; 2.27].33.33 3.47 corr(r f,r f ).51 [.37;.56].91.92.98 ( (NX/Y))/ ( y).7 [.67;.97].32.32.16 Notes: This table reports key moments for real consumption (C), output (Y ), the exchange rate (E), the risk-free rates (R f ), the net-export-to-output ratio (NX/Y), and the stochastic discount factor (M). Small letters refer to log-units; changes are denoted by ; foreign variables are marked by. We denote expectation, standard deviation, correlation, and first order auto-correlation by E,, corr, and ACF 1, respectively. The data refer to G-17 6 / 9

Volatility Pass-Through Index Back Using the VAR on σ t ( y i ) σ t ( y US ) y i y US Ỹ i,t = σ t ( c i ) σ t ( c US ), c i c US (NX/Y ) i (NX/Y ) US the VPTI is VPTI = 1 Σ 3,1 Σ 1,1 7 / 9

Volatility Pass-Through Index (cont d) Back Using the VAR on [ ] Ỹ i,t = σ t ( y US ),σ t ( y i ), y US, y }{{}}{{}}{{} i,σ }{{} t ( c US ),σ t ( c i ) }{{}}{{}, 1 2 3 4 5 6 VPTI from country i to US VPTI from US to country i VPTI = 1 Σ 6,2 Σ 5,2 Σ 2,2 VPTI = 1 Σ 5,2 Σ 6,2 Σ 1,1 8 / 9

Volatility shocks are priced Back Consider the case of ψ = 1, then U t = (1 δ)logc t + δθloge t exp { Ut+1 θ }, θ = 1/(1 γ) < A second order Taylor expansion about E t [U t+1 ] yields The SDF is U t (1 δ)logc t + δe t [U t+1 ] + δ 2θ Var t[u t+1 ] m t E t 1 [m t ] = ( c t E t 1 [ c t ]) + U t θ If Var t [U t+1 ] then U t and m t 9 / 9