Estimation Appendix to Dynamics of Fiscal Financing in the United States

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1 Estimation Appendix to Dynamics of Fiscal Financing in the United States Eric M. Leeper, Michael Plante, and Nora Traum July 9, 9. Indiana University. This appendix includes tables and graphs of additional results and diagnostics not included in the paper. For reference, the prior distributions for the parameters are recorded in table. Figures - display plot slices of the likelihood around the mode for the model in which all fiscal instruments adjust to debt. Similar plots exist for the alternative estimated models. We performed searches for the mode from different starting values to determine if more than one mode exists. In all the models, the searches for the posterior mode almost always converged to the same parameter values and likelihood value. The few exceptions were simply cases where the numerical optimization procedure failed to converge to any value. For the model in which all fiscal instruments respond to debt and the model in which only capital and labor taxes respond to debt, this occurred once in searches. For the model in which only government spending responds to debt, this occurred twice in searches. For the model in which only lump sum transfers respond to debt, this occurred in 6 of searches. The condition number of the Hessian at the mode for the model where all fiscal instruments adjust to debt is,; for the model where only government spending adjust to debt it is,9.9; for the model where only capital and labor taxes adjust to debt it is,77.; and for the model where only lump sum transfers adjust to debt it is,7.9. Tables - summarize estimated posterior means and standard deviations for parameters and reports the Geweke Separated Partial Means (GSPM) test p-values. Figures - 7 display trace plots for the model where all fiscal instruments adjust to debt. Figures 8 - The GSPM test determines whether the mean from the first % of the MCMC draws is identical to the

2 display trace plots for the model where only capital and labor taxes adjust to debt. Figures - display trace plots for the model where only government spending adjusts to debt. Figures - 6 display trace plots for the model where only lump sum transfers adjust to debt. Finally, figures 7 - display autocorrelation functions for the estimated parameters for each estimated model. Finally, we report some additional results not given in the main text. Table 7 gives variance decompositions using the mean estimates of posterior draws from the model where all fiscal instruments can adjust to debt. Tables 8 - give present value multipliers following a positive change in government spending, lump sum transfers, capital taxes, and labor taxes respectively using the mean estimates of posterior draws from the estimated models. We follow the measure in Mountford and Uhlig (9) to calculate present value multipliers. For example, the change in the present value of additional output over different horizons generated by a change in the present value of government spending is calculated as Present Value Multiplier(k) = E t k j= E t k j= ( j i= R t+i ( j i= R t+i ) ) Y t+j G t+j References Geweke, J. (): Contemporary Bayesian Econometrics and Statistics. John Wiley and Sons, Inc., Hoboken, NJ. Mountford, A., and H. Uhlig (9): What Are the Effects of Fiscal Policy Shocks?, Forthcoming in Journal of Applied Econometrics. mean of the final % of the draws. A Z-test of the hypothesis of equality of the two means is carried out and the corresponding chi-squared marginal significance is calculated (see Geweke (), pages 9- for more details). The GSPM test yields p-values above. for most parameters and above. for almost all parameters.

3 Table : Prior distributions Parameter Prior Distribution Name Domain Density Mean St. Dev. γ R + G.7. κ R + G.. h [,] B.. s R N.. δ R + G.7. ϕ g R + G.7. ϕ k R + G.. ϕ l R + G.. ϕ z R + G.. γ g R + G.. γ k R + G.. γ l R + G.. γ z R + G.. φ kl R N.. φ kc R N.. φ lc R N.. ρ a [,) B.7. ρ c [,) B.7. ρ l [,) B.7. ρ i [,) B.7. ρ g [,) B.7. ρ k [,) B.7. ρ l [,) B.7. ρ c [,) B.7. ρ z [,) B.7. σ a R + IG σ c R + IG σ l R + IG σ i R + IG σ g R + IG σ k R + IG σ l R + IG σ c R + IG σ z R + IG : The parameters for the Inverse Gamma distribution correspond to s and ν where f(x s, ν) = ν s Γ (s)x s exp ν x.

4 Table : Posterior Estimates for the model where only lump-sum transfers adjust to debt. Reports the posterior mode, mean, median, standard deviation, 9% credible interval, and the p-value for Geweke s Separated Partial Means test. Final acceptance rate: 7.%. In this case,,, draws were made, with the first, used as a burn-in period and every th thinned, leaving a sample size of,7. Parameter Prior Distribution Posterior Distribution Name Domain Density Mean St. Dev. Mode Mean Median 9% CI St. Dev. Geweke Chi-Sq. p γ R + Gamma (.6,.8) κ R + Gamma (.69,.8) h [,] Beta (.98,.88) s R Normal (.7,.89) δ R + Gamma (.9,.98).8.77 ρ a [,) Beta (.99,.98).6.96 ρ c [,) Beta (.6,.697) ρ l [,) Beta (.96,.99) ρ i [,) Beta (.888,.668).6.86 ρ g [,) Beta (.9,.98) ρ τ k [,) Beta (.87,.97).8.69 ρ τ l [,) Beta (.97,.99) ρ τ c [,) Beta (.8899,.9668) ρ z [,) Beta (.99,.979) γ z R + Gamma (.,.97) ϕ τ k R Normal (.97,.878) ϕ τ l R Normal (.68,.6).9.67 ϕ g R Normal (.9,.8)..769 ϕ z R Normal (.76,.7) σ a R + InvGamma (.76,.6766) σ c R + InvGamma (6., 7.777) σ l R + InvGamma (.,.68) σ i R + InvGamma (.778, 7.889).6.76 σ g R + InvGamma (.969,.68)..6 σ τ k R + InvGamma (.,.969) σ τ l R + InvGamma (.786,.).6.8 σ τ c R + InvGamma (.77,.696).7.77 σ z R + InvGamma (.9,.6) φ kl R Normal (.6,.) φ kc R Normal (-.,.99)..988 φ lc R Normal (-.79,.6).978.

5 Table : Posterior Estimates for the model where only government spending adjusts to debt. Reports the posterior mode, mean, median, standard deviation, 9% credible interval, and the p-value for Geweke s Separated Partial Means test. Final acceptance rate: 6.67%.,, draws were made, with the first, used as a burn-in period and every th thinned, leaving a sample size of,7. Parameter Prior Distribution Posterior Distribution Name Domain Density Mean St. Dev. Mode Mean Median 9% CI St. Dev. Geweke Chi-Sq. p γ R + Gamma (.8,.6).897. κ R + Gamma (.,.687).. h [,] Beta (.,.99).69. s R Normal (.,.887) δ R + Gamma (.978,.) ρ a [,) Beta (.978,.987)..9 ρ c [,) Beta (.687,.69) ρ l [,) Beta (.9699,.997) ρ i [,) Beta (.9,.66) ρ g [,) Beta (.98,.989) ρ τ k [,) Beta (.88,.99).8.9 ρ τ l [,) Beta (.96,.99) ρ τ c [,) Beta (.89,.9679) ρ z [,) Beta (.98,.9887) γ g R + Gamma (.6,.7) ϕ τ k R Normal (.98,.787) ϕ τ l R Normal (.6,.6).99.7 ϕ g R Normal (.6,.87).9.96 ϕ z R Normal (.,.).8.88 σ a R + InvGamma (.798,.676)..76 σ c R + InvGamma (6.7, ) σ l R + InvGamma (.,.) σ i R + InvGamma (.6, 6.97) σ g R + InvGamma (.87,.9) σ τ k R + InvGamma (.,.99) σ τ l R + InvGamma (.79,.) σ τ c R + InvGamma (.77,.) σ z R + InvGamma (.6,.697) φ kl R Normal (.8,.967).9.89 φ kc R Normal (-.,.97)..969 φ lc R Normal (-.77,.9).7.89

6 Table : Posterior Estimates for the model where taxes adjust to debt. Reports the posterior mode, mean, median, standard deviation, 9% credible interval, and the p-value for Geweke s Separated Partial Means test. Final acceptance rate:.87%.,, draws were made, with the first, used as a burn-in period and every th thinned, leaving a sample size of,7. Parameter Prior Distribution Posterior Distribution Name Domain Density Mean St. Dev. Mode Mean Median 9% CI St. Dev. Geweke Chi-Sq. p γ R + Gamma (.66,.7) κ R + Gamma (.,.6) h [,] Beta (.,.97) s R Normal (.6,.888) δ R + Gamma (.7,.97).6. ρ a [,) Beta (.98,.98)..66 ρ c [,) Beta (.6,.69) ρ l [,) Beta (.97,.9978) ρ i [,) Beta (.,.6) ρ g [,) Beta (.9,.9876).6.99 ρ τ k [,) Beta (.899,.97).8.76 ρ τ l [,) Beta (.966,.99) ρ τ c [,) Beta (.898,.9677).7. ρ z [,) Beta (.9,.9876)..986 γ τ k R + Gamma (.687,.).7.6 γ τ l R + Gamma (.,.9).79.9 ϕ τ k R Normal (.87,.).796. ϕ τ l R Normal (.,.8).96. ϕ g R Normal (.9,.77) ϕ z R Normal (.,.) σ a R + InvGamma (.78,.679) σ c R + InvGamma (6.8, 7.69)..87 σ l R + InvGamma (.6,.7) σ i R + InvGamma (.6, 6.96) σ g R + InvGamma (.98,.87).6.87 σ τ k R + InvGamma (.7,.7).79.9 σ τ l R + InvGamma (.7969,.) σ τ c R + InvGamma (.766,.7)..88 σ z R + InvGamma (.67,.76) φ kl R Normal (.9,.6)..69 φ kc R Normal (-.,.7).8.6 φ lc R Normal (-.7,.77)

7 Table : Posterior Estimates for the model where all instruments adjust to debt. Reports the posterior mode, mean, median, standard deviation, 9% credible interval, and the p-value for Geweke s Separated Partial Means test. Final acceptance rate:.98%.,, draws were made, with the first, used as a burn-in period and every th thinned, leaving a sample size of,7. Parameter Prior Distribution Posterior Distribution Name Domain Density Mean St. Dev. Mode Mean Median 9% CI St. Dev. Geweke Chi-Sq. p γ R + Gamma (.87,.68).7.99 κ R + Gamma (.7,.88) h [,] Beta (.7,.9) s R Normal (.8,.889) δ R + Gamma (.967,.66) ρ a [,) Beta (.988,.98).6. ρ c [,) Beta (.68,.69) ρ l [,) Beta (.967,.9966) ρ i [,) Beta (.667,.68).9.76 ρ g [,) Beta (.97,.988) ρ τ k [,) Beta (.897,.9699) ρ τ l [,) Beta (.9,.997) ρ τ c [,) Beta (.89,.968) ρ z [,) Beta (.97,.978).686. γ g R + Gamma (.,.7).98.8 γ τ k R + Gamma (.76,.7) γ τ l R + Gamma (.9,.9) γ z R + Gamma (.7,.899).88.8 ϕ τ k R Normal (.78,.7) ϕ τ l R Normal (.86,.6).8.8 ϕ g R Normal (.6,.87) ϕ z R Normal (.89,.7).7.77 σ a R + InvGamma (.776,.676).99.7 σ c R + InvGamma (6.8, 7.78) σ l R + InvGamma (.9,.7) σ i R + InvGamma (.687, 7.).8.77 σ g R + InvGamma (.8,.99)..8 σ τ k R + InvGamma (.69,.77).68.9 σ τ l R + InvGamma (.786,.69).8.88 σ τ c R + InvGamma (.789,.68) σ z R + InvGamma (.9,.6).7.69 φ kl R Normal (.,.6) φ kc R Normal..... (-.,.7) φ lc R Normal (-.77,.76)

8 Table 6: Posterior Estimates. Reports the posterior mean and standard deviation for the various models estimated. Parameter Prior Distribution Mean & St. Dev. of Posterior Distribution Name Domain Density Mean St. Dev. All Adjust Expenditures Adjust Tax Adjust Transfer Adjust γ R + Gamma (.) (.) (.) (.8) κ R + Gamma (.) (.9) (.88) (.) h [,] Beta (.7) (.78) (.) (.67) s R + Gamma (.8) (.9) (.799) (.9) δ R + Gamma (.7) (.9) (.) (.8) ρ a [,) Beta (.6) (.) (.) (.7) ρ c [,) Beta (.87) (.9) (.76) (.9) ρ l [,) Beta (.76) (.8) (.68) (.8) ρ i [,) Beta (.9) (.6) (.) (.) ρ g [,) Beta (.) (.6) (.6) (.6) ρ τ k [,) Beta (.8) (.8) (.8) (.8) ρ τ l [,) Beta (.8) (.) (.) (.6) ρ τ c [,) Beta (.8) (.87) (.7) (.88) ρ z [,) Beta (.69) (.7) (.) (.76) γ g R + Gamma (.9) (.99) - - γ τ k R + Gamma (.7) - (.7) - γ τ l R + Gamma (.76) - (.7) - γ z R + Gamma (.) (.) - (.69) ϕ τ k R Normal (.7) (.7) (.79) (.9) ϕ τ l R Normal (.) (.86) (.9) (.) ϕ g R Normal (.99) (.) (.8) (.) ϕ z R Normal (.7) (.9) (.98) (.9) σ a R + InvGamma (.) (.) (.7) (.) σ c R + InvGamma (.9) (.8) (.) (.9) σ l R + InvGamma (.97) (.) (.86) (.787) σ i R + InvGamma (.8) (.97) (.9) (.) σ g R + InvGamma (.) (.) (.6) (.) σ τ k R + InvGamma (.6) (.78) (.79) (.78) σ τ l R + InvGamma (.8) (.6) (.) (.6) σ τ c R + InvGamma (.8) (.) (.) (.7) σ z R + InvGamma (.7) (.6) (.86) (.79) φ kl R Normal (.8) (.8) (.) (.) φ kc R Normal (.) (.6) (.) (.) φ lc R Normal (.) (.7) (.7) (.9)

9 Table 7: Variance Decompositions using the mean estimates of posterior draws from the model where all fiscal instruments can adjust to debt. After Period ɛ a ɛ b ɛ h ɛ i ɛ g ɛ k ɛ l ɛ c ɛ z Tot Y C L I B G τ k τ l τ c After Periods ɛ a ɛ b ɛ h ɛ i ɛ g ɛ k ɛ l ɛ c ɛ z Tot Y C L I B G τ k τ l τ c After 6 Periods ɛ a ɛ b ɛ h ɛ i ɛ g ɛ k ɛ l ɛ c ɛ z Tot Y C L I B G τ k τ l τ c

10 Table 8: Present value multipliers following a positive change in government spending using the mean estimates of posterior draws from the estimated models. All Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( G) PV( C) PV( G) PV( I) PV( G) Tax Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( G) PV( C) PV( G) PV( I) PV( G) Transfers Adjust Variable quarter quarters quarters quarters PV( Y ) PV( G) PV( C) PV( G) PV( I) PV( G) Expenditures Adjust Variable quarter quarters quarters quarters PV( Y ) PV( G) PV( C) PV( G) PV( I) PV( G)

11 Table 9: Present value multipliers following a positive change in transfers using the mean estimates of posterior draws from the estimated models. All Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( Z) PV( C) PV( Z) PV( I) PV( Z) Tax Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( Z) PV( C) PV( Z) PV( I) PV( Z) Expenditures Adjust Variable quarter quarters quarters quarters PV( Y ) PV( Z) PV( C) PV( Z) PV( I) PV( Z)

12 Table : Present value multipliers following a positive change in capital taxes using the mean estimates of posterior draws from the estimated models. All Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T k ) PV( C) PV( T k ) PV( I) PV( T k ) Tax Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T k ) PV( C) PV( T k ) PV( I) PV( T k ) Transfers Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T k ) PV( C) PV( T k ) PV( I) PV( T k ) Expenditures Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T k ) PV( C) PV( T k ) PV( I) PV( T k )

13 Table : Present value multipliers following a positive change in labor taxes using the mean estimates of posterior draws from the estimated models. All Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T l ) PV( C) PV( T l ) PV( I) PV( T l ) Tax Instruments Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T l ) PV( C) PV( T l ) PV( I) PV( T l ) Transfers Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T l ) PV( C) PV( T l ) PV( I) PV( T l ) Expenditures Adjust Variable quarter quarters quarters quarters PV( Y ) PV( T l ) PV( C) PV( T l ) PV( I) PV( T l )

14 8 γ 88 κ 8 habit δ 8 s 8 ρ a ρ b ρ l ρ i Figure : Plots of slices of the likelihood around the posterior mode for the model where all fiscal instruments adjust to debt. The vertical line indicates the posterior mode.

15 ρ g ρ tk ρ tl ρ tc ρ z γ g γ tk γ tl γ z Figure : Plots of slices of the likelihood around the posterior mode for the model where all fiscal instruments adjust to debt. The vertical line indicates the posterior mode.

16 88. φ tk 88.9 φ tl 88.9 φ g φ z 8 σ a 8 σ b σ l 8 σ i 8 σ g Figure : Plots of slices of the likelihood around the posterior mode for the model where all fiscal instruments adjust to debt. The vertical line indicates the posterior mode. 6

17 σ tk σ tl σ tc σ z φ kl φ kc φ lc Figure : Plots of slices of the likelihood around the posterior mode for the model where all fiscal instruments adjust to debt. The vertical line indicates the posterior mode. 7

18 γ κ habit. δ x x x x 6. s ρ a.9 ρ b rho l x x x x ρ i ρ g ρ tk ρ tl x x x x Figure : Trace plots for the model where all fiscal instruments respond to debt. 8

19 ρ tc ρ z γ g γ tk x x x x γ tl γ z φ tk φ tl x x x x φ g φ z σ a σ b x x x x Figure 6: Trace plots for the model where all fiscal instruments respond to debt. 9

20 σ l σ i σ g σ tk x x x x σ tl σ tc σ z φ kl x x x x φ kc φ lc x x Figure 7: Trace plots for the model where all fiscal instruments respond to debt.

21 γ κ habit.8 δ x x x x 6. s ρ a.7 ρ b rho l x x x x ρ i ρ g ρ tk ρ tl x x x x Figure 8: Trace plots for the model where only capital and labor taxes respond to debt.

22 ρ tc ρ z γ tk γ tl x x x x φ tk φ tl φ g φ z x x x x σ a σ b σ l σ i x x x x Figure 9: Trace plots for the model where only capital and labor taxes respond to debt.

23 . σ g.. σ tk. σ tl... x σ tc.... x φ kc x... x σ z.... x φ lc x... x φ kl x Figure : Trace plots for the model where only capital and labor taxes respond to debt.

24 γ κ habit.8 δ x x x x 6. s ρ a.9 ρ b rho l x x x x ρ i ρ g ρ tk ρ tl x x x x Figure : Trace plots for the model where only government spending responds to debt.

25 ρ tc ρ z γ g φ tk x x x x φ tl φ g φ z σ a x x x x σ b σ l σ i σ g x x x x Figure : Trace plots for the model where only government spending responds to debt.

26 .. σ tk. σ tl.. σ tc... x σ z.... x φ lc..... x φ kl x.. x φ kc x.... x Figure : Trace plots for the model where only government spending responds to debt. 6

27 γ κ habit.8 δ x x x x 6. s ρ a.7 ρ b rho l x x x x ρ i ρ g ρ tk ρ tl x x x x Figure : Trace plots for the model where only lump sum transfers responds to debt. 7

28 ρ tc ρ z γ z φ tk x x x x φ tl φ g φ z σ a x x x x σ b σ l σ i σ g x x x x Figure : Trace plots for the model where only lump sum transfers responds to debt. 8

29 6 σ tk. σ tl.. σ tc.. x σ z..... x φ lc..... x φ kl x.. x φ kc x.... x Figure 6: Trace plots for the model where only lump sum transfers responds to debt. 9

30 γ ρ a ρ tk γ tk φ g σ i σ z κ ρ b ρ tl γ tl φ z σ g φ kl habit rho l ρ tc γ z σ a σ tk φ kc δ ρ i ρ z φ tk σ b σ tl φ lc s ρ g γ g φ tl σ l σ tc Figure 7: Autocorrelation functions for estimated parameters for the model where all fiscal instruments respond to debt.

31 γ ρ a ρ tk γ tl σ a σ tk φ kc κ ρ b ρ tl φ tk σ b σ tl φ lc habit rho l ρ tc φ tl σ l σ tc δ ρ i ρ z φ g σ i σ z s ρ g γ tk φ z σ g φ kl Figure 8: Autocorrelation functions for estimated parameters for the model where only capital and labor taxes respond to debt.

32 γ ρ a ρ tk φ tk σ b σ tl φ lc κ ρ b ρ tl φ tl σ l σ tc habit rho l ρ tc φ g σ i σ z δ ρ i ρ z φ z σ g φ kl s ρ g γ g σ a σ tk φ kc Figure 9: Autocorrelation functions for estimated parameters for the model where only government spending responds to debt.

33 γ ρ a ρ tk φ tk σ b σ tl φ lc κ ρ b ρ tl φ tl σ l σ tc habit rho l ρ tc φ g σ i σ z δ ρ i ρ z φ z σ g φ kl s ρ g γ z σ a σ tk φ kc Figure : Autocorrelation functions for estimated parameters for the model where only lump sum transfers respond to debt.

Macroeconomic Effects of Financial Shocks: Comment

Macroeconomic Effects of Financial Shocks: Comment Johannes Pfeifer (University of Cologne) 1st Research Conference of the CEPR Network on Macroeconomic Modelling and Model Comparison (MMCN) June 2, 217