We estimate the model with US quarterly data from 1983:Q1 to 2008:Q1 using Bayesian inference methods (see An and Schorfheide, 2007, for a survey). The sample period choice is driven by two stability considerations: (i) monetary policy is characterized by a Taylor rule (Taylor, 1993); and (ii) monetary policy is thought to be active and fiscal policy passive over this period (in the sense of Leeper, 1991).6
The estimation uses 12 observables, including real aggregate consumption, investment, labor, wages, nominal interest rate, gross inflation rate, and fiscal variables—capital, labor and consumption tax revenues, real government consumption and investment, and transfers.7 Details of the data construction, linkage to observables and estimation methods are available in the online Appendix.
3.1 Prior Distribution
We calibrate several parameters that are difficult to identify from the data. The discount factor β is set to 0.99, implying an annual steady-state real interest rate of 4%. The capital income share of total output α is 0.36, implying a labor income share of 0.64. The depreciation rate for private capital δ0 is set to 0.025 so that the annual depreciation rate is 10%. We set δG = 0.02, comparable to the calibrated value in DSGE models with productive investment Baxter and King (1993) and Kamps (2004). η p and ηw are set to 0.14 to have the steady-state markups in the product and labor markets be 14%, consistent with evidence that the average price markup of US firms is 10–15% (Basu and Fernald, 1995).Since there appears to be no consensus for the average markup in the labor market, we pick the same value for ηw by symmetry. The steady-state inflation rate, π, is assumed to be 1.8
The output elasticity to public capital, αG, cannot be identified without information on the capital stocks. The empirical literature has a wide range of values for αG, from a small negative number (Evans and Karras, 1994), to zero (Kamps, 2004), to near 0.4 (Pereira and de Frutos, 1999). For the baseline estimation, we make a conservative assumption on the productiveness of public capital and set αG = 0.05.9
The rest of the calibrated parameters are steady-state fiscal variables computed from the sample means. The federal government consumption to output share is 0.070, federal government investment to output share is 0.004, federal debt to annualized output share is 0.386, average federal labor tax rate is 0.209, capital tax rate is 0.196 and, finally, consumption tax rate is 0.015. When computing these shares, we use an output measure consistent with the model specification, namely, the sum of consumption, investment and total government purchases.
Columns 2–5 in Table 1 list the prior distributions. Our priors for common New Keynesian parameters are similar to those in Smets and Wouters (2007). One parameter less encountered is the share of non-savers, μ. Forni et al. (2009) and Iwata (2009) center the prior at 0.5 but obtain an estimate around 0.35. López-Salido and Rabanal's (2006) estimate using US data over a similar sample period is between 0.10 and 0.39. Thus we choose a beta prior with a mean of 0.3 and standard deviation of 0.1.
Table 1. Prior and posterior distributions for estimated parameters
|Mean||SD||90% int.||Mean||90% int.|
|Preference and technology|| || || || || || |
|100γ, steady state growth||N||0.5||0.03||[0.45, 0.55]||0.5||[0.45, 0.55]|
|κ, inverse Frisch labor elast.||G||2||0.5||[1.3, 2.9]||2.41||[1.65, 3.30]|
|θ, habit||B||0.5||0.2||[0.17, 0.83]||0.70||[0.63, 0.77]|
|μ, fraction of non-Ricar. households||B||0.3||0.1||[0.14, 0.48]||0.10||[0.06, 0.16]|
|Frictions|| || || || || || |
|ωw, wage stickiness||B||0.5||0.1||[0.34, 0.66]||0.28||[0.18, 0.43]|
|ωp, price stickiness||B||0.5||0.1||[0.34, 0.66]||0.70||[0.64, 0.77]|
|ψ, capital utilization||B||0.6||0.15||[0.35, 0.85]||0.75||[0.66, 0.84]|
|s, investment adj. cost||N||6||1.5||[3.5, 8.5]||5.78||[3.65, 8.11]|
|χw, wage partial indexation||B||0.5||0.15||[0.25, 0.75]||0.30||[0.20, 0.41]|
|χp, price partial indexation||B||0.5||0.15||[0.25, 0.75]||0.28||[0.13, 0.49]|
|Fiscal policy|| || || || || || |
|γGC, govt consumption resp. to debt||N||0.15||0.1||[-0.01, 0.3]||0.28||[0.15, 0.42]|
|γGI, govt investment resp. to debt||N||0.15||0.1||[-0.01, 0.3]||0.16||[0.01, 0.30]|
|γK, capital tax resp. to debt||N||0.15||0.1||[-0.01, 0.3]||0.20||[0.07, 0.33]|
|γL, labor tax resp. to debt||N||0.15||0.1||[-0.01, 0.3]||0.10||[0.02, 0.20]|
|γZ, transfers resp. to debt||N||0.15||0.1||[-0.01, 0.3]||0.02||[-0.09, 0.17]|
|ϕK, capital resp. to output||G||0.75||0.35||[0.28, 1.4]||0.51||[0.19, 0.93]|
|ϕL, labor resp. to output||G||0.40||0.15||[0.14, 0.76]||0.73||[0.24, 1.37]|
|Monetary policy|| || || || || || |
|ϕπ, interest rate resp. to inflation||N||1.5||0.25||[1.1, 1.8]||2.39||[2.12, 2.68]|
|ϕy, interest rate resp. to output||N||0.125||0.05||[0.04, 0.21]||0.04||[0.004, 0.07]|
|ρr, lagged interest rate resp.||B||0.5||0.2||[0.17, 0.83]||0.85||[0.82, 0.88]|
|Serial correl. in disturbances|| || || || || || |
|ρa, technology||B||0.5||0.2||[0.17, 0.83]||0.24||[0.12, 0.35]|
|ρb, preference||B||0.5||0.2||[0.17, 0.83]||0.91||[0.85, 0.95]|
|ρi, investment||B||0.5||0.2||[0.17, 0.83]||0.90||[0.84, 0.95]|
|ρw, wage markup||B||0.5||0.2||[0.17, 0.83]||0.92||[0.82, 0.97]|
|ρp, price markup||B||0.5||0.2||[0.17, 0.83]||0.95||[0.91, 0.98]|
|ρGC, government consumption||B||0.5||0.2||[0.17, 0.83]||0.97||[0.94, 0.99]|
|ρGI, government investment||B||0.5||0.2||[0.17, 0.83]||0.94||[0.89, 0.98]|
|ρK, capital tax||B||0.5||0.2||[0.17, 0.83]||0.87||[0.81, 0.92]|
|ρL, labor tax||B||0.5||0.2||[0.17, 0.83]||0.91||[0.83, 0.97]|
|ρC, consumption tax||B||0.5||0.2||[0.17, 0.83]||0.98||[0.97, 0.995]|
|ρZ, transfer||B||0.5||0.2||[0.17, 0.83]||0.91||[0.83, 0.98]|
|Standard deviation of shocks|| || || || || || |
|σa, technology||IG||0.1||2||[0.02, 0.28]||0.98||[0.86, 1.12]|
|σb, preference||IG||0.1||2||[0.02, 0.28]||2.26||[1.67, 3.18]|
|σm, monetary policy||IG||0.1||2||[0.02, 0.28]||0.14||[0.12, 0.16]|
|σi, investment||IG||0.1||2||[0.02, 0.28]||0.39||[0.32, 0.47]|
|σw, wage markup||IG||0.1||2||[0.02, 0.28]||0.26||[0.19, 0.37]|
|σp, price markup||IG||0.1||2||[0.02, 0.28]||0.14||[0.10, 0.18]|
|σGC, government consumption||IG||1||∞||[0.21, 2.8]||1.89||[1.68, 2.12]|
|σGI, government investment||IG||1||∞||[0.21, 2.8]||4.23||[3.77, 4.75]|
|σK, capital tax||IG||1||∞||[0.21, 2.8]||4.49||[3.99, 5.07]|
|σL, labor tax||IG||1||∞||[0.21, 2.8]||1.76||[1.57, 1.99]|
|σC, consumption tax||IG||1||∞||[0.21, 2.8]||3.25||[2.89, 3.65]|
|σZ, transfers||IG||1||∞||[0.21, 2.8]||2.40||[2.13, 2.69]|
|σKL, co-movement btw. K and L taxes||N||0.2||0.1||[0.036, 0.36]||0.24||[0.19, 0.29]|
The priors for the fiscal parameters are chosen to be fairly diffuse. To stabilize debt, government spending and transfers (taxes) should respond negatively (positively) to a debt increase. We assume normal distributions for the fiscal instruments' responses to debt with a mean of 0.15 and standard deviation of 0.1. While these priors place a larger probability mass in the regions of expected signs, some probability is allowed for the opposite signs. The prior ranges are guided by two considerations. First, when fiscal adjustment responses to debt are too high, overshooting occurs, resulting in oscillation patterns not observed in the data. Second, when adjustment responses are too low, under active monetary policy, there does not exist an equilibrium.
As capital and labor taxes are progressive, we restrict φK and φL to be positive. Since Social Security taxes are incorporated in our labor tax revenues, the labor tax rate elasticity is expected to be a value below the capital tax rate elasticity (since Social Security contributions have a cap and are regressive). The parameter measuring the co-movement between capital and labor tax rates (σKL) is assumed to have a normal distribution with a mean of 0.2 and astandard deviation of 0.1. The domain covers the range of past estimates for this parameter (see Yang, 2005; Leeper et al., 2010).
The online Appendix conducts prior predictive analysis about whether various deficit-financed fiscal shocks crowd out or crowd in investment. The results are compared to the same exercise from the posterior. The comparison shows substantial differences between prior and posterior multipliers. However, the priors do imply some restrictions on short-run multipliers, particularly for government consumption, transfers and capital taxes.
3.2 Posterior Estimates
The last two columns of Table 1 provide the mean and 90th percentiles from the posterior distributions.10 Overall the data are informative about the parameters estimated, as the 90% intervals are different from those implied by the priors, with the main exception of the technology growth rate γ. Our estimates for the common parameters in New Keynesian models are comparable to others estimated with postwar US data (Smets and Wouters, 2007; Del Negro et al., 2007; Fernandez-Villaverde et al., 2010), as shown in the online Appendix.
The mean estimate for the non-savers' fraction μ is 0.1. This low fraction suggests the importance of forward-looking behavior in the aggregate effects of fiscal policy. Although myopic behavior has been seen as important in understanding fiscal policy effects since Mankiw (2000), our estimate is much smaller than the commonly calibrated value of 0.5, based on single-equation estimation of a consumption function (Campbell and Mankiw, 1989; Galí et al., 2007). Previous studies model non-savers so that aggregate consumption can increase following a positive government spending shock. Given our mean estimates for the benchmark model, a fraction of 0.29 is required to deliver a positive short-term consumption response to an increase in government consumption, which falls outside the 90% interval [0.06, 0.16]. Our results are nonetheless consistent with vector autoregression (VAR) estimates of the same sample period. VARs with either federal government consumption alone or the sum of federal government consumption and investment find that, for 1983:Q1–2008:Q1, an increase in government spending does not have a positive effect on consumption.11
The 90% intervals of all fiscal instruments—except for transfers—have the expected signs for their responses to debt. We find that the federal government mainly relies on reducing government consumption and raising income taxes to stabilize debt, as in Leeper et al. (2010). The similar 90% intervals between prior and posterior estimations for the government investment response to debt (γGI) suggest that this parameter is less well identified.
3.3 Historical Decomposition
To illustrate what drove the debt dynamics historically, Figure 1 presents the historical decomposition of the model-implied dynamics of real primary deficits (defined as the sum of government consumption, investment and transfers less total tax revenues).12 The breakdowns of shocks are organized by monetary, fiscal (aggregating tax, government spending and transfers) and structural (aggregating all non-policy) shocks. The thick solid lines are the model-implied series, and the units on the y-axis are percentage deviations from the steady-state path.
Figure 1. Historical variance decomposition for model-implied primary deficits. Structural shocks are the aggregate of all non-policy shocks. Units for the y-axis are percentage deviation from the steady-state path
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The figure shows that fiscal and structural shocks are important in driving primary deficits. The fiscal position worsened in the early 1990s (continuing from the 1980s). To reduce deficits, the first Bush and Clinton administrations signed into law the Omnibus Budget Reconciliation Acts of 1990 and 1993, respectively, to increase the statutory tax rates on high income earners. Consequently, primary deficits fell below the steady-state trend path in the second half of the 1990s. The trend was reversed in the early 2000s mainly due to the recession, a series of tax cuts and increased defense spending, as captured by the dominance of structural and fiscal shocks shown in Figure 1.
Aside from fiscal shocks, monetary policy also affected deficit dynamics, particularly in the post-2000 period. The low interest rate environment following the 2001 recession stimulated economy activity and expanded the tax base. Thus monetary policy worked to reduce primary deficits relative to a scenario without interest rate cuts.