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State-dependent effects of fiscal policy

Steven M. Fazzari, James Morley and Irina Panovska

Abstract

We investigate the effects of government spending on US output with a threshold structural vector autoregressive model. We consider Bayesian model comparison and generalized impulse response analysis to test for nonlinearities in the responses of output to government spending. Our empirical findings support state-dependent effects of fiscal policy, with the government spending multiplier larger and more persistent whenever there is considerable economic slack. Based on capacity utilization as the preferred threshold variable, the estimated multiplier is large (1.6) for a low-utilization regime that accounts for more than half of the sample observations from 1967 to 2012 according to the estimated threshold level.

JEL Codes: C32; E32; E62

Corresponding author: Irina Panovska, Department of Economics, Lehigh University, 621 Taylor Street, Rauch Business Center, Bethlehem, PA 18015, USA, e-mail:

Acknowledgments

We thank Alan Auerbach, Rudi Bachmann, Jared Bernstein, Lutz Kilian, Raul Santaeulalia-Llopis, Jeffrey Sheen, Houston Stokes, John Taylor, Pao-Lin Tien, and conference and seminar participants at Deakin University, the International Monetary Fund, University of Melbourne, University of Otago, the 2012 VUW Macro Workshop, the 2012 MIFN Meetings, the 2012 SNDE Meetings, the 2012 CAMA Macro Workshop, the 2012 Berlin Conference of the Macroeconomics and Macroeconomic Policies (FMM) Research Network, the 2011 EC2 Conference, the 2011 SCE Meetings, the 2011 Midwest Econometrics Group Meetings, and the 2010 Washington University Graduate Student Conference for helpful comments. All errors are our own. We also thank Valerie Ramey for providing the narrative military spending data set. Fazzari and Panovska acknowledge the generous support of The Institute for New Economic Thinking. Panovska also gratefully acknowledges the support of the Graduate School of Arts and Sciences and the Olin Fellowship program at Washington University in St. Louis. Morley acknowledges financial support from the Australian Research Council (Discovery Grant DP130102950 on “Estimating the Effects of Fiscal Policy”).

Appendix

A Techical Appendix: Bayesian Estimation

A.1 Simulating from the posterior distributions

For the linear version of the baseline model, we assume that the prior for the conditional mean parameters is multivariate normal, the prior for the variance matrix is an inverse Wishart distribution, and the prior for the scale parameter λ is a Gamma distribution. Specifically, the linear model is simply Yt0+Φ(L)Yt–1+λtεt, where Φ(L) is an autoregressive matrix polynomial with roots strictly outside the unit circle, λt=1 for t = 1967q1,...,1983q4, λt=λ for t = 1984q1,... Tfinal, where Tfinal is the final observation, and εt is i.i.d. Gaussian random variable with mean 0 and variance- covariance matrix Ω that does not change over time. Then, letting Φ=vec0)∣vec1)…∣vecp), we assume that the prior for Φ is a normal distribution, truncated to the stationarity region, with mean equal to 0, and variance-covariance matrix equal to Vn. The scaling parameter λ is assumed to have an inverse gamma prior with parameters α/2 and β/2, and we impose an inverted Wishart prior with v0 degrees of freedom and a scale matrix R0. Letting xt=[1 yt–1,1yt–1, kytp, k]⊗Ip, it is straightforward to see that Φ∣Ω, λ, y is Gaussian with variance

V=(Vn1+t=p+1Txt(λtΩ)1xt)1

and mean

μ=V1(t=p+1Txt(λtΩ)1yt).

Similarly, Ω∣Φ, λ, y∼IW(v1, R1) where v1=v0+Tp and R1=[R01+t=p+1T(ytxtΦ)λt1(ytxtΦ)]1. The inverse Wishart distribution is a standard distribution, so we can sample Ω conditional on the other parameters directly. Conditional on the other parameters and the data, λ has an inverse gamma distribution with parameters α1=(α+Tp)/2 and β1=β+0.5t=t1T(ytxtΦ)Ω1(ytxtΦ) where t1=1984Q1, and Tp is the number of observations after 1984Q1. Under these assumptions, we can sample the model parameters directly using the Gibbs sampler.

For the threshold model in (1), it is straightforward to show that Φ∣Ω, λ, γ, y is Gaussian with mean and variance as before, except now xt=[xt*xt*I[qtd>c]]Ip, where xt*=[1yt1,1yt1,kytp,k] and the distribution is truncated such that the VAR model is stationary in each regime. The conditional distribution of Ω is inverse Wishart and the conditional posterior distribution of λ is gamma, as before.

Conditional on c and the threshold variable, the model is linear in Φ and Ω. Estimating the linear model by splitting the sample into two subsamples yields the conditional estimators Φ^ and Ω^. The estimated threshold value (conditional on the threshold variable and the delay lag) can be identified uniquely as

c^=argmaxcΓnlikn(c|q,d)

where Γ is approximated by a grid search on Γn=Γ∩{q1, q2, …, qn} and likn denotes the log likelihood. To ensure identification, the bottom and top 15% quantiles of the threshold variable are trimmed. We use the estimated value c^ for constructing the proposal for the first draw of the MH algorithm. Given a sufficiently large burn-in, the value of c^ does not affect the Bayesian estimates, but it provides us with a plausible starting value for the mode and it enables us to easily compare the Bayesian mode with the maximum likelihood estimate.

A potential issue is that the grid search makes it infeasible to obtain the variance of the estimate of c based on numerical derivatives. Instead, we follow the suggestion in Lo and Morley (2013) for constructing a proposal density for a threshold parameter. In particular, we obtain a measure of the curvature of the posterior with respect to c by inverting the likelihood ratio statistics for the threshold parameters based on the assumption that the parameter estimator is normally distributed and the LR statistics is χ2(1). We use the 95% CI for the likelihood ratio statistics to obtain a corresponding standard error for c, based on an asymptotic equivalence between the inverted LR and Wald-based confidence intervals. Even though the distributional assumption and equivalence is not correct due to the nonstandard distribution of the threshold parameter and related LR tests, this approach still provides a sense of the curvature of the posterior, which is all that is needed for the proposal distribution for the sampler. Specifically, at the ith iteration of sampler, the transition density for γ(i+1) is a Student-t distribution with mean equal to c(i) and variance equal to κσ^c2, where σ^c2 is obtained as described above. The parameter κ is calibrated on the fly to ensure acceptance rate between 20 and 60%.

To ensure that the results are robust to the choice of priors, we estimate the model by using different hyperparameters for the priors, and by using different functional forms for the priors (when the priors are not conjugate to the posteriors, we draw all parameters using a multi-block MH step). Also, to check for convergence for each combination of priors, we start the algorithm from different points, and we use a large burn-in for all runs of the MH algorithm. In particular, we use a burn-in sample of 20,000 draws and make inference based on an additional 50,000 MH iterations The results presented and discussed in the main text are based on the priors in Table 5.

Table 5

Priors.

DistributionMean/LocationVariance/Scale
ΦMultivariate Gaussian0100*Ik
ΩInverse Wishart[1000040000100001]25*μ
λGamma10.75
cUniform0.1650.3652

A.2 Marginal likelihoods and model comparison

When comparing two models, M1 versus M2, in a Bayesian setting, each model consists of the prior probability that we assign to that model p(Mi), which is tells us how likely we believe the model is ex-ante, the prior distribution for all of the parameters of the model, π(θi), and the likelihood function for that model conditional on the data and the parameters, f(data∣θi, Mi). To compare models, we can compute Prob(Midata), which is the probability that model i is the correct model, given the data. This probability can be computed using the Bayes theorem:

P(Mi|data)=P(Mi)f(data|Mi)f(y)=P(Mi)fi(y|θi,Mi)πi(θi)f(y)=P(Mi)fi(y|θi,Mi)πi(θi)jP(Mj)fi(y|θj,Mj)πj(θj)

where j=1, 2, …, N are all of the models under consideration. The integral

fi(y|θi,Mi)πi(θi)

is the marginal likelihood for model i. The marginal likelihood can be interepreted as the expected value of the likelihood function with respect to the prior distribution. The higher the odds ratio, the higher the support in favor of model M1. The Bayes factor is the ratio of the marginal likelihoods for two models (it gives us the ratio of the expected likelihoods, not taking into account any priors we may have put on the models ex-ante, before looking at the data). If two models are considered equally likely ex-ante, that is, if the researcher has no reason to believe that one model is more likely than another before looking at the data, the Bayes factor is simply the ratio of the marginal likelihoods. In that case, the distance between the marginal likelihoods tells us the probability of model 1 relative to model 2, given the data. If we have a model with marginal log likelihood lml1 and a model with marginal log likelihood lml2, and they are both equally likely ex-ante, the probability of modelM1 relative to model M2 is exp(lml1)/exp(lml2). If the researcher puts different prior probability on different models, the posterior odds ratio depends on the prior probabilities, but if lml1>lml2, this implies that the odds ratio in favor of M2 relative to M1 is large only if we are willing to put a really high ex-ante probability on M2 being the true model. It is, however, important to note that a large difference in the marginal likelihoods between the non-linear and the linear model does not directly imply that there is necessarily a difference in the size of the fiscal multipliers. It merely implies that there is strong evidence that at least one of the coefficients in the matrices Φ02 or Φ12 is different from zero. The model comparison is a useful first step that can help us evaluate whether there is any reason to use the nonlinear model at all. To compare whether this nonlinearity that is detected by the model comparison affects the spending multipliers, we look at the impulse response functions directly.

B Simulation-based impulse response function and impulse response comparison

The procedure for computing the generalized impulse response functions (GIRFs) follows Koop, Pesaran and Potter (1996), with the modification of considering an orthogonal structural shock, as in Kilian and Vigfusson (2011). The generalized impulse response is defined as the effect of a one-time shock on the forecasted level of variables in the model, and the response is compared against a baseline “no shock” scenario.

GIRFy(k,shockt,Ψt1)=[E[Yt+k|shockt,Ψt1]E[Yt+k|Ψt1]]

where k is the forecasting horizon, Ψt–1 denotes the initial values of the variables in the model. The impulse response is then computed by simulating the model. The shock to government spending is normalized to be equal to 1% of GDP (at the time the shock occurs). The GIRFy response for a given draw Θ(i) of the MH algorithm is generated using the following steps:

  1. Pick a history Ψt–1. The history is the actual value of the lagged endogenous variable at a particular date.

  2. Pick a sequence of forecast errors εt+k, k=0, 1, …, 20. The forecast errors are simulated assuming an independent Gaussian process with mean zero and variance-covariance matrix equal to λt(i)Ω(i).

  3. Using Ψt1r and εt+k, simulate the evolution of Yt+k over l+1 periods. Denote the resulting path Yt+k(εt+k, Ψt–1) for k=0, 1, …, l.

  4. Using the Cholesky decomposition of Ωt to orthogonalize the shocks, solve for the government spending shock at time t, replace it with a shock equal to 1% of GDP, and reconstruct the implied vector of forecast errors. Denote the implied vector of forecast errors as εtshockt, the sequence of forecast errors as εt+kshockt, and the resulting simulated evolution of Yt+k over l+1 periods as Yt+k(εt+kshockt,Ψt1) for k=0, 1, …, l.

  5. Construct a draw of a sequence of impulse responses as Yt+k(εt+kshockt,Ψt1)Yt+k(εt+k,Ψt1) for k=0, 1, …, l.

  6. Repeat steps 2 to 5 for B times, with B=500, and average the sequences of responses to obtain a consistent estimate of the impulse response function conditional on the history and the size of the shock.

  7. To obtain the average response for a subset of histories, repeat steps 1–6 for a the subset of histories of interest, and report the response averaged over all histories.

  8. In order to compare the responses for two types of shocks for a fixed history and, or the responses for two different histories, we construct the difference

ΔIRF=IRFy(k,shockt1,Ψt11)IRFy(k,shockt2,Ψt12).

Because the impulse responses are nonlinear functions of the parameters, their distribution of both the generalized impulse responses and the significance ΔIRF is nonstandard and it is not necessarily symmetric around the mean. In this case, reporting the median value is unlikely to be adequate, as the median may not be a valid measure of central tendency. In order to circumvent this problem, we adapt the approach proposed by Inoue and Kilian (2013). For a given history, we evaluate the impulse response function for each draw of the MH algorithm, drawing the entire impulse response function for periods 1 through 20. Then we average over the histories of interest, and we evaluate the posterior likelihood of the impulse response for that draw of the algorithm. The impulse response function with the highest average posterior likelihood is then used for inference. To construct the (1–α)*100% credibility interval, we order the posterior likelihood values, and we include the impulse responses whose posterior likelihood was in the upper (1–α)*100 percentile. This method results in a “credibility cloud” with a shot gun pattern because we draw entire impulse responses rather than responses for each individual point in time. For easy interpretation, we report only the outer points of the cloud. To convert the responses to dollar-for-dollar or jobs-for-dollar responses, all the impulse responses are converted to cumulative responses, and then scaled using the ratio Gt/Variable for every t.

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Supplemental Material

The online version of this article (DOI: 10.1515/snde-2014-0022) offers supplementary material, available to authorized users.

Published Online: 2014-11-5
Published in Print: 2015-6-1

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