Government Expenditures and Business Cycles—Policy Reaction and Surprise Shocks

Abstract

This article analyses the effects of government expenditures on German business cycle fluctuations by means of an estimated DSGE model based on low frequency oscillations. The results highlight that fiscal policy has a strong impact on the amplitude of fluctuations while hardly any on the duration of business cycles. To the extent that fiscal policy shocks are an important source for triggering economy-wide fluctuations, fiscal policy also strongly reacts to shocks originating elsewhere. Thus standard specifications of government expenditures in terms of simple AR(1) processes ignore the fact that there can indeed be a strong dependence of governmental variables on economic fluctuations.

JEL Classification: C13, C32, E3, E62

Keywords

Fiscal Policy Business Cycles DSGE Models Estimation

1. INTRODUCTION

This article seeks to understand the impact of government expenditures on business cycles once it is characterised as a responder to observed fluctuations versus a scenario where it acts as a source of fluctuations. To this end a dynamic stochastic general equilibrium (DSGE) model is formulated with an extended fiscal policy apparatus. There are two different types of policy mechanisms at work. On the one hand, the model specifies fiscal policy as reacting to changes in the economic stance. On the other hand, fiscal policy is extended with an exogenous stochastic term in order to elaborate on surprises in fiscal policy. The model is used to judge the importance of surprise fiscal policy shocks as well as to analyse the impact of its endogenous component. I do so by estimating the model using only low frequency oscillations of German time series data.

As far as the profoundness of the analysis of fiscal policy in DSGE models is concerned, up to now the literature surrounding this issue has received considerably less attention than the literature on monetary policy reaction functions. In DSGE models the specification of the fiscal authority typically involves some type of fiscal closure rule. Except for a few studies (Gali et al., 2007; Perez and Hiebert, 2004) this rule is still completely exogenous and the fiscal authority is not allowed to react to various shocks affecting the model economy. This is, of course, a rather limited description of the fiscal authorities’ reaction to economic fluctuations. In addition to this, the government is usually described in a very limited way. Rather than setting a profound government sector, it is usually preferred to stick to lumpsum taxes in a framework where there is no public debt.

The model that I construct has two key features. First it nests the now standard new Keynesian model, as for instance in Smets and Wouters (2007) and the Mortensen-Pissarides (Mortensen and Pissarides, 1994, Pissarides, 2000) model. This allows to disentangle the adjustment in the labour market into its extensive and intensive margins. Second, I model a tax and an unemployment payments structure to act as so called automatic stabilisers in order to separate fiscal policy fluctuations which are due to automatic stabilisers as opposed to those which are due to their discretionary component.

The key findings are as follows. First of all, fiscal policy has a strong impact on the amplitude of fluctuations, however, there is no effect on the duration of business cycles. In particular, fiscal policy can smooth cyclical fluctuations but there is no possibility of fundamentally disturbing cyclicalities in their lengths.

Second, relative to the other shocks, surprise fiscal policy shocks are a major source of fluctuations. Fiscal policy shocks explain a large amount of the fluctuations in inflation and unemployment for short horizons (up to 28 per cent), but for longer forecasting horizons this magnitude declines strongly. For output, fiscal policy shocks explain around 20–25 per cent of the fluctuations rather independently of the specific horizon considered.

Finally, as the estimates for the fiscal policy rule indicate, government expenditures significantly contribute to dampening cyclical oscillations. Considering a fiscal authority’s aim of dampening cyclical fluctuations indicates that especially for unemployment and output, countercyclical fiscal policy is a powerful tool, whereas for inflation its impact is limited due to the rather rigid fluctuations therein.

I pursue a particular limited information econometric strategy to estimate and evaluate the model. The basic idea behind this approach can be described as follows. DSGE models and especially new Keynesian and Real Business Cycle models are theoretical models aiming at explaining business cycles. However, usually the data used for estimation are a composition of cycles of different periodicities; there are long periodic cycles describing growth patterns, seasonal cyclicalities, purely random components and among them, business cycle patterns. Since the model is only aimed at explaining low cyclicalities and especially business cycle frequencies, I hence make only use of this part of information in the data. Carrying out this estimation procedure hence requires a frequency domain approach. I proceed by using a procedure proposed by Smith (1993) and Gourieroux et al. (1993) called Indirect Inference. I minimise a measure of distance between the model and empirical spectral density functions as proposed by Diebold et al. (1998) using only those fluctuations which correspond to business cycle fluctuations.

The article is structured as follows. Section 2 describes the model. Section 3 addresses the econometric methodology used to appropriately adjust the model to the data; Section 4 discusses the impact of the various fiscal policy rules and analyses the effects of different specifications on the fluctuations triggered by fiscal policy. Section 5 concludes.

2. THE MODEL

The present model has the key features that many authors have found useful for capturing the data; these include habit formation, investment adjustment costs and nominal price rigidities.

The key features in this model are in the labour market. Rather than considering aggregate hours worked irrespective of its dependence on the intensive and extensive margin, I introduce variation occurring at both margins. I do so by extending the new Keynesian model by the Mortensen-Pissarides model of search and matching.

The economy consists of two types of households, a continuum of firms, called the wholesale firms who produce differentiated intermediate goods, the retail firms, a fiscal authority and a central bank in charge of monetary policy. I use a representative family construct for each household type, similar as in Merz (1995), in order to introduce complete consumption insurance. The model features two important non-Ricardian elements: liquidity constraint agents and distortionary labour and capital taxation. The introduction of liquidity constrained agents is due to the results of Gali et al. (2007) who emphasise the importance of this friction in order to reasonably explain the effects of government expenditures shocks on private consumption.

2.1 Households

I assume a continuum of infinitely lived households, indexed byj ∈[0, 1]. A fraction 1 – ς of the total population has access to capital markets. These consumers are called Ricardian consumers. The remaining fraction does not own any assets as these households do not have access to capital markets. Hence they just consume their current labour income. These households are referred to as non-Ricardian consumers.

2.1.1 Ricardian Households

The starting point for the Ricardian consumers is based on the now conventional monetary DSGE model developed by Christiano et al. (2005), Smets and Wouters (2007) and others. Since the derivations here are standard, they are postponed to Appendix A.

2.1.2 Non-Ricardian Households

The number of family members of non-Ricardian households currently employed is N^r_t = ςN_t', where an r refers to non-Ricardian (that is, rule of thumb) consumers. Each family member employed offers $H_{t}^{r}$ hours of work and consumes $C_{t}^{r}$ amount of goods. These households solve a static optimisation problem; let their instantaneous utility function be given by u $(C_{t}^{r}, H_{t}^{r}) = \frac{1}{1 - σ_{c}} {(C_{t}^{r})}^{1 - σ_{c}} - υ (H_{t}^{r})$ which is v $(H_{t}^{r}) = χ \frac{{(H_{t}^{r})}^{1 + ϕ}}{1 + ϕ}$ , which is maximised given the following static budget constraint $C_{t}^{r} = (1 - τ^{N}) W_{t} N_{t}^{r} H_{t}^{r} + Γ U_{t}^{r}$ where τ^N is the labour tax rate, σ_c is the coefficient of relative risk aversion, ϕ is the inverse of the Frish labour supply elasticity, W_t is the real wage rate, Γ represents unemployment benefits and $U_{t}^{r}$ is the amount of unemployed household members.

Consumption is determined simply by after tax labour income. Hours worked $H_{t}^{r, s}$ is determined within the households’ and firms’ bargaining over hours worked and the real wage.

2.2 Matching of Vacancies and Unemployment

All unemployed workers at time t search for jobs. This implies that the pool of unemployed people U_t searching for a job at time t is given by the difference between the total labour force, which is normalised to unity and the amount of employed people at the end of period t –1: U_t = 1 – n_t_–1.

The function matching unemployed workers U_t and firms with a vacancy V_t is

m_{t} = ξ_{t}^{m} U_{t}^{ζ_{m}} V_{t}^{1 - ζ_{m}}

(1)

where $ξ_{t}^{m}$ is an exogenous shock in the matching process. ξ_m is the elasticity to unemployment and m_t is the total amount of matches. Let q_t denote the probability of a vacancy being filled by defining: $q_{t} = \frac{m_{t}}{V_{t}}$ . Similarly, the probability p_t a searching worker finds a job is given by: $p_{t} = \frac{m_{t}}{U_{t}} = ζ_{t}^{m} ϑ_{t}^{1 - ζ_{m}}$ , where $ϑ_{t} : = \frac{V_{t}}{U_{t}}$ measures the degree of labour market tightness. Firms and workers take q_t and p_t as given. The definitions of q_t and p_t show that there is an intricate connection between the process linking workers to jobs and the one linking jobs to workers. The total flow of hires for an individual firm (i) in period t + 1 is q_tV_t.

Finally, each period firms separate from a fraction s of their current workforce. Once a worker loses his job, he is not allowed to search until the next period. This restriction implies that fluctuations are triggered by cyclical fluctuations in hiring rather than due to fluctuations in separations.¹

2.3 Wholesale Firms

Wholesale firms and workers bargain over the real wage and the amount of hours worked based on an efficient Nash bargaining solution. Each output good Y_t(i) is produced by a firm i at time t using capital and labour as inputs. The firm considers workers as identical and independent of the corresponding household type and the output of job j at firm i at time t is:

e^{ξ_{t}^{A}} k_{r} {(i)}^{α} H_{t} {(j, i)}^{1 - α}

(2)

where $ξ_{t}^{A}$ is an aggregate AR(1) random disturbance term with mean zero. Firms rent a unit of capital from households at the capital rental rate $r_{t}^{K}$ . H_t(j, i) denotes the amount of hours worked by worker j at firm i. Total output at firm i is given by a production function including the aggregate disturbance, the amount of employment relationships n_t(i) and average per worker capital k_t(i), so that: $Y_{t} (i) = e^{ξ_{t}^{A}} k_{t} {(i)}^{α} \int_{0}^{n_{t} (i)} g (H_{t} (i),^{1 - α}$ , n) dn.

Following Pissarides (2000) and assuming identical work effort implies that $g (H, n) = \bar{g} (H) \bar{f} (n)$ ; since the mass of workers is uniformly distributed over [0, 1], $\bar{f} (n) = 1 \forall n \in [0, 1]$ . For $\bar{g} (H)$ I assume: $\bar{g} (H) = H$ , which implies that the production function simplifies to:

Y_{t} (i) = e_{t}^{ξ^{A}} k_{t} {(i)}^{α} n_{t} (i) H_{t} {(i)}^{1 - α}

(3)

Wholesale firms solve a profit maximisation problem based on the following expected stream of revenues net of expenses:²

\begin{array}{l} E_{t} [\sum_{s = 0}^{\infty} β^{s} Λ_{t, t + s} (n_{t + s} (i) . (μ_{t} e^{ξ_{t + s}^{A}} k_{t + s} {(i)}^{α} H_{t + s} {(i)}^{1 - α} - \\ r_{t + s}^{K} {\tilde{k}}_{t + s} (i) - W_{t + s} H_{t + s} (i)) - c_{υ} V_{t + S} (i))] \end{array}

(4)

where c_υ and μ_t describe the costs due to vacancy posting and marginal costs respectively, and Λ_t,t+1 is the households’ stochastic discount factor defined in equation (31), since households are the firms’ owners, profits are evaluated in terms of value attached to them. The employment flow equation of firm i is given by:

n_{t} (i) = (1 - s) n_{t - 1} (i) + q_{t - 1} V_{t - 1} (i)

(5)

The first order conditions for the firms optimization problem are:

α μ_{t} e^{ξ_{t}^{A}} {(\frac{H_{t} (i)}{k_{t} (i)})}^{1 - α} = r_{t}^{K}

(6)

k_{t} (i) = μ_{t} e^{ξ_{t}^{A}} k_{t} {(i)}^{α} H_{t} {(i)}^{1 - α} - r_{t}^{K} k_{t} (i) - W_{t} H_{t} (i) + (1 - s) β E_{t} [Λ_{t, t + 1} k_{t + 1} (i)]

(7)

\frac{c_{υ}}{q_{t}} = β E_{t} [Λ_{t, t + 1} k_{t + 1} (i)]

(8)

The Lagrange multiplier κ_t is the shadow value of employment. Equation (6) states that the real rental rate on capital equals the marginal product of capital. Combining equations (7) and (8) yields a job creation condition.³

2.4 Wage Bargaining

I proceed by following Pissarides (2000) and Mortensen and Pissarides (1994) and specifically Christoffel and Linzert (2005) by applying the efficient Nash bargaining solution. In this framework, the bargained real wage⁴ (W_t) and hours worked (H_t) are determined as the outcome of a Nash bargaining between workers and firms.

In what follows, I assume that each firm-worker pairing is equally productive so that the wage rate is the same everywhere. Firms consider workers independent of their corresponding household type. However, a worker’s valuation of his employment and unemployment status depends on the household type. Consider worker i of either household type. This worker’s value function in the employment $(Y_{t}^{E, h})$ and unemployment states $(Y_{t}^{U, h})$ satisfy:

\begin{array}{l} Y_{t}^{E, h} (i) = (1 + τ^{N}) W_{t} H_{t} - \frac{υ (H_{t})}{λ_{t}^{h}} \\ + β E_{t} [Λ_{t, t + 1} ((1 - s) Y_{t + 1}^{E, h} (i) + s Y_{t + 1}^{U, h} (i))] \forall h \in {o, r} \end{array}

(10)

\begin{array}{l} Y_{t}^{U, h} (i) = Γ + β E_{t} [Λ_{t, t + r} (p_{t + 1} Y_{t + 1}^{E, h} (i) + \\ (1 - p_{t + 1}) Y_{t + 1}^{U, h} (i))] \forall h \in {o, r} \end{array}

(11)

where υ(H_t) and $λ_{t}^{o}$ are defined as in Section 2.1 and $λ_{t}^{r} = {(C_{t}^{r})}^{- σ_{c}}$ . Note that in fact only the value function for employment depends on the consumer type due to a different valuation of marginal consumption. Γ is the amount of unemployment benefits given to an unemployed worker by the fiscal authority. The value of unemployment hence depends on the current flow value Γ and the likelihood of being employed versus unemployed in the next period. The value function of firm i for job j is given by equation (7) and the free entry condition by equation (8).

Due to the presence of Ricardian and non-Ricardian consumers, the Nash product is extended to take both kind of consumers into account. In particular, I assume that both groups bargain jointly with the firms where each group’s weight is given by its corresponding share in the population. The equilibrium real wage and hours worked are derived from the maximisation of the following Nash product:

\begin{array}{l} \max {((1 - ς) [Y^{E, o} - Y^{U, o}] + ς [Y^{E, r} - Y^{U, r}])}^{η} \cdot κ^{1 - η} \\ W, H \end{array}

(12)

where η ∈ (0, 1) measures the bargaining power of firms and households. The first order condition with respect to the real wage implies the following expression for the target wage W_t:

W_{t} H_{t} = \frac{1 - η}{1 - τ^{N}} (Γ + \frac{m r s_{t} H_{t}}{1 + ϕ}) + η (μ_{t} y_{t} (i, j) - r_{t}^{k} k_{t} (i) + c_{υ} ϑ_{t})

(13)

where the marginal rate of substitution is given by: $m r s_{t} = χ H_{t}^{ϕ} (\frac{(1 - ς)}{λ_{t}^{o}} + \frac{ς}{λ_{t}^{r}})$ and y_t(i, j) is the output in firm i of job j as defined in equation (2). The result is intuitive. The real wage workers get is increase in unemployment benefits (Γ), the relative bargaining strength (1 – η), the tightness in the labour market and the tax rate on households’ labour income. Workers get a weighted average of the unemployment benefits and the surplus, which consists of the output of job j minus the costs on capital needed for job j to be productive. Finally, the first order condition with respect to hours worked implies the following two optimality conditions:

W_{t} = (1 - α) μ_{t} e^{ξ_{t}^{A}} {(\frac{k_{t} (i)}{H_{t} (i)})}^{α} and (1 - τ^{N}) W_{t} = m r s_{t}

(14)

2.5 Retailers

Retailers are monopolistic competitors and set prices for their goods in a staggered fashion. They buy intermediate goods from the wholesale firms, transform them and sell them as either investment/capital or as a consumption good. The derivations follow Christiano et al. (2005) and Smets and Wouters (2007) closely and are therefore put into Appendix B.

2.6 Fiscal and Monetary Policy

Monetary policy is modelled by a simple Taylor type rule:

\frac{R_{t}}{R} = {(\frac{R_{t - 1}}{R})}^{ρ r} {[{(\frac{π_{t}}{π})}^{a_{π}} {(\frac{Y_{t}}{Y})}^{a_{Y}}]}^{1 - ρ_{r}} \cdot e^{\in_{t}^{M}}

(15)

where $\in_{t}^{M} \sim W N (0, σ_{M}^{2})$ and R, π, Y are the corresponding steady state variables of the nominal interest rate (or nominal bond yield), the inflation rate and output. The parameter ρ_r captures the degree of interest rate smoothing and a_π and a_Y capture the inflation and output elasticities.

The fiscal authority’s budget constraint satisfies:

T_{t} + R_{t}^{- 1} B_{t}^{*} = \frac{B_{t - 1}^{*}}{Π_{t}} + G_{t} + U_{t} Γ

(16)

where real government bonds satisfy $B_{t}^{*} = B_{t} / P_{t}$ and π_t = P_t/P_t_–1. Γ is the amount of unemployment benefits given to an unemployed worker as outlined in section 2.4. Hence the total amount of expenditures due to unemployment payments is U_tΓ. The total amount of taxes collected satisfies:

T_{t} = τ^{N} W_{t} N_{t} H_{t} + τ^{K} r_{t}^{K} K_{t - 1}

(17)

where τ^K is the capital tax rate. The government budget constraint implies a stable difference equation in nominal debt as long as R_t < 1 and in real debt as long as R_t/Π_t = < 1. Hence the fiscal policy sector has to be modified such as to control for the public debt in case the previous stability conditions are not satisfied.

2.7 The Fiscal Policy Rule

In the present model the fiscal authority controls government consumption by making use of the following fiscal policy rule (the variables expressed in log-deviations from the corresponding steady state):

{\hat{G}}_{t} = ϕ_{G}^{G} {\hat{G}}_{t - 1} + (1 - ϕ_{G}^{G}) (ϕ_{B}^{G} {\hat{B}}_{t}^{*} + ϕ_{Y}^{G} {\hat{Y}}_{t} + ϕ_{U}^{G} {\hat{U}}_{t}) + \in_{t}^{G},

\in_{t}^{G} \sim W N (0, σ_{G}^{2})

(18)

where it is assumed that $ϕ_{G}^{G} \in [0, 1), ϕ_{B}^{G} < 0, ϕ_{Y}^{G} < 0, ϕ_{U}^{G} > 0$ . The policy rule extends the standard model set up in terms of a higher degree of flexibility for the fiscal authority.

The rule for G_t is defined such that increases in debt trigger declines in government expenditures in order to control for the deviation of public debt from its long run level and hence to guarantee stability. Based on the log-linearised equilibrium equations outlined in Appendix F the condition for the government budget constraint to be a stable difference equation in $B_{t}^{*}$ requires the following to be satisfied: $b_{y} (R^{- 1} - Π^{- 1}) < γ_{G} (1 - ϕ_{G}^{G}) ϕ_{B}^{G}$ where b_y = B*/Y and γ_G = G/Y.

The current specification of the fiscal sector implies that there are two different types of fiscal policy mechanisms at work. On the one hand, the tax specification and the unemployment specification are two types of the so called automatic stabilisers. These automatic stabilisers change the fiscal policy stance automatically to altered economic situations and function without any specific actions taken by the government. In the current model set up, recessions triggering lower income and higher unemployment automatically imply lower tax revenues and higher expenditures due to higher unemployment payments which finally both lead to expansionary fiscal policy effects.

However, the model also has room for discretionary fiscal policy of two types. The fiscal policy rule implies that the government actively intervenes to changes in the economic stance both by relating its actions to current economic conditions and to an exogenous part. The reaction of the government to changes in unemployment and output determines that part of the rule which depends on the current economic stance, while ∈^G is a purely random intervention, that is, surprises in the fiscal authority’s actions. In Section 4 this rule is analysed in detail.

2.8 Equilibrium and Solution of the Model

Aggregate consumption (C_t) is specified by a weighted average of the relevant magnitudes for each household type: $C_{t} = ς C_{t}^{r} + (1 - ς) C_{t}^{o}$ , the amount of employed and unemployed family members of both household types is given by $N_{t} = (1 - ς) N_{t}^{o} + ς N_{t}^{r}$ and $U_{t} = (1 - ς) U_{t}^{o} + ς U_{t}^{r}$ . Similarly, aggregate investment and the aggregate capital stock are defined by: $I_{t} = (1 - ς) I_{t}^{o}$ and $K_{t} = (1 - ς) K_{t}^{o}$ .

Assuming representative firms next to representative households in equilibrium, the aggregate capital supply (K_t) in period t is equal to total capital demand so that: K_t = n_tk_t_–1. Moreover, labour supply (N_t) in period t is equal to labour demand (n_t): N_t = n_t. The aggregate resource constraint satisfies: C_t(19)I_tG_tc_υV_tY_t

and, by using equation (3) and the labour and capital market equilibrium conditions, total output supplied can be rewritten in the following way:

Y_{t} = e^{ξ_{t}^{A}} K_{t - 1}^{α} {(N_{t} H_{t})}^{1 - α}

(20)

In order to solve the model, I first log-linearise the model around the non-stochastic steady state, provided in Appendix E. Appendix F contains the complete log-linear model.

3. ECONOMETRIC METHODOLOGY

Recently, various different methodologies have been developed to estimate the underlying structural parameters of DSGE models given the reduced form representation. Here I proceed by using a procedure proposed by Smith (1993) and Gourieroux et al. (1993) called indirect inference. I minimise a measure of distance between the model and empirical spectral density functions as proposed by Diebold et al. (1998). Working with spectral densities enables a decomposition of variation across frequencies which is often useful and the multivariate focus facilitates the examination of cross-variable correlations and lead-lag relationships at those frequencies of interest. Specifically I only use those frequencies in the data which correspond to low cyclicalities including business cycle periodicities. In particular, the estimation methodology carried out here does not use all the information being contained in the data. This is intentional for the following reason. DSGE models and especially New Keynesian and Real Business Cycle models are theoretical models aiming at explaining business cycles. However, usually the data used for estimation are a composition of cycles of different periodicities; there are long periodic cycles describing growth patterns, seasonal cyclicalities, purely random components and among them, business cycle patterns. Since the model is only aimed at explaining low cyclicalities and especially business cycle frequencies, I hence make only use of this part of information in the data.⁵ Carrying out this estimation procedure hence requires a frequency domain approach.

Let $S (ω) \forall ω \in [0, π]$ denote the n × n matrix of spectral densities of observed time series (n denotes the number of variables). Consider a generic off-diagonal element of S(ω), s_i,j(ω). In polar form, the cross-spectral density is s_i_,j(ω) = ga_i_,j(ω) exp [i.ph_i_,j(ω)], where $g a_{i, j} (ω) = \sqrt{r e^{2} (s_{i, j} (ω)) + i m^{2} (s_{i, j} (ω))}$ refers to the amplitude and ph_i,j(ω) = arctan [im²(s_i,j(ω))/re²(s_i,j(ω))] is the phase shift function. In the following I proceed by using spectral densities of the individual variables and all coherences.⁶ For this I define a new matrix H(ω) as follows:

H (ω) = [\begin{matrix} s_{1, 1} (ω) & C o h_{1, 2} (ω) & \dots & C o h_{1, n} (ω) \\ C o h_{2, 1} (ω) & s_{2, 2} (ω) & ⋮ \\ ⋮ & ⋱ & C o h_{n - 1, n} (ω) \\ C o h_{n, 1} (ω) & \dots & C o h_{n, n - 1} (ω) & s_{n, n} (ω) \end{matrix}]

(21)

Denoted by H(θ, ω) the synthetic counterpart of H(ω) of artificial spectral densities generated by the DSGE model (see Appendix C for the details of the computations of the spectral density functions for the DSGE model). Then, the Indirect Inference estimator of θ, call it $\hat{θ}$ , is given by a vector which solves:

\hat{θ} = \arg \min_{θ} \int_{ω L}^{ω U} f (\hat{H} (ω), H (θ, ω)) ⊙ V (ω) d ω

(22)

where $⊙$ is the Hadamard-product and Hˆ (ω) is an estimator for H(ω) and will be discussed in the next section. In practice the integral is replaced with a sum over the frequencies ω ∈ [ω_L,ω_U] ⊂ [0, π]. In what follows, I specify a quadratic loss function with uniform weighting over all frequencies.⁷

3.1 Estimation

I use data for Germany from the OECD database on 10 variables for the estimation; these are: (a) real GDP, (b) real personal consumption of non-durables, (c) real private investment, (d) hours worked per employee, (e) inflation, measured by the percentage change of the GDP deflator, (f) the employment rate, (g) the nominal interest rate on 3-month government bonds, (h) the real wage rate which is defined here as the compensation per hour worked in the non-farm industry, (i) real government consumption, and (j) the number of vacancies posted. The GDP-deflator is used in all cases where nominal variables need to be converted to real ones. The sample is: 1990q1:2006q4. Within the estimation I do not include government debt due to the fact that it is only reported on an annual basis.

The empirical spectral densities are estimated based on a parametric estimation. In order to compute them, I estimate a stationary Bayesian Vector Autoregression (BVAR)⁸ using a non-informative prior density, four lags, a time trend and a constant term (see Uhlig (1994) and Canova (2007) for further details). The reason for sticking to a Bayesian VAR is due to the fact that this procedure facilitates the computation of standard errors of the estimated spectra as well as confidence intervals. All variables enter the BVAR in logarithmic terms except the nominal interest rate, inflation and the employment rate which are used in levels. A plot of the data used in the BVAR is given in Figure 1. Given these estimates, I transform the BVAR to get an estimator for the spectrum of the variables (see for instance Hamilton (1994), Chapter 10). Specifically, let the VMA of the corresponding VAR be given by: y (23)_ttLe_te_tN0tT

An estimator for the spectral density matrix S(ω) is then given by:

\hat{S} (ω) = \frac{1}{2 π} (\hat{Ψ} (e^{- i ω})) \sum^{^} {(\hat{Ψ} (e^{i ω}))}^{'}, \forall ω \in [0, π]

(24)

The presence of stationarity of the BVAR model is important here for at least two reasons. First of all, the derivation of the spectral density functions is based on a stationary VAR model. Secondly, since the DSGE model is stationary, the VAR model has to satisfy this property too in order to be a valid counterpart to the theoretical model. In order to determine the probability of stationarity of the BVAR model, I numerically draw 10,000 Monte Carlo replications of the posterior distribution of the BVAR coefficients. As it turns out, the probability of stationarity of the BVAR model is around 0.97.

The estimator for H(ω) is based on Ŝ(ω). In a similar fashion as before, I proceed by numerically drawing 10,000 Monte Carlo replications of the posterior density of the BVAR coefficients and taking the median across all draws. Figure 1 shows the spectra of all variables next to the data used in the BVAR. The graphs indicate that there are indeed lots of fluctuations which DSGE models are not able to account for. Especially hours worked, government expenditures and vacancies have many cyclicalities outside the typical business cycle interval. Since DSGE models are not able to explain fluctuations other than business cycles, I use only fluctuations which correspond to 2-year up to 15-year periodicities; this implies that the frequencies considered are: ω ∈ [2π/60, 2π/8]. In order to evaluate Ĥ(ω) and H(θ, ω) I use a grid of size N for the frequencies: ${ω_{i}}_{i = 1}^{N} \in [2 π / 60, 2 π / 8]$ .

Figure 1

Source: Author.

Table 1

Calibrated Parameters

	Parameter	Value
β	discount factor	0.98
δ	depreciation rate	0.025
α	capital share in output	0.33
τ^K	capital tax rate	0.35
τ^N	labour tax rate	0.40
σ_c	coefficient of relative risk aversion	1
b_y	steady state fiscal deficit	0.60
c_y	consumption share in output	0.55
γ_Γ	unemployment payments share in output	0.01
γ_G	government expenditures share in output	0.20

Source: Author.

There are a couple of parameters which are not estimated but calibrated instead. They are given in Table 1. These values are based on long run statistics as in Trabandt and Uhlig (2006), Pytlarczyk (2005) and Christoffel et al. (2007).

3.2 Estimation Results

The estimation results are presented in Table 2. In order to elaborate on the research questions stated in the introduction, I estimate three different versions of the previously specified DSGE model. The distinctions among them only refer to different fiscal policy specifications. Model 1 in Table 2 refers to a fiscal policy structure where fiscal policy is governed as outlined in Section 2.7. The second model defines fiscal policy as being completely exogenous to fluctuations other than its own ones. Specifically, I estimate $σ_{G}^{2}$ , $ϕ_{G}^{G}$ and $ϕ_{G}^{B}$ only. The third model is as Model 1 but omits $σ_{G}^{2}$ implying that no surprises are allowed in fiscal policy.

Table 2

Estimated Parameters

Parameter	Model 1		Model 2		Model 3
Parameter	Estimate t-value	Estimate t-value	Estimate t-value
p^A productivity shock	0.703	1.88	0.691	3.62	0.710	2.95
σ productivity shock	0.0942	1.15	0.1078	1.08	0.1088	1.56
σ discount factor shock	0.0892	1.06	0.1029	0.58	0.2236	1.10
σ government expenditures	0.2126	1.42	0.1060	1.29	–	–
σ monetary policy shock	0.1941	0.78	0.2129	0.94	0.2352	1.44
σ investment shock	0.0921	1.82	0.1326	1.23	0.1485	1.54
σ markup shock	0.1132	1.83	0.1162	1.88	0.1321	1.80
job separation rate (s)	0.039	2.26	0.042	3.14	0.042	3.11
elasticity of unemployment (ξ_m)	0.541	6.56	0.581	2.17	0.569	4.36
bargaining strength of workers (η)	0.502	2.67	0.508	3.04	0.510	2.94
vacancy posting cost (c_υ)	8.490	3.34	9.045	2.78	7.991	3.03
non-Ricardian share (ς)	0.221	1.27	0.210	1.40	0.216	1.21
habit persistence (h)	0.744	2.16	0.747	2.47	0.730	1.98
labour supply (ϕ)	4.010	2.06	4.097	2.07	4.090	1.76
investment adjustment cost (K^I)	3.049	1.94	2.985	1.81	2.958	1.92
Calvo prices (θ_P)	0.892	2.60	0.886	2.51	0.886	2.57
price indexation (ω_p)	0.976	3.02	0.970	3.44	0.980	3.70
government expenditures smoothing ( $ϕ_{G}^{G}$ )	0.574	2.62	0.612	3.13	0.599	3.52
government debt ( $ϕ_{B}^{G}$ )	–0.569	–1.79	–1.302	–1.79	–3.082	–1.89
output ( $ϕ_{Y}^{G}$ )	–1.997	–1.91	–	–	–2.073	–2.41
unemployment ( $ϕ_{U}^{G}$ )	2.450	1.81	–	–	–2.781	0.81
interest rate smoothing (p_r)	0.735	2.17	0.729	1.99	0.729	2.19
inflation in monetary policy rule (aπ)	1.618	2.94	1.691	3.38	1.690	2.31
output in monetary policy rule (a_Y)	0.040	0.97	–0.053	–0.13	–0.014	–0.26
SSR	134.20		1725.94		626.21
BK satisfied	yes		yes		yes
prob(BK satisfied)	0.82		0.59		0.66

Source: Author.

Notes: SSR denotes the sum of squared residuals, BK satisfied = yes if the Blanchard Kahn (1980) conditions are satisfied for the point estimates and prob(BK satisfied) gives the probability of these conditions being satisfied once having simulated the model 5,000 times based on the assumption that the structural parameters follow a normal distribution.

In order to elaborate on the way the current estimation procedure works and how well the model spectra fit the empirical ones, I use Model 1 to highlight a few noteworthy points. The analysis here only focuses on spectra and ignores coherences for simplicity. Figure 2 shows the empirical spectra with confidence intervals⁹ next to the model spectra. As can be seen, the DSGE model appropriately mimics the empirical spectra — the theoretical spectra are always inside the confidence intervals of the empirical ones. Moreover, the DSGE model also adequately describes the typical hump-shaped empirical pattern.

Figure 2

Source: Author.

The estimates of the structural parameters, are similar in size with commonly estimated values; specifically, the estimates of the parameters capturing the degree of price rigidity are similar in size as those obtained by Pytlarczyk (2005) and Christoffel, Kuester and Linzert (2007) using Bayesian estimation techniques. Moreover, the estimates of the interest rate smoothing parameter and the inflation parameter in the monetary policy rule closely reflect those of Pytlarczyk (2005) and Christoffel, Kuester and Linzert (2007). The estimates of the exogenous job separation rate highlight that there is a rather high exogenous degree of inertia in the labour market, especially along the extensive margin.

3.3 Robustness

The estimation results have been checked for robustness intensively. First of all, the decision regarding the chosen frequency band for the estimation might seem ad hoc. Despite the fact that business cycles are characterised by a recurrent pattern, the determination of their length is, however, rather difficult. The estimation results shown in Table 2 take business cycles into account from two to 15 years. The inclusion of cycles of a duration of more than 15 years does not change the results at all. However, in contrast to this, extending this window on the other side, that is, including cycles with a length of one year or less indeed changes the results. This is, however, not surprising. Fluctuations with a duration of one year or less represent seasonal patterns. Since the DSGE model is not intended to be able to explain these patterns, the estimation results will therefore depend on whether these cycles are taken into account or not. In order to circumvent this problem, the estimation is based on spectral densities, which provides a possibility to exclude these fluctuations entirely.

Second, the robustness of the results has been checked additionally with respect to the introduction of further shocks. In particular, a labour supply shock, a shock to the wage bargaining between workers and firms and several other reduced form shocks have been considered in order to judge the robustness of the results in Table 2 with respect to these extensions. As it turned out, these extensions did not change the key results at all. Since the two aforementioned structural shocks turned out to be rather unimportant from a quantitative point of view, they have been eliminated.

Finally, the optimisation problem has been solved by means of the differential evolution algorithm (see for instance Storn et al., 2005). In order to check the dependency of the estimates with respect to this procedure also gradient methods have been applied to minimise equation (22). As it turned out, the results shown in Table 2 did not change.

4. ANALYSIS

Using the previously estimated structural parameters, specifically those of Model 1, this section focuses on an analysis of the impact of the discretionary fiscal policy rule specified in Section 2.7 where the impact of each variable and coefficient is explored in isolation of the remaining parameters. The analysis focuses, among others, on second moments, impulse response functions, forecast error variance decompositions and spectral densities.

4.1 Variability

Fluctuations are at the core for governmental intervention to achieve less variability and smaller amplitudes of fluctuations. Figure 3 plots the standard deviations of inflation, unemployment and output as functions of the policy parameters of the fiscal policy rule for government expenditures (Equation (18)). While letting one specific parameter vary to investigate its impact on the standard deviations, the remaining coefficients are fixed at their estimated values, such that the mechanism at work in creating changes in the standard deviations can be assigned uniquely to the one analysed.

Figure 3

Source: Author.

The effect of the coefficient specifying the sensitivity of government expenditures to changes in debt ( $ϕ_{B}^{G}$ ) clearly shows that reactions to changes in debt imply a continuous decline in the standard deviation of inflation, unemployment and output for larger values of $ϕ_{B}^{G}$ . The intuition behind this is the following. A strong reaction of the government to deviations of public debt from its steady state value imply also large impacts in size on output, inflation and employment through the aggregate resource constraint. Hence the high variability of government expenditures due to debt fluctuations amounts into higher instability of those variables due to the nature of government expenditures as interventions on the demand side.

The effect of $ϕ_{U}^{G}$ also shows rather homogeneous effects on the standard deviations of the three variables under consideration. In all cases a declining pattern of the variability can be observed with increasing values of $ϕ_{U}^{G}$ . This is in line with countercyclical fiscal policy. It is apparent that only procyclical fiscal policy significantly increases instability, whereas a fiscal policy rule reacting positively to unemployment deviations or not at all does not imply large differences in the variability of inflation, unemployment and output.

The effect of stabilisation in terms of reacting to output fluctuations shows a rather clear pattern for all three variables. It is worth being noted that here, fiscal stabilisation policy in terms of a reaction to output indeed is a powerful tool for not only dampening fluctuations in output, but in unemployment and inflation too.

From a qualitative point of view, changes in the structural variance of surprise fiscal policy shocks as well as its degree of persistency described by the parameter $ϕ_{G}^{G}$ , do not imply any specific changes in the pattern of the standard deviations of unemployment, inflation and output outlined previously. What changes, however, is the size of the overall variance of unemployment, output and inflation in absolute terms. The calibrated model replicates empirical variances for these three variables appropriately. It has to be pointed out that changes in the various policy parameters of the fiscal policy reaction function cause only small changes in the value of the standard deviation of inflation in absolute terms, whereas, the range of variation is much larger for the other two variables. This implies that fiscal policy as such has a much stronger impact on output and unemployment than on inflation; more on this in Section 4.4.

4.2 Cyclical Interdependences

Since the various previously outlined fiscal policy specifications imply rather different patterns for the variability of the model economy, it is likely that also the cyclical pattern of fluctuations is influenced. Due to the fact that the area under the spectrum is the variance, the current section extends the previous one in elaborating on the changes in the variability of inflation, unemployment and output. Specifically the question is whether these changes are induced by changes in the amplitude or frequency of fluctuations. The important thing to note is that a change in the spectrum does not necessarily imply a change in the duration of cyclicalities since it can equally well affect the amplitude of fluctuations. To this extent a movement of the spectrum form one frequency band onwards to another amounts into a change in the duration of cycles while a general, not necessarily, proportional upward or downward movement would be in favour of a change in the amplitude of fluctuations.

Figure 4 shows the spectrum of inflation, unemployment and output for frequencies corresponding to business cycles of two to 15 years length ( $ω \in [2 π / 60 ≅ 0.10, 2 π / 8 ≅ 0.78]$ , for quarterly data). The figures, similarly as before, show the impact of one parameter of the fiscal policy rule for government expenditures in isolation of the remaining ones. The graphs show some interesting patterns.

Figure 4

Source: Author.

For the three variables at focus, the cyclicalities are affected rather homogeneously by various fiscal policy specifications. The graphs of the spectral density functions highlight that for any chosen parameter of the fiscal policy rule for government expenditures, the impact on the spectral densities is such that the new policy induces proportional changes in the spectra for a rather narrow frequency band only, while leaves the form of the spectral density functions outside this band completely unchanged. Across the parameter space, the spectrum does not change its shape, it only indicates a higher or lower value for specific calibrations for a rather narrow band across the frequency space. In all cases, the peak of the spectral density functions occurs at a frequency of ω ≃ 0.2. This implies that different fiscal policies primarily affect the amplitude of fluctuations rather than the duration of cyclicalities.

Hence in terms of the variability of the variables induced by the government only the amplitude of fluctuations is affected, however, hardly the duration of cycles. To this extent the effect of the reaction of government expenditures to inflation, unemployment and output affects the overall fluctuations in a similar way as surprise fiscal policy shocks which also only lead to proportional upward or downward movements of the spectral density functions.

4.3 The Adjustment Paths

In general, the impulse response functions due to a shock in government expenditures (shown in Figure 5 with an approximate 68 per cent confidence interval for the response functions¹⁰) are in line with general wisdom–inflation and output increase while unemployment declines, which is supported by various other empirical studies (see for instance Mountford and Uhlig (2008), Coenen and Straub (2005) and Blanchard and Perotti (2002) among others).

Figure 5

Source: Author.

Both output and unemployment display a rather strong effect initially which, however, dies out very quickly. The initial reaction of these two variables to this surprise shock is rather heterogeneous. Unemployment drops by around 0.2 per cent while output increases by around 0.2 per cent points. Eventhough the shock as such is rather persistent ( $ϕ_{G}^{G}$ = 0.55) its long-lasting effects on these two variables is negligible. Inflation displays a hump-shaped adjustment pattern with both positive and negative deviations. To the extent that these deviations are of rather equal length, their size is negligibly small.

Within the DSGE model, the source of the fiscal policy shock is uniquely defined. From an empirical point of view, however, this is everything else but clear. To this extent, there are three realistic sources: (a) changes in the relative weights defined by the fiscal authority when reacting to fluctuations in output, unemployment, etc., (b) imperfect information on the part of the fiscal authority about the current or future economic stance and (c) changes in the fiscal policy stance unrelated to the current or future economic stance as well as unrelated to the prevailing policy reaction function.

The first source of fiscal policy shocks refers to the decision making process within the fiscal authority. Different fiscal policy positions on how to set government expenditures are likely described by different preferences concerning the relative weights determining the reaction of those with respect to fluctuations in output, unemployment, etc. As a result, the decision making process itself can be random. In this case the random component in the fiscal policy reaction function corresponds to random fluctuations in the preferences of the fiscal authority.

The second source of fiscal policy shocks refers to measurement errors caused by lags in the collection of the data of those variables which are essential within the fiscal authority’s decision making process. The fiscal authority can observe the actual economic stance and reverse policy actions due to measurement errors only after the final data has become available. Hence, with a fiscal reaction function based on revised data due to previous misperceptions of the economic stance, all previous policy actions show up in the model as deviations from the rule, which can then be interpreted as unexpected fiscal policy shocks.

The third source refers to genuine surprise shocks which are indeed unrelated to the current fiscal policy reaction function as well as to the current or future economic stance.

In order to quantify the importance of these surprise shocks, the next section presents some evidence using forecast error variance decompositions.

4.4 Surprise Fiscal Policy Shocks

Surprise fiscal policy shocks are an important source of fluctuations. This can be best seen in Table 2 from the estimates of Model 1. In Model 1 all fiscal policy parameters are estimated including those of the variances ( $σ_{G}^{2}$ ). The estimates for the fiscal policy rule for government expenditures clearly indicate countercyclical fiscal policy with especially a rather strong reaction to output deviations. The estimate for the surprise in fiscal policy ( $σ_{G}^{2}$ ) has a rather large value compared to the other shocks. Now, the crucial thing to notice is that when $σ_{G}^{2}$ is omitted in the estimation (consider Model 3), specifically its value is set to zero, higher variability is introduced by a strong procyclical fiscal policy of government expenditures in terms of a procyclical behaviour with respect to unemployment. This implies that when ignoring surprise fiscal policy shocks, an important source of fluctuation is discarded such that a different fiscal policy stance has to account for this lack of variability.

Moreover, despite border-line significant effects of unemployment and output on government expenditures (consider the estimates of Model 1 in Table 2), ignoring those variables leads to a change in the variance of surprise fiscal policy shocks which henceforth captures the omitted effects of those variables. Specifically, the variance of surprise fiscal policy shocks is much larger in Model 1 where countercyclical policy is allowed in terms of a reaction to output and unemployment, while in Model 2 this damping factor is not prevalent and hence also the standard deviation of surprise fiscal shocks is much smaller. As it turns out, the drop in the standard deviation is rather severe: from 0.21 down to 0.10.

When focusing on the estimates of Model 1, it is apparent that fiscal policy is both an important force in contributing to fluctuations, due to its strong dependence on output and unemployment fluctuations, as well as an important source of fluctuations. To the extent that government expenditures cause rather different persistences across various variables, a surprise shock triggers important fluctuations in many of them. This is confirmed by the forecast error variance decomposition outlined in Table 3. As can be seen from the table, fiscal policy shocks explain a large amount of the fluctuations in inflation and unemployment for short horizons, but then this magnitude declines strongly. For output, fiscal policy shocks explain around 20–25 per cent of the fluctuations rather independently of the specific horizon considered. Relative to the other shocks, fiscal policy shocks are a major source of fluctuations.

Table 3

Forecast Error Variance Decomposition

Shock	σ^A	σ^u	σ^G	σ^M	σ^I	σ^R	σξ^M
Horizon: 4 quarters
Inflation	19.29	1.63	15.49	21.09	1.44	15.32	25.74
Unemployment	22.55	14.93	27.83	15.82	4.07	2.57	12.23
Output	38.44	2.74	23.09	11.87	9.95	3.12	10.79
Horizon: 16 quarters
Inflation	13.41	8.67	6.34	25.04	11.46	14.33	20.75
Unemployment	16.99	8.67	6.34	16.21	12.47	3.52	15.42
Output	24.16	6.11	26.00	13.22	8.22	8.01	14.28
Horizon: 32 quarters
Inflation	15.40	13.67	3.35	15.03	15.46	13.33	23.76
Unemployment	17.66	12.77	9.90	16.79	14.18	8.83	19.87
Output	13.79	6.70	25.59	16.66	11.27	9.21	16.78

Source: Author.

Notes: σ^A denotes the technology shock, σ^u the preference shock, σ^G the government expenditures shock, σ^M the monetary policy shock, σ^I the investment shock, σ^R the price markup shock, and σ^ξ^M the shock to the matching function in the labour market.

5. CONCLUSION

This study analysed the influence of discretionary fiscal policy based on an estimated DSGE model for Germany. Discretionary fiscal policy was designed to have government expenditures react to output and unemployment with an additional stochastic error term. In particular, the focus was to investigate, for output, unemployment and inflation, in how far fluctuations therein can be affected by fiscal policy. The important point here, among others, was to figure out the effect of fiscal policy once it is characterised as a force in contributing to the changing characteristics of cycles versus a scenario where it acts as a cause of fluctuations.

The following conclusions can be drawn from the analysis. Discretionary fiscal policy has a strong impact on the amplitude of fluctuations while basically none on the duration of business cycles. The results highlight that, similar as for monetary policy, fiscal policy can smooth cyclical fluctuations but there is no possibility of fundamentally disturbing cyclicalities in their lengths.

Apart from the fact that countercyclical discretionary fiscal policy can even be destabilising (Friedman, 1953) the results obtained here clearly indicate that countercyclical fiscal policy indeed contributes strongly to diminish the amplitude of fluctuations.

Fiscal policy shocks originating in government expenditures are an important source of economic fluctuations as especially the forecast error variance decomposition is displayed. To the extent that this technique indicates the importance of surprises in fiscal policy for explaining economic fluctuations, the applied econometric methodology strengthens this result since once the variance of the policy rule is omitted from the model, implying that there are no surprises in fiscal policy to be allowed, the discretionary fiscal policy stance changes significantly from a countercyclical to a strongly procyclical one so as to account for the fluctuations triggered by surprise shocks. Relative to the other shocks, fiscal policy shocks are a major source of fluctuations.

But besides the fact that fiscal policy is an important source for causing fluctuations, it is also an important responder such that, as the estimates for the fiscal policy rules indicate, government expenditures significantly contribute to dampening cyclical oscillations. To the extent that government expenditures play the most important role of governmental intervention to the economy, the prevalence of countercyclical fiscal policy can clearly be stated.

Considering a fiscal authority’s aim of dampening cyclical fluctuations indicates that for unemployment and output, countercyclical fiscal policy is a powerful tool, whereas for inflation its impact is limited due to the rather rigid fluctuations therein.

Moreover, the results highlight that once a researcher’s aim is to describe the fluctuations observed empirically by means of a DSGE model, a standard set up of the fiscal authority in terms of simple AR(1) processes ignores that there can indeed be a strong endogenous dependence of governmental variables on economic fluctuations.

Footnotes

APPENDIX

Notes

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