A. Description of the Model

The model consists of a continuum of households who value consumption, subject to external habit persistence, and leisure over an infinite life horizon. The utility function is subject to two exogenous shocks, one is labeled as a preference shock, the other as a labor supply shock.

Households trade bonds in a complete market, which allows them to share risk over time and guarantees that individual income is equalized within households despite imperfect competition and staggered wage setting à la Calvo (1983) in the labor market. Households rent capital services to firms and decide how much capital to accumulate given certain capital adjustment costs. As the rental price of capital goes up, utilization of capital changes according to a cost schedule à la King and Rebelo (2000). Firms produce differentiated goods, decide on labor and capital inputs, and set prices according to the Calvo model. The model is closed by a Taylor (1993) type of monetary policy rule. In the following we will describe the equilibrium conditions of the log-linearized model in some detail.

First, the equation governing the evolution of consumption can be summarized by:

where c_t is consumption, γ_c is the parameter determining the degree of habit persistence, σ is the degree of risk aversion, r_t is the nominal interest rate, π_t is the inflation rate defined as the the first difference of the aggregate price level p_t, and ${\epsilon }_t^b$ is an AR(1) shock to the intertemporal elasticity of substitution. As describe before, households have market power and act as wage setters in the labor market. Furthermore, households face nominal wage rigidity à la Calvo (1983) and are only able to reset wages with probability (1 – ξ_w). If they are not able to re-optimize, the nominal wage will grow at a rate equal partially to the nominal price inflation. As a result, the optimal wage setting of households leads to the following real wage equation:

where w_t is the real wage, β is the discount factor, λ_w is a parameter driving the degree of monopolistic competition in the labor market, γ_w determines the degree of partial wage indexation, ζ drives the disutility of labor in the utility function while ${\epsilon }_t^n$ stand for an AR(1) shock to labor supply.

Households own capital and rent to firms at rental rate $r_t^k$. Capital used in production can be modified by either changing the utilization rate or through investment. Adjusting both generates adjustment costs which are incurred by the households. Adjustment costs are only of second order and are assumed to be zero in steady state. Also, adjustment costs are subject to a stochastic AR(1) shock ${\epsilon }_t^i$, henceforth labelled as an investment shock. As a results the optimal investment i_t is determined by:

while the linearized capital accumulation is given by:

where γ is the inverse of the capital adjustment costs in the steady state, δ is the capital depreciation rate and q_t is the real value of capital and follows:

Notice that R^k stands for the steady state level of the real rate of capital which depends on the time preference rate β and the depreciation rate of capital.

There is a continuum of intermediate goods producers facing imperfect competition. Cost minimization of firms leads to the following equilibrium conditions, determining the evolution of the capital-labor ratio and the marginal cost function mc_t:

where n_t is hours worked and Ψ is the inverse of the elasticity of the capital utilization cost function and α is a parameter determining the evolution of the log-linearized Cobb-Douglas production function:

ϕ is 1 plus the share of fixed costs in steady state output, ${\epsilon }_t^a$ is an AR(1) technology shock common to all firms and z_t is the degree of capacity utilization. Firms face Calvo-type of nominal price rigidity with partial indexation and the degree of competition is subject to the AR(1) shock ${\epsilon }_t^p$. This leads to the following generalized New Keynesian Phillips curve:

where γ_p is the degree of price indexation and (1 – ξ_p) is the probability to reset prices in the actual period. We assume that fiscal policy is Ricardian and variations in government spending-output share g_t are governed by an AR (1) process. As a result, equilibrium in the goods market is determined by the following equation:

$\frac{C}{Y},\frac{I}{Y},\frac{K}{Y}$ are the corresponding steady state ratios of consumption, investment, and capital to output.

Finally, the model is closed by a generalized Taylor-rule, where $\overline{\pi }$ is the inflation target (which is implicitly assumed to be zero in the following) and ${\eta }_t^r$ an i.i.d. policy shock.

The output gap is calculated, consistently with New Keynesian type of DSGE models, as a difference between actual output and the level of output that would prevail under flexible prices and wages and in the absence of cost-push shocks. The parameter φ_r is the degree of interest rate smoothing, while φ_π, φ_y, φ_Δπ, φ_Δy are the elasticity of interest rate to inflation, output gap, inflation growth and the change in the output gap respectively. In the following section, we discuss the impulse response functions of the model.

B. Impulse Response Functions

In this section we present the impulse response functions of the model, from which a limit set is used later as sign restrictions in the VAR analysis, following a monetary policy, preference, government spending, investment, price markup, technology and labor supply shock. To test for robustness, we simulate the impulse responses of the model for a range of sensible parameter values. To delimit the range for the parameter values, we conducted a brief survey of the related empirical literature.⁵ For example, the papers by Smets and Wouters (2003 and 2004) estimate the posteriore distribution of the structural parameters of a NK DSGE model for the euro area and United States. Similar models by using alternative estimation techniques have been analyzed by Christiano et al. (2005), Altig et al. (2002), Onatski and Williams (2004) and Coenen and Straub (2005). We use the range of parameter values estimated in these papers as a benchmark. To choose the interval for the parameters governing investment adjustment cost, habit persistence, capacity utilization costs, we also consulted studies that focus on real models as King and Rebelo (2000) and Boldrin et al. (2001).

To calibrate the monetary policy rule, we took into account the confidence interval for the Taylor rule coefficients estimated in the papers by Judd and Rudebusch (1998), Sack (1998), Clarida et al. (1998) and Rotemberg and Woodford (1998). With regards to the persistence of the shock processes, we cover a wider range for the AR(1) coefficients than the 90 percent interval of the posteriore distribution of the Smets and Wouters (2003) estimates. Particularly, in contrast to Smets and Wouters (2003, 2004), we did not restrict the price markup shock to follow an i.i.d. process and included the entire range of possible values.⁶ We follow Smets and Wouters (2003) by setting some of the structural parameters to a fixed value from the start. In particular, we set α = 0.3, the subjective discount rate β = 0.99, the wage markup parameter λ_w = 0.5, and the depreciation rate δ = 0.025. The intervals for all other parameter values are reported in Table 1.⁷

Table 1:

Structural Parameters

Table 1:

Structural Parameters

β	0.99	ξp	[0.4 – 0.95]
σ	[1 – 4]	ξw	[0.4 – 0.95]
γc	[0 – 0.9]	γp	[0 – 0.99]
Ϛ	[1 – 3]	γw	[0 – 0.99]
α	0.3	φr	[0.6 – 0.99]
Φ	[1 – 1.8]	φy	[0 – 0.8]
λw	0.5	φπ	[1 – 4]
δ	0.025	φΔy	[0 – 0.2]
ϒ	[0.12 – 0.28]	φΔπ	[0 – 1]
Ψ	[2.8 – 10]	ρshock	[0.6 – 0.99]

View Table

Table 1:

Structural Parameters

β	0.99	ξp	[0.4 – 0.95]
σ	[1 – 4]	ξw	[0.4 – 0.95]
γc	[0 – 0.9]	γp	[0 – 0.99]
Ϛ	[1 – 3]	γw	[0 – 0.99]
α	0.3	φr	[0.6 – 0.99]
Φ	[1 – 1.8]	φy	[0 – 0.8]
λw	0.5	φπ	[1 – 4]
δ	0.025	φΔy	[0 – 0.2]
ϒ	[0.12 – 0.28]	φΔπ	[0 – 1]
Ψ	[2.8 – 10]	ρshock	[0.6 – 0.99]

After defining the range of sensible parameter values, we proceed with a simulation exercise. First, we assume that the parameters are uniformly distributed over the selected parameter range. Second, we draw a random value for each parameter from the presented intervals and calculate the corresponding impulse response functions of the model. This exercise is repeated for 100,000 simulations. The median, 16th and 84th percentiles of all conditional responses are shown in Figure 1. The corresponding signs are reported in Table 2.

Theoretical impulse responses of the DSGE model

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note; Median impulse response functions together with 16th and 84th percentiles

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Theoretical impulse responses of the DSGE model

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note; Median impulse response functions together with 16th and 84th percentiles

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Theoretical impulse responses of the DSGE model

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note; Median impulse response functions together with 16th and 84th percentiles

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Table 2:

All Sign Restrictions Derived from the NK DSGE Model

Table 2:

All Sign Restrictions Derived from the NK DSGE Model

	y	p	r	n	w	c	i
monetary policy	↑	↑	↓	↑	↑	↑	↑
preference	↑	↑	↑	↑	↑↓	↑	↓
government spending	↑	↑	↑	↑	↑↓	↓	↓
investment	↑	↑	↑	↑	↑↓	↓	↑
price markup	↑	↓	↓	↑	↑	↑	↑
technology	↑	↓	↓	↓	↑	↑	↑
labor supply	↑	↓	↓	↑	↓	↑	↑

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Table 2:

All Sign Restrictions Derived from the NK DSGE Model

	y	p	r	n	w	c	i
monetary policy	↑	↑	↓	↑	↑	↑	↑
preference	↑	↑	↑	↑	↑↓	↑	↓
government spending	↑	↑	↑	↑	↑↓	↓	↓
investment	↑	↑	↑	↑	↑↓	↓	↑
price markup	↑	↓	↓	↑	↑	↑	↑
technology	↑	↓	↓	↓	↑	↑	↑
labor supply	↑	↓	↓	↑	↓	↑	↑

Not surprisingly, the impulse response functions of the model are broadly in line with the results reported so far in the literature. An expansionary⁸ monetary policy shock, represented by an exogenous decrease in the nominal interest rate, results in an increase in consumption, investment, output, prices, hours, and real wages. Expansionary preference, government spending, and investment shocks all have a positive effect on output, prices and the nominal interest rate.⁹ A shift in preferences toward consumption today crowds out investment, but has a positive impact on consumption and output. In contrast, an expansionary investment shock, generated by a temporary reduction in the cost of installing capital, induces a negative correlation between output and consumption. An increase in nonproductive government spending implies a negative wealth effect and a corresponding fall in consumption and investment; the well-known crowding-out effect of exogenous government spending. The latter has a positive impact on both labor supply and labor demand resulting in an increase of hours worked and an undetermined effect on real wages. Most of impulse responses generated by the NK DSGE model turn out to be insensitive to variations in parameter values. The reaction of real wages following a preference, government spending, and investment shock depends, however, on the chosen parameterization. Specifically, for some values, the DSGE model predicts a positive and for others a negative response.¹⁰

Expansionary supply side shocks, such as the exogenous variation in technology, price markup and labor supply, have a positive impact on output, consumption, and investment, whilst the effect on prices and interest rates is negative. Notice that Calvo-type of price stickiness allows only for a restricted share of firms to adjust prices downwards in the short run. As a result, aggregate demand will initially rise less than aggregate supply following an increase in productivity so that hours worked needs to decrease in equilibrium. Notwithstanding its negative initial effect on marginal costs, a positive technology shock has a positive impact on real wages. The wealth effects induced by the increase in productivity and the corresponding labor supply shift in combination with wage stickiness dominates the negative impact of labor demand effects. On the other hand, an exogenous increase in labor supply has naturally a favorable impact on equilibrium hours worked, but a negative effect on the equilibrium real wage.¹¹ Finally, and in contrast to technology and labor supply shocks, expansionary price markup shocks generate a positive correlation between real wages and hours worked.¹²

Theoretically, the restrictions presented are sufficient to uniquely disentangle all seven shocks in an empirical VAR with sign restrictions. Specifically, for each shock the sign of the response of at least one variable is different from the sign of the response to another shock. If one fully believes in the derived restrictions from the New Keynesian model, the results of the SVAR estimations can be used to evaluate the magnitudes of the impulse responses and analyze variance and historical decompositions.

Although the previous exercise took into account the uncertainty surrounding the structural parameters of the model, the results presented are still conditional on the a priori choice of the model. For example, the response of hours worked is generally negative in the NK model following a technology shock. In contrast, RBC models predict a positive effect of technology shocks, also confirmed by empirical results presented in Christiano et al. (2004), Uhlig (2004) and Peersman and Straub (2004) among others. But the empirical literature is not unanimous (see for example Galí and Rabanal, 2004). The crowding-out of private consumption and investment following a government spending shock is similarly controversial. The majority of the empirical literature predicts a positive or insignificant impact of government spending on private consumption and investment. See the work by Blanchard and Perotti (2002), Fatás and Mihov (2001), Galí et al. (2004), and Mountford and Uhlig (2005).¹³ Finally, the lack of VAR evidence on the dynamic behavior of an economy following preference and investment adjustment cost shocks, give rise to caution in implementing the full set of sign restrictions derived from the NK DSGE model. We therefore propose a much general and less stringent approach to identify the shocks in the next subsection.

C. Relaxed Sign Restrictions

In a first step, we exclude those restrictions of the NK DSGE model that are not necessary to uniquely disentangle the structural shocks. Accordingly, we can already significantly reduce the number of restrictions and still identify all shocks. This does, however, not mean that all controversial restrictions are avoided. For instance, the only restriction to differentiate a price markup shock from a technology shock is the opposite sign of the reaction of hours worked. The latter is, however, exactly at the core of the debate between NK and RBC adherents.

In order to solve this problem, we avoid the use of controversial dynamic responses of the NK DSGE model in the identification procedure by tracking down commonly accepted and therefore more robust sign restrictions. In particular, we use restrictions on the signs of relative responses that are less stringent and less questionable than sign restrictions on the responses of the variables themselves. The signs of the latter can then be estimated. All restrictions used in the estimations described in Section III are presented in Table 3.

Table 3:

Relaxed Sign Restrictions

Table 3:

Relaxed Sign Restrictions

	y	p	r	n	w	C	i	yN – w	c – y	i – y
monetary policy	↑	↑	↓
preference	↑	↑	↑						↑
govern. spending	↑	↑	↑						↓	↓
investment	↑	↑	↑						↓	↑
price markup	↑	↓			↑			↓
technology	↑	↓			↑			↑
labor supply	↑	↓			↓

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Table 3:

Relaxed Sign Restrictions

	y	p	r	n	w	C	i	yN – w	c – y	i – y
monetary policy	↑	↑	↓
preference	↑	↑	↑						↑
govern. spending	↑	↑	↑						↓	↓
investment	↑	↑	↑						↓	↑
price markup	↑	↓			↑			↓
technology	↑	↓			↑			↑
labor supply	↑	↓			↓

An expansionary monetary policy shock is identified as a shock that has a negative effect on interest rates and a positive effect on output and prices. These restrictions are commonly accepted and sufficient to uniquely disentangle the shock from all other shocks. The data can then determine whether all other responses are still consistent with the DSGE model. Demand shocks, such as preference, government spending, and investment shock are assumed to have a positive impact on output, prices and the nominal interest rate. To differentiate them using the standard responses of the NK DSGE variables, we need to restrict the signs of the impulse response functions of consumption and investment. Specifically, the model predicts that a preference shock has a positive impact on consumption and a negative effect on investment. The opposite is true for an investment shock. On the other hand, both impulse responses are predicted to decrease following a government spending shock. However, the described responses, in particular those following government spending shocks, are disputed in the literature.¹⁴

Although RBC models also predict a negative impact of government spending on private consumption and investment, a majority of the empirical literature rejects this. Blanchard and Perotti (2002), Fatás and Mihov (2001), and Galí et al. (2004) find an increase in private consumption following an expansionary government spending shock, but confirm the crowding-out effects on investment. Edelberg et al. (1999) find that government spending crowds-in investment, while Perotti (2002) provides evidence using data on industrial economies that the impact of fiscal shocks on consumption and investment is insignificant in many countries and even negative in the post-1980 period for the United States. From the theoretical point of view, Galí et al. (2004) show that an extended NK DSGE model with rule-of-thumb consumers can account for a positive effect of government spending on consumption in specific circumstances. Consequently, to avoid these controversial restrictions, we introduce in what follows some less stringent constraints on the responses of the consumption-output ratio and investment-output ratio. The theoretical responses of these ratios, obtained from the DSGE model simulations, are shown in the second and third columns of Figure 2.

Extended theoretical impulse responses of the DSGE model

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: Median impulse response functions together with 16th and 84th percentiles

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Extended theoretical impulse responses of the DSGE model

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

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Extended theoretical impulse responses of the DSGE model

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Consider a positive shock to preferences. Since the model predicts a rise in consumption and a fall in all other components of output, the reaction of the consumption-output ratio must be positive. On the other hand, for the same reason, the reaction of the latter to a government spending and investment shock will be negative (not surprisingly because the model predicts for both shocks a rise in output and fall in consumption). More generally, preference shocks, that is a shift in preferences towards consumption today, will cause in most of the commonly accepted models a stronger response of consumption than output. Consequently, we restrict the reaction of the consumption-output ratio to be positive following a preference shock and negative after a shock to government spending and investment adjustment costs. Notice, however, that we still allow this way for a positive reaction of investment after a shock to preferences and a positive effect on private consumption following a government spending and investment shock, as long as the impact is smaller than the effect on total output.¹⁵ We use the same reasoning to discriminate between the investment and government spending shock; we assume that the investment-output ratio increases after the former and falls after the latter in the short-run. These restrictions are still consistent with a standard DSGE model, but are much more general and less stringent. The data will finally determine the exact signs of consumption and investment.

At the supply side (shocks with a negative correlation between output and prices), a labor supply shock is identified as a shock with a negative effect on real wages whilst this effect is assumed to be positive, in accordance with the model, for a technology and price markup shock. Peersman and Straub (2004) illustrate this for a large class of DSGE models, including RBC models. Moreover, Francis and Ramey (2002), and Fleischmann (1999) report a positive effect of technology shocks and a negative effect of labor supply shocks on real wages using an identification strategy in the spirit of Galí (1999).

Finally, the separation of price markup shocks from technology shocks appears to be difficult. The impulse response functions of the two shocks differ only with respect to the reaction of hours worked. Unfortunately, the true causal impact of technology shocks on hours worked is an area of controversy between RBC and sticky price models. The former expects a positive impact whilst the latter predicts a negative effect. Also the empirical evidence is mixed. Galí (1999), Shea (1998), Basu et al. (1999), Francis and Ramey (2002), and Francis et al. (2003) find a negative effect and Christiano et al. (2004), Peersman and Straub (2004), Uhlig (2004), Dedola and Neri (2004), and Canova and Gambetti (2004) find a positive impact. To discriminate between the two shocks, we follow Dedola and Neri (2004) and simulate the impact of both shocks on the difference between labor productivity, defined as y^N, and real wages. As shown in the first column of Figure 2, the reaction is clearly negative following a price markup shock and positive after a technology shock for the range of permissible parameter values.¹⁶ Accordingly, we assume that the impact of technology shocks on labor productivity is stronger than the one on real wages. In contrast, a negative shock to the price markup will have, through the corresponding fall in prices, a stronger effect on real wages than on labor productivity. No other restrictions are required and the data can determine the signs of all unconstrained responses.

A. Methodology

Consider the following specification for a vector of endogenous variables Y_t:

where c is a matrix of constants, A_i is an (n × n) matrix of autoregressive coefficients and ε_t is a vector of structural disturbances. The endogenous variables, Y_t, that we include in the VAR are real GDP (y_t), the GDP deflator (p_t), short-term nominal interest rate (r_t), hours (n_t), real wages (w_t), consumption (c_t) and investment (i_t). All variables are logs, except the interest rate which is in percentages. We estimate the VAR-model in levels with three lags.

Within the VAR, we identify seven types of shocks derived from the presented NK model: a monetary policy, preference, government spending, investment, price markup, technology, and labor supply shock respectively. In order to identify these shocks, we use the sign restrictions as shown in Table 3. The restrictions are sufficient to uniquely disentangle all seven shocks. Specifically, for each shock the sign of the response of at least one variable is different from the sign of the response to another shock. For the implementation of these restrictions, we refer to Peersman (2005). All restrictions are imposed as ≤ or ≥. For all variables, the time period over which the sign constraints are binding is set equal to four quarters.¹⁷ Following Uhlig (2005) and Peersman (2005), we use a Bayesian approach for estimation and inference. Our prior and posterior belong to the Normal-Wishart family used in the RATS manual for drawing error bands. Because there are an infinite number of admissible decompositions for each draw from the posterior when using sign restrictions, we use the following procedure. To draw the “candidate truths” from the posterior, we take a joint draw from the posterior for the usual unrestricted Normal-Wishart posterior for the VAR parameters as well as a uniform distribution for the rotation matrices. We then construct impulse response functions. If all the imposed conditions on the responses are satisfied, we keep the draw. If the decomposition does not match the criteria, the draw is rejected. This means that these draws receive zero prior weight. Based on the draws left, we calculate statistics and report the median responses, together with 84th and 16th percentiles error bands. We do not require the restrictions of all shocks to hold simultaneously. Specifically, if the impulse responses to an individual shock are consistent with the imposed conditions for this shock, the results for the specific shock are accepted. This implies that even if some restrictions for a certain shock are debatable, this has no effect on the estimation of the other shocks.¹⁸ Impulse responses and error bands are computed with a minimum of 1,000 solutions for each shock.

B. Results

Impulse response functions to all seven shocks are shown in Figures 3 and 4 for the United States and the euro area, respectively. The results are remarkably consistent across both areas. The effects of an exogenous change in monetary policy are in line with the results of the DSGE model. There is a hump-shaped response of output and the reaction of prices is persistent following a temporary rise in the short-term interest rate.¹⁹ Also consumption and investment fall after a monetary policy tightening. The reactions of hours and real wages are not significant. The responses to a government spending shock are also very comparable with the results of the benchmark DSGE model. The signs of output, prices and the interest rate are restricted according to the model’s impulse responses for the first four quarters. The reactions of real wages and hours worked are negative but not significant in both areas. Noticeable is that we still find the crowding-out effects of fiscal policy, that is there is a fall in investment and private consumption following an expansionary government spending shock. Both reactions were unconstrained in the estimations. Our finding is in contrast to many other studies using VAR methods such as Blanchard and Perotti (2002), Fatás and Mihov (2001), and Galí et al. (2004).

United States - Impulse responses of VAR with DSGE sign restrictions

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: median impulse responses with 84th and 16th percentiles error bands based on Monte Carlo integration, horizon is quarterly

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Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: median impulse responses with 84th and 16th percentiles error bands based on Monte Carlo integration, horizon is quarterly

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United States - Impulse responses of VAR with DSGE sign restrictions

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: median impulse responses with 84th and 16th percentiles error bands based on Monte Carlo integration, horizon is quarterly

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Euro area - Impulse responses of VAR with DSGE sign restrictions

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

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Euro area - Impulse responses of VAR with DSGE sign restrictions

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: median impulse responses with 84th and 16th percentiles error bands based on Monte Carlo integration, horizon is quarterly

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Euro area - Impulse responses of VAR with DSGE sign restrictions

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: median impulse responses with 84th and 16th percentiles error bands based on Monte Carlo integration, horizon is quarterly

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The conditional responses following preference and investment shocks are also conform with the predictions of the NK DSGE model for output, prices, interest rates, hours, and real wages respectively. We do find, however, a substantial difference in the response of investment and consumption following a shock to preferences and investment adjustment costs. A shift in preferences towards consumption today, that is a positive preference shock, has a significant positive impact on investment. Similarly, an exogenous fall in investment adjustment costs has a positive effect on private consumption. Crowding-in effects of both shocks seem to be more accepted by the data, which are at odds with the predictions of theoretical NK DSGE models. The described results are concurrently present in the data for both the United States and the euro area.

Concerning supply side shocks, the impulse responses following price markup and labor supply shocks do concur with the predictions of the DSGE model. Both shocks have a significant positive effect on consumption, investment and hours worked. The latter three responses were unconstrained in our estimations. The reaction of the nominal interest rate is mostly insignificant. For a technology shock, we do find discrepancies between the conditional moments of the data and the model. In particular, we find a positive effect on hours worked following an expansionary technology shock, while the corresponding response is predicted to be negative in standard NK models. As discussed in Section II, our results following technology shocks support the RBC hypothesis. Positive effects on hours worked following a technology shock are also found by Christiano et al. (2004), Uhlig (2004), Peersman and Straub (2004), and Canova and Gambetti (2004).

C. Does the Data Support the NK Hypothesis?

In this section, we analyze the likelihood and consequences of generating impulse responses that are in line with all sign conditions of the NK DSGE model. The results will give us further indication whether the data supports the hypothesis of the NK model and how likely it is that the economy follows a dynamic path predicted by the NK DSGE model.²⁰ In the previous section, we have only implemented a set of very general sign restrictions from the NK DSGE model; therefore, it is still possible that at least some of the accepted draws are also conform with all the conditional moments of the NK DSGE model, as described in Table 2. Furthermore, we evaluate whether variance decompositions of output and the magnitude of the impulse responses are sensitive to the inclusion of the additional restrictions.

The results are mixed. While imposing all sign restrictions from the DSGE model, we do find solutions for each shock indicating that the predictions of the NK DSGE model are not diametrically opposed by the data. However, for some shocks, it is very difficult to find admissible solutions, attesting that the probability of a realization of the corresponding responses is very low. Because the signs of all impulse responses are consistent with the theoretical model by construction, we do not report them. Table 4 contains the acceptance rates for all shocks. More specifically, the first column shows for each shock the average number of draws required to find a decomposition that match all the imposed sign conditions for our baseline identification strategy discussed in Section III. B (using relaxed, more general sign restrictions).²¹ For the United States, the number varies between 1.98 for an investment shock to 37.61 for a shock to preferences. For most of the shocks, it is relatively easy to reconcile the sign conditions with the conditional moments in the data. It appears to be somewhat difficult, however, to find a technology shock in the euro area (361 draws are required) using the general identification scheme.

Table 4:

Acceptance Rates for All shocks

Table 4:

Acceptance Rates for All shocks

	Relaxed restrictions	All restrictions	Relative acceptance (%)
United States
monetary policy	13.89	349.95	3.97
preference	37.61	17141.24	0.20
government spending	25.32	3911.39	0.64
investment	1.98	230.48	0.86
price markup	2.58	45.61	5.66
technology	2.08	78.92	2.64
labor supply	2.41	55.05	4.38
euro area
monetary policy	3.89	22.59	17.21
preference	20.92	7572392.0	0.00
government spending	17.22	36419.33	0.00
investment	3.57	1164.36	0.31
price markup	12.13	82.15	14.77
technology	361.20	46969.31	0.77
labor supply	3.17	17.51	18.10

View Table

Table 4:

Acceptance Rates for All shocks

	Relaxed restrictions	All restrictions	Relative acceptance (%)
United States
monetary policy	13.89	349.95	3.97
preference	37.61	17141.24	0.20
government spending	25.32	3911.39	0.64
investment	1.98	230.48	0.86
price markup	2.58	45.61	5.66
technology	2.08	78.92	2.64
labor supply	2.41	55.05	4.38
euro area
monetary policy	3.89	22.59	17.21
preference	20.92	7572392.0	0.00
government spending	17.22	36419.33	0.00
investment	3.57	1164.36	0.31
price markup	12.13	82.15	14.77
technology	361.20	46969.31	0.77
labor supply	3.17	17.51	18.10

Using all sign restrictions from the NK DSGE model makes the estimation procedure extremely difficult. For preference, government spending, investment, and technology shocks, it is very difficult to find admissible solutions in the data. The latter is also confirmed by the presented relative acceptance rates in the last column of Table 4. The relative acceptance rate is the percentage of accepted draws with more general restrictions that are also accepted using all sign constraints from the DSGE model. In the euro area, the ratio is below 1 percent for preference, government spending, investment, and technology shocks. In the United States, we get similar results for all but technology shocks.²²

In the next step, we evaluate the sensitivity of the variance decompositions and the impulse response functions of output to the inclusion of the additional restrictions. Figure 5 shows the impulse responses of output to all shocks in the United States and euro area. Black lines are median responses obtained from our baseline VAR with the relaxed restrictions, together with 16th and 84th percentiles error bands (dotted lines), as already reported in Figures 3 and 4. The grey lines are the median responses of output when all NK DSGE sign restrictions are imposed. For some shocks, the choice of the restrictions affects the magnitude of the output response. Introducing all restrictions from the NK DSGE model, the VAR underestimates the impact of investment and technology shocks on output in both areas, while preference and government spending shocks engender a subdued response of output only in the VAR using euro area data.²³ On the other hand, the reaction of output is rather overestimated after a price markup and, but less pronounced, following a monetary policy and labor supply shock.

Impulse responses of output

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: Black lines are median impulse response functions together with 16th and 84th percentiles obtained from VAR with relaxed OSGE restrictions Grey lines are median impulse response functions obtained from VAR with all DSGE restrictions

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Impulse responses of output

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: Black lines are median impulse response functions together with 16th and 84th percentiles obtained from VAR with relaxed OSGE restrictions Grey lines are median impulse response functions obtained from VAR with all DSGE restrictions

• Download Figure
• Download figure as PowerPoint slide

Impulse responses of output

Citation: IMF Working Papers 2006, 135; 10.5089/9781451863956.001.A001

Note: Black lines are median impulse response functions together with 16th and 84th percentiles obtained from VAR with relaxed OSGE restrictions Grey lines are median impulse response functions obtained from VAR with all DSGE restrictions

• Download Figure
• Download figure as PowerPoint slide

The previous results are confirmed by the forecast error variance decompositions of output. Figures are reported in Table 5 for the horizons of 0, 4, and 28 quarters. In the baseline with relaxed constraints, the forecast variance decomposition for the United States indicates that investment and technology shocks are the two most important shocks, explaining 31 and 36 percent of the output variance in the short run. In the long-run, the number even increases to 56 percent for a technology shock. When all restrictions from the NK DSGE model are imposed, the relative importance of both shocks are only 13 and 18 percent, respectively, while price markup shocks, in the same time, account for 40 percent of output fluctuations within one quarter. With the relaxed constraints, the contribution of price markup shocks to output fluctuations are reduced to around 12 percent. Similar conclusions are obtained for the euro area. Interestingly, however, preference shocks appear to play a much more important role in business cycle fluctuations than in the United States. Within one quarter, 16 percent of output fluctuations are explained by shocks to preferences in the baseline case.

Table 5:

Forecast Error Variance Decomposition of Output

Note: Figures are based on the median of the posterior distribution

Table 5:

Forecast Error Variance Decomposition of Output

	Relaxed restrictions			All restrictions
	0Q	4Q	28Q	0Q	4Q	28Q
United States
monetary policy	2.3	6.1	9.1	7.4	15.1	7.9
preference	1.3	3.2	1.4	1.1	2.1	0.8
government spending	5.3	1.7	0.5	4.7	1.1	0.4
investment	30.5	21.1	5.8	12.5	3.9	5.5
price markup	11.7	14.6	14.7	40.4	44.8	38.2
technology	35.7	41.5	55.9	18.0	13.9	29.6
labor supply	13.2	12.0	12.7	15.8	19.0	17.6
euro area
monetary policy	5.9	15.4	26.3	27.1	33.3	31.4
preference	16.1	9.0	6.9	0.4	1.8	0.5
government spending	7.1	9.3	7.4	2.6	4.1	3.3
investment	24.2	28.6	13.4	6.2	5.4	4.8
price markup	8.6	6.2	12.8	17.5	15.5	24.9
technology	29.7	20.9	21.3	20.9	8.7	10.4
labor supply	8.4	10.6	11.9	25.3	31.2	24.7

Note: Figures are based on the median of the posterior distribution

View Table

Table 5:

Forecast Error Variance Decomposition of Output

	Relaxed restrictions			All restrictions
	0Q	4Q	28Q	0Q	4Q	28Q
United States
monetary policy	2.3	6.1	9.1	7.4	15.1	7.9
preference	1.3	3.2	1.4	1.1	2.1	0.8
government spending	5.3	1.7	0.5	4.7	1.1	0.4
investment	30.5	21.1	5.8	12.5	3.9	5.5
price markup	11.7	14.6	14.7	40.4	44.8	38.2
technology	35.7	41.5	55.9	18.0	13.9	29.6
labor supply	13.2	12.0	12.7	15.8	19.0	17.6
euro area
monetary policy	5.9	15.4	26.3	27.1	33.3	31.4
preference	16.1	9.0	6.9	0.4	1.8	0.5
government spending	7.1	9.3	7.4	2.6	4.1	3.3
investment	24.2	28.6	13.4	6.2	5.4	4.8
price markup	8.6	6.2	12.8	17.5	15.5	24.9
technology	29.7	20.9	21.3	20.9	8.7	10.4
labor supply	8.4	10.6	11.9	25.3	31.2	24.7

Note: Figures are based on the median of the posterior distribution