A. Households

Each household supplies a differentiated type of labor, and is indexed by j ∊ [0,1]. They obtain utility from consuming the final good $(C_t^j)$, and from holding real money balances $(M_t^j/P_t)$, and obtain disutility from supplying hours of labor $N_t^j$. The lifetime utility function can be expressed as:

where the following functional forms are assumed:

E_t denotes the rational expectations operator using information up to time t. β∊ [0,1] is the discount factor. The instant utility function at any point in time depends on the quasi- difference between the levels of consumption in the current and last period, b∊ [0,1] denotes the importance of the habit stock. Habit formation in consumption has been suggested as a way to reconcile the behavior of consumption and asset prices.⁷ This paper includes this feature because purely forward-looking models have difficulty matching the persistence in output that we observe in the data. If b=0, then we go back to the baseline case where preferences on consumption are time separable. γ>0 is the inverse of the elasticity of intertemporal substitution, ξ>0 is the elasticity of real money balance holdings, and η>0 is the elasticity of labor supply with respect to the real wage.

Households maximize utility subject to the usual budget constraint:

where P_t is the price of the final good, $W_t^j$ is the nominal wage, and $B_t^j$ denotes holdings of a riskless bond that costs the inverse of the gross nominal interest rate (R_t > 1)and pays one unit of currency next period. $T_t^j$ denotes nominal transfers from (or lump-sum taxes paid to) the government. The last term of the right hand side of the previous expression denotes the profits from the monopolistically competitive intermediate goods producers firms, which are ultimately owned by households.

As is well known in this class of models (see Erceg, Henderson, and Levin, 2001), we assume that there exist state-contingent securities that insure households against variations in household specific labor income. In order to keep notation simple, the structure of the complete asset markets is not explicitly introduced. However, it is a well known result that with complete markets, households’ wealth is insured, and the consumption/savings decisions and the labor supply decision can be separated. Moreover, the consumption decision will be independent of current wealth and identical across households and states of nature. Therefore the j subscript in consumption can be safely ignored.

The first order conditions to the consumption/savings decision are:

where Λ_t is the Lagrange multiplier associated with the intertemporal budget constraint. Equation (3) simply states that the Lagrange multiplier equals the marginal utility of consumption. If there were no habit formation, we would get the usual condition that the marginal utility of consumption is decreasing. With habit formation, terms involving past and expected future consumption matter. Equation (4) reflects the intertemporal condition for holding bonds. Equation (5) reflects the money demand equation, and weights the utility of holding money with its opportunity costs.

Since the structure of the labor markets is monopolistic competition, households choose their labor supply schedule maximizing utility, and facing a downward sloping for their type of labor. Before we can derive such condition, we have to describe the technology of the intermediate and final good producers, so we leave the wage setting decision for a later section.

B. Final Good Producers

There is a continuum of final good producers, operating under perfect competition. The technology to produce the aggregate final good is:

where ε_t > 1 is the elasticity of substitution between types of goods, Y_t is the final good, and $Y_l^t$ are the intermediate goods. Since the price markup is related to the elasticity of substitution, we will have time varying price markups, as in Giannoni (2001). Profit maximizing from the final goods producers delivers the following demand for each intermediate good:

where P_t, the price of the final good, is obtained from the zero profit condition in the final goods sector:

and $P_t^i$ are the prices of intermediate goods.

C. Intermediate Goods Producers and the Cost Channel

The production function for intermediate goods producers is:

where A_t is an economy wide technology factor and N_i,t is the unit of effective labor input used by firm i.⁸ 0<δ<1 is the capital share of output, and the level of capital is fixed in the short run at a level $\overline{K}$.

In order to obtain one unit of effective labor, firms employ all types of labor from households, which are aggregated the following way:

Each firm chooses their labor demand schedules in order to obtain the optimal labor mix. As a result, aggregating across firms we obtain the following downward sloping demand for each type of labor j:

where N_t is an index of aggregate labor, and the aggregate wage index is defined as:⁹

In order to explicitly include a cost channel of monetary policy, it is assumed that intermediate goods producers indexed as i∊ [0,α] have to pay workers every period before they sell their product. These firms borrow at the riskless nominal interest rate. Hence, for a fraction α of firms the nominal wage bill is $R_t{\displaystyle {\int }_0^1{W}_t^j{N}_{i,t}^{j}dj}$, while for the remaining 1–α the nominal wage bill is simply $R_t{\displaystyle {\int }_0^1{W}_t^j{N}_{i,t}^{j}dj}$. This timing of events implies that for some firms there exists a cost channel of monetary policy, since an increase in the nominal interest will increase real unit labor costs.

D. Price and Wage Setting under Staggered Contracts

Prices and wages are set by intermediate goods producers and households in a staggered way. For convenience, it is assumed that prices and wages are set at random intervals as in the model of Calvo (1983). Agents can only adjust prices or wages whenever they receive a stochastic signal to do so. The probability of receiving this signal is independent of the past history of signals and across agents. This assumption greatly simplifies the aggregation of price and wage setting decisions.

We denote by θ_p the probability of the Calvo lottery for the price setters. Then, the average duration of price contracts is 1/(1–θ_p). When firms face a Calvo type restriction, they set prices maximizing the discounted sum of profits taking into account that the price that they fix today might not be reset optimally for some time. We assume that there is a degree ε∊ [0,1] of indexation to last period’s inflation rate (P_t–1/P_t–2), whenever firms are not allowed to reset prices optimally.

The first order condition for profit maximization is:

where

is the demand at t+τ assuming that the price set at t is not optimally reset, ${\mu }_t^p={\epsilon }_t/\left({\epsilon }_t-1\right)$ is the time varying price markup. The marginal cost of production depends on the labor’s share of output and the nominal interest rate, if applicable:

Finally, in the symmetric equilibrium, the evolution of the price level is:

where $\widehat{P_t^{*}}$ is the sum of optimal prices set at time t:

$\widehat{P_t^{*}}$ is the optimal price in the symmertic equilibrium for the “non-cost channel” firms and ${\tilde{\tilde{P}}}_t^{*}$ is the optimal price for the “cost channel” firms.

Households face the same restriction to set their wages. Let θ_W denote the probability of the Calvo lottery for the wage setters. The associated average duration of wage contracts is 1/(1–θ_W). Therefore, households’ labor supply schedule comes from choosing their wage to maximize utility facing a downward sloping demand for their type of labor. The first order condition is:

where

is the labor demand at t+τ assuming that the wage set at t is not optimally reset.

μ^w = φ/(φ – 1) is the wage markup. If wages were fully flexible, we would obtain the infratemporal condition that real wages equal the marginal rate of substitution between consumption and hours. However, with staggered contracts, agents know that they might not be able to reset their wage in the near future. Hence they take into account current and future expected deviations between the desired marginal rate of substitution and the real wage, using the corresponding probability as a weight.

In the symmetric equilibrium, the aggregate wage level evolves as:

E. Monetary and Fiscal Policy

As in Taylor (1993), it is assumed that the monetary authority conducts monetary policy with an interest rate rule. Then, the nominal amount of money is determined via the money demand equation. The government cannot run deficits or surpluses, so it finances transfers and other government spending (G_t) via money creation:

The role of transfers in this paper is to undo the inefficiency related to the cost channel of monetary policy. In steady state, for the cost channel firms, the wage bill is higher as long as $\overline{R}>1.$Different wage bills will lead to different steady state prices and production levels. In order to avoid these effects, we assume that the government undoes this inefficiency by subsidizing the “cost channel” firms.

Autonomous government spending is assumed to be a proportion of GDP:

such that the resource constraint can be written as:

or

For a given level of output, a decrease in government spending (increase in Γ_t) will increase private consumption.

A. The Data

This paper explains the comovement of output, price inflation, real wages and nominal interest rates for the United States and the euro area. Real variables are detrended using the Hodrick and Prescott (HP) filter, while nominal variables are treated as deviations from their mean. Data sources for the United States are the following: the Federal Funds rates is used as the relevant nominal interest rate. This series was obtained through the Federal Reserve Bank of Saint Louis database (FRED). The measure of output is the “Nonfarm Business Sector Output,” as published by the Bureau of Labor Statistics (BLS). The measure of prices is the associated price deflator. Finally, the measure of nominal wages is the “Hourly Compensation for the Nonfarm Business Sector,” also obtained through the BLS. The choice of variables is done for comparability with previous studies. The sample period is 1984:01 to 2002:03, at a quarterly frequency. The starting point of the sample reflects the fact that many authors¹¹ have suggested a structural break in the conduct of monetary policy by the Federal Reserve around 1982. In particular, changes involved the switch to an interest rate targeting regime instead of targeting nonborrowed reserves, and the strong anti-inflationary stance. However, it might be the case that it took some time before this change in policy was credible and accepted by the public.

Choices for the euro area are a little bit more difficult to make. First of all, finding data for the euro area as a whole is complicated. Even though member countries in the European Union have converged to a unified system of national accounts, the construction of retroactive data suffers from many complications. The most important are that before 1999, national currencies did in fact fluctuate between themselves, and that the weights of national currencies in the euro and in the ECU are not the same. The European Central Bank Econometric Modeling Unit has constructed a synthetic dataset for the euro area, which is detailed in Fagan, Henry and Mestre (2001).¹²

The starting date for the sample period in the euro area is also controversial. The strongest structural break is in fact the launch of the euro in January 1999. The operating procedures of the Federal Reserve and the German Bundesbank were quite different during the 1980s and 1990s. At the same time, other countries like Spain or Italy did not adopt an anti-inflationary stance until much later in the sample period. We make the choice that the starting date is also 1984:01, that is, we make the assumption that the euro area switched monetary policies around the same time as the United States did, and the public accepted and understood that change at the same time. Even though the conduct of monetary policy by the Bundesbank and the Federal Reserve was quite different, Smets and Wouters (2002) show that a Taylor rule would approximate the behavior of the “synthetic” European Central Bank conduct of policy quite well.

B. Prior Distributions

This section describes the prior distributions adopted for the estimation. Let $\Psi ={({\Psi }_1^{\prime },{\Psi }_2^{\prime },{\Psi }_3^{\prime },)}^{\prime }$ be the vector of all parameters that characterize the model, which is partitioned as follows: σ₁ = (ρ,b,θ_p,ω,θ_w,α,γ_p,γ_y,ρ_r),${\Psi }_2=(\beta ,\varphi ,\overline{\epsilon },\eta ,\delta ),$ and ψ₃ =(ρ_a, ρ_γ, σ_a, σ_γ, σ_m, σ_p) This partition is not arbitrary: while ψ₁ and ψ₂ contain the structural parameters of the model, ψ₃ contains the parameters that describe the model’s exogenous shocks. The parameters in ψ₁ will be estimated, while the parameters in ψ₂ will be calibrated, using degenerate priors.

There are three main reasons to proceed this way. First, we can see by looking at the expressions in the price and wage setting equation that many parameters are not identified. The choice here is to focus on the estimation of average price and wage durations, the cost channel elasticity, and the backward-looking parameter in price inflation. Hence, some parameters need to be fixed. Second, the values of the discount factor and the capital share of output would be indirectly estimated in a model without capital, as the one used in this paper. Third, in order to reduce the dimensionality of the parameter space, the parameters that relate to the price and wage markup, and the labor supply elasticity are held fixed.

Table 2 presents the prior distributions of the parameters. The prior for the inverse elasticity of intertemporal substitution is a Gamma distribution in order to stay in positive values. In theoretical papers, this parameter is usually calibrated at a value of one to be consistent with the existence of a balanced growth path. For this parameter, empirical values in the literature are set between 2 and 4.¹³ The mean of the prior is 2.5 and the standard deviation is large enough to incorporate enough uncertainty about this parameter. The habit formation in consumption parameter is a Normal distribution centered at 0.7, a value close to that suggested by Boldrin, Christiano and Fischer (2001), with a standard deviation of 0.05. The prior distribution is truncated at six standard deviations from the mean, so it can effectively take values between 0.4 and 1.

Table 2:

Prior Distributions

Table 2:

Prior Distributions

Parameter		Mean	Standard Dev.
σ −1	Gamma(2,1.25)	2.50	1.76
b	Normal(0.7,0.05)	0.70	0.05
1/(1−θp)−1	Gamma(2,l)	2.00	1.42
1/(1−θw)−1	Gamma(3,l)	3.00	1.76
ω	Uniform(0,l)	0.50	0.28
α	Uniform(0,l)	0.50	0.28
γ p	Normal(l.5,0.25)	1.50	0.25
γ y	Normal(0.5,0.125)	0.50	0.13
ρr	Uniform(0,l)	0.50	0.28
ρa, ρλ	Uniform(0,l)	0.50	0.28
σa, σλ, σp, σm	Gamma(4,0.25)	1.00	0.50

View Table

Table 2:

Prior Distributions

Parameter		Mean	Standard Dev.
σ −1	Gamma(2,1.25)	2.50	1.76
b	Normal(0.7,0.05)	0.70	0.05
1/(1−θp)−1	Gamma(2,l)	2.00	1.42
1/(1−θw)−1	Gamma(3,l)	3.00	1.76
ω	Uniform(0,l)	0.50	0.28
α	Uniform(0,l)	0.50	0.28
γ p	Normal(l.5,0.25)	1.50	0.25
γ y	Normal(0.5,0.125)	0.50	0.13
ρr	Uniform(0,l)	0.50	0.28
ρa, ρλ	Uniform(0,l)	0.50	0.28
σa, σλ, σp, σm	Gamma(4,0.25)	1.00	0.50

The duration of prices and wages is also a Gamma distribution. In order to keep the probabilities of the Calvo lottery between zero and one, the prior is specified in terms of “average duration minus one quarter.” These priors imply an average duration of three quarters for prices, and of four quarters for wages, following the informal evidence presented in Taylor (1999). Moreover, these priors incorporate the fact that we expect wages to be fixed for longer periods of time than prices. The coefficient on the cost channel has a prior uniform distribution between zero and one. The same prior is used for the coefficient on backward-looking price indexation.

The prior distributions of the coefficients of the Taylor rule are Normal distributions, centered at Taylor’s original values. The parameter space is censored to the region where the model has a unique, stable solution. Therefore, indeterminate solutions due to an interest rate rule that does not place enough weight on the coefficient of price inflation are ruled out. The interest rate smoothing parameter is allowed to take any value between zero and one, with a uniform prior. The priors of the coefficients on the autoregressive parameters are set to uniform distributions between zero and one, while the priors on the standard deviations of the shocks are Gamma distributions to stay in positive reals, with mean of 1 percent and standard deviation of 0.5 percent.

The following degenerate priors are used for the coefficients in ψ₂. The capital share of output is set to δ = 0.36, and the discount factor to β = 0.99, which are values fairly standard in the literature. The values for the elasticity of substitution between types of goods and labor are set to $\varphi =\overline{\epsilon }=6$, which implies steady state price and wage markups of 20 percent. The value for the inverse elasticity of labor supply with respect to the desired real wage is set to η = 1. This value is consistent with the estimates of Rabanal and Rubio- Ramirez (2002) for a model with staggered price and wage contracts. Below, we discuss some sensitivity analysis for these parameters.

C. The Law of Motion and the Likelihood Function

Let x_t = {∆p_t,y_t,r_t,w_t – p_t, mc_t, mrs_t, λ_t, n_t, c_t}^t be the vector of endogenous variables, $z_t={\{a_t,{\gamma }_t,m{p}_t,{\mu }_t^p\}}^t$, the vector of exogenous variables, and ${\zeta }_t={\{{\epsilon }_t^a,{\epsilon }_t^{\gamma },{\epsilon }_t^m,{\epsilon }_t^p\}}^t$, the vector of innovations. Then, the system of equations presented in Table 1, can be written as follows:

There are four exogenous shocks, so in order to avoid linearly dependent solutions, the likelihood function can only be written in terms of four observable variables. Let d_t = {Δp_t,y_t,r_t,w_t – p_t}′ be the vector of observable variables that we wish to explain. Then, the law of motion of the previous system can be obtained by using standard methods for solving linear rational expectations models, as in Uhlig (1999):

such that the solution to the system is written in state-space form. The first equation in (19) is the transition equation, while the second equation in (19) is the measurement equation. Finally, we can evaluate the likelihood function of the observable data conditional on the parameters $L({\left\{d_t\right\}}_{t=1}^T|\psi$, by applying the Kalman filter.¹⁴

D. Drawing from the Posterior and Computing the Bayes Factors

It becomes clear from the previous two subsections that it is not going to be possible to obtain an analytical solution for the posterior distribution. However, since we can evaluate numerically both the prior distribution and the likelihood function, we can apply a numerical algorithm to obtain a draw from the posterior distribution. Geweke (1998) discusses efficient methods to obtain draws from an unknown posterior distribution that can be evaluated numerically. The development of fast speed processors in recent years has removed the time constraint that many of these methods entail. Not too long ago, only well known, standard distributions could be evaluated or simulated.

In order to obtain a random draw of size N from the posterior distribution, a random walk Markov Chain using the Metropolis-Hastings algorithm is generated. The algorithm is implemented as follows:

1. Start with an initial value (ψ°). From that value, evaluate the product $L({\left\{d_t\right\}}_{t=1}^T|{\psi }^0)P({\psi }^0).$

2. For each i:

where ψ^i,* = ψ^i–t + νⁱ, νⁱ is follows a multivariate Normal distribution, and

The idea for this algorithm is that, regardless of the starting value, more draws will be accepted from the regions of the parameter space where the posterior density is high. At the same time, areas of the posterior support with low density (the tails of the distribution) are less represented, but will eventually be visited. The variance-covariance matrix of νⁱ is adopted such that the random draw has some desirable time series properties.¹⁵

Once we have obtained a random draw, for each model it is possible to obtain the marginal likelihood as follows:

The marginal likelihood averages all possible likelihoods across the parameter space, using the prior as a weight. When conveniently weighted, it can be interpreted as the probability of observing the data under a given model. The main shortcoming of the marginal likelihood is that it depends on the priors chosen by the researcher.

The marginal likelihood can be used to compare nested, as well as nonnested models.¹⁶ In the context of this paper, where nested models are compared, it is important to stress that the marginal likelihood already takes into account that the size of the parameter space for different models can be different. Hence, more complicated models will not necessarily rank better than simpler models, if the extra parameterization is unimportant.¹⁷ This is so because the marginal likelihood visits all the regions of the parameter space, and takes the average of both large and small values of the likelihood function.

Multiple integration is required to compute the marginal likelihood, making the exact calculation impossible. We follow Geweke (1998) to obtain an estimate from the random draw of the posterior distribution. Then, for two different models (1 and 2), the tool to compare them is the Bayes factor:

Note that the Bayes factor tells us how we would update our priors on which model is closer to the true one after observing the data. Therefore, the most preferred model when compared pair wise is the one that has the highest marginal likelihood. Moreover, it can be shown that as the sample size increases, $B_{12}({\left\{d_t\right\}}_{t=1}^T)$ goes to zero if model 2 is the “best” model.

A. Posterior Distributions of the Parameters

Figure 1 shows the posterior distribution for selected parameters of the model using data for the United States, while Table 3 contains the mean and standard deviation of the posterior distribution of all the model’s parameters.

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Posterior Distributions, United States

Citation: IMF Working Papers 2003, 149; 10.5089/9781451856910.001.A001

Source: Author’s estimates.

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Table 3:

Posterior Distributions, U.S.

Source: Author’s estimates.

Table 3:

Posterior Distributions, U.S.

Parameter	Mean	Sth. Dev.	Mean	Sth. Dev.	Mean	Sth. Dev.	Mean	Sth. Dev.
σ	0.30	0.16	0.33	0.19	0.30	0.11	0.31	0.09
b	0.85	0.06	0.86	0.05	0.88	0.05	0.89	0.04
Average Price Duration	3.51	0.24	3.41	0.22	3.21	0.22	3.12	0.19
Average Wage Duration	1.21	0.06	1.21	0.05	1.00	-	1.00	-
ω	0.58	0.08	0.61	0.10	0.53	0.14	0.55	0.11
α	0.24	0.19	0.00	-	0.22	0.11	0.00	-
γp	1.61	0.15	1.58	0.14	1.69	0.18	1.75	0.12
γy	0.34	0.06	0.35	0.06	0.34	0.06	0.35	0.05
ρ r	0.65	0.03	0.65	0.03	0.62	0.03	0.63	0.04
ρ a	0.55	0.07	0.56	0.06	0.69	0.04	0.69	0.04
ρ g	0.81	0.04	0.81	0.04	0.82	0.03	0.84	0.03
σ m	0.20	0.02	0.20	0.02	0.20	0.02	0.20	0.02
σ p	7.88	1.05	7.58	0.92	6.47	0.76	6.15	0.68
σ a	1.14	0.23	1.15	0.22	0.57	0.06	0.57	0.06
σ g	0.67	0.06	0.66	0.06	0.65	0.06	0.66	0.06
$\text{log}(\stackrel{⌢}{L})$	943.29		953.02		936.25		952.74

Source: Author’s estimates.

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Table 3:

Posterior Distributions, U.S.

Parameter	Mean	Sth. Dev.	Mean	Sth. Dev.	Mean	Sth. Dev.	Mean	Sth. Dev.
σ	0.30	0.16	0.33	0.19	0.30	0.11	0.31	0.09
b	0.85	0.06	0.86	0.05	0.88	0.05	0.89	0.04
Average Price Duration	3.51	0.24	3.41	0.22	3.21	0.22	3.12	0.19
Average Wage Duration	1.21	0.06	1.21	0.05	1.00	-	1.00	-
ω	0.58	0.08	0.61	0.10	0.53	0.14	0.55	0.11
α	0.24	0.19	0.00	-	0.22	0.11	0.00	-
γp	1.61	0.15	1.58	0.14	1.69	0.18	1.75	0.12
γy	0.34	0.06	0.35	0.06	0.34	0.06	0.35	0.05
ρ r	0.65	0.03	0.65	0.03	0.62	0.03	0.63	0.04
ρ a	0.55	0.07	0.56	0.06	0.69	0.04	0.69	0.04
ρ g	0.81	0.04	0.81	0.04	0.82	0.03	0.84	0.03
σ m	0.20	0.02	0.20	0.02	0.20	0.02	0.20	0.02
σ p	7.88	1.05	7.58	0.92	6.47	0.76	6.15	0.68
σ a	1.14	0.23	1.15	0.22	0.57	0.06	0.57	0.06
σ g	0.67	0.06	0.66	0.06	0.65	0.06	0.66	0.06
$\text{log}(\stackrel{⌢}{L})$	943.29		953.02		936.25		952.74

Source: Author’s estimates.

The first two columns of Table 3 show the estimates for the model presented in section 2. The estimates for the inverse elasticity of substitution suggest that it is much smaller than one, which would be the value consistent with a balanced growth path. The estimate we obtain is similar to that of Rabanal and Rubio-Ramirez (2002) and others.¹⁹ The parameter on habit formation in consumption is higher (0.85) than previous values found in the literature. The fact that the we do not incorporate capital accumulation and investment rigidities makes the estimate of this parameter higher than Christiano, Eichenbaum and Evans (2001) or Smets and Wouters (2002), because output persistence in this model only comes from the consumption side.

The proportion of firms that keep prices fixed for a given period delivers a mean posterior average duration of price contracts of between three and four quarters. This value is somewhat higher than previous estimates (usually in the range of two to three quarters), but still implies a reasonable duration of average interval price changes.

The result for the average duration of wage contracts is surprisingly low, just 1.2 quarters. It is surprising because our prior specification already assumes that wages are set for longer periods of time than prices are. By looking at the expression of k_w, we can see that the elasticity of substitution between types of labor, <p, interacts greatly with average wage duration. However, for reasonable wage markups, it is not possible to obtain much higher mean wage durations. An explanation of this result can be stated as follows: the average nominal wage series includes all wages in the economy—that is, wages set in multiperiod contracts as well as wages for temporary positions, which are in effect for very short periods of time. Hence, it may well be that the average duration of wage contracts is effectively that low.

The proportion of firms that follow a backward indexing rule is about one half, which means that in the hybrid equation of inflation a weight of one-third is given to the backward-looking inflation part, and two-thirds to expected inflation. The coefficient that effectively measures the cost channel gives us a point estimate of 0.24, with a standard deviation of 0.19. When taking a look at the posterior, we can see that it places more weight on smaller values of a. Hence, given our priors, the data do not seem to be that informative about this parameter. Putting it in another way, the prior and the posterior look quite similar because the likelihood function is relatively flat with respect to the value of a.

The coefficients on the Taylor rule are quite similar to what the literature traditionally assumes, with point estimates of around 1.6 for the reaction of the Taylor rule to deviations of inflation around its steady state value, and a value of around 0.35 for the reaction to deviations in the output gap. The coefficient interest rate smoothing is also high, with a point estimate of 0.65. Finally, the estimates for the shock processes suggest that the autoregressive component of the shocks is lower than usually assumed. This result can be attributed to the fact that, in the 1984–2002 period, the macroeconomic series for the United States exhibit less volatility and persistence than in the 1960–1984 period, and also because habit formation and backward-looking inflation imply significant departures from pure forward-looking behavior, and hence increase the endogenous propagation of exogenous shocks.

In the next two columns of Table 3, the estimates for the constrained model (i.e. α = 0) are presented. The estimates for the rest of parameters do not change significantly with respect to the previous case. So the question remains to identify the additional relevance of the cost channel of monetary policy. The log marginal likelihood for the model with cost channel is 953.02, while the log marginal likelihood for the model constrained to α = 0 is 943.29. The log Bayes factor in favor of the constrained model is 9.69, a result for which Kass and Raftery (1995) suggest that there is “decisive” evidence for the constrained model vis-à-vis the model with an explicit cost channel. Since we obtain evidence favoring the constrained model, we can conclude that the introduction of the cost channel parameter is unimportant.

Since the sticky wage component is estimated to be fairly, low, we reestimated the model assuming wages are flexible. In that case, we assume that θ_W= 0, which means that the relevant wage setting equation becomes:

Rabanal and Rubio-Ramirez (2002) found that, in a pure forward-looking model, sticky wages were an important addition to the sticky price model. This was the case because sticky nominal wages and sticky prices deliver sticky real wages, which is what we observe in the data. However, with habit formation in consumption, real wages are already persistent through the effect of persistent consumption in the marginal rate of substitution.

Columns 5–8 present the analogous estimates to columns with flexible wages. The most significant change is that the autoregressive component of the technology shock is estimated to be higher, with a posterior mean of 0.70. This is an indication that once the wage rigidity is removed, more exogenous persistence needs to be fed into the model to match the data. The parameter measuring the cost channel of monetary policy is estimated to be even smaller, with a posterior mean of 0.14. The remaining parameters do not change significantly. The log Bayes factor suggests that the constrained model is also better, when flexible wages are considered. Adding the cost channel simply implies an unimportant overparametrization.

We can also compare the models with sticky and flexible wages using the Bayes factor. The log Bayes factor (with cost channel) is about 7, while the log Bayes factor when the cost channel is eliminated is around 10. According to Kass and Raftery (1995), these magnitudes suggest there is “very strong” evidence of one model over the other. However, we feel that this result should be studied further, since the Bayes factor depends on the prior. It is an important result because it suggests that, once we introduce habit formation in consumption, and backward-looking inflation in the New Keynesian model, we only need small amounts of nominal wage stickiness to match the data.

Figure 2 shows the posterior distribution for selected parameters of the model using data for the euro area, while Table 4 contains the relevant moments of the posterior distributions for all the estimated parameters.

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Posterior Distributions, Euro Area

Citation: IMF Working Papers 2003, 149; 10.5089/9781451856910.001.A001

Source: Author’s estimates.

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Table 4:

Posterior Distributions, Euro Area

Source: Author’s estimates.

Table 4:

Posterior Distributions, Euro Area

Parameter	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
σ	0.26	0.11	0.30	0.15	0.37	0.11	0.38	0.13
b	0.86	0.06	0.87	0.06	0.91	0.03	0.91	0.04
Average Price Duration	2.96	0.21	2.92	0.19	2.77	0.20	2.69	0.19
Average Wage Duration	1.16	0.05	1.15	0.05	1.00	–	1.00	–
ω	0.44	0.11	0.46	0.11	0.54	0.13	0.49	0.09
α	0.17	0.13	0.00	–	0.10	0.09	0.00	–
γ p	1.49	0.14	1.48	0.12	1.54	0.12	1.53	0.17
γ y	0.34	0.08	0.35	0.08	0.37	0.06	0.36	0.07
ρr	0.75	0.03	0.75	0.03	0.74	0.04	0.73	0.03
ρa	0.65	0.06	0.67	0.06	0.77	0.04	0.76	0.04
ρg	0.77	0.04	0.78	0.04	0.81	0.03	0.82	0.03
σ m	0.14	0.02	0.14	0.02	0.15	0.02	0.15	0.02
σ p	6.25	0.88	6.09	0.84	5.67	0.78	5.28	0.68
σ a	0.98	0.22	0.93	0.21	0.47	0.07	0.47	0.07
σ g	0.63	0.06	0.64	0.06	0.63	0.06	0.64	0.06
$\text{log}(\widehat{L})$	807.19		832.86		812.08		826.14

Source: Author’s estimates.

View Table

Table 4:

Posterior Distributions, Euro Area

Parameter	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
σ	0.26	0.11	0.30	0.15	0.37	0.11	0.38	0.13
b	0.86	0.06	0.87	0.06	0.91	0.03	0.91	0.04
Average Price Duration	2.96	0.21	2.92	0.19	2.77	0.20	2.69	0.19
Average Wage Duration	1.16	0.05	1.15	0.05	1.00	–	1.00	–
ω	0.44	0.11	0.46	0.11	0.54	0.13	0.49	0.09
α	0.17	0.13	0.00	–	0.10	0.09	0.00	–
γ p	1.49	0.14	1.48	0.12	1.54	0.12	1.53	0.17
γ y	0.34	0.08	0.35	0.08	0.37	0.06	0.36	0.07
ρr	0.75	0.03	0.75	0.03	0.74	0.04	0.73	0.03
ρa	0.65	0.06	0.67	0.06	0.77	0.04	0.76	0.04
ρg	0.77	0.04	0.78	0.04	0.81	0.03	0.82	0.03
σ m	0.14	0.02	0.14	0.02	0.15	0.02	0.15	0.02
σ p	6.25	0.88	6.09	0.84	5.67	0.78	5.28	0.68
σ a	0.98	0.22	0.93	0.21	0.47	0.07	0.47	0.07
σ g	0.63	0.06	0.64	0.06	0.63	0.06	0.64	0.06
$\text{log}(\widehat{L})$	807.19		832.86		812.08		826.14

Source: Author’s estimates.