2.1 Preferences and production technologies

There is a double continuum of goods (i, j) in the economy, with i ∈ [0,1] representing a sector and j ∈ [0,1] indicating a particular variety within sector j. Each pair (i, j) represents both the good and the firm which produces it under monopolistic competition. The variables ${\tilde{y}}_t\left(i,j\right),y_t\left(i\right)$ and Y_t indicate, respectively, the log output of good (i, j), the log aggregate output of sector i, and the total log aggregate output of the economy at time t. Log prices are defined in the same fashion.

Demand functions for good (i, j) and for sector i’s overall bundle are derived from constant elasticity utility aggregators with elasticities given by ${\nu }_t^i$ and $\overline{\nu }$, respectively. Specifically, the consumer demand for good (i, j) at time t is given by

where the elasticity ${\nu }_t^i>1$ is time varying. For future reference, we define ${\tilde{\mu }}_t^i$ as the log sectoral target mark-up

Similarly, sectoral demands relative to aggregate demand, written in first difference, are defined as

where ${\pi }_t^i=p_t\left(i\right)-p_{t-1}\left(i\right)$ and π_t = P_t – P_t–1 are, respectively, sectoral and aggregate inflation and $\overline{\upsilon }>1$. Moreover, ${\omega }_t^i>0$ is an exogenous sectoral demand shifter, presented in more details below.

We turn to the definition of firms’ supply functions. We assume that the log nominal marginal cost is the same for all firms j within a given sector and is given by

The variable ${\tilde{z}}_t^i$ is the log of the sectoral TFP and the parameter δ_i indicates production real rigidities at the sectoral level. We usually think of δ_i as positive, but negative values are also allowed, e.g. if there are decreasing marginal cost at the sectoral level. In turn, the sets of K_m – 1 ≥ 1 sector-specific constants l_ki and variables ϕ_kt represent, respectively, sectoral loadings an aggregate factors which cause co-movements in sectoral log nominal marginal costs. By using the non-differenced version of (4) it is straightforward to derive the sectoral log real marginal cost $m_t^{ri}\equiv m_t^{ni}-p_t\left(i\right)$ as

where the K_m-th additional factor is ${\varphi }_{Kt}=-P_t-{\overline{\nu }}^{-1}{Y}_t$. Without loss of generality we take ${\displaystyle {\sum }_k^{K_m}{l}_{ki}}=1$. The form (6) for the log real marginal cost is a crucial equation of our model. It is important to point out that (6) is quite general and can be obtained, either directly or through log-linear approximations, from a variety of production functions. The constant δ_i controls the responsiveness of real marginal costs to output, as discussed in Rotemberg and Woodford (1999). The following example with Cobb-Douglas production function explains how increasing marginal costs at the sectoral level arise naturally, for instance, in the presence of fixed or quasi-fixed production factors.

Example. Normalize $\overline{\nu }=1$. Assume that, at the firm’s level, the log production function is specified by ${\tilde{y}}_t\left(i,j\right)=Z_t+{\tilde{z}}_t^i+{\tilde{n}}_t\left(i,j\right)$, where Z_t is aggregate log TFP and n_t(i, j) is the labor input. Using an appropriate Dixit-Stiglitz function, we assume that firms’ production in sector i is aggregated into a sectoral output y_t(i), which is given by y_t(i) = Z_t + n_t(i), with n_t(i) the corresponding sectoral aggregator for the labour input. The log labour supply schedule at the sectoral level is assumed to be $w_t^i=W_t+{\delta }_i\left[n_t\left(i\right)-N_t\right]$, with $w_t^i$ the sector-wide nominal log salary, while W_t and N_t are, respectively, the economy-wide nominal salary and employment. If we take δ_i > 0, then the labour supply schedule implies that higher than average employment in sector i translates into higher than average sectoral salaries. Salaries may vary across sectors as function of labour utilization because of, for instance, overtime wage premia, labor market segmentation across sectors or the presence of labour unions. It is then straightforward to show that the nominal log marginal cost in sector i is $m_t^{ni}=W_t-{\delta }_i{N}_t-\left(1+{\delta }_i\right)Z_t+{\delta }_i{y}_t\left(i\right)-\left(1+{\delta }_i\right){\tilde{z}}_t^i$. After an appropriate scaling of ${\tilde{z}}_t^i$, we obtain a formulation of the type (5). □

We conclude this section with the specification of the stochastic process for the exogenous variables of the model. In some instances, shocks are presented into a normalized form to yield less cluttered equilibrium conditions. The process for the sectoral demand shifter $w_t^i$ is the following,

The log sectoral mark-up ${\widehat{\mu }}_t^i$ is assumed to be the sum of an aggregate component ${\overline{\mu }}_t$ and an AR(1) idiosyncratic component ${\mu }_t^i$,

Finally, the process for the growth rate $\Delta {\tilde{z}}_t^i$ of sectoral TFP is determined by the change in a cyclical component $\Delta z_t^i$ and by a random walk stochastic trend ${\gamma }_t^i$

The auto-correlations ρ_ω, ρ_μ, p_z belong to the interval [0,1]. All the shocks are assumed to be mutually independent, and their variances are indicated, respectively, with ${\sigma }_{\omega }^2,{\sigma }_{\mu }^2,{\sigma }_z^2$.

2.2 Staggered prices

In every period t, each monopolistically competitive firm (i, j) can freely reset its price with probability 1 – θ_i. Prices are indexed at the sectoral level, i.e. firms that cannot optimally reset their price mechanically increase them at an exogenous and sector specific rate ${\overline{\pi }}_t^i$ given by³

where ${\overline{\pi }}_t$ is a stochastic process defining an aggregate variable used by agents for price indexation.⁴ More precisely, the aggregate indexation rate ${\overline{\pi }}_t$ can be seen as stemming from purely monetary phenomena, such as the time-varying inflation target in Cogley and Sbordone (2008), or the lag of aggregate inflation as in Yun (1996). The overall sectoral indexation ${\overline{\pi }}_t^i$ contains, in addition, a correction for the extra long-run deflation induced in sector i by its extra rate of long-run technical progress ${\gamma }_t^i$.

The assumptions made so far ensures that a set of sectoral NKPCs (1) is obtained using standard log-linearization techniques⁵. In particular, the slope λ_i of the sectoral Phillips curve is

2.3 Estimation challenges

To build some intuition about the challenges in estimating the NKPC we write, after some manipulations of (1),

The quantity $E_t\left[{\pi }_{t+1}^i\right]-{\pi }_{t+1}^i=e_{t+1}+e_{t+1}^i$ is the inflation forecasting error, where e_t+1 is the error part that is correlated across sectors and $e_{t+1}^i$ is instead i.i.d.⁶ Because of the assumption of rational expectation, inflation forecast errors are uncorrelated with time t variables. A similar separation between aggregate an idiosyncratic components holds for the mark-up shock ${\widehat{\mu }}_t^i$. The remaining element of (12) is the expected discounted change in the inflation indexation variable, which by virtue of (10) is split into an aggregate part ${\overline{e}}_t={\overline{\pi }}_t-\beta E_t\left[{\overline{\pi }}_{t+1}\right]$ and an idiosyncratic one $\left(1-\beta \right){\gamma }_t^i$.

An inspection of (12) immediately reveals two main estimation issues. First, all the components of the unobserved error that are dated t are correlated with the regressor $m_t^{ri}$. The LI literature has tried to solve this problem by estimating the above equation using IV-GMM methods, where lags of the observed variables are used as instruments. However, a pitfall of this strategy is that, in order for the lags to be valid instruments, mark-up shocks need to be assumed uncorrelated across time. A second issue is that the error term is likely non-stationary (Cogley and Sbordone, 2008). If, for instance, shifts in the monetary policy regime are modeled as a random walk process for ${\overline{\pi }}_t$, then ${\overline{e}}_t=\left(1-\beta \right){\overline{\pi }}_t$ is non-stationary (in addition to being correlated with $m_t^{ri}$). Estimating the NKPC in first difference might solve the non-stationarity problem. However, this would also worsen then endogeneity issue, since the first difference of the inflation forecast error $e_{t+1}+e_{t+1}^i$ would be correlated with $m_t^{ri}$.

3.1 Separating aggregate and idiosyncratic components

The central aspect of our analysis is the characterization of the stochastic process for the real marginal cost $m_t^{ri}$ and its associated real marginal cost gap ${\widehat{m}}_t^{ri}$, defined as

An important features of the stochastic process for ${\widehat{m}}_t^{ri}$ is its persistence in response to shocks, which can be characterized by an intrinsic and an extrinsic components. For idiosyncratic shocks, the extrinsic component is equivalent to the exogenous persistences ρ_ω, ρ_z and ρ_μ. The intrinsic component, instead, is generated purely by the presence of nominal (λ_i) and real (δ_i) rigidities. For the rest of our analysis we divide the sectors into groups for which the following homogeneity assumption in the way nominal and real rigidities interact holds,

Assumption 1. Consider a group of sectors for which the constants λ_i and δ_i satisfy

Assumption 1 is slightly milder than the requirement that λ_i and δ_i are the same across given group of sectors. For a group of sectors that share the same constant κ, the intrinsic persistence r is defined as follows,

Definition 1. For a group of sectors that satisfy Assumption 1, the intrinsic persistence r ∈ (0, 1) is given by the the smaller of the two characteristic roots of second order difference equation,

with χ₀ ≡ β^–1 and ${\chi }_1^i\equiv 1+{\chi }_0\left(1+\kappa \right)$.

Assumption 1 guarantees that the intrinsic persistence r is the same across all sectors belonging to a common group. While Appendix B provides a detailed characterization of the intrinsic and extrinsic persistences, we briefly present here a more intuitive discussion.

Equation (14) is the homogeneous part of the difference equation describing the evolution of marginal cost gaps in response to aggregate and idiosyncratic shocks. The equation thus provides information about dynamic persistence r of the gap’s stochastic process due only to the intrinsic, endogenous propagation of the shocks caused by the nominal and real rigidities. The extrinsic persistence, instead, is derived from the inhomogeneous part of the equation, whose process depends solely on the exogenous persistence of the underlying shocks. Note that the intrinsic persistence r is the same in response to both aggregate and idiosyncratic shocks.⁷

While for idiosyncratic shocks the extrinsic component equals the exogenous persistence ρ_s, for aggregate shocks such characterization doesn’t hold anymore. This can be easily seen in our Example in Section 2.1. An aggregate structural shock, like an increase in productivity Z_t, will in fact cause a cascade of additional general equilibrium “reduced-form” shocks to other factors, like the wage W_t. It follows that the overall extrinsic persistences generated by a structural shock to Z_t does not depend on the exogenous persistence of Z_t alone, but also on the endogenous persistence of the response of all the other factors. Our partial equilibrium approach aims exactly at side-stepping the task of modelling explicitly such complex endogenous propagation. Note also that since the root r lies within the unit circle, real marginal cost gaps, and hence sectoral inflation, could be subject to non-structural (sunspot) dynamics. This paper abstracts from the role of sunspot shocks, which could nonetheless be easily introduced into the model.

Define ι_s an indicator which equals –1 if s = z and 1 otherwise. We then have the following result,

Proposition 1. Assumption 1 is necessary and sufficient for the real marginal cost gap ${\widehat{m}}_t^{ri}$ to be linearly decomposed into aggregate and idiosyncratic components ${\widehat{m}}_t^{ri}={\widehat{x}}_t^i+{\widehat{m}}_t^i$ given by, respectively,

where ${\text{a}}_j^s\in \left(0,1\right)$ are functions only of ρ_s and r, while $\left\{{\widehat{f}}_{kt}\right\}$ is a set of K_m > 0 aggregate factors with sectoral loadings $\left\{{\widehat{\psi }}_{ik}\right\}$.

Proof. See Appendix B. □

According to Proposition 1, the real marginal cost gap can be decomposed into a component that linearly depends only on aggregate factors, and into an idiosyncratic part which is the sum of three MA(∞) processes. The explicit functional form for the weights ${\text{a}}_j^s$ is provided in Appendix B. Note also that an immediate consequence of Proposition 1 is that also the real marginal cost $m_t^{ri}$ can be decomposed into a linear combination $x_t^i$ of aggregate factors, plus an idiosyncratic part $m_t^i$ so that

The representation (16) makes clear one important distinction between idiosyncratic demand and supply shocks on one side, and idiosyncratic mark-up shocks on the other. Consider a positive unit impulse ${∊}_t^{is}{\iota }_s$ for a shock of type s ∈ {ω, z}. By (16), the shock raises the present and future values of ${\widehat{m}}_t^i$. Therefore, by virtue of (1), the shock increases current inflation ${\pi }_t^i$. In particular, the shock increases both ${\widehat{m}}_t^i$ and $m_t^i$ by an amount ${\text{a}}_0^s\in \left(0,1\right)$. Take instead a positive unit impulse ${∊}_t^{i\mu }$ to the mark-up. Again, the shock increases current inflation. However, while ${\widehat{m}}_t^i$ is raised by ${\text{a}}_0^{\omega }$, the current real marginal cost $m_t^i={\widehat{m}}_t^i-{\mu }_t$ decreases by $1-{\text{a}}_0^{\mu }$. In conclusion, for the purpose of characterizing the stochastic process of inflation and real marginal costs, demand and supply shocks are extremely similar, since they both cause inflation and real marginal costs to move in the same direction. Indeed, if we take ρ_ω = ρ_z, then ${\text{a}}_j^{\omega }={\text{a}}_j^z$ and the effects on ${\pi }_t^i$ and $m_t^i$ of two shocks are completely indistinguishable. Mark-up shocks, instead, introduce distinctive dynamics by causing inflation and real marginal costs to move in opposite directions. This is a standard result for New-Keynesian models, where only mark-up shocks generate a policy trade-off between output and inflation stabilization (Clarida et al. 1999). We will return to this point in the next section.

Similarly to what we have done with the marginal cost gap, define the inflation gap ${\tilde{\xi }}_t^i$ as

We then have the following Corollary to Proposition 1

Corollary 1. The process for the inflation gap ${\tilde{\xi }}_t^i$ can be decomposed into an aggregate and an idiosyncratic component ${\tilde{\xi }}_t^i={\tilde{x}}_t^i+{\xi }_t^i$ given by,

where ${\tilde{f}}_{kt}$ for k = 1, ..., K_ξ indicate a set of aggregate factors and ${\tilde{\psi }}_{ki}$ are a set of weights. The idiosyncratic component ${\xi }_t^i$ satisfies the system of equations

Proof. See Appendix B.

In empirical applications of our FIPE methodology, Proposition 1 and Corollary 1 are used as a preliminary step to separate aggregate and idiosyncratic components of the real marginal cost and inflation gaps data. In practice, this separation requires the use of econometric techniques, such as principal components methods. Assumption 1, by guaranteeing the validity of Corollary 1, provides necessary and sufficient conditions that justify the FAVAR model that Boivin et al. (2009) employ to study the inflation process at the sectoral level.

The results of this section can also be cast in the language of the panel literature. Every group of sectors to which Assumption 1 applies form a panel data. Corollary 1 implies that the dependent variable ${\pi }_t^i-\beta {\pi }_{t+1}^i$ in the linear model (12) has the multifactor error structure studied in Pesaran (2006). However, the estimation of (12) with the panel methodology in Pesaran (2006) is problematic since, as pointed out in Section 2.3, in our case the idiosyncratic error terms are correlated with the (idiosyncratic) sectoral regressor $m_t^{ri}$. The next section presents FIPE moments that exploit the model’s structural equations to take such correlation into account.

3.2 FIPE moments

In this section we establish the existence of closed form expressions for the projection of the inflation gaps ${\xi }_t^i$ on any lag (or lead) j of the real marginal cost $m_{t-j}^i$, with j an integer. We show that the projections are linear in the sectoral slope λ_i and non-linear in the intrinsic persistence r and in the relative variance R, to be defined below, of the idiosyncratic mark-up shocks. Projection coefficients are at the heart of many econometric methods, so our closed-form representations for the projection coefficients can be applied in a variety of estimation contexts (GMM and MLE are, for instance, natural candidates).

Corollary 2. Take ρ_ω = ρ_z ≡ ≡. Then, for a given value of ≡, the projections of ξ_t onto $m_{t-j}^i$, for j any integer number, can be expressed as

where each function b_j(r, R) has a closed form expression and R is the relative variance ofthe mark-up shock, i.e.

The representation (20) holds even when the projections are performed with variables in differences.

Proof. See Appendix B, which also provides the closed form expression for the functions b_j(r, R) at any (positive or negative) lag j. □

Corollary 2 is based on the assumption that ρ_ω = ρ_z, i.e. that exogenous demand and supply shocks have the same auto-correlation. In our context this is a mild assumption since, as noted in the discussion of Proposition 1, demand and supply shocks already play an almost identical role in defining the stochastic process for $m_t^i$ and ${\xi }_t^i$. The assumption ρ_ω = ρ_z exploits to the full extent the symmetric role of the two shocks by guaranteeing that ${\text{a}}_j^{\omega }={\text{a}}_j^z$. In this way, the stochastic processes for ${\xi }_t^i$ and $m_t^i$ depend only on ${∊}_t^{i\omega }-{∊}_t^{iz}$ and on ${∊}_t^{i\mu }$ separately. What matters for the projections is then just the relative variance R of the mark-up shock.

Note also that the formulation (20) holds even when the moments are calculated using lagged variables. This is a convenient result in light of the observation that the idiosyncratic error ${\mu }_t^i$ in (19) is in general non-stationary, an issue that can be tackled by estimating the NKPCs in first differences. In this case, the first differences $\Delta e_{t+1}^i$ of the forecast errors become an additional variable endogenous to the regressor $\Delta m_t^i$. This is not a problem, however, since the FIPE moments built according to Corollary 2 take into account this additional form of endogeneity of the error terms.

Finally, Corollary 2 allows us also to obtain explicit moments involving the auto-covariance of ${\xi }_t^i$. These conditions turn out to be especially useful when (19) is expressed in first differences $\Delta {\xi }_t^i$. The differencing operator, in fact, introduces auto-correlation in the shocks $\Delta u_t^i$. In this instance, the auto-covariances become a function also of the relative variance Γ of the trend shocks ${∊}_{t+1}^{i\gamma }$, which for our purposes can be treated as pure measurement error.⁸ For instance, define

so that Γ ∈ [0,1) is the contribution of the (normalized) variable $\Delta {\widetilde{∊}}_t^{i\gamma }\equiv \Delta {∊}_t^{i\gamma }/{\lambda }_i$ to the overall volatility of the forecast error $\Delta e_t^i$. We can then write the following additional moment conditions,

for any integer j and for given functions ${\widehat{b}}_j\left(r,R,\Gamma \right)$.⁹

3.3 Beyond the pure forward-looking model

In search for formulations that can better fit the data, the literature has departed from the pure forward-looking NK model and has proposed alternative versions of the Phillips curve and of the real marginal cost. A strength of the FIPE methodology is that it can be flexibly adapted to these cases.

For instance, while (1) already incorporates in ${\overline{\pi }}_t$ popular types of backward-looking inflation indexation, the framework can be extended to allow for indexation to past sectoral inflation ${\pi }_{t-1}^i$. This can be accomplished by adding a term ${\pi }_{t-1}^i$ to the right side side of (1), giving rise to a hybrid model (Galì and Gertler, 1999). Similarly, the real marginal cost function (6) could be made dependent also on lagged values y_t–1(i) of sectoral output to account for a delayed effect of sectoral wage bargaining or for other sectoral adjustment costs. For instance, Batini et al. (2005) show that the introduction of labor adjustment costs can improve the empirical performance of the Phillips curve.

For both these alternative formulations, the FIPE approach would be implemented as follows. First, the introduction of lagged variables implies that (14) is now of third order. Hence, Definition 1 might have to be modified to take into account the potential presence of multiple roots (i.e. multiple intrinsic persistences r) that lie within the unit circle. Correspondingly, Assumption 1 would also have to be adjusted to ensure that all such roots are the same within sector groups. With these modifications, the moving average decomposition in Proposition 1 is still valid. Of course, the weights ${\text{a}}_j^s$ in Proposition 1 may now depend on the multiple intrinsic persistences r. If this is the case, then the new moment conditions would also be a function of such multiple roots.

4.1 An ARMA(1,1) specification

A Bayesian approach to the use of the FIPE moments is particularly attractive because the projections b(r, R) depend on two parameters - the intrinsic persistence r and the relative variance R - which have intuitive interpretations and for which some prior knowledge can be elicited. With the aim of providing an even sharper intuition for the parameters (r, R) we make the assumption that all shocks have the same persistence, and follow either a random walk or of an i.i.d. process,

The assumptions above may appear restrictive, but they still allow us to move beyond the i.i.d. case typically analyzed in the LI literature. In particular, estimating the model under the two polar possibilities p ∈ {0, 1} can shed light on the sensitivity of the estimated NKPC’s slopes to different assumptions about the persistence of the underlying shocks. Moreover, the above calibration for the extrinsic persistences lead to a stochastic process for the forcing variable mt that is indeed quite general, i.e.

The real marginal cost gap then follows an ARMA(1,1). The intrinsic persistence r is identified as the coefficient of the AR(1) part, while the MA(1) part disappears in the random walk case ρ = 1. The coefficient ${\text{a}}_0^s$ now takes the same value for all types of shocks and is thus indicated simply with a₀. Moreover, the coefficient R for the relative variance takes an even more intuitive meaning: it is the share of variance of the forcing variable ${\widehat{m}}_t^i$ attributable to mark-up shocks. We can then use the expression (22) to elicit prior distributions for R and r in the following way.

First, the New Keynesian literature usually assigns a non-negligible role to mark-up shocks in explaining the variability of the aggregate forcing variable. Our prior is that this intuition carries over to the idiosyncratic component ${\widehat{m}}_t^i$. We then construct a loose prior distribution for R around a modal value of 1/3, where mark-up shocks are as important as the other two shocks in determining the variability of ${\widehat{m}}_t^i$.

Second, recall that, in the discussion about Assumption 1, it was pointed out that the intrinsic persistence r is the same in response to both aggregate and idiosyncratic shocks. Our prior for r is then constructed as a distribution centered around the estimated AR(1) coefficient of an ARMA(1,1) model, fitted to the time series of an aggregate proxy for ${\widehat{m}}_t^{ri}$.¹⁰ Specifically, as often done in the literature, the proxy is taken to be a measure of the output gap. ¹¹ More details about the prior distributions, including the prior for λ_i, are provided in the next section.

The likelihood of the data is obtained as follows. First of all, to address the problem of non-stationarity, we estimate (19) in first differences, namely we use projections of $\Delta {\xi }_t^i$ on leads and lags of $\Delta m_t^i$ and of $\Delta {\xi }_t^i$ itself. Specifically, we use a number 2j + 1 of symmetric leads and lags of $\Delta m_t^i$, and a number J of lags of $\Delta {\xi }_t^i$. We collect these variables into a column vector $X_t^i$,

Define Σ_a the covariance matrix of the elements of $X_t^i$, and Σ_b the row vector collecting the projections (20)-(21) scaled by λ_i,

The analytic expressions for Σ_a and Σ_b are provided in Appendix C. Define b(λ_i, r, R, Γ) the vector of regression coefficients of $\Delta {\xi }_t^i$ on $X_t^i$,

and call ${\overline{\sigma }}_i^2$ of the variance of the residuals of such regression. We estimate ${\overline{\sigma }}_i^2$ directly from the data.

Assume that the i.i.d. shocks ∊^ω, ∊^z, ∊^μ and ∊^γ are jointly normally distributed. Proposition 1 then implies that idiosyncratic real marginal costs are jointly normally distributed with respect to each other and with respect to the inflation gaps ξ_t. It follows that,

Suppose that we have a sample of T observations for each sector i. We approximate the sectoral likelihood L_i(·|·) of the data as

where f_i(·|·) is the probability density function of the normal distribution (24). The likelihood contains an approximation, since each f_i(·|·) is conditioned on $X_t^i$, which contains only a subset of the history, up to time t, of first differences of real marginal costs and of inflation gaps. This issue can be accommodated by choosing lags j and J large enough.

To increase the number of observations used in the estimation, we employ a pooled version of the sectoral likelihood. Assume that the slopes λ_i are drawn from a distribution with mean $\overline{\lambda }$. The draws are taken to be independent of all the idiosyncratic sectoral shocks. Then, by pooling sectors together, we can define a new likelihood as,

where Δξ and Δm represent the entire collection of time and sector observations of first differences of inflation gaps and real marginal costs. Note that the multiplication in L(·|·) of sectoral likelihoods is justified since, at this stage, all aggregate shocks have been removed, and hence the evolution of the observables are independent across sectors.

For given values of ${\overline{\sigma }}_i^2$, the posterior density $\text{p}\left(\overline{\lambda },r,R,\Gamma \right)$ of the parameters is given by,

where ${\text{p}}_0\left(\overline{\lambda },r,R,\Gamma \right)$ is the prior distribution of the parameters.

4.2 Data description and priors

The data source we employ is the EU KLEMS Database, March 2008 release. The database contains information for 3-digit industries at annual frequency for the period 1970-2005. We consider three groups of industries, corresponding to the categorization in Basu et al. (2006): Non-Durable manufacturing (sector group 1), Durable Manufacturing (sector group 2), Non-Manufacturing (sector group 3). Groups are of roughly equal size of 10 industries each. We posit that Assumption 1 holds within each group.

First differences of inflation gaps are constructed from sectoral log price of gross output, using a discount parameter calibrated to a standard annual value of β = 0.97. First differences in the real marginal cost are calculated from the log labor share in gross output. Our estimation thus tests the labor share-based, purely forward looking Phillips curve in Galì and Gertler (1999) and Cogley and Sbordone (2008). For each group of sectors, common factors are extracted from the series via standard principal components procedures, with the number of components selected according to the ER test in Bai and Ng (2002). The remaining idiosyncratic series, used in the estimation, typically explan 20-30% of the variance of the original series.

In the construction of the vectors $X_t^i$ we select j = J = 1. Standard deviations ${\overline{\sigma }}_i$ are calibrated using the preliminary regression described in the previous section. To assess whether a number J = 1 of lags is appropriate, we test whether the regression errors display auto-correlation and we conclude that they largely don’t.