Monetary Policy and the Relative Price of Durable Goods*

Alessandro Cantelmo; Giovanni Melina

doi:10.5089/9781484335451.001.A001

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Monetary Policy and the Relative Price of Durable Goods^*

Author: Mr. Alessandro Cantelmo and Mr. Giovanni Melina

Contributor Notes

Author’s e-mail address: acantelmo@imf.org; gmelina@imf.org.

Publication Date:: 22 Dec 2017

eISBN:: 9781484335451

Language:: English

Keywords:: WP; durable goods; monetary policy; mark-up nondurables; price markup shock; price deflator; house price inflation; durables price stickiness parameter; price mark-up nondurables

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In a SVAR model of the US, the response of the relative price of durables to a monetary contraction is either flat or mildly positive. It significantly falls only if narrowly defined as the ratio between new-house and nondurables prices. These findings are rationalized via the estimation of a two-sector New-Keynesian (NK) models. Durables prices are estimated to be as sticky as nondurables, leading to a flat relative price response to a monetary shock. Conversely, house prices are estimated to be almost flexible. Such results survive several robustness checks and a three-sector extension of the NK model. These findings have implications for building two-sector NK models with durable and nondurable goods, and for the conduct of monetary policy.

Abstract

1 Introduction

Whether monetary policy innovations create distortions in allocations across durable and nondurable goods boils down to the extent to which such shocks change their relative price. The importance of the response of the relative price of durables to monetary policy has been explored in a small number of theoretical contributions, but surprisingly largely neglected in the empirical literature.¹

In the context of optimal policy, Erceg and Levin (2006) show that the relative price of durables affects both the user cost and the demand of durable goods. A stable relative price of durables keeps output close to potential in both sectors and its role for the conduct of monetary policy is therefore non-negligible.² Petrella and Santoro (2011), in an economy with input-output structure, show that the relative price of services affects sectoral marginal costs and creates a channel through which the comovement between consumption of the two goods is attained. They claim that their results can be generalized to any sticky price model with two sectors. In fact, in a similar model featuring durable and nondurable goods, Sudo (2012) demonstrates that if the change in the relative price is small, the substitution effect between durables and nondurables is likewise small and the two goods comove in response to a monetary policy shock.

The comovement between durables and nondurables in response to monetary policy has indeed been a popular topic in the literature and has been documented in a number of papers employing recursive Structural Vector-Autoregressive (SVAR) models (see Bernanke and Gertler, 1995; Erceg and Levin, 2006; Monacelli, 2009; Sterk and Tenreyro, 2014; Di Pace and Hertweck, 2016, among others). Barsky et al. (2003) confirm this empirical result using Romer dates. However, Barsky et al. (2003, 2007, BHK henceforth) were the first to notice that a two-sector New-Keynesian (NK) model fails to replicate such a comovement, hence the so-called comovement puzzle. Consequently, several extensions of the baseline model have been explored in order to solve it.³

The crucial assumption that prevents the baseline model from generating the co-movement concerns sectoral price stickiness. In fact, BHK assume that prices of durable goods are flexible whereas prices of nondurables are sticky. This assumption is made for two reasons. First, durables prices such as houses are largely negotiated and most homes are priced for the first time when they are sold. Second, they appeal to microeconometric studies, such as Bils and Klenow (2004), documenting that durables are more flexible than nondurables. On these grounds, although durables price stickiness turn out to play a key role in the comovement issue (see Sterk, 2010), BHK and most of the subsequent papers assume that durables prices are completely flexible. In contrast, more recently, Nakamura and Steinsson (2008), Boivin et al. (2009), Klenow and Malin (2010) and Petrella and Santoro (2012) report microeconometric evidence of stickiness in many categories of durables other than houses (investment in housing represents about 23% of aggregate durables in US NIPA tables in the post-war period).

The assumption about sectoral price stickiness is closely related to the response of the relative price of durables to a monetary policy shock. In fact, when durables prices are assumed to be flexible, while nondurables prices are sticky, the relative price of durables necessarily falls following a monetary policy tightening, implying that monetary policy creates a distortion in sectoral allocations.

Quite surprisingly, little empirical analysis has focused specifically on this issue.⁴ Table 1 reports unconditional correlations between lags of changes in the federal funds rate (FFR) and changes in key macroeconomics variables over the main sample considered in the paper. The durables sector is defined as the sum of durable goods and residential investment and we report both the relative price of durables and the relative price of houses.⁵ As expected, changes in the FFR are negatively associated with changes in real GDP, durables, houses and nondurables with some lags. As regards inflation, it takes up to three years to detect a negative (though insignificant) correlation. Changes in the relative price of durables seem to be uncorrelated with changes in the FFR, this being in accordance with overall price stickiness not being dramatically different across the two sectors. The relative price of houses exhibits negative but insignificant correlation.

Table 1:

Correlations between lags of changes in the Federal funds rate (FFR) and changes in selected macroeconomic variables

Note: GDP, durables, houses and nondurables are first differences in log real per-capita variables. Inflation is the first difference in the log of the GDP deflator. The relative prices are the first difference of the ratios of the relevant price indices. More data details are available in the Appendix. Frequency: quarterly. Sample: 1969Q2-2007Q4. * denotes significance at a 5 percent level.

Table 1:

Correlations between lags of changes in the Federal funds rate (FFR) and changes in selected macroeconomic variables

						Rel. Price	Rel. Price
	GDP	Durables	Houses	Nondurables	Inflation
						Durables	Houses
FFR (-1)	0.0801	−0.3282*	−0.2534*	−0.3020*	0.1675*	0.0804	−0.0985
FFR (-4)	−0.1806*	−0.2865*	−0.3081*	−0.2411*	0.2230*	0.1110	−0.0049
FFR (-8)	−0.1810*	−0.0727	−0.0903	−0.0803	0.1392	0.0438	−0.0497
FFR (-12)	0.0318	−0.0533	0.1198	0.0634	−0.0070	0.0599	−0.0596

Table 1:

Correlations between lags of changes in the Federal funds rate (FFR) and changes in selected macroeconomic variables

						Rel. Price	Rel. Price
	GDP	Durables	Houses	Nondurables	Inflation
						Durables	Houses
FFR (-1)	0.0801	−0.3282*	−0.2534*	−0.3020*	0.1675*	0.0804	−0.0985
FFR (-4)	−0.1806*	−0.2865*	−0.3081*	−0.2411*	0.2230*	0.1110	−0.0049
FFR (-8)	−0.1810*	−0.0727	−0.0903	−0.0803	0.1392	0.0438	−0.0497
FFR (-12)	0.0318	−0.0533	0.1198	0.0634	−0.0070	0.0599	−0.0596

Given the important policy and modeling implications, this topic deserves more careful investigation. In the paper we exploit both SVAR and Dynamic Stochastic General Equilibrium (DSGE) models, in order to assess the effects of a monetary policy shock on the relative price of durables and the relative house price. The monetary policy shock in SVAR models is identified through recursive, sign restrictions and narrative approaches. Across subsamples and methodologies, the response of the relative price of durables is either flat or mildly positive, but it never falls, contrary to what most DSGE models imply under the assumption of flexible durables prices. A significant fall is found only if the relative price is narrowly defined as the ratio between house prices and nondurables prices, this being consistent with flexible house prices. The estimation of DSGE models corroborates and helps rationalizing the SVAR results. We build a two-sector NK model in which durable goods are used by credit-constrained impatient households as collateral to borrow funds from patient households. The Bayesian estimation unveils that the degree of price stickiness in the sector comprising all durable goods (housing and non-housing) is not significantly different from the nondurables sector. Thus the credible set of impulse responses of the relative price to a monetary policy shock includes zero. In contrast, when durables comprise only housing, house prices are estimated to be almost flexible whereas nondurables prices are substantially stickier. Only in this case, a monetary policy tightening affects the relative price of durable goods, namely the relative house price.

These results on price stickiness survive also modifications affecting sectoral Phillips curves, i.e. if we allow for imperfect labor mobility across sectors, or if we introduce sectoral price indexation to past inflation, and they hold true also in a generalization to a three-sector model.

Our DSGE analysis is related to recent contributions in the literature. Our results on price stickiness are broadly in line with those of Bouakez et al. (2009) who estimate price stickiness in a six-sector model. In their framework, however, there is full symmetry in modeling the various types of goods, which fully depreciate within one period. In contrast, we capture two important features of durable goods that distinguish them from nondurables. First, they yield utility over time rather than being completely consumed in one use. Second, they serve as collateral for borrowing purposes. In a simpler two-sector model Barsky et al. (2016) show that different degrees of durability have implications also for optimal monetary policy. Normative monetary policy implications in a two-sector model are drawn also in Petrella et al. (2017) who focus on input-output interactions. These last two papers, however, do not estimate price stickiness parameters as we do. Iacoviello and Neri (2010) estimate a two-sector model where durables comprise only housing and house prices are assumed a priori to be fully flexible. In contrast, we consider both housing and non-housing durables and estimate all price stickiness parameters.

Our SVAR and DSGE results have two important implications for modeling and policy. The first is that, when building a two-sector New-Keynesian, model it is desirable to assume that prices of durable goods are somewhat sticky, unless the model’s aim is to focus on the housing sector in isolation from other durables. A three-sector model is needed to fully capture the intrinsic differences between housing and non-housing durables, such as the type of goods that can be used as collateral and their different degree of durability. The second is that overall monetary policy innovations do not foster big distortions in sectoral allocations between durables and nondurables, this representing a desirable feature of the monetary policy conduct. Conversely, since monetary policy does affect the relative house price, it may potentially create allocative distortions between housing and non-housing goods.

The remainder of the paper is organized as follows. In Section 2 we perform the SVAR analysis. Section 3 presents the DSGE model, its extensions, and discusses the results of the Bayesian estimation. Section 4 concludes. An appendix complements the paper by providing details about the dataset, the theoretical model, and by reporting robustness checks.

2 Structural vector-autoregressive models

2.1 Methodology

As regards the estimation of the empirical model, we use quarterly, seasonally adjusted US data for the Federal funds rate, real GDP, real durable goods, real nondurable goods and services, the GDP deflator and the relative price of durables.⁶ In order to thoroughly investigate the effects of a monetary policy shock on the relative price of durables, we employ two alternative definitions of durables sector. We first follow Erceg and Levin (2006), Monacelli (2009), Sterk and Tenreyro (2014) and Di Pace and Hertweck (2016) in defining durables as the sum of durable goods consumption and residential investments.⁷ Then, we assume that durables comprise only houses. We label the former model baseline SVAR and the latter housing SVAR. Table 2 summarizes the various definitions of durables sector and relative prices used throughout the paper. Definitions I and II are used in the main analysis whereas III and IV serve for robustness checks. The algebraic details for the computation of all relative prices are reported in Appendix A.

The main analysis is performed over the sample 1969Q2-2007Q4. This choice is dictated by the availability of the narrative measure of monetary policy shocks constructed by Romer and Romer (2004, RR henceforth) and extended by Coibion et al. (2012) and Tenreyro and Thwaites (2016).

The vector of variables employed in the SVAR is the following:

\begin{matrix} x_{t} \equiv {[G D P_{t}, D_{t}, C_{t}, P_{t}, Q_{t}, F F R_{t}]}^{} & (1) \end{matrix}

where GDP_t denotes gross domestic product; D_t and C_t represent consumption of durable and nondurable goods, respectively; P_t is the GDP deflator; Q_t is the the relative price of durable goods; and FFR_t denotes the Federal funds rate. We take the natural logarithm of all variables except for the FFR, which is in levels.

Table 2:

Definitions of Relative Prices

Table 2:

Definitions of Relative Prices

	Definition	Description
I	Relative Price of Durables	Ratio of price deflator of durables and residential investment to price deflator of nondurables and services.
II	Relative House Price	Ratio of price deflator of new single and multifamily houses components of residential investment to price deflator of nondurables and services.
III	Relative Price of Durables and New Single Family Houses	Ratio of price deflator of durables and new single family houses components of residential investment to price deflator of nondurables and services.
IV	Relative Price of Durables and Broad Measure of Houses	Ratio of price deflator of durables and new single and multifamily houses components of residential investment to price deflator of nondurables and services.

Table 2:

Definitions of Relative Prices

	Definition	Description
I	Relative Price of Durables	Ratio of price deflator of durables and residential investment to price deflator of nondurables and services.
II	Relative House Price	Ratio of price deflator of new single and multifamily houses components of residential investment to price deflator of nondurables and services.
III	Relative Price of Durables and New Single Family Houses	Ratio of price deflator of durables and new single family houses components of residential investment to price deflator of nondurables and services.
IV	Relative Price of Durables and Broad Measure of Houses	Ratio of price deflator of durables and new single and multifamily houses components of residential investment to price deflator of nondurables and services.

For the sake of robustness, we take three different approaches to the identification of monetary policy shocks:⁸

i) recursive (Cholesky) approach in which we make the standard assumption that the monetary policy variable is ordered last, hence it has no contemporaneous effect on the other variables (see Bernanke and Mihov, 1998, among others);

ii) sign restrictions imposed on the impulse responses of the variables and derived from a DSGE model as in Canova (2002), Dedola and Neri (2007), Pappa (2009) and Bermperoglu et al. (2013), among others. Fry and Pagan (2011) critically review the sign restrictions approach arguing that if there is not enough information to discriminate among the various shocks, it may be problematic to correctly identify them. In principle, only if the researcher describes the sign pattern for each shock in the model it is possible to avoid this problem. In order to partially address this identification issue we proceed as follows. Following Peersman (2005), we first determine the sign pattern of two standard supply and demand shocks, and then we identify the monetary policy shock.⁹

Table 3 summarizes the set of sign restrictions imposed. A contractionary supply shock curbs output, nondurable and durable consumption, while increasing inflation, which leads the central bank to raise the nominal interest rate. A negative demand shock reduces both all the mentioned real variables and inflation, thus leading the central bank to cut the interest rate. Conversely, the monetary policy shock is characterized by an increase in the nominal interest rate, which leads to a decrease in output, nondurable consumption and inflation. As discussed in Appendix E, notwithstanding the lack of a robust response, in order to correctly identify the monetary policy shock, we assume that the nominal interest rate is positive in the first quarter. We remain agnostic on the response of the relative price and consumption of durables as it is the main objective of our investigation. The different restrictions imposed on the responses of the GDP deflator and the interest rate ensure the orthogonality between the disturbances and the correct identification of the monetary policy shock.

Table 3:

Sign restrictions

iii) recursive narrative approach: we follow Romer and Romer (2004), Coibion (2012) and Cloyne and Hurtgen (2016) and re-estimate the recursive SVAR model by replacing the FFR with the monetary policy shock constructed by RR and extended by Coibion et al. (2012) and Tenreyro and Thwaites (2016).

In the macro-fiscal literature, Mertens and Ravn (2013) and Mertens and Ravn (2014) challenge the use of narrative measures to identify fiscal shocks in SVAR models on the ground that such measures do not represent truly exogenous fiscal shocks. They therefore build on Stock and Watson (2012) and use narrative measures as external instruments for the identification of the structural shocks in the SVAR model (Proxy SVAR henceforth).¹⁰ However, as noted by Cloyne and Hurtgen (2016), while fiscal narrative measures are directly derived from historical sources, thus representing potential noisy proxies for the structural shocks, the monetary policy measure derived by RR is the result of a first-stage regression which yields a direct measure of the structural shock rather than a proxy. It is therefore reasonable to use such a measure directly in the VAR model rather than as an external instrument. In any case, for the sake of robustness, Figure C.14 in Appendix C.5 reports the impulse responses obtained with a Proxy SVAR model using the RR measure as external instrument to identify the structural shock linked to the monetary policy variable, i.e. the federal funds rate.¹¹ The relative prices exhibit the same sign patter to those reported in the main analysis.¹²

2.2 Results

The estimated impulse responses are presented in Figure 1. Rows refer to the variables of the model whereas columns refer to the three different identification approaches. The shock is a one standard deviation increase in the monetary policy measure. Solid lines depict the responses for the baseline SVAR model, and the shaded areas are the corresponding one-standard-deviation confidence bands. Dashed lines show the responses for the housing SVAR model, with dotted lines representing the corresponding one-standard-deviation confidence bands.¹³ The impulse responses show that results are broadly robust across models and identification approaches, with the exception of the relative price. There is evidence for the comovement between durables and non-durables, and the responses of durables are always larger than those of nondurables, a finding that is consistent with the empirical literature.¹⁴

Turning to the dynamic behavior of the relative price, the estimated responses to a monetary policy tightening are highly dependent on the definition of the durables sector adopted. If durables account for both consumption goods and residential investment, the response of the relative price is either flat or mildly positive, this being at odds with the assumption of flexible durable prices adopted in most of the theoretical literature. Conversely, a model in which the durables sector coincides exclusively with the housing sector, the relative price falls consistently with the notion of flexible new house prices. These results are confirmed by the responses of the relative price across seven subsamples.¹⁵ In Figure 2 rows plot the relative price responses for each subsample, whereas columns represent the three identification approaches. The relative price of durables never falls in the baseline SVAR model whereas it significantly decreases in the housing SVAR model, thus confirming the previous results across subsamples and identifications of the monetary policy shock. To sum up, this empirical evidence suggests that the definition of the durables sector is crucial. If durable goods are defined to include both non-housing goods and residential investment, these display dynamics consistent with a non-negligible degree of price stickiness. Conversely, durable goods defined to include only the housing sector exhibit a behavior compatible with flexible prices.

These results survive several robustness checks reported in Appendix C: (i) inclusion of a linear time-trend (C.1); (ii) alternative definitions of durables as described in Table 2 (C.2); (iii) subsample analysis (C.3); (iv) sign restrictions imposed for two, four and six quarters (C.4); (v) the Proxy SVAR approach (C.5); (vi) a three-sector SVAR model in which durables and housing are treated separately in the same model (C.6). It is noticeable from Section C.2 that when any measure of house prices is bundled with non-house durables prices, the relative price never falls in response to a monetary policy tightening.

3 New-Keynesian model

To rationalize the SVAR estimates, we analyze a two-sector New-Keynesian model in which households consume both durable and nondurable goods. Following Monacelli (2009), Sterk (2010) and Iacoviello and Neri (2010) we assume that impatient households obtain loans from patient ones using their durables stock as collateral, with the amount they can borrow tied to the value of the collateral, thus allowing for a further transmission mechanism of monetary policy beyond the standard interest-rate channel. ¹⁶ The economy is characterized by several frictions, the importance of which is empirically assessed. These are price and wage stickiness, investment adjustment costs in durable goods (IAC, henceforth) and habit formation in consumption of nondurable goods. Finally, the monetary authority sets the nominal interest rate according to a Taylor-type interest rate rule.

3.1 Households

The economy is populated by a continuum of two groups of infinitely-lived households (patient and impatient) each indexed by i ϵ [0,1] in which consumers derive utility from consumption of durable and nondurable goods and get disutility from supplying labor. Impatient households have a lower discount factor than patient ones (β’ < β) that is why they borrow in equilibrium. Throughout the paper, variables and parameters with a ‘refer to impatient households.

3.1.1 Patient households

Patient household’s lifetime utility is represented by

\begin{matrix} E_{0} \sum_{t = 0}^{\infty} e_{t}^{B} β^{t} U (X_{i, t}, N_{i, t}), & (2) \end{matrix}

where $e_{t}^{B}$ is a preference shock, $X_{i, t} = Z_{i, t}^{1 - α} D_{i, t}^{α}$ is a Cobb-Douglas consumption aggregator between nondurable (Z_it) and durable goods (D_it) with α ϵ [0, 1] representing the share of durable consumption on total expenditure, and N_i,t being the household’s labor supply. We assume that nondurable consumption is subject to external habit formation so that

\begin{matrix} Z_{i, t} = C_{i, t} - ζ S_{t - 1}, & (3) \end{matrix}

\begin{matrix} S_{t} = ρ_{c} S_{t - 1} + (1 - ρ_{c}) C_{t}, & (4) \end{matrix}

where C_i,t is the level of the household’s nondurable consumption; S_t, ς ϵ (0,1) and ρ_c ϵ (0,1) are the stock, the degree and the persistence of external habit formation, respectively, while C_t represents average consumption across households. Each household monopolistically supplies labor to satisfy the following demand function:

N_{i, t} = {(\frac{w_{i, t}}{w_{t}})}^{- e_{t}^{W} ?} N_{t},

where w_i,t is the real wage of each household whereas w_t is the average real wage in the economy. Parameter η is the intratemporal elasticity of substitution between labor services and $e_{t}^{W}$ is a wage markup shock. Finally, firms on average demand a quantity N_t of labor services. Nominal wages are subject to quadratic costs of adjustment à la Rotemberg (1982): $\frac{υ^{W}}{2} {(\frac{w_{i, t}}{w_{i, t - 1}} Π_{t}^{C} - Π^{C})}^{2} w_{t} N_{t}$ , where ν^W is the parameter governing the degree of wage stickiness, $Π_{t}^{C}$ is the gross rate of inflation in the non-durable sector, and Π^C is its steady-state level. The stock of durables evolves according to law of motion

\begin{matrix} D_{i, t + 1} = (1 - δ) D_{i, t} + e_{t}^{I} I_{i, t}^{D} [1 - S (\frac{I_{i, t}^{D}}{I_{i, t - 1}^{D}})], & (6) \end{matrix}

where δ is the depreciation rate of durables, $I_{i, t}^{D}$ is investment in durable goods that is subject to adjustment costs and $e_{t}^{I}$ represents an investment-specific shock. The adjustment costs function S (·) satisfies S (1) = S’ (1) = 0 and S” (1) > 0. In addition, each household purchases nominal bonds B_i,t, receives profits Q_i,t from firms and pays a lump-sum tax T_t so that the period-by-period real budget constraint reads as follows:

\begin{matrix} C_{i, t} + Q_{t} I_{i, t}^{D} + \frac{ϑ^{W}}{2} {(\frac{w_{i, t}}{w_{i, t - 1}} Π_{t}^{C} - Π^{C})}^{2} w_{t} N_{t} + \frac{R_{t} B_{i, t - 1}}{Π_{t}^{C}} = \frac{B_{i, t}}{P_{t}^{C}} + \frac{W_{i, t}}{P_{t}^{C}} N_{i, t} + Ω_{i, t} - T_{t}, & (7) \end{matrix}

where $Q_{t} \equiv \frac{P_{D, t}}{P_{C, t}}$ is the relative price of durables, R_t is the gross nominal interest rate and W_i,t is the nominal wage. Households choose $Z_{i, t}, B_{i, t}, D_{i, t + 1}, I_{i, t}^{D}, w_{i, t}$ to maximize (2) subject to (3), (4), (5), (6) and (7). At the symmetric equilibrium, the patient household’s optimality conditions are:

\begin{matrix} 1 = E_{t} [Λ_{t, t + 1} \frac{R_{t}}{Π_{t + 1}^{C}}], & (8) \end{matrix}

\begin{matrix} \begin{matrix} Q_{t} ψ_{t} = \frac{U_{D, t}}{U_{Z, t}} + (1 - δ) E_{t} [Λ_{t + t + 1} Q_{t + 1} ψ_{t + 1}], \end{matrix} & (9) \end{matrix}

\begin{matrix} 1 = & ψ_{t} e_{t}^{I} [1 - S (\frac{I_{t}^{D}}{I_{t - 1}^{D}}) - S^{'} (\frac{I_{t}^{D}}{I_{t - 1}^{D}}) \frac{I_{t}^{D}}{I_{t - 1}^{D}}] + \\ + E_{t} {Λ_{t, t + 1} ψ_{t + 1} \frac{Q_{t + 1}}{Q_{t}} e_{t + 1}^{I} [S^{'} (\frac{I_{t + 1}^{D}}{I_{t}^{D}}) (\frac{I_{t + 1}^{D}}{I_{t}^{D}})^{2}]}, & (10) \end{matrix}

\begin{matrix} 0 = & [1 - e_{t}^{W} η] + \frac{e_{t}^{W} η}{μ_{t}} - υ^{W} (Π_{t}^{W} - Π^{C}) Π_{t}^{W} + \\ + E_{t} [Λ_{t, t + 1} υ^{W} (Π_{t}^{W} - Π^{C}) Π_{t + 1}^{W} \frac{w_{t + 1} N_{t + 1}}{w_{t} N_{t}}] . & (11) \end{matrix}

Equation (8) is a standard Euler equation with $Λ_{t, t + 1} \equiv β \frac{U_{Z, t + 1}}{U_{Z, t}} \frac{e_{t + 1}^{B}}{e_{t}^{B}}$ representing the stochastic discount factor and U_z,t denoting the marginal utility of habit-adjusted consumption of nondurable goods. Equation (9) represents the asset price of durables, where U_D,t is the marginal utility of durables consumption and ψ_t is the Lagrange multiplier attached to constraint (6). Equation (10) is the optimality condition with respect to investment in durable goods. Finally, equation (11) is the wage setting equation in which $μ_{t} \equiv \frac{w_{t}}{M R S_{t}}$ is the wage markup, $M R S_{t} = - \frac{U_{N, t}}{U_{Z, t}}$ is the marginal rate of substitution between consumption and leisure, U_N,t is the marginal disutility of work and $Π_{t}^{W} = \frac{w_{t}}{w_{t - 1}} Π_{t}^{C}$ is the gross wage inflation rate.

3.1.2 Impatient households

Impatient households solve a maximization problem analogous to patient households, with the additional assumption that the former are limited in the amount they can borrow from the latter by the value of their durables stock according to the following borrowing constraint:

\begin{matrix} B_{i, t}^{'} \leq m E_{t} (\frac{Q_{t + 1} D_{i, t}^{'} Π_{t + 1}^{C}}{R_{t}}), & (12) \end{matrix}

where m represents the loan-to-value (LTV) ratio.¹⁷ At the symmetric equilibrium, the impatient household’s optimality conditions are:

\begin{matrix} λ_{t}^{'} = e_{t}^{B} U_{Z', t}, & (13) \end{matrix}

\begin{matrix} λ_{t}^{'} = β' E_{t} [λ_{t + 1}^{'} \frac{R_{t}}{Π_{t + 1}^{C}}] + λ_{t}^{B C} R_{t}, & (14) \end{matrix}

\begin{matrix} Q_{t} ψ_{t}^{'} = \frac{U_{D', t}}{U_{Z' t}} + β' (1 - δ) E_{t} [\frac{λ_{t + 1}^{'}}{λ_{t}^{'}} ψ_{t + 1}^{'} Q_{t + 1}] + \frac{λ_{t}^{B C}}{U_{Z' t}} m E_{t} [Q_{t + 1} Π_{t + 1}^{C}], & (15) \end{matrix}

\begin{matrix} 1 & = ψ_{t}^{'} e_{t}^{I} [1 - S (\frac{I_{t}^{D'}}{I_{t - 1}^{D'}}) - S' (\frac{I_{t}^{D'}}{I_{t - 1}^{D'}}) \frac{I_{t}^{D'}}{I_{t - 1}^{D'}}] + \\ + β' E_{t} {\frac{λ_{t + 1}^{'} Q_{t + 1}}{λ_{t}^{'} Q_{t}} ψ_{t + 1}^{'} e_{t + 1}^{I} [S' (\frac{I_{t + 1}^{D'}}{I_{t}^{D'}}) (\frac{I_{t + 1}^{D'}}{I_{t}^{D'}})^{2}]}, & (16) \end{matrix}

\begin{matrix} 0 & = [1 - e_{t}^{W} η] + \frac{e_{t}^{W} η}{μ_{t}^{'}} - υ^{W} (Π_{t}^{W'} - Π^{C}) Π_{t}^{W'} + \\ + β' E_{t} [\frac{λ_{t + 1}^{'}}{λ_{t}^{'}} υ^{W} (Π_{t + 1}^{W'} - Π^{C}) Π_{t + 1}^{W'} \frac{w_{t + 1}^{'} N_{t + 1}^{'}}{w_{t}^{'} N_{t}^{'}}] . & (17) \end{matrix}

Variables $λ_{t}^{'}$ and $λ_{t}^{B C}$ are the Lagrangian multipliers attached to the budget and borrowing constraints, respectively. Notice that (14) is a modified version of the typical Euler equation due to the presence of the borrowing constraint. Equations (15) and (16) show the optimal decisions about the stock and flow of durables whereas (17) is the wage equation. Here, $Π_{t}^{W'} = \frac{w_{t}^{'}}{w_{t - 1}^{'}} Π_{t}^{C}$ is the gross wage inflation rate of impatient households.

3.2 Firms

Firms face quadratic costs of changing prices as in Rotemberg (1982): $\frac{υ_{j}}{2} (\frac{P_{ω, t}^{j}}{P_{ω, t - 1}^{j}} - 1)^{2} Y_{t}^{j}$ , where υ_j is the parameter of sectoral price stickiness. Each firm produces differentiated goods according to a constant returns to scale production function,

\begin{matrix} Y_{ω, t}^{j} = e_{t}^{A} {(N_{ω, t}^{j})}^{\tilde{ψ}} {({N'}_{ω, t}^{j})}^{1 - \tilde{ψ}}, & (18) \end{matrix}

where ω ϵ [0, 1] and j = C, D are indices for firms and sectors respectively, $\tilde{ψ} \in [0, 1]$ denotes the share of the patient household and $e_{t}^{A}$ is a labor augmenting shock.¹⁸ Firms maximize the present discounted value of profits,

\begin{matrix} E_{t} {Σ_{t = 0}^{\infty} Λ_{t, t + 1} [\frac{P_{ω, t}^{j}}{P_{t}^{j}} Y_{ω, t}^{j} - \frac{W_{ω, t}}{P_{t}^{j}} N_{ω, t}^{j} - \frac{W_{ω, t}^{'}}{P_{t}^{j}} {N'}_{ω, t}^{j} - \frac{υ_{j}}{2} (\frac{P_{ω, t}^{j}}{P_{ω, t - 1}^{j}} - 1)^{2} Y_{t}^{j}]}, & (19) \end{matrix}

subject to production function (18) and a standard Dixit-Stiglitz demand equation $Y_{ω, t}^{j} = {(\frac{P_{ω, t}^{j}}{P_{t}^{j}})}^{- e_{t}^{j} \in j} Y_{t}^{j}$ , where ϵ_j and $e_{t}^{j}$ are the sectoral intratemporal elasticities of substitution across goods and the sectoral price markup shocks, respectively. At the symmetric equilibrium, the price setting equations for the two sectors read as

\begin{matrix} (1 - e_{t}^{C} \in_{c}) + e_{t}^{C} \in_{c} M C_{t}^{C} = & υ_{c} (Π_{t}^{C} - 1) Π_{t}^{C} - \\ - υ_{c} E_{t} [Λ_{t, t + 1} \frac{Y_{t + 1}^{C}}{Y_{t}^{C}} (Π_{t}^{C} - 1) Π_{t + 1}^{C}], & (20) \end{matrix}

\begin{matrix} (1 - e_{t}^{D} \in_{d}) + e_{t}^{D} \in_{d} {M C}_{t}^{D} = & υ_{d} (Π_{t}^{D} - 1) Π_{t}^{D} - \\ - υ_{d} E_{t} [Λ_{t, t + 1} \frac{Q_{t + 1}}{Q_{t}} \frac{Y_{t + 1}^{D}}{Y_{t}^{D}} (Π_{t}^{D} - 1) Π_{t + 1}^{D}] . & (21) \end{matrix}

If υ_j = 0 prices are flexible and are set as constant markups over the marginal costs.

3.3 Fiscal and monetary policy

Every period, a lump-sum tax equates government spending so that the government budget is balanced. Government spending $e_{t}^{G}$ follows an exogenous process and, as in Erceg and Levin (2006), we assume that the government purchases only nondurable goods and services. Monetary policy is set according to the following Taylor rule:

\begin{matrix} \log (\frac{R_{t}}{R}) = ρ_{r} \log (\frac{R_{t - 1}}{R}) + (1 - ρ_{r}) [ρ_{π} \log (\frac{{\tilde{Π}}_{t}}{\tilde{Π}}) + ρ_{y} \log (\frac{Y_{t}}{Y})] + e_{t}^{R}, & (22) \end{matrix}

where ρ_r is the interest rate smoothing parameter, ρ_π and ρ_y are the monetary policy responses to the deviations of the inflation aggregator and output from their respective steady states, and $e_{t}^{R}$ represents the exogenous innovation to the monetary policy rule. ${\tilde{Π}}_{t} \equiv {(Π_{t}^{C})}^{1 - τ} {(Π_{t}^{D})}^{τ}$ is an aggregator of the gross rates of inflation in the two sectors with τ ϵ [0,1] representing the weight of durables. Different monetary policy rules have been used in two-sector NK models with no difference in their main implications.¹⁹

3.4 Market clearing conditions and exogenous processes

In equilibrium all markets clear and the model is closed by the following identities:

\begin{matrix} Y_{t} = Y_{t}^{C} + Q_{t} Y_{t}^{D} + \frac{υ^{W}}{2} {(Π_{t}^{W} - Π^{C})}^{2} w_{t} N_{t} + \frac{υ^{W}}{2} {(Π_{t}^{W'} - Π^{C})}^{2} w_{t}^{'} N_{t}^{'}, & (23) \end{matrix}

\begin{matrix} Y_{t}^{C} = C_{t} + C_{t}^{'} + G_{t} + \frac{υ_{c}}{2} (Π_{t}^{C} - Π^{C})^{2} Y_{t}^{C}, & (24) \end{matrix}

\begin{matrix} Y_{t}^{D} = [D_{t} - (1 - δ) D_{t - 1}] + [D_{t}^{'} - (1 - δ) D_{t - 1}^{'}] + \frac{υ_{c}}{2} (Π_{t}^{D} - Π^{D})^{2} Y_{t}^{D}, & (25) \end{matrix}

\begin{matrix} 0 = B_{t} + B_{t}^{'}, & (26) \end{matrix}

\begin{matrix} N_{t} = N_{t}^{C} + N_{t}^{D}, & (27) \end{matrix}

\begin{matrix} N_{t}^{'} = {N'}_{t}^{C} + {N'}_{t}^{D} . & (28) \end{matrix}

As in Smets and Wouters (2007), the wage markup and the price markup shocks follow ARMA(1,1) processes:

\begin{matrix} \log (\frac{𝛞_{t}}{𝛞}) = ρ_{𝛞} \log (\frac{𝛞_{t - 1}}{𝛞}) + \in_{t}^{𝛞} - θ_{i} \in_{t - 1}^{𝛞}, & (29) \end{matrix}

with 𝛞 = [e^W, e^C, e^D], i = [W, C, D], whereas all other shocks follow an AR(1) process:

\begin{matrix} \log (\frac{κ_{t}}{κ}) = ρ_{κ} \log (\frac{κ_{t - 1}}{κ}) + \in_{t}^{κ}, & (30) \end{matrix}

where κ = [e^B, e^I, e^R, e^A, e^G] is a vector of exogenous variables, ρ_𝛞 and ρ_κ are the autoregressive parameters, θ_i are the moving average parameters, $\in_{t}^{𝛞}$ and $\in_{t}^{κ}$ are i.i.d shocks with zero mean and standard deviations σ_𝛞 and σ_κ.²⁰

3.5 Functional forms

The utility function is additively separable and logarithmic in the consumption aggregator: $U (X_{t}, N_{t}) = \log (X_{t}) - ν \frac{N_{t}^{1 + φ}}{1 + φ}$ is a scaling parameter for hours worked and ψ is the inverse of the Frisch elasticity of labor supply. Following Chris-tiano et al. (2005), we assume quadratic adjustment costs in durables investment:

$S (\frac{I_{t}^{D}}{I_{t - 1}^{D}}) = \frac{ϕ}{2} {(\frac{I_{t}^{D}}{I_{t - 1}^{D}} - 1)}^{2}$ , with ϕ > 0 representing the degree of adjustment costs. The same functional forms are assumed for the impatient households, with the preference and investment adjustment cost parameters specific to them.

3.6 Bayesian estimation

The model is estimated with Bayesian methods. The Kalman filter is used to evaluate the likelihood function, which combined with the prior distribution of the parameters yields the posterior distribution. Then, the Monte-Carlo-Markov-Chain Metropolis-Hastings (MCMC-MH) algorithm with two parallel chains of 150,000 draws each is used to generate a sample from the posterior distribution in order to perform inference. We estimate the model over the sample 1969Q2-2007Q4, the same as in the SVAR analysis. We use eight observables: GDP, investment in durable goods, consumption of nondurable goods, real wage, hours worked, inflation in the nondurables sector, inflation in the durables sector and the nominal interest rate, using US data. Similarly to the SVAR analysis, we first define the durables sector as the sum of durable goods and residential investments and label this model as the baseline DSGE. Then, we estimate the model by assuming that durables comprise only houses and we will refer to it as the housing DSGE. This model becomes then very close to Iacoviello and Neri (2010) who, however do not estimate the price stickiness parameter in the housing sector and assume that prices are flexible.²¹ The following measurement equations link the data to the endogenous variables of the model:

\begin{matrix} Δ Y_{t}^{o} = γ + {\hat{Y}}_{t} - {\hat{Y}}_{t - 1}, & (31) \end{matrix}

\begin{matrix} Δ I_{D, t}^{o} = γ + {\hat{I}}_{D, t}^{*} - {\hat{I}}_{D, t - 1}^{*}, & (32) \end{matrix}

\begin{matrix} Δ C_{t}^{o} = γ + {\hat{C}}_{t}^{*} - {\hat{C}}_{t - 1}^{*}, & (33) \end{matrix}

\begin{matrix} Δ W_{t}^{o} = γ + {\hat{W}}_{t}^{*} - {\hat{W}}_{t - 1}^{*}, & (34) \end{matrix}

\begin{matrix} N_{t}^{o} = {\hat{N}}_{t}^{*}, & (35) \end{matrix}

\begin{matrix} Π_{C, t}^{o} = {\bar{π}}_{C} + {\hat{Π}}_{t}^{C}, & (36) \end{matrix}

\begin{matrix} Π_{D, t}^{o} = {\bar{π}}_{D} + {\hat{Π}}_{t}^{D}, & (37) \end{matrix}

\begin{matrix} R_{t}^{o} = \bar{r} + {\hat{R}}_{t} . & (38) \end{matrix}

Variables with a ^ are in log-deviations from their own steady state while * denotes that the variable has been aggregated between the patient and impatient households (i.e. $x_{t}^{*} = x_{1} + x_{t}^{'}$ ).²² γ is the common quarterly trend growth rate of GDP, investment of durables, consumption of nondurables and the real wage; ${\bar{π}}_{C}$ and ${\bar{π}}_{D}$ are the average quarterly inflation rates in nondurable and durable sectors respectively; $\bar{r}$ is the average quarterly Federal funds rate. Hours worked are demeaned so no constant is required in the corresponding measurement equation (35).

3.6.1 Calibration and priors

The structural parameters and steady state values presented in Table 4 are calibrated at a quarterly frequency. As in Iacoviello and Neri (2010), the discount factors β and β′ are 0.99 and 0.97, respectively. Following Monacelli (2009), the depreciation rate of durable goods δ is calibrated at 0.010 amounting to an annual depreciation of 4%, and the durables share of total expenditure α is set at 0.20. The sectoral elasticities of substitution across different varieties ϵ_c and ϵ_d equal 6 in order to target a steady-state gross mark-up of 1.20. The elasticity of substitution in the labor market η is set equal to 21 as in Zubairy (2014), implying a 5% steady-state gross wage mark-up. The preference parameters ν and ν′ are set to target steady-state hours of work of 0.33 for both households. The government-output ratio g_y is calibrated at 0.20, in line with the data. Finally, we follow Iacoviello and Neri (2010) and set the loan-to-value ratio m to 0.85 and the share of patient households $\tilde{ψ}$ at their estimated value 0.79.²³

Table 5 summarizes the prior and posterior distributions of the parameters and the shocks. The choice of priors correspond to a large extent to those in previous studies of the US economy. We set the prior mean of the inverse Frisch elasticities ψ and ψ′ to 0.5, in line with Smets and Wouters (2007, SW henceforth) who estimate a Frisch elasticity of 1.92. We also follow SW in setting the prior means of the habit parameter, ς and ς′, to 0.7, the interest rate smoothing parameter, ρ_r, to 0.80 and in assuming a stronger response of the central bank to inflation than output. As far as the the constants in the measurement equations are concerned, we set the prior means equal to the average values in the dataset. In general, we use the Beta (B) distribution for all parameters bounded between 0 and 1. We use the Inverse Gamma (IG) distribution for the standard deviation of the shocks for which we set a loose prior with 2 degrees of freedom. Kim and Katayama (2013) are the only authors who jointly estimate the price and wage stickiness parameters whereas all the other studies calibrate them such that prices of nondurable goods are sticky whereas prices of durable goods are flexible. However, they define Calvo parameters for prices and a Rotemberg parameter for wages. Our model features Rotemberg parameters for both prices and wages and we choose a Gamma (G) distribution, given that these are non-negative. One of our main interests is to assess whether the durables price stickiness parameter is close to zero, or whether it tends towards values closer to those estimated for the nondurables sector. This is crucial in order to assess whether the response of the relative price of durables is significantly different from zero or not. To this aim, we assign a prior whereby durables prices are as sticky as nondurables prices and both degrees of price stickiness are low (corresponding to firms resetting prices around 2.3 quarters on average in a Calvo world).²⁴ Then, we let the data decide whether and to what extent these should depart from one another.

Table 4:

Calibrated parameters

Table 4:

Calibrated parameters

Parameter		Value
Discount factor patient households	β	0.99
Discount factor impatient households	β’	0.97
Durables depreciation rate	δ	0.010
Durables share of total expenditure	α	0.20
Elasticity of substitution nondurable goods	ϵ_c	6
Elasticity of substitution durable goods	ϵ_d	6
Elasticity of substitution in labor	η	21
Preference parameters	ν, ν’	set to target N = N’ = 0.33
Loan-to-value ratio	m	0.85
Share of patient households	$\tilde{ψ}$	0.79
Government share of output	g_y	0.20

Table 4:

Calibrated parameters

Parameter		Value
Discount factor patient households	β	0.99
Discount factor impatient households	β’	0.97
Durables depreciation rate	δ	0.010
Durables share of total expenditure	α	0.20
Elasticity of substitution nondurable goods	ϵ_c	6
Elasticity of substitution durable goods	ϵ_d	6
Elasticity of substitution in labor	η	21
Preference parameters	ν, ν’	set to target N = N’ = 0.33
Loan-to-value ratio	m	0.85
Share of patient households	$\tilde{ψ}$	0.79
Government share of output	g_y	0.20

3.6.2 Estimation results

Table 5 also reports the posterior mean with 90% probability intervals in square brackets of the baseline and the housing DSGE models. The posterior means suggest that various frictions are supported by the data in both models.²⁵ Impatient households

Table 5:

Prior and posterior distributions of estimated parameters

(90% confidence bands in square brackets)

Table 5:

Prior and posterior distributions of estimated parameters

(90% confidence bands in square brackets)

Parameter			Prior		Posterior Mean	Posterior Mean
		Distr.	Mean	Sd/df	Baseline DSGE	Housing DSGE
Structural
Inv. Frisch elasticity patients	φ	N	0.50	0.10	0.5504 [0.4010;0.6986]	0.6448 [0.4933;0.7942]
Inv. Frisch elasticity impatients	φ′	N	0.50	0.10	0.6468 [0.4952;0.8028]	0.6860 [0.5300;0.8431]
Habits patients	ς	B	0.70	0.10	0.6505 [0.5979;0.7036]	0.6615 [0.6188;0.6965]
Habits. impatients	ς ‘	B	0.70	0.10	0.9336 [0.9240;0.9442]	0.9404 [0.9338;0.9465]
Habit persist. patients	^ρc	B	0.70	0.10	0.5068 [0.3964;0.6206]	0.6399 [0.5412;0.7436]
Habit persist. impatients	$ρ_{c}^{'}$	B	0.70	0.10	0.2195 [0.1564;0.2809]	0.3221 [0.2366;0.4142]
Price stickiness nondurables	ϑ_c	G	15.0	5.00	23.38 [15.82;30.61]	26.06 [18.56;33.99]
Price stickiness durables	ϑ_d	G	15.0	5.00	24.45 [16.09;33.26]	1.79 [1.13;2.43]
Wage stickiness	ϑ_W	G	100.0	10.00	152.39 [136.15;169.71]	168.06 [158.30;177.30]
IAC durables patients	ϕ	N	1.5	0.50	3.4738 [2.8002;4.1114]	3.7908 [3.2240;4.4022]
IAC durables impatients	ϕ’	N	1.5	0.50	1.9112 [1.2022;2.5902]	1.7710 [1.0228;2.4684]
Share of durables inflation	τ	B	0.20	0.10	0.1440 [0.0519;0.2299]	0.0516 [0.0367;0.0672]
Inflation -Taylor rule	ρ_π	N	1.50	0.20	1.4042 [1.2298;1.5702]	1.7285 [1.5062;1.9437]
Output -Taylor rule	ρ_y	G	0.10	0.05	0.0175 [0.0056;0.0291]	0.0221 [0.0059;0.0368]
Interest rate smoothing	ρ_r	B	0.80	0.10	0.7088 [0.6657;0.7545]	0.7681 [0.7314;0.8054]
Averages
Trend growth rate	γ	N	0.49	0.10	0.4017 [0.3678;0.4343]	0.4054 [0.3752;0.4356]
Inflation rate nondurables	${\bar{π}}_{C}$	G	1.05	0.10	1.0135 [0.9120;1.1146]	1.0427 [0.9348;1.1438]
Inflation rate durables	${\bar{π}}_{D}$	G	0.37	0.10	0.4324 [0.3200;0.5495]	0.3462 [0.2180;0.4767]
Interest rate	$\bar{r}$	G	1.65	0.10	1.6140 [1.4898;1.7467]	1.6404 [1.4980;1.7754]
Exogenous processes
Technology	ρ_eA	B	0.50	0.20	0.9775 [0.9574;0.9970]	0.9555 [0.9223;0.9903]
	σ_eA	IG	0.10	2.0	0.6933 [0.6196;0.7607]	0.7483 [0.6678;0.8308]
Monetary Policy	ρ_eR	B	0.50	0.20	0.1019 [0.0258;0.1774]	0.0777 [0.0173;0.1320]
	σ_eR	IG	0.10	2.0	0.2880 [0.2566;0.3180]	0.2803 [0.2503;0.3109]
Investment Durables	ρ_eI	B	0.50	0.20	0.4915 [0.3072;0.6710]	0.9205 [0.8872;0.9543]
	σ_eI	IG	0.10	2.0	6.1724 [4.0832;8.2543]	6.1915 [5.4443;6.9462]
Preference	ρ_eB	B	0.50	0.20	0.7465 [0.6805;0.8167]	0.8641 [0.8198;0.9103]
	σ_eB	IG	0.10	2.0	1.8512 [1.5433;2.1479]	2.1900 [1.7840;2.5739]
Price mark-up nondurables	ρ_eC	B	0.50	0.20	0.9053 [0.8544;0.9595]	0.9098 [0.8490;0.9728]
	θ_C	B	0.50	0.20	0.5643 [0.4506;0.6733]	0.6449 [0.5262;0.7698]
	σ_eC	IG	0.10	2.0	2.0840 [1.6533;2.4938]	2.2972 [1.7984;2.7678]
Price mark-up durables	ρ_eD	B	0.50	0.20	0.9869 [0.9768;0.9976]	0.9888 [0.9778;0.9994]
	θ_D	B	0.50	0.20	0.7037 [0.6025;0.8045]	0.8903 [0.8561;0.9265]
	σ_eD	IG	0.10	2.0	4.3290 [3.0803;5.5682]	24.451 [20.311;28.074]
Wage mark-up	ρ_eW	B	0.50	0.20	0.9481 [0.9210;0.9761]	0.9557 [0.9346;0.9791]
	θ_W	B	0.50	0.20	0.5544 [0.4511;0.6632]	0.5661 [0.4608;0.6780]
	σ_eW	IG	0.10	2.0	5.2236 [4.3012;6.1217]	5.5840 [4.6366;6.5235]
Government spending	ρ_eG	B	0.50	0.20	0.9658 [0.9458;0.9879]	0.9565 [0.9283;0.9857]
	σ_eG	IG	0.10	2.0	3.5099 [3.1807;3.8336]	3.3376 [3.0351;3.6504]

Table 5:

Prior and posterior distributions of estimated parameters

(90% confidence bands in square brackets)

Parameter			Prior		Posterior Mean	Posterior Mean
		Distr.	Mean	Sd/df	Baseline DSGE	Housing DSGE
Structural
Inv. Frisch elasticity patients	φ	N	0.50	0.10	0.5504 [0.4010;0.6986]	0.6448 [0.4933;0.7942]
Inv. Frisch elasticity impatients	φ′	N	0.50	0.10	0.6468 [0.4952;0.8028]	0.6860 [0.5300;0.8431]
Habits patients	ς	B	0.70	0.10	0.6505 [0.5979;0.7036]	0.6615 [0.6188;0.6965]
Habits. impatients	ς ‘	B	0.70	0.10	0.9336 [0.9240;0.9442]	0.9404 [0.9338;0.9465]
Habit persist. patients	^ρc	B	0.70	0.10	0.5068 [0.3964;0.6206]	0.6399 [0.5412;0.7436]
Habit persist. impatients	$ρ_{c}^{'}$	B	0.70	0.10	0.2195 [0.1564;0.2809]	0.3221 [0.2366;0.4142]
Price stickiness nondurables	ϑ_c	G	15.0	5.00	23.38 [15.82;30.61]	26.06 [18.56;33.99]
Price stickiness durables	ϑ_d	G	15.0	5.00	24.45 [16.09;33.26]	1.79 [1.13;2.43]
Wage stickiness	ϑ_W	G	100.0	10.00	152.39 [136.15;169.71]	168.06 [158.30;177.30]
IAC durables patients	ϕ	N	1.5	0.50	3.4738 [2.8002;4.1114]	3.7908 [3.2240;4.4022]
IAC durables impatients	ϕ’	N	1.5	0.50	1.9112 [1.2022;2.5902]	1.7710 [1.0228;2.4684]
Share of durables inflation	τ	B	0.20	0.10	0.1440 [0.0519;0.2299]	0.0516 [0.0367;0.0672]
Inflation -Taylor rule	ρ_π	N	1.50	0.20	1.4042 [1.2298;1.5702]	1.7285 [1.5062;1.9437]
Output -Taylor rule	ρ_y	G	0.10	0.05	0.0175 [0.0056;0.0291]	0.0221 [0.0059;0.0368]
Interest rate smoothing	ρ_r	B	0.80	0.10	0.7088 [0.6657;0.7545]	0.7681 [0.7314;0.8054]
Averages
Trend growth rate	γ	N	0.49	0.10	0.4017 [0.3678;0.4343]	0.4054 [0.3752;0.4356]
Inflation rate nondurables	${\bar{π}}_{C}$	G	1.05	0.10	1.0135 [0.9120;1.1146]	1.0427 [0.9348;1.1438]
Inflation rate durables	${\bar{π}}_{D}$	G	0.37	0.10	0.4324 [0.3200;0.5495]	0.3462 [0.2180;0.4767]
Interest rate	$\bar{r}$	G	1.65	0.10	1.6140 [1.4898;1.7467]	1.6404 [1.4980;1.7754]
Exogenous processes
Technology	ρ_eA	B	0.50	0.20	0.9775 [0.9574;0.9970]	0.9555 [0.9223;0.9903]
	σ_eA	IG	0.10	2.0	0.6933 [0.6196;0.7607]	0.7483 [0.6678;0.8308]
Monetary Policy	ρ_eR	B	0.50	0.20	0.1019 [0.0258;0.1774]	0.0777 [0.0173;0.1320]
	σ_eR	IG	0.10	2.0	0.2880 [0.2566;0.3180]	0.2803 [0.2503;0.3109]
Investment Durables	ρ_eI	B	0.50	0.20	0.4915 [0.3072;0.6710]	0.9205 [0.8872;0.9543]
	σ_eI	IG	0.10	2.0	6.1724 [4.0832;8.2543]	6.1915 [5.4443;6.9462]
Preference	ρ_eB	B	0.50	0.20	0.7465 [0.6805;0.8167]	0.8641 [0.8198;0.9103]
	σ_eB	IG	0.10	2.0	1.8512 [1.5433;2.1479]	2.1900 [1.7840;2.5739]
Price mark-up nondurables	ρ_eC	B	0.50	0.20	0.9053 [0.8544;0.9595]	0.9098 [0.8490;0.9728]
	θ_C	B	0.50	0.20	0.5643 [0.4506;0.6733]	0.6449 [0.5262;0.7698]
	σ_eC	IG	0.10	2.0	2.0840 [1.6533;2.4938]	2.2972 [1.7984;2.7678]
Price mark-up durables	ρ_eD	B	0.50	0.20	0.9869 [0.9768;0.9976]	0.9888 [0.9778;0.9994]
	θ_D	B	0.50	0.20	0.7037 [0.6025;0.8045]	0.8903 [0.8561;0.9265]
	σ_eD	IG	0.10	2.0	4.3290 [3.0803;5.5682]	24.451 [20.311;28.074]
Wage mark-up	ρ_eW	B	0.50	0.20	0.9481 [0.9210;0.9761]	0.9557 [0.9346;0.9791]
	θ_W	B	0.50	0.20	0.5544 [0.4511;0.6632]	0.5661 [0.4608;0.6780]
	σ_eW	IG	0.10	2.0	5.2236 [4.3012;6.1217]	5.5840 [4.6366;6.5235]
Government spending	ρ_eG	B	0.50	0.20	0.9658 [0.9458;0.9879]	0.9565 [0.9283;0.9857]
	σ_eG	IG	0.10	2.0	3.5099 [3.1807;3.8336]	3.3376 [3.0351;3.6504]

display a higher degree of habits in nondurables consumption (ς < ς’), as found by Iacoviello and Neri (2010) but with a lower persistence $(ρ_{c} > ρ_{c}^{'})$ . In addition, patient households face larger costs of adjusting their durables stock (ϕ > ϕ’). The posterior mean of the inverse Frisch elasticities of labor supply in both models are higher than the prior and are well identified in the data (as can be seen from comparing prior and posterior distributions in Figure G.1, Appendix G). Estimates of the Taylor rule parameters show a high degree of policy inertia, and a stronger response to inflation than to output, a likely consequence of estimating the model over a sample including the Great Moderation. Overall, estimates from both models are quite close to each other.

As regards price stickiness in the two sectors, when we employ the broad measure of durable goods (baseline DSGE) the posterior means are very similar - with confidence intervals almost entirely overlapping. The point estimates of durables and nondurables price stickiness (ϑ_d = 24.45, ϑ_c = 23.38) correspond to Calvo probabilities of resetting the price of 35.9% and 36.5% and an average price duration of 2.8 and 2.7 quarters respectively. Conversely, in the housing DSGE the posterior mean of house prices (ϑ_d = 1.79) is dramatically lower than that of nondurables (ϑ_c = 26.06) corresponding to Calvo probabilities of resetting the price of 78.1% and 35% and average price durations of 1.3 and 2.8 quarters respectively. In addition, confidence intervals never overlap. Here, the estimated degree of wage stickiness (ϑ^W = 168.06) guarantees that the comovement between consumption in the two sectors is still attained despite house prices being estimated to be quasi-flexible.²⁶

Figure 3 shows the prior and posterior distributions of the price stickiness parameters in both models. First, we notice that the data is informative as the posterior distributions are rather apart from the prior. In the baseline DSGE (left box), the two distributions almost entirely overlap thus pointing to a negligible difference in the price stickiness across the nondurables and durables sectors. Conversely, in the housing DSGE (right box) the posterior distribution of the housing price stickiness moves towards zero and in opposite direction with respect to the posterior distribution of nondurables prices. Such estimates highlight that when a broad measure of durables is employed, then prices display the same degree of stickiness with respect to non-durables whereas, if the durables sector coincides with the housing sector, then prices are estimated to reset almost every quarter.

This result that durables prices are as sticky as nondurables contrasts with Kim and Katayama (2013) who find that prices of durables are substantially more flexible than prices of nondurables, in a model with homogenous households and no role for durables as collateral, fewer shocks and different observables.²⁷ We try and be as close as possible to mainstream estimated models as far as shocks and observables are concerned, with the natural addition of observables related to durables consumption and durables inflation. Moreover, such results are closer to the latest microeconometric evidence. In particular, Nakamura and Steinsson (2008), Klenow and Malin (2010) and Petrella and Santoro (2012) use highly disaggregated data and find no decisive evidence that categories of nondurables are stickier than durables. In addition, Boivin et al. (2009) argue that inflation in sectors with high price stickiness display a high autocorrelation and low volatility. They estimate that durables inflation has higher autocorrelation and lower volatility than nondurables inflation hence it is possible to infer that prices of durables are stickier than nondurables.

3.6.3 Impulse response functions

In order to investigate the dynamic properties of the models, Figure 4 displays the estimated impulse responses of the variables of interest to a one standard-deviation increase in the nominal interest rate across the baseline and housing DSGE models.²⁸ As the estimated parameters are very similar across the two models, the mean responses do not show large differences. Taking into account the 68%, 90% and the 95% confidence bands (see Figures F.1 and F.2 in Appendix F) further highlights these similarities. An increase in the monetary policy rate leads to an output contraction and a decrease in overall and sectoral inflations. Furthermore, the presence of wage and price stickiness generates the desired comovement between durables and nondurables. The only noticeable difference between the two models concerns the response of the relative price, due to the different degree of estimated price stickiness. In the baseline model, prices of durables and nondurables are equally sticky hence the response of the relative price is flat whereas in the housing model, house prices are almost flexible and prices of nondurables are sticky hence the relative price falls in response to a monetary policy contraction. Figure 5 highlights the Bayesian impulse responses of the relative prices in the two models together with the 68%, 90% and the 95% confidence bands. The credible set of estimated impulse responses in the baseline model does not exclude zero at any of the confidence levels considered whereas it is significantly negative in the housing DSGE. Such dynamic properties of the models are consistent with the findings of the SVAR models estimated in Section 2 (see Figures 1 and 2) and represent the main novel contribution of our paper.²⁹

To sum up, we have estimated prices of durables (defined as the sum of durable goods and residential investments) to be as sticky as nondurables which is at odds with the assumption made in most two-sectors New-Keynesian models that they are fully flexible. We have then demonstrated that such assumption is consistent only with a narrow definition of durables sector that coincide with exclusively with residential investments.

3.7 Estimated sectoral price stickiness in extended models

The two-sector DSGE model estimated in the previous section builds mainly on Barsky et al. (2007) with the addition of several frictions and of the collateral constraint as in Monacelli (2009), Iacoviello and Neri (2010) and Sterk (2010). In this section we extend the model to account for two additional features affecting sectoral Phillips curves, namely imperfect sectoral labor mobility and price indexation. We re-estimate the DSGE model jointly with the additional parameters. Then, we further generalize our analysis and estimate a three-sector DSGE model in which housing and non-housing durables are treated separately and display heterogeneity in terms of rate of depreciation, adjustment costs and degree of substitutability with nondurable goods. Table 6 reports the estimated sectoral price stickiness across the extended models whereas the full set of estimated parameters is in Appendix I.

3.7.1 Imperfect sectoral labor mobility

Households in two-sector models are allowed to optimally choose the quantity of labor to supply in each sector according to their preferences. Standard two-sector models typically assume either that labor is perfectly mobile hence sectoral wages are equalized across the two sectors or assume no mobility at all.³⁰ However, more recent contributions have emphasized the importance of limited labor mobility in accounting for the behavior of the economy in multi-sector models. Indeed, Bouakez et al. (2009) argue that imperfect labor mobility affects the dispersion of hours across sectors whereas Bouakez et al. (2011) show that accounting for limited labor mobility jointly with inter-sectoral linkages solves the comovement puzzle. Moreover, Iacoviello and Neri (2010) find evidence of limited labor mobility across the consumption and housing sectors.³¹

In the context of our main two-sector model above, perfect labor mobility implies that the production structure and thus marginal costs are always the same across the two sectors. Therefore, it seems sensible to modify it to allow for limited labor mobility, and hence for different dynamics of wages and the marginal costs across the two sectors, and to check the robustness of our results as regards the estimation of the price stickiness parameters. Following the above mentioned contributions, limited labor mobility is introduced by specifying a constant elasticity of substitution (CES) aggregator between sectoral hours for each household:

\begin{matrix} N_{t} = {[(χ^{C})^{- \frac{1}{λ}} (N_{t}^{C})^{\frac{1 + λ}{λ}} + {(1 - χ^{C})}^{- \frac{1}{λ}} {(N_{t}^{D})}^{\frac{1 + λ}{λ}}]}^{\frac{λ}{1 + λ}}, & (39) \end{matrix}

where the intra-temporal elasticity of substitution λ ϵ (0, ∞) governs the degree of labor mobility.³² Note that λ → 0 denotes the case of labor immobility, while as λ → ∞ labor can be freely reallocated and all workers earn the same wage at the margin. For λ < ∞ the economy displays a limited degree of labor mobility and sectoral wages are not equal. Moreover, χ^C = N^C/N represents the steady-state share of labor supply in the nondurables sector.

We then replace the labor market clearing conditions (27) and (28) with the CES aggregators and bring the model to the data. Typically, limited labor mobility is calibrated at a value of λ =1 (see Bouakez et al. 2009, Petrella and Santoro 2011 and Petrella et al. 2017), except Bouakez et al. (2011) who explore values between 0.5 and 1.5 whereas Iacoviello and Neri (2010) estimate values of 1.51 and 1.03 for savers and borrowers, respectively.³³ Accordingly, we set the prior mean of the labor mobility parameters at the posterior estimates of Iacoviello and Neri (2010) and bring the model to the data.

The third and fourth columns of Table 6 report the estimated price stickiness in the baseline and housing DSGE models with imperfect sectoral labor mobility and show that the results of our main model (first and second columns of Table 6) continue to hold. In the baseline DSGE, price stickiness is similar across the two sectors with 90% confidence intervals widely overlapping. Conversely, in the housing DSGE prices of nondurables are significantly stickier than house prices, which are quasi-flexible. The top panel of Figure 6 plots the posterior distribution of the price stickiness parameters in the baseline and housing DSGE models with imperfect sectoral labor mobility. While in the baseline DSGE the two posterior distributions widely overlap, in the housing DSGE the posterior distributions are rather apart from each other thus implying a significant difference between the two sectoral price stickiness parameters. In addition, labor mobility in the housing DSGE is estimated to be somewhat lower than in the baseline DSGE (see Table I.1 Appendix I).³⁴

3.7.2 Price indexation

The price setting behavior of firms specified in equations (20) and (21) yield purely forward-looking sectoral Phillips curves. In this section we introduce a backward-looking component of the Phillips curves by estimating the degree of sectoral indexation to past inflation and verify that the results reported in Section 3.6.2 as regards price stickiness are not driven by the absence of indexation. Following Ireland (2007; 2011) and Ascari et al. (2011) we introduce indexation in the Rotemberg price adjustment cost specification, which now read as:

\begin{matrix} \frac{υ_{j}}{2} (\frac{P_{i, t}^{j}}{Π_{t - 1}^{j, ζ_{j}} P_{i, t - 1}^{j}} - 1)^{2} Y_{t}^{j}, & (40) \end{matrix}

where ς_j ϵ [0,1] determines the degree of indexation to past inflation and j = C, D.³⁵

When bringing the extended model to the data, we set the prior mean of the sectoral indexation parameters as in Smets and Wouters (2007) and Iacoviello and Neri (2010) at 0.50 and standard deviation 0.20. Table 6 (fifth and sixth columns) shows that the estimated sectoral price stickiness is very similar across sectors in the baseline DSGE, whereas in the housing DSGE nondurables prices are much stickier than house prices, which are virtually flexible. This confirms the results of the main model. Looking at the posterior distributions of the price stickiness parameters in the baseline and housing DSGE models with sectoral price indexation (middle panel of Figure 6) leads to the same inference as in the main model. The estimated degrees of price rigidity are not significantly different in the baseline DSGE whereas in the housing DSGE house prices are significantly more flexible than nondurable prices. Finally, we estimate a low degree of sectoral price indexation (see Table I.2, Appendix I).³⁶

3.7.3 Three-sector model

We have so far demonstrated that the definition of the durables sector plays a crucial role in the estimation of the sectoral price stickiness. In this section, we verify that our results continue to hold when we generalize the model to a three-sector economy producing nondurables, housing and non-housing durables, where only housing goods serve as collateral. Non-housing and housing durables display several sources of heterogeneity with respect to each other: (i) different depreciation rates, i.e. different degrees of durability; (ii) different adjustment costs in investment in these two goods; (iii) different degree of substitutability between housing and non-housing durables with nondurable goods. In particular, here parameter δ denotes the depreciation rate only of non-housing durables, while parameter δ^H ≠ δ denotes the depreciation of housing goods. Similarly, while parameters ϕ and ϕ’ refer to investment adjustment costs of non-housing durables, parameters ϕ^H and ϕ’^H refer to adjustment costs of housing goods. Finally, accounting for different degrees of substitutability between housing and non-housing durables with nondurable goods requires a generalization of the consumption aggregator, which we specify as a nested constant-elasticity-of-substitution (CES) function of the three goods as follows:

\begin{matrix} X_{t} = {[(1 - α) {\tilde{C}}_{t}^{\frac{ρ - 1}{ρ}} + α H_{t}^{\frac{ρ - 1}{ρ}}]}^{\frac{ρ - 1}{ρ}}, & (41) \end{matrix}

\begin{matrix} {\tilde{C}}_{t} = {[(1 - \tilde{α}) Z_{t}^{\frac{\tilde{ρ} - 1}{\tilde{ρ}}} + \tilde{α} D_{t}^{\frac{\tilde{ρ} - 1}{\tilde{ρ}}}]}^{\frac{\tilde{ρ}}{\tilde{ρ} - 1}}, & (42) \end{matrix}

where parameters $ρ \tilde{ρ} \in (0, \infty)$ represent the elasticities of substitution between non-housing (durables and nondurables) and housing goods and between nondurable and non-housing durable goods, respectively. The resulting degree of substitutability between non-durables and housing and between non-durables and non-housing durables is then a function of these two elasticities and is allowed to be different.³⁷

Housing goods are used as collateral by impatient households, hence the borrowing constraint (12) now reads as:

\begin{matrix} B_{t}^{'} \leq m E_{t} (\frac{Q_{t + 1}^{H} H_{t}^{'} Π_{t + 1}^{C}}{R_{t}}), & (43) \end{matrix}

with $Q_{t}^{H} \equiv \frac{P_{t}^{H}}{P_{t}^{C}}$ being the relative house price. Firms in the housing sector behave as firms in the nondurables and durables sector, as outlined in Section 3.2: they maximize profits and are subject to quadratic costs of adjusting prices, with three different price stickiness parameters: ϑ_c, ϑ_d, ϑ_h ϵ [0, ∞), denoting price stickiness in the non-durables, non-housing durables and housing durables sector, respectively. Finally, distinguishing between housing and non-housing investment requires making a distinction between (i) housing and non-housing investment-specific shocks; and (ii) housing and non-housing durables price markup shocks. These are assumed to follow AR(1) and ARMA(1,1) processes, respectively, in line with the analysis above. This means that the extended model features two more shocks relative to that in Section 3.

Consistently to this new structure, we bring the model to the data by distinguishing between residential and non-residential investment in durable goods, as well as inflation in the housing and non-housing durables sectors. Since now the observables of durables investment and inflation exclude housing goods, we need to add the following two measurement equations for residential investment and house price inflation respectively:

\begin{matrix} Δ I_{H, t}^{o} = γ + {\hat{I}}_{H, t}^{*} - {\hat{I}}_{H, t - 1}^{*}, & (44) \end{matrix}

\begin{matrix} Π_{H, t}^{o} = {\bar{π}}_{H} + {\hat{Π}}_{t}^{H} . & (45) \end{matrix}

In addition to the parameters calibrated in Table 4, we set the elasticity of substitution in the housing sector ϵ_h to 6 and the distributional parameters of the CES consumption aggregators $α, \tilde{α} \in [0, 1]$ to match the sectoral expenditure shares over the sample considered. The calibration of depreciation rates of the non-housing durables δ and housing goods δ^H deserves more attention. These parameters are crucial for the property of quasi-constancy of the shadow-value of long-lived goods, as demonstrated by Barsky et al. (2007) and Barsky et al. (2016). The literature has used a variety of values, ranging from a quarterly depreciation of 0.01 (see, among others, Monacelli, 2009; Sterk, 2010; Iacoviello and Neri, 2010; Chen and Liao, 2014), to 0.025 (see Erceg and Levin, 2006; Carlstrom and Fuerst, 2010; Petrella and Santoro, 2011; Sudo, 2012).³⁸ In accordance with the microeconometric evidence (see, e.g. Fraumeni, 1997) and the literature just mentioned, we assume that non-housing durables display a higher depreciation rate than housing goods. We thus calibrate δ = 0.025 and δ^H = 0.01. Price stickiness and investment adjustment costs parameters are estimated using the same priors as outlined in Section 3.6.1 whereas we set the prior mean of the consumption elasticities of substitution ρ and $\tilde{ρ}$ to 1 (which imply a nested Cobb-Douglas aggregator), and a standard deviation of 0.1.

Table 6:

Estimated price stickiness parameters in extended models

(90% confidence bands in square brackets)

Table 6:

Estimated price stickiness parameters in extended models

(90% confidence bands in square brackets)

	Main model		Imperfect Labor Mobility		Price Indexation		Three sector
	Baseline	Housing	Baseline	Housing	Baseline	Housing
ϑ_c	23.38	26.06	25.72	51.08	20.58	23.87	33.37
	[15.82;30.61]	[18.56;33.99]	[18.07;33.85]	[45.73;56.06]	[13.63;27.65]	[16.55;31.31]	[24.07;42.82]
ϑ_d	24.45	1.79	27.02	0.72	22.05	1.26	46.13
	[16.09;33.26]	[1.13;2.43]	[17.95;35.59]	[0.56;0.85]	[13.77;30.25]	[0.69;1.79]	[34.99;57.06]
ϑ_h							4.70
	^\	^\	^\	^\	^\	^\
							[2.34;7.09]

Table 6:

Estimated price stickiness parameters in extended models

(90% confidence bands in square brackets)

	Main model		Imperfect Labor Mobility		Price Indexation		Three sector
	Baseline	Housing	Baseline	Housing	Baseline	Housing
ϑ_c	23.38	26.06	25.72	51.08	20.58	23.87	33.37
	[15.82;30.61]	[18.56;33.99]	[18.07;33.85]	[45.73;56.06]	[13.63;27.65]	[16.55;31.31]	[24.07;42.82]
ϑ_d	24.45	1.79	27.02	0.72	22.05	1.26	46.13
	[16.09;33.26]	[1.13;2.43]	[17.95;35.59]	[0.56;0.85]	[13.77;30.25]	[0.69;1.79]	[34.99;57.06]
ϑ_h							4.70
	^\	^\	^\	^\	^\	^\
							[2.34;7.09]

It turns out that point estimates of investment adjustment cost parameters differ across sectors, and confidence intervals of both consumption elasticities of substitution do not exclude the Cobb-Douglas case (Tables I.3 and I.4, Appendix I). The last column of Table 6 reports the estimated price stickiness parameters across the three sectors. Although the distance between the point estimates of non-housing durables and nondurables price stickiness is larger than that existing between overall durables and nondurable (see two-sector model estimates, Table 6, column 1), their confidence intervals overlap at any conventional confidence level, as it can also be seen by inspecting the posterior distributions (bottom panel of Figure 6). This means that they are not significantly different from each other in a statistical sense. It is not surprising that, with respect to our main model, the posterior distribution of the non-housing durable price stickiness moves further to the right, as we have deducted housing from the relevant observables. Indeed, house prices robustly continue to be the most flexible component of durables prices also in the three-sector model. Confidence intervals of house price stickiness still never overlap with the other two, given that its posterior distribution moves rather apart from the others towards zero (see bottom panel of Figure 6).

4 Concluding remarks

Several papers engaged in building a two-sector New-Keynesian model able to generate the comovement between durable and nondurable goods following a monetary policy shock, as documented by the SVAR literature. This paper contributes to the existing literature by focusing on a less studied but equally important issue: the effects of a monetary policy innovation on the relative price of durables.

We show that, robustly across identifications and subsamples, in SVAR models the response of the relative price of durables to monetary policy shocks crucially depends on the definition of the durables sector. If durables include both non-housing durable goods and residential investment (as common in the literature), the relative price marginally increases or stays flat in response to a monetary policy contraction. Conversely, employing a narrow measure of durable goods that includes only new houses generates a fall in the relative price.

To rationalize the SVAR results, we build a rather canonical two-sector DSGE model in which impatient households borrow from patient households against the value of their durables collateral. We bring the model to the data using Bayesian methods employing, first, the broad definition of durables (non-housing durables and residential investments) and then the narrow measure of durables including only residential investment. Similarly to the most recent microeconomic evidence, we estimate the degree of price stickiness to be almost the same when non-housing durables are bundled with residential investment. It follows that the credible set of responses of the relative price of durables to a monetary policy shock includes zero. Conversely, durables -defined as including only residential investment- display a much lower stickiness than nondurables hence the credible set of responses of the relative price of durables to a monetary policy shock is significantly negative. Such results not only agree, but also rationalize our SVAR estimates. The results regarding the estimation of price stickiness parameters survive extensions of the DSGE model affecting sectoral Phillips curves and a three-sector generalization.

The importance of these findings is twofold. First, when building a two-sector New-Keynesian model it is desirable to assume that prices of durable goods are sticky, unless the aim is modeling the housing sector in isolation from other durables. When this is the case, the comovement puzzle is no longer an issue: if prices are sticky in both sectors, durables and nondurables will move in the same direction in response to monetary innovations. A three-sector model is needed to fully capture the intrinsic differences between housing and non-housing durables, such as the type of goods that can be used as collateral and their different degree of durability. Second, from a policy viewpoint, while the central bank is not likely to create big allocative distortions between the durables and nondurables sector, it may indeed create allocative distortions regarding the housing sector. Whether this has large welfare implications, and the optimal monetary policy design in this context, are beyond the scope of this paper, but these issues should certainly be investigated in future research.

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Appendix

A Data: sources and transformations

Table A.1:

Data Sources

Table A.1:

Data Sources

Series	Definition	Source	Mnemonic
DUR^N	Nominal Durable Goods	BEA	Table 2.3.5 Line 3
RI^N	Nominal Residential Investment	BEA	Table 1.1.5 Line 13
ND^N	Nominal Nondurable Goods	BEA	Table 2.3.5 Line 8
S^N	Nominal Services	BEA	Table 2.3.5 Line 13
P_DUR	Price Deflator, Durable Goods	BEA	Table 1.1.9 Line 4
P_RI	Price Deflator, Residential Investment	BEA	Table 1.1.9 Line 13
P_ND	Price Deflator, Nondurable Goods	BEA	Table 1.1.9 Line 5
P_S	Price Deflator, Services	BEA	Table 1.1.9 Line 6
Y_N	Nominal GDP	BEA	Table 1.1.5 Line 1
P_Y	Price Deflator, GDP	BEA	Table 1.1.9 Line 1
FFR	Effective Federal Funds Rate	FRED	FEDFUNDS
N	Nonfarm Business Sector: Average Weekly Hours	FRED	PRS85006023
W	Nonfarm Business Sector: Compensation Per Hour	FRED	COMPNFB
POP	Civilian Non-institutional Population, over 16	FRED	CNP16OV
CE	Civilian Employment, 16 over	FRED	CE16OV
NH^N	Nominal New-single family houses	BEA	Table 5.3.5 Line 23
P_NH	Price Deflator, New-single family houses	BEA	Table 5.3.4 Line 23
MH^N	Nominal Multifamily houses	BEA	Table 5.3.5 Line 24
^PMH	Price Deflator, Multifamily houses	BEA	Table 5.3.4 Line 23

Table A.1:

Data Sources

Series	Definition	Source	Mnemonic
DUR^N	Nominal Durable Goods	BEA	Table 2.3.5 Line 3
RI^N	Nominal Residential Investment	BEA	Table 1.1.5 Line 13
ND^N	Nominal Nondurable Goods	BEA	Table 2.3.5 Line 8
S^N	Nominal Services	BEA	Table 2.3.5 Line 13
P_DUR	Price Deflator, Durable Goods	BEA	Table 1.1.9 Line 4
P_RI	Price Deflator, Residential Investment	BEA	Table 1.1.9 Line 13
P_ND	Price Deflator, Nondurable Goods	BEA	Table 1.1.9 Line 5
P_S	Price Deflator, Services	BEA	Table 1.1.9 Line 6
Y_N	Nominal GDP	BEA	Table 1.1.5 Line 1
P_Y	Price Deflator, GDP	BEA	Table 1.1.9 Line 1
FFR	Effective Federal Funds Rate	FRED	FEDFUNDS
N	Nonfarm Business Sector: Average Weekly Hours	FRED	PRS85006023
W	Nonfarm Business Sector: Compensation Per Hour	FRED	COMPNFB
POP	Civilian Non-institutional Population, over 16	FRED	CNP16OV
CE	Civilian Employment, 16 over	FRED	CE16OV
NH^N	Nominal New-single family houses	BEA	Table 5.3.5 Line 23
P_NH	Price Deflator, New-single family houses	BEA	Table 5.3.4 Line 23
MH^N	Nominal Multifamily houses	BEA	Table 5.3.5 Line 24
^PMH	Price Deflator, Multifamily houses	BEA	Table 5.3.4 Line 23

A.1 Durables and Residential Investments

Sum nominal series: DU R^N + RI^N = DR^N
Calculate sectoral weights of deflators: $ω^{D} = \frac{D U R^{N}}{D R^{N}}; ω^{R I} = \frac{R I^{N}}{D R^{N}}$
Calculate Deflator: PD = ω^DP_DUR + ω^RIP_RI
Calculate Real Durable Consumption: $D = \frac{D U R^{N} + R I^{N}}{P_{D}}$

A.2 Nondurables and Services

Sum nominal series: ND^N + S^N = NS^N
Calculate sectoral weights of deflators: $ω^{N D} = \frac{N D^{N}}{N S^{N}}; ω^{S} = \frac{S^{N}}{N S^{N}}$
Calculate Deflator: P_C = ω^NDP_ND + ω^SP_S
Calculate Real Nondurable Consumption: $C = \frac{N D^{N} + S^{N}}{P_{C}}$

A.3 Only broad measure of houses

Sum nominal series: NH^N + MH^N = DR^N
Sectoral weights of deflators: $ω^{N H} = \frac{N H^{N}}{D R^{N}}; ω^{M H} = \frac{{M H}^{N}}{D R^{N}}$
Calculate Deflator: PD = ω^NHP_NH + ω^MHP_MH
Calculate Real Durable Consumption: $D = \frac{N H^{N} + {M H}^{N}}{P_{D}}$

A.4 Durable goods and New-single family houses

1. Sum nominal series: DUR^N + NH^N = DR^N

2. Calculate sectoral weights of deflators: $ω^{D} = \frac{D U R^{N}}{D R^{N}}; ω^{N H} = \frac{N H^{N}}{D R^{N}}$

3. Calculate Deflator: P_D = ω^DP_DUR + ω^NHP_NH

4. Calculate Real Durable Consumption: $D = \frac{D U R^{N} + N H^{N}}{P_{D}}$

A.5 Durable goods and broad measure of houses

1. Sum nominal series: DUR^N + NH^N + MH^N = DR^N

2. Sectoral weights of deflators: $ω^{D} = \frac{D U R^{N}}{D R^{N}}; ω^{N H} = \frac{N H^{N}}{D R^{N}}; ω^{M H} = \frac{M H^{N}}{D R^{N}}$

3. Calculate Deflator: P_D = ω^DP_DUR + ω^NHP_NH + ω^MHP_MH

4. Calculate Real Durable Consumption: $D = \frac{D U R^{N} + N H^{N} + M H^{N}}{P_{D}}$

A.6 Data transformation for Bayesian estimation

Table A.2:

Data transformation - Observables

Table A.2:

Data transformation - Observables

Variable	Description	Construction
POP_index	Population index	$\frac{P O P}{P O P_{2009 : 1}}$
CE_index	Employment index	$\frac{C E}{{C E}_{2009 : 1}}$
Y°	Real per capita GDP	$\ln (\frac{\frac{Y^{N}}{P_{Y}}}{P O P_{i n d e x}}) 100$
$I_{D}^{o}$	Real per capita investment: durables	$\ln (\frac{I_{D}}{P O P_{i n d e x}}) 100$
$I_{H}^{o}$	Real per capita investment: houses	$\ln (\frac{I_{H}}{P O P_{i n d e x}}) 100$
C°	Real per capita consumption: nondurables	$\ln (\frac{C}{P O P_{i n d e x}}) 100$
W°	Real wage	$\ln (\frac{W}{P_{Y}}) 100$
N °	Hours worked per capita	$\ln (\frac{H \times C E_{i n d e x}}{{P O P}_{i n d e x}}) 100$
$Π_{C}^{o}$	Inflation: nondurables sector	Δ (In P_C) 100
$Π_{D}^{o}$	Inflation: durables sector	Δ (In P_D) 100
$Π_{H}^{o}$	Inflation: housing sector	Δ (In P_H ) 100
R°	Quarterly Federal Funds Rate	$\frac{F F R}{4}$

Table A.2:

Data transformation - Observables

Variable	Description	Construction
POP_index	Population index	$\frac{P O P}{P O P_{2009 : 1}}$
CE_index	Employment index	$\frac{C E}{{C E}_{2009 : 1}}$
Y°	Real per capita GDP	$\ln (\frac{\frac{Y^{N}}{P_{Y}}}{P O P_{i n d e x}}) 100$
$I_{D}^{o}$	Real per capita investment: durables	$\ln (\frac{I_{D}}{P O P_{i n d e x}}) 100$
$I_{H}^{o}$	Real per capita investment: houses	$\ln (\frac{I_{H}}{P O P_{i n d e x}}) 100$
C°	Real per capita consumption: nondurables	$\ln (\frac{C}{P O P_{i n d e x}}) 100$
W°	Real wage	$\ln (\frac{W}{P_{Y}}) 100$
N °	Hours worked per capita	$\ln (\frac{H \times C E_{i n d e x}}{{P O P}_{i n d e x}}) 100$
$Π_{C}^{o}$	Inflation: nondurables sector	Δ (In P_C) 100
$Π_{D}^{o}$	Inflation: durables sector	Δ (In P_D) 100
$Π_{H}^{o}$	Inflation: housing sector	Δ (In P_H ) 100
R°	Quarterly Federal Funds Rate	$\frac{F F R}{4}$

B SVAR methodologies

B.1 Recursive approach

Let Σ_ε be the variance-covariance matrix of the reduced-form shocks of the SVAR model. Under the recursive approach, the structural shocks are identified through a Cholesky decomposition of Σ_ε. Consequently, the order of the variables in vector x_t matters for the identification of the monetary disturbance. Indeed, at time t one variable is affected by the previous but not from those which follow. In our estimation, we make the standard assumption that the monetary policy variable is ordered last hence it has no contemporaneous effect on the other variables (see Bernanke and Mihov, 1998, among others). Our SVAR model includes a vector of constant terms and four lags, as commonly assumed in the literature for a monetary SVAR with quarterly frequency.

B.2 Sign restrictions approach

The second approach we employ is the pure sign restrictions proposed by Uhlig (2005). This method implies that shocks are identified when they follow specific and unique patterns by imposing restrictions on the impulse response functions (IRFs) of the SVAR model. Several orthogonal matrices linking the reduced-form and the structural shocks are drawn, where we retain those generating impulse responses that satisfy the set of restrictions while discarding the others.³⁹ We employ the model-based methodology outlined by Canova (2002) and applied in Dedola and Neri (2007), Pappa (2009) and Bermperoglu et al. (2013), among others, according to which the restrictions are extracted from a theoretical model. We can summarize the procedure in three main steps:

Build a nested DSGE model in which nominal and real frictions can be removed via appropriate parametrizations. We do this in ^{Section 3}, where our two-sector model encompasses a continuum of models featuring different subsets of frictions.
Define ranges for the structural parameters, generate thousands of random draws of the parameter values from their support and obtain IRFs for each draw.⁴⁰
Use the robust IRFs to impose sign restrictions on the IRFs of the SVAR model.

B.3 Narrative approach

The third approach we employ is based on the contribution of Romer and Romer (2004). RR develop a new measure of U.S. monetary policy shock that is somewhat immune to two problems embedded in monetary policy variables such as the actual FFR. Indeed, RR argue that such measures suffer from endogeneity and anticipatory movements. In particular, the former implies that the FFR moves with changes in economic conditions hence not with changes in the conduct of monetary policy. The latter implies that movements in the FFR represent responses to information about future events in the economy. As a result, RR argue that such measures of monetary policy do not really represent exogenous shocks and they derive a new measure that enables the researcher to overcome these shortcomings.

The derivation of the alternative monetary policy variable consists of two main steps. RR first derive a series of intended FFR changes around meetings of the Federal Open Market Committee (FOMC) of the Federal Reserve (Fed). They rely on a combination of narrative and quantitative evidence in order to retrieve the direction and the magnitude of such intended changes. This step eliminates the endogeneity between the interest rate and economic conditions thus solving the first of the two shortcomings outlined above. The second step consists of controlling for the Fed’s internal forecasts in order to disentangle the effects of information about future economic developments. RR then regress the change in the intended FFR on its level, on the level and the changes of forecasts about GDP growth and the GDP deflator, and forecasts about the unemployment rate. Then they take the residuals of this regression as the new measure of monetary policy shocks. Consequently, the resulting series gains a higher degree of exogeneity with respect to the FFR since it represents movements in the monetary policy measure not stemming from forecasts about inflation, GDP growth and unemployment.