## 2.1  Methodology

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### Research question and motivation
- Whether monetary policy innovations create distortions in allocations across durable and nondurable goods depends on the extent to which such shocks change their relative price.
- Prior theoretical work cited: Erceg and Levin (2006); Aoki (2001).
- Empirical comovement documented in Bernanke and Gertler (1995); Monacelli (2009); Sterk and Tenreyro (2014); Di Pace and Hertweck (2016); Barsky et al. (2003); but Barsky, House and Kurlat (BHK) note a comovement puzzle where a two-sector New-Keynesian model fails to replicate observed comovement.

### Key assumptions under investigation
- Crucial assumption: sectoral price stickiness.
  - Common assumption (BHK and follow-on): durables prices completely flexible; nondurables prices sticky.
  - Justifications: durables (e.g., houses) negotiated, microeconometric evidence (Bils and Klenow (2004)) shows durables more flexible.
  - Counter-evidence: Nakamura and Steinsson (2008); Boivin et al. (2009); Klenow and Malin (2010); Petrella and Santoro (2012) document stickiness in many durables categories other than houses.
- Investment in housing represents about 23% of aggregate durables in US NIPA in the post-war period.
- Implication: if durables flexible and nondurables sticky, a monetary tightening lowers the relative price of durables and creates sectoral allocation distortions.

### Empirical strategy (overview)
- Models employed:
  - Structural Vector-Autoregressive (SVAR) identified via recursive, sign restrictions, and narrative approaches.
  - Dynamic Stochastic General Equilibrium (DSGE) two-sector New-Keynesian model with extensions: imperfect sectoral labor mobility, price indexation, three-sector generalization.
- Data and definitions:
  - Durables sector definitions: baselineSVAR = durable goods consumption + residential investment; housingSVAR = houses only.
  - Relative price measures: Relative Price of Durables; Relative House Price; alternative definitions for robustness.
  - Frequency: quarterly. Sample: 1969Q2-2007Q4.
  - Variables: GDP, durables, houses and nondurables are first differences in log real per-capita variables; inflation is first difference in log GDP deflator; relative prices are first differences of ratios of price indices.
  - SVAR variable vector: x_t ≡[GDP_t, D_t, C_t, P_t, Q_t, FFR_t]′ (FFR in levels; other variables in natural logs).

### Identification of monetary policy shocks
- Three approaches for robustness:
  i) Recursive (Cholesky) approach: monetary policy variable ordered last (Bernanke and Mihov, 1998).
  ii) Sign restrictions approach:
    - Sign restrictions derived from DSGE; proceed as in Peersman (2005) to first determine supply and demand shocks then identify monetary shock.
    - Imposed sign patterns (Table 3):
      - Supply shock: GDP <0; D <0; C <0; P >0; Q none; FFR >0.
      - Demand shock: GDP <0; D <0; C <0; P <0; Q none; FFR <0.
      - Monetary policy shock: GDP <0; D none; C <0; P <0; Q none; FFR >0.
    - Report median with 16th and 84th percentiles for confidence bands.
    - Additional assumption: nominal interest rate positive in first quarter to identify monetary shock; remain agnostic on relative price and durables consumption responses.
  iii) Recursive narrative approach:
    - Replace FFR with Romer and Romer (2004, RR) monetary policy shock (extended by Coibion et al. (2012) and Tenreyro and Thwaites (2016)).
- Proxy SVAR note: RR measure can be used directly or as an external instrument; Proxy SVAR robustness reported in Appendix C.5 yields same relative price sign pattern.

### Empirical diagnostics (sample correlations)
- Unconditional correlations (sample: 1969Q2-2007Q4) between lags of changes in FFR and changes in macro variables:
  - FFR (-1):
    - GDP: 0.0801
    - Durables (Rel. Price Durables): -0.3282*
    - Houses (Rel. Price Houses): -0.2534*
    - Nondurables: -0.3020*
    - Inflation: 0.1675*
    - Rel. Price Durables–Houses: 0.0804-0.0985
  - FFR (-4):
    - GDP: -0.1806*
    - Durables (Rel. Price Durables): -0.2865*
    - Houses (Rel. Price Houses): -0.3081*
    - Nondurables: -0.2411*
    - Inflation: 0.2230*
    - Rel. Price Durables–Houses: 0.1110-0.0049
  - FFR (-8):
    - GDP: -0.1810*
    - Durables (Rel. Price Durables): -0.0727
    - Houses (Rel. Price Houses): -0.0903-
    - Nondurables: -0.08030.13920.0438-0.0497
  - FFR (-12):
    - GDP: 0.0318
    - Durables (Rel. Price Durables): -0.0533
    - Houses (Rel. Price Houses): 0.11980.0634-0.0070
    - Nondurables: 0.0599-0.0596
- Note: * denotes significance at a 5 percent level.

### SVAR and DSGE complementarity — main methodological findings
- SVAR results (across subsamples and identification methods):
  - Response of relative price of durables is either flat or mildly positive; it never falls except when narrowly defined as house price relative to nondurables — consistent with flexible house prices.
- DSGE estimation:
  - Two-sector NK model estimated with durables used by credit-constrained impatient households as collateral.
  - Bayesian estimation finds price stickiness in the sector comprising all durable goods (housing and non-housing) is not significantly different from nondurables — credible set of impulse responses of the relative price to a monetary shock includes zero.
  - When durables comprise only housing, house prices estimated nearly flexible while nondurables substantially stickier; monetary tightening then affects relative house price.

### Data construction and transformations (exact procedures preserved)
- Data sources and series mnemonics listed (BEA tables and FRED series).
- Construction procedures for relative price deflators and real series (A.1–A.5) detailed: e.g., DUR_N + RI_N = DR_N; ω_D = DUR_N / DR_N; P_D = ω_D P_DUR + ω_RI P_RI; D = (DUR_N + RI_N) / P_D.
- Bayesian estimation observables transformations (A.6) include:
  - Y_o: ln( (Y_N / P_Y) / POP_index ) × 100
  - ∆I^o_{D,t}, ∆C^o_t, ∆W^o_t definitions preserve γ and log-deviations; R_o = FFR / 4.

### Selected calibration and priors (exact values)
- β = 0.99; β′ = 0.97.
- δ = 0.010 (annual depreciation 4%).
- α = 0.20.
- ϵ_c = 6; ϵ_d = 6.
- η = 21.
- Target N = N′ = 0.33 via ν, ν′.
- m = 0.85.
- ˜ψ = 0.79.
- g_y = 0.20.

### Key posterior estimates (selected posterior means and intervals)
- Inv. Frisch elasticity patients φ:
  - Baseline DSGE 0.5504 [0.4010;0.6986]; Housing DSGE 0.6448 [0.4933;0.7942].
- Habits patients ζ:
  - Baseline 0.6505 [0.5979;0.7036]; Housing 0.6615 [0.6188;0.6965].
- Price stickiness nondurables θ_c:
  - Baseline 23.38 [15.82;30.61]; Housing 26.06 [18.56;33.99].
- Price stickiness durables θ_d:
  - Baseline 24.45 [16.09;33.26]; Housing 1.79 [1.13;2.43].
- Wage stickiness θ_W:
  - Baseline 152.39 [136.15;169.71]; Housing 168.06 [158.30;177.30].
- Share of durables inflation τ:
  - Baseline 0.1440 [0.0519;0.2299]; Housing 0.0516 [0.0367;0.0672].
- Monetary policy parameters:
  - ρ_π: Baseline 1.4042 [1.2298;1.5702]; Housing 1.7285 [1.5062;1.9437].
  - ρ_y: Baseline 0.0175 [0.0056;0.0291]; Housing 0.0221 [0.0059;0.0368].
  - ρ_r: Baseline 0.7088 [0.6657;0.7545]; Housing 0.7681 [0.7314;0.8054].

### Interpretation of estimation results
- Frictions supported by data: habits, price and wage stickiness, investment adjustment costs.
- Price stickiness across sectors:
  - Baseline DSGE: θ_d = 24.45 and θ_c = 23.38 — very similar; Calvo reset probabilities correspond to 35.9% (durables) and 36.5% (nondurables); average price durations 2.8 and 2.7 quarters.
  - Housing DSGE: θ_d = 1.79 (houses) vs θ_c = 26.06 (nondurables) — house prices much more flexible; Calvo resetting: 78.1% (houses) vs 35% (nondurables); average price durations 1.3 and 2.8 quarters.
- Wage stickiness role: high θ_W helps generate comovement between durables and nondurables even when house prices quasi-flexible.
- Heterogeneous households: impatient households have higher habits ζ′ and lower persistence; patient households face larger durables adjustment costs.
- Monetary policy: stronger response to inflation than output; substantial policy inertia consistent with Great Moderation sample.

### Impulse response dynamics (to one standard-deviation increase in nominal interest rate)
- General: tightening leads to output contraction and decreases in overall and sectoral inflation; wage and price stickiness produce comovement between durables and nondurables.
- Model differences:
  - Baseline DSGE (similar sectoral stickiness): relative price response approximately flat.
  - Housing DSGE (quasi-flexible house prices): relative price falls; credible set significantly negative in housing DSGE but includes zero in baseline DSGE.
- Volatility: durables more volatile than nondurables and output, consistent with SVAR.

### Robustness and model extensions
- Extensions estimated: imperfect sectoral labor mobility; price indexation; three-sector model (nondurables, non-housing durables, housing durables).
- Imperfect labor mobility:
  - Labor mobility parameter λ introduced via CES aggregator; prior mean set to Iacoviello and Neri (2010) estimates.
  - Baseline DSGE with imperfect mobility: price stickiness similar across sectors.
  - Housing DSGE with imperfect mobility: nondurable prices significantly stickier than house prices.
- Price indexation:
  - Sectoral indexation parameters ς_j estimated (priors mean 0.50, sd 0.20); estimated degree of indexation low.
  - Price stickiness results mirror main inference.
- Three-sector model:
  - Distinct depreciation rates δ = 0.025 (non-housing durables) and δ_H = 0.01 (housing).
  - Nested CES consumption aggregator with ρ, ̃ρ prior mean 1.
  - Findings: house prices remain most flexible; non-housing durables and nondurables price stickiness estimates larger but confidence intervals overlap; house price stickiness confidence intervals do not overlap with others.

### Robust IRFs and simulation methodology
- Robust-IRF procedure: draw θ vectors from parameter ranges; impose restriction θ_c ≥ θ_d; performed 22000 draws; 10034 accepted.
  - 92% of discarded draws violated θ_c ≥ θ_d; 7% violated Blanchard-Kahn conditions.
- Robustness criterion: signs of 84th and 16th percentiles in impact period must agree.
- Simulation findings:
  - With wage stickiness unrestricted, on impact output, nondurable consumption, durable consumption and inflation exhibit robust negative responses; relative price bounded below zero by construction.
  - With fully-flexible wages, wage rigidities crucial to obtain comovement; flexible wages yield parameter combinations where durable consumption increases after tightening.
- Parameter ranges (selected): β ∈ [0.985,0.995]; β′ ∈ [0.96,0.984]; δ ∈ [0.0025,0.025]; α ∈ [0.05,0.35]; ϵ_c,ϵ_d ∈ [4,11]; η ∈ [4,25]; φ,φ′ ∈ [0.3,3]; θ_c,θ_d ∈ [0,58]; θ_W ∈ [0,180]; m ∈ [0.55,0.95]; ˜ψ ∈ [0.60,0.90]; ρ_π ∈ [1.05,5]; ρ_y ∈ [0,0.5]; ρ_R ∈ [0,0.9].

### Model comparison and the importance of frictions
- Likelihood race (Table H.1) — Log-marg. likelihoods and Kass-Raftery (KR):
  - Baseline: −1472.494
  - Flexible Wages θW = 0: −1672.300; KR 399.612
  - Flexible Durables Prices θd = 0: −1538.150; KR 131.312
  - No IAC φ=φ′=0: −1970.003; KR 995.018
  - No Habit ζ=ζ′=0: −1698.053; KR 451.118
  - No Durables Inflation τ=0: −1473.396; KR 1.804
- Interpretation: decisive evidence in favor of baseline over models removing key frictions (θW=0, θd=0, φ=φ′=0, ζ=ζ′=0); slight evidence favoring baseline over τ=0.
- Restricted-model price-stickiness estimates (Table H.2):
  - Baseline: θc = 23.38 [15.82;30.61]; θd = 24.45 [16.09;33.26].
  - Flexible Wages θW = 0: θc = 1.2032 [0.5643;1.7338]; θd = 2.4006 [1.4801;3.3098].
  - No IAC φ=φ′=0: θc = 47.135 [32.832;62.022]; θd = 51.378 [37.533;65.994].

### Policy and modeling implications
- Modeling guidance:
  - Two-sector New-Keynesian models should assume durable goods prices are somewhat sticky unless model focuses exclusively on housing.
  - A three-sector model preferred to capture differences between housing and non-housing durables (collateral use, durability, depreciation, investment adjustment costs).
- Policy conclusion:
  - Overall monetary policy innovations do not foster large distortions in allocations between durables and nondurables when durables defined broadly.
  - Monetary policy can affect the relative house price and thus potentially create allocative distortions between housing and non-housing goods.

_Italic: Source: wp17290 - 2.1 Methodology (page 11) and related sections from the provided PDF content._

### 2.1  Methodology  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   11

### 2.1  Methodology

### Research question and motivation
- Whether monetary policy innovations create distortions in allocations across durable and nondurable goods depends on the extent to which such shocks change their relative price.
- Prior theoretical work (Erceg and Levin (2006); Aoki (2001)) shows the relative price of durables affects both the user cost and demand of durable goods and therefore the conduct of monetary policy.
- Empirically, comovement between durables and nondurables in response to monetary policy has been documented (Bernanke and Gertler, 1995; Monacelli, 2009; Sterk and Tenreyro, 2014; Di Pace and Hertweck, 2016; Barsky et al., 2003), but Barsky, House and Kurlat (BHK) note a comovement puzzle: a two-sector New-Keynesian model fails to replicate the observed comovement.

### Key assumptions under investigation
- The crucial assumption concerns sectoral price stickiness:
  - BHK and many follow-on papers assume durables prices are completely flexible and nondurables prices are sticky.
  - Justifications: durable prices (e.g., houses) are often negotiated and priced only at sale; microeconometric studies (Bils and Klenow (2004)) document durables are more flexible than nondurables.
  - Counter-evidence: Nakamura and Steinsson (2008), Boivin et al. (2009), Klenow and Malin (2010), Petrella and Santoro (2012) document 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.
- Implication: If durables prices are flexible while nondurables are sticky, a monetary policy tightening implies the relative price of durables falls and monetary policy creates sectoral allocation distortions.

### Empirical strategy (methodological overview)
- Models employed:
  - Structural Vector-Autoregressive (SVAR) models identified through recursive, sign restrictions, and narrative approaches.
  - Dynamic Stochastic General Equilibrium (DSGE) two-sector New-Keynesian model with extensions (imperfect sectoral labor mobility, price indexation, three-sector generalization).
- Data choices:
  - Durables sector defined as the sum of durable goods and residential investment; relative price measures include both the relative price of durables and the relative price of houses.
  - Frequency: quarterly. Sample: 1969Q2-2007Q4.
  - Variables: 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 (further data details in the Appendix).

### Empirical diagnostics (Table 1 correlations)
- Unconditional correlations between lags of changes in the Federal funds rate (FFR) and changes in selected macroeconomic variables (sample: 1969Q2-2007Q4):
  - FFR (-1):
    - GDP: 0.0801
    - Durables (Rel. Price Durables): -0.3282*
    - Houses (Rel. Price Houses): -0.2534*
    - Nondurables: -0.3020*
    - Inflation: 0.1675*
    - Rel. Price Durables–Houses: 0.0804-0.0985
  - FFR (-4):
    - GDP: -0.1806*
    - Durables (Rel. Price Durables): -0.2865*
    - Houses (Rel. Price Houses): -0.3081*
    - Nondurables: -0.2411*
    - Inflation: 0.2230*
    - Rel. Price Durables–Houses: 0.1110-0.0049
  - FFR (-8):
    - GDP: -0.1810*
    - Durables (Rel. Price Durables): -0.0727
    - Houses (Rel. Price Houses): -0.0903-
    - Nondurables: -0.08030.13920.0438-0.0497
  - FFR (-12):
    - GDP: 0.0318
    - Durables (Rel. Price Durables): -0.0533
    - Houses (Rel. Price Houses): 0.11980.0634-0.0070
    - Nondurables: 0.0599-0.0596
- Note: * denotes significance at a 5 percent level.

### Main methodological findings (SVAR and DSGE complementarity)
- SVAR results (across subsamples and identification methods):
  - 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 in relative price is found only when the relative price is narrowly defined as the ratio between house prices and nondurables prices — consistent with flexible house prices.
- DSGE estimation:
  - A two-sector NK model is estimated where durable goods are used by credit-constrained impatient households as collateral to borrow from patient households.
  - Bayesian estimation finds the degree of price stickiness in the sector comprising all durable goods (housing and non-housing) is not significantly different from the nondurables sector — the credible set of impulse responses of the relative price to a monetary policy shock includes zero.
  - When durables comprise only housing, house prices are estimated to be almost flexible while nondurables prices are substantially stickier; only then does a monetary policy tightening affect the relative price of durables (the relative house price).

### Robustness and model extensions
- Estimated price stickiness results persist when:
  - Allowing for imperfect labor mobility across sectors.
  - Introducing sectoral price indexation to past inflation.
  - Extending the model to a three-sector framework.
- Relation to other literature:
  - Results on price stickiness broadly align with Bouakez et al. (2009) who estimate stickiness in a six-sector model; key difference is modeling of durability and collateral use.
  - Differences from Iacoviello and Neri (2010): they estimate a model where durables comprise only housing and assume house prices fully flexible a priori; this paper estimates price stickiness parameters across housing and non-housing durables.

### Policy and modeling implications
- Modeling:
  - Two-sector New-Keynesian models should assume that prices of durable goods are somewhat sticky unless the model focuses exclusively on housing.
  - A three-sector model is preferable to capture intrinsic differences between housing and non-housing durables (collateral use, degree of durability).
- Policy:
  - Overall monetary policy innovations do not foster large distortions in allocations between durables and nondurables.
  - Monetary policy can affect the relative house price and thus potentially create allocative distortions between housing and non-housing goods.

_Italic: Source: wp17290 - 2.1  Methodology (page 11) from the provided PDF content._

### 2.1  Methodology

### 2.1  Methodology

### Data and variables
- Quarterly, seasonally adjusted US data are used for:
  - Federal funds rate
  - Real GDP
  - Real durable goods
  - Real nondurable goods and services
  - The GDP deflator
  - The relative price of durables
- Sample used in the main analysis: 1969Q2-2007Q4.
- Vector of variables employed in the SVAR:
  - x_t ≡[GDP_t, D_t, C_t, P_t, Q_t, FFR_t]′
    - 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 relative price of durable goods
    - FFR_t denotes the Federal funds rate
- The natural logarithm is taken of all variables except for the FFR, which is in levels.

### Definitions of the durables sector and relative prices
- Two alternative definitions of durables are employed:
  - baselineSVAR: durables = durable goods consumption + residential investment (follows Erceg and Levin (2006), Monacelli (2009), Sterk and Tenreyro (2014), Di Pace and Hertweck (2016))
  - housingSVAR: durables = houses only
- Table 2 (definitions used throughout the paper):
  - 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.
- Definitions I and II are used in the main analysis; III and IV serve for robustness checks.
- Algebraic details for computation of all relative prices are reported in Appendix A.

### Identification of monetary policy shocks — three approaches
- For robustness, three identification approaches are taken:
  i) Recursive (Cholesky) approach
    - Standard assumption: monetary policy variable is ordered last and has no contemporaneous effect on other variables (see Bernanke and Mihov, 1998).
  ii) Sign restrictions approach
    - Sign restrictions are imposed on impulse responses of variables and derived from a DSGE model (following Canova (2002), Dedola and Neri (2007), Pappa (2009), Bermperoglu et al. (2013)).
    - To mitigate identification ambiguity, proceed as in Peersman (2005): first determine sign pattern of two standard supply and demand shocks, then identify the monetary policy shock.
    - Set of sign restrictions imposed (Table 3):
      - Supply shock:
        - GDP: <0
        - D: <0
        - C: <0
        - P: >0
        - Q: none
        - FFR: >0
      - Demand shock:
        - GDP: <0
        - D: <0
        - C: <0
        - P: <0
        - Q: none
        - FFR: <0
      - Monetary Policy shock:
        - GDP: <0
        - D: none
        - C: <0
        - P: <0
        - Q: none
        - FFR: >0
    - Interpretation:
      - Contractionary supply shock: curbs output, nondurable and durable consumption, increases inflation, central bank raises nominal interest rate.
      - Negative demand shock: reduces real variables and inflation, central bank cuts interest rate.
      - Monetary policy shock: increase in nominal interest rate leads to decrease in output, nondurable consumption and inflation.
    - Additional assumption: despite lack of robust response, assume the nominal interest rate is positive in the first quarter to correctly identify the monetary policy shock. Remain agnostic on response of relative price and consumption of durables.
    - Confidence bands: in the sign restrictions approach, report the median together with the 16th and the 84th percentiles.
  iii) Recursive narrative approach
    - Re-estimate the recursive SVAR replacing FFR with the monetary policy shock constructed by Romer and Romer (2004, RR) and extended by Coibion et al. (2012) and Tenreyro and Thwaites (2016).
- Proxy SVAR discussion
  - In macro-fiscal literature, narrative measures have been used as external instruments (Proxy SVAR).
  - The RR monetary policy measure is the result of a first-stage regression yielding a direct measure of the structural shock rather than a proxy; hence it is considered reasonable to use it directly in the VAR.
  - For robustness, a Proxy SVAR using the RR measure as external instrument is reported in Appendix C.5; the relative prices exhibit the same sign pattern as in main analysis.
  - Note: Proxy SVAR does not impose timing restrictions; hence impact responses are not zero by construction as in recursive identification.

### Treatment of uncertainties and confidence bands
- One-standard-deviation confidence bands in recursive approaches are computed by Monte Carlo methods based on 2000 draws.
- In the sign restrictions approach, the distribution of impulse responses is constructed and the 16th and 84th percentiles are used to form comparable confidence bands.

*Source: wp17290 - 2.1  Methodology (PDF chapter).*

### 3.2  Firms

### 3.2 Firms

### Pricing and production
- Firms face quadratic costs of changing prices as in Rotemberg (1982): θ_j/2 (P^j_{ω,t}/P^j_{ω,t−1} − 1)^2 Y^j_t, where θ_j is the parameter of sectoral price stickiness.
- Each firm produces differentiated goods with constant returns to scale:
  - Y^j_{ω,t} = e^{A_t} (N^j_{ω,t})^{˜ψ} (N′^j_{ω,t})^{1−˜ψ}, where ω ∈ [0,1], j = C, D index firms and sectors, ˜ψ ∈ [0,1] denotes the share of the patient household and e^{A_t} is a labor-augmenting shock.
- Firms maximize the present discounted value of profits:
  - E_t { Σ_{t=0}^∞ Λ_{t,t+1} [ P^j_{ω,t}/P^j_t Y^j_{ω,t} − W_{ω,t}/P^j_t N^j_{ω,t} − W′_{ω,t}/P^j_t N′^j_{ω,t} − θ_j/2 (P^j_{ω,t}/P^j_{ω,t−1} − 1)^2 Y^j_t ] } subject to production and Dixit-Stiglitz demand: Y^j_{ω,t} = (P^j_{ω,t}/P^j_t)^{−e^j_t} ϵ^j Y^j_t.
- Symmetric equilibrium price-setting equations:
  - (1−e^C_t ϵ_c) + e^C_t ϵ_c MC^C_t = θ_c (Π^C_{t−1})/Π^C_t − θ_c E_t[ Λ_{t,t+1} Y^C_{t+1}/Y^C_t (Π^C_{t+1}−1)/Π^C_{t+1} ].
  - (1−e^D_t ϵ_d) + e^D_t ϵ_d MC^D_t = θ_d (Π^D_{t−1})/Π^D_t − θ_d E_t[ Λ_{t,t+1} Q_{t+1}/Q_t Y^D_{t+1}/Y^D_t (Π^D_{t+1}−1)/Π^D_{t+1} ].
- If θ_j = 0 prices are flexible and set as constant markups over marginal costs.

### Fiscal and monetary policy
- Government budget is balanced every period via lump-sum tax; government spending G_t is exogenous and purchases only nondurable goods and services.
- Monetary policy follows a Taylor rule:
  - log(R_t/R) = ρ_r log(R_{t−1}/R) + (1−ρ_r) [ ρ_π log(˜Π_t/˜Π) + ρ_y log(Y_t/Y) ] + e^R_t,
    where ˜Π_t ≡ (Π^C_t)^{1−τ} (Π^D_t)^τ and τ ∈ [0,1] is durables' weight.
- Parameters: ρ_r is interest rate smoothing; ρ_π and ρ_y are responses to inflation aggregator and output deviations; e^R_t is monetary policy innovation.

### Market clearing conditions and exogenous processes
- Aggregate and sectoral identities:
  - Y_t = Y^C_t + Q_t Y^D_t + θ_W/2 (Π^W_t − Π^C)^2 w_t N_t + θ_W/2 (Π^{W′}_t − Π^C)^2 w′_t N′_t.
  - Y^C_t = C_t + C′_t + G_t + θ_c/2 (Π^C_t − Π^C)^2 Y^C_t.
  - Y^D_t = [D_t − (1−δ) D_{t−1}] + [D′_t − (1−δ) D′_{t−1}] + θ_d/2 (Π^D_t − Π^D)^2 Y^D_t.
  - 0 = B_t + B′_t.
  - N_t = N^C_t + N^D_t; N′_t = N′^C_t + N′^D_t.
- Wage markup and price markup shocks follow ARMA(1,1):
  - log(κ_t/ ̄κ) = ρ_κ log(κ_{t−1}/ ̄κ) + ϵ^κ_t − θ_i ϵ^κ_{t−1}, with κ = [e^W, e^C, e^D], i = [W,C,D].
- All other shocks follow AR(1):
  - log(κ_t/ ̄κ) = ρ_κ log(κ_{t−1}/ ̄κ) + ϵ^κ_t, with κ = [e^B, e^I, e^R, e^A, e^G]. Shocks ϵ^κ_t are i.i.d. with standard deviations σ_κ.

### Functional forms
- Utility: U(X_t, N_t) = log(X_t) − ν N_t^{1+φ}/(1+φ), where ν is a scaling parameter and φ is inverse Frisch elasticity.
- Durables investment adjustment cost:
  - S(I^D_t / I^D_{t−1}) = φ/2 (I^D_t / I^D_{t−1} − 1)^2, with φ > 0.
- Same functional forms for impatient households; preference and investment adjustment cost parameters are household-specific.

### Bayesian estimation: sample, observables, measurement
- Estimation method: Bayesian with Kalman filter likelihood and MCMC-MH (two parallel chains of 150,000 draws each).
- Sample: 1969Q2-2007Q4.
- Eight observables (US data): GDP, investment in durable goods, consumption of nondurable goods, real wage, hours worked, inflation in nondurables sector, inflation in durables sector, nominal interest rate.
- Two model variants:
  - baseline DSGE: durables = durable goods + residential investment.
  - housing DSGE: durables comprise only houses.
- Measurement equations (variables with ˆ are log-deviations; ∗ denotes aggregation across households):
  - ∆Y^o_t = γ + ˆY_t − ˆY_{t−1}.
  - ∆I^o_{D,t} = γ + ˆI^*_D,t − ˆI^*_D,t−1.
  - ∆C^o_t = γ + ˆC^*_t − ˆC^*_{t−1}.
  - ∆W^o_t = γ + ˆW^*_t − ˆW^*_{t−1}.
  - N^o_t = ˆN^*_t.
  - Π^o_{C,t} = ̄π_C + ˆΠ^C_t.
  - Π^o_{D,t} = ̄π_D + ˆΠ^D_t.
  - R^o_t = ̄r + ˆR_t.
- γ is common quarterly trend growth rate for GDP, durables investment, nondurables consumption, real wage; ̄π_C and ̄π_D are average quarterly inflation in nondurable and durable sectors; ̄r is average quarterly Federal funds rate.

### Calibration and priors (selected calibrated values)
- Discount factor patient households β = 0.99.
- Discount factor impatient households β′ = 0.97.
- Durables depreciation rate δ = 0.010 (annual depreciation 4%).
- 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 (implying 5% steady-state gross wage mark-up).
- Preference parameters ν, ν′ set to target N = N′ = 0.33.
- Loan-to-value ratio m = 0.85.
- Share of patient households ˜ψ = 0.79.
- Government share of output g_y = 0.20.

### Estimation results — posterior means and key parameter estimates (selected)
- Structural and behavioral parameters (posterior means; 90% intervals shown in Table 5):
  - Inv. Frisch elasticity patients φ: Baseline DSGE 0.5504 [0.4010;0.6986]; Housing DSGE 0.6448 [0.4933;0.7942].
  - Inv. Frisch elasticity impatients φ′: Baseline 0.6468 [0.4952;0.8028]; Housing 0.6860 [0.5300;0.8431].
  - Habits patients ζ: Baseline 0.6505 [0.5979;0.7036]; Housing 0.6615 [0.6188;0.6965].
  - Habits impatients ζ′: Baseline 0.9336 [0.9240;0.9442]; Housing 0.9404 [0.9338;0.9465].
  - Habit persist. patients ρ_c: Baseline 0.5068 [0.3964;0.6206]; Housing 0.6399 [0.5412;0.7436].
  - Habit persist. impatients ρ′_c: Baseline 0.2195 [0.1564;0.2809]; Housing 0.3221 [0.2366;0.4142].
  - Price stickiness nondurables θ_c: Baseline 23.38 [15.82;30.61]; Housing 26.06 [18.56;33.99].
  - Price stickiness durables θ_d: Baseline 24.45 [16.09;33.26]; Housing 1.79 [1.13;2.43].
  - Wage stickiness θ_W: Baseline 152.39 [136.15;169.71]; Housing 168.06 [158.30;177.30].
  - IAC durables patients φ (investment adjustment cost): Baseline 3.4738 [2.8002;4.1114]; Housing 3.7908 [3.2240;4.4022].
  - IAC durables impatients φ′: Baseline 1.9112 [1.2022;2.5902]; Housing 1.7710 [1.0228;2.4684].
  - Share of durables inflation τ: Baseline 0.1440 [0.0519;0.2299]; Housing 0.0516 [0.0367;0.0672].
  - Inflation-Taylor rule ρ_π: Baseline 1.4042 [1.2298;1.5702]; Housing 1.7285 [1.5062;1.9437].
  - Output-Taylor rule ρ_y: Baseline 0.0175 [0.0056;0.0291]; Housing 0.0221 [0.0059;0.0368].
  - Interest rate smoothing ρ_r: Baseline 0.7088 [0.6657;0.7545]; Housing 0.7681 [0.7314;0.8054].
- Exogenous process estimates (selected):
  - Technology persistence ρ_{e_A}: Baseline 0.9775 [0.9574;0.9970]; Housing 0.9555 [0.9223;0.9903].
  - σ_{e_A}: Baseline 0.6933 [0.6196;0.7607]; Housing 0.7483 [0.6678;0.8308].
  - Investment durables persistence ρ_{e_I}: Baseline 0.4915 [0.3072;0.6710]; Housing 0.9205 [0.8872;0.9543].
  - σ_{e_I}: Baseline 6.1724 [4.0832;8.2543]; Housing 6.1915 [5.4443;6.9462].
  - Price mark-up durables persistence ρ_{e_D}: Baseline 0.9869 [0.9768;0.9976]; Housing 0.9888 [0.9778;0.9994].
  - σ_{e_D}: Baseline 4.3290 [3.0803;5.5682]; Housing 24.451 [20.311;28.074].

### Key empirical findings and interpretation
- Frictions supported by the data: posterior means indicate the presence of various frictions (habits, price and wage stickiness, investment adjustment costs).
- Price stickiness across sectors:
  - Baseline DSGE (durables broad): posterior means θ_d = 24.45 and θ_c = 23.38 — very similar with almost entirely overlapping confidence intervals.
    - These point estimates correspond to Calvo reset probabilities of 35.9% (durables) and 36.5% (nondurables) and average price durations of 2.8 and 2.7 quarters respectively.
  - Housing DSGE (durables = houses): posterior mean θ_d = 1.79 for house prices vs θ_c = 26.06 for nondurables — dramatically lower for houses; confidence intervals do not overlap.
    - Corresponds to Calvo probabilities of resetting: 78.1% (houses) vs 35% (nondurables) and average price durations of 1.3 and 2.8 quarters respectively.
- Wage stickiness importance:
  - High estimated wage stickiness (θ_W = 168.06 in housing DSGE) helps ensure comovement between consumption in durables and nondurables even when house prices are quasi-flexible.
- Heterogeneous households:
  - Impatient households display higher habits in nondurables consumption (ζ < ζ′) with lower persistence (ρ_c > ρ′_c).
  - Patient households face larger durables adjustment costs (φ > φ′).
- Monetary policy estimates:
  - Stronger response to inflation than to output; high policy inertia (ρ_r substantial), consistent with sample spanning the Great Moderation.

### Impulse response dynamics (to a one standard-deviation increase in the nominal interest rate)
- General dynamics (baseline and housing DSGE mean responses):
  - Monetary policy tightening leads to output contraction and decreases in overall and sectoral inflation.
  - Wage and price stickiness generate comovement between durables and nondurables.
- Difference across models:
  - Baseline DSGE: similar price stickiness ⇒ relative price response is approximately flat.
  - Housing DSGE: quasi-flexible house prices and sticky nondurables prices ⇒ relative price falls following a monetary contraction; the credible set for the relative-price response is significantly negative in housing DSGE but does not exclude zero in baseline DSGE.
- Volatility ranking:
  - Durables are estimated to be more volatile than nondurables and output, consistent with SVAR results.

### Summary conclusion (from the estimates)
- When durables are broadly defined (durable goods plus residential investment), durables prices are estimated to be as sticky as nondurables — at odds with common two-sector NK model assumptions that durables prices are fully flexible.
- If durables are narrowly defined as housing alone, house prices are estimated to be nearly flexible, producing materially different dynamics for relative prices.
- The data therefore support two possibilities depending on the durables definition: (i) durables price stickiness similar to nondurables (broad durables), or (ii) quasi-flexible house prices (housing-only durables), with wage stickiness playing a key role in generating cross-sector comovement.

*Source: wp17290 - 3.2 Firms (chapter/section from the provided PDF content).*

### 3.7  Estimated sectoral price stickiness in extended models

### 3.7  Estimated sectoral price stickiness in extended models

### 3.7.1  Imperfect sectoral labor mobility — setup and priors
- Limited labor mobility introduced via a CES aggregator between sectoral hours with intra-temporal elasticity of substitution λ ∈ (0,∞) governing labor mobility:
  - λ → 0 denotes labor immobility; λ → ∞ denotes perfect mobility.
- Literature calibration and estimates referenced:
  - Typical calibration: λ = 1.
  - Bouakez et al. (2011) explore λ between 0.5 and 1.5.
  - Iacoviello and Neri (2010) estimates map to λ = 1/0.66 = 1.51 and λ = 1/0.97 = 1.03 for savers and borrowers respectively.
- In estimation here:
  - Prior mean of labor mobility parameters set at the posterior estimates of Iacoviello and Neri (2010).
- Empirical result summary:
  - In the baseline DSGE with imperfect labor mobility, estimated price stickiness is similar across nondurables and durables with 90% confidence intervals widely overlapping.
  - In the housing DSGE with imperfect labor mobility, nondurable prices are significantly stickier than house prices; house prices are quasi-flexible.
  - Labor mobility is estimated to be somewhat lower in the housing DSGE than in the baseline DSGE.
  - Note: confidence bands of estimated elasticities do not overlap for patient households, but they overlap for impatient households.

### 3.7.2  Price indexation — specification, priors, and results
- Indexation introduced in the Rotemberg price adjustment cost specification; sectoral indexation parameter ς_j ∈ [0,1] determines degree of indexation to past inflation (j = C, D).
- Priors used when estimating indexation:
  - Prior mean of sectoral indexation parameters: 0.50.
  - Prior standard deviation: 0.20.
- Estimated outcomes:
  - Estimated sectoral price stickiness remains very similar across sectors in the baseline DSGE.
  - In the housing DSGE, nondurable prices are much stickier than house prices, which are virtually flexible.
  - Posterior distributions of price stickiness mirror the main-model inference: no significant sectoral difference in baseline DSGE; significant difference in housing DSGE with house prices more flexible.
  - Estimated degree of sectoral price indexation is low.

### 3.7.3  Three-sector model — structure, calibration, and findings
- Three sectors: nondurables, non-housing durables, housing durables. Only housing goods serve as collateral for impatient households.
- Key heterogeneities across housing and non-housing durables:
  - Different depreciation rates: δ denotes depreciation rate of non-housing durables; δ_H ≠ δ denotes depreciation of housing goods.
  - Different investment adjustment costs: φ and φ′ for non-housing durables; φ_H and φ′_H for housing goods.
  - Different degrees of substitutability with nondurable goods via a nested CES consumption aggregator:
    - X_t = [ (1−α) C̃_t^(ρ−1)/ρ + α H_t^(ρ−1)/ρ ]^(ρ/(ρ−1))
    - C̃_t = [ (1− ̃α) Z_t^(̃ρ−1)/̃ρ + ̃α D_t^(̃ρ−1)/̃ρ ]^(̃ρ/(̃ρ−1))
    - ρ, ̃ρ ∈ (0,∞) are elasticities of substitution.
- Collateral and borrowing constraint:
  - B′_t ≤ m E_t( Q_H,t+1 H′_t / (Π_C,t+1 R_t) ), with Q_H,t ≡ P_H,t / P_C,t.
- Shocks and observables:
  - Two additional shocks relative to Section 3: housing vs non-housing investment-specific shocks and housing vs non-housing durables price markup shocks.
  - Measurement equations added for residential investment and house price inflation:
    - ∆I^o_H,t = γ + Î^*_H,t − Î^*_H,t−1
    - Π^o_H,t = ̄π_H + ̂Π_H,t
- Calibration choices and priors:
  - Elasticity of substitution in the housing sector ϵ_h set to 6.
  - Consumption distributional parameters α, ̃α ∈ [0,1] set to match sectoral expenditure shares.
  - Depreciation calibrations (in line with microeconometric evidence and literature):
    - δ = 0.025 for non-housing durables.
    - δ_H = 0.01 for housing goods.
  - Consumption elasticities of substitution ρ and ̃ρ prior mean set to 1 (implying nested Cobb-Douglas); prior standard deviation 0.1.
  - Price stickiness and investment adjustment cost parameters estimated using priors from Section 3.6.1.
- Empirical findings in the three-sector model:
  - Point estimates of investment adjustment cost parameters differ across sectors.
  - Confidence intervals for consumption elasticities of substitution do not exclude the Cobb-Douglas case.
  - House prices remain the most flexible component of durables prices; house price stickiness posterior moves toward zero relative to the others.
  - Although the distance between point estimates of non-housing durables and nondurables price stickiness is larger than in two-sector estimates, their confidence intervals overlap at conventional levels — they are not significantly different statistically.
  - Confidence intervals of house price stickiness do not overlap with those of the other two sectors.

### Table 6 (as reported in the source)
- Table 6 reports estimated price stickiness parameters in extended models and 90% confidence bands (square brackets). The full numerical table and the full set of estimated parameters are reported in Appendix I of the source.

### Implications highlighted in the section
- Definition of the durables sector crucially affects estimated sectoral price stickiness:
  - When non-housing durables are bundled with residential investment, price stickiness estimates across durables and nondurables are similar.
  - When durables are narrowly defined to include only residential investment, house prices are much less sticky than nondurables.
- Robustness: The main results on sectoral price stickiness survive extensions that introduce imperfect labor mobility, price indexation, and the three-sector generalization.

*Source: IMF Working Paper — Section 3.7 "Estimated sectoral price stickiness in extended models" (figures, equations, and table references as in the provided content).*

### 1253. jae.2490.

### wp17290 - 1253. jae.2490.

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### Data: sources, series, and transformation procedures
- Table A.1: Data Sources (series definitions and mnemonics as in source):
  - 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
  - FF: 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
  - P_MH: Price Deflator, Multifamily houses — BEA Table 5.3.4 Line 23

- Construction procedures (exact formulas and steps preserved):
  - A.1 Durables and Residential Investments
    1. Sum nominal series: DUR_N + RI_N = DR_N
    2. Calculate sectoral weights of deflators: ω_D = DUR_N / DR_N ; ω_RI = RI_N / DR_N
    3. Calculate Deflator: P_D = ω_D P_DUR + ω_RI P_RI
    4. Calculate Real Durable Consumption: D = (DUR_N + RI_N) / P_D
  - A.2 Nondurables and Services
    1. Sum nominal series: ND_N + S_N = NS_N
    2. Calculate sectoral weights of deflators: ω_ND = ND_N / NS_N ; ω_S = S_N / NS_N
    3. Calculate Deflator: P_C = ω_ND P_ND + ω_S P_S
    4. Calculate Real Nondurable Consumption: C = (ND_N + S_N) / P_C
  - A.3 Only broad measure of houses
    1. Sum nominal series: NH_N + MH_N = DR_N
    2. Sectoral weights of deflators: ω_NH = NH_N / DR_N ; ω_MH = MH_N / DR_N
    3. Calculate Deflator: P_D = ω_NH P_NH + ω_MH P_MH
    4. Calculate Real Durable Consumption: D = (NH_N + MH_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 = DUR_N / DR_N ; ω_NH = NH_N / DR_N
    3. Calculate Deflator: P_D = ω_D P_DUR + ω_NH P_NH
    4. Calculate Real Durable Consumption: D = (DUR_N + NH_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 = DUR_N / DR_N ; ω_NH = NH_N / DR_N ; ω_MH = MH_N / DR_N
    3. Calculate Deflator: P_D = ω_D P_DUR + ω_NH P_NH + ω_MH P_MH
    4. Calculate Real Durable Consumption: D = (DUR_N + NH_N + MH_N) / P_D

- A.6 Data transformation for Bayesian estimation (observables and exact constructions):
  - POP_index: Population index — POP / POP_2009:1
  - CE_index: Employment index — CE / CE_2009:1
  - Y_o: Real per capita GDP — ln( (Y_N / P_Y) / POP_index ) × 100
  - I_o^D: Real per capita investment: durables — ln( I_D / P / POP_index ) × 100
  - I_o^H: Real per capita investment: houses — ln( I_H / P / POP_index ) × 100
  - C_o: Real per capita consumption: nondurables — ln( C / P / POP_index ) × 100
  - W_o: Real wage — ln( W / P_Y ) × 100
  - N_o: Hours worked per capita — ln( (H × CE_index) / POP_index ) × 100
  - Π_o^C: Inflation: nondurables sector — ∆ (ln P_C) × 100
  - Π_o^D: Inflation: durables sector — ∆ (ln P_D) × 100
  - Π_o^H: Inflation: housing sector — ∆ (ln P_H) × 100
  - R_o: Quarterly Federal Funds Rate — FFR / 4

### SVAR methodologies and identification strategies
- B.1 Recursive approach (Cholesky identification)
  - Structural shocks identified via Cholesky decomposition of Σ_ε (the variance-covariance matrix of the reduced-form shocks).
  - Order of variables in vector x_t matters: at time t one variable is affected by previous but not by those which follow.
  - Standard assumption used: the monetary policy variable is ordered last so it has no contemporaneous effect on other variables (see Bernanke and Mihov, 1998).
  - Model includes a vector of constant terms and four lags (quarterly frequency).

- B.2 Sign restrictions approach (pure sign restrictions per Uhlig (2005))
  - Identification by imposing restrictions on impulse response functions (IRFs) and retaining orthogonal matrices that generate IRFs satisfying those restrictions.
  - Methodology follows Canova (2002) and applications in Dedola and Neri (2007), Pappa (2009), Bermperoglu et al. (2013).
  - Procedure summarized in three steps:
    1. Build a nested DSGE model in which nominal and real frictions can be removed via appropriate parametrizations (two-sector model in Section 3 encompasses a continuum of models).
    2. Define ranges for the structural parameters, generate thousands of random draws of the parameter values from their support and obtain IRFs for each draw.
    3. Draw several orthogonal matrices linking reduced-form and structural shocks; retain those matrices whose IRFs satisfy the imposed sign restrictions and discard others.
  - Note: the process is repeated until 500 draws are accepted. See section E of the Appendix for details on choice of ranges and IRF dynamics (as referenced in source).

*Source: wp17290 - 1253. jae.2490. (IMF Working Paper PDF content provided)*

### 3. Use the robust IRFs to impose sign restrictions on the IRFs of the SVAR model.

### 3. Use the robust IRFs to impose sign restrictions on the IRFs of the SVAR model.

### B.3 Narrative approach
- Objective: construct an alternative measure of U.S. monetary policy shocks that is less affected by endogeneity and anticipatory movements inherent in actual FFR series (Romer and Romer, 2004).
- Two-step derivation:
  - Step 1: derive a series of intended FFR changes around FOMC meetings using narrative and quantitative evidence to retrieve direction and magnitude of intended changes — eliminates endogeneity between the interest rate and economic conditions.
  - Step 2: control for the Fed’s internal forecasts to remove effects of information about future economic developments. Specifically, regress the change in the intended FFR on:
    - its level,
    - the level and the changes of forecasts about GDP growth and the GDP deflator,
    - forecasts about the unemployment rate.
  - Take residuals of this regression as the new monetary policy shock measure — these residuals represent movements not stemming from forecasts about inflation, GDP growth and unemployment, thus gaining a higher degree of exogeneity.

### C Robustness checks for the SVAR model
- SVAR variations and robustness checks presented:
  - C.1 SVAR Models with trend: comparison of baseline model without trend (bold lines) and with trend (dashed lines); one-standard-deviation confidence bands shown.
  - C.2 Alternative definitions of durables: impulse responses for samples 1969Q2-2007Q4 with variants (durable goods and new single family houses; durable goods and broad measure of houses; new single family houses).
  - C.3 Subsample analysis: impulse responses across multiple subsamples:
    - Sample: 1969Q2-1993Q1
    - Sample: 1971Q4-1995Q3
    - Sample: 1974Q2-1998Q1
    - Sample: 1976Q4-2000Q3
    - Sample: 1979Q2-2003Q1
    - Sample: 1981Q4-2005Q3
    - Sample: 1984Q2-2007Q4
    - In each case, bold lines = all durable goods; dashed lines = only houses; shaded areas and dotted lines = one-standard-deviation confidence bands.
  - C.4 Sign restrictions: sign restrictions imposed for 2, 4 and 6 quarters against 1 quarter; Sample: 1969Q2-2007Q4; comparisons made for baseline model and model with broad measure of houses as durables.
  - C.5 Proxy SVAR approach: impulse responses to a one standard deviation increase in the monetary policy measure; Sample: 1969Q2-2007Q4; models with all durable goods vs only houses compared.
  - C.6 Three-sector SVAR model: SVAR impulse responses in a three-sector model; Sample: 1969Q2-2007Q4.

### D The DSGE models — structure and symmetric equilibrium
- Model components and definitions (selected highlights preserving notation and equations as in source):
  - Patient households (D.1.1): key equations include
    - X_t = Z_t^{1−α} D_t^{α} (D.1)
    - Z_t = C_t − ζ S_{t−1} (D.2)
    - U(X_t,N_t) = log(X_t) − ν N_t^{1+φ}/(1+φ) (D.4)
    - Λ_{t,t+1} ≡ β (U_{Z,t+1}/U_{Z,t}) e^{B_{t+1}}/e^{B_t} (D.8)
    - 1 = ψ_t e^{I_t}[1 − S(I_{D,t}/I_{D,t−1}) − S′(I_{D,t}/I_{D,t−1})(I_{D,t}/I_{D,t−1})] + E_t{Λ_{t,t+1} ψ_{t+1} Q_{t+1}/Q_t e^{I_{t+1}}[S′(...) (...)^2]} (D.12)
    - 1 = E_t[Λ_{t,t+1} R_t / Π_{C,t+1}] (D.15)
  - Impatient households (D.1.2), Firms (D.1.3), Monetary policy and market clearing (D.1.4): full system of equilibrium conditions listed (e.g., (D.16)-(D.46), (D.31)-(D.41)).
- Steady state (D.2):
  - In steady state x_t = x_{t+1} = x and stochastic shocks absent.
  - Given parameters and steady-state conditions, variables solved recursively. Selected steady-state relations:
    - Λ = β (D.47)
    - R = 1/β (D.48)
    - ψ = 1 (D.49)
    - μ = η/(η−1) (D.50)
    - MC_C = ϵ_c^{−1}/ϵ_c (D.53)
    - MC_D = ϵ_d^{−1}/ϵ_d (D.54)
    - N = N′ = 0.33 (steady state labor shares); ν and ν′ are endogenized to match N,N′ ∈ [0.2,0.5] as specified elsewhere.
  - Detailed steady-state expressions for D, D′, B′, Y_D, wages, consumptions, and other aggregates provided ((D.55)-(D.82) etc.).
- Extensions (D.3):
  - D.3.1 Imperfect sectoral labor mobility: replace (D.45)-(D.46) with CES aggregators (D.83)-(D.84); labor supply schedules (D.85)-(D.86); sectoral wages replaced by (D.87)-(D.90).
  - D.3.2 Price indexation: amend sectoral price setting equations to include indexation (D.91)-(D.92); market clearing updated (D.93)-(D.94).
  - D.3.3 Three-sector model: full set of three-sector equilibrium conditions and definitions ((D.95)-(D.154)), including household and firm conditions, investment laws of motion ((D.102)-(D.103)), aggregate definitions (D.104)-(D.111), and extended price/wage/market clearing relations.

### E Robust impulse responses — methodology and simulation results
- Parameter sampling methodology:
  - Let θ be an N×1 vector of structural parameters; each parameter i is uniformly distributed over range Θ_i; Θ = ∏_i Θ_i.
  - Ranges set around quarterly-calibrated U.S. economy values; lengths chosen to include reasonable values and avoid indeterminacy.
  - Restriction imposed: θ_c ≥ θ_d (price stickiness in nondurables at least as large as in durables). This allows θ_c = θ_d = 0 (fully flexible prices) in draws.
  - Main simulations: randomly draw θ_m^i, i=1,...,N; m=1,...,10034 from each Θ_i (m is number of random draws).
  - Draw outcomes:
    - Performed 22000 draws; 10034 were accepted.
    - 92% of discarded draws violated the price stickiness restriction.
    - 7% of discarded draws did not satisfy the Blanchard-Kahn conditions (indeterminacy).
  - Robustness criterion for impulse responses:
    - For each accepted draw construct K×1 vector h(y_t(θ^m|u_t)) of impulse responses to structural shocks u_t.
    - Function h_K(y_t(θ|u_t)) is considered robust if in the impact period the signs of the 84th and 16th percentiles are the same:
      - sign[h_K^U(y_t(θ|u_t))] = sign[h_K^L(y_t(θ|u_t))], where h_U and h_L are the 84th and 16th percentiles respectively.
- Simulation evidence (Figure E.1 summary, exact labels preserved):
  - Figure E.1 plots 68% probability bands of impulse responses to a 1% increase in the nominal interest rate for two sets of simulations:
    - First set: wage stickiness parameter unrestricted (blue dashed lines).
    - Second set: wages fully flexible (red dotted lines).
  - Main empirical findings from these simulations:
    - In the unrestricted-wage simulations (blue dashed lines), on impact:
      - output, nondurable consumption, durable consumption, and inflation exhibit robust negative responses.
      - model frictions (wage and price rigidities) help solve the comovement puzzle across many parameter combinations.
      - by construction the response of the relative price of durables is bounded below zero.
      - the nominal interest rate response is not robust and can be counter-intuitive in some draws (a recognized issue in two-sector NK models).
    - In the fully-flexible wages simulations (red dotted lines):
      - wage rigidities play a crucial role in solving the comovement puzzle.
      - with flexible wages there exist parameter combinations where durable consumption increases following a monetary policy tightening.
      - fewer parameter combinations generate comovement between durables and nondurables when wages are flexible.
  - The counter-intuitive nominal interest rate response is explained by near constancy of the shadow value of durables making their real return constant and thus forcing the nominal interest rate to track expected inflation in the durable goods sector (as in BHK and Sterk (2010)).
- Calibration and simulation details (selected numerical and range information preserved from Table E.1):
  - Parameter ranges (Table E.1):
    - Patient households’ discount factor β ∈ [0.985,0.995]
    - Impatient households’ discount factor β′ ∈ [0.96,0.984]
    - Durables depreciation rate δ ∈ [0.0025,0.025]
    - Durables share of total expenditure α ∈ [0.05,0.35]
    - Elasticity of substitution in nondurables ϵ_c ∈ [4,11]
    - Elasticity of substitution in durables ϵ_d ∈ [4,11]
    - Elasticity of substitution in labor η ∈ [4,25]
    - Inverse Frisch elasticities φ, φ′ ∈ [0.3,3]
    - Disutilities of labor ν, ν′: N,N′ ∈ [0.2,0.5]
    - Habits degree parameters ζ, ζ′ ∈ [0,0.9]
    - Habits persistence parameters ρ_c, ρ′_c ∈ [0,0.9]
    - Price stickiness in nondurables θ_c ∈ [0,58] (*)
    - Price stickiness in durables θ_d ∈ [0,58] (*)
    - Nominal wage rigidities θ_W ∈ [0,180]
    - Investment adjustment cost parameters φ, φ′ ∈ [0,5]
    - Loan-to-value ratio m ∈ [0.55,0.95]
    - Share of patient households ˜ψ ∈ [0.60,0.90]
    - Share of durables inflation in inflation aggregator τ ∈ [0,1]
    - Steady state government share of output g_y ∈ [0.1,0.3]
    - Monetary policy to inflation ρ_π ∈ [1.05,5]
    - Monetary policy to output gap ρ_y ∈ [0,0.5]
    - Interest rate smoothing ρ_R ∈ [0,0.9]
    - Persistence of monetary policy shock ρ_{e_R} ∈ [0,0.95]
    - Persistence of business cycle shock ρ_{e_A} ∈ [0,0.95]
    - Persistence of preference shock ρ_{e_B} ∈ [0,0.95]
    - Persistence of durables investment shock ρ_{e_I} ∈ [0,0.95]
    - Persistence of wage markup shock ρ_{e_W} ∈ [0,0.95]
    - Persistence of nondurables price markup shock ρ_{e_C} ∈ [0,0.95]
    - Persistence of durables price markup shock ρ_{e_D} ∈ [0,0.95]
    - Persistence of government consumption shock ρ_{e_G} ∈ [0,0.95]
  - Note: * denotes parameters subject to restriction θ_c ≥ θ_d.
  - Additional calibration note: θ_W is calibrated at 2 in some robustness draws; authors estimate θ_W = 153 in the baseline model and θ_W = 168 in the housing DSGE (text notes these values are “extremely larger than 2”).
  - Simulation quantities:
    - Performed 22000 draws; 10034 accepted; aim to generate about 10000 sets of impulse response functions.
    - 92% of discarded draws violated θ_c ≥ θ_d; 7% violated Blanchard-Kahn conditions.

### F Bayesian impulse responses
- The paper proceeds to plot Bayesian impulse responses of the estimated models (section begins F; figures referenced).
- Example figure description: Figure F.1 shows Bayesian impulse responses of relative prices to a contractionary monetary policy shock in the baseline DSGE:
  - Bold lines are mean responses.
  - Dark-shaded areas are 68% confidence bands.
  - Medium and lighter shaded areas represent 90% and 95% confidence bands respectively.

*Source: wp17290 - 3. Use the robust IRFs to impose sign restrictions on the IRFs of the SVAR model (IMF working paper content).*

### Section 3.6 together with the 68%, 90% and 95% confidence bands. Figure F.1 refers

### wp17290 - Section 3.6 together with the 68%, 90% and 95% confidence bands. Figure F.1 refers

### F. Bayesian impulse responses and confidence bands
- Figure F.2 (housing DSGE) and Figure F.1 (baseline DSGE) show that:
  - The comovement between durables and nondurables is attained due to the presence of prices and wages stickiness.
  - The only noticeable difference between the models concerns the response of the relative prices, as discussed in the main text.
- Figure F.2 caption (exact phrasing): "Bayesian impulse responses of relative prices to a contractionary monetary policy shock in the housing DSGE (bold lines are mean responses, dark-shaded areas are 68% confidence bands, medium and lighter shaded areas represent 90% and 95% confidence bands respectively)".

### G. Posterior distributions of Inverse Frisch Elasticities (Figure G.1)
- Prior and posterior densities are displayed for:
  - Baseline DSGE: distributions for Patient, Impatient, and Prior (left box).
  - Housing DSGE: distributions for Patient, Impatient, and Prior (right box).
- Axes and plotting ranges shown in figure: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (horizontal) and 0 to 4.5 (vertical) for both boxes.

### H. Models comparison — likelihood race and impulse-response comparison
- Approach:
  - Likelihood race between the baseline and five restricted models (each with one friction removed).
  - Plot impulse responses of baseline and restricted models to a contractionary monetary policy shock.
- Table H.1 (exact entries):
  - Baseline: Log-marg. likelihood −1472.494
  - Flexible Wages θW = 0: Log-marg. likelihood −1672.300; Kass-Raftery 399.612
  - Flexible Durables Prices θd = 0: Log-marg. likelihood −1538.150; Kass-Raftery 131.312
  - No IAC φ=φ′=0: Log-marg. likelihood −1970.003; Kass-Raftery 995.018
  - No Habit ζ=ζ′=0: Log-marg. likelihood −1698.053; Kass-Raftery 451.118
  - No Durables Inflation τ=0: Log-marg. likelihood −1473.396; Kass-Raftery 1.804
- Interpretation:
  - The KR statistic decisively favors the baseline model.
  - Slight evidence in favor of baseline relative to the model with τ=0 (central bank responds only to inflation in nondurables).
  - Very strong evidence against models with θd=0, θW=0, φ=φ′=0, and ζ=ζ′=0.
  - Conclusion: the frictions considered are important when the theoretical model is brought to the data.
- Table H.2: Estimated price stickiness parameters in restricted models (exact entries):
  - Baseline: Price stickiness nondurables θc = 23.38 [15.82;30.61]; Price stickiness durables θd = 24.45 [16.09;33.26]
  - Flexible Wages θW = 0: θc = 1.2032 [0.5643;1.7338]; θd = 2.4006 [1.4801;3.3098]
  - No IAC φ=φ′=0: θc = 47.135 [32.832;62.022]; θd = 51.378 [37.533;65.994]
  - No Habit ζ=ζ′=0: θc = 27.629 [19.122;35.731]; θd = 30.482 [19.540;41.270]
  - No Durables Inflation τ=0: θc = 22.338 [15.209;28.933]; θd = 25.961 [16.311;35.075]
- Impulse-response comparisons (Figure H.1) — key findings:
  - Black-solid line: baseline impulse responses (same as Figure 4).
  - Blue-dashed line: model with flexible wages — responses close to baseline; comovement between durables and nondurables still attained.
  - Red-dotted line (flexible durables prices, sticky wages): comovement survives; relative price response differs (almost flat in baseline, decreases in restricted case).
  - Excluding habit formation in nondurable consumption leads to considerably larger fall in nondurables and output — including this friction is crucial to obtain reasonable sizes in responses of nondurables consumption and output.
  - IACs in durable goods are crucial: in absence of IACs, at the trough, durables fall by almost 7% whereas output falls by about 0.4% — "the maximum fall in durables is about 17.5 times larger than the maximum fall of output", an implausible result according to SVAR estimates.
- Calibration note: impulse responses are rescaled to generate a 1% increase in the policy rate. Responses of model with τ=0 are not plotted because they overlap with others.

### H.1 The importance of the income share of patient households
- Baseline assumption: income share of patient households = 79% (as estimated by Iacoviello and Neri (2010)); Jappelli (1990) reports 80% for savers.
- Exercise: calibrated baseline DSGE using posterior mean of parameters from Section 3.6 with alternative values for income share (denoted ̃ψ).
- Findings (Figure H.2):
  - Qualitatively, dynamic responses of baseline DSGE are not affected by changes in the income share.
  - Quantitatively:
    - Increasing the share of impatient households (blue-dashed and red-dotted lines) exacerbates negative effects of a monetary policy shock — more households are credit constrained, so durables investment and nondurables consumption fall more.
    - This highlights the importance of the transmission channel of monetary policy through the collateral constraint.
    - Lowering the share of impatient households (black-rounded line) mitigates the effects of a monetary policy shock.

### I. Posterior estimates of extended models — selected structural and process estimates
- Table I.1: Prior and posterior distributions (models with imperfect labor mobility; 90% confidence bands in square brackets). Selected posterior means and 90% bands (Baseline DSGE / Housing DSGE where provided):
  - Inv. Frisch elasticity patients φ:
    - Baseline DSGE: 0.5262 [0.3807;0.6765]
    - Housing DSGE: 0.4996 [0.6441;0.9352]
  - Inv. Frisch elasticity impatients φ′:
    - Baseline DSGE: 0.6599 [0.5044;0.8172]
    - Housing DSGE: 0.7803 [0.5300;0.8431]
  - Habits patients ζ:
    - Baseline DSGE: 0.6761 [0.6281;0.7238]
    - Housing DSGE: 0.7022 [0.6532;0.7557]
  - Habits impatients ζ′:
    - Baseline DSGE: 0.9346 [0.9259;0.9443]
    - Housing DSGE: 0.9487 [0.9445;0.9526]
  - Price stickiness nondurables θc:
    - Baseline DSGE: 25.72 [18.07;33.85]
    - Housing DSGE: 51.08 [45.73;56.06]
  - Price stickiness durables θd:
    - Baseline DSGE: 27.02 [17.95;35.59]
    - Housing DSGE: 0.72 [0.56;0.85]
  - Wage stickiness θW:
    - Baseline DSGE: 159.09 [145.64;176.10]
    - Housing DSGE: 172.37 [166.50;177.30]
  - IAC durables patients φ:
    - Baseline DSGE: 3.0043 [2.3778;3.6516]
    - Housing DSGE: 2.5507 [1.9744;3.1002]
  - IAC durables impatients φ′:
    - Baseline DSGE: 1.6987 [0.9303;2.4394]
    - Housing DSGE: 0.0018 [0.0010;0.0027]
  - Share of durables inflation τ:
    - Baseline DSGE: 0.2018 [0.1077;0.2887]
    - Housing DSGE: 0.0534 [0.0359;0.0700]
  - Inflation -Taylor rule ρπ:
    - Baseline DSGE: 1.4099 [1.2523;1.5715]
    - Housing DSGE: 1.7766 [1.5348;2.0045]
  - Interest rate smoothing ρr:
    - Baseline DSGE: 0.7052 [0.6656;0.7488]
    - Housing DSGE: 0.7978 [0.7653;0.8323]
  - Trend growth rate γ:
    - Baseline DSGE: 0.3957 [0.3606;0.4315]
    - Housing DSGE: 0.4127 [0.3881;0.4403]
  - Selected exogenous process posterior means (Baseline DSGE / Housing DSGE):
    - Technology ρeA: 0.9792 [0.9603;0.9973] / 0.9255 [0.8722;0.9765]
    - σeA: 0.7053 [0.6304;0.7743] / 0.8059 [0.7240;0.8878]
    - Monetary Policy ρeR: 0.1052 [0.2592;0.3215] / 0.0674 [0.0141;0.1151]
    - σeR: 0.2908 [0.0252;0.1762] / 0.2675 [0.2403;0.2930]
    - Investment Durables ρeI: 0.3021 [0.1457;0.4544] / 0.9292 [0.8937;0.9658]
    - σeI: 6.8359 [4.9939;8.6048] / 7.4079 [6.6757;8.1498]
- Table I.2: Prior and posterior distributions (models with price indexation; 90% confidence bands). Selected posterior means:
  - Price stickiness nondurables θc:
    - Baseline DSGE: 20.58 [13.63;27.65]
    - Housing DSGE: 23.87 [16.55;31.31]
  - Price stickiness durables θd:
    - Baseline DSGE: 22.05 [13.77;30.25]
    - Housing DSGE: 1.26 [0.69;1.79]
  - IAC durables patients φ:
    - Baseline DSGE: 3.5149 [2.8751;4.1782]
    - Housing DSGE: 3.7729 [3.2011;4.3405]
  - Share of durables inflation τ:
    - Baseline DSGE: 0.1535 [0.0646;0.2463]
    - Housing DSGE: 0.0517 [0.0362;0.0671]
  - Trend growth rate γ:
    - Baseline DSGE: 0.4075 [0.3749;0.4412]
    - Housing DSGE: 0.4055 [0.3740;0.4364]
- Table I.3 and I.4: Three-sector model posterior highlights (90% confidence bands):
  - Price stickiness nondurables θc = 33.37 [24.07;42.82]
  - Price stickiness durables θd = 46.13 [34.99;57.06]
  - Price stickiness housing θh = 4.70 [2.34;7.09]
  - IAC durables patients φ = 2.7309 [2.0306;3.4376]
  - IAC housing patients φH = 3.9123 [3.3607;4.4951]
  - Weight in inflation aggregator τ = 0.2308 [0.1464;0.3156]; ̃τ = 0.0918 [0.0330;0.1476]
  - Selected exogenous process σeH = 16.336 [12.201;20.351] (investment housing)
  - Wage stickiness θW = 0.6874 [0.6134;0.7675] and wage mark-up ρeW = 0.9674 [0.9493;0.9856]

*Source: wp17290 - Section 3.6 together with the 68%, 90% and 95% confidence bands. Figure F.1 refers*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17290.pdf_