## 3.1  Optimizing households

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### Household preferences, production of effective consumption, and budget constraint
- Effective consumption Xot is a CES index of private consumption Cot and government spending Gt:
  - Xot = [κ(Cot)^(ν−1)/ν + (1−κ)(Gt)^(ν−1)/ν]^(ν/(ν−1)).
  - κ is the weight of private consumption; ν>0 is the elasticity of substitution between private consumption and government spending.
- Representative optimizing household chooses {Cot, Bot, Hot, Kot+1, Iot} to maximize:
  - E0 Σt=0^∞ β^t ιt [ (Xot)^(1−σ)/(1−σ) − μ(Hot)^(1+φ)/(1+φ) ].
  - Parameters: β∈(0,1), σ>0, φ>0, δ∈(0,1).
- Period budget constraint (nominal):
  - Pt Cot + Pt Iot + Bot Rt ≤ (1−τt) Wt Hot + (1−τt) Rkt Kot + Bot−1 + (1−τt) Dot + Trt.
- Capital accumulation with adjustment cost:
  - Kot+1 = (1−δ) Kot + [AI,t − Γ(Iot/Iot−1)] Iot,
  - Γ(1)=0, Γ′(1)=0, χ = Γ′′(1) > 0.
- Exogenous shocks:
  - ιt = (ιt−1)^{ρι} exp(ει,t), 0<ρι<1, ει,t ∼ N(0,σι).
  - AI,t = (AI,t−1)^{ρI} exp(εI,t), 0<ρI<1, εI,t ∼ N(0,σI).

### First-order conditions for optimizing households
- Consumption marginal condition:
  - Λt = κ ιt (Xot)^(−σ) (Xot/Cot)^{1/ν}.
- Euler equation for bonds:
  - Λt = β Et Λt+1 Rt / πt+1.
- Intratemporal labor condition:
  - Λt (1−τt) wt = ιt μ (Hot)^φ.
- Tobin’s q and capital return:
  - Λt qt = β Et Λt+1 [ (1−τt+1) rkt+1 + (1−δ) qt+1 ].
- Investment optimality (Tobin’s q relation with adjustment cost):
  - Λt qt = q_t Λt [ AI,t − Γ(Iot/Iot−1) − Iot/Iot−1 Γ′(Iot/Iot−1) ] + β Et qt+1 Λt+1 [ (Iot+1/Iot)^2 Γ′(Iot+1/Iot) ].
- Definitions and normalizations:
  - Λt is the Lagrange multiplier on the budget constraint.
  - rkt = Rkt/Pt, wt = Wt/Pt, qt is Tobin’s q.

### Rule-of-thumb households
- Rule-of-thumb households:
  - Do not hold bonds or invest in physical capital.
  - Do not optimize intertemporally or intratemporally.
  - Consumption equals labor income net of taxes plus transfers:
    - Pt Crt = (1−τt) Wt Hrt + Trt.
  - Labor supply: Hrt = Hot ≡ Ht.

### Firms and production structure
- Final output Yt is a CES aggregate of differentiated finished goods Yt(j), j∈(0,1), with elasticity θ.
- Demand for finished good j:
  - Yt(j) = (Pt(j)/Pt)^{−θ} Yt.
- Production function for differentiated goods:
  - Yt(j) = At Kt(j)^α Ht(j)^α.
- Technology shock:
  - At = At−1 uA,t, uA,t = (uA,t−1)^{ρA} exp(εA,t), 0<ρA<1, εA,t ∼ N(0,σA). (I(1) process: permanent impact on real variables.)
- Firms:
  - Price-takers in input markets, monopolistic competitors in product markets.
  - Calvo pricing: fraction (1−θ) of firms can change prices each period.
- First-order conditions for inputs (firm j):
  - wt = (1−α) ζt(j) Yt(j) / Ht(j).
  - rt = α ζt(j) Yt(j) / Kt(j).
- Optimal price for firms allowed to change prices:
  - Ṕt(j) = (σ/(σ−1)) [ Et Σ_{q=0}^∞ (β θ)^q (Λt+q/Λt) ζt(j) Yt+q(j) ] / [ Et Σ_{q=0}^∞ (β θ)^q (Λt+q/Λt) Yt+q(j) (1/Pt+q) ].
- Aggregate price index (symmetric equilibrium):
  - Pt = [ θ P_{t−1}^{1−θ} + (1−θ) Ṕt^{1−θ} ]^{1/(1−θ)}.

### Government and monetary policy
- Government budget constraint (nominal):
  - Pt Gt + Bt−1 + Trt = τt Pt Yt + Bt Rt.
- Fiscal rule (log-linear) for lump-sum transfers deviations:
  - ˆtrt = φg ˆgt + φb ˆbt,
  - where ˆtrt = trt − tr_y, ˆgt = gt − g_y, ˆbt = bt − b_y,
  - and normalization: y_t = Yt / At, tr_t = Trt / At, g_t = Gt / At, b_t = Bt / (At Pt).
- Stationary government spending process:
  - g_t = g^{1−ρg} (g_{t−1})^{ρg} exp(εg,t), 0<ρg<1, εg,t ∼ N(0,σg).
- Monetary policy: Taylor rule with interest rate smoothing:
  - Rt = (Rt−1)^{ρR} (πt/π)^{ρπ} (y_t/y)^{ρy} exp(εR,t),
  - εR,t serially uncorrelated, εR,t ∼ N(0,σR).

### Aggregation across household types
- Aggregate per-capita consumption:
  - Ct = (1−γ) Cot + γ Crt.
- Only optimizing households hold assets; aggregate holdings:
  - Bt = (1−γ) Bot.
  - Kt = (1−γ) Kot.
  - It = (1−γ) Iot.
  - Dt = (1−γ) Dot.

### Theoretical results, transmission channels, and parameter calibration
- Two channels through which government spending affects private consumption:
  - Wealth effect: higher spending affects wages and taxes, directly impacting private demand (crowding-out in standard RBC).
  - Utility effect: government spending enters utility via effective consumption; affects marginal utility of consumption.
- Log-linearized partial derivative of marginal utility with respect to government spending:
  - ∂λt / ∂gt = (1/ν − σ) (1−κ) (g/x)^{(ν−1)/ν},
  - where g and x are steady-state values of government spending and effective consumption, respectively.
  - Necessary condition for government spending to increase marginal utility (and potentially crowd-in private consumption): ν < 1/σ (private and public spending are Edgeworth complements).
  - Lower η and σ in combination can offset negative wealth effect and produce crowding-in after a government spending shock.
- Calibration (quarterly period):
  - β = 0.99.
  - Share of private consumption in effective consumption κ = 0.7.
  - Inverse Frisch elasticity of labor supply φ = 0.5.
  - Preference parameter η chosen so steady-state hours = 0.25.
  - Depreciation rate δ = 0.025 (annual depreciation implied = 10 percent).
  - Capital share α = 0.3.
  - Elasticity of substitution between intermediary goods θ = 6.
  - Steady-state shares: government spending = 0.2 (relative to total output); debt = 4×0.6 (relative to total output).
  - Average income tax rate τ = 0.2.
  - Autocorrelation of government spending ρg = 0.9.
  - Taylor rule parameters: ρR = 0.5, ρπ = 1.5, ρy = 0.2.
- Sensitivity analysis conducted on ν, σ, γ, and θ to characterize shock transmission.
- Key theoretical insights from contour analyses:
  - Wealth effect magnitude depends on price stickiness θ and share of rule-of-thumb consumers γ; sticky prices and high γ can produce consumption crowding-in and multipliers > 1.
  - Utility effect visualization: when ν is sufficiently low (below 0.5), consumption is crowded in regardless of σ; higher ν and σ can produce negative aggregate output responses under extreme values.

*Source: wp1749 - 3.1  Optimizing households*

### 4.3  Time varying impulse-response functions

### Assessment approach and simulation
- The model is simulated based on the posterior distributions of the estimated parameters in each rolling window.
- Figure 9 reports the 50th percentile of the impulse-response functions per window to assess how changes in deep parameters affect endogenous propagation mechanisms of a government spending shock.
- Estimation approach (summary):
  - Bayesian estimation on log-linearized model using rolling 36-year windows (1950:I–1980:IV initial window), shifting first and last observations by one year.
  - Posterior simulated via Metropolis-Hastings with 150,000 replications.
  - Observables: output, consumption growth, government spending share to output; plus 3-month treasury bill rate and consumer price inflation.
  - Structural shocks: (i) discount rate; (ii) investment productivity; (iii) trending technology; (iv) government spending; (v) monetary policy. Measurement errors included.

### Drivers of crowding-in and crowding-out
- Theoretical crowding-in requires:
  - a high share of rule-of-thumb consumers, γ;
  - a low elasticity of substitution between private and public consumption, ν;
  - a high elasticity of intertemporal substitution, 1/σ.
- Early-sample posterior averages generate:
  - an increasing reaction of private consumption owing to a high share of rule-of-thumb consumers;
  - mild substitutability between private and public consumption (declining ν over time contributes to stronger consumption responses up to late 1970s).
- Subsequent crowding-out in late samples associated with:
  - a drop in the elasticity of intertemporal substitution (1/σ declining).

### Key impulse-response findings
- Consumption:
  - Consumption responds more sharply as the posterior average of ν declines over time, becoming clearly positive up to late 1970s.
  - Non-monotonic change in impulse-response functions is qualitatively captured by the estimated theoretical model: initial crowding-in linked to declining ν and higher γ; later crowding-out linked to lower 1/σ.
- Output multiplier (impact response of output divided by impact response of government spending):
  - Multiplier was higher than 1 in the early samples, with a peak of 1.5 during 1969:I–1999:IV.
  - Multiplier dropped below 1 at the very end of the sample, with a trough of 0.9 during 1985:I–2015:IV.
- Real wages:
  - Response primarily sensitive to degree of price rigidity, θ, and inverse Frisch elasticity, φ.
  - Estimated posterior average labor supply elasticity ranges between 0.24 and 0.48.
  - If labor supply elasticity is sufficiently low, real wages decline to compensate for higher marginal utility of consumption; decline in real wages is exacerbated when crowding-in peaks.

### Statistical significance of impulse responses
- Output:
  - Output reactions to a government spending shock do not seem to be significantly different across sampling windows.
  - The model cannot generate a negative public spending multiplier in late windows but replicates the decline in the response of output to government spending shocks.
- Consumption:
  - Consumption reaction slowly increases in early samples, then switches sign in a statistically significant fashion in the latest periods.
  - Structural change in degree of financial inclusion, elasticity of intertemporal substitution, and elasticity between private and public consumption qualitatively account for the change in consumption response.

### Counterfactual exercises (Section 4.4)
- Counterfactual design:
  - Impulse-response functions computed assuming γ, ν, and σ are alternatively stable over sampling windows.
  - Parameters γ, ν, and σ alternatively set to initial posterior averages corresponding to 0.62, 0.68, and 1.65, respectively.
- Counterfactual findings:
  - Until periods centered around the 1990s, impact multipliers are not very sensitive to small changes in γ and σ.
  - In late samples, changes in γ and σ exacerbate the revenue effect, leading to an abrupt decline in consumption response and further curbing the impact of government spending shocks.
  - Holding ν at its initial posterior average (0.68) produces virtually negative consumption reactions during the whole sample, in contrast to the benchmark declining-ν case.
- Counterfactual conclusions:
  - The size of consumption and output responses initially increases owing to rising complementarity (declining ν) between public and private spending.
  - The decline in the impact of government spending in recent years can be primarily attributed to increasing access to financial services (declining γ) and declining elasticity of intertemporal substitution (σ rising / 1/σ falling).

### Back to the BVAR estimation and robustness
- Cautionary note:
  - Comparing impulse-response functions from a BVAR versus a theoretical model requires care because of possible discrepancies (number of lags, set of endogenous and state variables, nature and identification of structural shocks).
- Robustness exercise:
  - The same BVAR specification is estimated using simulated data from the model (simulated series of 1,000 observations using posterior averages).
  - The three-variable BVAR is estimated on simulated data and impulse-response functions computed under the same identification strategy as for observed data.
- BVAR results:
  - When model parameters are set to their posterior averages for each rolling window, the model can mimic the hump shape and magnitude of BVAR dynamic reactions of consumption to a government spending shock.
  - The crowding-in effect gradually increases to reach its maximum around the sampling window 1970:I–2010:IV, then vanishes afterwards, leading to negative co-movement during recent periods with a trough of about −0.1 percent on impact during 1985:I–2015:IV.

### Conclusion (summary of implications)
- Empirical finding: Responses of output and consumption to government spending shocks are unstable over time and exhibit two episodes:
  - Early sampling windows: increasing crowding-in, surge in U.S. fiscal multipliers, Keynesian behavior of private consumption centered around the 1970s.
  - Later sampling windows: declining correlation, negative reaction of consumption, decline in public spending multiplier.
- Model explanation:
  - A model with households limited access to financial services and non-separable private–public spending can generate both crowding-in and crowding-out depending on deep parameters.
  - The decline in elasticity of substitution between private and public spending (ν) accounts for much of the early increase in responses.
  - Increasing access to financial services (γ declining) and declining elasticity of intertemporal substitution (1/σ falling) substantially reduce expansionary effects of government purchases in recent periods.

*Source: wp1749 - 4.3  Time varying impulse-response functions*

### 3.1  Optimizing households

### 3.1  Optimizing households

### Household preferences, production of effective consumption, and budget constraint
- Effective consumption Xot is a CES index of private consumption Cot and government spending Gt:
  - Xot = [κ(Cot)^(ν−1)/ν + (1−κ)(Gt)^(ν−1)/ν]^(ν/(ν−1)).
  - κ is the weight of private consumption; ν>0 is the elasticity of substitution between private consumption and government spending.
- Representative optimizing household chooses {Cot, Bot, Hot, Kot+1, Iot} to maximize:
  - E0 Σt=0^∞ β^t ιt [ (Xot)^(1−σ)/(1−σ) − μ(Hot)^(1+φ)/(1+φ) ].
  - Parameters: β∈(0,1), σ>0, φ>0, δ∈(0,1).
- Period budget constraint (nominal):
  - Pt Cot + Pt Iot + Bot Rt ≤ (1−τt) Wt Hot + (1−τt) Rkt Kot + Bot−1 + (1−τt) Dot + Trt.
- Capital accumulation with adjustment cost:
  - Kot+1 = (1−δ) Kot + [AI,t − Γ(Iot/Iot−1)] Iot,
  - Γ(1)=0, Γ′(1)=0, χ = Γ′′(1) > 0.
- Exogenous shocks:
  - ιt = (ιt−1)^{ρι} exp(ει,t), 0<ρι<1, ει,t ∼ N(0,σι).
  - AI,t = (AI,t−1)^{ρI} exp(εI,t), 0<ρI<1, εI,t ∼ N(0,σI).

### First-order conditions for optimizing households
- Consumption marginal condition:
  - Λt = κ ιt (Xot)^(−σ) (Xot/Cot)^{1/ν}.
- Euler equation for bonds:
  - Λt = β Et Λt+1 Rt / πt+1.
- Intratemporal labor condition:
  - Λt (1−τt) wt = ιt μ (Hot)^φ.
- Tobin’s q and capital return:
  - Λt qt = β Et Λt+1 [ (1−τt+1) rkt+1 + (1−δ) qt+1 ].
- Investment optimality (Tobin’s q relation with adjustment cost):
  - Λt qt = q_t Λt [ AI,t − Γ(Iot/Iot−1) − Iot/Iot−1 Γ′(Iot/Iot−1) ] + β Et qt+1 Λt+1 [ (Iot+1/Iot)^2 Γ′(Iot+1/Iot) ].
- Definitions and normalizations:
  - Λt is the Lagrange multiplier on the budget constraint.
  - rkt = Rkt/Pt, wt = Wt/Pt, qt is Tobin’s q.

### Rule-of-thumb households
- Rule-of-thumb households:
  - Do not hold bonds or invest in physical capital.
  - Do not optimize intertemporally or intratemporally.
  - Consumption equals labor income net of taxes plus transfers:
    - Pt Crt = (1−τt) Wt Hrt + Trt.
  - Labor supply: Hrt = Hot ≡ Ht.

### Firms and production structure
- Final output Yt is a CES aggregate of differentiated finished goods Yt(j), j∈(0,1), with elasticity θ.
- Demand for finished good j:
  - Yt(j) = (Pt(j)/Pt)^{−θ} Yt.
- Production function for differentiated goods:
  - Yt(j) = At Kt(j)^α Ht(j)^α.
- Technology shock:
  - At = At−1 uA,t, uA,t = (uA,t−1)^{ρA} exp(εA,t), 0<ρA<1, εA,t ∼ N(0,σA). (I(1) process: permanent impact on real variables.)
- Firms:
  - Price-takers in input markets, monopolistic competitors in product markets.
  - Calvo pricing: fraction (1−θ) of firms can change prices each period.
- First-order conditions for inputs (firm j):
  - wt = (1−α) ζt(j) Yt(j) / Ht(j).
  - rt = α ζt(j) Yt(j) / Kt(j).
- Optimal price for firms allowed to change prices:
  - Ṕt(j) = (σ/(σ−1)) [ Et Σ_{q=0}^∞ (β θ)^q (Λt+q/Λt) ζt(j) Yt+q(j) ] / [ Et Σ_{q=0}^∞ (β θ)^q (Λt+q/Λt) Yt+q(j) (1/Pt+q) ].
- Aggregate price index (symmetric equilibrium):
  - Pt = [ θ P_{t−1}^{1−θ} + (1−θ) Ṕt^{1−θ} ]^{1/(1−θ)}.

### Government and monetary policy
- Government budget constraint (nominal):
  - Pt Gt + Bt−1 + Trt = τt Pt Yt + Bt Rt.
- Fiscal rule (log-linear) for lump-sum transfers deviations:
  - ˆtrt = φg ˆgt + φb ˆbt,
  - where ˆtrt = trt − tr_y, ˆgt = gt − g_y, ˆbt = bt − b_y,
  - and normalization: y_t = Yt / At, tr_t = Trt / At, g_t = Gt / At, b_t = Bt / (At Pt).
- Stationary government spending process:
  - g_t = g^{1−ρg} (g_{t−1})^{ρg} exp(εg,t), 0<ρg<1, εg,t ∼ N(0,σg).
- Monetary policy: Taylor rule with interest rate smoothing:
  - Rt = (Rt−1)^{ρR} (πt/π)^{ρπ} (y_t/y)^{ρy} exp(εR,t),
  - εR,t serially uncorrelated, εR,t ∼ N(0,σR).

### Aggregation across household types
- Aggregate per-capita consumption:
  - Ct = (1−γ) Cot + γ Crt.
- Only optimizing households hold assets; aggregate holdings:
  - Bt = (1−γ) Bot.
  - Kt = (1−γ) Kot.
  - It = (1−γ) Iot.
  - Dt = (1−γ) Dot.

### Theoretical results, transmission channels, and parameter calibration
- Two channels through which government spending affects private consumption:
  - Wealth effect: higher spending affects wages and taxes, directly impacting private demand (crowding-out in standard RBC).
  - Utility effect: government spending enters utility via effective consumption; affects marginal utility of consumption.
- Log-linearized partial derivative of marginal utility with respect to government spending:
  - ∂λt / ∂gt = (1/ν − σ) (1−κ) (g/x)^{(ν−1)/ν},
  - where g and x are steady-state values of government spending and effective consumption, respectively.
  - Necessary condition for government spending to increase marginal utility (and potentially crowd-in private consumption): ν < 1/σ (private and public spending are Edgeworth complements).
  - Lower η and σ in combination can offset negative wealth effect and produce crowding-in after a government spending shock.
- Calibration (quarterly period):
  - β = 0.99.
  - Share of private consumption in effective consumption κ = 0.7.
  - Inverse Frisch elasticity of labor supply φ = 0.5.
  - Preference parameter η chosen so steady-state hours = 0.25.
  - Depreciation rate δ = 0.025 (annual depreciation implied = 10 percent).
  - Capital share α = 0.3.
  - Elasticity of substitution between intermediary goods θ = 6.
  - Steady-state shares: government spending = 0.2 (relative to total output); debt = 4×0.6 (relative to total output).
  - Average income tax rate τ = 0.2.
  - Autocorrelation of government spending ρg = 0.9.
  - Taylor rule parameters: ρR = 0.5, ρπ = 1.5, ρy = 0.2.
- Sensitivity analysis conducted on ν, σ, γ, and θ to characterize shock transmission.
- Key theoretical insights from contour analyses:
  - Wealth effect magnitude depends on price stickiness θ and share of rule-of-thumb consumers γ; sticky prices and high γ can produce consumption crowding-in and multipliers > 1.
  - Utility effect visualization: when ν is sufficiently low (below 0.5), consumption is crowded in regardless of σ; higher ν and σ can produce negative aggregate output responses under extreme values.

### Estimation strategy and empirical findings
- Estimation approach:
  - Bayesian estimation on log-linearized model using rolling 36-year windows (1950:I–1980:IV initial window), shifting first and last observations by one year as in BVAR estimation.
  - Posterior simulated via Metropolis-Hastings with 150,000 replications.
  - Observables: output, consumption growth, government spending share to output; plus 3-month treasury bill rate and consumer price inflation.
  - Structural shocks in the model: (i) discount rate; (ii) investment productivity; (iii) trending technology; (iv) government spending; (v) monetary policy. Measurement errors included.
- Estimation results highlights (recursive estimation over 36-year rolling samples):
  - Degree of price stickiness θ: posterior average stable between 0.7 and 0.8.
  - Persistence of government spending ρg: stable between 0.9 and 0.95.
  - Inverse intertemporal elasticity σ: fluctuates substantially, increasing from about 1.45 (trough during 1970:I–2000:IV) to about 2.24 (peak during 1985:V–2015:IV).
  - Share of private spending in consumption bundle κ: relatively flat, shifts from early-sample average 0.72 to about 0.76 in a later blip.
  - Elasticity between private and public consumption ν: falls until late 1970s to around 0.44, peaks around 0.68 toward end of sample, falls to about 0.52 in last rolling window.
  - Share of rule-of-thumb consumers γ: falls steadily from about 0.62 in 1950:I–1980:IV to about 0.36 in early 2000s; posterior average increases by about 10 percent at very end of sample with inclusion of financial crisis data (reflecting deterioration of access to financial services).
- Timing and interpretation:
  - Changes in σ and ν begin around estimation window 1975:I–2005:IV, roughly 5 years after the break identified in BVAR estimation.
  - Decline in γ seems synchronized with the break in empirical impulse-response functions.
- Model comparison and model selection:
  - A version of the model without rule-of-thumb consumers was estimated for comparison.
  - Bayes factor strongly favors the benchmark model with rule-of-thumb consumers: marginal data density is 14.57 larger on a log-scale in favor of the benchmark, translating into a posterior odds ratio largely supporting the rule-of-thumb assumption.

*Source: wp1749 - 3.1  Optimizing households*

### 4.3  Time varying impulse-response functions

### 4.3  Time varying impulse-response functions

### Assessment approach
- The model is simulated based on the posterior distributions of the estimated parameters in each rolling window.
- Figure 9 reports the 50th percentile of the impulse-response functions per window to assess how changes in deep parameters affect endogenous propagation mechanisms of a government spending shock.

### Drivers of crowding-in and crowding-out
- The theoretical crowding-in effect arises from:
  - a high share of rule-of-thumb consumers, γ;
  - a low elasticity of substitution between private and public consumption, ν;
  - a high elasticity of intertemporal substitution, 1/σ.
- Early-sample posterior averages generate:
  - an increasing reaction of private consumption owing to a high share of rule-of-thumb consumers;
  - mild substitutability between private and public consumption (declining ν over time contributes to stronger consumption responses up to late 1970s).
- The subsequent crowding-out effect in late samples is associated with:
  - a drop in the elasticity of intertemporal substitution (1/σ declining).

### Key impulse-response findings
- Consumption:
  - Consumption responds more sharply as the posterior average of ν declines over time, becoming clearly positive up to late 1970s.
  - The non-monotonic change in impulse-response functions is qualitatively captured by the estimated theoretical model: initial crowding-in linked to declining ν and higher γ; later crowding-out linked to lower 1/σ.
- Output multiplier (defined as the ratio of the impact response of output to the impact response of government spending):
  - Multiplier was higher than 1 in the early samples, with a peak of 1.5 during 1969:I–1999:IV.
  - Multiplier dropped below 1 at the very end of the sample, with a trough of 0.9 during 1985:I–2015:IV.
- Real wages:
  - Response is primarily sensitive to the degree of price rigidity, θ, and the inverse of Frisch elasticity, φ.
  - Estimated posterior average labor supply elasticity ranges between 0.24 and 0.48.
  - If labor supply elasticity is sufficiently low, real wages decline to compensate for higher marginal utility of consumption; the decline in real wages is exacerbated during periods where crowding-in peaks.

### Statistical significance of impulse responses (Figure 10)
- Output:
  - Output reactions to a government spending shock do not seem to be significantly different across sampling windows.
  - The model cannot generate a negative public spending multiplier in late windows but replicates the decline in the response of output to government spending shocks.
- Consumption:
  - Consumption reaction slowly increases in early samples, then switches sign in a statistically significant fashion in the latest periods.
  - Structural change in degree of financial inclusion, elasticity of intertemporal substitution, and elasticity between private and public consumption qualitatively account for the change in consumption response.

### Counterfactual exercises (Section 4.4)
- Counterfactual design:
  - Impulse-response functions are computed assuming γ, ν, and σ are alternatively stable over sampling windows.
  - The parameters γ, ν, and σ are alternatively set to their initial posterior average corresponding to 0.62, 0.68, and 1.65, respectively.
- Counterfactual findings:
  - Until periods centered around the 1990s, impact multipliers are not very sensitive to small changes in γ and σ, consistent with their posterior averages.
  - In late samples, changes in γ and σ exacerbate the revenue effect, leading to an abrupt decline in consumption response and further curbing the impact of government spending shocks.
  - Holding ν at its initial posterior average (0.68) produces virtually negative consumption reactions during the whole sample, in contrast to the benchmark declining-ν case.
- Summary of counterfactual conclusions:
  - The size of consumption and output responses initially increases owing to rising complementarity (declining ν) between public and private spending.
  - The decline in the impact of government spending in recent years can be primarily attributed to increasing access to financial services (declining γ) and declining elasticity of intertemporal substitution (σ rising / 1/σ falling).

### Back to the BVAR estimation (Section 4.5)
- Cautionary note:
  - Comparing impulse-response functions from a BVAR versus a theoretical model requires care because of possible discrepancies (number of lags, set of endogenous and state variables, nature and identification of structural shocks).
- Robustness exercise:
  - The same BVAR specification (same endogenous variables and number of lags) is estimated using simulated data from the model.
  - The model is simulated using posterior averages of estimated parameters and series of 1,000 observations for endogenous variables.
  - The three-variable BVAR is estimated on simulated data and impulse-response functions are computed under the same identification strategy as for observed data.
- BVAR results:
  - When the model parameters are set to their posterior averages for each rolling window, the model can mimic the hump shape and magnitude of BVAR dynamic reactions of consumption to a government spending shock.
  - The crowding-in effect gradually increases to reach its maximum around the sampling window 1970:I–2010:IV, then vanishes afterwards, leading to negative co-movement during recent periods with a trough of about −0.1 percent on impact during 1985:I–2015:IV.

### Conclusion (summary of implications)
- Empirical finding: Responses of output and consumption to government spending shocks are unstable over time and exhibit two episodes:
  - Early sampling windows: increasing crowding-in, surge in U.S. fiscal multipliers, Keynesian behavior of private consumption centered around the 1970s.
  - Later sampling windows: declining correlation, negative reaction of consumption, decline in public spending multiplier.
- Model explanation:
  - A model with households limited access to financial services and non-separable private–public spending can generate both crowding-in and crowding-out depending on deep parameters.
  - The decline in elasticity of substitution between private and public spending (ν) accounts for much of the early increase in responses.
  - Increasing access to financial services (γ declining) and declining elasticity of intertemporal substitution (1/σ falling) substantially reduce expansionary effects of government purchases in recent periods.

*Source: wp1749 - 4.3  Time varying impulse-response functions*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp1749.pdf_