## _wp13105

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### Introduction — purpose and unifying concept
- Purpose:
  - Discusses several popular methods to estimate the “output gap” using available macroeconomic data.
  - Provides a unified concept and demonstrates decomposition of the output gap into contributions of observed data on output, inflation, unemployment, and other variables.
  - Demonstrates interpretation of data revisions and new data releases for output gap estimates (“news effects”) and real-time properties of estimators.
- Unifying concept:
  - The theory of linear filters: most output gap estimation methods can be represented as a moving average of observed data — a linear filter.
  - Filter weights are determined by model structure and parameters and reveal which variables (and at what frequencies) contribute to estimated unobservables.
- Scope and limits:
  - Methods analyzed range from (i) univariate/multivariate statistical filters, (ii) simple multivariate filters with economic theory (Phillips/IS curves), (iii) production function approach, to (iv) DSGE models.
  - The paper does not attempt to judge which estimator is “most sensible” or “optimal”; emphasis is on defining the concept and understanding model-based estimates for policy use.
- Illustrative state-space example (selected equations as presented):
  - y_t = x_t + τ_t  (Eq. (1))
  - τ_t − τ_{t−1} = τ_{t−1} − τ_{t−2} + ε_τt  (Eq. (2))
  - x_t = ρ x_{t−1} − κ π̂_t + ε_xt  (Eq. (3))
  - π̂_t = λ π̂_{t−1} + θ x_t + ε_πt  (Eq. (4))
  - Penalized least squares and equivalent two-sided linear filter representation (Eq. (6)):
    - x_{t|∞} = A(L) y_t + B(L) π̂ = Σ_{i=−∞}^{∞} A_i y_{t+i} + Σ_{i=−∞}^{∞} B_i π̂_{t+i}
- Equivalence emphasized:
  - Penalized Least-Squares, Wiener-Kolmogorov filtering, and Kalman filter/smoother are equivalent under general conditions for these models.

### Methods — linear-filter formulation and decomposition
- Core finite-sample and doubly-infinite representations:
  - Finite-sample multivariate linear filter (Eq. (7)):
    - x_{t|T} = Σ_{i=0}^{T−t} w_{1,t} y_{1,t−i} + … = Σ_{k=1}^{n} ξ_k
  - Doubly-infinite multivariate moving average (Eq. (8)):
    - X_t = Σ_{i=−∞}^{∞} W_i Y_{t−i} = B(L) Y_t
- Practical uses:
  - Understand weighting scheme behind estimates.
  - Chart output gap as a function of underlying data.
  - Quantify sources of revisions when sample is extended or revised.
- Benefits:
  - Decomposition into observables clarifies “what is in the output gap.”
  - Transfer functions enable frequency-domain analysis (gain, coherence, phase); focus in the paper is on time-domain weights and decomposition.
- Real-time revision analysis:
  - Two-sided (smoother) filter highlights endpoint problems and “news effects” when extending sample or receiving new data.

### A. State-space and DSGE formulation as linear filters
- State-space setup (Eqs. (9)–(10)):
  - X_t = T X_{t−1} + G ε_t  (Eq. (9))
  - Y_t = Z X_t + H ε_t  (Eq. (10))
  - ε ∼ N(0, Σ_ε), Σ_ε = I; impose G Σ_ε H′ = 0 to separate structural shocks from measurement errors.
- Smoothing objective:
  - X_{t|T} = E[X_t | Y_0,...,Y_T], emphasis on smoother (two-sided) estimates.
- Doubly-infinite filter representation (Eq. (12)):
  - X_{t|∞} = Ω(L) Y_t = Σ_{i=−∞}^{∞} Ω_i Y_{t+i}

### Weights of the filter — transfer function and computation
- Wiener–Kolmogorov expression:
  - Ω(z) = Γ_{XY}(z) Γ_Y(z)^{−1}
  - Γ_{XY}(z) and Γ_{YY}(z) (Eqs. (14)–(15)):
    - Γ_{XY}(z) = (I − T z)^{−1} G Σ_ε G′ (I − T′ z^{−1})^{−1} Z′  (Eq. (14))
    - Γ_{YY}(z) = Z (I − T z)^{−1} G Σ_ε G′ (I − T′ z^{−1})^{−1} Z′ + H Σ_ε H′  (Eq. (15))
- Numerical computation:
  - Analytical weights are difficult except for simple models; numerical methods (inverse Fourier transform; Koopman and Harvey (2003) recursive methods; Gomez (2006) time-domain formulas) obtain weights.
- Gomez (2006) time-domain expressions for model (9)–(10):
  - Ω_0 = P (Z′ Σ^{−1} − L R_{|∞} ) K  (Eq. (16))
  - Ω_j = (I − P R_{|∞} ) L^{−j−1} K  for j < 0  (Eq. (17))
  - Ω_j = P L′^j (Z′ Σ^{−1} − L′ R_{|∞} )  for j > 0  (Eq. (18))
  - Definitions: L ≡ T − K Z; K = steady-state Kalman gain; P solves DARE; R_{|∞} solves a Lyapunov equation; Σ = Z (Z P Z′ + H Σ_ε H′).

### Practical implementation of observable decomposition
- Two approaches:
  - Compute time-varying weights directly (Koopman and Harvey (2003)) — requires storing large objects and state-space tooling.
  - Shortcut: exploit linearity of Kalman smoother X = Ômega Y. For two observable sets {Y_A, Y_B} with common initial state:
    - X_A − X_B = Ômega (Y_A − Y_B)
    - Decompose differences by successive Kalman smoother runs with different inputs; no need to compute Ômega explicitly.
  - Applications: data-revision analysis, new observations, counterfactuals, missing observations, direct-observations ("expert judgement").
- Time-varying finite-sample weights:
  - Kalman smoother produces time-varying finite-sample weights approximating Ω(z); re-weighting reflects optimal backcasts/forecasts implied by the finite-sample implementation.

### 2. Univariate filters — representations and properties
- Univariate filter specification forms:
  - Time-domain weights, frequency-domain transfer function, state-space model, penalized least squares (e.g., Hodrick-Prescott, exponential smoothing).
- General linear filter form (equation (19)):
  - x_t = F(L) y_t = Σ_{i=−∞}^{∞} w_i y_{t+i}
- Two literature concerns:
  - (i) approximation of the ideal band-pass filter;
  - (ii) revision properties as sample size increases.
- Class of filters includes HP/Leser, Butterworth (Gomez (2001)), rational square wave (Pollock (2000)), multiresolution Haar scaling — all admit representation (19).
- Revision properties depend on finite-sample approximation quality and the data-generating process (Christiano and Fitzgerald (2003); Schleicher (2003)).

### 3. Structural VARs — output gap extraction
- Reduced-form VAR (equation (20)):
  - A(L) Y_t = (I − Σ_{j=1}^p A_j) Y_t = ε,  E[ε ε^T] = Σ
- Structural form with identification (e.g., Blanchard and Quah (1989)): Y_t = S(L) η_t.
- If the j-th component is GDP growth Δy_t (Y_{j,t} = Δy_t):
  - Δy_t = S_{11}(L) η_{1,t} + S_{12}(L) η_{2,t} + … + S_{1n}(L) η_{n,t}  (equation (21))
  - Output gap x_t is the part not affected by permanent shocks (equation (22)).
- Practical points:
  - Identification schemes are case-specific.
  - Investigate spectral properties of S(z): which observed-series frequencies drive estimates.
  - SVAR-based estimates can be outliers versus other methods.

### 4. Production Function Approach — linear filtering view
- Cobb-Douglas specification (equation (23)):
  - y_t = a_t + (1−α) k_t + α l_t
- Trend TFP a*_t obtained by smoothing a_t with A(L) (equation (24)):
  - a*_t = A(L) a_t = A(L) y_t − (1−α) A(L) k_t − α A(L) l_t
- Equilibrium employment l*_t = E(L) l_t; output gap (equation (25)):
  - x_t = y_t − y*_t = y_t − (a*_t + (1−α) k*_t + α l*_t)
- Rearranged as multivariate linear filter (equation (27)):
  - x_t = Σ_{i=−∞}^{∞} w_{y,i} y_{t+i} + Σ_{i=−∞}^{∞} w_{l,i} l_{t+i} + Σ_{i=−∞}^{∞} w_{k,i} k_{t+i}
- If E(L) = A(L) employment contribution is eliminated; capital gaps often small at business cycle frequencies, so PF can approximate a univariate filter.
- Using HP transfer functions for A(L), K(L) yields PF estimates resembling HP and inheriting HP-related issues (e.g., Epstein and Macciarelli (2010)).
- Non-causal filters in PF affect real-time revision properties; linear filter revision analysis applies.

### B. Revising properties — data revisions and news effects
- Importance:
  - Revision properties are crucial for policymakers — large real-time revisions increase uncertainty and can lead to policy errors.
- Sources of revisions:
  - (i) historical data revisions (revised GDP releases);
  - (ii) new information (arrival of new data changing interpretation of past estimates).
- Linear filter framework:
  - For finite-sample implementations, penalized least squares, Kalman filter/smoother, and Wiener-Kolmogorov filtering yield equivalent results.
  - Finite-sample representation (equation (28)):
    - X_{t|T} = Σ_{j=t}^T Ω_{j|t} Y_j
- Decomposition of revisions into contributions of new data helps assess informativeness of observations.

### 1. Decomposing effects of data revision
- Revision decomposition (equation (29)):
  - R_{t|T} = X^A_{t|T} − X^B_{t|T} = Σ_{τ=t}^T Ω_{t|τ} [Y^A_τ − Y^B_τ]
- In matrix form for Kalman filter (X̂ = Ω̂ Y):
  - X^A − X^B = Ω̂ (Y^A − Y^B), given Ω̂ fixed.
- Requirement: keep Ω̂ fixed — identical observation cross-section and time dimensions needed.
- Applications: output gap estimates, revised technology/preference estimates, structural shocks in DSGE frameworks.

### 2. News effects and end-point bias
- Non-causal filters implemented on finite samples suffer endpoint problems and news effects; state-space models with Kalman smoother are not immune.
- Infinite-sample formulation (equation (30)):
  - X_{t|∞} = ∑_{i=−∞}^{∞} Ω_i Y_{t+i}
- Revision due to availability of new observations after period T (equation (31)):
  - N_{t|T} = ∑_{j=T}^{∞} Ω_j (Y_j − Y_{j|T})
  - Interpretation: N_{t|T} is a weighted average of prediction errors conditioned on information up to T.
- Intuition:
  - Smaller weight on future observations and better predictions of future values => smaller revision variance.
  - Population revision variance computable given the filter and data-generating process.
- Optimal finite-sample implementation (equation (32)):
  - min_{̂Ω_{t,j}, j∈[t0,...,T]} || X_{t|∞} − X_{t|T} ||^2
  - Equivalent frequency-domain expression (equation (34)):
    - ∫_{−π}^{π} ||Ω(e^{−iω}) − ̂Ω_t(e^{−iω})||^2 S_Y(ω) dω
- Equivalence and padding:
  - Solution is equivalent to applying infinite filter to available sample padded with forecasts/backcasts (pinned by the auto-covariance function).
  - Padding with model-based forecasts/backcasts is the optimal finite-sample implementation (practitioner heuristic justified).
  - State-space Kalman smoother and penalized least-squares implicitly implement optimal finite-sample approximation via re-weighting.
- Caveats:
  - Forecasts depend exclusively on the covariance generating function of the model; model-data mismatch can produce poor forecasting properties leading to larger revision variance and end-point bias.
  - Chain-rule of projections causes end-of-sample observations to be implicitly counted multiple times when applying a doubly-infinite filter to a padded sample (larger effective weight on end observations).
- HP/Leser example:
  - HP/Leser filter often cited for poor revision properties and large end-point bias.
  - HP/Leser requires many periods to converge — more than 20 data points on both sides unless factorization is provided (Kaiser and Maravall (1999)).
  - HP/Leser implies ARIMA(0,2,2) for output; output gap uncorrelated white noise and potential output growth as a random walk — often economically implausible.
- Practical news-effect decomposition using state-space models:
  - New observations create news effects only if they are unpredictable from past data.
  - Representation (31) gives a method to compute and decompose news effects into components of newly observed data.
  - Padding the sample with conditional expectations makes prediction error zero for padded observations; Kalman update example: X_{t+1|t+1} = T X_{t|t} + K (Y_{t+1} − Y_{t+1|t}); for padded data (Y_{t+1} − Y_{t+1|t}) = 0.
  - Implementation: convert different sample-size problems to identical sample-size problems by padding with model-based forecasts and running successive Kalman smoother runs.

### III. Applications — decomposition into observables & news effects

A. Variants of HP/Leser and local linear trend models
- HP/Leser penalized least-squares (doubly-infinite sample) (equation (35)); implies ARIMA(0,2,2) for y_t.
- HP as linear time-invariant filter (equation (36)):
  - x_t = C(L) y_t = [λ(1−L)^2/(1−L^{−1})^2] / [1 + λ(1−L)^2/(1−L^{−1})^2] y_t = Σ_{k=−∞}^{∞} w_{x,k} y_{t−k}
- Unobserved components representation (equations (37)–(39)), signal-to-noise ratio √λ = σ_g / σ_x.
- Modified HP (allowing persistence and mean-reverting potential growth) (equations (40)–(43)):
  - Key feature: potential output growth g_t is mean-reverting and persistent (not a random walk), improving revision properties relative to standard HP.
- Example parametrization and revision comparison (US, sample 1967:1–2010:3):
  - Parametrization:
    - ρ_1 = 0.70
    - ρ_g = 0.95
    - σ_{̄y} = 0
    - σ_x = 1/(1−ρ_1)
    - σ_g = √(1/λ) × [1/(1−ρ_g)]
    - λ = 1600
  - Steady-state growth of potential output assumed to be 2 %.
  - Revision standard deviations (final estimate versus real-time estimates):
    - Standard HP filter revision standard deviation = 1.489.
    - Modified filter revision standard deviation = 0.858.
    - Modified filter revision SD is 57.6% of the standard HP revision SD.
  - Both filter weights and forecasting properties contribute to this reduction.

B. Multivariate semi-structural filter (Benes and others (2010))
- Model features:
  - Backward-looking output gap equation, Phillips curve, Okun’s law, capacity utilization as additional cyclical measurement.
  - Year-over-year inflation π^4_t follows a unit root process anchored by long-term inflation expectations π^{4LTE}_t.
- Model equations (44)–(55) (selected):
  - y_t = Y_t − ̄Y_t; c_t = C_t − ̄C_t; u_t = ̄U_t − U_t. (44)
  - y_t = ρ y_{t−1} − ̃ρ_2 (π^4_{t−1} − π^{4LTE}_{t−1}) + ε^y. (45)
  - π^4_t = π^4_{t−1} + β y_t + Ω(y_t − y_{t−1}) + ε^{π4}_t. (46)
  - ̄Y_t = ̄Y_{t−1} + ḠY_t/4 + θ(̄U − ̄U_{t−1}) − (1−θ)(̄U − ̄U_{t−20})/19 + ε_{̄Y,t}. (50)
  - ḠY_t = τ ḠY_{SS} + (1−τ) ḠY_{t−1} + ε_{ḠY,t}. (51)
  - π^{4LTE}_t = π^{4LTE}_{t−1} + ε^{π4LTE}_t. (49)
- All innovation processes ε_{i,t} are uncorrelated Gaussian with zero mean; standard deviations of innovations determine the model’s transfer function and revision properties.
- Analytical uses:
  - Output gap depends on innovations to output, inflation, and inflation expectations.
  - Filter representation permits decomposition into observables and quantification of their contributions.

### Decomposition into Observables — key empirical findings (Benes et al. applications)
- Contributions to the ‘Great Recession’:
  - Contribution of GDP growth to the episode starting in 2007 is smaller than during the largest recession in the sample (1980–1985).
  - Contribution of the unemployment series is the greatest ever for that episode.
  - Inflation data provide limited information for the output gap in the parametrization used.
- Parameter/identification specifics noted:
  - Loading coefficient ̃ρ2 = 0.005 (very small).
  - Parameters determining output gap effect on inflation β, Ω are small.
- Behavioural interpretation:
  - Unemployment contribution is large and persistent, consistent with US “jobless recovery” since the 1990s.
  - NAIRU is modelled to change more slowly than potential output; both NAIRU and potential output growth vary less than filters tuned to cycles of frequency 6–32 periods.
- Transfer-function insights:
  - Gain from inflation is flat and very small across the spectrum.
  - Gain from GDP growth and capacity utilization: large weight at lower frequencies, some weight at business cycle frequencies (6-32 quarters), little weight at high frequencies.
  - Level-to-output gain requires integration filter with transfer function 1/(2 − 2 cos(ω)).
  - Gain from level of unemployment shows weight on low frequencies and gradual decline with periodicity, implying spillover of longer cycles into the output gap.
- Univariate approximation:
  - A univariate HP filter with a large smoothing parameter λ can closely approximate the model’s output gap when using unemployment as input (examples include λ = 1600 and λ = 540000 as illustration).
- News effects decomposition (case examples):
  - 2009Q1: new data further deepened the output gap, a complete news relative to model-implied recovery.
  - Largest contributors to the 2009Q1 news: GDP growth (largest), then capacity utilization, then unemployment; inflation/inflation expectations contributed modestly and persistently.
  - Proper interpretation requires understanding both the filter derivative to new data and the prediction error (forecast vs realization).

### Natural output gap in DSGE (Smets and Wouters (2007))
- Decomposition of the “flexible-price equilibrium output gap” into observables:
  - Hours worked, consumption, and output are important at high and business-cycle frequencies.
  - Inflation and interest rates contribute mainly to low-frequency dynamics.
  - The model extracts information on the natural rate of output while placing little weight on observed inflation, similarly to the semi-structural model.

### Methodological conclusions
- Many commonly used potential output estimation methods can be cast as linear filter problems, enabling frequency- and time-domain analysis and clarifying revision properties as properties of two-sided moving averages.
- Revision properties and end-point bias relate to the spread of filter weights and forecasting quality of the filter-implied model: more spread weights and worse forecasting properties imply larger real-time revision variance.
- No a priori reason multivariate state-space filters must have better revision properties than univariate filters; the critical factor is the model structure linking economic theory to optimal signal extraction.

### Appendix highlights — Benes et al. (2010) parameter estimates (selected entries)
- Estimation sample: United States data for period 1967:1–2010:2.
- Selected prior mode, prior dispersion, posterior mode, posterior dispersion (Table 1, selected rows):
  - ḠY S S — Prior Mode: 2.1400 Dispersion: 0.300 — Posterior Mode: 2.143 Dispersion: 0.041
  - U S S — Prior Mode: 5.8000 Dispersion: 0.300 — Posterior Mode: 5.801 Dispersion: 0.042
  - θ — Prior Mode: 0.7000 Dispersion: 0.030 — Posterior Mode: 0.700 Dispersion: 0.004
  - α — Prior Mode: 0.9000 Dispersion: 0.150 — Posterior Mode: 0.895 Dispersion: 0.022
  - β — Prior Mode: 0.4000 Dispersion: 0.300 — Posterior Mode: 0.176 Dispersion: 0.024
  - Ω — Prior Mode: 0.5000 Dispersion: 0.300 — Posterior Mode: 0.265 Dispersion: 0.038
  - ρ — Prior Mode: 0.8000 Dispersion: 0.150 — Posterior Mode: 0.821 Dispersion: 0.019
  - 100 ̃ρ2 — Prior Mode: 5.000 Dispersion: 3.000 — Posterior Mode: 5.145 Dispersion: 0.418
  - κ1 — Prior Mode: 0.1000 Dispersion: 0.600 — Posterior Mode: 0.353 Dispersion: 0.037
  - σπ4LTE — Prior Mode: 0.300 Dispersion: 0.300 — Posterior Mode: 0.081 Dispersion: 0.008
  - σ̄Y — Prior Mode: 0.250 Dispersion: 0.100 — Posterior Mode: 0.217 Dispersion: 0.013

### Not-for-publication appendices — illustrative results
- Appendix B: equivalence between two HP state-space representations; both yield identical results to standard matrix HP implementations.
- Appendix C: three representations (state-space, Wiener-Kolmogorov filter, penalized least squares) of a simple multivariate filter; equivalence highlighted and time-domain weights derivable by inverse z-transform.
- Appendix D: variance reduction via multiple noisy measurements — adding relevant measurements reduces MSE and revision variance; simple formulas (e.g., equation (68)) and AR(1) measurement examples provided.
- Appendix E: graphical illustrations referenced (HP transfer function for λ = 1600 and λ = 414400; detrended series ygap, ugap, capugap).

*Source: _wp13105 (selected sections: Introduction; Methods; Univariate filters; Structural VARs; Production Function Approach; Revising properties; Applications; Appendices) — PDF content unit _wp13105.*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### _wp13105 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Introduction — purpose and unifying concept
- Purpose:
  - Discusses several popular methods to estimate the “output gap” using available macroeconomic data.
  - Provides a unified concept and demonstrates decomposition of the output gap into contributions of observed data on output, inflation, unemployment, and other variables.
  - Demonstrates interpretation of data revisions and new data releases for output gap estimates (“news effects”) and real-time properties of estimators.
- Unifying concept:
  - The theory of linear filters: most output gap estimation methods can be represented as a moving average of observed data — a linear filter.
  - Filter weights are determined by model structure and parameters and reveal which variables (and at what frequencies) contribute to estimated unobservables.
- Scope and limits:
  - Methods analyzed range from (i) univariate/multivariate statistical filters, (ii) simple multivariate filters with economic theory (Phillips/IS curves), (iii) production function approach, to (iv) DSGE models.
  - The paper does not attempt to judge which estimator is “most sensible” or “optimal”; emphasis is on defining the concept and understanding model-based estimates for policy use.
- Illustrative example (state-space representation of an extended HP-style multivariate model):
  - Observed aggregate output: y_t = x_t + τ_t  (Eq. (1))
  - Potential output driftless random walk specification: τ_t − τ_{t−1} = τ_{t−1} − τ_{t−2} + ε_τt  (Eq. (2))
  - Output gap dynamics: x_t = ρ x_{t−1} − κ π̂_t + ε_xt  (Eq. (3))
  - Phillips-type inflation deviation: π̂_t = λ π̂_{t−1} + θ x_t + ε_πt  (Eq. (4))
  - Penalized least squares formulation shown (Eq. (5)) and equivalent two-sided linear filter representation:
    - x_{t|∞} = A(L) y_t + B(L) π̂ = Σ_{i=−∞}^{∞} A_i y_{t+i} + Σ_{i=−∞}^{∞} B_i π̂_{t+i}  (Eq. (6))
- Equivalence emphasized:
  - Penalized Least-Squares, Wiener-Kolmogorov filtering, and Kalman filter/smoother are equivalent under general conditions for these models.

### Methods — linear-filter formulation and decomposition
- Core representation:
  - Finite-sample multivariate linear filter expressing an unobservable x_{t|T} as weighted averages of observed series (Eq. (7)):
    - x_{t|T} = Σ_{i=0}^{T−t} w_{1,t} y_{1,t−i} + … = Σ_{k=1}^{n} ξ_k
  - Doubly-infinite multivariate moving average form (Eq. (8)):
    - X_t = Σ_{i=−∞}^{∞} W_i Y_{t−i} = B(L) Y_t
- Practical uses of filter representation:
  - Understand weighting scheme behind estimates.
  - Chart output gap as a function of underlying data.
  - Quantify sources of revisions when sample is extended or revised.
- Benefits stressed:
  - Decomposition into observables clarifies “what is in the output gap.”
  - Filter transfer functions enable frequency-domain analysis (gain, coherence, phase), though paper focuses on time-domain weights and decomposition.
- Real-time revision analysis:
  - The two-sided (smoother) filter representation highlights endpoint problems and “news effects” when extending sample or receiving new data.

### A. Formulating Potential Output Estimates as Linear Filters — state-space and DSGE
- State-space setup for analysis (Eqs. (9)–(10)):
  - X_t = T X_{t−1} + G ε_t  (Eq. (9))
  - Y_t = Z X_t + H ε_t  (Eq. (10))
  - ε ∼ N(0, Σ_ε), Σ_ε = I; structural shocks uncorrelated with measurement errors by imposing G Σ_ε H′ = 0.
- Smoothing focus:
  - Uses notation X_{t|T} = E[X_t | Y_0,...,Y_T], focusing primarily on smoother (two-sided) estimates.
- Representation in terms of structural shocks and cumulative effects (Eq. (11)):
  - Y_t = [Z A(L) + H] ε̂_t,  X_{t|∞} = A(L) ε̂_t = Σ_{j=0}^{T} T^j G ε̂_{t−j}
- Inverted question addressed:
  - “What observed data drive each unobserved state?” — motivates expressing X_{t|∞} as Ω(L) Y_t with doubly-infinite sample (Eq. (12)):
    - X_{t|∞} = Ω(L) Y_t = Σ_{i=−∞}^{∞} Ω_i Y_{t+i}

### Weights of the filter — transfer function and computation
- Wiener–Kolmogorov expression for the filter (transfer function):
  - Ω(z) = Γ_{XY}(z) Γ_Y(z)^{−1}
  - Γ_{XY}(z) and Γ_{YY}(z) given by Eqs. (14) and (15):
    - Γ_{XY}(z) = (I − T z)^{−1} G Σ_ε G′ (I − T′ z^{−1})^{−1} Z′  (Eq. (14))
    - Γ_{YY}(z) = Z (I − T z)^{−1} G Σ_ε G′ (I − T′ z^{−1})^{−1} Z′ + H Σ_ε H′  (Eq. (15))
- Interpretations:
  - The transfer function Ω(z) is the key ingredient for frequency-domain analysis and for understanding most relevant frequencies of observed series for the output gap estimate.
- Numerical computation:
  - Analytical weights are difficult for all but simple models; weights can always be obtained numerically (e.g., inverse Fourier transform, Koopman and Harvey (2003) recursive methods, Gomez (2006) time-domain formulas).
- Gomez (2006) time-domain weight expressions for model (9)–(10):
  - Ω_0 = P (Z′ Σ^{−1} − L R_{|∞} ) K  (Eq. (16))
  - Ω_j = (I − P R_{|∞} ) L^{−j−1} K  for j < 0  (Eq. (17))
  - Ω_j = P L′^j (Z′ Σ^{−1} − L′ R_{|∞} )  for j > 0  (Eq. (18))
  - Definitions:
    - L ≡ T − K Z, K denotes steady-state Kalman gain.
    - P is steady-state solution for state error covariance (DARE).
    - R_{|∞} solves Lyapunov equation R_{|∞} = L′ R_{|∞} L + Z (Z P Z′ + H Σ_ε H′)^{−1} Z′.
    - Σ = Z (Z P Z′ + H Σ_ε H′).

### Practical implementation of observable decomposition
- Two practical approaches:
  - Compute time-varying weights directly using recursive methods (Koopman and Harvey (2003)) and implement moving-average calculations — requires storing large objects and state-space tooling.
  - Shortcut: exploit linearity of Kalman smoother as X = F(Y), where X and Y are (T×n) and (T×m) matrices and X = Ômega Y.
    - For two different observable sets {Y_A, Y_B} with common initial state, X_A − X_B = Ômega (Y_A − Y_B). Therefore:
      - Decompose differences by successive runs of the Kalman smoother with different inputs; no need to compute Ômega explicitly.
    - This method is applicable for data-revision analysis, new observations, counterfactuals (steady-state paths, unconditional forecasts), missing observations, and direct-observations implementations of linear restrictions (“expert judgement”).
- Time-varying weights:
  - Kalman smoother produces time-varying finite-sample weights that approximate the doubly-infinite optimal filter Ω(z); re-weighting reflects optimal backcasts/forecasts implied by finite sample implementation.

*Source: _wp13105 - References (selected sections: Introduction; Methods; Appendix references and equations as presented in the source PDF).*

### 2.   Univariate filters – Band-Pass, Hodrick-Prescott, etc.

### 2.   Univariate filters – Band-Pass, Hodrick-Prescott, etc.

### Univariate filter representations and properties
- Univariate filters are specified in one of four ways: weights in the time domain, transfer function in the frequency domain, state-space model, or as a penalized least squares problem (e.g. Hodrick-Prescott filter, exponential smoothing).
- Any such specification admits a time-domain linear filter representation:
  - x_t = F(L) y_t = Σ_{i=-∞}^{∞} w_i y_{t+i}  (equation (19))
  - w_i are the filter weights.
- Discussion typically focuses on spectral properties implied by F(z); filter weights are also discussed in the literature (e.g. Harvey and Trimbur (2008)).
- Two main literature concerns:
  - (i) approximation of the ideal band-pass filter;
  - (ii) revision properties of filters as sample size increases.
- The ideal band-pass filter (perfectly rectangular gain) is a benchmark for assessing “sharpness”: leakage, strength of the Gibbs effect, and finite-sample implementation ease.
- The class of linear filters includes many designs: Hodrick-Prescott/Leser filter, Butterworth filters (Gomez (2001)), rational square wave filters (Pollock (2000)), multiresolution Haar scaling filters — all fit representation (19).
- Revision properties of a possibly infinite F(z) hinge on the quality of its finite-sample approximation and on the data generating process (see Christiano and Fitzgerald (2003), Schleicher (2003)).

### Implications for multivariate and semi-structural models
- Univariate filters often form parts of semi-structural multivariate models or production function approaches to the output gap; hence understanding their properties is essential.
- Finite-sample implementation and optimal approximation of univariate filters are relevant for their role inside multivariate frameworks.

### Key takeaways for practitioners
- Spectral analysis of chosen F(z) is important to understand which frequencies are being used and potential outlier behavior relative to other methods.
- Non-causal or symmetric filters incorporated into larger estimation procedures directly affect real-time and revision properties.

---

### 3.   Structural VARs

### SVAR representation and output gap extraction
- An estimated reduced-form VAR of order p is:
  - A(L) Y_t = (I − Σ_{j=1}^p A_j) Y_t = ε,  E[ε ε^T] = Σ  (equation (20))
  - Reduced-form residuals ε_t are linked to structural shocks η_t via ε_t = Q η_t; Y_t is n-dimensional.
- With identification (e.g. long-run restrictions following Blanchard and Quah (1989)), the structural form is:
  - Y_t = B(L) Q Q^{-1} ε_t = S(L) η_t.
- If the j-th component is GDP growth Δy_t (Y_{j,t} = Δy_t), then:
  - Δy_t = S_{11}(L) η_{1,t} + S_{12}(L) η_{2,t} + … + S_{1n}(L) η_{n,t}  (equation (21))
  - The output gap x_t is the part not affected by permanent shocks:
    - x_t = S_{12}(L) η_{2,t} + … + S_{1n}(L) η_{n,t}  (equation (22)), assuming Σ_{k=0}^∞ S_{1i}(k) η_{i,t−k} = 0 for permanent components.
- Structural shocks are estimated from reduced-form residuals via η_t = Q^{-1} ε_t = Q^{-1} A(L) Y_t, allowing recovery of the estimated output gap as a function of observations.

### Practical considerations and recommendations
- Identification schemes are case-specific; for SVAR estimates of the output gap, concurrent and final estimates coincide unless parameters are re-estimated with an extended sample.
- Investigate spectral properties of S(z) because:
  - It is not ex-ante clear which observed-series frequencies drive estimates.
  - SVAR-based estimates often appear as outliers versus other methods (see McNellis and Bagsic (2007), Cayen and van Norden (2005), Scott (2000)).
- Decomposing the output gap into contributions from observables is feasible using the expressions above; the gap is ultimately a function of structural shocks, themselves functions of observed data.

---

### 4.   Production Function Approach

### Formulation as a linear filtering problem
- Starting from a Cobb-Douglas production function in logs:
  - y_t = a_t + (1−α) k_t + α l_t  (equation (23))
  - a_t is the Solow residual; k_t and l_t are logs of capital stock and hours worked.
- Trend total factor productivity a*_t is obtained by smoothing a_t with a filter A(L), typically a variant of the Hodrick-Prescott or other symmetric moving average:
  - a*_t = A(L) a_t = A(L) y_t − (1−α) A(L) k_t − α A(L) l_t  (equation (24))
- Equilibrium employment often estimated as the trend of employment via a univariate filter: l*_t = E(L) l_t.
- Output gap x_t is:
  - x_t = y_t − y*_t = y_t − (a*_t + (1−α) k*_t + α l*_t)  (equation (25))
  - Rearranged:
    - x_t = (1−A(L)) y_t − (1−α) (1−A(L)) k_t − α (E(L) − A(L)) l_t  (equation (26))
    - Or as a multivariate linear filter:
      - x_t = Σ_{i=-∞}^{∞} w_{y,i} y_{t+i} + Σ_{i=-∞}^{∞} w_{l,i} l_{t+i} + Σ_{i=-∞}^{∞} w_{k,i} k_{t+i}  (equation (27))
- If E(L) = A(L), the employment contribution is eliminated. Capital stock typically has little variance at business cycle frequencies, so its gaps are small; as they shrink, the PF approach approximates a simple univariate filter for the output gap.

### Links to Hodrick-Prescott and implications
- When A(L) and K(L) are HP transfer functions (common practice), PF results tend to resemble HP estimates and inherit HP-related problems (see Epstein and Macciarelli (2010)).
- Expressing PF estimates as linear functions of inputs (output, labor, capital) and decomposing contributions is desirable; finite-sample versions are straightforward when processes are linear.
- Incorporating non-causal filters into PF calculations affects real-time revision properties; linear filter analysis of revisions and news effects applies to PF estimates.

---

### B.   Analyzing Revision Properties – Data Revisions and News Effects

### Importance for policy and estimation
- Revision properties of output gap estimators are crucial for policymakers: large real-time revisions increase uncertainty and can lead to policy errors.
- Revisions arise from:
  - (i) historical data revisions (e.g. revised GDP releases);
  - (ii) new information (arrival of new data that changes interpretation of past estimates).
- Many studies highlight real-time unreliability of output gap methods and propose remedies (e.g. Cayen and van Norden (2005), Orphanides and van Norden (2002)).
- The linear filter framework is the natural approach to understand sources of revisions and to decompose revisions into contributions of new data.

### Analytical framework and practical usefulness
- For finite-sample implementations, penalized least squares, Kalman filter/smoother, and Wiener-Kolmogorov filtering yield equivalent results.
- Key finite-sample representation used for analysis:
  - X_{t|T} = Σ_{j=t}^T Ω_{j|t} Y_j  (equation (28)), a finite-sample implementation of Ω(z).
- The analysis applies to both data revisions and news effects; characteristics of the process Y_t are crucial for optimal finite-sample implementation of Ω(z).
- Decomposition of revisions into contributions of new data helps assess informativeness of available observations.

---

### 1.   Decomposing Effects of Data Revision

### Decomposition formula and interpretation
- Data revisions refer to revisions of past data releases (most often GDP).
- If filtering is applied to the same sample and same number of observed series before and after revision (Y^A, Y^B), the revision decomposition is:
  - R_{t|T} = X^A_{t|T} − X^B_{t|T} = Σ_{τ=t}^T Ω_{t|τ} [Y^A_τ − Y^B_τ]  (equation (29))
- In stacked (matrix) form, for the Kalman filter (a least-squares solution) X̂ = Ω̂ Y, so:
  - X^A − X^B = Ω̂ (Y^A − Y^B), given Ω̂ fixed.
- Requirement: keep the structure of Ω̂ fixed — identical observation cross-section and time dimensions are needed.

### Applications and implications
- The decomposition is useful not only for output gap estimates but also for understanding revised estimates of technology, preferences, and structural shocks in DSGE forecasting frameworks (see Andrle and others (2009b)).
- Example implication: interpretation of inflationary pressures can change when domestic absorption is revised, even if CPI inflation estimates remain unchanged.

*Source: _wp13105 - 2.   Univariate filters – Band-Pass, Hodrick-Prescott, etc.*

### 2.   News Effects and End-Point Bias

### 2.   News Effects and End-Point Bias

### Non-causal filters, finite samples, and news effects
- Implementation of a doubly-infinite, non-causal filter Ω(z) is problematic because all economic applications feature finite samples; newly available data lead to revisions of past estimates of unobserved variables.
- All non-causal filters suffer from the finite-sample problem; state-space models using the Kalman smoother are not exempt. The claim that state-space signal extraction techniques “do not suffer from what is often referred to as end-of-sample bias” is incorrect.
- Infinite-sample formulation (starting point for analysis of news effects):
  - X_{t|∞} = ∑_{i=−∞}^{∞} Ω_i Y_{t+i}. (30)
- Revision due to availability of new observations after period T (chronological):
  - N_{t|T} = ∑_{j=T}^{∞} Ω_j (Y_j − Y_{j|T}). (31)
  - Interpretation: N_{t|T} is a weighted average of prediction errors (conditioned on information up to T) given the constant-weights filter.
- Intuition on revision variance:
  - The smaller the weight on future observations and the better the predictions of future values, the smaller the revision variance.
  - Population revision variance can be computed given the filter and the data-generating process (prediction errors determined by the latter).

### Optimal finite-sample implementation and equivalence with padding
- Optimal finite-sample approximation problem:
  - min_{̂Ω_{t,j}, j∈[t0,...,T]} || X_{t|∞} − X_{t|T} ||^2. (32)
  - Equivalent expressions:
    - Difference of infinite and finite sums (33).
    - Frequency-domain representation: ∫_{−π}^{π} ||Ω(e^{−iω}) − ̂Ω_t(e^{−iω})||^2 S_Y(ω) dω. (34)
- Solution delivers a sequence of filters ̂Ω_t(z) that depend on position in the sample t and on the auto-covariance function of the data.
- Equivalence: solution to (32) is equivalent to applying the time-invariant infinite filter to the available sample padded with forecasts and backcasts by the j-step ahead (j chosen so the filter converges). Forecasts are pinned down by the auto-covariance function of the data-generating process.
- Practical implication: a common practitioner heuristic (padding with model-based forecasts/backcasts) is the optimal finite-sample implementation.
- State-space models with Kalman smoother and Penalized Least-Squares provide implicitly the optimal finite-sample approximation by re-weighting time-invariant weights — equivalent to providing backcasts and forecasts and padding the sample.
- Caveat: forecasts are based exclusively on the covariance generating function of the underlying model; mismatch between model and data can produce poor forecasting properties and hence larger revision variance and end-point bias.
- When filter weights are spread across many periods, poor forecasting properties translate into large revisions and end-point bias.
- Chain-rule of projections causes end-of-sample observations to be implicitly counted many times when applying a doubly-infinite filter to a padded sample, effectively giving larger weight to end observations.

### Hodrick-Prescott (HP)/Leser filter example and convergence
- HP/Leser filter is often cited for poor revision properties and large “end-point” bias.
- Unless factorization is provided (e.g., Kaiser and Maravall (1999)), the HP filter takes many periods to converge — more than 20 data points on both sides.
- Viewing the HP/Leser filter as a smooth low-pass filter: optimal finite-sample implementation suggests re-weighting filter weights by the auto-covariance function or extending the sample with best linear backcasts and forecasts.
- Model interpretation: HP/Leser implies ARIMA(0,2,2) for output; output gap is uncorrelated white noise and potential output growth follows a random walk — an economically implausible data-generating process for most countries.
- State-space (Kalman smoother) and penalized least-squares formulations using HP assumptions yield equivalent results to the HP model.

### Practical news-effect decomposition using state-space models
- New observations create “news effects” only if they carry information not predictable from past data.
- Representation (31) provides a practical method to compute and decompose news effects into components of newly observed data.
- Padding the sample with conditional expectations or model-based forecasts does not change estimates because prediction error is zero for padded observations; information sets are identical.
  - Kalman update example: X_{t+1|t+1} = T X_{t|t} + K (Y_{t+1} − Y_{t+1|t}); for padded data the prediction error (Y_{t+1} − Y_{t+1|t}) = 0.
- Implementation: convert different sample-size problems to identical sample-size problems by padding with model-based forecasts and running successive Kalman smoother runs; coding is simple and usually faster than computing weights explicitly.
- Decomposition allows judgement-free forecast investigation and distribution of prediction error into revision of past unobserved shocks; it easily accounts for judgement imposed on the filter via dummy-observations.

---

### III. Applications – Decomposition into Observables & News Effects

#### A. Variants of Hodrick-Prescott/Leser Filter and Local Linear Trend Models

- HP/Leser PLS form (doubly-infinite sample):
  - min_{ { ̄y_t } } ∑_{t=−∞}^{∞} (y_t − ̄y_t)^2 + ∑_{t=−∞}^{∞} λ [ ( ̄y_t − ̄y_{t−1}) − ( ̄y_{t−1} − ̄y_{t−2}) ]^2. (35)
  - Doubly-infinite model (35) implies ARIMA(0,2,2) for y_t.
- HP filter as linear time-invariant filter:
  - x_t = C(L) y_t = [λ(1−L)^2/(1−L^{−1})^2] / [1 + λ(1−L)^2/(1−L^{−1})^2] y_t = ∑_{k=−∞}^{∞} w_{x,k} y_{t−k}. (36)
- Unobserved components representation (infinite-sample HP):
  - y_t = ̄y_t + x_t. (37)
  - x_t = ε^x_t, ε^x_t ∼ N(0, σ_x^2). (38)
  - ̄y_t − ̄y_{t−1} = ̄y_{t−1} − ̄y_{t−2} + ε^g_t, ε^g_t ∼ N(0, σ_g^2). (39)
  - Signal-to-noise ratio: √λ = σ_g / σ_x.
- Stationary state-space form: observe growth rates ∆y_t = ∆x_t + g_t with g_t = g_{t−1} + ε^g_t; integration filter gives x_t = [C(L)/(1−L)] ∆y_t.

- Modified HP (allowing persistence and mean-reverting potential growth)
  - Model:
    - y_t = ̄y_t + x_t. (40)
    - x_t = ρ_x x_{t−1} + ε^x_t, ε^x_t ∼ N(0, σ_x). (41)
    - ̄y_t = ̄y_{t−1} + g_t + ε_{̄y,t}, ε_{̄y,t} ∼ N(0, σ_{̄y}). (42)
    - g_t = ρ_g g_{t−1} + (1−ρ_g) g_{ss} + ε^g_t, ε^g_t ∼ N(0, σ_g). (43)
  - Key feature: potential output growth g_t is mean-reverting and persistent (not a random walk), improving revision properties.
  - Many semi-structural models use steady-state assumptions for potential output; such steady-states improve revision properties relative to standard HP.

- Example parametrization and revision comparison (US, sample 1967:1–2010:3)
  - Parametrization used for modified HP:
    - ρ_1 = 0.70
    - ρ_g = 0.95
    - σ_{̄y} = 0
    - σ_x = 1/(1−ρ_1)
    - σ_g = √(1/λ) × [1/(1−ρ_g)]
    - λ = 1600
  - Steady-state growth of potential output assumed to be 2 %.
  - Revision standard deviations (final estimate versus real-time estimates):
    - Standard HP filter revision standard deviation = 1.489.
    - Modified filter revision standard deviation = 0.858.
    - Modified filter revision SD is 57.6% of the standard HP revision SD.
  - Both filter weights and forecasting properties contribute to this reduction.

#### B. Output Gap Estimation using a Multivariate Semi-Structural Filter (Benes and others (2010) model)
- The model is a state-of-the-art multivariate filter for US output gap estimation; it has exceptionally small revision variance and very good forecasting properties.
- Structure: backward-looking output gap equation, Phillips curve, Okun’s law, capacity utilization series as an additional cyclical measurement. Year-over-year inflation π^4_t follows a unit root process anchored by long-term inflation expectations π^{4LTE}_t.
- Model equations (44)–(55):
  - y_t = Y_t − ̄Y_t; c_t = C_t − ̄C_t; u_t = ̄U_t − U_t. (44)
  - y_t = ρ y_{t−1} − ̃ρ_2 (π^4_{t−1} − π^{4LTE}_{t−1}) + ε^y. (45)
  - π^4_t = π^4_{t−1} + β y_t + Ω(y_t − y_{t−1}) + ε^{π4}_t. (46)
  - u_t = φ_1 u_{t−1} + φ_2 y_t + ε^u_t. (47)
  - c_t = κ_1 u_{t−1} + κ_2 y_t + ε^c_t. (48)
  - π^{4LTE}_t = π^{4LTE}_{t−1} + ε^{π4LTE}_t. (49)
  - ̄Y_t = ̄Y_{t−1} + ḠY_t/4 + θ(̄U − ̄U_{t−1}) − (1−θ)(̄U − ̄U_{t−20})/19 + ε_{̄Y,t}. (50)
  - ḠY_t = τ ḠY_{SS} + (1−τ) ḠY_{t−1} + ε_{ḠY,t}. (51)
  - ̄U_t = ̄U_{t−1} + ḠU_t − ̃ω y_{t−1} − ̃λ(̄U_{t−1} − U_{SS}) + ε_{̄U,t}. (52)
  - ḠU_t = (1−α) ḠU_{t−1} + ε_{ḠU,t}. (53)
  - ̄C_t = ̄C_{t−1} + ḠC_t + ε_{̄C,t}. (54)
  - ḠC_t = (1−δ) ḠC_{t−1} + ε_{ḠC,t}. (55)
- All innovation processes ε_{i,t} are uncorrelated and Gaussian with zero mean and variances specified in Benes and others (2010). Standard deviations of innovations are crucial to the model’s transfer function.
- Analytical uses:
  - The model’s output gap depends on its own innovations and innovations to inflation and inflation expectations.
  - More interestingly, the filter representation permits decomposition into observables and analysis of how observed data contribute to the output gap estimate.

_Italic: Source — Chapter “2. News Effects and End-Point Bias” from PDF content unit _wp13105 - 2.   News Effects and End-Point Bias_._

### 1.   Decomposition into Observables

### 1.   Decomposition into Observables

### Overview
- Presents a method to decompose an unobservable output gap into contributions from observed inputs (GDP, y/y inflation, capacity utilization, unemployment) by exploiting the model’s state-space form.
- Complements Benes and others (2010) and illustrates methods on a semi-structural multivariate filter and a DSGE model (Smets and Wouters (2007)).

### Contribution of Observables to the Output Gap
- Key empirical findings:
  - The contribution of GDP growth to the ‘Great Recession’ estimates started in 2007 and is smaller than during the largest recession in the sample (1980–1985).
  - The contribution of the unemployment series is the greatest ever for that episode.
  - Information extracted from inflation data is limited; the model parameterization attributes a negligible role for inflation in identification of the output gap.
  - The loading coefficient of the ‘inflation gap’ to the output gap is ̃ρ2 = 0.005 and parameters determining output gap effect on inflation β, Ω are small.
- Behavioural interpretation:
  - Unemployment’s contribution is large and more persistent than GDP and capacity utilization, consistent with US “jobless recovery” since the 1990s (GDP and capacity utilization recover faster than unemployment; inflation expectations well anchored).
  - Model is parametrized so NAIRU takes more time to change than potential output. Both NAIRU and potential output growth vary less than most models based on cycles with frequency 6–32 periods, contributing to realistic magnitudes of the output gap.

### Transfer Function and Frequency-Domain Analysis
- Transfer-function insights (Fig. 3):
  - Gain from inflation is rather flat and very small across the whole spectrum; it does not distinguish between business cycle and high frequency dynamics.
  - Gain of output with respect to GDP growth and changes in capacity utilization: large weight at lower frequencies, some weight at business cycle frequencies (6-32 quarters), little weight at high frequencies.
  - First-difference filtering boosts high frequencies; level-to-output gain requires integration filter with transfer function 1/(2 − 2 cos(ω)).
  - Gain profile from the level of unemployment to the output gap shows weight on low frequencies and only a gradual decline with increasing periodicity, implying large spillover of longer cycles into the output gap.
- Practical implication:
  - The model draws more cyclical information from unemployment than from GDP; unemployment is used to back out the output gap to match the model’s estimate.

### Univariate Approximation Using Unemployment
- A univariate approximation using the HP filter with a large smoothing parameter λ (i.e., smooth trend) can closely approximate the model’s output gap when using unemployment.
- Example parameters cited:
  - HP filter λ = 1600 (used for low-pass filtering example; footnote notes there is nothing magical about 1600).
  - HP λ = 540000 shown in Figure 4 as an alternative extreme smoothing.

### News Effects Decomposition
- Defines a news effect as a projection error; decomposition quantifies contributions of observables to output gap revisions.
- Case studies: 2007Q2 and 2009Q1 (both relevant to the ‘Great Recession’).
- Findings for 2009Q1:
  - Data arriving in 2009Q1 resulted in a further deepening of the output gap; the drop was a complete news relative to model-implied recovery.
  - Largest contribution to the news: GDP growth, followed by capacity utilization, and unemployment.
  - Observed data on inflation or inflation expectations contribute only modestly and revise the estimate persistently.
- Interpretation guidance:
  - Proper interpretation requires understanding both the derivative of the filter to new data and the differential between the forecast of observables and the actual outcome. In 2009Q1 GDP growth, capacity utilization, and unemployment were all lower than predicted.

### Natural Output Gap in a DSGE Model (Smets and Wouters (2007))
- Decomposes the ‘flexible-price equilibrium output gap’ into observables within a medium-sized DSGE model estimating the US economy.
- Definition: output gap = deviation of the ‘natural rate of output’ from actual output (natural rate defined absent nominal wage and price rigidities and with no ‘mark-up’ shocks).
- Observable contributions in this DSGE decomposition:
  - Some variables (hours worked, consumption, output) are important at high and business cycle frequencies.
  - Inflation and interest rates contribute mainly to low frequency dynamics of the output gap.
  - The model extracts information about the natural rate of output while placing little weight on observed inflation data, similarly to the semi-structural model.

### Conclusions and Methodological Implications
- The decomposition answers what observed variables and at which periodicity contribute to estimates of the output gap and allows quantification of news effects that generate revisions.
- Many commonly used potential output estimation methods can be cast as linear filter problems, enabling frequency- and time-domain analysis and clarifying revision properties as properties of two-sided moving averages (filter representations).
- Revision properties and end-point bias relate to the spread of filter weights and forecasting quality of the filter-implied model: more spread weights and worse forecasting properties imply larger real-time revision variance.
- There is no a priori reason multivariate filters in state-space form must have better revision properties than univariate filters; the critical factor is the model structure linking economic theory to optimal signal extraction.

*Source: _wp13105 - 1.   Decomposition into Observables*

### References

### References

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### Appendix A — Parameter Estimates from Beneš et al. (2010)
- Model estimated with United States data for period 1967:1–2010:2.
- Estimation approach: Bayesian-likelihood described as ‘regularized maximum likelihood’ with independent joint prior from truncated-Normal distribution.

- Table 1. Estimated parameters for Beneš et al. (2010) — Prior and Posterior (Mode, Dispersion)
  - ḠY S S — Prior Mode: 2.1400 Dispersion: 0.300 — Posterior Mode: 2.143 Dispersion: 0.041
  - U S S — Prior Mode: 5.8000 Dispersion: 0.300 — Posterior Mode: 5.801 Dispersion: 0.042
  - θ — Prior Mode: 0.7000 Dispersion: 0.030 — Posterior Mode: 0.700 Dispersion: 0.004
  - α — Prior Mode: 0.9000 Dispersion: 0.150 — Posterior Mode: 0.895 Dispersion: 0.022
  - β — Prior Mode: 0.4000 Dispersion: 0.300 — Posterior Mode: 0.176 Dispersion: 0.024
  - Ω — Prior Mode: 0.5000 Dispersion: 0.300 — Posterior Mode: 0.265 Dispersion: 0.038
  - ρ — Prior Mode: 0.8000 Dispersion: 0.150 — Posterior Mode: 0.821 Dispersion: 0.019
  - 100 ̃ρ2 — Prior Mode: 5.000 Dispersion: 3.000 — Posterior Mode: 5.145 Dispersion: 0.418
  - κ1 — Prior Mode: 0.1000 Dispersion: 0.600 — Posterior Mode: 0.353 Dispersion: 0.037
  - κ2 — Prior Mode: 1.500 Dispersion: 1.500 — Posterior Mode: 1.421 Dispersion: 0.070
  - φ1 — Prior Mode: 0.8000 Dispersion: 0.150 — Posterior Mode: 0.729 Dispersion: 0.018
  - φ2 — Prior Mode: 0.3000 Dispersion: 0.150 — Posterior Mode: 0.241 Dispersion: 0.014
  - τ — Prior Mode: 0.1000 Dispersion: 0.150 — Posterior Mode: 0.113 Dispersion: 0.021
  - δ — Prior Mode: 0.5000 Dispersion: 0.150 — Posterior Mode: 0.496 Dispersion: 0.021
  - 100 ̃ω — Prior Mode: 3.000 Dispersion: 1.500 — Posterior Mode: 2.796 Dispersion: 0.207
  - λ — Prior Mode: 2.000 Dispersion: 3.000 — Posterior Mode: 2.078 Dispersion: 0.371
  - σY — Prior Mode: 1.000 Dispersion: 0.300 — Posterior Mode: 0.888 Dispersion: 0.039
  - σG — Prior Mode: 1.000 Dispersion: 0.300 — Posterior Mode: 0.921 Dispersion: 0.042
  - σu — Prior Mode: 0.500 Dispersion: 0.300 — Posterior Mode: 0.198 Dispersion: 0.024
  - σ̄U — Prior Mode: 0.100 Dispersion: 0.150 — Posterior Mode: 0.079 Dispersion: 0.018
  - σḠU — Prior Mode: 0.100 Dispersion: 0.150 — Posterior Mode: 0.087 Dispersion: 0.017
  - σu (second entry) — Prior Mode: 0.400 Dispersion: 0.300 — Posterior Mode: 0.482 Dispersion: 0.033
  - σc — Prior Mode: 0.250 Dispersion: 0.150 — Posterior Mode: 0.288 Dispersion: 0.021
  - σḠC — Prior Mode: 0.075 Dispersion: 0.030 — Posterior Mode: 0.076 Dispersion: 0.004
  - σπ4 — Prior Mode: 0.500 Dispersion: 0.300 — Posterior Mode: 0.554 Dispersion: 0.030
  - σπ4LTE — Prior Mode: 0.300 Dispersion: 0.300 — Posterior Mode: 0.081 Dispersion: 0.008
  - σ̄Y — Prior Mode: 0.250 Dispersion: 0.100 — Posterior Mode: 0.217 Dispersion: 0.013

### Appendix B — Difference between two representations of the HP filter
- HP filter state-space form presented:
  - yt = ̄yt + xt (equation (56))
  - xt = εxt, εxt ∼ N(0, σ2x) (equation (57))
  - ̄yt = ̄yt−1 + βt−1 (equation (58))
  - βt = βt−1 + εgt, εgt ∼ N(0, σ2g) (equation (59))
- Statement: This representation is equivalent to the HP filter state-space representation included in the text; results from both state-space implementations are identical and identical to standard matrix formulas of HP filter implementations.

### Appendix C — Example: Simple Multivariate Filter – Three Representations
- Presents a semi-structural multivariate filter represented as:
  - (i) state-space model
    - yt = xt + τt (equation (60))
    - τt − τt−1 = τt−1 − τt−2 + ετt (equation (61))
    - xt = ρxt−1 − κπgapt + εxt (equation (62))
    - πgapt = λπgapt−1 + θxt + επt (equation (63))
  - (ii) Wiener-Kolmogorov filter
    - Implied Wiener-Kolmogorov estimate of output gap xt:
      - xt = (1 − Wy(L)) yt − Wπ(L) πgapt = (1 − Wy(L)/(1 − L)) Δyt − Wπ(L) πgapt (equation (65))
    - Wy(L) and Wπ(L) expressed as z-transform weights (equations (66) and (67)).
  - (iii) penalized least squares
    - Minimize over {τ}0T: Σt=0T [1/σ2x (εxt)2 + 1/σ2τ (ετt)2 + 1/σ2τ (επt)2] (equation (64))
    - For ρ = κ = θ = 0 and λ = (σx/στ)2, problem (64) is equivalent to penalized least squares formulation of HP filter.
- Note: Time domain weights can be obtained from state-space representation or from Wy(L) and Wπ(L) by inverse z-transform.

### Appendix D — Not for publication: Variance Reduction via Common Component and Multiple Measures
- Key STAT-101 intuition: Adding multiple noisy but relevant measurements reduces variance of estimates and lowers revision variance by concentrating filter weights.
- Simple deterministic example:
  - z1 = μ + u1, z2 = μ + u2 with u1 ∼ N(0, σ21) and u2 ∼ N(0, σ22).
  - Single measurement estimate: μ̂ = z1 with variance σ21.
  - Two-measurement estimate:
    - μ̂ = (σ22/(σ21 + σ22)) z1 + (σ21/(σ21 + σ22)) z2
    - MSE(μ̂) = (1/σ21 + 1/σ22)−1 (equation (68)).
- AR(1) signal with two measurements:
  - xt = ρxt−1 + ν (equation (69))
  - y1,t = xt + ε1,t (equation (70))
  - y2,t = φ xt + ε2,t (equation (71))
- Two-sided filter weight representation:
  - xt|∞ = c1∞ Σi=−∞ λ|i| X t−i (equation (72)), where X is convex combination of observables following weighting scheme (68).
- Special cases:
  - Single measurement (φ = 0): xt = q / (q + |1 − ρL|2) y1,t (equation (73)). For ρ = 0, xt = q/(1 + q) = σ2x/(σ2x + σ21) yt.
  - Two measurements, ρ = 0: xt = [q1/(q1 + φ q2) + |1 − ρL|2] y1,t + [φ q2/(q1 + φ q2) + |1 − ρL|2] y2,t (equation (74)).
- Numerical example: Figure 8 depicts weights for ρ = 0.50 and ρ = 1.0, given φ = 1 and σy1 = σy2 = 0.9.

### Appendix E — Not for publication: Additional Graphs
- Figure 9. HP filter transfer function plotted for:
  - λ = 1600
  - λ = 414400
- Figure 10. Output, unemployment (+1Q) and cap. utilization detrended for period markers shown (e.g., 1966:4 1971:4 1976:4 1981:4 1986:4 1991:4 1996:4 2001:4 2006:4), series:
  - ygap
  - ugap
  - capugap

*Content derived from _wp13105 - References (PDF).*

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