## 2.1 Overview

## Source details

**Canonical URL:** [2.1 Overview](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025134-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025134-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025134-print-pdf.pdf.json)

---

### Data and model scope
- Real monthly GDP is modeled as the sum of N components:
  - gdp_t = sum_{i=1}^N y_{i t}
- Baseline specification:
  - N = 8
  - Components: goods consumption, services consumption, residential investment, non-residential investment, imports, exports, government consumption & investment, changes in inventories.
- Treatment of BEA chain-linking aggregation error:
  - Aggregation error omitted in baseline results; included in robustness checks and data made available with the paper (note in footnote 5).
- Conditioning set I_T consists of observations t = 1,...,T from three sources:
  1. Quarterly NIPA data Y_{i t} with Y_t = (Y_{1 t},...,Y_{N t})′ and, for t = 3,6,9,...:
     - Y_{i t} = sum_{l=0}^2 y_{i t−l}
  2. Exact monthly data for components i = 1,...,K denoted x_{i t} with x_t = (x_{1 t},...,x_{K t})′ (e.g., monthly BEA income and outlays report for consumption).
  3. Other monthly indicators correlated with y_{i t} denoted q_{i t} with q_t = (q_{1 t}′,...,q_{N t}′)′ and total number of indicators P.
- Data-structure summary (as presented in Table 1):
  - For i = 1,...,K: Quarterly NIPA data ✓, Exact monthly data ✓, Monthly indicators ✓ (possibly)
  - For i = K+1,...,N: Quarterly NIPA data ✓, Exact monthly data ✗, Monthly indicators ✓

### Monthly GDP conditional moments
- Target quantities:
  - μ_t = E(y_t | I_T)
  - Σ_t = Var(y_t | I_T)
- Conditional mean and variance for monthly GDP given component estimates:
  - E(gdp_t | I_T) = 1′_N μ̂_t
  - Var(gdp_t | I_T) = 1′_N Σ̂_t 1_N

### The Kalman form
- State-space representation used:
  - ξ_t = F_t ξ_{t−1} + V_t,    cov(V_t) = Q_t
  - Z_t = H_t ξ_t + W_t,        cov(W_t) = R_t
- Key assumption: state and observation errors V_t and W_t are each serially uncorrelated.
- Section maps model elements to ξ_t and Z_t and defines matrices F_t, H_t, R_t, Q_t.

### The state equations
- Component-level stochastic representation:
  - y_{i t} = ȳ_{i t} (1 + ŷ_{i t})
  - log ȳ_{i t} = sum_{m=0}^M γ_{i m} t^m
  - ŷ_{i t} = sum_{l=1}^L ρ_{i l} ŷ_{i t−l} + w_{i t}
- Trend choice and motivation:
  - Polynomial trend chosen to capture differing trends across GDP components.
  - Using ratios (ŷ_{i t} defined as growth ratios) preferred over logs for quarterly adding-up constraints and to allow negative changes in inventories.
- Error structure:
  - w_{i t} serially uncorrelated but cross-correlated across components: Cov(w_{i t}, w_{j t}) ≡ σ^2_{i,j} may be non-zero.
  - Cross-correlations are important information sources (example: high consumption correlated with increased imports and reduced inventories).
- Mapping to Kalman form:
  - ξ_t is the L stacked lags of ŷ_t
  - F_t is matrix of ρ_i coefficients
  - Q_t is an LN × LN matrix, zero except for top-left L × L block equal to covariance matrix of the w_{i t}
- Moving average representation assumed invertible:
  - ŷ_{i t} = sum_{l=0}^∞ a_l w_{i t−l}
  - Cov(ŷ_{i t}, ŷ_{i t−l}) = θ_{i l}, l = 0,...,∞

### The observation equations
- Observables Z_t, loading H_t, and observation covariance R_t partitioned into three parts corresponding to Y_t (quarterly NIPA), x_t (exact monthly), and q_t (monthly indicators):
  - Z_t ((N+K+P)×1) = [Z^Y_t (N×1); Z^x_t (K×1); Z^q_t (P×1)]
  - H_t ((N+K+P)×LN) = [H^Y_t (N×LN); H^x_t (K×LN); H^q_t (P×LN)]
  - R_t block-structured with R^Y_t (N×N), R^x_t (K×K), R^q_t (P×P) on diagonal and zeros off-diagonal blocks as specified.

- Quarterly NIPA data:
  - Aggregation constraint for deviations from trend:
    - Ŷ_{i t} = sum_{l=0}^2 [ȳ_{i t−l} / Ȳ_{i t}] ŷ_{i t−l}
    - Ŷ_t = Y_t / Ȳ_t−1
  - Observables: Z^Y_t = Ŷ_t with missing observations for two months in every quarter (handled by Kalman filter).
  - H^Y_t entries given by ȳ_{i t−l} / Ȳ_{i t}.
  - R^Y_t is zero in theory; numerical stability uses R^Y_t = η I_{N×N} with η a very small number.

- Exact monthly data:
  - x_{i t} / ȳ_{i t−1} = ŷ_{i t}
  - Observables: Z^x_t = (x_{i t} / ȳ_{1 t−1},..., x_{N t} / ȳ_{N t−1})′
  - H^x_t has ones on the primary diagonal, zeroes elsewhere.
  - R^x_t is zero.

- Other monthly indicators:
  - Fit polynomial trend for each indicator and form departures:
    - log ̄q_{i,j t} = sum_{m=0}^M η_{i,j m} t^m
    - ˆq_{i,j t} = q_{i,j t} / ̄q_{i,j t−1}
  - Remove serial correlation via ARIMA (lag length by AIC) and denote residuals ε_{i,j t}.
  - Statistical relation: ε_{i,j t} = κ_{i,j} w_{i t} + u_{i,j t}, with u_{i,j t} white noise and cov(w_{i t}, u_{i,j t}) = 0.
  - Rewriting in terms of ŷ_{i t}:
    - ε_{i,j t} = κ_{i,j} ŷ_{i t} − sum_{l=1}^L κ_{i,j} ρ_{i l} ŷ_{i t−l} + u_{i,j t}
  - Z^q_t is stacked vector of ε_{i,j t}; H^q_t contains κ_{i,j} or κ_{i,j} ρ_{i l} terms.
  - κ_{i,j} and full set of entries of R^q_t are estimated by GMM.
  - R^q_t constrained block-diagonal: cross-correlation of monthly indicators for different components restricted to zero.

- Discussion of alternative specifications:
  - Alternative w_{i t} = sum_j β_{i,j} ε_{i,j t} + ũ_{i,j t} is feasible but offers no clear advantage.
  - Simpler specification ε_{i,j t} = α_{i,j} + β_{i,j} ŷ_{i t} + ũ_{i t} produces serially correlated residuals, violating Kalman assumptions and mis-allocating timing of information.
  - Justification for using ε_{i,j t} = κ_{i,j} w_{i t} + u_{i,j t} includes better handling of missing observations and consistency with zero serial correlation of observation errors.

### Data
- Quarterly NIPA data available back to 1950Q1; starting monthly series: January 1950.
- BEA publishes monthly data for two GDP components: consumption of goods and services (direct monthly analogues).
- Eight further monthly series used as indicators for remaining components (summarized in Table 2).
- Indicator-selection guidance:
  - High bar for including extra series: adding series can worsen estimated monthly GDP because it reduces filter uncertainty but increases estimation uncertainty.
  - Exception: strong a priori substitutes for GDP components (e.g., Census monthly import/export series, BEA monthly change in inventories).

- Selected monthly series and start dates (as presented in Table 2):
  - Goods Consumption: x_{i t} ✓, Starting Jan 1959
  - Services Consumption: x_{i t} ✓, Starting Jan 1959
  - Non-residential fixed investment: Monthly indicator Non-residential construction, Starting Jan 2002
  - Residential fixed investment: Monthly indicators Housing starts (Starting Jan 1960), Single-family housing starts (Starting Jan 1959)
  - Government spending: Monthly indicators None
  - Exports: Monthly indicator Census export volume, Starting Jan 1992
  - Imports: Monthly indicator Census import volume, Starting Jan 1992
  - Inventories: Monthly indicators Change in manufacturing inventories (Starting Jan 1950) and Change in inventories (Starting Jan 1997)
  - Aggregation error: Quarterly ✓, Monthly indicators None

### Estimation strategy
- Trend estimation:
  - Estimate ̃γ_{i m} by regressing log Y_{i t} on polynomial time trend.
  - Set log ȳ_{i t} = (1/3) sum_{m=0}^M ̃γ_{i m} (t+1)^m.
  - For baseline M = 2, inversion:
    - γ_{i 0} = (1/3) [̃γ_{i 0} + ̃γ_{i 1} + ̃γ_{i 2}]
    - γ_{i 1} = (1/3) [̃γ_{i 1} + 2 ̃γ_{i 2}]
    - γ_{i 2} = (1/3) ̃γ_{i 2}

- State-parameter estimation (GMM moments from quarterly averages):
  - Use relationships between autocovariances of Ŷ_{i t} and ŷ_{i t} approximating ȳ_{i t} ≃ (1/3) Ȳ_t:
    - Var(Ŷ_{i t}) ≃ (1/9) [3 θ_{i 0} + 4 θ_{i 1} + 2 θ_{i 2}]
    - Cov(Ŷ_{i t}, Ŷ_{i t−3k}) ≃ (1/9) [θ_{i 3k−2} + 2 θ_{i 3k−1} + 3 θ_{i 3k} + 2 θ_{i 3k+1} + θ_{i 3k+2}], k ≥ 1
  - Cross-covariance moment condition:
    - Cov(Ŷ_{i t}, Ŷ_{j t}) = (1/9)[a_{i 0} a_{j 0} + (a_{i 0}+a_{i 1})(a_{j 0}+a_{j 1}) + sum_{s=0}^∞ (sum_{l=0}^2 a_{i s+l})(sum_{l=0}^2 a_{j s+l})] σ^2_{i,j}
  - Replace population covariances with sample analogues to form GMM moment conditions.

- Observation-equation parameter estimation (GMM moments):
  - Construct quarterly-equivalent residual E_{i,j t} = (1/3) sum_{l=0}^2 ε_{i,j t−l}.
  - Covariance relation yields κ_{i,j} via:
    - Cov(E_{i,j t}, Ŷ_{i t}) = (1/9) [3 a_{i 0} + 2 a_{i 1} + a_{i 2}] κ_{i,j} σ^2_i
  - Residual covariance of u's:
    - Cov(u_{i,j t}, u_{k,l t}) = Cov(ε_{i,j t}, ε_{k,l t}) − κ_{i,j} κ_{k,l} σ^2_{i,k}
  - Replace population covariances with sample analogues to form GMM moment conditions.

- Two-stage implementation:
  1. Estimate σ^2_i, ρ_{i 1},...,ρ_{i L} by regular GMM applied to equations for each i.
  2. Given stage-1 estimates, solve directly for remaining parameters by inverting sample analogues of equations (17)-(19).
- Advantages:
  - Two-stage yields same estimates as one big GMM but is much quicker and easier than maximum likelihood.
- Parameter dimensionality in baseline:
  - Baseline L = 1, N = 9, K = 2, P = 9 → 60 parameters to optimize if using full likelihood.

### Measuring uncertainty via the delta method
- Two sources of uncertainty:
  - Kalman prediction error, conditional on parameters (filter uncertainty).
  - Parameter estimation uncertainty.
- Kalman outputs:
  - Kalman mean: ˆE(y_i_t | I_T, Ψ)
  - Kalman prediction error variance: ˆV(y_i_t | I_T, Ψ)
- Total variance decomposition:
  - Var(y_t | I_T) = ∫ ˆV(y_t | I_T, Ψ) dF_n(Ψ) + ∫ (ˆE(y_t | I_T, Ψ*) − E(y_t | I_T, Ψ))^2 dF_n(Ψ)
- Delta-method computation:
  - Numerically compute derivatives of Kalman means with respect to Ψ by perturbing Ψ in each dimension.
  - Σ_t = ˆV(y_t | I_T, Ψ*) ︳ Filter uncertainty
    + (d ˆE(y_t | I_T, Ψ*) / dΨ)' Var Ψ* (d ˆE(y_t | I_T, Ψ*) / dΨ) ︳ Estimation uncertainty
- Empirical implications (numerical findings):
  - Confidence intervals based solely on filter uncertainty are too narrow because they omit parameter uncertainty.
  - Standard errors for GDP:
    - 0.3 percent of trend or less since the 1960s.
    - Less than 0.2 percent of trend since the late 1990s (COVID is a notable exception).
  - Introduction of monthly consumption data in 1959:
    - Direct effect: standard errors for the two consumption series reduce to zero.
    - Spillover effect: uncertainty for other components (notably the two investment series and imports) declines; imports fall by almost one half due to strong residual correlation with consumption.
  - Introduction of monthly exports and inventories data in the 1990s dramatically reduces uncertainty for those components.
  - By onset of COVID (March 2020), many more monthly indicators are available and series become much more precise; the abrupt halt in activity is precisely identified as starting in March 2020.
- Decomposition dynamics:
  - Filter uncertainty is relatively constant over intervals with no change in the information set and drops when new monthly data become available.
  - Estimation uncertainty varies over time and increases in volatile periods because estimated coefficients scale the data and their uncertainty.
  - Adding monthly indicators reduces filter uncertainty but increases estimation uncertainty; net effect depends on balance between these forces.

- Signal-to-noise ratios (decadal averages, standard deviations in percentage of trend from Table 5):
  - 1950s: (1) 3.87, (2) 1.32, (3) 0.56, (1)/(3) 6.92, (2)/(3) 2.3
  - 1960s: (1) 3.23, (2) 1.20, (3) 0.30, (1)/(3) 10.7, (2)/(3) 3.9
  - 1970s: (1) 2.17, (2) 1.16, (3) 0.27, (1)/(3) 8.14, (2)/(3) 4.4
  - 1980s: (1) 2.81, (2) 1.16, (3) 0.24, (1)/(3) 11.54, (2)/(3) 4.7
  - 1990s: (1) 1.86, (2) 1.17, (3) 0.20, (1)/(3) 9.4, (2)/(3) 5.9
  - 2000s: (1) 2.43, (2) 1.22, (3) 0.15, (1)/(3) 16.0, (2)/(3) 8.0
  - 2010s: (1) 0.72, (2) 1.30, (3) 0.15, (1)/(3) 4.78, (2)/(3) 8.5
  - 2020s: (1) 2.79, (2) 1.40, (3) 0.18, (1)/(3) 15.6, (2)/(3) 7.8
- Interpretation:
  - Preferred signal-to-noise ratio implies one-month-ahead variation in estimated monthly GDP is about 2 times the standard error in the 1950s and about 7 times in the 2010s.
  - On average over the whole sample, the monthly GDP estimate’s standard error is about one-fourth to one-fifth of the one-month growth volatility (about 4.5 times signal-to-noise on average).
  - Appendix A (Table A2) reports component-level results: imports have the highest signal-to-noise ratio (except consumption where exact data exist), inventories the lowest.

- Practical notes:
  - Numerical derivatives for the delta method obtained by perturbing Ψ.
  - Appendix B summarizes steps to produce final monthly national accounts estimates.
  - Model-selection tradeoff: adding monthly indicators reduces filter uncertainty but increases estimation uncertainty; net effect must be evaluated when choosing which monthly indicators to include.

### Recession dating (Section 5.4)
- NBER definition: a recession is “a significant decline in economic activity that is spread across the economy and that lasts more than a few months”.
- Monthly-GDP-based dating methodology:
  - peak_j = argmax_{t < trough_{j} - 1} gdp_t
  - trough_j = argmin_{t > peak_{j} + 1} gdp_t
  - Peaks and troughs cannot be in consecutive months.
  - Monthly GDP series include 95 percent confidence intervals for peak/trough dates (range of dates where point estimates are within two standard deviations of the estimated peak or trough).
- Comparison of eleven NBER recessions (NBER dates vs monthly GDP point estimates with 95 percent CIs):
  1. Early 1950s — NBER: Jul 1953 to May 1954; Monthly GDP: May 1953 to Jun 1954 (Monthly CI: (Mar 1953,Aug 1953) peak; (Nov 1953,Aug 1954) trough)
  2. Late 1950s — NBER: Aug 1957 to Apr 1958; Monthly GDP: Sep 1957 to Feb 1958 (Monthly CI: (Dec 1956,Dec 1957) peak; (Jan 1958,Apr 1958) trough)
  3. Early 1960s — NBER: Apr 1960 to Feb 1961; Monthly GDP: Mar 1960 to May 1960 (Monthly CI: (Mar 1960,Apr 1960) peak; (May 1960,Dec 1960) trough)
  4. Late 1960s — NBER: Dec 1969 to Nov 1970; Monthly GDP: Aug 1969 to Apr 1970 (Monthly CI: (Aug 1969,Oct 1969) peak; (Nov 1969,May 1970) trough)
  5. Oil Crisis — NBER: Nov 1973 to Mar 1975; Monthly GDP: Nov 1973 to Apr 1975 (Monthly CI: (May 1973,Nov 1973) peak; (Mar 1975,Apr 1975) trough)
  6. 1980 recession — NBER: Jan 1980 to Jul 1980; Monthly GDP: Jan 1980 to May 1980 (Monthly CI: (Jan 1980,Jan 1980) peak; (May 1980,Sep 1980) trough)
  7. Early 1980s — NBER: Jul 1981 to Nov 1982; Monthly GDP: Aug 1981 to Mar 1982 (Monthly CI: (Jan 1981,Sep 1981) peak; (Feb 1982,Mar 1982) trough)
  8. Early 1990s — NBER: Jul 1990 to Mar 1991; Monthly GDP: Jul 1990 to Jan 1991 (Monthly CI: (Jan 1990,Sep 1990) peak; (Jan 1991,Jan 1991) trough)
  9. Dot-com bubble — NBER: Mar 2001 to Nov 2001; Monthly GDP: May 2001 to Sep 2001 (Monthly CI: (May 2001,Aug 2001) peak; (Sep 2001,Sep 2001) trough)
  10. Great Recession — NBER: Dec 2007 to Jun 2009; Monthly GDP: Jun 2008 to Apr 2009 (Monthly CI: (May 2008,Jul 2008) peak; (Feb 2009,Sep 2009) trough)
  11. COVID-19 — NBER: Feb 2020 to Apr 2020; Monthly GDP: Feb 2020 to Apr 2020 (Monthly CI: (Dec 2019,Feb 2020) peak; (Apr 2020,Apr 2020) trough)
- Main findings:
  - Start dates: generally close alignment; three episodes have statistically different start dates (late 1960s, 2001 dot-com, Great Recession) associated with "stuttering" starts.
  - End dates: larger and more systematic differences; five of eleven recessions have statistically significantly different end dates across methods, usually with earlier end dates using monthly GDP.
  - Interpretation: NBER’s requirement for broader confirmation across indicators (including lagging labor market indicators) can delay official dating of recession ends relative to monthly GDP troughs.

### Robustness to model specification (Section 6.4)
- Four alternative specifications estimated:
  - Use efficient GMM rather than unweighted GMM.
  - Fit the model to more dynamic moments, matching 12 autoregressive equations.
  - Consider a higher-order (third-order) polynomial trend.
  - Model the aggregation error in the national account specifically as a ninth component of GDP.
- Key findings:
  - In all cases except the specification that includes the aggregation error, differences from the baseline are minimal.
  - When aggregation error is included as a separate monthly series:
    - Differences for the GDP series are larger because headline quarterly GDP now includes an explicit aggregation error.
    - Inclusion of the aggregation error has no noticeable effect on any of the other components of GDP; it "drives a wedge between the components and the total."
  - Data associated with the paper report two series for GDP: one including the aggregation error and one excluding it.
    - Users can choose to include aggregation error to match headline GDP exactly, or drop it to have components sum exactly.
- Robustness numeric summaries (average absolute difference versus baseline, including aggregation error: GDP by decade):
  - 1950s: 13.304
  - 1960s: 7.730
  - 1970s: 4.742
  - 1980s: 5.251
  - 1990s: 4.419
  - 2000s: 2.700
  - 2010s: 2.488
  - 2020s: 1.864
- Efficient GMM alternative (average absolute differences for GDP by decade):
  - 1950s: 0.034
  - 1960s: 0.042
  - 1970s: 0.034
  - 1980s: 0.042
  - 1990s: 0.037
  - 2000s: 0.039
  - 2010s: 0.036
  - 2020s: 0.102
- Fitting 12 AR equations and third-order trend specifications similarly yield small average absolute differences for GDP in most decades (examples):
  - Fitting 12 AR equations: GDP 2020s: 0.028
  - Third-order trend: GDP 2020s: 0.054
- Implications:
  - Baseline monthly series are robust to alternative specifications other than explicit inclusion of aggregation error.
  - Main practical decision for users mirrors quarterly data choice: include aggregation error to match headline GDP or exclude it so components sum exactly; component-level dynamics are not materially altered by this choice.

_Italic: Source: wpiea2025134-print-pdf - 2.1 Overview (canonical PDF provided)._

### 2.1    Overview

### 2.1 Overview

### Data and model scope
- Real monthly GDP is modeled as the sum of N components:
  - gdp_t = sum_{i=1}^N y_{i t} (equation (1))
- Baseline specification:
  - N = 8
  - Components: goods consumption, services consumption, residential investment, non-residential investment, imports, exports, government consumption & investment, changes in inventories.
- Treatment of BEA chain-linking aggregation error:
  - Aggregation error omitted in baseline results; included in robustness checks and data made available with the paper (note in footnote 5).
- Conditioning set I_T consists of observations t = 1,...,T from three sources:
  1. Quarterly NIPA data Y_{i t} with Y_t = (Y_{1 t},...,Y_{N t})′ and, for t = 3,6,9,...:
     - Y_{i t} = sum_{l=0}^2 y_{i t−l} (equation (2))
  2. Exact monthly data for components i = 1,...,K denoted x_{i t} with x_t = (x_{1 t},...,x_{K t})′ (e.g., monthly BEA income and outlays report for consumption).
  3. Other monthly indicators correlated with y_{i t} denoted q_{i t} with q_t = (q_{1 t}′,...,q_{N t}′)′ and total number of indicators P.
- Data structure summary (as presented in Table 1):
  - For i = 1,...,K: Quarterly NIPA data ✓, Exact monthly data ✓, Monthly indicators ✓ (possibly)
  - For i = K+1,...,N: Quarterly NIPA data ✓, Exact monthly data ✗, Monthly indicators ✓

### Monthly GDP conditional moments
- Aim: estimate conditional means and variances for component vector y_t = (y_{1 t},...,y_{N t})′:
  - μ_t = E(y_t | I_T)
  - Σ_t = Var(y_t | I_T)
- Conditional mean and variance for monthly GDP given component estimates:
  - E(gdp_t | I_T) = 1′_N μ̂_t
  - Var(gdp_t | I_T) = 1′_N Σ̂_t 1_N (equation (3))

---

### 2.2 The Kalman form

### Filtering framework and assumptions
- Use of the Kalman filter to estimate μ_t and Σ_t.
- Standard Kalman state-space representation:
  - ξ_t = F_t ξ_{t−1} + V_t,    cov(V_t) = Q_t
  - Z_t = H_t ξ_t + W_t,        cov(W_t) = R_t
- Key assumption: state and observation errors V_t and W_t are each serially uncorrelated.
- Section devoted to mapping model elements to ξ_t and Z_t and defining matrices F_t, H_t, R_t, Q_t.

---

### 2.3 The state equations

### Component-level stochastic representation
- Each component y_{i t} modeled as departures from a long-run polynomial growth path with an AR(L) process for the stochastic deviation:
  - y_{i t} = ȳ_{i t} (1 + ŷ_{i t}) (equation (4))
  - log ȳ_{i t} = sum_{m=0}^M γ_{i m} t^m (equation (5))
  - ŷ_{i t} = sum_{l=1}^L ρ_{i l} ŷ_{i t−l} + w_{i t} (equation (6))
- Trend choice and motivation:
  - Polynomial trend chosen to capture differing trends across GDP components.
  - Using ratios (ŷ_{i t} defined as growth ratios) preferred over logs for quarterly adding-up constraints and to allow negative changes in inventories (footnote 7).
- Error structure:
  - w_{i t} serially uncorrelated (Kalman requirement) but cross-correlated across components:
    - Cov(w_{i t}, w_{j t}) ≡ σ^2_{i,j} may be non-zero.
  - Cross-correlations are important information sources (example: high consumption correlated with increased imports and reduced inventories).
- Mapping to Kalman form:
  - ξ_t is the L stacked lags of ŷ_t
  - F_t is matrix of ρ_i coefficients
  - Q_t is an LN × LN matrix, zero except for top-left L × L block equal to covariance matrix of the w_{i t}
- Moving average representation assumed invertible:
  - ŷ_{i t} = sum_{l=0}^∞ a_l w_{i t−l} (equation (7))
  - Covariances: Cov(ŷ_{i t}, ŷ_{i t−l}) = θ_{i l}, l = 0,...,∞ (equation (8))

---

### 2.4 The observation equations

### Partitioning observations and matrices
- Observables Z_t, loading H_t, and observation covariance R_t partitioned into three parts corresponding to Y_t (quarterly NIPA), x_t (exact monthly), and q_t (monthly indicators):
  - Z_t ((N+K+P)×1) = [Z^Y_t (N×1); Z^x_t (K×1); Z^q_t (P×1)]
  - H_t ((N+K+P)×LN) = [H^Y_t (N×LN); H^x_t (K×LN); H^q_t (P×LN)]
  - R_t ((N+K+P)×(N+K+P)) block-structured with R^Y_t (N×N), R^x_t (K×K), R^q_t (P×P) on diagonal and zeros off-diagonal blocks as specified.

### 2.4.1 Quarterly NIPA data
- Aggregation constraint:
  - Y_t = sum_{l=0}^2 [ȳ_{i t−l} / Ȳ_{i t}] (1 + ŷ_{i t−l}) ⇒ Ŷ_{i t} = sum_{l=0}^2 [ȳ_{i t−l} / Ȳ_{i t}] ŷ_{i t−l}
  - Ŷ_t = Y_t / Ȳ_t−1 is deviation of quarterly NIPA from trend, with Ȳ_{i t} = sum_{l=0}^2 ȳ_{i t−l}
- Observables: Z^Y_t = Ŷ_t with missing observations for two months in every quarter (handled by Kalman filter).
- H^Y_t entries given by ȳ_{i t−l} / Ȳ_{i t}.
- R^Y_t is zero in theory, but in practice numerical stability uses R^Y_t = η I_{N×N} with η a very small number.

### 2.4.2 Exact data
- Exact monthly observations satisfy:
  - x_{i t} / ȳ_{i t−1} = ŷ_{i t}
- Observables: Z^x_t = (x_{i t} / ȳ_{1 t−1},..., x_{N t} / ȳ_{N t−1})′
- H^x_t has ones on the primary diagonal, zeroes elsewhere.
- R^x_t is zero.

### 2.4.3 Other monthly indicators
- For indicator q_{i,j t}, fit its own polynomial trend ̄q_{i,j t}:
  - log ̄q_{i,j t} = sum_{m=0}^M η_{i,j m} t^m (equation (9))
- Departures from trend: ˆq_{i,j t} = q_{i,j t} / ̄q_{i,j t−1}
- To eliminate serial correlation, estimate ARIMA for each ˆq_{i,j t} (lag length chosen by AIC) and denote serially uncorrelated residuals by ε_{i,j t}.
- Statistical relation between indicator residuals and component innovations:
  - ε_{i,j t} = κ_{i,j} w_{i t} + u_{i,j t} (equation (10))
    - u_{i,j t} white noise, cov(w_{i t}, u_{i,j t}) = 0
- Rewriting in terms of ŷ_{i t}:
  - ε_{i,j t} = κ_{i,j} ŷ_{i t} − sum_{l=1}^L κ_{i,j} ρ_{i l} ŷ_{i t−l} + u_{i,j t} (equation (11))
- Z^q_t is stacked vector of ε_{i,j t}; H^q_t contains κ_{i,j} or κ_{i,j} ρ_{i l} terms.
- κ_{i,j} and full set of entries of R^q_t are estimated by GMM (Section 4).
- R^x_t constrained block-diagonal to avoid parameter explosion; cross-correlation of monthly indicators for different components restricted to zero.

### Discussion of alternative specifications
- Alternative observation equation placing state on left (w_{i t} = sum_j β_{i,j} ε_{i,j t} + ũ_{i,j t}, equation (12)) is feasible but offers no clear advantage.
- Specification ε_{i,j t} = α_{i,j} + β_{i,j} ŷ_{i t} + ũ_{i t} (equation (13)) is simpler to estimate but produces serially correlated residuals, violating Kalman assumptions and mis-allocating timing of information.
- Justification for equation (10):
  - Handles missing observations better (can drop only missing indicators rather than entire periods).
  - Maintains consistency with zero serial correlation of observation errors.

---

### 3 Data

### 3.1 Quarterly National Income and Product Accounts
- Quarterly NIPA data available back to 1950Q1.
- This defines starting data for monthly series: January 1950.
- Figure 1 shows quarterly NIPA data with estimated trends; sample indicated as 1050Q1-2025Q1 (as labeled in the source).

### 3.2 Monthly indicators
- BEA publishes monthly data for two GDP components: consumption of goods and services (direct monthly analogues of corresponding quarterly NIPA series).
- Eight further monthly series used as indicators for remaining components summarized in Table 2 and shown in Figure 2; residualized versions in Figure A1 (Appendix A).
- Guidance on indicator selection:
  - High bar for including extra series: adding series can worsen estimated monthly GDP because it reduces filter uncertainty but increases estimation uncertainty.
  - Exception: strong a priori substitutes for GDP components (e.g., Census monthly import/export series, BEA monthly change in inventories).
- Table 2 listing monthly data series used (as presented):
  - Goods Consumption: x_{i t} ✓, Starting Jan 1959
  - Services Consumption: x_{i t} ✓, Starting Jan 1959
  - Non-residential fixed investment: Quarterly ✓, Exact monthly ✗, Monthly indicator Non-residential construction, Starting Jan 2002
  - Residential fixed investment: Quarterly ✓, Exact monthly ✗, Monthly indicators Housing starts, Single-family housing starts, Starting Jan 1960 and Jan 1959 respectively
  - Government spending: Quarterly ✓, Exact monthly ✗, Monthly indicators None
  - Exports: Quarterly ✓, Exact monthly ✗, Monthly indicator Census export volume, Starting Jan 1992
  - Imports: Quarterly ✓, Exact monthly ✗, Monthly indicator Census import volume, Starting Jan 1992
  - Inventories: Quarterly ✓, Exact monthly ✗, Monthly indicators Change in manufacturing inventories (Starting Jan 1950) and Change in inventories (Starting Jan 1997)
  - Aggregation error: Quarterly ✓, Exact monthly ✗, Monthly indicators None

---

### 4 Estimation Strategy

### 4.1 Trend estimation
- Estimate γ_{i} trend parameters by regressing quarterly NIPA Y_{i t} on a polynomial time trend:
  - log Y_{i t} = sum_{m=0}^M ̃γ_{i m} t^m + e_{i t}
- Set:
  - log ȳ_{i t} = (1/3) sum_{m=0}^M ̃γ_{i m} (t+1)^m (equation (14))
- For baseline M = 2, inversion yields:
  - γ_{i 0} = (1/3) [̃γ_{i 0} + ̃γ_{i 1} + ̃γ_{i 2}]
  - γ_{i 1} = (1/3) [̃γ_{i 1} + 2 ̃γ_{i 2}]
  - γ_{i 2} = (1/3) ̃γ_{i 2}

### 4.2 State-parameter estimation (GMM moments from quarterly averages)
- Use relationships between autocovariances of Ŷ_{i t} (quarterly deviations) and those of ŷ_{i t} (monthly deviations), approximating ȳ_{i t} ≃ (1/3) Ȳ_t:
  - Var(Ŷ_{i t}) = (1/9) Var(sum_{j=0}^2 ŷ_{i t−j}) ≃ (1/9) [3 θ_{i 0} + 4 θ_{i 1} + 2 θ_{i 2}] (equation (15))
  - Cov(Ŷ_{i t}, Ŷ_{i t−3k}) = (1/9) [θ_{i 3k−2} + 2 θ_{i 3k−1} + 3 θ_{i 3k} + 2 θ_{i 3k+1} + θ_{i 3k+2}], k ≥ 1 (equation (16))
- These provide over-identifying moment conditions for σ^2_i, ρ_{i 1},...,ρ_{i L}.
- Cross-covariance moment condition from quarterly averages:
  - Cov(Ŷ_{i t}, Ŷ_{j t}) = (1/9)[a_{i 0} a_{j 0} + (a_{i 0}+a_{i 1})(a_{j 0}+a_{j 1}) + sum_{s=0}^∞ (sum_{l=0}^2 a_{i s+l})(sum_{l=0}^2 a_{j s+l})] σ^2_{i,j} (equation (17))
- Replacing population covariances with sample analogues yields GMM moment conditions.

### 4.3 Observation-equation parameter estimation (GMM moments)
- Construct quarterly-equivalent residual for indicator q_{i,j t}:
  - E_{i,j t} = (1/3) sum_{l=0}^2 ε_{i,j t−l}
- Covariance relation leading to κ_{i,j}:
  - Cov(E_{i,j t}, Ŷ_{i t}) ≃ (1/9) [3 a_{i 0} + 2 a_{i 1} + a_{i 2}] Cov(ε_{i,j t}, w_{i t})
  - Given Cov(w_{i t}, u_{i,j t}) = 0 and κ_{i,j} = Cov(ε_{i,j t}, w_{i t}) / σ^2_i, obtain:
    - Cov(E_{i,j t}, Ŷ_{i t}) = (1/9) [3 a_{i 0} + 2 a_{i 1} + a_{i 2}] κ_{i,j} σ^2_i (equation (18))
- For residual covariances of u_{i,j t}:
  - Cov(ε_{i,j t}, ε_{k,l t}) = κ_{i,j} κ_{k,l} σ^2_{i,k} + Cov(u_{i,j t}, u_{k,l t})
  - Hence Cov(u_{i,j t}, u_{k,l t}) = Cov(ε_{i,j t}, ε_{k,l t}) − κ_{i,j} κ_{k,l} σ^2_{i,k} (equation (19))
- Replace population covariances with sample analogues to form moment conditions for GMM.

### 4.4 Two-stage implementation and computational considerations
- Moment conditions (equations (15), (16), (17), (18), (19)) are sufficient to identify all Kalman parameters F_t, H_t, R_t, Q_t.
- Two-stage estimation approach:
  1. Estimate σ^2_i, ρ_{i 1},...,ρ_{i L} by regular GMM applied to sample analogues of equations (15) and (16) separately for each i.
  2. Given estimates from stage 1, solve directly for remaining parameters by inverting sample analogues of equations (17)-(19) (just-identified conditional on stage-1 estimates).
- Advantages of two-stage approach:
  - Yields exactly same estimates as one big GMM using all moment conditions.
  - Much quicker and easier than maximum likelihood, which requires likelihood evaluation for each candidate parameter vector and is computationally intensive.
- Parameter dimensionality in baseline:
  - Baseline L = 1, N = 9, K = 2, P = 9 → 60 parameters to optimize if using full likelihood (noting the rapid growth of parameters with square of number of components and data series).

_Italic: Source: wpiea2025134-print-pdf - 2.1 Overview (canonical PDF provided)_.

### 4.5    Measuring uncertainty via the delta method

### 4.5    Measuring uncertainty via the delta method

### Conceptual decomposition of uncertainty
- Two sources of uncertainty around the estimated monthly series:
  - The Kalman prediction error, conditional on parameters (filter uncertainty).
  - The parameter estimation uncertainty.
- Kalman filter outputs for each i:
  - The Kalman mean: ˆE(y_i_t | I_T, Ψ)
  - The Kalman prediction error variance:
    ˆV(y_i_t | I_T, Ψ) = E[(y_i_t − ˆE(y_i_t | I_T, Ψ))^2 | I_T, Ψ]
- Consistency result for the Kalman mean (equation (20)):
  - If Ψ* → Ψ0 then ˆE(y_i_t | I_T, Ψ*) → E(y_i_t | I_T)
- Total variance integrates parameter uncertainty over the sampling distribution F_n(Ψ):
  - Var(y_t | I_T) = ∫ ˆV(y_t | I_T, Ψ) dF_n(Ψ) + ∫ (ˆE(y_t | I_T, Ψ*) − E(y_t | I_T, Ψ))^2 dF_n(Ψ)
  - Intuition: error variance including parameter uncertainty = average Kalman pred. error variance + variance of the Kalman mean over parameters.

### Delta-method adjustment for parameter uncertainty
- The paper computes the second term (variance of the Kalman mean over parameters) using the delta method:
  - Numerically compute sequence of derivatives of Kalman means in each period with respect to the parameter vector Ψ by perturbing Ψ in each dimension and computing marginal changes in the Kalman mean.
- Covariance for the estimated national accounts components (equation (21)):
  - Σ_t = ˆV(y_t | I_T, Ψ*) ︳ Filter uncertainty
    + (d ˆE(y_t | I_T, Ψ*) / dΨ)' Var Ψ* (d ˆE(y_t | I_T, Ψ*) / dΨ) ︳ Estimation uncertainty
- Consistency: with consistent Ψ*, Σ_t is a consistent estimator of Var(y_t | I_T).

### Empirical implications and numerical findings
- Confidence intervals based solely on ˆV(· | I_T, Ψ) (filter-only) will be too narrow because they omit parameter uncertainty.
- Time evolution of uncertainty (summary of Figures and text):
  - Standard errors for GDP are 0.3 percent of trend or less since the 1960s.
  - Standard errors for GDP are less than 0.2 percent of trend since the late 1990s (COVID is a notable exception).
  - Introduction of monthly consumption data in 1959:
    - Direct effect: standard errors for the two consumption series reduce to zero.
    - Spillover effect: uncertainty for other components (notably the two investment series and imports) declines; imports fall by almost one half due to strong residual correlation with consumption.
  - Introduction of monthly exports and inventories data in the 1990s dramatically reduces uncertainty for those components.
  - By onset of COVID (March 2020), many more monthly indicators are available and series become much more precise; the abrupt halt in activity is precisely identified as starting in March 2020.
- Decomposition dynamics (filter vs estimation uncertainty):
  - Filter uncertainty is relatively constant over intervals with no change in the information set and drops when new monthly data become available.
  - Estimation uncertainty varies over time and increases in volatile periods because estimated coefficients (multipliers) scale the data and hence scale their own uncertainty.
  - Adding monthly indicators reduces filter uncertainty but increases estimation uncertainty because new data introduce additional parameters; net effect depends on the balance between these forces (model-selection tradeoff).

### Signal-to-noise ratios (Table 5)
- Definitions:
  - Preferred signal variance: one-step-ahead prediction variance of detrended GDP, Var_{t−1} μ_gdp_t.
  - Signal-to-noise ratio reported as (1)/(3) and (2)/(3) in Table 5.
- Decadal averages (standard deviations in percentage of trend):
  - 1950s: (1) 3.87, (2) 1.32, (3) 0.56, (1)/(3) 6.92, (2)/(3) 2.3
  - 1960s: (1) 3.23, (2) 1.20, (3) 0.30, (1)/(3) 10.7, (2)/(3) 3.9
  - 1970s: (1) 2.17, (2) 1.16, (3) 0.27, (1)/(3) 8.14, (2)/(3) 4.4
  - 1980s: (1) 2.81, (2) 1.16, (3) 0.24, (1)/(3) 11.54, (2)/(3) 4.7
  - 1990s: (1) 1.86, (2) 1.17, (3) 0.20, (1)/(3) 9.4, (2)/(3) 5.9
  - 2000s: (1) 2.43, (2) 1.22, (3) 0.15, (1)/(3) 16.0, (2)/(3) 8.0
  - 2010s: (1) 0.72, (2) 1.30, (3) 0.15, (1)/(3) 4.78, (2)/(3) 8.5
  - 2020s: (1) 2.79, (2) 1.40, (3) 0.18, (1)/(3) 15.6, (2)/(3) 7.8
- Interpretation:
  - Preferred signal-to-noise ratio (rightmost column) implies the one-month-ahead variation in estimated monthly GDP is about 2 times the standard error in the 1950s and about 7 times in the 2010s.
  - On average over the whole sample, the monthly GDP estimate’s standard error is about one-fourth to one-fifth of the one-month growth volatility (about 4.5 times signal-to-noise on average).
  - Appendix A (Table A2) reports component-level results: imports have the highest signal-to-noise ratio (except consumption where exact data exist), inventories the lowest.

### Practical notes on implementation
- Numerical calculation of derivatives for the delta method is done by perturbing Ψ in each dimension and computing marginal changes in Kalman means.
- Appendix B summarizes the steps to produce final monthly national accounts estimates.
- The analysis highlights a model-selection tradeoff: adding monthly indicators reduces filter uncertainty but increases estimation uncertainty because of additional parameters; the net effect must be evaluated when choosing which monthly indicators to include.

*Source: Section 4.5 and related exposition in the provided content unit.*

### 5.4    Recession Dating

### 5.4    Recession Dating

### NBER definition and role of monthly GDP
- The NBER business cycle dating committee (BCDC) defines a recession as “a significant decline in economic activity that is spread across the economy and that lasts more than a few months”.
- The BCDC identifies recessions using a broad set of indicators with considerable latitude on the importance placed on each.
- Given its importance as an aggregate summary measure of economic activity, a monthly GDP data would likely be a key input into this process.
- Sample references in the chapter: Jan 1960 - Dec 2019 (Figure 8); Jan 1950-Mar 2025 (Table 6).

### Monthly-GDP-based dating methodology
- Peaks and troughs are defined using the estimated monthly GDP series:
  - peak_j = argmax_{t < trough_{j} - 1} gdp_t
  - trough_j = argmin_{t > peak_{j} + 1} gdp_t
  - (Formula numbered (22) in source.)
- Imposition: peaks and troughs cannot be in consecutive months.
- Monthly GDP series include 95 percent confidence intervals for peak/trough dates, computed as the range of dates where point estimates are within two standard deviations of the estimated peak or trough.

### Recession dates: NBER versus monthly GDP (summary of Table 6)
- The eleven NBER recessions compared (NBER dates vs monthly GDP point estimates with 95 percent CIs):
  1. Early 1950s — NBER: Jul 1953 to May 1954; Monthly GDP: May 1953 to Jun 1954 (Monthly CI: (Mar 1953,Aug 1953) for peak; (Nov 1953,Aug 1954) for trough)
  2. Late 1950s — NBER: Aug 1957 to Apr 1958; Monthly GDP: Sep 1957 to Feb 1958 (Monthly CI: (Dec 1956,Dec 1957) peak; (Jan 1958,Apr 1958) trough)
  3. Early 1960s — NBER: Apr 1960 to Feb 1961; Monthly GDP: Mar 1960 to May 1960 (Monthly CI: (Mar 1960,Apr 1960) peak; (May 1960,Dec 1960) trough)
  4. Late 1960s — NBER: Dec 1969 to Nov 1970; Monthly GDP: Aug 1969 to Apr 1970 (Monthly CI: (Aug 1969,Oct 1969) peak; (Nov 1969,May 1970) trough)
  5. Oil Crisis — NBER: Nov 1973 to Mar 1975; Monthly GDP: Nov 1973 to Apr 1975 (Monthly CI: (May 1973,Nov 1973) peak; (Mar 1975,Apr 1975) trough)
  6. 1980 recession — NBER: Jan 1980 to Jul 1980; Monthly GDP: Jan 1980 to May 1980 (Monthly CI: (Jan 1980,Jan 1980) peak; (May 1980,Sep 1980) trough)
  7. Early 1980s — NBER: Jul 1981 to Nov 1982; Monthly GDP: Aug 1981 to Mar 1982 (Monthly CI: (Jan 1981,Sep 1981) peak; (Feb 1982,Mar 1982) trough)
  8. Early 1990s — NBER: Jul 1990 to Mar 1991; Monthly GDP: Jul 1990 to Jan 1991 (Monthly CI: (Jan 1990,Sep 1990) peak; (Jan 1991,Jan 1991) trough)
  9. Dot-com bubble — NBER: Mar 2001 to Nov 2001; Monthly GDP: May 2001 to Sep 2001 (Monthly CI: (May 2001,Aug 2001) peak; (Sep 2001,Sep 2001) trough)
  10. Great Recession — NBER: Dec 2007 to Jun 2009; Monthly GDP: Jun 2008 to Apr 2009 (Monthly CI: (May 2008,Jul 2008) peak; (Feb 2009,Sep 2009) trough)
  11. COVID-19 — NBER: Feb 2020 to Apr 2020; Monthly GDP: Feb 2020 to Apr 2020 (Monthly CI: (Dec 2019,Feb 2020) peak; (Apr 2020,Apr 2020) trough)
- Note: NBER dates are reported in bold in the source table when they fall outside the equivalent monthly GDP confidence intervals.

### Main findings and statistical differences
- Start dates:
  - In general, recession start dates are fairly closely aligned between NBER and monthly-GDP dating.
  - Of the eleven recessions, start dates for three episodes are statistically different: the late 1960s, the dot-com bubble in 2001, and the Great Recession.
  - These three episodes feature "stuttering" starts to the downturn (gradual onset), in contrast to sudden onset recessions (early 1960s, COVID-19).
  - For episodes with stuttering starts, there can be multiple reasonable start-date choices (example: Great Recession could be dated to late 2007 or late Summer 2008).
- End dates:
  - Dating of recession end dates shows larger and more systematic differences across the two methods.
  - Five of the eleven recessions have statistically significantly different end dates across the two methods, usually with earlier end dates using monthly GDP.
  - Troughs in the monthly GDP series are almost always clear low points; NBER appears slightly conservative, waiting for pick-up across a broad swathe of indicators (including lagging labor market indicators) before calling cycle ends.

### Interpretation and implications
- Monthly GDP can produce different assessments of the timing of business cycle peaks and troughs, especially for troughs (recession ends).
- The monthly estimates’ uncertainty matters: 95 percent confidence intervals around monthly GDP peak/trough dates are used to assess statistical alignment with NBER dates.
- One interpretation: the BCDC’s practice of requiring broader confirmation across indicators (some of which lag real activity) can delay official dating of recession ends relative to monthly GDP troughs.

*Source: Excerpt “5.4    Recession Dating” from the provided PDF content (samples and tables reported within the excerpt).*

### 6.4    Robustness to model specification

### 6.4    Robustness to model specification

### Approach
- Four alternative model specifications were estimated to assess sensitivity of the baseline monthly national accounts series:
  - Use efficient GMM rather than unweighted GMM.
  - Fit the model to more dynamic moments, matching 12 autoregressive equations.
  - Consider a higher-order (third-order) polynomial trend.
  - Model the aggregation error in the national account specifically as a ninth component of GDP.
- Tables A4 and A5 in Appendix D report the average and average absolute deviations from the baseline for these alternatives. Sample: Jan 1959-Mar 2025.

### Key findings
- In all cases except the specification that includes the aggregation error, the differences from the baseline are minimal.
- When the aggregation error is included as a separate monthly series:
  - Differences for the GDP series are naturally larger because "GDP itself in the quarterly data is now different, and includes an explicit aggregation error."
  - The inclusion of the aggregation error has no noticeable effect on any of the other components of GDP; it "drives a wedge between the components and the total."
- The data associated with the paper report two series for GDP: one including the aggregation error and one excluding it.
  - Users therefore face the same choice as with the quarterly data: either include the aggregation error and match headline GDP exactly, or drop it and have the components sum exactly.

### Robustness results (selected numeric summaries from Appendix D)
- Average absolute difference versus baseline (Including aggregation error: GDP by decade):
  - 1950s: 13.304
  - 1960s: 7.730
  - 1970s: 4.742
  - 1980s: 5.251
  - 1990s: 4.419
  - 2000s: 2.700
  - 2010s: 2.488
  - 2020s: 1.864
- Alternative specifications produce much smaller average absolute differences for GDP (selected examples, Efficient GMM: GDP by decade):
  - 1950s: 0.034
  - 1960s: 0.042
  - 1970s: 0.034
  - 1980s: 0.042
  - 1990s: 0.037
  - 2000s: 0.039
  - 2010s: 0.036
  - 2020s: 0.102
- Fitting 12 AR equations and third-order trend specifications similarly yield small average absolute differences for GDP in most decades (examples, Fitting 12 AR equations: GDP 2020s: 0.028; Third-order trend: GDP 2020s: 0.054).

### Implications for users and interpretation
- The baseline monthly series are robust to alternative model specifications other than the explicit inclusion of the aggregation error.
- The main practical decision for prospective users is identical to that faced with the quarterly data:
  - Include the aggregation error to match headline (quarterly) GDP exactly; or
  - Exclude the aggregation error so that component series sum exactly to GDP (no aggregation error).
- Because the inclusion of aggregation error affects only the relationship between the sum of components and headline GDP (and not the component series themselves), researchers can choose the series that best fits their application without concern that component-level dynamics are materially altered by that choice.

*Source: Estimated Monthly National Accounts for the United States, Working Paper No. WP/2025/134 — Section 6.4 Robustness to model specification*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025134-print-pdf.pdf_