## wpiea2024190-print-pdf

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### Methodology: Definition, Model, and Estimation
- Observables and latent structure:
  - Yt is an N×1 vector of observables; yi,t are individual series.
  - ft is a q×1 vector of unobservable factors; et are “possibly serially correlated” innovations.
  - εt and vt are normally distributed i.i.d shocks, t=1,2,...,T.
- Dynamic factor model (DFM) form:
  - Measurement (dynamic) and state (VAR) equations with regime-dependent coefficients: ci, ai, and μ0[st].
  - Regime indicator st ∈ {1,...,M} with transition probabilities πi,j forming matrix Z.
  - Exact vs approximate DFM distinction retained; approximate DFM allows serially and cross-sectionally correlated et and is represented in static augmented form.
- Estimation targets and approach:
  - Jointly estimate parameters θ and latent variables [Ft, st].
  - Use Expectation-Maximization (EM) algorithm iterating a modified Kalman filter and smoother conditional on parameters and maximizing the expected complete-data likelihood until joint convergence.
  - Paper derives missing auto-covariance terms required to complete the EM expectation step and provides closed-form maximization solutions to avoid heavy numerical optimization.
- Filtering and smoothing details:
  - Modified one-step-ahead prediction, prediction error, and filter update are computed conditional on transitions st−1 = i → st = j.
  - To contain the exponential growth of regime-paths, the generalized Pseudo Bayesian algorithm of order r=2 (Kim and Nelson (1999)) is used to collapse paths:
    - Collapsed filtered factor Ft|t j and collapsed covariance Pt,t j computed by weighting Ft|t(i,j) and Pt,t(i,j) by Pr(st−1 = i | st = j, Ωt).
  - Smoothed recursions are implemented for each transition and then collapsed across st+1 to obtain Ft,T j and Pt,T j.
  - The required auto-covariance term for EM is defined and expressed via Pt,t+1,T (j,k) = Pt+1,T k Jt+1,t (j,k)′ and integrated across regimes to obtain Pt,t+1,T k, with an approximation Ft+1,T k ≈ Ft,t+1,T (j,k) reducing the expression to Pt,t+1,T k = ∑ Pr(st = j | st+1 = k, ΩT) Pt,t+1,T (j,k).
- Maximization step and closed-form estimators:
  - Model written as Yt = C[st] Ft + et and Ft = μ[st] + A[st] Ft−1 + v[st], with ε[st] ∼ N(0, R[st]) and v[st] ∼ N(0, Q[st]).
  - Parameter set θ = {(Aj, Qj, Cj, Rj, μj): 1 < j < M; πi,j ∈ Z}.
  - EM target θ̂ = argmaxθ Eθ[ logLc(θ) | ΩT ] with Q-function given in the paper.
  - First-order conditions yield closed-form estimators:
    - A[st = j] given by weighted sums of E[ Ft (Ft−1)′ | ΩT ] and E[ Ft−1 (Ft−1)′ | ΩT ] with a μ[st = j] correction.
    - μ[st = j] solved as weighted averages of smoothed Ft and Ft−1.
    - C[st = j] = [ ∑ Pr(st = j | ΩT) E[ Yt (Ft j)′ | ΩT ] ] × [ ∑ Pr(st = j | ΩT) E[ Ft j (Ft j)′ | ΩT ] ]^−1.
    - R[st = j] and Q[st = j] given by weighted sums involving Yt Yt′ and smoothed factor covariances.
    - Transition probabilities πi,j = [ ∑ Pr(st−1 = i, st = j | ΩT) ] / [ ∑ Pr(st = i | ΩT) ] (sums over t=1..T).
  - Special case M = 1 reduces Q-function to the single-regime DFM expected log-likelihood equivalent to Banbura and Modugno (2014).
  - Note: explicit solutions for estimators are provided in the Online Appendix referenced in the source.

### Modifications for Mixed Frequency and Missing Data
- Mixed-frequency (quarterly-to-monthly) treatment:
  - Use Mariano and Murasawa (2003) approximation: quarterly series yt is geometric mean of underlying monthly ŷt.
  - log yt = 1/3 × (log ŷt + log ŷt−1 + log ŷt−2).
  - Quarterly difference in logs Δ3 yt = 1/3 × (Δ1 ŷt + 2 Δ1 ŷt−1 + 3 Δ1 ŷt−2 + 2 Δ1 ŷt−3 + Δ1 ŷt−4).
  - DFM augmented accordingly and retains form to which EM applies.
- Missing data handling:
  - Modify estimators to sum only over observed periods t ∈ t*:
    - C[st = j] = ∑_{t ∈ t*} [ Pr(st = j | ΩT) E[ Yt (Ft j)′ | ΩT ] ] / ∑_{t ∈ t*} [ Pr(st = j | ΩT) E[ Ft j (Ft j)′ | ΩT ] ].
  - R[st = j] estimation follows analogous observed-data summation.

### Performance and Replication Exercises
- Replication setup:
  - Replicate two seminal procedures: (i) one-step numerical maximization (Kim and Nelson (2006)) and (ii) two-step / shortcut one-step (Camacho et al. (2018)); apply the EM methodology and compare.
- Computation time comparisons (total computation time in seconds):
  - One-step Approach: 56.1
  - EM Approach: 41.0
  - Two-step Approach: 110.2
  - EM Approach: 71.5
- Replication outcomes:
  - One-step vs EM (Figure 1a): smoothed recession probabilities from one-step (blue solid) and EM (red dashed) give similar recession probability estimates; EM provides a speed advantage (Table 1).
  - Two-step vs EM (Figure 1b): smoothed recession probabilities mostly agree and resemble NBER dating; EM outperforms two-step in the 1997 and 2003 recessions where two-step mostly misses; EM provides a speed advantage (Table 2).
  - Estimates in these replication exercises are not real-time vintage series; Section 4 demonstrates real-time performance.
- Additional materials:
  - Online Appendix contains mapping from single-regime DFM to regime-switching model, derivations for smoothing covariance terms, replication details, and Monte Carlo simulations.

### Main methodological contributions (concise)
- Derives missing terms to complete Kim & Nelson (1999) filter and smoother design for EM application.
- Recommends an iterative EM process with closed-form maximization solutions for efficient and accurate estimation without heavy numerical routines.
- Shows that the common two-step or “short-cut” one-step approach (extract factors under linear model then estimate regimes) can be too strong and may miss turning points.

---

### Application to US Vintage Data: Model Implementation and Nowcasting Results
- Model and data setup:
  - Extends the New York FED nowcasting model to a regime-switching DFM.
  - Original NY FED model uses 22 monthly indicators and three quarterly indicators; common components classified into four blocks: global, soft, real, and labor.
  - Every indicator loads on the global factor; non-price-level variables load on one additional factor.
  - Data sourced from ALFRED vintage series; total span January 2000 to March 2022.
  - Training sample: the first seven years of data; nowcasting exercise starts from January 2007.
  - All nowcasts produced in real-time for more than 2000 data releases.
- Nowcasting target and evaluation:
  - Forecast target: the first flash estimate of US real GDP.
  - Relative performance metric: relative root mean square error (RMSE) comparing the regime-switching model to the single-regime New York FED model.
  - Expanding-window subsample used to compute relative RMSE over real-time data releases; figures show relative RMSE (red line), shaded real-time smoothed recession probabilities, solid horizontal line at one (benchmark), and dashed horizontal line at mean relative RMSE.
- Key empirical findings:
  - Sample contains only two recessions: the global financial crisis of 2008 and the COVID-19 pandemic.
  - Overall performance:
    - Regime-switching model is not substantially better during the early sample (it trains on early observations).
    - Forecasting performance becomes notably superior starting in 2013.
    - Superior performance is particularly evident during the onset of the COVID-19 pandemic, with the RMSE dropping substantially.
  - Indicator-level performance:
    - Regime-switching model shows clear superior forecasting performance for:
      - nonfarm payroll employment
      - consumer price index
      - durable goods orders
      - unemployment rate
    - Some indicators produce noisier information during recession regimes, reflecting indicator sensitivity.
  - Recession dating and probabilities:
    - Real-time recession probabilities from the regime-switching model sufficiently overlap with NBER recession dating for both recessions in the sample, despite less real-time information.
    - Backcasting (using information up to one year out) brings model-based dating and NBER dating closer for the COVID-19 period.
    - Recession end dates are fully covered with a lag of three to six months, which falls within the publication range of the following quarter GDP.
- Interpretation and policy relevance:
  - Regime-switching nowcasting model outperforms the single-regime NY FED model in relative terms and closely matches NBER recession dating in real time.
  - Provides an automated recession dating tool alternative and a robust, efficient solution for nowcasting with big data under regime changes.
  - Real-time probabilistic recession dating can inform timely policy stance adjustments and allow preemptive assessment of recession or inflationary risks.

### Empirical implications and performance (summary)
- EM algorithm yields satisfactory performance relative to other methods, offering a good trade-off between accuracy and speed for real-time, large-dimensional applications.
- Regime-switching modification improves nowcasting performance even after the COVID-19 break compared with the NY FED model that suspended releases.
- Model can closely match NBER recession dating with less information than actual committee meetings, enabling potential preemptive policy action.
- Backcasting shows model–recession fit improves as more information becomes available.

### Practical recommendations and extensions
- Use the EM estimation approach for regime-switching DFMs when:
  - Real-time applications require efficient estimation with large-dimensional data.
  - Mixed-frequency and missing-data problems are present (quarterly-to-monthly aggregation and incomplete releases).
- Avoid relying solely on two-step or short-cut one-step procedures when regime detection and turning-point accuracy are critical.
- Consider the regime-switching DFM framework as broadly applicable to any state-space representable model beyond nowcasting; further research may expand automated real-time recession dating applications.

*Source: wpiea2024190-print-pdf*

### 5. Conclusion ..........................................................................................................

### 5. Conclusion

### Main findings
- Proposes the expectation maximization (EM) approach to estimate regime-switching state-space models that can accommodate mixed frequency data and missing data problems.
- The EM algorithm provides satisfactory performance relative to other estimation methods and delivers a good trade-off between accuracy and speed, making it especially useful for real-time applications involving large dimensional data.
- The regime-switching modification improves nowcasting performance even after the COVID-19 break, compared with the NY FED nowcasting model that suspended its releases.
- The regime-switching nowcasting model is able to closely match the recession dating of the NBER despite having less information than actual committee meetings, enabling policymakers to act preemptively.
- In a backcasting exercise, the fit between actual NBER recession dates and model estimates of recession becomes more apparent with the additional available information.

### Methodological contributions
- Derives the required missing terms to complete filter and smoother design of Kim & Nelson (1999) for the EM application.
- Recommends an iterative process with closed-form solutions to provide an efficient and accurate estimation methodology without relying on numerical routines.
- Shows that the common two-step or “short-cut” one-step approach (extracting factors under a linear model assumption and then estimating regimes) might be too strong and may miss turning points.

### Empirical implications and performance
- Application to vintage US data demonstrates superior nowcasting performance for the regime-switching model, particularly when key economic indicators are released.
- The model’s ability to approximate NBER recession dating with less information suggests usefulness for preemptive policy actions.
- Backcasting results indicate that model-recession fit improves as more information becomes available.

*Source: wpiea2024190-print-pdf - 5. Conclusion*

### 3. Methodology

### 3. Methodology

### 3.1. Definition and Notation
- Observables:
  - Let Yt be an N×1 vector of observables where each element, yi,t, represents a different time series.
- Latent factors and shocks:
  - ft is a q×1 vector of unobservable factors.
  - et is a vector of “possibly serially correlated” innovations.
  - εt and vt are normally distributed i.i.d shocks, t=1,2,...,T.
- Dynamic factor model (DFM) (measurement and state equations):
  - Measurement equation (dynamic form):
    - Yt = c0[st] ft + c1[st] ft−1 + ⋯ + cs[st] ft−s + et  (equation (1))
    - ci are N×q factor loadings.
  - State equation (factors follow a VAR):
    - ft = μ0[st] + a1[st] ft−1 + a2[st] ft−2 + ⋯ + ah[st] ft−h + vt  (equation (2))
    - ai are q×q vector autoregressive coefficients; μ0[st] is regime-dependent shift.
- Regimes:
  - st is a discrete regime indicator taking values 1,...,M.
  - Transition probabilities: Pr(st = j | st−1 = i) = πi,j forming transition matrix Z (equation (3)).
- Exact vs approximate DFM:
  - Exact DFM (Stock & Watson (2016)): et uncorrelated cross-sectionally and serially.
  - Approximate DFM allows serially correlated and cross-sectionally correlated et.
  - Static approximate DFM representation (with lag-augmented Ft = [1 ft ft−1 ⋯ ft−p]′) and AR structure for et: see equation (4).

### 3.2. Estimation of Regime-Switching DFMs
- Estimation targets:
  - Two sets: model parameters θ and latent variables [Ft, st].
  - Regime-switching adds latent variable st to the single-regime DFM estimation problem.
- Approach:
  - Use Expectation-Maximization (EM) algorithm iterating a modified Kalman filter and smoother conditional on parameters and maximizing likelihood conditional on latent variables until joint convergence.
  - Kim & Nelson (2006) modify single-regime Kalman filter and smoother to regime-switching state-space models; their numerical likelihood maximization avoids computing lagged auto-covariance of factors.
  - This paper derives the missing auto-covariance terms to complete the EM expectation step and provides closed-form maximization solutions.

### 3.2.1. The Expectation Step (filtering and smoothing)
- Modified Kalman filter for each possible transition st−1 = i → st = j:
  - One-step ahead predicted factor and covariance:
    - Ft|t−1(i,j) = Atj Ft−1|t−1 i  (equation (5))
    - Pt|t−1(i,j) = Atj Pt−1|t−1 i (Atj)′ + Qj  (equation (6))
  - Prediction error and its variance:
    - ηt|t−1(i,j) = Yt − Cj Ft|t−1(i,j)  (equation (7))
    - ft|t−1(i,j) = Cj Pt|t−1(i,j) Cj′ + Rj  (equation (8))
  - Filter update for factors and their covariance:
    - Ft|t(i,j) = Ft|t−1(i,j) + Pt|t−1(i,j) Cj′ [ft|t−1(i,j)]−1 ηt|t−1(i,j)  (equation (9))
    - Pt|t(i,j) = (I − Pt|t−1(i,j) Cj′ [ft|t−1(i,j)]−1 Cj) Pt|t−1(i,j)  (equation (10))
- Curse of dimensionality and approximation:
  - Considerable growth of regime-paths requires collapsing approximations.
  - Use generalized Pseudo Bayesian algorithm of order r=2 (Kim and Nelson (1999)):
    - Collapsed filtered factor conditional on st = j:
      - Ft|t j = ∑ Pr(st−1 = i | st = j, Ωt) Ft|t(i,j)  (equation (11))
    - Collapsed covariance:
      - Pt,t j = ∑ Pr(st−1 = i | st = j, Ωt)[ Pt,t(i,j) + (Ft,t j − Ft,t(i,j))(Ft,t j − Ft,t(i,j))′ ]  (equation (12))
- Smoothing (retrospective whole-sample inference):
  - Smoothed recursion for each transition st−1 = i → st = j conditional on full information ΩT:
    - Jt+1,t(j,k) = Pt,t j A k t ′ [Pt+1,t (j,k)]−1  (equation (13))
    - Ft,T (j,k) = Ft,t j + Jt+1,t(j,k) (Ft+1,T k − Ak t Ft,t j)  (equation (14))
    - Pt,T (j,k) = Pt,t j + Jt+1,t(j,k) (Pt+1,T k − Pt+1,t (j,k)) Jt+1,t(j,k)′  (equation (15))
  - Collapse smoothed outputs due to dimensionality:
    - Ft,T j = ∑ Pr(st+1 = k, st = j | ΩT) Ft,T (j,k)  (equation (16))
    - Pt,T j = ∑ Pr(st+1 = k, st = j | ΩT)[ Pt,T (j,k) + (Ft,T j − Ft,T (j,k))(Ft,T j − Ft,T (j,k))′ ]  (equation (17))
- Auto-covariance term required for EM:
  - Define Pt,t+1,T (j,k) = E[(Ft+1 − Ft+1,t (j,k))(Ft+1 − Ft+1,t (j,k))′ | ΩT, st = j, st+1 = k].
  - Extend single-regime smoother covariance (Byron et al. (2004)):
    - Pt,t+1,T (j,k) = Pt+1,T k Jt+1,t (j,k)′  (equation (18))
  - Integrate out Pt,t+1,T (j,k) to obtain Pt,t+1,T k:
    - Pt,t+1,T k = ∑ Pr(st = j, st+1 = k | ΩT) Pr(st+1 = k | ΩT)−1 (Pt,t+1,T (j,k) + (Ft+1,T k − Ft,t+1,T (j,k))(Ft+1,T k − Ft,t+1,T (j,k))′)  (equation (19))
  - Approximation Ft+1,T k ≈ Ft,t+1,T (j,k) reduces expression to:
    - Pt,t+1,T k = ∑ Pr(st = j | st+1 = k, ΩT) Pt,t+1,T (j,k)  (equation (20))

### 3.2.2. The Maximization Step
- General model form used for maximization (explicit intercept on factors):
  - Yt = C[st] Ft + et
  - Ft = μ[st] + A[st] Ft−1 + v[st]
  - ε[st] ∼ N(0, R[st]), v[st] ∼ N(0, Q[st])  (equation (21))
- Parameter set:
  - θ = {(Aj, Qj, Cj, Rj, μj): 1 < j < M; πi,j ∈ Z}
- EM maximization target:
  - θ̂ = argmaxθ Eθ[ logLc(θ) | ΩT ]  (equation (22))
  - Q-function expression provided in equation (23) (expected log-likelihood involving smoothed variables and regime probabilities).
- Special case (Corollary 1):
  - For M = 1, μ = 0, Pr(st = j, st−1 = i | ΩT) = Pr(st = j | ΩT) = 1 ∀ t ∈ [1,...,T], Q reduces to expression (24) equivalent to Banbura and Modugno (2014) single-regime DFM expected log-likelihood.
- First-order conditions and closed-form estimators:
  - Solve ∂ Eθ̂[ logLc(θ) | ΩT ] / ∂θ [st = j] = 0  (equation (25)).
  - Estimator for A[st = j]:
    - A[st = j] = [ ∑ Prθ̂(st = j | ΩT) Eθ̂[ Ft j (Ft−1 j)′ | ΩT ] ]_{t=1..T} × [ ∑ Prθ̂(st = j | ΩT) Eθ̂[ Ft−1 j (Ft−1 j)′ | ΩT ] ]_{t=1..T}^−1  minus term involving μ[st = j] (equation (26)).
    - μ[st = j] is given by expression (27) (sample averages of smoothed Ft and Ft−1 weighted by Pr).
  - Estimator for C[st = j]:
    - C[st = j] = [ ∑ Prθ̂(st = j | ΩT) Eθ̂[ Yt (Ft j)′ | ΩT ] ]_{t=1..T} × [ ∑ Prθ̂(st = j | ΩT) Eθ̂[ Ft j (Ft j)′ | ΩT ] ]_{t=1..T}^−1  (equation (28)).
  - Variance matrices:
    - R[st = j]′ and Q[st = j]′ given by expressions (29) and (30) respectively involving weighted sums of Yt Yt′ and smoothed factor covariances.
  - Transition probabilities:
    - πi,j = [ ∑ Prθ̂(st−1 = i, st = j | ΩT) ]_{t=1..T} / [ ∑ Prθ̂(st = i | ΩT) ]_{t=1..T}  (equation (31)).
  - Note: Solutions for estimators C[st = j], R[st = j], A[st = j], Q[st = j], and μ[st = j] are given in Online Appendix.

### 3.2.3. Modifications for Mixed and Missing Data
- Mixed frequency (quarterly-to-monthly) using Mariano and Murasawa (2003) approximation (Banbura and Modugno (2014); Bok et al. (2018)):
  - Quarterly series yt is geometric mean of underlying monthly series ŷt:
    - yt = (ŷt × ŷt−1 × ŷt−2)1/3  (equation (32))
  - Log transformation yields:
    - log yt = 1/3 × (log ŷt + log ŷt−1 + log ŷt−2)  (equation (33))
  - Quarterly difference in logs (Δ3 yt) expressed as weighted sum of monthly first differences Δ1 ŷ at different lags (equation (34)):
    - Δ3 yt = 1/3 × (Δ1 ŷt + 2 Δ1 ŷt−1 + 3 Δ1 ŷt−2 + 2 Δ1 ŷt−3 + Δ1 ŷt−4)
  - The DFM can be augmented to include these relations and remain in the form of equation (21); EM algorithm continues to apply.
- Missing data:
  - Modify estimator for C[st = j] by summing only over observed periods t ∈ t*:
    - C[st = j] = ∑_{t ∈ t*} [ Prθ̂(st = j | ΩT) Eθ̂[ Yt (Ft j)′ | ΩT ] ] / ∑_{t ∈ t*} [ Prθ̂(st = j | ΩT) Eθ̂[ Ft j (Ft j)′ | ΩT ] ]  (equation (35))
  - Estimation of R[st = j] follows similar modification summing over observed t ∈ t*.

### 3.3. Performance of the Methodology (replication exercises)
- Replication setup:
  - Two seminal papers replicated: (i) one-step numerical maximization approach (Kim and Nelson (2006)) and (ii) two-step / shortcut one-step procedure (Camacho et al. (2018)).
  - For each original methodology, also apply the EM methodology in this paper and compare.
- Computation time comparisons (reported total computation time in seconds):
  - Table 1 (one-step vs EM):
    - One-step Approach: 56.1
    - EM Approach: 41.0
  - Table 2 (two-step vs EM):
    - Two-step Approach: 110.2
    - EM Approach: 71.5
- Replication outcomes:
  - Figure 1a (one-step vs EM):
    - Smoothed recession probabilities from one-step (blue solid line) and EM (red dashed line) give similar recession probability estimates.
    - EM approach provides a speed advantage (Table 1).
  - Figure 1b (two-step vs EM):
    - Smoothed recession probabilities from two-step (blue solid line) and EM (red dashed line) mostly agree and closely resemble NBER recession dates (gray shaded areas).
    - EM methodology outperforms two-step in 1997 and 2003 recessions where two-step mostly misses.
    - EM approach provides a speed advantage (Table 2).
- Notes:
  - Estimates in these replication exercises are not real-time vintage series.
  - Section 4 will demonstrate real-time performance of regime-switching model estimated with the EM approach against NBER dating.
- Additional details:
  - Online Appendix contains: estimation procedure mapping single-regime DFM to regime-switching model (3.2 note), derivations for smoothing covariance terms, replication exercise details, and Monte Carlo simulations on regime-switching DFM using the proposed estimation methodology.

*Source: wpiea2024190-print-pdf - 3. Methodology*

### 4. An Application to US Vintage Data

### 4. An Application to US Vintage Data

### Model and Data Setup
- Extension of the New York FED nowcasting model to allow regime-switching in dynamic factor models (DFMs).
- Original New York FED model (as used on the NY FED website) consists of 22 monthly indicators and three quarterly indicators.
- Common components classified into four blocks: global, soft, real, and labor.
- Every indicator loads on the global factor; non-price-level variables load on one additional factor.
- Dataset extended forward and backward using vintage data from ALFRED (St. Louis FED archival database).
- Total data span: January 2000 to March 2022.
- Training sample: the first seven years of data; nowcasting exercise starts from January 2007.
- All nowcasts are produced in real-time, i.e., for more than 2000 data releases.

### Nowcasting Target, Evaluation, and Procedure
- Forecast target: the first flash estimate of US real GDP.
- Relative performance metric: relative root mean square error (RMSE) comparing regime-switching model against the single-regime New York FED model (see equation (36) in source).
- Expanding-window subsample used to compute relative RMSE over real-time data releases.
- Presentation elements in figures:
  - Relative RMSE (red line) over data releases.
  - Shaded areas: combinations of real-time (smoothed) recession probabilities from the regime-switching model.
  - Solid horizontal line: benchmark level of one (both models perform equally well).
  - Dashed horizontal line: mean of the relative RMSE.
- Note: each horizontal-axis point corresponds to a data release (irregular spacing in calendar time).

### Key Empirical Findings
- Sample contains only two recessions: the global financial crisis of 2008 and the COVID-19 pandemic.
- Overall performance:
  - The regime-switching model is not substantially better during the early sample (it trains on early observations).
  - Regime-switching model’s forecasting performance becomes notably superior starting in 2013.
  - Superior performance is particularly evident during the onset of the COVID-19 pandemic, with the RMSE dropping substantially.
- Indicator-level performance:
  - The regime-switching model shows clear superior forecasting performance for the following indicators:
    - nonfarm payroll employment
    - consumer price index
    - durable goods orders
    - unemployment rate
  - Some indicators produce noisier information during recession regimes, reflecting indicator sensitivity.
- Recession dating and probabilities:
  - Real-time recession probabilities from the regime-switching model sufficiently overlap with NBER recession dating for both recessions in the sample, despite the model having less information in real time.
  - Backcasting exercise allows using information up to one year out relative to the period being dated.
  - For the COVID-19 period, model-based dating and NBER dating become closer as more information is available.
  - Recession end dates are fully covered with a lag of three to six months, which falls within the publication range of the following quarter GDP.

### Interpretation and Practical Utility
- The regime-switching nowcasting model:
  - Outperforms the single-regime New York FED nowcasting model in relative terms.
  - Matches NBER recession dating closely in real time, despite information disadvantages.
  - Holds promise as an alternative automated recession dating tool.
- Policy-relevant implications:
  - Provides a robust and efficient solution for nowcasting with big data in the presence of regime-switching.
  - Improved nowcasts, especially for key indicators, can provide more accurate and timely forecasts crucial for policy-making and economic planning.
  - Real-time probabilistic recession dating can inform timely policy stance adjustments (e.g., assessing whether to act more hawkish or dovish).
  - By adapting the regime-switching structure, central bankers can evaluate risks of upcoming recessions or inflationary regimes and act preemptively.

### Methodological Contribution and Extensions
- Demonstrates implementation of the EM algorithm to estimate regime-switching DFMs and its application to nowcasting with big data.
- The dynamic factor model form is flexible and applicable beyond nowcasting to any model representable in state-space form.
- Opens avenues for future research on regime-switching DFMs and automated real-time recession dating.

*Source: 4. An Application to US Vintage Data (chapter from wpiea2024190-print-pdf).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024190-print-pdf.pdf_