## _wp16241

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---

### I. INTRODUCTION — key messages and contributions
- Research questions:
  - What are the key business cycle stylized facts, how strong is the co-movement of real and nominal variables, and what are the implications for structural models?
- Main empirical claim:
  - Most business cycle fluctuations in advanced and some emerging economies appear driven by a single dominant factor, labeled the “demand shock,” because it explains strong and predictable co-movement of real and nominal variables over the business cycle.
- Evidence on co-movement:
  - Positive co-movement of real output and inflation supports a demand-driven explanation rather than a technology-driven explanation.
  - The first dynamic principal component explains up to 80% of business cycle variation in real and nominal macroeconomic aggregates across a variety of countries.
- Frequency focus:
  - Analysis focuses exclusively on business cycle frequencies, defined as fluctuations between 6-32 or 0-32 quarters, and does not aim to explain long-run trends or high-frequency variations.
- Relation to literature and methodological lineage:
  - Follows the spirit of Sargent and Sims (1977), Burns and Mitchell (1946), and Kydland and Prescott (1990).
  - Differs from some prior work by arguing for strong output–inflation co-movement where some papers argue little or no co-movement.
- Implications for structural models:
  - At business cycle frequencies, second-order moments should possess a clear factor structure with a dominant factor explaining most variation.
  - Implication: impulse-response functions to shocks need to be rather similar for shocks with larger variance, or be dominated by one shock with strong real and nominal co-movement.
  - Authors do not claim everything is driven by a single shock; various shocks can have similar effects at business cycle frequencies and may be distinguishable only via low- or high-frequency dynamics.
- Three original contributions:
  1. Documentation of great regularities in Post-War business-cycle co-movements across multiple economies; DPCA and transformation techniques identify one dominant factor typically explaining more than two thirds of cyclical fluctuations.
  2. Analysis of both real variables and inflation reveals tight co-movement and motivates labeling the dominant component a “demand factor”; use of inflation (instead of price level) and its deviations from trend or long-term expectations is key.
  3. Agnostic empirical results imply new constraints and implications for theoretical models regarding the number and properties of dominant structural shocks.
- Caveats and limitations:
  - The approach is statistical and not driven by a specific model, though guided by economic reasoning in variable selection and transformation.
  - Focus is on cyclical frequencies; the authors do not claim this is the best definition of the business cycle nor that trend components are “potential” or “equilibrium” values.
  - Low-frequency dynamics do not show as much co-movement; distinct shocks may have similar cyclical dynamics and be distinguishable only at other frequencies.
- Paper structure:
  - Section II describes methods; Section III presents U.S. results and summarizes other countries; Section IV assesses implications for macro modeling; Section V concludes; appendices contain non-core graphs, sensitivity, and robustness checks.

### II. EMPIRICAL MODELS AND METHODS — methods and technical specifics
- Main methodological tool:
  - Principal component analysis (PCA) for dimensionality reduction; Dynamic PCA (DPCA) is the default choice.
- Decomposition representation:
  - Observed series x_{i,t} = χ_t + ξ_{i,t}, where χ_t is the low-dimensional common component spanned by principal components, and ξ_{i,t} is idiosyncratic noise uncorrelated with χ_t.
- Static vs. Dynamic PCA:
  - Static PCA (SPCA): eigenvalue decomposition of the covariance matrix; does not account for lead-lag relationships.
  - Dynamic PCA (DPCA): eigenvalue decomposition of the spectral density matrix; accounts for lead-lag relationships and can be applied in time and frequency domains.
  - DPCA preferred because it accounts for lead-lag relationships; SPCA used as robustness check and yields qualitatively unchanged implications with slightly lower fit.
- Time-domain approach and fit statistic:
  - Cycles isolated using the band-pass filter (Fitzgerald-Christiano) and high-pass Hodrick-Prescott filter with conventional parameter values for quarterly frequencies.
  - PCA applied to isolated cycles; use Stock and Watson (2002) goodness-of-fit statistic:
    - R^2(k) ≔ 1 − (∑_{t=1}^T (x_{i t} − χ^k_{i t})^2) ∕ (∑_{t=1}^T (x_{i t} − x̄_i)^2), where x̄_i is the sample mean of x_{i,t}.
- Frequency-domain DPCA and co-movement statistic:
  - Estimate multivariate spectral density Σ_x(ω) of observed process x_t.
  - Select dominant eigenvalues λ^{(i)}(ω) of Σ_x(ω) at each frequency ω to obtain spectral density Σ_χ(ω) of the common component.
  - Co-movement statistic:
    - S_x(ω,k) ≔ (∑_{i=1}^k λ^{(i)}(ω)) ∕ (∑_{i=1}^n λ^{(i)}(ω)), the percentage of variability explained by k principal components at frequency ω.
  - Frequency-domain analysis avoids criticisms of pre-filtering.
- Treatment of non-stationary data and spectral estimation:
  - For non-stationary macro variables, use the non-parametric Bartlett approach on first log differences (when meaningful), rendering series stationary.
  - Invariance: S_Y(ω,k) = S_{ΔY}(ω,k) for all frequencies ω such that both sides are defined; thus statistic can be estimated for first differences for ω ≠ ±2π n for n ∈ ℕ^+.
- Practical estimation choices and notation:
  - Use same Bartlett non-parametric approach and smoothing window as Forni et al. (2000).
  - Terms “factor” and “component” used interchangeably.
  - DPCA introduced by Brillinger (1981); two-sided time-domain representation by Forni and others (2000).
- Note on filters:
  - The claim that HP or similar filters always cause spurious cycles is challenged; Pollock (2013) formally proves “this idea is largely mistaken.”

### Coherence and data preprocessing
- Coherence invariance:
  - Coherence is invariant to first-differencing: C_x,y(ω) = C_Δx,Δy(ω) for all ω for which both are defined (Koopman (1974, pp. 149)).
- Focus:
  - Analysis focuses explicitly on business cycle frequencies.
- Variables considered for each country:
  - Real GDP, real consumption, real investment, real exports, real imports, the unemployment rate, and the short-term interest rate.
- Inflation treatment:
  - Inflation is treated separately as the deviation from its trend (the inflation cycle) and compared to the first dynamic component (time domain) and to output (time domain).
  - In the frequency domain, coherence is computed between inflation and output, and between inflation and the isolated first dynamic component.

### Inflation measurement and sample limitations
- Preferred inflation measure:
  - Trimmed-mean inflation (Cleveland’s FED trimmed mean inflation) to eliminate outliers and reduce high-frequency variation.
  - Advantage: not dependent on past and future observations; can be computed in real-time with zero revisions.
- Trimmed-mean inflation data availability:
  - Available pre-built for the U.S. and Australia.
  - For most other countries, trimmed-mean inflation measures were constructed using Haver Analytics data starting only from the early 90’s.
- Practical consequence:
  - Inflation is generally not included directly in the DPCA estimation because trimmed-mean inflation spans a smaller sample than other macro variables, which would restrict analysis.
  - Exception: USA, where principal components are estimated jointly including trimmed inflation.

### United States — empirical results and robustness
- Dominance of the first dynamic principal component:
  - Virtually every variable, with the exception of real exports and short-term interest rates, is explained by more than 80% using a single dynamic principal component.
- Time-domain evidence:
  - The first dynamic principal component explains a great portion of U.S. business cycle variation.
  - Strong co-movement of output and the dominant factor with the cyclical dynamics of median inflation; the short-term “Phillips Curve” appears active.
- Notable deviations from the common cycle:
  - Private consumption slowdowns in 1992 and 1997.
  - Short-term interest rate deviations in late 1980s related to monetary policy under Chairman Volcker.
  - Exports deviations explained by abroad-implied dynamics (U.S. exports approximated by trade-weighted combination of partners’ imports).
- Frequency-domain evidence:
  - Without pre-filtering, the first two dynamic principal components explain a large portion of spectral density across frequencies.
  - One principal component fits business-cycle spectral density especially well for imports and investment.
  - Exports require the second principal component for an almost perfect fit over business cycles.
- Robustness to filter choice and sample extension:
  - Results hold for both Christiano-Fitzgerald band-pass and Hodrick-Prescott (HP) filters.
  - Extending the sample back to 1966 (sample starts in 1966Q2 and ends in 2015Q4) yields similar co-movement conclusions:
    - Relative explanatory power of the first principal component changes little.
    - Expected deterioration of short-term interest rate fit before 1985 due to volatile policy rates and oil shocks.
    - First principal component changes its variance but filter loadings (coefficients) remain constant — relative variances among real variable cycles have not changed significantly across Great Moderation and Great Recession periods.
  - Using growth rates (first-differences) instead of band-pass cycles deteriorates DPCA fit (transfer function of 1−L amplifies high frequencies), but comovement among real variables remains detectable.
- Inclusion of trimmed-mean inflation directly in DPCA for the U.S.:
  - When included, the first principal component produces an excellent fit for output, consumption, investment, and unemployment.
  - For exports, the short-term interest rate, and trimmed inflation, the first principal component explains about 50% of volatility (lower explanatory power driven by high volatility in the 1960s and 1970s; filter loadings maintain the same sign).

### Co-movement of real and nominal variables; interpretation
- Main empirical observation:
  - Strong and stable co-movement between cyclical components of main macroeconomic variables and inflation over the business cycle.
  - Inflation lags the output cycle in a relatively stable and predictable way.
- Conceptual points:
  - It is deviations of inflation from its target (inflation cycle), not the overall level of inflation, that relate to the output cycle—consistent with inflation-targeting frameworks.
  - Low-frequency inflation movements are driven by perceptions of the inflation target embodied in long-term inflation expectations or long-term nominal bond yields.
  - Ten-years-ahead long-term inflation expectations (from Survey of Professional Forecasters, SPF) can be used as an alternative means to remove trend from inflation.
- Statistical inference:
  - Coherence estimates between trimmed inflation and output (and between trimmed inflation and the first estimated dynamic component) are reported with 95 percent confidence intervals (computed using wild bootstrap, per Wu (1986)).
- Economic interpretation:
  - The dominant first dynamic principal component, which co-moves positively with inflation cycle and output, is labeled a ‘demand factor’ or demand shock.
  - The analysis does not identify specific events causing demand shocks; co-movement is the empirical basis for the label.

### Data transformations, filter choice, and policy-relevant implications
- Data transformation matters:
  - Band-pass filters highlight business-cycle co-movement more clearly than simple growth rates (first-differences).
  - Choice between Christiano-Fitzgerald and HP filters affects inclusion/exclusion of high frequencies; conclusions are robust across these filters.
- Policy relevance:
  - Tight co-movement of inflation deviation from target with output suggests demand-driven inflation dynamics, supporting frameworks that link inflation-gap and output-gap (Okun’s law tightness implies output or unemployment specifications of Phillips Curve are nearly equivalent).
  - Distinguishing long-term inflation expectations (inflation target) from cyclical inflation dynamics is crucial when relating inflation to the cyclical stance; without knowledge of the inflation target, such relations are meaningless for countries undergoing disinflation.
- Econometric implication:
  - Acknowledging distinct volatility across sample subperiods (volatile pre–mid-1980s period vs. Great Moderation and post-2007 Great Recession) is important for models with time-varying coefficients, but the dynamics driving relative variance and co-movement appear essentially time invariant.

### Summary statistics and sample coverage
- Dataset:
  - Quarterly data for a set of advanced and several emerging market countries.
- Country sample:
  - Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Hungary, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey, the U.K., and the U.S.
- U.S. benchmark sample:
  - Sample used for extended-sample robustness runs from 1966Q2 to 2015Q4.

### Choice of 1985 breakpoint, empirical findings, and quantitative stylized facts
- Motivation for 1985 breakpoint:
  - Choice of 1985 is motivated by the change in relative volatilities of inflation and real activity (Great Moderation) in developed countries around the mid-1980s.
- DPCA empirical findings (cross-country summary):
  - For most countries, the first dynamic principal component explains most of the dynamics in output, investment, imports, and unemployment.
  - The first two dynamic principal components explain a high share of the dynamics in all variables.
  - Largest dispersion of percentage explained across countries is for exports, short-term real rate, and consumption.
  - Co-movement between inflation and real variables is relatively strong for all countries in the sample.
  - For each country, there exists a lag k ∈ (0,...,4) for which correlation between cyclical inflation and the cyclical component of output is positive and significantly different from zero at the 5% level.
- Quantitative stylized facts and required model properties (U.S. examples):
  - At business cycle frequency, empirical moments imply strict requirements on models:
    - Investment volatility relative to output must be around four.
    - The ‘Okun’s coefficient’ for unemployment is around half.
    - The relative variance of the consumption cycle to output is slightly lower than one, with an identical direction of the response (consumption and output respond in the same direction).
  - The first dynamic principal component clearly dominates in terms of explained variance and satisfies the sign restriction expected from a broadly-understood “demand shock”: positive co-movement of output and inflation.

### Implications for structural macroeconomic models and specification checks
- Model requirements:
  - Empirically successful models should be able to mimic the correlation structure consistent with a dominant principal component for the variables considered at business cycle frequencies.
  - If a model does not feature a structural shock that dominates the cyclical frequencies of consumption, investment, output, hours worked, and inflation, the model is likely misspecified.
    - Misspecification manifests as remaining structural shocks being cross-correlated.
    - Cross-correlated estimated ‘structural shocks’ give the appearance that multiple shocks regularly offset each other.
- Requirements beyond variance:
  - Strict requirements on the direction of co-movement of relevant variables and the shape (amplitude and phase) of impulse-response functions at cyclical frequencies.
- Suggested empirical ‘smell tests’ and specification checks:
  - Ex-ante: check whether the model-induced principal-component space is close to the principal-component space of the data.
  - Ex-post: check cross-correlation of estimated shocks.
  - Verify that impulse-responses ‘make sense’ in light of robust stylized facts on co-movement.
- Companion work:
  - Andrle, Brůha, and Solmaz (2016) discusses implications and proposes misspecification tests.

### Conclusions
- Business cycle dynamics of key macroeconomic data can be largely explained by a single source of variation — the dominant unobserved principal component (the “demand factor”).
- The demand factor explains positive co-movement of output cycle and inflation.
- Structural economic models have great difficulties delivering structural shocks resembling the robustly-estimated dominant principal component.
- The study provides a model-independent test for empirical models: shocks considered plausible sources of business cycles must generate principal-component behavior consistent with the data.

*Source: _wp16241 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22

### _wp16241 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22

### I. INTRODUCTION — key messages and contributions
- Research questions: What are the key business cycle stylized facts, how strong is the co-movement of real and nominal variables, and what are the implications for structural models?
- Main empirical claim: Most business cycle fluctuations in advanced and some emerging economies appear driven by a single dominant factor, labeled the “demand shock,” because it explains strong and predictable co-movement of real and nominal variables over the business cycle.
- Evidence on co-movement:
  - Positive co-movement of real output and inflation (reminiscent of the “Phillips Curve”) supports a demand-driven explanation rather than a technology-driven explanation.
  - The first dynamic principal component explains up to 80% of business cycle variation in real and nominal macroeconomic aggregates across a variety of countries.
- Frequency focus: Analysis focuses exclusively on business cycle frequencies, defined as fluctuations between 6-32 or 0-32 quarters, and does not aim to explain long-run trends or high-frequency variations.
- Time-domain presentation: Despite frequency-domain methods, most results are presented in the time domain using simple charts.
- Relation to literature:
  - Methodological lineage: follows the spirit of Sargent and Sims (1977), Burns and Mitchell (1946), and Kydland and Prescott (1990).
  - Difference from some literature: argues for strong output–inflation co-movement where some prior papers argue little or no co-movement.
- Implications for DSGE and structural models:
  - At business cycle frequencies, second-order moments should possess a clear factor structure with a dominant factor explaining most variation.
  - Implication: impulse-response functions to shocks need to be rather similar for shocks with larger variance, or be dominated by one shock with strong real and nominal co-movement.
  - Clarification: authors do not claim everything is driven by a single shock; rather, various shocks can have similar effects at business cycle frequencies and may be distinguishable only via low- or high-frequency dynamics.
- Three original contributions:
  1. Documentation of great regularities in Post-War business-cycle co-movements across multiple economies; DPCA and transformation techniques identify one dominant factor typically explaining more than two thirds of cyclical fluctuations.
  2. Analysis of both real variables and inflation reveals tight co-movement and motivates labeling the dominant component a “demand factor”; use of inflation (instead of price level) and its deviations from trend or long-term expectations is key.
  3. Agnostic empirical results imply new constraints and implications for theoretical models regarding the number and properties of dominant structural shocks.
- Caveats and limitations:
  - The approach is statistical and not driven by a specific model, though guided by economic reasoning in variable selection and transformation.
  - Focus is on cyclical frequencies; the authors do not claim this is the best definition of the business cycle nor that trend components are “potential” or “equilibrium” values.
  - Low-frequency dynamics do not show as much co-movement; distinct shocks may have similar cyclical dynamics and be distinguishable only at other frequencies.
- Structure of the paper: Section II describes methods; Section III presents U.S. results and summarizes other countries; Section IV assesses implications for macro modeling; Section V concludes; appendices contain non-core graphs, sensitivity, and robustness checks.

### II. EMPIRICAL MODELS AND METHODS — methods and technical specifics
- Main methodological tool: principal component analysis (PCA) for dimensionality reduction.
- Decomposition representation:
  - Observed series x_{i,t} = χ_t + ξ_{i,t}, where χ_t is the low-dimensional common component spanned by principal components, and ξ_{i,t} is idiosyncratic noise uncorrelated with χ_t.
  - A set of K time series is fully explained by K principal components; often a small number of components explains most dynamics.
- Static vs. Dynamic PCA:
  - Static PCA (SPCA): eigenvalue decomposition of the covariance matrix; does not account for lead-lag relationships.
  - Dynamic PCA (DPCA): based on eigenvalue decomposition of the spectral density matrix; accounts for lead-lag relationships and can be applied in time and frequency domains.
  - DPCA is the default choice because it accounts for lead-lag relationships; SPCA is used as a robustness check.
- Time-domain approach:
  - Cycles isolated using the band-pass filter (Fitzgerald-Christiano) and high-pass Hodrick-Prescott filter with conventional parameter values for quarterly frequencies.
  - PCA applied to isolated cycles; use Stock and Watson (2002) goodness-of-fit statistic:
    - R^2(k) ≔ 1 − (∑_{t=1}^T (x_{i t} − χ^k_{i t})^2) ∕ (∑_{t=1}^T (x_{i t} − x̄_i)^2), where x̄_i is the sample mean of x_{i,t}.
  - DPCA preferred to account for lead-lag relationships (e.g., time shift of unemployment with respect to output, Okun’s law, and leads/lags of inflation and interest rates).
  - Applying SPCA yields qualitatively unchanged implications with slightly lower fit.
- Frequency-domain DPCA:
  - Estimate multivariate spectral density Σ_x(ω) of observed process x_t.
  - Select dominant eigenvalues λ^{(i)}(ω) of Σ_x(ω) at each frequency ω to obtain spectral density Σ_χ(ω) of the common component.
  - Co-movement statistic:
    - S_x(ω,k) ≔ (∑_{i=1}^k λ^{(i)}(ω)) ∕ (∑_{i=1}^n λ^{(i)}(ω)), the percentage of variability explained by k principal components at frequency ω.
  - Frequency-domain analysis avoids criticisms of pre-filtering.
- Treatment of non-stationary data and spectral estimation:
  - For non-stationary macro variables, spectral estimates require modification.
  - Use the non-parametric Bartlett approach on first log differences (when meaningful), rendering series stationary.
  - Measure S_Y(ω,k) is invariant to first-differencing: S_Y(ω,k) = S_{ΔY}(ω,k) for all frequencies ω such that both sides are defined.
  - This invariance implies that for non-stationary I(1) series, statistic (2) can be estimated for first differences for all ω ≠ ±2π n for n ∈ ℕ^+.
- Practical estimation choices:
  - The authors use the same Bartlett non-parametric approach and the same smoothing window setting as suggested by Forni and others (2000).
  - Note on filters: the claim that HP or similar filters always cause spurious cycles is challenged; Pollock (2013) formally proves “this idea is largely mistaken.”
- Notation and terminology:
  - Terms “factor” and “component” are used interchangeably.
  - DPCA introduced by Brillinger (1981); two-sided time-domain representation by Forni and others (2000).

*Source: _wp16241 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22*

### 0. Moreover, some other statistics of interest, such as coherence,

### _wp16241 - 0. Moreover, some other statistics of interest, such as coherence,

### Coherence and data preprocessing
- Coherence is invariant to first-differencing: C_x,y(ω) = C_Δx,Δy(ω) for all ω for which both are defined (Koopman (1974, pp. 149)).
- Analysis focuses explicitly on business cycle frequencies.
- Variables considered for each country: real GDP, real consumption, real investment, real exports, real imports, the unemployment rate, and the short-term interest rate.
- Inflation is treated separately as the deviation from its trend (the inflation cycle) and compared to:
  - the first dynamic component (time domain), and
  - output (time domain).
- In the frequency domain, coherence is computed and reported between:
  - inflation and output, and
  - inflation and the isolated first dynamic component.

### Inflation measurement and sample limitations
- Preferred inflation measure: trimmed-mean inflation (Cleveland’s FED trimmed mean inflation).
  - Rationale: eliminates outliers and lowers high-frequency variation without ex-ante eliminating particular components of the consumer basket.
  - Advantage over linear filters: not dependent on past and future observations; can be computed in real-time with zero revisions.
- Trimmed-mean inflation data availability:
  - Available pre-built for the U.S. and Australia.
  - For most other countries, trimmed-mean inflation measures were constructed using Haver Analytics data starting only from the early 90’s.
- Practical consequence:
  - Inflation is generally not included directly in the DPCA estimation because trimmed-mean inflation spans a smaller sample than other macro variables, which would restrict analysis.
  - Exception: USA, where principal components are estimated jointly including trimmed inflation.

### United States — empirical results and robustness
- Dominance of the first dynamic principal component:
  - Virtually every variable, with the exception of real exports and short-term interest rates, is explained by more than 80% using a single dynamic principal component.
- Time-domain evidence:
  - The first dynamic principal component explains a great portion of U.S. business cycle variation (Figure 1 referenced).
  - Strong co-movement of output and the dominant factor with the cyclical dynamics of median inflation; the short-term “Phillips Curve” appears active (Figure 3 referenced).
- Notable deviations from the common cycle:
  - Private consumption slowdowns in 1992 and 1997.
  - Short-term interest rate deviations in late 1980s related to monetary policy under Chairman Volcker.
  - Exports deviations explained by abroad-implied dynamics (U.S. exports approximated by trade-weighted combination of partners’ imports; see Figure 17).
- Frequency-domain evidence:
  - Without pre-filtering, the first two dynamic principal components explain a large portion of spectral density across frequencies (Figure 4 referenced).
  - One principal component fits business-cycle spectral density especially well for imports and investment.
  - Exports require the second principal component for an almost perfect fit over business cycles.
- Robustness to filter choice and sample extension:
  - Results hold for both Christiano-Fitzgerald band-pass and Hodrick-Prescott (HP) filters (HP does not exclude high frequencies and has less sharp cutoff).
  - Extending the sample back to 1966 (sample starts in 1966Q2 and ends in 2015Q4) yields similar co-movement conclusions:
    - Relative explanatory power of the first principal component changes little.
    - Expected deterioration of short-term interest rate fit before 1985 due to volatile policy rates and oil shocks.
    - First principal component changes its variance but filter loadings (coefficients) remain constant — relative variances among real variable cycles have not changed significantly across Great Moderation and Great Recession periods.
  - Using growth rates (first-differences) instead of band-pass cycles deteriorates DPCA fit (transfer function of 1−L amplifies high frequencies), but comovement among real variables remains detectable (Figure 5 and Figure 16 referenced).
- Inclusion of trimmed-mean inflation directly in DPCA for the U.S.:
  - When included, the first principal component produces an excellent fit for output, consumption, investment, and unemployment.
  - For exports, the short-term interest rate, and trimmed inflation, the first principal component explains about 50% of volatility (lower explanatory power driven by high volatility in the 1960s and 1970s; filter loadings maintain the same sign).

### Co-movement of real and nominal variables; interpretation
- Main empirical observation:
  - Strong and stable co-movement between cyclical components of main macroeconomic variables and inflation over the business cycle.
  - Inflation lags the output cycle in a relatively stable and predictable way.
- Conceptual points:
  - It is deviations of inflation from its target (inflation cycle), not the overall level of inflation, that relate to the output cycle—consistent with inflation-targeting frameworks.
  - Low-frequency inflation movements are driven by perceptions of the inflation target embodied in long-term inflation expectations or long-term nominal bond yields.
  - For consistency across countries, cyclical inflation components are obtained using band-pass and HP filters in baseline calculations.
  - Ten-years-ahead long-term inflation expectations (from Survey of Professional Forecasters, SPF) can be used as an alternative means to remove trend from inflation.
- Statistical inference:
  - Coherence estimates between trimmed inflation and output (and between trimmed inflation and the first estimated dynamic component) are reported with 95 percent confidence intervals (computed using wild bootstrap, per Wu (1986)).
- Economic interpretation:
  - The dominant first dynamic principal component, which co-moves positively with inflation cycle and output, is labeled a ‘demand factor’ or demand shock.
  - The analysis does not identify specific events causing demand shocks; co-movement is the empirical basis for the label.

### Data transformations, filter choice, and policy-relevant implications
- Data transformation matters:
  - Band-pass filters highlight business-cycle co-movement more clearly than simple growth rates (first-differences).
  - Choice between Christiano-Fitzgerald and HP filters affects inclusion/exclusion of high frequencies; conclusions are robust across these filters.
- Policy relevance:
  - Tight co-movement of inflation deviation from target with output suggests demand-driven inflation dynamics, supporting frameworks that link inflation-gap and output-gap (Okun’s law tightness implies output or unemployment specifications of Phillips Curve are nearly equivalent).
  - Distinguishing long-term inflation expectations (inflation target) from cyclical inflation dynamics is crucial when relating inflation to the cyclical stance; without knowledge of the inflation target, such relations are meaningless for countries undergoing disinflation.
- Econometric implication:
  - Acknowledging distinct volatility across sample subperiods (volatile pre–mid-1980s period vs. Great Moderation and post-2007 Great Recession) is important for models with time-varying coefficients, but the dynamics driving relative variance and co-movement appear essentially time invariant.

### Summary statistics and sample coverage
- Dataset: quarterly data for a set of advanced and several emerging market countries.
- Country list provided in the sample: Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Hungary, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey, the U.K., and the U.S.
- Benchmark analysis start year unspecified in the excerpt; U.S. sample used for extended-sample robustness runs from 1966Q2 to 2015Q4.

*Source: _wp16241 - 0. Moreover, some other statistics of interest, such as coherence,*

### 1985. The choice of this year is motivated by the change in relative volatilities of inflation and

### _wp16241 - 1985. The choice of this year is motivated by the change in relative volatilities of inflation and

### Data, sample choice, and methodology
- The choice of 1985 as a primary break year is motivated by the change in relative volatilities of inflation and real activity (Great Moderation) in developed countries around the mid-1980s.
- For countries with longer available samples, the exercise is also carried out with a longer sample; the co-movement among real variables remains stable when a larger sample is used.
- Dynamic principal component analysis (DPCA) is employed to analyze key real and nominal macroeconomic data for OECD countries.
- Two commonly used filters are applied for decomposition of cyclical components: the Christiano-Fitzgerald band-pass filter (CF Bandpass) and the Hodrick–Prescott (HP) filter.
- Boxplots are used to summarize model fit: central mark = median, edges = 25th and 75th percentiles, whiskers extend to most extreme non-outlier points, outliers plotted individually; observations are outliers if larger than Q75 + 1.5(Q75 − Q25) or smaller than Q25 − 1.5(Q75 − Q25), where Q25 and Q75 are the 25th and 75th percentiles.

### Empirical findings: fit of DPCA and co-movement
- For most countries, the first dynamic principal component explains most of the dynamics in:
  - output
  - investment
  - imports
  - unemployment
- The first two dynamic principal components explain a high share of the dynamics in all variables.
- The largest dispersion of percentage explained across countries is for:
  - exports
  - short-term real rate
  - consumption
- The co-movement between inflation and real variables is relatively strong for all countries in the sample.
- Summary measures reported:
  - Coherence and cross-correlation between the inflation cycle and output, and between inflation and the first dynamic principal component, indicate relatively high co-movement between inflation and the real economy over the business cycle.
- Significance of lagged correlations:
  - For each country in the sample, there exists a lag k ∈ (0,...,4) for which correlation between cyclical inflation and the cyclical component of output is positive and significantly different from zero at the 5% level.

### Quantitative stylized facts (examples and required model properties)
- At business cycle frequency, empirical moments imply strict requirements on models:
  - In the United States, investment volatility relative to output must be around four.
  - The ‘Okun’s coefficient’ for unemployment is around half.
  - The relative variance of the consumption cycle to output is slightly lower than one, with an identical direction of the response (consumption and output respond in the same direction).
- The first dynamic principal component clearly dominates in terms of explained variance; other components are not explicitly analyzed or identified.
- The dominant component satisfies the sign restriction expected from a broadly-understood “demand shock”: positive co-movement of output and inflation. The component is labeled the “demand factor.”

### Implications for structural macroeconomic models
- Empirically successful models should be able to mimic the correlation structure consistent with a dominant principal component for the variables considered at business cycle frequencies.
- If a model does not feature a structural shock that dominates the cyclical frequencies of consumption, investment, output, hours worked, and inflation, the model is likely misspecified.
  - Misspecification manifests as remaining structural shocks being cross-correlated.
  - Cross-correlated estimated ‘structural shocks’ give the appearance that multiple shocks regularly offset each other.
- The requirements are strict not only in terms of the variance contribution of the shock but also the direction of co-movement of relevant variables and the shape (amplitude and phase) of impulse-response functions at cyclical frequencies.
- Suggested empirical ‘smell tests’ and specification checks:
  - An ex-ante test based on the principal-component space of the model: check whether the model-induced principal-component space is close to the principal-component space of the data.
  - An ex-post test based on cross-correlation of estimated shocks.
  - Verify that impulse-responses ‘make sense’ in light of robust stylized facts on co-movement.
- Companion work (Andrle, Brůha, and Solmaz (2016) referenced in text) discusses implications and proposes misspecification tests.

### Interpretation of real-nominal relationship and frequency considerations
- The apparent absence of a real-nominal dichotomy is attributed to variable definitions and frequency-domain focus:
  - Positive co-movement of output and inflation is found at business cycle frequencies when using cyclical components (inflation gap relative to target or long-term expectations).
  - Using first differences of inflation (to render it stationary) with demeaned GDP growth or gap can over-difference and amplify high-frequency disturbances, weakening the detectable positive relationship.
- The analysis is invariant to both time- and frequency-domain techniques and does not rely on time-domain filtering.

### Conclusions
- Business cycle dynamics of key macroeconomic data can be largely explained by a single source of variation — the dominant unobserved principal component (the “demand factor”).
- The demand factor explains positive co-movement of output cycle and inflation.
- Structural economic models have great difficulties delivering structural shocks resembling the robustly-estimated dominant principal component.
- The study provides a model-independent test for empirical models: shocks considered plausible sources of business cycles must generate principal-component behavior consistent with the data.

*Source: Excerpt from the provided IMF working paper content (post-1985 sample analysis, DPCA results, implications, and conclusions).*

### REFERENCES

### REFERENCES

### Key cited works
- Adelman, I., and F. Adelman, 1959, “The Dynamic Properties of the Klein-Goldberger Model,” Econometrica, No. 4, pp. 596–625.
- Andrle, M., 2012, “Cheers to Good Health of the US Phillips Curve: 1960–2012,” Techn. rep., International Monetary Fund.
- Andrle, Michal, Jan Brůha, and Serhat Solmaz, 2016, “On the Sources of Business Cycles: Implications for DSGE Models,” Czech National Bank Working Paper No. 3/2016.
- Andrle, Michal, Jan Bruha, and Serhat Solmaz, 2013, “Inflation and Output Comovement in the Euro Area: Love at Second Sight?” IMF Working Papers 13/192, International Monetary Fund.
- Brillinger, D.R., 1981, Time Series: Data Analysis and Theory (San Francisco: Holden-Day).
- Burns, A.F., and W.C. Mitchell, 1946, Measuring Business Cycles (New York: NBER).
- Cochrane, J., 1994, “Shocks,” Carnegie-Rochester Conference Series on Public Policy, Vol. 41, pp. 295–364.
- Forni, Mario, Marc Hallin, Marco Lippi, and Lucrezia Reichlin, 2000, “The Generalized Dynamic-Factor Model: Identification And Estimation,” The Review of Economics and Statistics, Vol. 82, No. 4, pp. 540–554.
- Justiniano, A., G.E. Primiceri, and A. Tambalotti, 2010, “Investment Shock and Business Cycles,” Journal of Monetary Economics, Vol. 57, No. 2, pp. 132–145.
- Kindleberger, Ch. P., and R.Z. Aliber, 2005, Manias, Panics and Crashes, Fifth Ed. (New York: Palgrave MacMillan).
- Koopman, L.H., 1974, The Spectral Analysis of Time Series (San Diego, CA: Academic Press).
- Koopmans, T.C., 1957, “Measurement without Theory,” Review of Economic Statistics, Vol. 29, No. August, pp. 161–172.
- Kydland, F.E., and E.C. Prescott, 1990, “Business Cycles: Real Facts and a Monetary Myth,” Federal Reserve Bank of Minneapolis Quarterly Review, Vol. Sping, pp. 3–18.
- Meyer, Brent, and Saeed Zaman, 2013, “It’s not just for inflation: The usefulness of the median CPI in BVAR forecasting,” Working Paper 1303, Federal Reserve Bank of Cleveland, URL http://ideas.repec.org/p/fip/fedcwp/1303.html.
- Pollock, D.S.G., 2013, “Cycles, Syllogisms and Semantics: Examining the Idea of Spurious Cycles,” Journal of Time Series Econometrics, Vol. 6, No. 1, pp. 81–102.
- Sargent, T.J., and C.A. Sims, 1977, “Business cycle modeling without pretending to have too much a-priori economic theory,” in New Methods in Business Cycle Research, ed. by C. Sims et al. (FRB of Minneapolis, Minneapolis).
- Stock, James, and Marc W. Watson, 2002, “Macroeconomic forecasting using diffusion indexes,” Journal of Business and Economic Statistics, Vol. 20, pp. 147–162.
- Summers, L.H., 1986, “Some Skeptical Observations on real business cycle theory,” Federal Reserve Bank of Minneapolis Quarterly Review, Vol. Fall, pp. 23–27.
- Wu, C.F.J., 1986, “Jackknife, bootstrap and other resampling methods in regression analysis (with discussions),” Annals of Statistics, Vol. 14, pp. 1261–1350.

### Role of cited literature (by theme)
- Time series theory and spectral analysis: Brillinger (1981); Koopman (1974); Wu (1986).
- Business cycle measurement and history: Burns and Mitchell (1946); Kydland and Prescott (1990); Sargent and Sims (1977); Summers (1986).
- Dynamic-factor models and diffusion indexes: Forni et al. (2000); Stock and Watson (2002).
- Identification of shocks and investment-related business cycle work: Cochrane (1994); Justiniano, Primiceri, and Tambalotti (2010).
- Practical forecasting and median/trimmed measures of inflation: Andrle (2012); Meyer and Zaman (2013).

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### APPENDIX A. ADDITIONAL GRAPHS

### Figure inventory and descriptive highlights
- Figure 8. Inflation Components – Decomposition
  - Panels include: Headline and Core (headline, median); Trend/Implicit Target (Trend, SPF 10Y Ahead); Cyclical Components: Inflation and Output (inflation cycle, output cycle); High-Frequency Component.
  - Time span labels shown include: 1960:1 1970:1 1980:1 1990:1 2000:1 2010:1.
  - Caption: Source: own computations.

- Figure 9. Cyclical components: data and fit with the DPCA (Christiano-Fitzgerald filter) – the U.S. (full sample)
  - Variables plotted: Real GDP (Y); Real Consumption (C); Real Investment (I); Real Exports (X); Real Imports (M); Unemployment Rate (UR); Short term Interest rate (IR).
  - Series/legend entries: Data; Fit with the first principal component; Fit with the first two principal components.
  - Variance explained panel: YCIXMURIR, with One factor, Two factors, Three factors.
  - Time labels present: 66 76 86 96 06 (as compacted cell labels).

- Figure 10. Cyclical components (Hodrick-Prescott filter): data and fit with the DPCA – the U.S. (1966Q2–2015Q4)
  - Same variables and legend entries as Figure 9.
  - Period explicitly: 1966Q2–2015Q4.

- Figure 11. Spectral density of HP-filtered series: data and fit using the DPCA – the U.S.
  - Panels: Spectral Density of Real GDP (Y); Real Consumption (C); Real Investment (I); Real Exports (X); Real Imports (M); Unemployment Rate (UR); Short term Interest rate (IR).
  - Legend entries: Estimated spectral density; Spectral density of the first common component; Spectral density of the first two common components.
  - Horizontal axis labeled: Periods (examples 2 4 6 8 10 12 16 32 64).

- Figure 12. The DPCA in time domain for all variables together (HP cycles) – the U.S.
  - Variables plotted over 1970–2010: Real GDP (Y); Real Consumption (C); Real Investment (I); Real Exports (X); Real Imports (M); Unemployment Rate (UR); Short term Interest rate (IR); Trimmed inflation.
  - Legend entries: Data; Fit with the first principal component; Fit with first two principal components.
  - Variance explained panel shows: YCIXMURIRPI with The first principal component, Two principal components, Three principal components.

- Figures 13–14. Cyclical components (HP filter) – Japan and Germany
  - Figure 13: Japan panels include Real GDP (Y), Real Consumption (C), Real Investment (I), Real Exports (X), Real Imports (M), Unemployment Rate (UR), Short term Interest rate (IR). Legend as above.
  - Figure 14: Germany panels include Real GDP (Y), Real Consumption (C), Real Investment (I), Real Exports (X), Real Imports (M), Unemployment Rate (UR), Short term Interest rate (IR). Legend as above.

- Figure 15. The boxplot summary statistics (the whole sample)
  - Panels display % explained for YCIXMURIR under: First Factor (BP filter); Two Factors (BP filter); Three Factors (BP filter); First Factor (HP filter); Two Factors (HP filter); Three Factors (HP filter).

- Figure 16. Growth Rates of Macroeconomic Data Consistent with Justiniano, Primiceri, and Tambalotti (2010)
  - Panels (1952:1 1962:1 1972:1 1982:1 1992:1 2002:1 2012:1): Consumption vs. Investment Growth (q/q, ann.); Output vs. Hours Worked Growth (q/q, ann.); Output vs. Investment Growth (q/q, ann.); Consumption vs. Hours Worked (q/q, ann.).
  - Note: Investment contain durable consumption; consumption consists of non-durable consumption only. The series are normalized to equal variance. Source: Haver Analytics.

- Figure 17. US Export Cycle – Actual and Trade-Weighted Foreign Demand
  - Time span labels include: 1995:1 1997:1 1999:1 2001:1 2003:1 2005:1 2007:1 2009:1 2011:1 2013:1.
  - Series: Implied; Actual.
  - Caption: Source: own computations based on IMF Global Projection Model database.

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### APPENDIX B. DATA SOURCES AND SPECIFICATION

### General notes on data and processing
- All computations were performed in Matlab by MathWorks.
- Data and codes for the paper are available upon request.
- Trimmed-mean inflation:
  - EU countries: computed using EUROSTAT data as represented in the Haver Analytics database, Level 3.
  - United States: weighted median inflation from FRB Cleveland.
  - Australia: trimmed-mean inflation from the Reserve Bank of Australia website.
- Ten-years-ahead inflation expectations: Survey of Professional Forecasters (SPF) by FRB of Philadelphia.
- The ‘PTR’ variable (proxy for inflation target) of FRB/US model: kindly provided by Bob Tetlow.
- Seasonal adjustment: provided by source authority when available; otherwise default Bureau of Census X12/ARIMA algorithm was applied in its default setting.

### Table 1. OECD Data (variables collected per country / data source)
- Euro Area15: Private final consumption expenditure, value, GDP expenditure approach.
- Australia, Austria, Belgium, Canada, Finland, France, Germany, Ireland, Italy, Japan, Korea, Luxemburg, Mexico, Netherland, New Zealand, Norway, Poland, Portugal, Spain, Sweden, Switzerland, United Kingdom, United States of America: assorted series including Private final consumption expenditure, Gross domestic product (value and volume), Gross fixed capital formation (value and volume), Imports/Exports of goods and services (value and volume), Core inflation index, Unemployment rate, Short-term interest rate (specific series vary by country).
- Data source noted: OECD Economic Outlook No. 94.

### Tables 2–3. National source data (selection of entries as specified)
- Czech Republic (Czech Statistical Office): multiple series including GDP: Final Consumption Expenditure: Households (SWDA, Mil.CZK); Gross Domestic Product (SWDA, Mil.CZK and Mil.Chn.2005.CZK); Gross Fixed Capital Formation; Imports and Exports (SA, Mil.CZK and SA, Mil.Chn.2005.CZK); Unemployment Rate (SA, %); PRIBOR: 3 Month (Avg, %) from Czech National Bank.
- Denmark (Danmarks Statistik / Danmarks Nationalbank / Statistical Office of the European Communities): Private Consumption Expenditure; Gross Domestic Product; Gross Fixed Capital Formation; Imports/Exports; Harmonized Unemployment Rate (SA, %); Interbank Offered Rate: 3-months (AVG, %).
- Greece (Hellenic Statistical Authority (ELSTAT)): Private Consumption; Gross Domestic Product; Gross Fixed Capital Formation; Imports/Exports; Labor Force Survey: Unemployment Rate (SA, %).
- Hungary (Central Statistical Office / National Bank of Hungary): Final Consumption Expenditure: Private; Gross Domestic Product; Gross Fixed Capital Formation; Imports/Exports; Unemployment Rate (SA, %); Yield on 3-Month Government Debt Securities (EOP, % per annum).
- Slovakia (Statistical Office of the Slovak Republic / Central Office of Labour, Social Affairs and Family / National Bank of Slovakia): GDP components, Unemployment Rate [Registered] (SA, %), New Household Deposits: Redeemable at Notice: Up to 3 Months (%).
- Slovenia (Statistical Office of the Republic of Slovenia / International Monetary Fund / IFS): GDP components; Unemployment Rate (%); Money Market Rate (% per annum).
- Turkey (Turkish Statistical Institute / Central Bank of the Republic of Turkey): Res/Nonresident HHs Final Consumption Expenditure (SA, Thous.TL and SA, Thous.98.TL); Gross Domestic Product (SA, Thous.TL and SA, Thous.98.TL); Gross Fixed Capital Formation; Exports/Imports (SA, Thous.TL and SA, Thous.98.TL); Unemployment Rate (SA, % of Labor Force); Weighted Average Interest Rates for TL Deposits: Up to 3 Months (% p.a.).

*Source: _wp16241 - REFERENCES (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16241.pdf_