## 6. Coincident Indicator and Common Component of Each Sector

## Source details

**Canonical URL:** [6. Coincident Indicator and Common Component of Each Sector](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0469.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0469.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0469.pdf.json)

---

### Introduction
- Objective: construct a reliable short-term economic indicator for France using INSEE monthly and quarterly business survey data.
- Approach: extract a coincident index (common component) from survey opinions using the generalized dynamic factor model (GDFM) of Forni and others (2000).
- Application: use the coincident index to forecast GDP growth and compare forecasting power of GDFM with the random walk (RW) models in out-of-sample tests.
- Sample: 189 quarterly seasonally adjusted French surveys spanning from 1991Q1 until 2003Q1.

### Literature review and methodological context
- Common composite-index approach: X = Σ_{i=1,n} w_i I_i (equation (2.1)).
- Stock and Watson (1989) Kalman-Filter single latent variable approach described (equation (2.2)); relies on a priori distinction between coincident and leading variables and assumes uncorrelated idiosyncratic components.
- Principal components approach: multiple latent factors (Stone 1947); no a priori distinction between variable types.
- GDFM (Forni and others, 2000): reconciles dynamic principal components and dynamic factor models; allows estimation without a priori classification of series; permits mild cross-correlation among idiosyncratic components; coincident index is the weighted average of common components.

### Methodology (GDFM specification)
- Each series decomposed into common components driven by q < n common shocks and idiosyncratic components:
  - x_it = α_i(L) u_t + ε_it  (equation (3.1)).
- u_t is a (q×1) vector of mutually orthogonal common shocks with zero mean and unit variance.
- ε_it are idiosyncratic shocks; GDFM allows autocorrelation among ε_it and assumes their variances are bounded as n → ∞.
- Key notion: as n grows large, idiosyncratic components vanish by law of large numbers, leaving the common components for the business cycle extraction.

### Data transformation
- Data: quarterly, seasonally adjusted INSEE business surveys aggregated under demand, labor market, orders, price, production, services; sectors subdivided into agro-food, consumption, automobile, intermediate, manufacturing, total industry products.
- Panel size reported in appendix: 218 variables in total from 1991Q1 to 2003Q1 (note sample of 189 quarterly surveys used for main estimation).
- Stationarity testing: DFGLS (Elliott, Rothenberg, and Stock, 1996); optimal lag by MAIC criterion.
- Transformation: nonstationary series differenced (transformation code 0 = no transformation, 1 = first difference). All variables normalized by subtracting mean and dividing by standard deviation.
- Moving averages: centered 5-quarter moving averages used for visualization.

### Empirical results — construction and properties of the coincident indicator
- Constructed indicator (COM): average of common components extracted via GDFM.
- Relationship with GDP growth:
  - Visual: moving averages of COM and economic growth exhibit same turning points in 1991Q3, 1992Q2, 1993Q3, 1997Q1, 1999Q3; COM has a quarter lead at 1998Q2 and lags at 1995Q3 and 2001Q1.
  - Correlation: statistically significant positive correlation coefficient of 0.86 between COM and GDP growth.
- Regression of GDP growth on COM (in-sample 1991Q1–2001Q4):
  - Equation (4.1): GROWTH_t = 1.92 − 0.33 GROWTH_{t−1} + 0.90 COM_{t−1} + 0.98 e_{t−2}
    - t-statistics: (14.23) (8.79) (27.83) (37.22) as reported for (4.1).
    - Reported statistics: 95.0 = R^2, 84.0 = DW (as presented).
  - After Newey-West correction for autocorrelation, coefficients reported in (4.2) with t-statistics: (8.72) (6.64) (25.30) (47.09).
  - Interpretation: approximately 95 percent of variation in growth rate is explained by variables in the model (as stated).
  - Diagnostic: White test p-value of F-statistic noted as 0 (greater than 0.05 interpretation presented in text indicates absence of heteroscedasticity as reported).
- Coincidence vs. timing: constructed indicator is "coincident" with GDP changes but, since surveys are on average available 6 weeks before GDP release, the indicator can function as a leading signal relative to the quantitative GDP release.

### Benchmark models: Random Walk specifications
- Two RW specifications used as benchmarks:
  - Optimistic RW (RWO): GROWTH_t = GROWTH_{t−1} (equation (4.3))
    - t-statistic reported: (30.69)
    - 88.0 = R^2 (as reported).
  - Realistic RW (RWR) (publication delays motivate two-period lag): GROWTH_t = GROWTH_{t−2} (equation (4.4))
    - t-statistic reported: (15.73)
    - 57.0 = R^2 (as reported).
- Visual findings: RW models lag turning points and are not prudent for predicting turning points ahead of time.

### Forecast evaluation (in-sample and out-of-sample)
- Evaluation metrics:
  - RMSE: RMSE = sqrt(Σ_t (y_t − ŷ_t)^2 / n) (equation (4.5) as presented).
  - Theil’s U: U = Σ_t (y_t − ŷ_t)^2 / Σ_t ŷ_t^2 (equation (4.6) formatted as in source).
  - Lower RMSE and U closer to zero indicate better forecast performance.
- Sample split:
  - In-sample: 1991Q1 to 2001Q4.
  - Out-of-sample: 2002Q1 to 2003Q1.
- Forecast performance statistics (Table 1):
  - In-sample
    - COM: RMSE = 0.298, U = 0.125
    - RWO: RMSE = 0.499, U = 0.209
    - RWR: RMSE = 0.927, U = 0.389
  - Out-of-sample
    - COM: RMSE = 0.233, U = 0.195
    - RWO: RMSE = 0.298, U = 0.250
    - RWR: RMSE = 0.689, U = 0.578
- Result: GDFM (COM) outperforms both RW benchmarks in-sample and out-of-sample; RW models exhibit a lag in predicting turning points (example: COM predicts 2002Q1 turning point in last quarter of 2001 while RW predicts it one quarter later).

### Sectoral common components: correlations and interpretation
- Computed common components for each aggregate using equation (3.1).
- Correlation coefficients between GROWTH, COM, and sector common components (Table 2 values):
  - Demand: GROWTH 0.94 ; COM 0.87
  - Labor Market: GROWTH 0.94 ; COM 0.82
  - Orders: GROWTH -0.77 ; COM -0.70
  - Prices: GROWTH 0.26 ; COM -0.08
  - Production: GROWTH -0.89 ; COM -0.88
  - Service Sector: GROWTH 0.94 ; COM 0.80
  - Stocks: GROWTH -0.50 ; COM -0.28
- Interpretation of signs:
  - Significant positive correlations: demand, labor market, services with both GDP growth and COM.
  - Orders, production, stocks show negative correlation coefficients; negative sign identifies variables that are leading or lagging by an average of two periods or more rather than implying a negative economic relationship.
  - Prices and stocks display non-significant correlations with COM.
  - Production is identified as a potential leading indicator.
- Visual patterns (Figure 6 summary): demand, services, and labor market variables appear contemporaneous with COM; orders and stocks relatively constant across sample.

### Conclusions and suggested extensions
- Main conclusion: survey-based coincident indicator constructed via GDFM predicts economic growth in France with a relatively high degree of accuracy, implying high predictive power of agents’ opinions.
- Suggested further work from the paper:
  - Improve short-term prediction of output growth using monthly surveys and national account series (because most economic variables are quarterly).
  - Apply the approach to prices and other economic variables.
  - Use business surveys to analyze behavior of economic agents (e.g., computing latent factors for common opinion to assess rationality in decision-making).

*Source: _wp0469 - 6. Coincident Indicator and Common Component of Each Sector (IMF working paper content unit).*

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

### _wp0469 - References

### Tables
- 1. Forecast Performance Statistics ...........................................................................................11
- 2. Correlation Coefficients.......................................................................................................12

### Figures
- 1. Moving Average of Common Components and Economic Growth Rate .............................7
- 2. Moving Average of Economic Growth Rate and Regression Line .......................................8
- 3. Random Walk Optimistic ......................................................................................................9
- 4. Random Walk Realistic .......................................................................................................10
- 5. GDFM and Random Walk Model........................................................................................11

*Source: _wp0469 - References (PDF).*

### 6. Coincident Indicator and Common Component of Each Sector..........................................12

### 6. Coincident Indicator and Common Component of Each Sector

### Introduction
- Objective: construct a reliable short-term economic indicator for France using INSEE monthly and quarterly business survey data.
- Approach: extract a coincident index (common component) from survey opinions using the generalized dynamic factor model (GDFM) of Forni and others (2000).
- Application: use the coincident index to forecast GDP growth and compare forecasting power of GDFM with the random walk (RW) models in out-of-sample tests.
- Sample: 189 quarterly seasonally adjusted French surveys spanning from 1991Q1 until 2003Q1.

### Literature review and methodological context
- Common composite-index approach: X = Σ_{i=1,n} w_i I_i (equation (2.1)).
- Stock and Watson (1989) Kalman-Filter single latent variable approach described (equation (2.2)); relies on a priori distinction between coincident and leading variables and assumes uncorrelated idiosyncratic components.
- Principal components approach: multiple latent factors (Stone 1947); no a priori distinction between variable types.
- GDFM (Forni and others, 2000): reconciles dynamic principal components and dynamic factor models; allows estimation without a priori classification of series; permits mild cross-correlation among idiosyncratic components; coincident index is the weighted average of common components.

### Methodology (GDFM specification)
- Each series decomposed into common components driven by q < n common shocks and idiosyncratic components:
  - x_it = α_i(L) u_t + ε_it  (equation (3.1)).
- u_t is a (q×1) vector of mutually orthogonal common shocks with zero mean and unit variance.
- ε_it are idiosyncratic shocks; GDFM allows autocorrelation among ε_it and assumes their variances are bounded as n → ∞.
- Key notion: as n grows large, idiosyncratic components vanish by law of large numbers, leaving the common components for the business cycle extraction.

### Data transformation
- Data: quarterly, seasonally adjusted INSEE business surveys aggregated under demand, labor market, orders, price, production, services; sectors subdivided into agro-food, consumption, automobile, intermediate, manufacturing, total industry products.
- Panel size reported in appendix: 218 variables in total from 1991Q1 to 2003Q1 (note sample of 189 quarterly surveys used for main estimation).
- Stationarity testing: DFGLS (Elliott, Rothenberg, and Stock, 1996); optimal lag by MAIC criterion.
- Transformation: nonstationary series differenced (transformation code 0 = no transformation, 1 = first difference). All variables normalized by subtracting mean and dividing by standard deviation.
- Moving averages: centered 5-quarter moving averages used for visualization.

### Empirical results — construction and properties of the coincident indicator
- Constructed indicator (COM): average of common components extracted via GDFM.
- Relationship with GDP growth:
  - Visual: moving averages of COM and economic growth exhibit same turning points in 1991Q3, 1992Q2, 1993Q3, 1997Q1, 1999Q3; COM has a quarter lead at 1998Q2 and lags at 1995Q3 and 2001Q1.
  - Correlation: statistically significant positive correlation coefficient of 0.86 between COM and GDP growth.
- Regression of GDP growth on COM (in-sample 1991Q1–2001Q4):
  - Equation (4.1): GROWTH_t = 1.92 − 0.33 GROWTH_{t−1} + 0.90 COM_{t−1} + 0.98 e_{t−2}
    - t-statistics: (14.23) (8.79) (27.83) (37.22) as reported for (4.1).
    - Reported statistics: 95.0 = R^2, 84.0 = DW (as presented; note formatting preserved from source).
  - After Newey-West correction for autocorrelation, coefficients reported in (4.2) with t-statistics: (8.72) (6.64) (25.30) (47.09).
  - Interpretation: approximately 95 percent of variation in growth rate is explained by variables in the model (as stated).
  - Diagnostic: White test p-value of F-statistic noted as 0 (greater than 0.05 interpretation presented in text indicates absence of heteroscedasticity as reported).
- Coincidence vs. timing: constructed indicator is "coincident" with GDP changes but, since surveys are on average available 6 weeks before GDP release, the indicator can function as a leading signal relative to the quantitative GDP release.

### Benchmark models: Random Walk specifications
- Two RW specifications used as benchmarks:
  - Optimistic RW (RWO): GROWTH_t = GROWTH_{t−1} (equation (4.3))
    - t-statistic reported: (30.69)
    - 88.0 = R^2 (as reported).
  - Realistic RW (RWR) (publication delays motivate two-period lag): GROWTH_t = GROWTH_{t−2} (equation (4.4))
    - t-statistic reported: (15.73)
    - 57.0 = R^2 (as reported).
- Visual findings: RW models lag turning points and are not prudent for predicting turning points ahead of time.

### Forecast evaluation (in-sample and out-of-sample)
- Evaluation metrics:
  - RMSE: RMSE = sqrt(Σ_t (y_t − ŷ_t)^2 / n) (equation (4.5) as presented).
  - Theil’s U: U = Σ_t (y_t − ŷ_t)^2 / Σ_t ŷ_t^2 (equation (4.6) formatted as in source).
  - Lower RMSE and U closer to zero indicate better forecast performance.
- Sample split:
  - In-sample: 1991Q1 to 2001Q4.
  - Out-of-sample: 2002Q1 to 2003Q1.
- Forecast performance statistics (Table 1, preserved exactly):
  - In-sample
    - COM: RMSE = 0.298, U = 0.125
    - RWO: RMSE = 0.499, U = 0.209
    - RWR: RMSE = 0.927, U = 0.389
  - Out-of-sample
    - COM: RMSE = 0.233, U = 0.195
    - RWO: RMSE = 0.298, U = 0.250
    - RWR: RMSE = 0.689, U = 0.578
- Result: GDFM (COM) outperforms both RW benchmarks in-sample and out-of-sample; RW models exhibit a lag in predicting turning points (example: COM predicts 2002Q1 turning point in last quarter of 2001 while RW predicts it one quarter later).

### Sectoral common components: correlations and interpretation
- Computed common components for each aggregate using equation (3.1).
- Correlation coefficients between GROWTH, COM, and sector common components (Table 2 reproduced values):
  - Demand: GROWTH 0.94 ; COM 0.87
  - Labor Market: GROWTH 0.94 ; COM 0.82
  - Orders: GROWTH -0.77 ; COM -0.70
  - Prices: GROWTH 0.26 ; COM -0.08
  - Production: GROWTH -0.89 ; COM -0.88
  - Service Sector: GROWTH 0.94 ; COM 0.80
  - Stocks: GROWTH -0.50 ; COM -0.28
- Interpretation of signs:
  - Significant positive correlations: demand, labor market, services with both GDP growth and COM.
  - Orders, production, stocks show negative correlation coefficients; negative sign identifies variables that are leading or lagging by an average of two periods or more rather than implying a negative economic relationship.
  - Prices and stocks display non-significant correlations with COM.
  - Production is identified as a potential leading indicator.
- Visual patterns (Figure 6 summary): demand, services, and labor market variables appear contemporaneous with COM; orders and stocks relatively constant across sample.

### Conclusions and suggested extensions
- Main conclusion: survey-based coincident indicator constructed via GDFM predicts economic growth in France with a relatively high degree of accuracy, implying high predictive power of agents’ opinions.
- Suggested further work from the paper:
  - Improve short-term prediction of output growth using monthly surveys and national account series (because most economic variables are quarterly).
  - Apply the approach to prices and other economic variables.
  - Use business surveys to analyze behavior of economic agents (e.g., computing latent factors for common opinion to assess rationality in decision-making).

*Source: _wp0469 - 6. Coincident Indicator and Common Component of Each Sector (IMF working paper content unit).*

### References

### References

### Cited works
- Burns, A.F., and W.C. Mitchell, 1946, Measuring Business Cycles, (New York: National Bureau of Economic Research).
- Elliott, G., T.J. Rothenberg, and J.H. Stock, 1996, “Efficient Tests for Autoregressive Unit Root,” Econometrica, Vol. 64, No. 4, pp. 813–36.
- Forni, M., M. Hallin, M. Lippi, and L. Reichlin, 2000, “The Generalized Factor Model: Identification and Estimation,” Review of Economic and Statistics, Vol. 82, No. 4, pp. 540–54.
- Fukuda, S., and T. Onodera, 2001, “A New Composite Index of Coincident Economic Indicators in Japan: How Can We Improve the Forecast Performance?” International Journal of Forecasting, Vol. 17, pp. 483–98.
- Geweke, J., 1977, “The Dynamic Factor Analysis of Economic Time Series,” in Latent Variables in Socio-Economic Models ed. by D.J. Aigner and A.S. Golberger (Amsterdam: North-Holland), sp. 19.
- Klein, L.R. and S. Ozmucur, 2003, “The Estimation of China’s Economic Growth Rate,” in Journal of Economic and Social Measurement, forthcoming.
- Nadal-De Simone, F., 2002, “Common and Idiosyncratic Components in Real Output Further: International Evidence,” Working Paper No. 20/229 (Washington: International Monetary Fund).
- Sargent, T.J. and C.A. Sims, 1977, “Business Cycle Modelling without Pretending to have too much a Priori Economic Theory”, in New Methods in Business Research, ed. by C.A. Sims (Mineapolis: Federal Reserve Bank of Mineapolis).
- Stock, J.H. and M.H. Watson, 1989, “New Indexes of Coincident and Leading Economic Indicators,” NBER Macroeconomic Annual Report 1989, pp 351–94.
- Stone, J.R.N., 1947, “On the Interdependence of Blocks of Transactions,” Supplement of Journal of the Royal Statistical Society, Chapter VIII, Part 1, pp. 1–32.

*Source: _wp0469 - References*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0469.pdf_