## _wp10203 — Section 2: Timing conventions of WEO data and related empirical findings

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---

### Timing conventions
- Timing conventions follow Timmerman (2005) for compatibility and comparison with existing research.
- Because reported variables are subject to revision, the paper uses the first-available data in the Spring WEO issue of year t + 1 to measure the outcome of the predicted variable in period t (labeled x_t).
- Next-year forecasts for period t + 1 are compared to the realized values for year t + 1 (x_{t+1}) reported in the Fall WEO issue of year t + 2.
- Forecast notation:
  - Two sets of current-year forecasts: ˆx_{tt}^{Spring} and ˆx_{tt}^{Fall}.
  - Two sets of next-year forecasts: ˆx_{t,t+1}^{Spring} and ˆx_{t,t+1}^{Fall}.
  - First subscript indicates the period being predicted; second subscript indicates the year when the forecast was generated.
  - Superscript indicates the WEO issue (Spring or Fall) where the WEO forecast was reported.

### Definitions of forecast errors
- Spring current year forecast error:
  - e_{tt}^{Spring} = x_{tt} - ˆx_{tt}^{Spring}
- Fall current year forecast error:
  - e_{tt}^{Fall} = x_{tt} - ˆx_{tt}^{Fall}
- Spring next year forecast error:
  - e_{t,t+1}^{Spring} = x_{t+1,t+1} - ˆx_{t,t+1}^{Spring}
- Fall next year forecast error:
  - e_{t,t+1}^{Fall} = x_{t+1,t+1} - ˆx_{t,t+1}^{Fall}

### Properties of optimal forecasts (Diebold and Lopez (1995))
- Four key properties stated in the source:
  1. Optimal forecast errors have a zero mean (they are unbiased).
  2. 1-step-ahead optimal forecast errors are white noise.
  3. k-step-ahead optimal forecast errors are at most MA(k-1).
  4. k-step-ahead optimal forecast error variance is non-decreasing in k.

### Property: k-step-ahead optimal forecast error variance is non-decreasing in k — implications
- Optimal forecasts that use more information should have smaller conditional variance than those using less information.
- Example implication: Spring WEO forecasts for the current year should have greater variances than Fall WEO forecasts for the same year because Fall forecasts are conditional on more information.
- The variance inequality is stated in the source as equation (5).

### Covariance structure and correlations of multivariate optimal forecast errors
- Intra-temporal correlations between forecast errors of different economic variables may be nonzero due to common aggregate shocks; thus intra-temporal correlations need not equal zero.
- Both conditions in equation (6) of the source are permissible for optimal forecasts.
- The source notes violations of the no-correlation condition (6) in empirical data, suggesting adaptive expectations and no systematic role for data dissemination standards in these violations.

### Data set, exclusions, and cleaning
- Data: Spring and Fall WEO real GDP growth rates and CPI inflation rate forecasts for the period 1999-2007.
- Observations dropped:
  - Realized absolute value of real GDP growth exceeding 15 percent per annum were dropped.
  - Realized CPI inflation rate exceeding 200 percent per annum were dropped.
  - The third WEO of year 2001 is excluded.
  - Countries not existing for the entire period of observation were omitted.
- Sample size notes:
  - Minimum of 170 observations in 1999 and maximum of 349 in 2007.
  - Country coverage after 2002 nearly complete; approximately 180 member countries with two observations per annum.
  - Number of observations per region ranges from 422 (Region 2) to 677 (Region 3).
  - The number of all observations exceeds two thousand.

### Descriptive statistics of forecast errors (selected key figures preserved)
- From Table 1 (All row):
  - Real GDP growth forecast error, Same year: Mean 0.30, Std. Dev. 1.75, Obs. 2524
  - Real GDP growth forecast error, One year ahead: Mean 0.27, Std. Dev. 2.96, Obs. 2175
  - Inflation forecast error, Same year: Mean 0.18, Std. Dev. 3.50, Obs. 2522
  - Inflation forecast error, One year ahead: Mean 1.07, Std. Dev. 5.85, Obs. 2173
- From Table 3 (descriptive statistics):
  - GDP_{t+1}: Mean 0.26; Median 0.30; Maximum 27.20; Minimum -35.30; Std. Dev. 2.96; Skewness -0.64; Kurtosis 21.55; Jarque-Bera statistic 31275.7; Jarque-Bera p-value 0.0000
  - GDP_{t}: Mean 0.30; Median 0.10; Maximum 10.20; Minimum -9.10; Std. Dev. 1.79; Skewness 0.03; Kurtosis 7.42; Jarque-Bera statistic 1767.6; Jarque-Bera p-value 0.0000
  - π_{t+1}: Mean 1.07; Median 0.40; Maximum 60.00; Minimum -90.50; Std. Dev. 5.85; Skewness -1.83; Kurtosis 67.16; Jarque-Bera statistic 373723.2; Jarque-Bera p-value 0.0000
  - π_{t}: Mean 0.17; Median 0.00; Maximum 60.00; Minimum -81.00; Std. Dev. 3.65; Skewness -2.14; Kurtosis 160.03; Jarque-Bera statistic 2233172.0; Jarque-Bera p-value 0.0000
- Distributional findings:
  - Forecast errors are unconditionally unbiased (t-values well below standard critical values).
  - Three out of four forecast errors exhibit skewed distributions; all forecast errors have fat tails (kurtosis well above 3).
  - Normality strongly rejected for all forecast errors (Jarque-Bera p-values equal to 0.0000).
  - Same period forecast errors have lower standard deviations than one period ahead forecast errors.

### Parametric modeling of forecast error distributions
- Empirical density of same period real GDP forecast error differs strongly from normality; higher mode and fatter tails.
- The four-parameter Pearson family is identified as the most suitable parametric family for forecast errors.
- Table 4 Pearson family estimates:
  - GDP_{t}: Pearson type 4; b2 0.4466; b1 0.0758; b0 0.1845
  - GDP_{t+1}: Pearson type 4; b2 0.1816; b1 -0.3068; b0 0.4553
  - π_{t}: Pearson type 1, four-parameter beta; b2 0.5884; b1 -0.7652; b0 -23.676
  - π_{t+1}: Pearson type 6, F with additional location and scale parameters; b2 0.1883; b1 1.7997; b0 0.4351
- Conclusion: Normal (Gaussian) distribution is a poor probabilistic model of forecast errors; the Pearson family is recommended.

### Empirical tests and results (mean-variance, variance equality, correlations)
- Mean-variance analysis (Tables 5–8):
  - The variance inequality (more information → lower variance) is satisfied for all reported cases.
  - Sample means of forecast real GDP errors and of inflation forecast errors are generally not statistically different from zero.
- Variance equality tests — real GDP growth rate forecast errors (Table 9):
  - Levene test, Same period: (2, 2521) Value 0.52; Probability 0.5940
  - Brown-Forsythe test, Same period: (2, 2521) Value 0.32; Probability 0.7270
  - Levene test, One period ahead: (2, 2172) Value 2.78; Probability 0.0620
  - Brown-Forsythe test, One period ahead: (2, 2172) Value 2.55; Probability 0.0782
  - Group statistics (Same period): SDDS Count 1117; Std. Dev. 1.69. GDDS Count 1119; Std. Dev. 1.80. None Count 288; Std. Dev. 1.77.
  - Group statistics (One period ahead): SDDS Count 989; Std. Dev. 2.63. GDDS Count 943; Std. Dev. 3.05. None Count 243; Std. Dev. 3.73.
  - Interpretation: Tests reject the null for one period ahead forecasts (variance of SDDS subscribers smaller than GDDS participants) but not for same period forecasts.
- Variance equality tests — inflation forecast errors (Table 10):
  - Levene test, Same period: (2, 2519) Value 24.49; Probability 0.0000
  - Brown-Forsythe test, Same period: (2, 2519) Value 22.03; Probability 0.0000
  - Levene test, One period ahead: (2, 2170) Value 13.76; Probability 0.0000
  - Brown-Forsythe test, One period ahead: (2, 2170) Value 12.74; Probability 0.0000
  - Group statistics (Same period): SDDS Count 1115; Std. Dev. 3.65. GDDS Count 1120; Std. Dev. 3.01. None Count 287; Std. Dev. 4.51.
  - Group statistics (One period ahead): SDDS Count 987; Std. Dev. 5.57. GDDS Count 944; Std. Dev. 6.04. None Count 242; Std. Dev. 5.99.
  - Interpretation: Null hypothesis of equal variances strongly rejected for both same-year and one-year-ahead inflation forecasts; no uniform ordering on forecast quality for inflation across dissemination standards.
- Robustness check:
  - Tests of equality of variances of the forecasted variables themselves (real GDP growth and CPI inflation) between SDDS subscribers and GDDS participants could not reject equality; hence differential forecast variances are not driven by differences in variances of the underlying variables.
- Correlation analysis (Table 11 and text):
  - Table 11 reports covariance/correlation analysis for All (N=2202), SDDS (N=989), GDDS (N=964), No diss. std. (N=249).
  - There appear to be no significant systematic variations in the correlation structure across SDDS, GDDS, no data dissemination standard, and the combined sample; in all four cases there are three statistically significant correlations that violate identity (5).
  - Interpretation: Suggests some form of adaptive expectations used in forecasting and that data dissemination standards do not play a systematic role in these correlation violations.

### Main conclusions and policy implications (as stated in the source)
- Graduating from GDDS to SDDS broadly improves statistical quality of forecasts of macroeconomic variables.
  - For the WEO: variances of real GDP growth rate forecast errors are smaller in SDDS subscribers than in GDDS participants for one year ahead forecasts, but not for same year forecasts.
  - Statistical quality of WEO inflation forecasts is significantly better for SDDS subscribers than for GDDS participants (tests reject equal variances).
- Conditional correlations analysis shows frequent violations of the no-correlation condition (6) for same period and one period ahead real GDP and inflation forecasts, with no systematic variation in these violations across data dissemination standards.
- Parametric estimates show real GDP forecast errors cannot be modeled by standard probability distributions; normal distribution is a poor model.
- Practical implication: IMF’s data standard initiative positively affects WEO forecast statistical quality and thereby improves necessary conditions for providing member countries with high quality policy advice.

*Source: _wp10203 — Section 2 describes the timing conventions of WEO data; plus Section 4 property that k-step-ahead optimal forecast error variance is non-decreasing in k.*

### Section 2 describes the timing conventions of WEO data,

### _wp10203 - Section 2 describes the timing conventions of WEO data,

### Timing conventions
- Timing conventions follow Timmerman (2005) for compatibility and comparison with existing research.
- Because reported variables are subject to revision, the paper uses the first-available data in the Spring WEO issue of year t + 1 to measure the outcome of the predicted variable in period t (labeled x_t).
- Next-year forecasts for period t + 1 are compared to the realized values for year t + 1 (x_{t+1}) reported in the Fall WEO issue of year t + 2.
- Forecast notation:
  - Two sets of current-year forecasts: ˆx_{tt}^{Spring} and ˆx_{tt}^{Fall}.
  - Two sets of next-year forecasts: ˆx_{t,t+1}^{Spring} (written in source as 1,ˆSpring_tt x+ ?) and ˆx_{t,t+1}^{Fall} (notation in source: 1,ˆFall_tt x+ ?).
  - First subscript indicates the period being predicted; second subscript indicates the year when the forecast was generated.
  - Superscript indicates the WEO issue (Spring or Fall) where the WEO forecast was reported.

### Definitions of forecast errors
- Four forecast errors are defined using the timing and notation above:

  - Spring current year forecast error:
    - e_{tt}^{Spring} = x_{tt} - ˆx_{tt}^{Spring}
    - (source notation: , , ˆSpringSpring t ttt t exx = - . Equation (1))

  - Fall current year forecast error:
    - e_{tt}^{Fall} = x_{tt} - ˆx_{tt}^{Fall}
    - (source notation: , , ˆFallFall t ttt t exx = - . Equation (2))

  - Spring next year forecast error:
    - e_{t,t+1}^{Spring} = x_{t+1,t+1} - ˆx_{t,t+1}^{Spring}
    - (source notation: 1,11, ˆSpringSpring ttttt exx +++ = - . Equation (3))

  - Fall next year forecast error:
    - e_{t,t+1}^{Fall} = x_{t+1,t+1} - ˆx_{t,t+1}^{Fall}
    - (source notation: 1,11, ˆFallFall ttttt exx +++ = - . Equation (4))

### Properties of optimal forecasts (Diebold and Lopez (1995))
- Four key properties stated:
  1. optimal forecast errors have a zero mean, that is, they are unbiased;
  2. 1-step-ahead optimal forecast errors are white noise;
  3. k-step-ahead optimal forecast errors are at most MA(k-1);
  4. (text in source continues beyond provided excerpt)

### Implications noted in Section 2
- The choice of vintage (first-available Spring issue for outcomes in period t and Fall issue two years later for next-year outcomes) underpins comparability with prior research.
- The paper uses these timing conventions to define and compute the four forecast errors above, which feed into subsequent empirical analysis and statistical testing.

*Source: _wp10203 - Section 2 describes the timing conventions of WEO data*

### 4.      k-step-ahead optimal forecast error variance is non-decreasing in k.

### _wp10203 - 4.      k-step-ahead optimal forecast error variance is non-decreasing in k.

### Property: k-step-ahead optimal forecast error variance is non-decreasing in k
- A direct consequence of the fourth property of optimal forecasts is a decreasing variance of forecast errors: optimal forecasts that use more information should have smaller conditional variance than those using less information.
- Example implication: Spring WEO forecasts for the current year should have greater variances than Fall WEO forecasts for the same year because Fall forecasts are conditional on more information than Spring forecasts.
- The text states this mathematically as equation (5) (referenced in the source).

### Covariance structure and correlation properties of optimal multivariate forecast errors
- The covariance structure of optimal multivariate forecast errors possesses important properties for analyzing statistical quality.
- Intra-temporal correlations between forecast errors of different economic variables may be nonzero due to common aggregate shocks; therefore, intra-temporal correlations between forecast errors of different economic variables need not equal zero.
- Both conditions in equation (6) are permissible for optimal forecasts (equation (6) as presented in the source).

### Data dissemination standards and hypothesis on forecast statistical quality
- The IMF’s data dissemination standards, GDDS and SDDS, impose requirements on periodicity and timeliness of macroeconomic and socio-economic data.
- Broad characterization: the SDDS in most cases requires higher frequency (periodicity) and less delay (timeliness) in data dissemination than the GDDS.
- Hypothesis stated in the source:
  - Unbiased forecasts of macroeconomic variables that are disseminated by SDDS subscribers should have smaller variances than unbiased forecasts disseminated by GDDS participants only if SDDS periodicity and timeliness requirements are stricter than GDDS requirements.
- Examples from the text:
  - Real GDP forecasts for SDDS subscribers should have smaller variances than real GDP forecasts for GDDS participants because SDDS requires quarterly dissemination with a quarterly lag while GDDS only requires annual dissemination of national accounts data.
  - One should not expect significant differences in variances of inflation forecasts because both SDDS and GDDS require dissemination of inflation data with monthly periodicity and monthly timeliness.

### The data set (period, exclusions, and cleaning)
- Data set: Spring and Fall WEO real GDP growth rates and CPI inflation rates forecasts for the period 1999-2007.
- Forecast errors computed as described in Section II of the source.
- Observations dropped to reduce outlier influence:
  - All observations where realized absolute value of real GDP growth exceeded 15 percent per annum were dropped.
  - All observations where realized CPI inflation rate exceeded 200 percent per annum were dropped.
  - The third WEO of year 2001 is excluded (one-off due to 9/11).
  - Data on countries that did not exist for the entire period of observation were omitted.
- Sample size notes:
  - Minimum of 170 observations in 1999 and maximum of 349 in 2007.
  - Country coverage after 2002 nearly complete; approximately 180 member countries with two observations per annum.
  - Number of observations per region ranges from 422 (Region 2) to 677 (Region 3).
  - The number of all observations exceeds two thousand.

### Descriptive statistics of forecast errors (key summary statistics preserved)
- From Table 1 (All row):
  - Real GDP growth forecast error, Same year: Mean 0.30, Std. Dev. 1.75, Obs. 2524
  - Real GDP growth forecast error, One year ahead: Mean 0.27, Std. Dev. 2.96, Obs. 2175
  - Inflation forecast error, Same year: Mean 0.18, Std. Dev. 3.50, Obs. 2522
  - Inflation forecast error, One year ahead: Mean 1.07, Std. Dev. 5.85, Obs. 2173
- From Table 3 (descriptive statistics):
  - GDP_{t+1}: Mean 0.26; Median 0.30; Maximum 27.20; Minimum -35.30; Std. Dev. 2.96; Skewness -0.64; Kurtosis 21.55; Jarque-Bera statistic 31275.7; Jarque-Bera p-value 0.0000
  - GDP_{t}: Mean 0.30; Median 0.10; Maximum 10.20; Minimum -9.10; Std. Dev. 1.79; Skewness 0.03; Kurtosis 7.42; Jarque-Bera statistic 1767.6; Jarque-Bera p-value 0.0000
  - π_{t+1}: Mean 1.07; Median 0.40; Maximum 60.00; Minimum -90.50; Std. Dev. 5.85; Skewness -1.83; Kurtosis 67.16; Jarque-Bera statistic 373723.2; Jarque-Bera p-value 0.0000
  - π_{t}: Mean 0.17; Median 0.00; Maximum 60.00; Minimum -81.00; Std. Dev. 3.65; Skewness -2.14; Kurtosis 160.03; Jarque-Bera statistic 2233172.0; Jarque-Bera p-value 0.0000
- Distributional findings:
  - Forecast errors are unconditionally unbiased (t-values well below standard critical values).
  - Three out of four forecast errors exhibit skewed distributions; all forecast errors have fat tails (kurtosis well above 3).
  - Normality strongly rejected for all forecast errors (Jarque-Bera p-values equal to 0.0000 to four decimal places).
  - Same period forecast errors have lower standard deviations than one period ahead forecast errors.

### Parametric modeling of forecast error distributions
- Empirical density of same period real GDP forecast error differs strongly from normality; higher mode and fatter tails.
- The four-parameter Pearson family is identified as the most suitable parametric family for forecast errors.
- Table 4 parametric estimates (Pearson family coefficients preserved):
  - GDP_{t}: Pearson type 4, not related to any standard distribution; b2 0.4466; b1 0.0758; b0 0.1845
  - GDP_{t+1}: Pearson type 4, not related to any standard distribution; b2 0.1816; b1 -0.3068; b0 0.4553
  - π_{t}: Pearson type 1, four-parameter beta; b2 0.5884; b1 -0.7652; b0 -23.676
  - π_{t+1}: Pearson type 6, F with additional location and scale parameters; b2 0.1883; b1 1.7997; b0 0.4351
- Conclusion drawn: normal (Gaussian) distribution is a poor probabilistic model of forecast errors; the Pearson family is recommended for modeling statistical properties of forecast errors.

### Empirical results: mean-variance (variance inequality), variance equality tests, and correlation analysis
- Mean-variance analysis (Tables 5–8):
  - The variance inequality (5) is satisfied for all cases reported: more information (Fall vs Spring) leads to forecasts with lower variances.
  - Sample means of forecast real GDP errors and of inflation forecast errors are generally not statistically different from zero (unbiased).
  - Because cross-sectional means are not significantly different from zero, forecast statistical quality can be assessed primarily via variances.
- Variance equality tests (Table 9 and Table 10):
  - For annual real GDP growth rate forecast errors (Table 9):
    - Levene test, Same period: (2, 2521) Value 0.52; Probability 0.5940
    - Brown-Forsythe test, Same period: (2, 2521) Value 0.32; Probability 0.7270
    - Levene test, One period ahead: (2, 2172) Value 2.78; Probability 0.0620
    - Brown-Forsythe test, One period ahead: (2, 2172) Value 2.55; Probability 0.0782
    - Interpretation from source: Tests reject the null for the one period ahead forecasts, but not for the same period forecasts. Variance of SDDS subscribers is smaller than GDDS participants for one-year ahead forecasts, but difference is not significant for same-year forecasts.
    - Group statistics (Same period): SDDS Count 1117; Std. Dev. 1.69. GDDS Count 1119; Std. Dev. 1.80. None Count 288; Std. Dev. 1.77.
    - Group statistics (One period ahead): SDDS Count 989; Std. Dev. 2.63. GDDS Count 943; Std. Dev. 3.05. None Count 243; Std. Dev. 3.73.
  - For annual inflation rate forecast errors (Table 10):
    - Levene test, Same period: (2, 2519) Value 24.49; Probability 0.0000
    - Brown-Forsythe test, Same period: (2, 2519) Value 22.03; Probability 0.0000
    - Levene test, One period ahead: (2, 2170) Value 13.76; Probability 0.0000
    - Brown-Forsythe test, One period ahead: (2, 2170) Value 12.74; Probability 0.0000
    - Group statistics (Same period): SDDS Count 1115; Std. Dev. 3.65. GDDS Count 1120; Std. Dev. 3.01. None Count 287; Std. Dev. 4.51.
    - Group statistics (One period ahead): SDDS Count 987; Std. Dev. 5.57. GDDS Count 944; Std. Dev. 6.04. None Count 242; Std. Dev. 5.99.
    - Interpretation from source: Null hypothesis of equal variances strongly rejected for both same-year and one-year-ahead inflation forecasts. No uniform ordering on forecast quality for inflation across data dissemination standards because many countries report inflation monthly regardless of standard.
  - A robustness check reported: test of equality of variances of the forecasted variables themselves (real GDP growth and CPI inflation rate) between SDDS subscribers and GDDS participants could not reject equality; therefore differential forecast variances are not driven by differences in variances of the underlying variables.
- Correlation analysis (Table 11 and text):
  - Optimal forecasts should satisfy a no-correlation condition (6), but due to common aggregate shocks nonzero correlations between forecast errors for different macro aggregates are likely.
  - Table 11 reports covariance/correlation analysis for All (N=2202), SDDS (N=989), GDDS (N=964), No diss. std. (N=249).
  - The source notes: there appear to be no significant systematic variations in the correlation structure across SDDS, GDDS, no data dissemination standard, and the combined sample; in all four cases there are three statistically significant correlations that violate identity (5).
  - Interpretation from source: This suggests some form of adaptive expectations used in forecasting and that data dissemination standards do not play a systematic role in these correlation violations.

### Main conclusions and policy implications (as stated)
- Graduating from GDDS to SDDS broadly improves statistical quality of forecasts of macroeconomic variables.
  - For the WEO: variances of real GDP growth rate forecast errors are smaller in SDDS subscribers than in GDDS participants for one year ahead forecasts, but not for same year forecasts.
  - Statistical quality of WEO inflation forecasts is significantly better for SDDS subscribers than for GDDS participants (tests reject equal variances).
- Conditional correlations analysis shows frequent violations of the no-correlation condition (6) for same period and one period ahead real GDP and inflation forecasts, with no systematic variation in these violations across data dissemination standards.
- Parametric estimates show real GDP forecast errors cannot be modeled by standard probability distributions; normal distribution is a poor model.
- Practical implication: IMF’s data standard initiative positively affects WEO forecast statistical quality and thereby improves necessary conditions for providing member countries with high quality policy advice.

*Source: _wp10203 - 4.      k-step-ahead optimal forecast error variance is non-decreasing in k.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10203.pdf_