## 3.1 Setup (wpiea2020078-print-pdf)

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---

### Background and motivation
- Predictive densities and fan charts are widely used by central banks and policy institutions.
- IMF practices referenced:
  - Since 2006, WEO’s global GDP growth projections accompanied by a fan chart summarizing the balance of risks.
  - Since 2017, GFSR has published assessments of unlikely but possible global GDP growth collapses, focusing on the left tail.
- Goal: quantify probabilistically the uncertainty around point forecasts and estimate probabilities of specified events (for example, outcomes associated with a five percent probability).

### Aggregation challenges for predictive densities
- Combining individual predictive densities is often superior to single-model densities, but aggregation raises problems:
  - Country-specific data-generating processes differ and are not i.i.d.
  - Country aggregates are subject to tail dependence.
  - Linear combinations of country-specific densities do not account for interdependencies and are not viable for consistent global aggregates.
- Standard density-combination approaches (linear opinion pool, logarithmic opinion pool) have shortcomings: loss of calibration, incoherence, assigning zero probability to events zero in any component.
- Direct estimation from aggregate data addresses interdependencies but has limitations:
  - Relies on shorter series and may neglect periods with extreme events.
  - Quantile regression-based approaches may characterize tails with very few data points.
  - Survey-based approaches can be unstable due to small samples of forecasters.
  - Averaging country-level information can dilute informative country-specific signals.

### Proposed four-step bottom-up probabilistic approach
- Objective: estimate and combine single-country predictive densities to obtain the predictive density of global GDP growth while avoiding curse of dimensionality.
- Country-sample split and inter-dependency structure:
  - Other economies depend on large economies and their respective regional leaders.
  - Regional leaders depend on large economies.
  - Regional leaders and other economies have no impact on large economies.
- Four-step procedure:
  1. Generate joint predictive density for large economies (US, EA, CHN) via multivariate conditional kernel density estimation; assign probability to combinations of their future GDP growth rates conditional on country-specific factors and contemporaneous growth rates of the other two large economies.
  2. For each conditioning triplet of large-economy GDP growth rates, estimate each regional leader’s kernel density and sample from it to construct regional predictive densities.
  3. For each tuple of large economies’ and regional leaders’ GDP growth rates, estimate other economies’ predictive density by drawing from a normal distribution with historical mean and standard deviation.
  4. Aggregate each sampled vector of GDP growth rates using a PPP-weighted average to construct global GDP growth predictive draws.
- Introducing judgment (re-centering the mode):
  - Rejection sampling algorithm, or
  - Fit a parametric distribution minimizing the Kolmogorov-Smirnov distance between empirical and parametric distributions.

### Definition and aggregation
- Global GDP growth defined as PPP-weighted average: y_Wt = Σ_{i=1}^N ω_{i,t-1} y_{i,t}, where ω_{i,t-1} = Y_{i,t-1} / Σ_{j=1}^N Y_{j,t-1}.
- Sample is non-i.i.d.; estimate joint distribution f(y^1_t, …, y^N_t) to capture cross-country inter-dependencies.
- To overcome dimensionality, partition into three groups and use sequential estimation (see equation (4) in source).

### Country grouping and shares (end-2018, IMF WEO PPP weights)
- Large economies:
  - United States (US), the euro area (EA), and China (CHN).
  - Combined share of world GDP: 45 percent.
- Regional leaders:
  - South Africa (Africa); India and Japan (Asia); Russia and the United Kingdom (Europe); Brazil and Mexico (Western Hemisphere).
  - Combined share of world GDP: 22 percent.
  - Note: Africa has only one country in the sample due to quarterly data limitations.
- Other economies:
  - Argentina, Australia, Belarus, Canada, Chile, Colombia, Croatia, Czech Republic, Denmark, Hong Kong SAR China, Hungary, Indonesia, Israel, Jordan, Kazakhstan, Korea, Malaysia, Moldova, Norway, Peru, Philippines, Poland, Romania, Serbia, Singapore, Sweden, Switzerland, Taiwan Province of China, Thailand, Turkey, Ukraine, Venezuela, and Vietnam.
  - Combined share of world GDP: 19.5 percent.
- Appendix A lists countries with relative share of world GDP in PPP terms; IMF WEO regional classification is used.

### Key modeling assumptions (H1–H3)
- H1 (Other economies): other economies’ predictive densities depend on large economies and respective regional leaders and country-specific factors X_o; conditional density f(y_o | X_o, y_US, y_EA, y_CHN, …).
- H2 (Regional leaders): regional leaders’ predictive densities depend on large economies and country-specific factors X_r; conditional density f(y_r | X_r, y_US, y_EA, y_CHN); inter-dependencies among regional leaders are ruled out.
- H3 (Large economies): regional leaders and other economies have no impact on large economies; predictive densities of large economies depend contemporaneously only on the other large economies (e.g., f(y_US | X_US, y_EA, y_CHN)).

### Sequential empirical strategy (overview)
- Step 1: Estimate joint density of three large economies (US, EA, CHN).
- Step 2: Estimate predictive densities of regional leaders conditional on Step 1.
- Step 3: Estimate predictive densities of other economies conditional on Steps 1–2.
- Step 4: Aggregate predictive densities to obtain predictive density for global GDP growth.
- Rolling window: excludes the 12 quarters following the quarter being forecast.

---

### 4.1 Estimation Results (selected findings)

### Sample, covariates, and specification
- Sample composition:
  - 43 countries and one country group of 19 countries (the euro area), representing 87 percent of world output.
- Main covariate categories:
  - Financial conditions (financial conditions index).
  - Leading indicators (industrial production, purchasing managers’ index).
  - External variables (import demand, real effective exchange rate, commodity terms of trade).
  - Proxies of uncertainty (business confidence, economic policy uncertainty, geopolitical risk).
  - Demographics (population growth).
- Data-frequency notes:
  - Joint distribution for three large economies limited by China’s GDP growth data starting in 1995q1.
  - If covariates available monthly, include lead of average monthly observations over a quarter.
- Country-specific variable choices based on data availability and expert judgment.
- Parsimony: baseline excludes financial conditions index for EA and China due to high correlation with US financial conditions index.

### Joint predictive density of large economies: stylized dynamics and crisis periods
- Visualization via contour plots for selected quarters (2006q1, 2008q4, 2009q1, 2009q2, 2009q3, 2012q3).
- Key empirical patterns:
  - 2006q1: probability concentrated around about 3 percent for the US and 2 percent for the euro area.
  - 2008q4: mode shifts toward lower growth; US shows relatively high probability of negative growth and longer tails.
  - 2009q1: joint distribution displays two distinguishable modes.
  - 2009q2: predicted distribution assigns higher probability to the bad outcome.
  - 2009q3: bi-modality disappears; probability of a deep recession in both economies increases considerably.
  - 2012q3 (post Euro-zone debt crisis): prevalent good-growth mode but with non-negligible probability attached to lower growth.

### Country-specific predictive densities and interval forecasts
- Interval forecast evaluation for large economies:
  - Figure 3 reports estimated 5–95 and 25–75 percentile ranges, mode, and realized GDP growth since 2005 for US, EA, China.
  - Actual GDP growth falls within the 25–75 percentile range most of the time.
  - China: model captures negative trend since 2007; 50 percent interval sometimes appears wide.
  - US and EA: larger probability mass to the left of the mode during global financial crisis and Euro-zone debt crisis.
- Regional leaders (four-quarter ahead forecasts): Brazil, India, Japan, Mexico, Russia, United Kingdom, South Africa.
  - Predicting normal times easier; mode close to realization.
  - During large swings and long horizon, predictive densities move consistently.
  - Some densities appear jagged due to bandwidth choices; bandwidth adjustments avoided to limit discretion.
- Other economies (illustrative): Switzerland and Turkey reported; assumption of Gaussianity for other economies does not significantly affect forecast accuracy.

### Global predictive density, tail risks, and slowdown probabilities
- Interval forecast evaluation for global GDP growth (four-quarter ahead):
  - Since 2006 the global 5–95 percentile range ranges between 1.8 percent and 6.4 percent, with exceptions during crisis periods.
  - During the global financial crisis the 5th percentile reached -2.6 percent.
  - During the euro-zone crisis the 5th percentile reached -2 percent.
  - Model struggles to predict the global financial crisis in real time and tends to detect it with a lag.
  - Significant downward skewness observed in correspondence with the euro-zone crisis.
- Slowdown threshold:
  - Threshold set at the global GDP growth rate corresponding to the 25th percentile of the historical distribution, which is about 3 percent.
- Probability of global growth below 3 percent (four-quarter ahead):
  - High probability during the global financial crisis (end of 2007 and onwards).
  - Probability spiked to about 60 percent in 2011 and 2012, and again in 2015.
  - Since 2017 the probability crawled upwards and reached about 43 percent in 2020.

### Forcing central tendency: options to align predictive mode with external baseline
- Rationale: users may wish to force the mode to an alternative central tendency (e.g., IMF WEO baseline).
- Two approaches:
  1. Parametric fit with constraint on the mode:
     - Fit parametric distribution constrained to a chosen mode.
     - Use minimum Kolmogorov-Smirnov distance as selection criterion.
     - Candidate distributions: skewed-t, asymmetric-t, beta, skew normal, Weibull.
  2. Rejection sampling algorithm:
     - Draw points from original sampled predictive density and reject points with probability proportional to distance to desired mode, producing new sample with desired mode while preserving moments where possible.
- Illustrative example (2019q4 estimated with data up to 2018q4):
  - Sampled density varies between one and six percent; mode of 3.2 percent.
  - Beta distribution achieved smallest Kolmogorov-Smirnov distance in this example and overlaps kernel fit.
  - Forcing mean to IMF WEO projection of 2.9 percent:
    - Forcing mode/mean affects tails and asymmetry.
    - Rejection sampling led to modest change in symmetry; sampled density became slightly skewed to the right in the example.
    - Re-fitting parametric distributions after forcing found beta distribution best capturing adjusted sampled density in this example.

---

### 4.2 Evaluating the Accuracy of Density Forecasts (selected methods and results)

### 4.2.1 Probability Integral Transform (PIT)
- PIT definition: cumulative probability evaluated at realized value; assigns values between 0 and 1.
- Theoretical property: for correctly specified forecasts, PIT values should be uniform, independent, and identically distributed.
- Empirical test: plotted empirical CDF of PIT for world GDP growth four quarters ahead against 45-degree line and confidence bands based on Rossi and Sekhposyan (2019).
- Key finding:
  - "The empirical distribution of the PIT is within the confidence band for all quantiles, even for the most conservative confidence band based on 10 percent critical values."
  - Implication: satisfactory out-of-sample accuracy and robust predictive distributions that capture upside and downside risks.

### 4.2.2 Entropy Measures (downside/upside entropy)
- Purpose: compare probability assigned to extreme growth outcomes by conditional density relative to unconditional density.
- Downside entropy L_D_t and upside entropy L_U_t defined relative to ˆf_{y_{t+h}|x_t} and ˆg_{y_{t+h}} (see equations (12) and (13) in source).
- Empirical approach: fit parametric distribution minimizing Kolmogorov-Smirnov distance.
- Findings:
  - Conditioning factors contributed to higher probability of worse global GDP growth outcomes in run-up to global financial crisis.
  - When crisis erupted, conditional density shows fatter tails and increased uncertainty: mode shifted to worse outcomes while conditional density shows increased probability of both return to pre-crisis growth and further worsening.
  - During euro area crisis, downside entropy remained high, implying conditioning factors predicted higher downside risks than unconditional density.
  - Downside entropy is more volatile than upside entropy, though they comove to some extent.

### 4.2.3 Expected Shortfall and Longrise
- Definitions:
  - Expected shortfall SF_{t+h} = (1/π) ∫_{0}^{π} ˆF^{-1}_{y_{t+h}|x_t}(τ|x_t) dτ (equation (14)).
  - Expected longrise LR_{t+h} = (1/π) ∫_{1-π}^{1} ˆF^{-1}_{y_{t+h}|x_t}(τ|x_t) dτ (equation (15)).
- Empirical results (four-quarter ahead):
  - Expected shortfall and longrise computed for 5th and 95th percentiles respectively.
  - Expected shortfall moves significantly when downside entropy spikes (global financial crisis); similar co-movement at euro area crisis.
  - Reveals non-Gaussian behavior of predictive density.

### 4.2.4 Log Scores
- Log score definition: S(ˆf_t, y_{t+h}) = log ˆf_t(y_{t+h}) (equation (16)).
- Test for equal predictive performance uses score differential and Diebold and Mariano (1995) test statistic.
- Empirical finding:
  - Average logscore difference between conditional predictive density and unconditional one is 0.7.
  - Null hypothesis of equal performance rejected at the 95 percent confidence interval.
  - Implication: conditional predictive density performs better in relative terms than unconditional density.

### Summary of evaluation evidence
- Multiple tests applied: PIT uniformity, upside/downside entropy, expected shortfall/longrise, log scores.
- Collective findings:
  - Conditional predictive density generally provides accurate forecasts of central tendency and characterizes risks.
  - Evaluation exercises indicate satisfactory out-of-sample performance and robustness in capturing upside and downside risks.
  - Conditional densities capture increases in tail risk around major crises and exhibit non-Gaussian features.

*Source: wpiea2020078-print-pdf (IMF Working Paper sections 3.1 and 4.1–4.2).*

### 3.1    Setup   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  

### 3.1    Setup

### Background and motivation
- Predictive densities and fan charts are widely used by central banks and policy institutions (example: Bank of England has published a fan chart for inflation forecasts since 1996).
- As of 2006, the IMF World Economic Outlook (WEO)’s global GDP growth projections have been accompanied by a fan chart summarizing the balance of risks.
- Since 2017 the IMF Global Financial Stability Report (GFSR) has published assessments of unlikely but possible global GDP growth collapses, focusing on the left tail of the predictive density.
- The goal is to quantify probabilistically the uncertainty around point forecasts and to estimate probabilities of specified events (for example, outcomes associated with a five percent probability).

### Aggregation challenges for predictive densities
- Combining individual predictive densities is often superior to single-model densities, but aggregating densities for different objects (e.g., GDP growth across countries) raises specific problems:
  - Country-specific data-generating processes differ and are not identically and independently distributed (not i.i.d.).
  - Country aggregates are subject to tail dependence: negative shocks in one country can increase the likelihood of negative shocks elsewhere.
  - Linear combinations of country-specific densities do not account for interdependencies and therefore are not viable for constructing consistent global aggregates.
- Standard density-combination approaches (e.g., linear opinion pool and logarithmic opinion pool) have theoretical and practical shortcomings, including loss of calibration, incoherence, and assigning zero probability to events zero in any component.
- Directly estimating the global predictive density from aggregate data (status quo in IMF WEO and GFSR) addresses interdependencies by construction but has limitations:
  - Relies on shorter series and may neglect periods with extreme events.
  - Quantile regression-based approaches may characterize tails with very few data points.
  - Survey-based approaches can be unstable due to small samples of forecasters.
  - Averaging country-level information can dilute informative country-specific signals.

### Proposed four-step bottom-up probabilistic approach
- The paper proposes a bottom-up probabilistic model that estimates and combines single-country predictive densities to obtain the predictive density of global GDP growth.
- To avoid the curse of dimensionality, the sample is split into three groups based on share of world output and an inter-dependency structure is assumed:
  - Other economies depend on large economies and their respective regional leaders.
  - Regional leaders depend on large economies.
  - Regional leaders and other economies have no impact on large economies.
- The approach mixes parametric and non-parametric features and proceeds in four steps:
  1. Generate the joint predictive density for a set of large economies via multivariate conditional kernel density estimation (reference to Li and Racine 2007). A certain probability is assigned to all combinations of future GDP growth rates in the three large economies (the US, the euro area, and China), conditional on current country-specific factors and contemporaneous growth rates of the other two large economies.
  2. For each conditioning triplet of large-economy GDP growth rates obtained in step 1, estimate each regional leader’s kernel density and sample from it to construct regional predictive densities.
  3. For each tuple of large economies’ and regional leaders’ GDP growth rates, estimate the other economies’ predictive density by drawing from a normal distribution with historical mean and standard deviation.
  4. Aggregate each sampled vector of GDP growth rates using a PPP-weighted average to construct global GDP growth predictive draws.
- If judgment is to be introduced (for example, to re-center the mode), the methodology proposes:
  - A rejection sampling algorithm, or
  - Fitting a parametric distribution that minimizes the Kolmogorov-Smirnov distance between the empirical distribution and the parametric one.

### Empirical observations from four-quarter ahead predictive densities (2005–2019)
- The four-quarter ahead predictive density of global GDP growth is constructed over 2005–2019 using the proposed methodology.
- Notable empirical patterns observed:
  - The central tendency of the distribution shifts significantly over time and tracks actual global GDP growth realizations relatively well.
  - The distribution is often close to symmetric for the world in “normal” times, consistent with the central limit theorem (upsides in some economies offset downsides in others).
  - During the global financial crisis, the collapse in GDP growth coincided with an upward-skewed predictive density, signalling higher probability of growth above the mode in the aftermath (consistent with mean-reverting processes).
  - During the Euro area crisis the density becomes significantly flatter with a predominant left tail, indicating large downside risks.
- Density evaluation exercises based on out-of-sample metrics confirm the accuracy of the proposed approach.

### Advantages of the proposed approach
- Characterizes the probability of each state of the world while allowing:
  - Inter-dependencies across large economies.
  - Potential amplification effects from regional dynamics.
  - Endogenous reactions of other economies.
- Circumvents density-aggregation problems by using sampling techniques that preserve consistency needed for ex-post aggregation.
- Estimates country-specific specifications rather than treating the world as a single aggregate, avoiding dilution of conditioning-factor signals when averaging across countries.
- Compared with quantile-regression-based approaches, kernel density estimators impose fewer constraints and do not require linearity assumptions, which can yield accuracy gains when forecasting higher moments of the distribution.

*Source: wpiea2020078-print-pdf (section 3.1, Setup).*

### 3.1  Setup

### 3.1  Setup

### Definition and aggregation
- Global GDP growth is defined as the PPP-weighted average of N individual economies’ real GDP growth: y_Wt = Σ_{i=1}^N ω_{i,t-1} y_{i,t}, where ω_{i,t-1} = Y_{i,t-1} / Σ_{j=1}^N Y_{j,t-1} and Y_jt is real GDP of economy j.  
- The sample is non-i.i.d., so the paper estimates the joint distribution f(y^1_t, …, y^N_t) to capture cross-country inter-dependencies.  
- To overcome the dimensionality curse, countries are partitioned into three groups and a sequential estimation strategy is implemented (see equation (4) for the density chain-rule decomposition).

### Country grouping and shares (end-2018, IMF WEO PPP weights)
- Large economies:
  - United States (US), the euro area (EA), and China (CHN).
  - Combined share of world GDP: 45 percent.
- Regional leaders:
  - South Africa (Africa); India and Japan (Asia); Russia and the United Kingdom (Europe); Brazil and Mexico (Western Hemisphere).
  - Combined share of world GDP: 22 percent.
  - Note: Africa has only one country in the sample due to quarterly data limitations.
- Other economies:
  - Argentina, Australia, Belarus, Canada, Chile, Colombia, Croatia, Czech Republic, Denmark, Hong Kong SAR China, Hungary, Indonesia, Israel, Jordan, Kazakhstan, Korea, Malaysia, Moldova, Norway, Peru, Philippines, Poland, Romania, Serbia, Singapore, Sweden, Switzerland, Taiwan Province of China, Thailand, Turkey, Ukraine, Venezuela, and Vietnam.
  - Combined share of world GDP: 19.5 percent.
- Appendix A lists countries with relative share of world GDP in PPP terms; IMF WEO regional classification is used.

### Key modeling assumptions (H1–H3)
- H1 (Other economies):
  - Other economies’ predictive densities depend on large economies’ and their respective regional leaders’, as well as country-specific factors X_o.
  - Conditional density for an economy o: f(y_o | X_o, y_US, y_EA, y_CHN, …).
  - This dramatically reduces dimensionality by avoiding joint estimation of all other economies.
- H2 (Regional leaders):
  - Regional leaders’ predictive densities depend on large economies’ GDP growth rates and country-specific factors X_r.
  - Conditional density for regional leader r: f(y_r | X_r, y_US, y_EA, y_CHN).
  - Inter-dependencies among regional leaders are ruled out; only spillovers from US, EA, CHN are considered.
- H3 (Large economies):
  - Regional leaders and other economies have no impact on large economies.
  - Predictive densities of large economies depend contemporaneously only on the other large economies.
  - Example for US: f(y_US | X_US, y_EA, y_CHN).

### Sequential empirical strategy (overview)
- Step 1: Estimate the joint density of the three large economies (US, EA, CHN).
- Step 2: Estimate predictive densities of regional leaders, conditional on Step 1 results.
- Step 3: Estimate predictive densities of other economies, conditional on Steps 1–2 results.
- Step 4: Aggregate predictive densities to obtain the predictive density for global GDP growth.
- Rolling window: excludes the 12 quarters following the quarter being forecast.

*Source: IMF Working Paper, chapter section 3.1.*

### 4.1  Estimation Results

### 4.1  Estimation Results

### Sample, covariates, and specification
- Sample composition:
  - 43 countries and one country group of 19 countries (i.e., the euro area), representing 87 percent of the world output.
- Main covariate categories included in each country’s predictive density:
  - Financial conditions (proxied by the financial conditions index).
  - Leading indicators of economic activity (industrial production and purchasing managers’ index).
  - External variables (import demand, real effective exchange rate, and commodity terms of trade).
  - Proxies of uncertainty (business confidence, economic policy uncertainty, and geopolitical risk).
  - Demographics factors (population growth).
- Data-frequency and sample issues:
  - Estimation period varies by country depending on data availability.
  - Joint distribution for the three large economies is limited by China’s GDP growth data, which start in 1995q1.
  - If covariates are available at monthly frequency, the lead of the average of monthly observations over a quarter is included.
- Country-specific variable choices are based on data availability and expert judgment rather than automated selection criteria; examples of baseline specifications for large economies:
  - US: past GDP growth, financial condition index, industrial production, business confidence.
  - Euro area: past GDP growth, industrial production, real effective exchange rate.
  - China: past GDP growth, real effective exchange rate, import demand, commodity terms of trade, population growth.
- Parsimony note:
  - Baseline excludes the financial conditions index for the euro area and China due to high correlation with the US financial conditions index; including them does not significantly change results.

### Joint predictive density of large economies: stylized dynamics and crisis periods
- Visualization approach:
  - Contour plots of joint predictive distributions (e.g., US on x-axis and euro area on y-axis), with brighter colors indicating higher probability.
  - Four-quarter ahead joint distributions shown at selected horizons (2006q1, 2008q4, 2009q1, 2009q2, 2009q3, 2012q3).
- Key empirical patterns:
  - 2006q1: probability concentrated around relatively high growth rates — about 3 percent for the US and 2 percent for the euro area.
  - 2008q4 (approaching global financial crisis): mode shifts toward lower growth for both economies; US shows relatively high probability of negative growth and longer tails (greater uncertainty).
  - 2009q1: predictive joint distribution displays two distinguishable modes (good and bad growth outcomes).
  - 2009q2: predicted distribution assigns higher probability to the bad outcome compared to the good one.
  - 2009q3: bi-modality disappears; probability of a deep recession in both economies increases considerably.
  - 2010–2011 (Euro-zone debt crisis): predictive joint density reflects downside risks associated with sovereign debt fears; 2012q3 shows a prevalent good-growth mode but with non-negligible probability attached to lower growth.

### Country-specific predictive densities and interval forecasts
- Interval forecast evaluation for large economies:
  - Figure 3 reports estimated 5–95 and 25–75 percentile ranges, the mode of the forecast density, and realized GDP growth since 2005 for the US, euro area, and China.
  - Actual GDP growth falls within the 25–75 percentile range most of the time, indicating reasonable specification for joint distribution estimation.
  - For China (panel [c]), the model captures a negative trend in GDP growth since 2007; the 50 percent interval sometimes appears wide.
  - US and euro area show larger probability mass to the left of the mode during the global financial crisis and the Euro-zone debt crisis.
- Predictive density behavior for regional leaders (four-quarter ahead forecasts):
  - Countries reported: Brazil, India, Japan, Mexico, Russia, United Kingdom, South Africa.
  - Kernel fits plotted for a quarter of ‘normal times’ (2006q1) and a quarter of deceleration (country-specific dates).
  - Findings:
    - Predicting normal times is easier; mode of predictive density close to realization.
    - During large swings and long forecast horizon (4 quarters ahead), predictive densities move consistently.
    - Some densities can appear jagged due to bandwidth choices (Silverman’s rule of thumb); bandwidth adjustments were avoided to limit discretion.
- Illustrative other economies:
  - Switzerland and Turkey reported examples; predictive densities move during bad times and modes are relatively close to actual realizations.
  - Assumption of Gaussianity for other economies does not significantly affect forecast accuracy.

### Global predictive density, tail risks, and slowdown probabilities
- Interval forecast evaluation for global GDP growth (four-quarter ahead):
  - Since 2006 the global 5–95 percentile range ranges between 1.8 percent and 6.4 percent, with exceptions during crisis periods.
  - During the global financial crisis the 5th percentile reached -2.6 percent.
  - During the euro-zone crisis the 5th percentile reached -2 percent.
  - The model struggles to predict the global financial crisis in real time and tends to detect it with a lag.
  - Significant downward skewness observed in correspondence with the euro-zone crisis.
- Defining a slowdown threshold:
  - Threshold set at the global GDP growth rate corresponding to the 25th percentile of the historical distribution, which is about 3 percent.
- Probability of global growth below 3 percent (four-quarter ahead):
  - High probability during the global financial crisis (end of 2007 and onwards).
  - Probability spiked to about 60 percent in 2011 and 2012, and again in 2015.
  - Since 2017 the probability crawled upwards and reached about 43 percent in 2020.

### Forcing central tendency: options to align predictive mode with external baseline
- Motivations:
  - Users may wish to force the mode of the predictive density to an alternative central tendency (e.g., IMF WEO baseline).
  - IMF WEO baseline for global GDP growth is a PPP-weighted average of individual country projections, incorporating different models and informed judgement.
- Two proposed approaches to force the mode or central tendency:
  1. Parametric fit with constraint on the mode:
     - Fit a parametric distribution constrained to a chosen mode.
     - Requires selecting across a wide family of parametric distributions.
     - Minimum Kolmogorov-Smirnov distance between empirical and parametric distribution can be used as a selection criterion to limit arbitrariness.
     - Candidate distributions mentioned: skewed-t distribution, asymmetric-t distribution, beta distribution, skew normal distribution, Weibull distribution.
  2. Rejection sampling algorithm:
     - Draw points from the original sampled predictive density and reject points with probability proportional to the distance of drawn points to the desired mode, producing a new sample with the desired mode while preserving moments as much as possible.

- Illustrative example (global predictive density for 2019q4, estimated with data up to 2018q4):
  - Sampled density for 2019q4 varies between one and six percent and is broadly symmetric with a mode of 3.2 percent.
  - Parametric fits:
    - Beta distribution achieved the smallest Kolmogorov-Smirnov distance in this example and overlaps with the kernel fit.
    - In other cases, skewed-t may perform best; the preferred parametric form can be chosen case-by-case.
  - Forcing the mean to the IMF WEO projection of 2.9 percent:
    - Forcing the mode/mean affects tails and asymmetry.
    - Rejection sampling led to a modest change in symmetry; the sampled density became slightly skewed to the right in the example.
    - Re-fitting parametric distributions after forcing again found the beta distribution as best capturing the adjusted sampled density in this example.

*Source: IMF Working Paper — Section 4.1 Estimation Results (figures and text as provided).*

### 4.2  Evaluating the Accuracy of Density Forecasts

### 4.2  Evaluating the Accuracy of Density Forecasts

### 4.2.1 Probability Integral Transform
- Definition: PIT is "the cumulative probability evaluated at the actual, realized value of the target variable" and assigns values between 0 and 1 (e.g., left-tail 1 percent → PIT = 0.01; median → PIT = 0.5).
- Theoretical property: For correctly specified probability forecasts, PIT values should be uniform, independent, and identically distributed (Diebold et al., 1999).
- Empirical test implemented:
  - Plotted empirical cumulative distribution function (CDF) of PIT for world GDP growth four quarters ahead against the theoretical 45-degree line and confidence bands based on Rossi and Sekhposyan (2019).
- Key finding:
  - "The empirical distribution of the PIT is within the confidence band for all quantiles, even for the most conservative confidence band based on 10 percent critical values." 
  - Implication: The approach produces satisfactory out-of-sample accuracy and robust predictive distributions that capture upside and downside risks.

### 4.2.2 Entropy Measures
- Purpose: Compare probability assigned to extreme growth outcomes by the conditional density relative to the unconditional density; entropy measures are the extra probability mass in tails relative to unconditional density.
- Definitions (as given in the source):
  - Downside entropy L_D_t and upside entropy L_U_t defined relative to ˆf_{y_{t+h}|x_t} and ˆg_{y_{t+h}} (equations (12) and (13) in source).
  - ˆF^{-1}_{y_{t+h}|x_t}(0.5|x_t) is the conditional median used to split tails.
- Empirical approach:
  - Fit parametric distribution that minimizes the Kolmogorov-Smirnov distance between the empirical and parametric distributions.
- Key findings:
  - Conditioning factors contributed to a higher probability of worse global GDP growth outcomes in the run-up to the global financial crisis.
  - When the global financial crisis erupted, the conditional density shows fatter tails and increased uncertainty: mode shifted to worse outcomes and the conditional density shows both increased probability of a return to pre-crisis growth and increased probability of further worsening.
  - During the euro area crisis, downside entropy remained high, implying conditioning factors predicted higher downside risks than the unconditional density.
  - Downside entropy is more volatile than upside entropy, though they comove to some extent.

### 4.2.3 Expected Shortfall and Longrise
- Purpose: Provide absolute measures of tail behavior of predictive density (contrast with relative entropy measures).
- Definitions (as given in the source):
  - Expected shortfall SF_{t+h} = (1/π) ∫_{0}^{π} ˆF^{-1}_{y_{t+h}|x_t}(τ|x_t) dτ (equation (14))
  - Expected longrise LR_{t+h} = (1/π) ∫_{1-π}^{1} ˆF^{-1}_{y_{t+h}|x_t}(τ|x_t) dτ (equation (15))
- Empirical results (four-quarter ahead global GDP growth):
  - Expected shortfall and longrise computed for the 5th and 95th percentiles respectively.
  - When downside entropy spikes (global financial crisis), expected shortfall also moves significantly; similar co-movement observed at the euro area crisis.
  - These similarities reveal non-Gaussian behavior of the global GDP growth predictive density.

### 4.2.4 Log Scores
- Definition: Likelihood score S(ˆf_t, y_{t+h}) = log ˆf_t(y_{t+h}) evaluates the probability assigned ex-ante to the realized outcome (equation (16)).
- Test for equal predictive performance between two densities ˆf_t and ˆg_t:
  - Score differential d^{ls}_{t+1} = S(ˆf_t, y_{t+1}) − S(ˆg_t, y_{t+1}).
  - Use average ̄d_{m,n} over quarters m to n and Diebold and Mariano (1995) test statistic t_{m,n} = ̄d_{m,n} / sqrt(ˆσ^2_{m,n}/n) where ˆσ^2_{m,n} is HAC estimator.
- Empirical finding:
  - The average logscore difference between the conditional predictive density and the unconditional one is 0.7.
  - The null hypothesis of equal performance is rejected at the 95 percent confidence interval.
  - Implication: The conditional predictive density performs better in relative terms than the unconditional density.

### Summary of Evaluation Evidence
- Multiple complementary tests applied: PIT uniformity, upside/downside entropy, expected shortfall/longrise, and log scores.
- Collective findings:
  - The conditional predictive density provides generally accurate forecasts of the central tendency of global GDP growth and describes risks around it.
  - Density evaluation exercises point to a satisfactory out-of-sample performance and robustness in capturing upside and downside risks.
  - Conditional densities capture increases in tail risk around major crises (global financial crisis, euro area crisis) and exhibit non-Gaussian features.

*Source: IMF Working Paper section 4.2, "Evaluating the Accuracy of Density Forecasts" (extracted content).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020078-print-pdf.pdf_