## Assessing IMF Lending: A Model of Sample Selection — Sections 1–2

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---

### Introduction and motivation
- Extends previous work on determinants of IMF lending by modeling both selection (which countries enter arrangements) and size (how large individual arrangements are) within a unified econometric framework.
- Methodological innovation: exploit correlated country effects along the selection and size dimensions using a panel-data extension of Heckman’s selection correction in the spirit of Mundlak (correlated random effects).
- Rationale: unobserved heterogeneity at the country level can create an additional, permanent channel for sample selection that, if unaccounted for, can bias inferences about determinants of IMF lending.
- Focus: non-concessional lending from General Resource Account (GRA) resources only; programs financed by the Poverty Reduction and Growth Trust (PRGT) are outside the scope.
- Objective: estimate parameters to simulate a wide spectrum of global crisis scenarios and generate a matrix of potential aggregate calls on Fund resources to inform discussions on IMF resource adequacy.

### Stylized facts on arrangement sizes and aggregate patterns
- Average IMF arrangement size has increased over time in both nominal terms and as a percent of the requesting member’s GDP.
- Distributional change:
  - 21 percent of all arrangements were larger than the average during 1957-1979.
  - 34 percent of arrangements were larger than the average during 2008-16.
- Trend toward larger and more dispersed arrangements started in the 1990s with large capital account crisis arrangements; exceptional access framework approved in 2002 recognized increased importance of capital account crises.
- On an aggregate level, correlation between IMF credit outstanding and global risk aversion (proxied by the VIX) has been high.

### Empirical methodology (model structure and identification)
- Target quantity: conditional density of aggregate Fund resources given covariates (parameters represent partial correlations, not causal effects).
- Model specification (Heckman-type with panel correlated effects):
  - Size equation (observed only if selection indicator equals 1):
    - y_it = exp(x_i1,t-1 β_1 + x_i2t β_2 + c_i + u_it)
  - Selection equation:
    - s_it = 1(z_i1,t-1 γ_1 + z_i2t γ_2 + d_i + v_it ≥ 0)
  - (c_i, d_i) denote potentially correlated unobserved country effects; covariates x_it ≡ (x_i1,t-1, x_i2t), z_it ≡ (z_i1,t-1, z_i2t).
  - u_it and v_it are assumed joint normal conditional on Z_i1,t-1 and Z_i2t.
  - Non-negativity of program size addressed via exponential form of the size equation.
- Identification strategy:
  - Do not seek causal parameters; adopt exclusion restrictions to improve identification of partial correlations.
  - Model selection and exclusion restrictions are chosen using AIC and BIC criteria.
  - Monte Carlo simulation used with estimated parameters to recover conditional distribution f_{y|S} of aggregate resources for scenario S (realization of Z_2t).

### Estimation sample and specification choices
- Data: panel of 92 advanced, emerging, and frontier market economies over 1992-2014, covering 119 GRA arrangements (dataset from Poulain and Reynaud).
- EFN per GDP (external financing needs per GDP) used as the only “Mundlak variable” (appears along both selection and size dimensions).
- Specification selected based on information criteria (AIC/BIC).

### Key estimation results (benchmark model)
- Sample and model fit:
  - Countries/observations: 1293741
  - Used: 921533
  - Uncensored/censored: 1191414
  - AIC/BIC: 9181,025
  - R^2 Adjusted: 0.30
- Size equation coefficients (β estimates and significance):
  - EFN per GDP: 3.77 (|z_H| 4.60, |p_H| 0.000***)
  - GDP per capita: 0.28 (|z_H| 2.45, |p_H| 0.014**)
  - Exchange rate variation: -1.25 (|z_H| 3.05, |p_H| 0.002***)
  - Growth: -0.04 (|z_H| 2.38, |p_H| 0.017**)
  - VIX (β_2): 0.04 (|z_H| 3.59, |p_H| 0.000***)
  - EFN per GDP (-) (β_3): listed as present in table (Mundlak appearance)
- Selection equation coefficients (γ estimates and significance):
  - EFN per GDP: 2.53 (|z_H| 3.21, |p_H| 0.001***)
  - GDP per capita: -0.42 (|z_H| 3.92, |p_H| 0.000***)
  - Government Stability: -0.16 (|z_H| 3.41, |p_H| 0.000***)
  - Interconnectedness: -0.33 (|z_H| 2.02, |p_H| 0.043**)
  - Past Arrangement: 0.13 (|z_H| 2.78, |p_H| 0.005***)
  - Credit-to-GDP gap: 0.01 (|z_H| 2.46, |p_H| 0.014**)
  - Exchange rate variation: -0.92 (|z_H| 2.21, |p_H| 0.027**)
  - Growth: -0.04 (|z_H| 2.86, |p_H| 0.004***)
  - VIX (γ_2): 0.04 (|z_H| 3.18, |p_H| 0.001***)
  - EFN per GDP (-) (γ_3): listed as present in table (Mundlak appearance)
- Error and heterogeneity parameters:
  - Sample selection parameter (ρ_uv): 0.49 (|z_H| 2.38, |p_H| 0.017**)
  - σ_u (shock variation): 0.83
  - σ_c (effect variation): 0.70
  - σ_d (effect variation): 0.23
- Sign and interpretation highlights:
  - Selection variable signs align with Poulain and Reynaud [11].
  - Size equation signs align with economic intuition:
    - Higher external financing needs associated with larger idiosyncratic arrangement size (EFN per GDP positive in size).
    - Lower GDP growth associated with larger arrangement size (growth negative).
    - Increased global risk aversion (VIX) associated with larger arrangement size (VIX positive).
    - Larger exchange rate depreciation associated with larger arrangement size (exchange rate variation negative coefficient in size equation implies higher depreciation leads to larger size given construction).
  - GDP per capita: negative in selection (richer countries less likely to require Fund financing) but positive in size (if richer country requires financing, it may need larger access).
- Specification decision:
  - Information criteria favor a model where government stability, interconnectedness, past arrangements, and the credit-to-GDP gap enter only the selection equation, not the size equation.

### Exclusion restrictions and identification
- Both exclusion criteria used are statistical artifacts unrelated to the economic phenomenon and require critical assessment using economic intuition.
- It is acceptable that some factors may influence the decision to lend (e.g. spillovers, government stability) but not the size of the arrangement, which is primarily driven by the balance of payment need of the member country.
- The framework does not allow for separate identification of supply and demand, complicating intuitive evaluation (an issue noted by Ghosh et al. [4]).
- A potential covariate may constitute both a demand and a supply shifter without affecting observed outcomes in a statistically significant way.

### Interpretation of specific covariates
- Interconnectedness
  - A priori expected to enter with the same sign in both selection and size equations because the IMF aims to prevent spillovers (see Poulain and Reynaud [11]).
  - Higher interconnectedness may also mirror greater availability of private finance and thus smaller need for IMF resources if access to capital markets is not completely shut down.
- Government stability
  - Higher government stability is presumably mirrored by lower demand, but also less constrained supply.
- Past arrangement
  - Many member countries have a history of protracted financial engagement with the Fund, mirroring deeply-rooted economic problems.
  - Conversely, Fund policies governing access to resources (exceptional access, conditionality) may constrain supply in some cases.
- Credit-to-GDP gap
  - A large credit-to-GDP gap has been identified as an early warning signal of banking crisis, which may increase demand for Fund resources, but also potentially reduce supply if adequate safeguards are not in place.

### Mundlak variable and sample selection channel
- The Mundlak variable aims to capture permanent cross-country differences and is intuitively related to Reinhart, Rogoff, and Savastano’s notion of “debt (in)tolerance” [12]; its signs must be interpreted with utmost caution.
- To avoid inaccurate "within country" interpretation, the table does not show the estimated parameter’s value.
- The Mundlak variable carries the same sign across both selection and size dimensions, indicating the existence of a sample selection channel at the country level: permanently higher likelihoods of selection are mirrored by larger arrangements per GDP.
- There is limited evidence of idiosyncratic sample selection as measured by the estimate ˆρ_uv.

### Model fit — Selection
- Model accuracy tested by creating bins of ex ante selection probabilities (fitted by the model) and computing the proportion of arrangements in each bin.
- Table 2 - Selection performance (fitted probability bins and observed proportions)
  - Fitted probability [0,5%] — Proportion with actual arrangement 1%
  - Fitted probability [5,10%] — Proportion with actual arrangement 10%
  - Fitted probability [10,15%] — Proportion with actual arrangement 12%
  - Fitted probability [15,20%] — Proportion with actual arrangement 18%
  - Fitted probability [20,25%] — Proportion with actual arrangement 22%
  - Fitted probability [25,30%] — Proportion with actual arrangement 33%
  - Fitted probability [30,35%] — Proportion with actual arrangement 47%
  - Fitted probability [35,40%] — Proportion with actual arrangement 33%
  - Fitted probability [40,45%] — Proportion with actual arrangement 46%
  - Fitted probability [45,50%] — Proportion with actual arrangement 50%
- Note: Number of observations in the higher probability bins is too small for the law of large numbers to apply; overall performance should be viewed as satisfactory at worst.

### Model fit — Size
- Figure 4 depicts both the log moment E[log(y_it)|s_it = 1] fitted by maximum likelihood and the level moment E[y_it|s_it = 1]; the second and third rows are related (the third is an enlarged version of the second).
- Even an untruncated lognormal random variable’s expected value exceeds its median by factor exp(σ^2/2).
- Although y_it|s_it = 1 is neither lognormal nor truncated lognormal, more mass is expected below the mean so number of dots is not equal on both sides of the black line.
- The sum’s density closely resembles a normal distribution, so y_it|s_it = 1 may approximately be thought of as a lognormal random variable.
- Residuals have a negative mode and large mass below zero (figure 5).
- Two large negative residuals at the left tail correspond to two post-GFC large loans largely financed by non-IMF sources; when accounting for these sizable contributions, the two residuals shift substantially to the right (blue observations).

### Scenarios — simulating potential aggregate IMF lending
- Methodology: simulate aggregate IMF lending in the first post-sample year under global volatility shocks by varying VIX levels and holding all other variables unchanged.
- A global volatility shock would likely be accompanied by a drop in GDP growth and an increase in external financing needs (which would increase potential Fund financing), while some non-modeled considerations could work in the opposite direction (e.g. stigma, conditionality).
- Two approaches to simulation:
  - First approach (Monte Carlo style)
    - Vectors (c, d, u_t, v_t) simulated given their estimated joint density as implied by the model while Z_{2t} is held fixed to some realization Z^1_{2t}.
  - Second approach (rank-based threshold)
    - A country is selected if its estimated selection probability exceeds a threshold given some realization Z^2_{i2t}; size of each arrangement is pinned down by u^2_it = ˆρ_uv ˆσ_u v^2_it = −ˆρ_uv ˆσ_u z_it γ.
    - Characterized by a fixed ordinal ranking of country selection and more intuitive interpretation.
- Both approaches generate a similar range of conceivable aggregate access and exhibit a substantial right tail.
- Distributional features (first approach): sizeable wedge between the conditional expectation and the upper quantiles.
- Illustrative global shock: VIX = 30 on average for a year (only slightly below the average level of 32 reached during the GFC).
  - Under the first approach:
    - Potential needs estimated to reach about SDR 280 billion at the 85th quantile.
    - Potential needs estimated to reach about SDR 560 billion at the 95th quantile.
  - Under the second approach:
    - Potential needs estimated to reach about SDR 400 billion at the 7.5 percent threshold (equal to the frequency of events in the sample).
    - Potential needs estimated to reach about SDR 825 billion at the 4 percent threshold.

### Conclusion — implications for Fund resource adequacy
- Methodology corrects for sample selection bias by extending Heckman’s selection correction model in the spirit of Mundlak [9], exploiting correlated country effects along selection and size dimensions.
- Estimation starts from the selection model of Poulain and Reynaud [11]; selection-equation coefficients match signs found by Poulain and Reynaud [11], and size-equation coefficients carry signs supported by economic intuition.
- In particular, higher external financing needs, exchange rate depreciation, global risk aversion, and lower GDP growth are mirrored by higher idiosyncratic arrangement size.
- Scenario simulations suggest that most crises could be accommodated with a lending capacity of about SDR 600-800 billion.
- The distribution of aggregate access exhibits a substantial right tail; a more extreme crisis could conceivably lead to significantly larger needs.
- At current low levels of global risk aversion, expected aggregate access would be much lower, consistent with the global financial cycle being a major driver of IMF lending cycles.
- The approach may be one of many inputs for assessing adequacy of Fund resources; the issue requires informed judgment based on the risk tolerance of the international community and political, governance, and financial considerations.

### Appendix A — Likelihood (methodological details)
- Contemporary estimation of heckit models is traditionally implemented via full information maximum likelihood (FIML) rather than Heckman’s original two-step.
- With unobserved heterogeneity, the likelihood approach allows for integrating out corresponding effects.
- Following Wooldridge, a fixed effect’s key feature is its non-zero correlation with covariates; Chamberlain [3] and Mundlak [9] linearly project unobserved heterogeneity onto covariates (correlated random effects, CRE).
- The likelihood is constructed by conditioning on and subsequently integrating out the unobserved effects:
  - L(y|Z;θ) = sum_{i=1}^N log(∫∫ exp(∑_{t=1}^T l_it) dF(d_i) dF(ε^c_i))
  - l_it denotes the conditional log-likelihood of observing (or not observing) y_it given covariates, parameter vector θ, and unobserved effects d_i and ε^c_i.
- The conditional log-likelihood l_it is:
  - l_it ≡ l(y_it|x_it, z_it, d_i, ε^c_i; θ)
  - = 1(s_it = 0) log[1 − Φ(z_it γ + d_i)]
    + 1(s_it = 1){ log[ Φ( (z_it γ + d_i + (ρ_uv/σ_u)(log(y_it) − x_it β − ρ_cd (σ_c/σ_d) d_i − ε^c_i) / (1 − ρ^2_uv)^{0.5} ) ) ] + log[ φ( (log(y_it) − x_it β − ρ_cd (σ_c/σ_d) d_i − ε^c_i) / σ_u ) ] − log(σ_u) }
  - φ and Φ denote standard normal density and cumulative density functions respectively.
- Joint normality implies c_i = ρ_cd (σ_c/σ_d) d_i + ε^c_i with ε^c_i ∼ N(0, (1 − ρ^2_cd) σ^2_c).
- The log entry of y_it is the primary difference from Heckman’s linear model.
- Maximization:
  - ˆθ_MLE = argmax_{θ∈Θ} L(y|Z;θ)
  - Inference conducted using asymptotic z-statistic from the negative Hessian at ˆθ_MLE.
  - Gauss-Hermite quadrature may be used to integrate out unobserved effects; the authors use fminunc in Matlab.

### Appendix B — Data (variables and sample)
- Key variables and sources:
  - New arrangement — IMF — Indicator equals one when new arrangement starts
  - New arrangement size — IMF — Size of new arrangement, percent of GDP
  - Past arrangement — IMF — 5-year moving average of active arrangement indicator
  - EFN — WEO — External financing needs; see [11]
  - GDP Growth — WEO — In percent
  - GDP per capita — WEO — log of level
  - GDP — WEO — log of level
  - Credit-to-GDP gap — BIS; WDI; WEO — Deviation of credit-to-GDP from 5-year moving average
  - Exchange rate variation — WEO — Variation of bilateral nominal exchange rate vs. USD
  - Government Stability — ICRG — Government unity, legislative strength, popular support
  - Interconnectedness — WEO; DOTS — See [11]
  - 3-month interest rate — WEO — In percent
  - VIX — CBOE — Measure of risk aversion
  - Oil price — WEO — Deviation from 5-year moving average
- Countries in the sample (selected listing):
  - Albania, Algeria, Angola, Argentina, Armenia, Austria, Australia, Azerbaijan, The Bahamas, Bahrain, Belarus, Belgium, Botswana, Brazil, Canada, China, Colombia, Costa Rica, Croatia, Cyprus, Czech Republic, Denmark, Dominican Republic, Ecuador, El Salvador, Estonia, Finland, France, Gabon, Germany, Greece, Guatemala, Guyana, Hungary, Iceland, India, Indonesia, Iran, Iraq, Ireland, Israel, Italy, Jamaica, Japan, Kazakhstan, Korea, Kuwait, Latvia, Lebanon, Libya, Luxembourg, Malaysia, Malta, Mexico, Morocco, Namibia, Netherlands, New Zealand, Norway, Oman, Pakistan, Panama, Peru, Philippines, Poland, Portugal, Qatar, Romania, Russia, Saudi Arabia, Singapore, Slovak Republic, Slovenia, South Africa, Spain, Sri Lanka, Suriname, Sweden, Switzerland, Syria, Thailand, Trinidad and Tobago, Tunisia, Turkey, Ukraine, United Arab Emirates, United Kingdom, Uruguay, Venezuela

*Source: IMF Working Paper WP/19/157 — Sections 1–2*

### Section 1

### Assessing IMF Lending: A Model of Sample Selection — Section 1

### Introduction and motivation
- The paper extends previous work on determinants of IMF lending by modeling both selection (which countries enter arrangements) and size (how large individual arrangements are) within a unified econometric framework.
- Key methodological innovation: exploit correlated country effects along the selection and size dimensions using a panel-data extension of Heckman’s selection correction in the spirit of Mundlak (correlated random effects).
- Rationale: unobserved heterogeneity at the country level can create an additional, permanent channel for sample selection that, if unaccounted for, can bias inferences about determinants of IMF lending.
- Focus: non-concessional lending from General Resource Account (GRA) resources only; programs financed by the Poverty Reduction and Growth Trust (PRGT) are outside the scope.
- Objective: estimate parameters to simulate a wide spectrum of global crisis scenarios and generate a matrix of potential aggregate calls on Fund resources to inform discussions on IMF resource adequacy.

### Stylized facts on arrangement sizes and aggregate patterns
- Average IMF arrangement size has increased over time in both nominal terms and as a percent of the requesting member’s GDP.
- Distributional change: the distribution of arrangement sizes has become wider with a fatter right tail over time.
  - 21 percent of all arrangements were larger than the average during 1957-1979.
  - 34 percent of arrangements were larger than the average during 2008-16.
- Trend toward larger and more dispersed arrangements started in the 1990s with large capital account crisis arrangements; exceptional access framework approved in 2002 recognized increased importance of capital account crises.
- On an aggregate level, correlation between IMF credit outstanding and global risk aversion (proxied by the VIX) has been high.

### Empirical methodology (model structure and identification)
- Target quantity: conditional density of aggregate Fund resources given covariates (parameters represent partial correlations, not causal effects).
- Model specification (Heckman-type with panel correlated effects):
  - Size equation (observed only if selection indicator equals 1):
    - y_it = exp(x_i1,t-1 β_1 + x_i2t β_2 + c_i + u_it)
  - Selection equation:
    - s_it = 1(z_i1,t-1 γ_1 + z_i2t γ_2 + d_i + v_it ≥ 0)
  - (c_i, d_i) denote potentially correlated unobserved country effects; covariates x_it ≡ (x_i1,t-1, x_i2t), z_it ≡ (z_i1,t-1, z_i2t).
  - u_it and v_it are assumed joint normal conditional on Z_i1,t-1 and Z_i2t.
  - Non-negativity of program size addressed via exponential form of the size equation.
- Identification strategy:
  - Do not seek causal parameters; adopt exclusion restrictions to improve identification of partial correlations.
  - Model selection and exclusion restrictions are chosen using AIC and BIC criteria.
  - Monte Carlo simulation used with estimated parameters to recover conditional distribution f_{y|S} of aggregate resources for scenario S (realization of Z_2t).

### Estimation sample and specification choices
- Data: panel of 92 advanced, emerging, and frontier market economies over 1992-2014, covering 119 GRA arrangements (dataset from Poulain and Reynaud).
- EFN per GDP (external financing needs per GDP) used as the only “Mundlak variable” (appears along both selection and size dimensions).
- Specification selected based on information criteria (AIC/BIC).

### Key estimation results (benchmark model — Table 1 summary)
- Sample and model fit:
  - Countries/observations: 1293741
  - Used: 921533
  - Uncensored/censored: 1191414
  - AIC/BIC: 9181,025
  - R^2 Adjusted: 0.30
- Size equation coefficients (β estimates and significance):
  - EFN per GDP: 3.77 (|z_H| 4.60, |p_H| 0.000***)
  - GDP per capita: 0.28 (|z_H| 2.45, |p_H| 0.014**)
  - Exchange rate variation: -1.25 (|z_H| 3.05, |p_H| 0.002***)
  - Growth: -0.04 (|z_H| 2.38, |p_H| 0.017**)
  - VIX (β_2): 0.04 (|z_H| 3.59, |p_H| 0.000***)
  - EFN per GDP (-) (β_3): listed as present in table (Mundlak appearance)
- Selection equation coefficients (γ estimates and significance):
  - EFN per GDP: 2.53 (|z_H| 3.21, |p_H| 0.001***)
  - GDP per capita: -0.42 (|z_H| 3.92, |p_H| 0.000***)
  - Government Stability: -0.16 (|z_H| 3.41, |p_H| 0.000***)
  - Interconnectedness: -0.33 (|z_H| 2.02, |p_H| 0.043**)
  - Past Arrangement: 0.13 (|z_H| 2.78, |p_H| 0.005***)
  - Credit-to-GDP gap: 0.01 (|z_H| 2.46, |p_H| 0.014**)
  - Exchange rate variation: -0.92 (|z_H| 2.21, |p_H| 0.027**)
  - Growth: -0.04 (|z_H| 2.86, |p_H| 0.004***)
  - VIX (γ_2): 0.04 (|z_H| 3.18, |p_H| 0.001***)
  - EFN per GDP (-) (γ_3): listed as present in table (Mundlak appearance)
- Error and heterogeneity parameters:
  - Sample selection parameter (ρ_uv): 0.49 (|z_H| 2.38, |p_H| 0.017**)
  - σ_u (shock variation): 0.83
  - σ_c (effect variation): 0.70
  - σ_d (effect variation): 0.23
- Sign and interpretation highlights:
  - Selection variable signs align with Poulain and Reynaud [11].
  - Size equation signs align with economic intuition:
    - Higher external financing needs associated with larger idiosyncratic arrangement size (EFN per GDP positive in size).
    - Lower GDP growth associated with larger arrangement size (growth negative).
    - Increased global risk aversion (VIX) associated with larger arrangement size (VIX positive).
    - Larger exchange rate depreciation associated with larger arrangement size (exchange rate variation negative coefficient in size equation implies higher depreciation leads to larger size given construction).
  - GDP per capita: negative in selection (richer countries less likely to require Fund financing) but positive in size (if richer country requires financing, it may need larger access).
- Specification decision:
  - Information criteria favor a model where government stability, interconnectedness, past arrangements, and the credit-to-GDP gap enter only the selection equation, not the size equation.

### Simulation and policy relevance
- Using estimated parameters, Monte Carlo simulation of a wide spectrum of global shock scenarios is employed to produce distributions of potential aggregate IMF lending.
- Simulations suggest the distribution of potential aggregate IMF lending exhibits a substantial right tail (i.e., possibility of large aggregate calls on Fund resources).
- Policy implication: the approach provides inputs for broader policy discussions on the adequacy of IMF resources by mapping a matrix of potential calls under different global scenarios.

*Source: IMF Working Paper WP/19/157 — Section 1*

### Section 2

### Section 2

### Exclusion restrictions and identification
- Both exclusion criteria used are statistical artifacts unrelated to the economic phenomenon and require critical assessment using economic intuition.
- It is acceptable that some factors may influence the decision to lend (e.g. spillovers, government stability) but not the size of the arrangement, which is primarily driven by the balance of payment need of the member country.
- The framework does not allow for separate identification of supply and demand, complicating intuitive evaluation (an issue noted by Ghosh et al. [4]).
- A potential covariate may constitute both a demand and a supply shifter without affecting observed outcomes in a statistically significant way.

### Interpretation of specific covariates
- Interconnectedness
  - A priori expected to enter with the same sign in both selection and size equations because the IMF aims to prevent spillovers (see Poulain and Reynaud [11]).
  - Higher interconnectedness may also mirror greater availability of private finance and thus smaller need for IMF resources if access to capital markets is not completely shut down.
- Government stability
  - Higher government stability is presumably mirrored by lower demand, but also less constrained supply.
- Past arrangement
  - Many member countries have a history of protracted financial engagement with the Fund, mirroring deeply-rooted economic problems.
  - Conversely, Fund policies governing access to resources (exceptional access, conditionality) may constrain supply in some cases.
- Credit-to-GDP gap
  - A large credit-to-GDP gap has been identified as an early warning signal of banking crisis, which may increase demand for Fund resources, but also potentially reduce supply if adequate safeguards are not in place.

### Mundlak variable and sample selection channel
- The Mundlak variable aims to capture permanent cross-country differences and is intuitively related to Reinhart, Rogoff, and Savastano’s notion of “debt (in)tolerance” [12]; its signs must be interpreted with utmost caution.
- To avoid inaccurate "within country" interpretation, the table does not show the estimated parameter’s value.
- The Mundlak variable carries the same sign across both selection and size dimensions, indicating the existence of a sample selection channel at the country level: permanently higher likelihoods of selection are mirrored by larger arrangements per GDP.
- There is limited evidence of idiosyncratic sample selection as measured by the estimate ˆρ_uv.

### Model fit - Selection
- Model accuracy tested by creating bins of ex ante selection probabilities (fitted by the model) and computing the proportion of arrangements in each bin.
- If predicted likelihoods are accurate and bin sizes large, the proportion of actual arrangements must fall into the bin’s probability interval.
- Table 2 - Selection performance (fitted probability bins and observed proportions)
  - Fitted probability [0,5%] — Proportion with actual arrangement 1%
  - Fitted probability [5,10%] — Proportion with actual arrangement 10%
  - Fitted probability [10,15%] — Proportion with actual arrangement 12%
  - Fitted probability [15,20%] — Proportion with actual arrangement 18%
  - Fitted probability [20,25%] — Proportion with actual arrangement 22%
  - Fitted probability [25,30%] — Proportion with actual arrangement 33%
  - Fitted probability [30,35%] — Proportion with actual arrangement 47%
  - Fitted probability [35,40%] — Proportion with actual arrangement 33%
  - Fitted probability [40,45%] — Proportion with actual arrangement 46%
  - Fitted probability [45,50%] — Proportion with actual arrangement 50%
- Note: Number of observations in the higher probability bins is too small for the law of large numbers to apply; overall performance should be viewed as satisfactory at worst.

### Model fit - Size
- Figure 4 depicts both the log moment E[log(y_it)|s_it = 1] fitted by maximum likelihood and the level moment E[y_it|s_it = 1]; the second and third rows are related (the third is an enlarged version of the second).
- Even an untruncated lognormal random variable’s expected value exceeds its median by factor exp(σ^2/2).
- Although y_it|s_it = 1 is neither lognormal nor truncated lognormal, more mass is expected below the mean so number of dots is not equal on both sides of the black line.
- The sum’s density closely resembles a normal distribution, so y_it|s_it = 1 may approximately be thought of as a lognormal random variable.
- Residuals have a negative mode and large mass below zero (figure 5).
- Two large negative residuals at the left tail correspond to two post-GFC large loans largely financed by non-IMF sources; when accounting for these sizable contributions, the two residuals shift substantially to the right (blue observations).

### Scenarios - simulating potential aggregate IMF lending
- Methodology: simulate aggregate IMF lending in the first post-sample year under global volatility shocks by varying VIX levels and holding all other variables unchanged.
- A global volatility shock would likely be accompanied by a drop in GDP growth and an increase in external financing needs (which would increase potential Fund financing), while some non-modeled considerations could work in the opposite direction (e.g. stigma, conditionality).
- Two approaches to simulation:
  - First approach (Monte Carlo style)
    - Vectors (c, d, u_t, v_t) simulated given their estimated joint density as implied by the model while Z_{2t} is held fixed to some realization Z^1_{2t}.
  - Second approach (rank-based threshold)
    - A country is selected if its estimated selection probability exceeds a threshold given some realization Z^2_{i2t}; size of each arrangement is pinned down by u^2_it = ˆρ_uv ˆσ_u v^2_it = −ˆρ_uv ˆσ_u z_it γ.
    - Characterized by a fixed ordinal ranking of country selection and more intuitive interpretation.
- Both approaches generate a similar range of conceivable aggregate access and exhibit a substantial right tail.
- Distributional features (first approach): sizeable wedge between the conditional expectation and the upper quantiles.
- Illustrative global shock: VIX = 30 on average for a year (only slightly below the average level of 32 reached during the GFC).
  - Under the first approach:
    - Potential needs estimated to reach about SDR 280 billion at the 85th quantile.
    - Potential needs estimated to reach about SDR 560 billion at the 95th quantile.
  - Under the second approach:
    - Potential needs estimated to reach about SDR 400 billion at the 7.5 percent threshold (equal to the frequency of events in the sample).
    - Potential needs estimated to reach about SDR 825 billion at the 4 percent threshold.

### Conclusion - implications for Fund resource adequacy
- The paper provides a methodology to correct for sample selection bias in estimating potential need for Fund financing by extending Heckman’s selection correction model in the spirit of Mundlak [9].
- The approach exploits correlated country effects along selection and size dimensions, allowing for an additional, permanent channel of sample selection.
- Estimation starts from the selection model of Poulain and Reynaud [11]; selection-equation coefficients match signs found by Poulain and Reynaud [11], and size-equation coefficients carry signs supported by economic intuition.
- In particular, higher external financing needs, exchange rate depreciation, global risk aversion, and lower GDP growth are mirrored by higher idiosyncratic arrangement size.
- Scenario simulations suggest that most crises could be accommodated with a lending capacity of about SDR 600-800 billion.
- The distribution of aggregate access exhibits a substantial right tail; a more extreme crisis could conceivably lead to significantly larger needs.
- At current low levels of global risk aversion, expected aggregate access would be much lower, consistent with the global financial cycle being a major driver of IMF lending cycles.
- The approach may be one of many inputs for assessing adequacy of Fund resources; the issue requires informed judgment based on the risk tolerance of the international community and political, governance, and financial considerations.

### Appendix A - Likelihood (methodological details)
- Contemporary estimation of heckit models is traditionally implemented via full information maximum likelihood (FIML) rather than Heckman’s original two-step.
- With unobserved heterogeneity, the likelihood approach allows for integrating out corresponding effects.
- Following Wooldridge, a fixed effect’s key feature is its non-zero correlation with covariates; Chamberlain [3] and Mundlak [9] linearly project unobserved heterogeneity onto covariates (correlated random effects, CRE).
- The likelihood is constructed by conditioning on and subsequently integrating out the unobserved effects:
  - L(y|Z;θ) = sum_{i=1}^N log(∫∫ exp(∑_{t=1}^T l_it) dF(d_i) dF(ε^c_i))
  - l_it denotes the conditional log-likelihood of observing (or not observing) y_it given covariates, parameter vector θ, and unobserved effects d_i and ε^c_i.
- The conditional log-likelihood l_it is:
  - l_it ≡ l(y_it|x_it, z_it, d_i, ε^c_i; θ)
  - = 1(s_it = 0) log[1 − Φ(z_it γ + d_i)]
    + 1(s_it = 1){ log[ Φ( (z_it γ + d_i + (ρ_uv/σ_u)(log(y_it) − x_it β − ρ_cd (σ_c/σ_d) d_i − ε^c_i) / (1 − ρ^2_uv)^{0.5} ) ) ] + log[ φ( (log(y_it) − x_it β − ρ_cd (σ_c/σ_d) d_i − ε^c_i) / σ_u ) ] − log(σ_u) }
  - φ and Φ denote standard normal density and cumulative density functions respectively.
- Joint normality implies c_i = ρ_cd (σ_c/σ_d) d_i + ε^c_i with ε^c_i ∼ N(0, (1 − ρ^2_cd) σ^2_c).
- The log entry of y_it is the primary difference from Heckman’s linear model.
- Maximization:
  - ˆθ_MLE = argmax_{θ∈Θ} L(y|Z;θ)
  - Inference conducted using asymptotic z-statistic from the negative Hessian at ˆθ_MLE.
  - Gauss-Hermite quadrature may be used to integrate out unobserved effects; the authors use fminunc in Matlab.

### Appendix B - Data (variables and sample)
- Key variables and sources (as listed)
  - New arrangement — IMF — Indicator equals one when new arrangement starts
  - New arrangement size — IMF — Size of new arrangement, percent of GDP
  - Past arrangement — IMF — 5-year moving average of active arrangement indicator
  - EFN — WEO — External financing needs; see [11]
  - GDP Growth — WEO — In percent
  - GDP per capita — WEO — log of level
  - GDP — WEO — log of level
  - Credit-to-GDP gap — BIS; WDI; WEO — Deviation of credit-to-GDP from 5-year moving average
  - Exchange rate variation — WEO — Variation of bilateral nominal exchange rate vs. USD
  - Government Stability — ICRG — Government unity, legislative strength, popular support
  - Interconnectedness — WEO; DOTS — See [11]
  - 3-month interest rate — WEO — In percent
  - VIX — CBOE — Measure of risk aversion
  - Oil price — WEO — Deviation from 5-year moving average
- Countries in the sample (selected listing as provided)
  - Albania, Algeria, Angola, Argentina, Armenia, Austria, Australia, Azerbaijan, The Bahamas, Bahrain, Belarus, Belgium, Botswana, Brazil, Canada, China, Colombia, Costa Rica, Croatia, Cyprus, Czech Republic, Denmark, Dominican Republic, Ecuador, El Salvador, Estonia, Finland, France, Gabon, Germany, Greece, Guatemala, Guyana, Hungary, Iceland, India, Indonesia, Iran, Iraq, Ireland, Israel, Italy, Jamaica, Japan, Kazakhstan, Korea, Kuwait, Latvia, Lebanon, Libya, Luxembourg, Malaysia, Malta, Mexico, Morocco, Namibia, Netherlands, New Zealand, Norway, Oman, Pakistan, Panama, Peru, Philippines, Poland, Portugal, Qatar, Romania, Russia, Saudi Arabia, Singapore, Slovak Republic, Slovenia, South Africa, Spain, Sri Lanka, Suriname, Sweden, Switzerland, Syria, Thailand, Trinidad and Tobago, Tunisia, Turkey, Ukraine, United Arab Emirates, United Kingdom, Uruguay, Venezuela

*Source: wpiea2019157 - Section 2 (wpiea2019157 - Section 2).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019157.pdf_