## _wp0974

## Source details

**Canonical URL:** [_wp0974](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0974.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0974.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0974.pdf.json)

---

### I. Introduction and problem statement
- Model uncertainty arises when lack of clear theoretical guidance and tradeoffs on regressors produce many possible specifications; unsystematic searches can yield overconfident and fragile inferences.
- Bayesian Model Averaging (BMA) bases inferences on a weighted average of the full model space rather than a single selected model and thus incorporates uncertainty in predictions and parameter estimates.
- Literature gaps:
  - Most BMA work uses static models and cross-section analyses, ignoring dynamics.
  - Prior models do not allow inclusion of endogenous variables as regressors in the dynamic panel context.
- Contribution:
  - Propose methodology LIBMA for model uncertainty in panel data with short T where lagged dependent and endogenous variables appear as regressors; refines limited information BMA (Tsangarides (2004)) and constructs likelihood by data transformation and Bayesian arguments.

### II. Theoretical framework and priors
- Standard setup:
  - Linear regression Y = Z θ + u with universe of k possible regressors yields 2^k models.
  - Posterior model probability p(M_j | D) = p(D | M_j) p(M_j) / Σ_l p(D | M_l) p(M_l).
  - BMA for quantity Γ: p(Γ | D) = Σ_j p(Γ | D, M_j) p(M_j | D).
- Priors and implications:
  - Gaussian models: conjugate Normal-Gamma prior for (θ, σ_u^2).
  - Zellner’s g-prior: p(θ | σ_u^2) ∼ N(0, σ_u^2 g (Z′Z)^{-1}).
  - Common approaches noted:
    - Diffuse prior for σ_u with density p(σ_u) ∝ 1/σ_u (equivalent to γ = λ = 0).
    - BACE procedure asymptotically equivalent to g = 1/(NT).
    - BIC approximation described for likelihood ratio.
  - Model priors: uniform over model space or Mitchell–Beauchamp structure giving prior p(M_j) as in (10) with prior odds in (11).

### III. LIBMA: limited-information BMA for dynamic panels with endogeneity
- Model setup:
  - Dynamic panel: y_it = α y_{i,t-1} + x_{it} θ_x + w_{it} θ_w + η_i + v_it, i = 1,...,N, t = 1,...,T, with T small relative to N.
  - Regressor counts: lagged dependent variable (indexed 1), m exogenous variables (X), q endogenous variables (W); total k = 1 + m + q.
  - Assumptions: E[v_it] = 0, no serial correlation in v_it; exogenous: E[x_{is} v_{it}] = 0 for all s,t; endogenous: E[w_{is} v_{it}] ≠ 0 for s = t and zero otherwise.
- Estimation and instruments (GMM / system GMM):
  - First-difference instruments for lagged dependent: y_{i,t-2}, y_{i,t-3}, ... leading to moment conditions.
  - Exogenous variables: use Δ x_{it} or x_{it} as instruments depending on assumptions.
  - Endogenous variables: lags w_{i,t-2}, w_{i,t-3}, ... as instruments.
  - Moment counts summarized in text:
    - FD provides (1/2) (T−1) moments for lagged dependent, m (T−1) for exogenous, (1/2 q (T−2)(T−1)) for endogenous.
    - Levels provide (T−1) for lagged dependent, m T for exogenous, 2 q (T−1) for endogenous.
  - Moment conditions compactly written as E[G_i′ U_i] = 0 with G defined in (28) and U_i containing u_i and Δ v_i.
- Constructing limited-information likelihood and criterion:
  - Full parametric likelihood unavailable under GMM; use asymptotic Normal approximation:
    - Lemma 1: √N g_N(θ_0) →d N(0, S).
    - Likelihood for θ ∝ exp(−(1/2) N g_N(θ)′ S^{-1} g_N(θ)).
  - Laplace approximation around GMM estimator θ̂ with weighting matrix Ŝ yields model likelihood approximation and BIC-like posterior odds (equation (37)).
  - Iterative GMM with consistent weighting matrix approximates Bayes factors; limited information criterion resembles BIC and underpins LIBMA.

### IV. Monte Carlo simulation design
- Universe: 6 exogenous variables, 2 endogenous variables, and lagged dependent variable.
- Fixed T = 4; N ∈ {200, 500, 1000, 2000}.
- Exogenous variables generation:
  - First four exogenous variables: x_{it,1:4} ∼ N(0, 4 I) + specified constants (equation (38)).
  - Variables 5 and 6 correlated with 1 and 2 via (39).
- Endogenous variables w_{it} generated per (40) with autoregressive structure and normal v_it.
- Dependent variable generation:
  - t = 0: y_{i0} by (41) with θ_x′ = (0.07, 0, 0, −0.09, 0, 0.1), θ_w′ = (0, −0.1), w_{i0} = (1, 2)′, η_i ∼ N(0, σ_η^2) with σ_η^2 = 0.10 and v variance σ_v^2 varying.
  - t ≥ 1: y_{it} by (42) with α ∈ {0.95, 0.50, 0.30}.
- Robustness check: non-Gaussian v_it constructed by discrete support sampling (uniform on [−1, 1], exponential weights) to produce zero-mean non-normal v_it.
- Simulation parameters:
  - 100 Monte Carlo replications per design.
  - σ_v ∈ {0.05, 0.10, 0.50} with σ_η = 0.10 in Gaussian-error experiments; analogous non-Gaussian-error designs.
  - Report medians, means, variances, and quartiles for posterior model probabilities, inclusion probabilities, parameter estimates, and recovery rates.

### V. Monte Carlo results — model selection performance (key statistics)
- Posterior probability of the true model (from Table 1):
  - Mean posterior probabilities for N = 200, 500, 1000, 2000 are about 0.31, 0.46, 0.57, and 0.65.
  - Median posterior model probabilities: 0.32, 0.50, 0.62, and 0.69.
  - Distributions skew toward 1 as N increases; quartiles and density plots show long left tails.
- Posterior probability ratio (true model / best among others) (Table 2):
  - Average ratios increase with N: about 2.26 (N = 200) up to 7.09 (N = 2000).
- Recovery rates (Table 3) — % times true model attains highest posterior probability across 100 replications:
  - N = 200: varies from 65 percent to 83 percent.
  - N = 500: ranges from 82 percent to 93 percent.
  - N ≥ 1000: stays over 90 percent, reaching 97 percent in some cases.
- Example detailed entries (Table 3):
  - N=200  % Correct 74 78 74 83 73 65 76 76 65
  - N=500  % Correct 91 87 83 83 84 83 85 82 87
  - N=1000 % Correct 91 92 90 90 91 91 91 92 90
  - N=2000 % Correct 94 93 97 92 94 92 92 94 97

### VI. Monte Carlo results — model averaging and inclusion probabilities
- Prior inclusion probability per variable implied by priors used is 0.5.
- Median posterior inclusion probabilities for relevant explanatory variables:
  - Median values exceed 0.942 in all cases considered; for N > 500 medians are practically 1.
- Posterior inclusion probabilities for irrelevant variables decrease with sample size; upper bound less than 0.076 for N = 2000 (Gaussian case).
- Detailed medians and variances (Table 4 examples):
  - For N=200 and α = 0.95, σ_ν = 0.05:
    - y t-1 True Value 1 Median 0.99999 Variance 0.00007
    - x1 True Value 1 Median 1.00000 Variance 0.00005
    - x2 True Value 0 Median 0.18080 Variance 0.02581
    - x3 True Value 0 Median 0.17971 Variance 0.02050
    - x4 True Value 1 Median 1.00000 Variance 0.00000
    - x5 True Value 0 Median 0.17414 Variance 0.02745
    - x6 True Value 1 Median 1.00000 Variance 0.00002
    - w1 True Value 0 Median 0.18608 Variance 0.02415
    - w2 True Value 1 Median 1.00000 Variance 0.00001
  - For N=1000 and α = 0.50, σ_ν = 0.10:
    - x2 True Value 0 Median 0.08698 Variance 0.01251
    - x5 True Value 0 Median 0.09657 Variance 0.02133
    - w2 True Value 1 Median 1.00000 Variance 0.00000

### VII. Monte Carlo results — parameter estimation performance
- Table 5 reports median parameter estimates (averaged over 100 replications) compared to true values; estimates are very close to true parameter values.
- Variance across replications often less than 5 × 10^{−10} in many cases (textual statement).
- Box plots (Figure 3, Appendix A) show variance decreases and medians converge to true values as N increases (example case: α = 0.95, σ_v = 0.1).
- Selected numeric summaries (Tables 5 and 10 — sample snapshots):
  - Sample N=200 (selected entries)
    - Mean: 0.344, 0.318, 0.240, 0.349, 0.328, 0.262, 0.332, 0.319, 0.252
    - Variance: 0.017, 0.015, 0.017, 0.014, 0.021, 0.017, 0.017, 0.018, 0.019
    - Median: 0.379, 0.321, 0.244, 0.370, 0.355, 0.278, 0.360, 0.349, 0.253
  - Sample N=500 (selected entries)
    - Mean: 0.476, 0.470, 0.437, 0.474, 0.460, 0.479, 0.478, 0.501, 0.470
    - Variance: 0.025, 0.020, 0.026, 0.019, 0.027, 0.018, 0.022, 0.017, 0.021
    - Median: 0.523, 0.499, 0.463, 0.515, 0.518, 0.534, 0.528, 0.538, 0.510
  - Sample N=1000 (selected entries)
    - Mean: 0.532, 0.538, 0.549, 0.549, 0.564, 0.553, 0.564, 0.578, 0.567
    - Variance: 0.031, 0.031, 0.028, 0.026, 0.023, 0.022, 0.027, 0.028, 0.020
    - Median: 0.603, 0.578, 0.595, 0.597, 0.605, 0.587, 0.624, 0.642, 0.610
  - Sample N=2000 (selected entries)
    - Mean: 0.598, 0.633, 0.606, 0.626, 0.657, 0.667, 0.651, 0.668, 0.660
    - Variance: 0.039, 0.019, 0.028, 0.034, 0.021, 0.016, 0.020, 0.014, 0.018
    - Median: 0.669, 0.669, 0.657, 0.699, 0.710, 0.717, 0.708, 0.706, 0.699

### VIII. Robustness to non-Gaussian idiosyncratic errors
- Non-Gaussian v_it results (Tables 6-10) closely mirror Gaussian-case results:
  - Mean and median posterior model probabilities approx. same as Gaussian-case (Table 6).
  - Posterior probability ratio remains > 1 across cases and increases with N (starts ~1.66 at N = 200 and reaches 7.29 at N = 2000 in Table 7).
  - Recovery rates similar to Gaussian case:
    - N = 200 recovery rate 65–82 percent.
    - N = 500 success rate 81–91 percent.
    - N = 2000 recovery rate stays over 88 percent, reaching 97 percent for α = 0.30 (Table 8).
  - Posterior inclusion probabilities and parameter-estimate medians and variances in Tables 9 and 10 match closely Gaussian-case results; for N > 500 inclusion probabilities for relevant variables practically 1 and upper bounds for irrelevant variables less than 0.078 for N = 2000.
  - Box plots (Figure 6, Appendix A) for Table 10 parameters (case α = 0.95, σ_v = 0.1) show variance decreases and medians converge to true values.

### IX. Figures and diagnostic displays (preserved labels and axes)
- Figures 1 and 2: posterior density histograms for probabilities in Tables 1 and 2 across α ∈ {0.95, 0.50, 0.30}, σ_ν ∈ {0.05, 0.10, 0.50}, and N ∈ {200, 500, 1000, 2000}.
  - Frequency axes and tick labels preserved in reporting.
- Box plot numeric ticks preserved for example panels:
  - y lag ticks: .930 .935 .940 .945 .950 .955 .960 .965
  - x1 ticks (Figure 3): .062 .064 .066 .068 .070 .072 .074 .076
  - x4 ticks (Figure 3): -.10 5 -.10 0 -.09 5 -.09 0 -.08 5 -.08 0 -.07 5
  - x6 ticks (Figure 3): .088 .092 .096 .100 .104 .108 .112
  - w2 ticks: -.12 -.11 -.10 -.09 -.08 -.07
- Figures repeat across nine α and σ_ν combinations (listed in section IV above).
- Note under figures: "The error terms are constructed using discrete distributions (see section IV.A.)."

### X. Conclusions and implications
- LIBMA integrates GMM-based limited-information estimation with BMA to handle dynamic panel models with endogenous regressors and lagged dependent variables.
- Key advantages:
  - Operates under limited information (GMM) rather than requiring full stochastic specification.
  - Explicitly controls for endogeneity via instruments and moment conditions.
  - Extends BMA framework to panel settings with short T and large N.
- Monte Carlo evidence:
  - LIBMA performs very well asymptotically for model selection, model averaging (inclusion probabilities), and parameter estimation.
  - Robust to non-Gaussian idiosyncratic errors as constructed in simulations.
- Suggested future work: explore LIBMA applications in contexts with constrained sample sizes, such as cross-country growth studies where N or T limitations are binding.

*Source: _wp0974 - References  ............................................................................................................*

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

### References

### Tables
- 1.      Posterior Probability of the True Model  ................................................................... 27
- 2.      Posterior Probability Ratio of True Model/Best among the Other Models  ............... 28
- 3.     Probability of Retrieving the True Model  ................................................................. 29
- 4.     Model Recovery: Medians and Variances of Posterior Inclusion Probability for Each 
  Variable  .................................................................................................................... 30
- 5.     Model Recovery: Medians and Variances of Estimated Parameter Values ................. 31
- 6.      Posterior Probability of the True Model (Non-Gaussian Case)  ................................. 32
- 7.     Posterior Probability Ratio: True Model/best among the Other Models (Non-Gaussian 
  Case) .........................................................................................................................33
- 8.     Probability of Retrieving the True Model (Non-Gaussian Case)................................. 34
- 9.     Model Recovery: Medians and Variances of Posterior Inclusion Probability for Each 
  Variable (Non-Gaussian Case)  .................................................................................. 35
- 10.    Model Recovery: Medians and Variances of Estimated Parameter Values (Non-
  Gaussian Case) .......................................................................................................... 36

### Appendix A Figures
- 1.     Posterior Densities for the Probabilities in Table 1  ................................................... 37
- 2.     Posterior Densities for the Probabilities in Table 2 .................................................... 38
- 3.      Box Plots for Parameters in Table 5  ......................................................................... 39
- 4.     Posterior Densities for the Probabilities in Table 6  ................................................... 40
- 5.     Posterior Densities for the Probabilities in Table 7  ................................................... 41

*Source: _wp0974 - References  ............................................................................................................*

### 6.      Box Plots for Parameters in Table 10  ....................................................................... 42

### 6.      Box Plots for Parameters in Table 10  ....................................................................... 42

### I. Introduction and problem statement
- Model uncertainty arises when lack of clear theoretical guidance and tradeoffs on regressors produce many possible specifications; unsystematic searches can yield overconfident and fragile inferences.
- Bayesian Model Averaging (BMA) bases inferences on a weighted average of the full model space rather than a single selected model and thus incorporates uncertainty in predictions and parameter estimates.
- Literature gaps identified:
  - Most BMA work uses static models and cross-section analyses, ignoring dynamics.
  - To the authors’ knowledge, prior models do not allow inclusion of endogenous variables as regressors in the dynamic panel context.
- Contribution: propose a methodology (LIBMA) for dealing with model uncertainty in panel data with short T where lagged dependent and endogenous variables appear as regressors; refines limited information BMA (Tsangarides (2004)) and constructs likelihood by data transformation and Bayesian arguments.

### II. Bayesian model uncertainty and BMA (theoretical framework)
- Standard linear regression model: Y = Z θ + u (equation (1)); universe of k possible regressors yields 2^k models.
- Posterior probability of model M_j given data D: p(M_j | D) = p(D | M_j) p(M_j) / Σ_l p(D | M_l) p(M_l) (equation (3)).
- Marginal likelihood: p(D | M_j) = ∫ p(D | M_j, θ) p(θ | M_j) dθ (equation (4) and (8)).
- Bayesian Model Averaging for a quantity Γ: p(Γ | D) = Σ_j p(Γ | D, M_j) p(M_j | D) (equation (5)).
- Posterior mean and variance under BMA given by (6) and (7).

### III. Choice of priors and implications for Bayes factors
- For Gaussian models: conjugate Normal-Gamma prior for (θ, σ_u^2) yields closed-form posteriors.
- Zellner’s g-prior for θ conditional on σ_u^2: p(θ | σ_u^2) ∼ N(0, σ_u^2 g (Z′Z)^{-1}).
- Likelihood/Bayes factor sensitivity to prior parameters {θ_0, V, γ, λ} noted; common approaches:
  - Diffuse prior for σ_u with density p(σ_u) ∝ 1/σ_u (equivalent to γ = λ = 0).
  - BACE procedure asymptotically equivalent to g = 1/(NT).
  - BIC approximation described for likelihood ratio.
- Model priors: uniform over model space or Mitchell–Beauchamp structure with equal inclusion probability leading to prior p(M_j) as in (10) and prior odds in (11).

### IV. LIBMA: Limited Information Bayesian Model Averaging for dynamic panels with endogeneity
A. Model setup
- Dynamic panel with lagged dependent variable (indexed 1), m exogenous variables (X), q endogenous variables (W); total regressors k = 1 + m + q.
- Model (12): y_it = α y_{i,t-1} + x_{it} θ_x + w_{it} θ_w + η_i + v_it, observed for i = 1,...,N and t = 1,...,T, with T small relative to N.
- Assumptions: E[v_it] = 0, no serial correlation in v_it; exogenous variables satisfy E[x_{is} v_{it}] = 0 for all s,t; endogenous variables satisfy E[w_{is} v_{it}] ≠ 0 for s = t and zero otherwise as in (14).

B. Estimation and moment conditions (GMM / system GMM)
- First-difference (FD) equations (15) and corresponding moment conditions for instruments provided:
  - Lagged dependent instruments for FD: y_{i,t-2}, y_{i,t-3}, ... leading to moment conditions (16).
  - Exogenous variables: use Δ x_{it} or x_{it} as instruments depending on assumptions, moment conditions (17).
  - Endogenous variables: lags w_{i,t-2}, w_{i,t-3}, ... as instruments, moment conditions (18).
- Level equation instruments and moments summarized in (19)-(21).
- Counting of moment conditions:
  - FD provides (1/2) (T−1) moment conditions for lagged dependent variable, (m (T−1)) for exogenous variables, and (1/2 q (T−2)(T−1)) for endogenous variables (as described in text).
  - Levels provide (T−1) for lagged dependent variable, m T for exogenous, and 2 q (T−1) for endogenous variables (as described).
- Moment conditions aggregated into matrices: Y_i, W_i, ΔY_i, ΔW_i, X_i, ΔX_i with dimensions and block structures given in (22)-(25).
- Moment conditions compactly written as E[G_i′ U_i] = 0 with G defined in (28) and U_i containing u_i and Δ v_i (equations (26)-(28)).

C. Constructing a limited-information likelihood and criterion
- Full parametric likelihood unavailable under GMM; construct a likelihood based on moment conditions.
- Under standard regularity (stationarity, ergodicity, existence of S = Var(√N g_N(θ_0))), Lemma 1: √N g_N(θ_0) →d N(0, S).
- For individual i moment vector g_i as in (30)-(33) and average g_N = (1/N) Σ_i g_i, the asymptotic Normal approximation yields likelihood for θ proportional to exp(−(1/2) N g_N(θ)′ S^{-1} g_N(θ)) (equation (34)).
- Using Laplace approximation around the GMM estimator θ̂ (with weighting matrix Ŝ), model likelihood approximated by expression in (35).
- For model M_j with k_j nonzero θ elements and estimate θ̂_j, model likelihood in (36) leads to posterior odds with a BIC-like form (equation (37)).
- Iterative GMM with consistent estimate of the weighting matrix is used to approximate Bayes factors; the limited information criterion resembles BIC and provides basis for LIBMA.

### V. Monte Carlo simulation design
A. Data generating process
- Universe: 6 exogenous variables, 2 endogenous variables, and lagged dependent variable.
- Fixed T = 4 and varying N ∈ {200, 500, 1000, 2000}.
- Exogenous variables:
  - First four exogenous variables for each i,t: x_{it,1:4} ∼ N(0, 4 I) + specified constants (equation (38) indicates generation with 4I and random draws r_t).
  - Variables 5 and 6 correlated with 1 and 2 via (39).
- Endogenous variables w_{it} generated per (40) with specified autoregressive structure and normal v_it; for t ≥ 1 detailed dynamics provided.
- Dependent variable:
  - For t = 0: y_{i0} generated by (41) with θ_x′ = (0.07, 0, 0, −0.09, 0, 0.1), θ_w′ = (0, −0.1), w_{i0} = (1, 2)′ (as described), η_i ∼ N(0, σ_η^2) with σ_η^2 = 0.10 and v variance σ_v^2 varying.
  - For t ≥ 1: y_{it} generated by (42) with lag coefficient α taking values 0.95, 0.50, and 0.30.
- Robustness: non-Gaussian v_it constructed by sampling support v_S uniformly from [−1, 1], drawing ω_k ∼ Exponential(1), forming probabilities p_k = ω_k / Σ_k ω_k, translating support to zero mean; used to test LIBMA robustness to non-normal errors.

B. Simulation parameters and experiments
- 100 Monte Carlo replications for each design.
- Consider σ_v ∈ {0.05, 0.10, 0.50} with σ_η = 0.10 in Gaussian-error experiments; analogous designs for non-Gaussian-error experiments.
- Report medians, means, variances, and quartiles for posterior model probabilities, inclusion probabilities, parameter estimates, and recovery rates.

### VI. Monte Carlo results — key findings
Model selection performance
- Posterior probability of the true model:
  - Mean posterior probabilities of the true model for N = 200, 500, 1000, 2000 are about 0.31, 0.46, 0.57, and 0.65, respectively (reported from Table 1).
  - Median posterior model probabilities slightly higher: 0.32, 0.50, 0.62, and 0.69.
  - Distributions become more skewed toward 1 as N increases; quartiles and density plots show long left tails.
- Relative measure (ratio of posterior probability of true model to highest posterior probability of other models):
  - Average ratios increase with sample size: about 2.26 for N = 200 up to 7.09 for N = 2000 (Table 2).
- Recovery rates (how often true model attains highest posterior probability across 100 replications):
  - For N = 200, recovery rate varies from 65 percent to 83 percent.
  - For N = 500, success rate ranges from 82 percent to 93 percent.
  - For N ≥ 1000, recovery rate stays over 90 percent, reaching 97 percent in some cases (Table 3).

Model averaging and inclusion probabilities
- Prior inclusion probability per variable implied by priors used is 0.5.
- Median posterior inclusion probabilities for relevant explanatory variables:
  - Median values exceed 0.942 in all cases considered; for N > 500 medians are practically 1 (Table 4).
- Posterior inclusion probabilities for irrelevant variables decrease with sample size; upper bound less than 0.076 for N = 2000.

Parameter estimation performance
- Table 5 reports median parameter estimates (averaged over 100 replications) compared to true values.
- Estimates are very close to true parameter values; variance across replications is often less than 5 × 10^{−10} in many cases (textual statement).
- Box plots (Figure 3, Appendix A) show variance decreases and medians converge to true values as N increases (example case: α = 0.95, σ_v = 0.1).

Robustness to non-Gaussian errors
- Results under non-Gaussian v_it (Tables 6-10) are very similar to Gaussian-case results:
  - Mean and median posterior model probabilities approx. same as in Table 1; distributions skew more toward 1 with larger N (Table 6).
  - Ratio of posterior probability of true model to highest among others remains > 1 across cases and increases with N (starts ~1.66 at N = 200 and reaches 7.29 at N = 2000 in Table 7).
  - Recovery rates similar to Gaussian case: for N = 200 recovery rate 65–82 percent; for N = 500 success rate 81–91 percent; for N = 2000 recovery rate stays over 88 percent, reaching 97 percent for α = 0.30 (Table 8).
  - Posterior inclusion probabilities and parameter-estimate medians and variances in Tables 9 and 10 match closely the Gaussian-case results; for N > 500 inclusion probabilities for relevant variables practically 1 and upper bounds for irrelevant variables less than 0.078 for N = 2000.
  - Box plots (Figure 6, Appendix A) for Table 10 parameters (case α = 0.95, σ_v = 0.1) show variance decreases and medians converge to true values.

### VII. Conclusions and implications
- LIBMA (Limited Information Bayesian Model Averaging) integrates GMM-based limited-information estimation with BMA to handle dynamic panel models with endogenous regressors and lagged dependent variables.
- Key advantages relative to standard BMA:
  - Operates under limited information (GMM) rather than requiring full stochastic specification.
  - Explicitly controls for endogeneity via instruments and moment conditions.
  - Extends the BMA framework to panel settings (short T, large N).
- Monte Carlo evidence:
  - Asymptotically LIBMA performs very well for model selection, model averaging (inclusion probabilities), and parameter estimation.
  - Robust to non-Gaussian idiosyncratic errors as constructed in simulations.
- Suggested future work: explore LIBMA applications in contexts with constrained sample sizes, such as cross-country growth studies where N or T limitations are binding.

*Source: IMF working paper content (section and figures summarized as provided).*

### REFERENCES

### _wp0974 - REFERENCES

### References
- Andrews, D. W. K., and B. Lu, 2001, “Consistent Model and Moment Selection Procedures for  GMM  Estimation  with Application  to  Dynamic  Panel  Data  Models,”  Journal  of Econometrics, Vol. 101, No. 1, pp. 123—64.
- Aragones,  E.,  I.  Gilboa,  A.  Postlewaite,  and  D.  Schmeidler,  2005,    “Fact-Free  Learning,”    American Economic Review Vol. 95, No. 5, pp. 1355—68.
- Back,   K.   and   D.   P.   Brown,      1993,   “Implied   Probabilities   in   GMM   Estimators”,   Econometrica, Vol. 61, No. 4, pp. 971—975.
- Blundell,  R.,  and  S.  Bond,  1998,  “Initial  Conditions  and  Moment  Restrictions  in  Dynamic  Panel Data Models,” Journal of Econometrics, Vol. 87, No. 1, pp. 114—43.
- Brock,  W.,  and  S.  Durlauf,  2001,  “Growth  Empirics  and  Reality,”  World  Bank  Economic  Review, 15, pp. 229—272.
- Chernozhukov,  V.,  and  H.  Hong,  2003,  “An  MCMC  Approach  to  Classical  Estimation,”  Journal of Econometrics, Vol. 115, No. 2, pp. 293—346.
- Chipman,  H.,  E.I.  George,  and  R.E.  McCulloch,  2001,  “The  Practical  Implementation  of  Bayesian  Model  Selection,”  (with  discussion)  in  P.  Lahiri  (ed),  Model  Selection  IMS Lecture Notes, Vol. 38, pp. 70—134.
- Durlauf, S., and D. Quah, 1999, “The New Empirics of Economic Growth,” in J. B. Taylor and M. Woodford (eds), Handbook of Macroeconomics Vol. IA (North Holland).
- Fernàndez  C.,  E.  Ley,  and  M.  Steel,  2001a,  “Model  Uncertainty  in  Cross-Country  Growth  Regressions,” Journal of Applied Econometrics, Vol. 16, pp. 563—76.
- Fernàndez  C.,  E.  Ley  and  M.F.J.  Steel,  2001b,  “Benchmark  Priors  for  Bayesian  Model  Averaging,” Journal of Econometrics, Vol. 100, pp. 381—427.
- Hall, A.R., 2005, Generalized Method of Moments (New York: Oxford University Press).
- Hoeting,  J.A.,  D.  Madigan,  A.E.  Raftery,  and  C.T.  Volinsky,  1999,  “Bayesian  Model  Averaging: A Tutorial,” Statistical Science, Vol. 14, No. 4, pp. 382—417.
- Hong,  H.,  and  B.  Preston,  2008,  “Bayesian  Averaging,  Prediction  and  Nonnested  Model  Selection,” NBER Working Papers 14284, National Bureau of Economic Research, Inc.
- Jacobson  T.,  and  S.  Karlsson,  2004,  “Finding  Good Predictors  for  Inflation:  A  Bayesian  Model Averaging Approach”, Journal of Forecasting, Vol. 23, pp. 476—496.
- Kass,  R.,  and  A.  Raftery,  1995,  “Bayes  Factors,”  Journal  of  the  American  Statistical  Association, Vol. 90, No. 430, pp. 773—95.
- Kass, R., and L. Wasserman, 1995, “A Reference Bayesian Test for Nested Hypotheses and Its Relationship to the Schwarz Criterion,” Journal of the American Statistical Association, Vol. 90, No. 431, pp. 928—34.
- Kim,  J.  Y.,  2002,  “Limited  Information  Likelihood  and  Bayesian  Analysis,”  Journal  of Econometrics, Vol. 107, No. 1-2, pp. 175—93.
- Koop,  G.,  and  L.  Tole,  2004,  “Measuring  the  Health  Effects  of  Air  Pollution:  to  What  Extent  Can  We  Really  Say  that  People  are  Dying  from  Bad  Air?”  Journal  of Environmental Economics and Management 47, January, pp. 30—54.
- Leamer,  E.,  1978,  Specification  Searches:  Ad  Hoc  Inference  with  Non-experimental  Data  (New York: Wiley).
- Leamer, E., 1983, “Let’s Take the Con Out of Econometrics,” American Economic Review, Vol. 73, pp. 31—43.
- Ley  E.,  and  M.  Steel,  2008,  “On  the  Effect  of  Prior Assumptions  in  Bayesian  Model  Averaging  with  Applications  to  Growth  Regression,” Journal  of  Applied  Econometrics, forthcoming.
- Madigan,  D.M.  and  Raftery,  A.E.,  1994,    “Model  Selection  and  Accounting  for  Model  Uncertainty  in  Graphical  Models  using  Occam’s  Window,”  Journal  of  the  American  Statistical Association, Vol. 89, pp. 1335—1346.
- Mitchell,   T.J.,   and   J.J.   Beauchamp,   1988,   “Bayesian   Variable   Selection   in   Linear   Regression,” Journal of the American Statistical Association Vol. 83, pp. 1023—1032.
- Moral-Benito,  E.,  2007,  “Determinants  of  Economic  Growth:  A  Bayesian  Panel  Data  Approach,” CEMFI Working Papers Paper 0719.
- Morales,  K.H.,  J.G.  Ibrahim,  C.  Chen,  and  L.M.  Ryan,  2006,  “Bayesian  Model  Averaging  With  Applications  to  Benchmark  Dose  Estimation  for  Arsenic  in  Drinking  Water,” Journal of the American Statistical Association 101, Vol. 473, pp. 9—17.
- Moulton,  B.R.,  1991,  “A  Bayesian  Approach  to  Regression  Selection  and  Estimation  with  Application to a Price Index for Radio Services,” Journal of Econometrics, Vol. 49, pp. 169—93.
- Raftery,   A.   E.,   1995,   “Bayesian   Model   Selection   in   Social   Research,”   Sociological Methodology, Vol. 25, pp. 111—163.
- Raftery, A. E., 1996, “Approximate Bayes Factors and Accounting for Model Uncertainty in Generalized Linear Models,” Biometrika, Vol. 83, pp. 251—66.
- Raftery, A.E., D. Madigan, and J. A. Hoeting, 1997, “Bayesian Model Averaging for Linear Regression Models,” Journal of the American Statistical Association, 92, pp. 179—191.
- Ragusa, G., 2008, “Bayesian Likelihoods for Moment Condition Models,” mimeo.
- Sala-i-Martin,  X.,  Doppelhofer,  G.,  R.,  and  I.  Miller,  2004,  “Determinants  of  Long-Term  Growth:  A  Bayesian  Averaging  of  Classical  Estimates  (BACE)  Approach,”  American Economic Review, Vol. 94, No. 4, pp. 813—35.
- Schennach,  S.C.,  2005,  “Bayesian  Exponentially  Tilted  Empirical  Likelihood,”  Biometrika, Vol. 92, pp. 31—46.
- Tsangarides, C., 2004, “A Bayesian Approach to Model Uncertainty,” IMF Working Paper No. 04/68 (Washington: International Monetary Fund).
- Wintle,  B.A.,  M.A.  McCarthy,  C.T.  Volinsky,  and  R.P.  Kavanagh,  2003,  “The  Use  of  Bayesian  Model  Averaging  to  Better  Represent  Uncertainty  in  Ecological  Models,”  Conservation Biology 17, December, pp. 1579—1590.
- Yeung,  K.Y.,  R.E.  Bumgarner,  and  A.  E.  Raftery,  2005,  “Bayesian  Model  Averaging:  Development  of  an  Improved  Multi-Class,  Gene  Selection  and  Classification  Tool  for Micro array Data,” Bioinformatics 21, Vol. 10, pp. 2394—2402.

### Tables and Notes
- Table 1. Posterior Probability of the True Model
  - Caption: Summary statistics using LIBMA estimation for various n, a, and sν 0.950.50
  - Notes:
    1. For the idiosyncratic error term, ηi ~ N(0,ση2) where ση = 0.10.
    2. The error term is normally distributed νit ~ N(0,σν2).
- Table 2. Posterior Probability Ratio of True Model/Best Among the Other Models
  - Caption: Summary statistics using LIBMA estimation for various n, α, and σν 0.950.50
  - Notes:
    1. For the idiosyncratic error term, ηi ~ N(0,ση2) where ση = 0.10.
    2. The error term is normally distributed νit ~ N(0,σν2).
- Table 3. Probability of Retrieving the True Model
  - Caption: Summary statistics using LIBMA estimation for various n, α, and σν 0.500.300.95
  - Example values (as presented):
    - N=200  % Correct 74 78 74 83 73 65 76 76 65
    - N=500  % Correct 91 87 83 83 84 83 85 82 87
    - N=1000 % Correct 91 92 90 90 91 91 91 92 90
    - N=2000 % Correct 94 93 97 92 94 92 92 94 97
  - Notes:
    1. For the idiosyncratic error term, ηi ~ N(0,ση2) where ση = 0.10.
    2. The error term is normally distributed νit ~ N(0,σν2).
- Table 4. Model Recovery: Medians and Variances of Posterior Inclusion Probability for Each Variable
  - Caption: True model vs BMA posterior inclusion probability for various n, α, and σν 0.95 0.50
  - Example structure and entries (selected exact values as shown):
    - For N=200 and α = 0.95, σν = 0.05:
      - y t-1 True Value 1 Median 0.99999 Variance 0.00007
      - x1 True Value 1 Median 1.00000 Variance 0.00005
      - x2 True Value 0 Median 0.18080 Variance 0.02581
      - x3 True Value 0 Median 0.17971 Variance 0.02050
      - x4 True Value 1 Median 1.00000 Variance 0.00000
      - x5 True Value 0 Median 0.17414 Variance 0.02745
      - x6 True Value 1 Median 1.00000 Variance 0.00002
      - w1 True Value 0 Median 0.18608 Variance 0.02415
      - w2 True Value 1 Median 1.00000 Variance 0.00001
    - For N=1000 and α = 0.50, σν = 0.10:
      - x2 True Value 0 Median 0.08698 Variance 0.01251
      - x5 True Value 0 Median 0.09657 Variance 0.02133
      - w2 True Value 1 Median 1.00000 Variance 0.00000
  - Notes:
    1. For the idiosyncratic error term, ηi ~ N(0,ση2) where ση = 0.10.
    2. The error term is normally distributed νit ~ N(0,σν2).

*References and tables reproduced as presented in the source content.*

### 2. The error term is normally distributed

### 2. The error term is normally distributed

### Key assumptions (Appendix A)
- For the idiosyncratic error term, η_i ~ N(0, σ_η^2) where σ_η = 0.10.
- The error term is normally distributed ν_it ~ N(0, σ_ν^2).

### Model recovery — medians and variances of estimated parameter values (Tables 5 and 10)
- Sample N=200 (selected entries)
  - Mean: 0.344, 0.318, 0.240, 0.349, 0.328, 0.262, 0.332, 0.319, 0.252
  - Variance: 0.017, 0.015, 0.017, 0.014, 0.021, 0.017, 0.017, 0.018, 0.019
  - Median: 0.379, 0.321, 0.244, 0.370, 0.355, 0.278, 0.360, 0.349, 0.253
  - Q10: 0.273, 0.223, 0.127, 0.284, 0.231, 0.145, 0.229, 0.212, 0.146
  - Q30: 0.454, 0.414, 0.342, 0.449, 0.453, 0.356, 0.435, 0.433, 0.366
- Sample N=500 (selected entries)
  - Mean: 0.476, 0.470, 0.437, 0.474, 0.460, 0.479, 0.478, 0.501, 0.470
  - Variance: 0.025, 0.020, 0.026, 0.019, 0.027, 0.018, 0.022, 0.017, 0.021
  - Median: 0.523, 0.499, 0.463, 0.515, 0.518, 0.534, 0.528, 0.538, 0.510
  - Q10: 0.410, 0.397, 0.378, 0.393, 0.373, 0.386, 0.388, 0.454, 0.402
  - Q30: 0.604, 0.594, 0.567, 0.585, 0.588, 0.580, 0.585, 0.603, 0.584
- Sample N=1000 (selected entries)
  - Mean: 0.532, 0.538, 0.549, 0.549, 0.564, 0.553, 0.564, 0.578, 0.567
  - Variance: 0.031, 0.031, 0.028, 0.026, 0.023, 0.022, 0.027, 0.028, 0.020
  - Median: 0.603, 0.578, 0.595, 0.597, 0.605, 0.587, 0.624, 0.642, 0.610
  - Q10: 0.406, 0.451, 0.509, 0.493, 0.492, 0.497, 0.516, 0.525, 0.518
  - Q30: 0.664, 0.675, 0.668, 0.662, 0.672, 0.657, 0.674, 0.692, 0.669
- Sample N=2000 (selected entries)
  - Mean: 0.598, 0.633, 0.606, 0.626, 0.657, 0.667, 0.651, 0.668, 0.660
  - Variance: 0.039, 0.019, 0.028, 0.034, 0.021, 0.016, 0.020, 0.014, 0.018
  - Median: 0.669, 0.669, 0.657, 0.699, 0.710, 0.717, 0.708, 0.706, 0.699
  - Q10: 0.506, 0.580, 0.513, 0.562, 0.624, 0.627, 0.598, 0.630, 0.638
  - Q30: 0.742, 0.734, 0.737, 0.756, 0.753, 0.752, 0.742, 0.751, 0.746

### Posterior probability of the true model (Table 6)
- Sample N=200 (selected statistics)
  - Mean: 2.282, 2.040, 1.659, 2.323, 2.242, 1.798, 2.175, 2.080, 1.820
  - Variance: 1.981, 2.000, 1.606, 1.726, 2.321, 1.629, 1.803, 1.932, 1.822
  - Median: 2.111, 1.823, 1.353, 2.196, 1.908, 1.522, 2.152, 2.047, 1.461
  - Q1: 1.230, 0.867, 0.690, 1.304, 0.949, 0.730, 1.047, 0.873, 0.707
  - Q3: 3.502, 3.050, 2.417, 3.348, 3.466, 2.475, 3.213, 3.064, 2.690
- Sample N=500 (selected statistics)
  - Mean: 3.738, 3.567, 3.175, 3.563, 3.596, 3.681, 3.704, 3.882, 3.504
  - Variance: 5.286, 4.471, 4.485, 4.499, 5.716, 4.507, 4.530, 4.605, 4.756
  - Median: 3.809, 3.196, 2.959, 3.128, 3.430, 3.899, 3.860, 3.682, 3.570
  - Q1: 1.784, 1.809, 1.526, 2.031, 1.564, 1.797, 1.651, 2.021, 1.572
  - Q3: 5.677, 5.719, 4.581, 5.223, 5.279, 5.189, 5.422, 5.816, 5.368
- Sample N=1000 (selected statistics)
  - Mean: 4.482, 4.651, 4.645, 4.578, 5.034, 4.693, 4.995, 5.686, 4.969
  - Variance: 9.738, 10.456, 8.166, 8.924, 9.303, 9.957, 8.401, 11.604, 8.523
  - Median: 4.213, 3.746, 4.760, 3.775, 4.945, 4.029, 5.106, 5.878, 4.703
  - Q1: 1.634, 1.813, 2.436, 2.232, 2.226, 2.468, 2.657, 2.674, 2.826
  - Q3: 6.376, 7.137, 6.650, 6.789, 7.405, 6.868, 7.220, 8.211, 7.094
- Sample N=2000 (selected statistics)
  - Mean: 6.061, 5.997, 5.740, 6.569, 7.008, 7.180, 6.699, 7.293, 6.669
  - Variance: 18.867, 14.406, 16.746, 19.530, 15.723, 15.063, 14.792, 16.717, 13.738
  - Median: 5.228, 5.302, 5.160, 6.278, 6.977, 7.028, 6.701, 6.917, 6.427
  - Q1: 2.663, 2.953, 1.912, 3.268, 4.008, 4.345, 3.288, 3.937, 3.740
  - Q3: 9.679, 8.340, 8.646, 9.696, 9.771, 10.033, 9.571, 10.203, 9.015

### Posterior probability ratio and percent correct identification (Table 7)
- % Correct (True model vs best among other models) by sample size and settings (selected)
  - N=200: 78, 70, 65, 82, 72, 71, 76, 70, 66
  - N=500: 87, 87, 81, 90, 84, 88, 85, 91, 83
  - N=1000: 82, 88, 89, 88, 91, 87, 87, 90, 92
  - N=2000: 88, 94, 89, 89, 92, 92, 96, 97, 96

### Probability of retrieving the true model — detailed medians and variances per variable (Table 8)
- Many "true" regressors show medians equal or extremely close to 1.00000 (high retrieval probability) for larger N:
  - For variables with True = 1: medians often reported as 1.00000 with variances 0.00000 or tiny (examples across N=200, 500, 1000, 2000).
  - Examples (N=200): y_{t-1} median 1.00000 variance 0.00645; x4 median 1.00000 variance 0.00001; x6 median 1.00000 variance 0.00264; w2 median 1.00000 variance 0.00004.
  - Examples (N=500 and larger): y_{t-1}, x1, x4, x6, w2 medians 1.00000 with variance 0.00000 across many configurations.
- Non-true regressors (True = 0) have medians substantially below 1:
  - Examples (N=200): x2 median 0.17454 variance 0.02173; x3 median 0.17667 variance 0.02116; w1 median 0.16520 variance 0.01047.
  - As N increases these medians decline: e.g., x2 median 0.06555 (N=2000), x3 median 0.07132 (N=2000), w1 median 0.06935 (N=2000).

### Posterior inclusion probability for each variable (Table 9)
- True inclusion probabilities (reported "True Value"):
  - y_{t-1}: 0.95; observed medians around 0.95000 to 0.95641 (variance examples 0.00586, 0.00024, 0.00136).
  - For an alternative true value 0.50: medians around 0.50875 (N=500), 0.50535 (N=1000), 0.50489 (N=2000) with small variances (examples 0.00023, 0.00012, 0.00004).
  - For true value 0.30: medians around 0.30944 (N=500), 0.30580 (N=1000), 0.30273 (N=2000).
- Small true-value regressors (True = 0.07, 0.00, -0.09, 0.10, -0.10) show medians matching those true values closely with very small variances for larger N:
  - x1 True 0.07: medians ~0.07003, 0.07012, 0.07005, 0.07026 with variances often 0.00000–0.00004.
  - x2 True 0.00: medians near 0.00002 to 0.00000 with variances near 0.00000.
  - x4 True -0.09: medians around -0.09019, -0.09016, -0.09038, -0.09020 with variances ~0.00000–0.00079.
  - x6 True 0.10: medians ~0.09953, 0.10020, 0.10008, 0.10012 with variances ~0.00003–0.00000.
  - w2 True -0.10: medians around -0.09635, -0.09861, -0.09959, -0.09988 with variances small (examples 0.00127, 0.00001).
- Trends:
  - Posterior inclusion medians converge toward true values as N increases (variances decline).
  - Variables with True = 1 (core regressors) show high posterior inclusion medians (close to 1.00000) and negligible variance for moderate-to-large N.

### Figures (posterior densities)
- Figures 1 and 2 present posterior density histograms for the probabilities reported in Tables 1 and 2 across combinations of α ∈ {0.95, 0.50, 0.30} and σ_ν ∈ {0.05, 0.10, 0.50} for sample sizes N ∈ {200, 500, 1000, 2000}.
  - Frequency axes shown across bins (e.g., 0.0 to 1.0 in Figure 1; 0 to 20 in Figure 2) illustrate distributional concentration and dispersion for each configuration.
  - Visual patterns correspond to tabulated medians and variances: tighter concentration (narrower densities) for larger N and smaller σ_ν.

*Notes: 1. The error terms are constructed using discrete distributions (see section IV.A.).*

### 1. For the idiosyncratic error term, η

### _wp0974 - 1. For the idiosyncratic error term, η

### Error term specifications
- For the idiosyncratic error term, η_i ~ N(0,σ_η^2) where σ_η = 0.10.
- The error term is normally distributed ν_it ~ N(0,σ_ν^2).

### Experimental settings and parameter values reported
- α values reported: α = 0.95; α = 0.50; α = 0.30.
- σ_ν values reported in figures: σ_ν = 0.05; σ_ν = 0.10; σ_ν = 0.50.
- Sample sizes N reported: N = 200; N = 500; N = 1000; N = 2000.
- Figures label BMA coefficients' estimated values for α = 0.95, σ_ν = 0.1 and various N.

### Variables shown in Figures (box plots and densities)
- Variables/coefficients listed in box plots and figures:
  - y lag
  - x1
  - x2
  - x3
  - x4
  - x5
  - x6
  - w1
  - w2

### Numerical axis ticks and ranges appearing in figures (preserved exactly)
- Figure 3 / Figure 6 vertical axis ranges and ticks shown for the first panels:
  - .930 .935 .940 .945 .950 .955 .960 .965
  - .932 .936 .940 .944 .948 .952 .956 .960 .964
- Example numeric ranges/ticks for specific variables across figures:
  - y lag: .055 .060 .065 .070 .075 .080 .085
  - x1 (Figure 3): .062 .064 .066 .068 .070 .072 .074 .076
  - x2 (Figure 3): -.0 04 -.0 03 -.0 02 -.0 01 .000 .001 .002 .003 .004 .005
  - x2 (Figure 6): -.0016 -.0012 -.0008 -.0004 .0000 .0004 .0008 .0012 .0016 .0020
  - x3 (Figure 3): -.0 06 -.0 04 -.0 02 .000 .002 .004 .006
  - x3 (Figure 6): -.0015 -.0010 -.0005 .0000 .0005 .0010 .0015 .0020 .0025
  - x4 (Figure 3): -.10 5 -.10 0 -.09 5 -.09 0 -.08 5 -.08 0 -.07 5
  - x4 (Figure 6): -.098 -.096 -.094 -.092 -.090 -.088 -.086 -.084
  - x5 (Figure 3): -.0 04 -.0 03 -.0 02 -.0 01 .000 .001 .002 .003 .004
  - x5 (Figure 6): -.0 02 -.0 01 .000 .001 .002 .003
  - x6 (Figure 3): .088 .092 .096 .100 .104 .108 .112
  - x6 (Figure 6): .092 .094 .096 .098 .100 .102 .104 .106 .108
  - x6 alternate ticks (Figure 3): -.00 6 -.00 4 -.00 2 .000 .002 .004 .006 .008 .010
  - x6 alternate ticks (Figure 6): -.004 -.003 -.002 -.001 .000 .001 .002 .003 .004
  - w2 (Figure 3 / 6): -.12 -.11 -.10 -.09 -.08 -.07

### Posterior density and frequency displays (preserved labels and axes)
- Posterior density figures show Frequency on the vertical axis with horizontal axis tick labels "05 10 15 20 25" in multiple panels.
- Frequency axis labeled with ticks: 0.0 0.2 0.4 0.6 0.8 1.0 in posterior density panels for combinations of α and σ_ν.
- Figures repeat panels across N = 200, N = 500, N = 1000, N = 2000 for α and σ_ν combinations:
  - α = 0.95, σ_ν = 0.05
  - α = 0.95, σ_ν = 0.10
  - α = 0.95, σ_ν = 0.50
  - α = 0.50, σ_ν = 0.05
  - α = 0.50, σ_ν = 0.10
  - α = 0.50, σ_ν = 0.50
  - α = 0.30, σ_ν = 0.05
  - α = 0.30, σ_ν = 0.10
  - α = 0.30, σ_ν = 0.50

### Notes from figures
- Note 1 repeated under figures: "The error terms are constructed using discrete distributions (see section IV.A.)."

*Source: _wp0974 - 1. For the idiosyncratic error term, η (IMF PDF content provided).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0974.pdf_