## _wp09268

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### I. Contribution and framing
- Revisits cross-country growth empirics using a limited-information Bayesian Model Averaging (LIBMA) methodology for a dynamic panel data growth model with endogenous regressors.
- LIBMA constructed as a small-sample counterpart of Chen, Mirestean, and Tsangarides (2009).
- Methodological features:
  - Limited information technique based on Generalized Method of Moments (GMM) moment estimation.
  - Obtains posteriors through a Bayesian procedure leveraging the linear structure of the model.
- Motivating problems addressed:
  - Model uncertainty from ad hoc model selection.
  - Endogeneity of many candidate regressors.
  - Short time dimension panels (T small relative to N).

### II. Theoretical model and estimation framework
- Canonical dynamic panel specification:
  - g = Zβ + u
  - yit = ρ yi,t−1 + x'it βx + w'it βw + p'it βp + uit ; uit = ηi + vit, j with |ρ| < 1; i = 1,…,N; t = 1,…,T. yit = log(Yit).
- Regressor partitioning:
  - lagged dependent variable indexed by 1;
  - exogenous variables X (m variables);
  - predetermined variables P (p variables);
  - endogenous variables W (q variables).
  - Total k = m + q + p + 1.
- First-difference moment conditions (typical instruments):
  - Lagged dependent instruments: E(yi,t−s Δvit) = 0 for t = 2,…,T; s = 2,…,t (T ≥ 2).
  - Exogenous: E(xℓit Δvit) = 0 for t = 2,…,T; ℓ = 1,…,m.
  - Predetermined: E(pℓ,i,t−s Δvit) = 0 for t = 2,…,T; s = 1,…,t−1.
  - Endogenous: E(wℓ,i,t−s Δvit) = 0 for t = 3,…,T; s = 2,…,t−1 (T ≥ 3).
- Moment-counts (first-difference):
  - Lagged dependent variable: T(T−1)/2 moment conditions.
  - Exogenous variables: m(T−1) moment conditions.
  - Endogenous variables: q(T−2)(T−1)/2 moment conditions.
- Levels equation (system GMM) moment conditions summarized:
  - Lagged dependent variable: (T−1) moment conditions.
  - Exogenous variables: m(T−1) moment conditions.
  - Endogenous variables: q(T−2) moment conditions.
  - Predetermined variables: p(T−1) moment conditions.
  - Additional homoskedasticity linear moments: (T−1) additional conditions under assumptions.

### III. Limited-information Bayesian Model Averaging (LIBMA) using moments
- Model space size: K = 2^k models.
- Bayesian updating and BMA formulas preserved:
  - p(Mj | D) = p(D | Mj) p(Mj) / Σl p(D | Ml) p(Ml).
  - Marginal likelihoods constructed from moment information only (no distributional assumptions).
  - Unit information prior yields a BIC-like marginal likelihood (equations (8) and (9) in text).
  - Posterior inclusion probability for regressor Zi: p(Zi | D) = Σj I(Zi ∈ Mj) p(Mj | D).
  - Posterior mean and variance aggregation formulas used (equations (12) and (13)).
- Prior choices:
  - Uniform prior over model space: p(M1) = … = p(MK) = 1/K.
  - Prior probability of inclusion for each variable: 0:50.

### IV. Instrument proliferation and practical reductions
- Problem: instrument count grows quadratically with T; too many instruments cause overfitting and imprecise GMM weighting matrix estimation.
- Reduction strategies implemented and evaluated:
  - Collapsing the instrument matrix (Roodman (2009)).
  - Limiting maximum lag length L used for instruments.
  - Transformations to make instrument count linear in T.
- Collapsing examples and moment-count reductions (representative numeric entries preserved exactly):
  - For T = 6, m = 5, q = 6:
    - Uncollapsed full: 119
    - Uncollapsed 2 lags: 95
    - Uncollapsed 1 lag: 73
    - Collapsed full: 46
    - Collapsed 3 lags: 38
    - Collapsed 2 lags: 31
    - Collapsed 1 lag: 24
  - For T = 6, m = 6, q = 12:
    - Uncollapsed full: 205
    - Collapsed full: 77
    - Collapsed 2 lags: 35
    - Collapsed 1 lag: 28
  - Illustrative statement: for 19 regressors (6 exogenous, one predetermined, 12 endogenous) and 6 time periods, collapsing reduces instruments from 205 to 77; collapsing and limiting lag length to 2 reduces to 50.
- Chosen practical approach in the paper: collapsed instruments using 2 lags for the primary investigation.

### V. Data, variable universe, and panel construction
- Universe: 42 growth determinants grouped into 10 categories (Solow-Swan determinants, human capital, macro stability, financial development, trade regime, external environment, internal environment, institutions, violence/conflict, geography/fixed factors, regional characteristics).
- Key Solow-Swan and human-capital proxies:
  - Physical capital: ratios of real investment to GDP (investment).
  - Human capital: life expectancy; school enrollment/education measures.
  - Population growth: population growth rates.
- Panel averaging windows and sample:
  - Five-year panels (eight panels): 1961-1965, 1966-1970, 1971-1975, 1976-1980, 1981-1985, 1986-1990, 1991-1995, 1996-2000.
  - Eight-year panels (five panels): 1961-1968, 1969-1976, 1977-1984, 1985-1992, 1993-2000.
  - Countries with less than two observations filtered out for endogenous variables (need at least three observations to have useful moment conditions).
- Model universe sizes reported:
  - For 5-year averaged data: 2^22 (4;194;304) possible models.
  - For 8-year averaged data: 2^19 (524;288) possible models.
- Baseline estimation sample: 107 countries with 593 observations over the period 1960-2005, with an average of5:5observations per country.
- Variable definitions and sources provided (e.g., Penn World Table 6.2 for DIFFY; Barro and Lee for education; World Development Indicators for life expectancy; IFS/IMF for inflation).

### VI. Main empirical findings (LIBMA robustness analysis)
- Robust determinants (5-year averaged data):
  - initial income
  - investment
  - population growth
  - life expectancy
  - debt
  - openness
- Robust determinants (8-year averaged data):
  - initial income
  - investment
  - population growth
  - inflation (enters as new robust variable in the 8-year sample)
  - life expectancy has inclusion probability very close to 0:50 (reported 0.47)
  - debt and openness have inclusion probabilities less than 0:50 in the 8-year sample
- Strongest evidence for robustness found for:
  - initial income
  - investment
  - life expectancy
  - population growth
- Additional robust evidence:
  - debt, openness, and inflation (inflation more robust in 8-year sample)
- Specific numeric diagnostics (selected table entries preserved exactly):
  - Table 1 (5-year / 8-year examples):
    - 1 Log(initial income): 0.87, -0.211, 0.204; 0.97, -0.202, 0.066
    - 2 Log(investment): 0.84, 0.147, 0.184; 0.73, 0.212, 0.077
    - 3 Log(population growth): 0.66, -0.322, 0.497; 0.61, -0.208, 0.522
    - 5 Life expectancy: 0.70, 0.007, 0.017; 0.47, 0.006, 0.001
    - 9 Openness: 0.85, 0.103, 0.217; 0.29, -0.011, 0.039
  - Estimated rate of convergence: in the range of 3-4.5 percent.
    - Implication noted: for convergence rates of 2, 3, and 4.5 percent, a country needs approximately 35, 23, and 15 years, respectively, to cover half the distance to steady state.
- Interpretation:
  - Accounting for dynamics and endogeneity materially affects which determinants are identified as robust.
  - Several commonly used regressors (macroeconomic stability proxies, institutions, geography, political environment) have inclusion probabilities below the 0:50 cutoff and are less robust once model uncertainty and endogeneity are addressed.

### VII. Comparisons with alternative BMA approaches
- Cross-section BMA (FLS; BACE) findings:
  - Also find initial income, investment, life expectancy, and debt as strong determinants.
  - Openness and population growth—robust in LIBMA—have low inclusion probabilities in cross-section FLS/BACE.
  - Overvaluation index enters with high inclusion probability in cross-section FLS/BACE.
- Panel analogues (panel FLS and panel BACE):
  - Identify nine and seven variables as robust for the 5-year and 8-year data sets, respectively.
  - Identify additional robust variables relative to LIBMA (e.g., population growth, inflation, openness, terms of trade).
  - Many time effects flagged as robust by panel FLS/BACE but not by LIBMA.
- Conclusion from comparison:
  - Accounting for endogeneity (as LIBMA does) leads to a different set of robust determinants than methods that do not fully account for endogeneity; LIBMA may better distinguish true effects from proxies captured by time dummies.

### VIII. Monte Carlo evidence on instrument reduction and model recovery (Appendix I & II)
- Monte Carlo design (Appendix II):
  - Universe: 12 variables (5 exogenous, 2 time invariant; 6 endogenous; lagged dependent variable).
  - T = 6; N varied: 50, 75, 90, 100.
  - Instrument sets examined: (i) all lags, (ii) 2 lags, (iii) 1 lag; both collapsed and non-collapsed instruments.
  - For full (collapsed) forms, moment conditions: all lags: 119 (51); 2 lags: 95 (36); 1 lag: 73 (29).
  - 500 Monte Carlo replications; v_it ~ N(0, ^2_v) with ^2_v = 0:10; eta_i ~ N(0, ^2_eta) with ^2_eta = 0:10.
  - Prior probability of inclusion = 0:50.
- Main Monte Carlo findings:
  - Collapsing instruments improves inclusion probabilities for included and non-included variables, especially for smaller N (improvement often by factor of 1.5 or more).
  - As N increases, posterior inclusion probabilities approach 1 for relevant variables; medians for non-relevant variables decrease with N.
  - Among collapsed forms, using 1 or 2 lags rather than full lag set generally yields higher inclusion probabilities for endogenous variables and for lower N; no clear distinction between 1 or 2 lags.
  - Collapsing instruments reduces G/N (ratio instruments to sample size), improving results; further reducing lags yields diminishing returns once G/N is small.
  - For the paper's investigation, collapsed instruments with 2 lags were selected.
- Model recovery statistics (selected Appendix excerpts preserved exactly):
  - Table A3: Probability of retrieving the true model (% Correct) reported across configurations; selected entries:
    - N=50: % Correct = 0 2 16 13
    - N=75: % Correct = 11 15 13 11 3
    - N=90: % Correct = 3 16 11 20 6 21
    - N=100: % Correct = 7 20 12 27 8 23
  - Posterior probability of the true model (example N=100, one configuration reported verbatim):
    - Mean 0.017 0.169 0.064 0.282 0.029 0.218
    - Variance 0.341 0.548 0.464 0.681 0.340 0.573
    - Median 0.017 0.169 0.064 0.282 0.029 0.218
    - Q3 0.111 0.730 0.340 1.061 0.197 0.915
- Parameter estimation and selection:
  - Collapsing instruments reduces bias and variance of parameter estimates; improvements rise with sample size.
  - Even when model selection performance is imperfect, BMA differentiates relevant from non-relevant variables.

### IX. Methodological and policy implications
- Methodological:
  - LIBMA demonstrates the importance of combining dynamic panel estimation, limited-information GMM, instrument-count control (collapsing/lag limits), and Bayesian Model Averaging to address endogeneity and model uncertainty in short panels.
  - Choice of instrument strategy materially affects inclusion probabilities and model recovery; collapsed instruments with limited lags provide a practical trade-off.
- Policy-relevant implications:
  - Robust empirical support for factors aligned with the augmented Solow-Swan framework:
    - initial income (conditional convergence),
    - investment,
    - population growth,
    - life expectancy (health).
  - Trade openness and debt emerge as important considerations (inclusion probabilities vary by averaging window).
  - Many commonly used regressors may be less robust once dynamics, endogeneity, and model uncertainty are jointly addressed, suggesting caution in policy inferences drawn from ad hoc specifications.

_Italic: Content derived solely from _wp09268 - References (PDF)._

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

### _wp09268 - References

### I. Introduction: Motivation and framing
- Economic growth and human development topics have been examined since the beginning of recorded history.
- Over the last two decades, philosophical rhetoric has emphasized the primacy of human development as the ultimate objective of economic pursuits.
- Empirical work seeks to explain why some countries have experienced rapid long-term growth rates in income while others have not.
- Economic growth has been described as “the part of macroeconomics that really matters,” not least because relatively small differences in growth rates when cumulated over time can have major consequences for standards of living.
- Despite many cross-country growth studies following Barro (1991) and Mankiw, Romer, and Weil (1992), there is little consensus on the mechanics of economic growth.
- A fundamental problem is the lack of an explicit theory identifying the determinants of growth, with surveys (Durlauf and Quah (1999); Durlauf, Johnson, and Temple (2005)) identifying more variables partially correlated with growth than the number of countries for which data are available.
- The neoclassical Solow-Swan (1956) growth model and endogenous growth models by Romer (1986) and Lucas (1988) are described as “open-ended” (see Brock and Durlauf (2001)), admitting a vast range of logical and testable extensions and many possible specifications.

### II. Problems with ad hoc model selection and the turn to BMA
- Unsystematic searches of “ad hoc” growth model configurations may result in overconfident and fragile inferences — even contradictory conclusions — and fundamentally ignore model uncertainty.
- Growing number of growth researchers use Bayesian Model Averaging (BMA) to address model uncertainty.
- Key methodological developments and applications cited:
  - Raftery (1995)
  - Fern·ndez, Ley and Steel (2001)
  - Brock and Durlauf (2001)
  - Sala-i-Martin, Doppelhofer and Miller (2004)
  - Brock, Durlauf, and West (2003)
  - Durlauf, Kourtelos and Tan (2008)
  - Ley and Steele (2007, 2009)
  - Doppelhofer and Weeks (2009)
- Extensions and refinements discussed in the literature:
  - Testing for growth theories instead of particular variables (Brock, Durlauf, and West (2003); Durlauf, Kourtelos and Tan (2008)).
  - Quantifying the degree to which development determinants act “jointly” to affect growth (Ley and Steele (2007, 2009); Doppelhofer and Weeks (2009)).
- Limitations of most existing BMA work in growth empirics:
  - Focus on cross section analysis with data averaged over the time dimension, thus ignoring dynamic relationships among variables and the dynamic evolution of the growth process.
  - Very few methodologies allow for inclusion of variables that are endogenous in a statistical sense (i.e., correlated with the disturbance term).
- Noted methodological contributions addressing endogeneity and dynamics:
  - Durlauf, Kourtelos and Tan (2008) construct instruments for variables endogenously determined in the economic sense and introduce a model-averaged version of two-stage Least Squares (2SLS).
  - Eicher, Lenkoski and Raftery (2009) introduce an instrumental variable BMA (IVBMA) approach.
  - Moralñ-Benito (2009) considers a panel data model where the lagged dependent variable is correlated with the individual effects but not correlated with the error term.

### III. Contribution of the paper: LIBMA and limited-information BMA for dynamic panels
- The paper revisits the cross-country growth empirics debate using a proposed limited information BMA methodology to address model uncertainty in a dynamic panel data growth model with endogenous regressors.
- Constructed a small sample counterpart of the LIBMA developed by Chen, Mirestean, and Tsangarides (2009).
- The proposed BMA methodology:
  - Is a limited information technique based on Generalized Method of Moments (GMM) moment estimation.
  - Obtains posteriors through a simple Bayesian procedure leveraging the linear structure of the model.

### IV. Main empirical findings (as stated)
- Once endogeneity and model uncertainty are accounted for, various economic factors are robustly correlated with economic growth.
- Strongest evidence for robustness found for:
  - initial income
  - investment
  - life expectancy
  - population growth
- Also strong evidence that:
  - debt
  - openness
  - inflation
  are robust growth determinants.
- The set of robust growth determinants identified differs from sets found by other studies that incorporate model uncertainty.
- These results suggest the importance of investigating growth empirics in a setting that explicitly accounts for dynamics and endogeneity.

### V. Structure of the paper (organization)
- Section 2: presents the theoretical specification, discusses estimation issues, and describes the estimator used for the robustness analysis.
- Section 3: presents the data and identifies the growth determinants.
- Supporting materials listed in the document include:
  - References (page 21)
  - Tables:
    - 1. Robustness of Growth Determinants (page 24)
    - 2. LIBMA Comparison with BACE and FLS (page 25)
    - 3. LIBMA Comparison with BACE and FLS (Panel Data Format) (page 26)
  - Appendices (page 27) with:
    - Appendix I: Monte Carlo experiment results (page 27)
    - Appendix II: Monte Carlo data generating process (page 29)
  - Appendix I Tables (pages 30–32) listing:
    - 1. True model vs BMA posterior inclusion probability (page 30)
    - 2. Model Recovery: Medians and Variances of Estimated Parameter Values (page 31)
    - 3. Posterior Probabilities (page 32)
  - Appendix III Table:
    - 1. Sample Data (page 33)

*Source: _wp09268 - References*

### Section 4 summarizes the results, and section 5 concludes.

### _wp09268 - Section 4 summarizes the results, and section 5 concludes.

### II. Theoretical Considerations — model setup
- Canonical cross-country growth regression representation:
  - g = Zβ + u(1)
  - g is growth rate of output per worker Y between period t and t−1.
  - Z is an n×k matrix of growth regressors; β = (β1 … βk)' unknown parameters; u is the error term.
- Universe of k possible explanatory variables indexed by U = {1, 2, …, k}. For model Mj ⊆ U, a k×k diagonal choice matrix CMj selects included variables so that Z = Z CMj.
- Partition of potential regressors:
  - lagged dependent variable indexed by 1;
  - exogenous variables X (m variables);
  - predetermined variables P (p variables);
  - endogenous variables W (q variables).
- Dynamic panel model (for model Mj):
  - yit = ρ yi,t−1 + x'it βx + w'it βw + p'it βp + ui t
  - uit = ηi + vit, j with |ρ| < 1; i = 1,…,N; t = 1,…,T.
  - yit = log(Yit).
- Assumptions on moments and endogeneity:
  - E(vit) = 0; vit not serially correlated.
  - Exogenous: Elit(xit vit−s) = 0 ∀ i,t,s for xℓit ∈ xit.
  - Predetermined: E(pℓit vit−s) ≠ 0 for s < t, = 0 otherwise.
  - Endogenous: E(wℓit vit−s) ≠ 0 for s < t, = 0 otherwise.
- Total number of possible explanatory variables: k = m + q + p + 1. Observed across N countries and T periods, with T small relative to N.

### A. A Dynamic Growth Model with Endogenous Regressors — instruments and moment conditions (first-difference equation)
- First-difference equation:
  - Δyit = ρ Δyi,t−1 + Δxit βx + Δwit βw + Δpit βp + Δvit, for t = 2,…,T.
- Typical instruments and moment conditions for first-difference equation:
  - Lagged dependent variable yi,t−s instruments: E(yi,t−s Δvit) = 0 for t = 2,…,T; s = 2,…,t, for T ≥ 2.
  - Exogenous xℓit: E(xℓit Δvit) = 0 for t = 2,…,T; ℓ = 1,…,m.
  - Predetermined pℓ,i,t−1: E(pℓ,i,t−s Δvit) = 0 for t = 2,…,T; s = 1,…,t−1; l = 1,…,p; for T ≥ 2.
  - Endogenous wℓ,i,t−2: E(wℓ,i,t−s Δvit) = 0 for t = 3,…,T; s = 2,…,t−1; l = 1,…,q; for T ≥ 3.
- Moment-count summary (first-difference equation):
  - Lagged dependent variable: T(T−1)/2 moment conditions.
  - Exogenous variables: m(T−1) moment conditions.
  - Endogenous variables: q(T−2)(T−1)/2 moment conditions.
- Table A (specified instruments → moment conditions) — variables and instruments:
  - Δyi,t−1 instruments: yi,t−2; …; yi,0 ⇒ E(yi,t−s Δvit) = 0; t = 2,…,T; s = 2,…,t.
  - xℓit instruments: xℓit; …; xℓi1 ⇒ E(xℓit Δvit) = 0; t = 2,…,T; l = 1,…,m.
  - pℓit instruments: pℓ,i,t−1; …; pℓ,i1 ⇒ E(pℓ,i,t−s Δvit) = 0; t = 2,…,T; s = 1,…,t−1; l = 1,…,p.
  - wℓit instruments: wℓ,i,t−2; …; wℓ,i1 ⇒ E(wℓ,i,t−s Δvit) = 0; t = 3,…,T; s = 2,…,t−1; l = 1,…,q.

### B. Estimation and moment conditions — levels equation and system GMM
- Levels equation (original dynamic model) combined with first-difference equation in system GMM (Arellano and Bover (1995); Blundell and Bond (1998)).
- Instruments and moment conditions for levels equation:
  - E(Δyi,t−1 uit) = 0; t = 2,…,T (lagged dependent variable in levels equation).
  - E(Δxℓit uit) = 0; t = 2,…,T; ℓ = 1,…,m.
  - E(Δpℓit uit) = 0; t = 2,…,T; ℓ = 1,…,p (predetermined; pℓ,i,1 must be observable; holds for T ≥ 2).
  - E(Δwℓi,t−1 uit) = 0; t = 3,…,T; ℓ = 1,…,q (endogenous; wℓ,i,1 must be observable; holds for T ≥ 3).
- Moment-count summary (levels equation):
  - Lagged dependent variable: (T−1) moment conditions.
  - Exogenous variables: m(T−1) moment conditions.
  - Endogenous variables: q(T−2) moment conditions.
  - Predetermined variables: p(T−1) moment conditions.
- Additional homoskedasticity linear moment conditions (Ahn and Schmidt (1995)):
  - E(yt u t − yt−1 u t−1) = 0; t = 2,…,T; i = 1,…,N; yields (T−1) additional linear moment conditions if vit homoskedastic through time and E(Δy i1 ui2) = 0.
- Matrix formulation of full set of moment conditions:
  - Define ui (T×1) and Dvi ((T−1)×1); Ui = [ui' Dvi']'.
  - Full set: E(Gi' Ui) = 0 (equation (4)).
  - Gi is a (2T−1) × [dimension depending on T, m, p, q] matrix (see text for exact dimension and structure given in equation (5)).

### C. Model uncertainty — limited-information Bayesian model averaging using moments
- Model space: K = 2^k models M = (M1,…,MK).
- Bayesian updating:
  - Posterior model probability: p(Mj | D) = p(D | Mj) p(Mj) / Σl p(D | Ml) p(Ml).
  - Marginal likelihood: p(D | Mj) = ∫ p(D | θj, Mj) p(θj | Mj) dθj.
  - Posterior odds ratio between M1 and M2:
    - p(M1 | D) / p(M2 | D) = [p(M1)/p(M2)] × [p(D | M1) / p(D | M2)].
- Use of limited information Bayesian model averaging (Chen, Mirestean, and Tsangarides (2009)):
  - Construct marginal likelihoods based only on information from moment conditions, no distributional assumptions required.
  - Unit information prior yields a BIC-like form for model likelihood.
  - Model likelihood for model Mj with kj nonzero elements of β:
    - Likelihood ∝ exp{ −1/2 N b̄' N [b̂0;j]' S−1 [b̂0;j] b̄ N [b̂0;j] − (kj/2) logN } (expression as given in text; equation (8) describes form).
  - Posterior odds in BIC-like exponential form (equation (9)). Uniform prior over model space is assumed: p(M1) = p(M2) = … = p(MK) = 1/K.
- Bayesian Model Averaging (BMA) inference:
  - Posterior for quantity of interest θ: p(θ | D) = Σj p(θ | D, Mj) p(Mj | D) (equation (10)).
  - Inclusion probability for regressor Zi: p(Zi | D) = Σj I(Zi ∈ Mj) p(Mj | D) (equation (11)).
  - Posterior mean and variance for parameter θℓ:
    - E(θℓ | D) = Σj E(θℓ | D, Mj) p(Mj | D) (equation (12)).
    - Var(θℓ | D) = Σj p(Mj | D) Var(θℓ | D, Mj) + Σj p(Mj | D) [E(θℓ | D, Mj) − E(θℓ | D)]^2 (equation (13)).

### D. Reducing the number of moment conditions — instrument proliferation and practical reductions
- Problem: number of instruments grows quadratically with T; sensitivity of GMM estimators to instrument count when T rises and N is small. Too many instruments can cause overfitting and imprecise estimates of the GMM weighting matrix (Donald, Imbens, and Newey (2008); Roodman (2009)).
- Approaches to reduce instrument count:
  - Collapsing the instrument matrix (Roodman (2009)).
  - Limiting maximum lag length L used for instruments.
  - Transformations make instrument count linear in T.
- First-difference equation instrument collapsing:
  - Original count for lagged dependent variable: T(T−1)/2 moment conditions. Collapsed stacking reduces count to (T−1).
  - Example stacked instrument matrices:
    - Ya i: stacked all possible lags (T−1 × T−1) structure shown.
    - Y1 i, Y2 i, Y3 i: stacked matrices using at most 1, 2, or 3 lags respectively; dimensions (T−1)×(T−1), (T−1)×(T−2), (T−1)×(T−3).
  - Predetermined variables follow same collapsing logic; if L denotes max lags used, number of moment conditions for predetermined variables = L p.
  - Endogenous variables: original q(T−2)(T−1)/2 moments; stacking and limiting lags reduces count to L q (e.g., W1 i, W2 i matrices produce q and 2q moment counts respectively).
  - Exogenous variables: reduce from m(T−1) to m by stacking into Xi (T−1)×m.
- Levels equation instrument collapsing:
  - Stack instruments to reduce lagged-dependent variables moment conditions from (T−1) to 1.
  - Define matrices:
    - DY i: T×(T−1) matrix of first differences of dependent variable.
    - DW i: T×q instrument matrix of first differences of endogenous variables.
    - DX i: T×m matrix of first-differenced exogenous variables.
    - Ȳ i: T×(T−1) instrument matrix from Ahn and Schmidt (1995) homoskedasticity restriction.
- Moment condition matrices with maximum lag L:
  - General GLa form: GL i has dimension (2T−1) × (T + (L+1) q + m + L).
  - Special cases:
    - Ga i (all lags): (2T−1) × (m + 1 + (2+q)(T−1)) matrix (exact form in text).
    - G1 i (1 lag): (2T−1) × (T + 2q + m + 1).
    - G2 i (2 lags): (2T−1) × (T + 3q + m + 2).
- Numerical illustration (Table C) — instrument counts for various options of T, m, q and instrument strategies:
  - Parameters considered in table: T = 6 and T = 10; exogenous m = 5 or 6; endogenous q = 6, 8, 12. One predetermined variable (lagged dependent variable) assumed.
  - Reported instrument counts (selected entries):
    - T = 6, m = 5, q = 6:
      - Uncollapsed full: 119
      - Uncollapsed 2 lags: 95
      - Uncollapsed 1 lag: 73
      - Collapsed full: 46
      - Collapsed 3 lags: 38
      - Collapsed 2 lags: 31
      - Collapsed 1 lag: 24
    - T = 6, m = 5, q = 8:
      - Uncollapsed full: 147
      - Uncollapsed 2 lags: 117
      - Uncollapsed 1 lag: 86
      - Collapsed full: 65
      - Collapsed 3 lags: 46
      - Collapsed 2 lags: 37
      - Collapsed 1 lag: 28
    - T = 6, m = 6, q = 12:
      - Uncollapsed full: 205
      - Uncollapsed 2 lags: 163
      - Uncollapsed 1 lag: 131
      - Collapsed full: 77
      - Collapsed 3 lags: 50
      - Collapsed 2 lags: 35
      - Collapsed 1 lag: 28
    - T = 10, m = 5, q = 6:
      - Uncollapsed full: 337
      - Uncollapsed 2 lags: 181
      - Uncollapsed 1 lag: 133
      - Collapsed full: 96
      - Collapsed 3 lags: 67
      - Collapsed 2 lags: 54
      - Collapsed 1 lag: 41
    - T = 10, m = 5, q = 8:
      - Uncollapsed full: 425
      - Uncollapsed 2 lags: 322
      - Uncollapsed 1 lag: 165
      - Collapsed full: 133
      - Collapsed 3 lags: 70
      - Collapsed 2 lags: 41
      - Collapsed 1 lag: 32
    - T = 10, m = 6, q = 12:
      - Uncollapsed full: 603
      - Uncollapsed 2 lags: 323
      - Uncollapsed 1 lag: 231
      - Collapsed full: 203
      - Collapsed 3 lags: 67
      - Collapsed 2 lags: 54
      - Collapsed 1 lag: 41
  - Illustrative statement in text: for 19 regressors (6 exogenous, one predetermined, 12 endogenous) and 6 time periods, collapsing reduces instruments from 205 to 77; collapsing and limiting lag length to 2 reduces to 50.

### III. The Data — growth determinants
- The study assembles a sample of 42 growth determinants grouped into 10 categories to capture proxies for growth theories, policies, institutions, and exogenous factors.
- Variables include those suggested by the augmented neoclassical Solow-Swan model and additional variables from empirical growth literature (Durlauf and Quah (1999); Durlauf, Johnson and Temple (2005)). Further details and motivation for variable choice are described in Tsangarides (2004) (text indicates more detail follows beyond excerpt).

*Source: _wp09268 - Section 4 summarizes the results, and section 5 concludes.*

### 1. Solow-Swan determinants and human capital

### 1. Solow-Swan determinants and human capital

### Key theoretical variables (augmented neoclassical Solow-Swan model)
- Physical capital: captured through ratios of real investment to GDP.
- Human capital: captured through measures of health and educational status (such as life expectancy and school enrollment rates).
- Population: captured through population growth rates.

### Other candidate determinant categories (as used in the analysis)
- Macroeconomic stability: captured by (i) inflation and its volatility, (ii) the government budget balance, (iii) government consumption relative to GDP, (iv) debt to GDP ratios, (v) indices of exchange rate overvaluation, and (vi) exchange rate regime classification.
- Financial development: measured by the ratio of broad money to GDP and by the ratio of assets of deposit money banks to total bank assets.
- Trade regime: captured by the degree of openness and exogenous terms-of-trade changes.
- External environment: improvements in terms of trade; external regime volatility; foreign direct investment to GDP; capital flows to GPD; foreign aid as percent of GDP.
- Internal environment: agricultural productivity proxied by ratio of arable land to total area; measures of ethnic heterogeneity and ethno-linguistic diversity.
- Institutions and governance: democracy and autocracy variables; indices of civil liberties.
- Violence, war, and conflict: proxies for war prevalence, domestic conflict, and regional conflict.
- Geography and fixed factors: distance to coastline or navigable river; percentage of land area in tropics; being landlocked.
- Regional characteristics / unobserved heterogeneity: regional dummies (e.g., sub-Saharan Africa, Latin America), resource-rich country groups, country income groups.

### Data structure and variable universe
- Annual data primarily from Summers and Heston (Penn World Tables, version 6.2) and other sources.
- Averaging windows used to create panels:
  - Five-year panels (eight panels): 1961-1965, 1966-1970, 1971-1975, 1976-1980, 1981-1985, 1986-1990, 1991-1995, and 1996-2000.
  - Eight-year panels (five panels): 1961-1968, 1969-1976, 1977-1984, 1985-1992, and 1993-2000.
- Filtering and sample construction:
  - Countries with less than two observations filtered out for endogenous variables (need at least three observations to have useful moment conditions).
  - Resulting in an unbalanced, regularly spaced panel set.
- Model universe sizes (as reported):
  - For 5-year averaged data: 2^22 (4;194;304) possible models.
  - For 8-year averaged data: 2^19 (524;288) possible models.
- Baseline estimation sample: 107 countries with 593 observations over the period 1960-2005, with an average of5:5observations per country.

### Priors and time effects
- Prior probability of inclusion for each variable: 0:50.
- Posterior inclusion probability threshold to label a variable "robust": p(Zi|D)0:50 (i.e., inclusion probability above the prior).
- Time variables included as candidate regressors (rather than always demeaning) so time effects can be selected like other variables.

---

### Results — Robustness analysis using LIBMA (dynamic panel + Bayesian Model Averaging)
- Main robust determinants (5-year averaged data): initial income; investment; population growth; life expectancy; debt; openness.
- Main robust determinants (8-year averaged data): initial income; investment; population growth; (inflation enters as a new robust variable in the 8-year sample); debt and openness have inclusion probabilities less than 0:50 in the 8-year sample (life expectancy is very close with 0.47).
- Specific findings and diagnostics:
  - Initial income: negative elasticity with respect to per capita growth; strong evidence for conditional convergence.
  - Investment, population growth, and life expectancy: high inclusion probabilities in the 5-year sample (three of the remaining four Solow-Swan determinants).
  - Education: weak evidence as a second proxy for human capital.
  - Trade openness and debt-to-GDP: identified as robust growth determinants in baseline results.
  - Time variable "panel 1990" is robust with a positive coefficient for 1986-1990 (suggesting a time effect).
  - Estimated rate of convergence: in the range of 3-4.5 percent (implying, as noted in the source, that a country will need approximately 35, 23, and 15 years, respectively, to cover half the distance between its initial position and its steady state for convergence rates of 2, 3, and 4.5 percent).
- Overall interpretation:
  - Strongest evidence for the robustness of four determinants: initial income, investment, population growth, and life expectancy.
  - Strong evidence for the robustness of inflation, debt, and trade openness (though inclusion probabilities vary across averaging windows).
  - Several widely used determinants in ad hoc regressions (macroeconomic stability proxies, institutions, geography, political environment) enter with lower inclusion probabilities than the 0:50 cutoff and are therefore less robust in this analysis.

### Model uncertainty and fragility of "ad hoc" specifications
- Adding variables to the Solow determinants changes the significance and sometimes the sign of coefficients.
- Parameter significance fluctuates across studies that use different subsets of control variables.
- The analysis confirms that lessons from ad hoc specifications can be problematic and underscores the importance of incorporating model uncertainty.

### Comparisons with other methodologies (FLS and BACE; panel analogues)
- Cross-section Bayesian methods applied: Fern·ndez, Ley, and Steel (FLS, 2001a) and BACE (Sala-i-Martin, Doppelhofer, and Miller, 2004).
  - Both FLS and BACE (cross-section) also find initial income, investment, life expectancy, and debt as strong determinants.
  - Openness and population growth, robust in LIBMA, have low inclusion probabilities in the cross-section FLS/BACE results.
  - The overvaluation index enters with high inclusion probability in the cross-section FLS/BACE.
  - Differences between LIBMA and cross-section BMA methods are attributed to accounting for dynamics and endogeneity in LIBMA.
- Panel analogues ("panel FLS" and "panel BACE"):
  - Identify nine and seven variables as robust for the 5-year and 8-year data sets, respectively, and several time effects as robust.
  - Compared to LIBMA, panel FLS and panel BACE identify additional robust variables: population growth, inflation, openness, and terms of trade are identified when applying panel versions of FLS/BACE.
  - Comparing Table 3 (panel FLS/BACE) to Table 1 (LIBMA) yields six "wrongly" identified variables in Table 3 relative to Table 1; inflation, investment, openness (all endogenous) and terms of trade are examples flagged as differing.
  - Panel FLS and panel BACE methods identify many time effects as robust determinants when LIBMA does not.
  - Conclusion from comparisons: accounting for endogeneity (as LIBMA does) leads to identification of a different set of robust determinants than methods that do not fully account for endogeneity; LIBMA may better identify true effects versus proxies captured by time dummies.

---

### Methodological contributions and implications
- The LIBMA methodology combines dynamic panel estimation with Bayesian Model Averaging to address:
  - Endogeneity,
  - Omitted variable bias,
  - Model uncertainty.
- Accounting simultaneously for dynamics and endogeneity (in addition to model uncertainty) materially affects which growth determinants are identified as robust.
- Policy-relevant implication: robust empirical evidence supports emphasis on factors related to the augmented Solow-Swan framework (initial income convergence, investment, population growth, and health/life expectancy) and highlights trade openness and debt as important considerations, while many other commonly used regressors may be less consistently robust once model uncertainty and endogeneity are addressed.

*Source: _wp09268 - 1. Solow-Swan determinants and human capital (PDF chapter), IMF working paper content provided.*

### References

### _wp09268 - References

### References (bibliographic list)
- Ahn, S., and P. Schmidt, 1995, “E¢ cient Estimation of a Model with Dynamic Panel Data,” Journal of Econometrics, Vol. 68, pp. 5–27.
- Anand, S., and A. Sen, 2000, “Human Development and Economic Sustainability,” World Development, Vol. 28, pp. 2029–49.
- Arellano, M., 2003, Panel Data Econometrics, Oxford University Press.
- Arellano, M., and O. Bover, 1995, “Another Look at the Instrumental-Variable Estimation of Error Components Models,” Journal of Econometrics, Vol. 68, pp. 29–52.
- Arellano, M., 2002, “Modelling Optimal Instrumental Variables for Dynamic Panel Data Models,” CEMFI Working Paper no. 0310.
- Barro, R., 1991, “Economic Growth in a Cross Section of Countries,” Quarterly Journal of Economics, Vol. 106, pp. 223–51.
- Blundell, R., and S. Bond, 1998, “Initial Conditions and Moment Restrictions in Dynamic Panel Data Models,” Journal of Econometrics, Vol. 87, pp. 114–343.
- Brock, W., and S. Durlauf, 2001, “Growth Economics and Reality,” World Bank Economic Review, Vol. 15, pp. 229–72.
- Brock W., Durlauf S.N., and K. West, 2003, “Policy Evaluation in Uncertain Economic Environments,” Brookings Papers on Economic Activity 1, pp. 235–322.
- Chen, H., A. Mirestean, and C. Tsangarides, 2009, “Limited Information Bayesian Model Averaging for Dynamic Panels with Short Time Periods,” IMF Working Papers 09/74.
- Donald, S., G. Imbens, and W. Newey, 2008, “Empirical Likelihood Estimation and Consistent Tests with Conditional Moment Restrictions,” mimeo.
- Doppelhofer, G. and M. Weeks, 2009, “Jointness of Growth Determinants,” Journal of Applied Econometrics, Vol. 24, pp. 209–244.
- 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 (Amsterdam: North Holland).
- Durlauf, S., P. Johnson, and J. Temple, 2005, “Growth Econometrics,” in P. Aghion and S. Durlauf (eds), Handbook of Economic Growth Vol. IA (Elsevier: North Holland).
- Durlauf, S.N., Kourtellos, A., and C.M. Tan, 2008, “Are Any Growth Theories Robust?” Economic Journal, Vol. 118, pp. 329–346.
- Fern·ndez, C., E. Ley, and M. Steel, 2001a, “Model Uncertainty in Cross-Country Growth Regressions,” Journal of Applied Econometrics, Vol. 16, pp. 563–76.
- Fern·ndez, C., E. Ley and M.F.J. Steel, 2001b, “Benchmark Priors for Bayesian Model Averaging,” Journal of Econometrics, Vol. 100, pp. 381–427.
- Hayashi, F., 2009, Econometrics, Princeton University Press.
- 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, pp. 928–34.
- Ley, E. and M.F.J. Steel, 2007, “Jointness in Bayesian Variable Selection with Applications to Growth Regression,” Journal of Macroeconomics, Vol. 29, pp. 476–493.
- Ley, E., Steel, and M.F.J. Steel, 2009, “On the E§ect of Prior Assumptions in Bayesian Model Averaging with Applications to Growth Regression,” Journal of Applied Econometrics, forthcoming.
- Lucas, R., 1988, “On the Mechanics of Economic Development,” Journal of Monetary Economics, Vol. 22, pp. 3–42.
- Mankiw, G., D. Romer, and D. Weil, 1992, “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, Vol. 107, pp. 407–37.
- Mehrho§, J., 2009, “A Solution to the Problem of Too Many Instruments in Dynamic Panel Data GMM,” Deutsche Bundesbank Discussion Papers 2009/31.
- Moral-Benito, E., 2009, “Determinants of Economic Growth: a Bayesian Panel Data Approach,” World Bank Policy Research Working Paper Number 4830.
- Masanjala, W. and C. Papageorgiou, 2008, “Rough and Lonely Road to Prosperity: A Reexamination of the Sources of Growth in Africa using Bayesian Model Averaging,” Journal of Applied Econometrics, Vol. 23, pp. 671–682.
- Raftery, A.E., 1995, “Bayesian Model Selection in Social Research,” Sociological Methodology, Vol. 25, pp. 111–63.
- Romer, P. M., 1986, “Increasing Returns and Long Run Growth,” Journal of Political Economy, Vol. 94, pp. 1002–37.
- Roodman, D., 2009, “A Note on the Theme of Too Many Instruments,” Oxford Bulletin of Economics and Statistics, Vol. 71, pp. 135–58.
- 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, pp. 813–35.
- Solow, R. M., 1956, “A Contribution to the Theory of Economic Growth,” Quarterly Journal of Economics, Vol. 70, pp. 65–94.
- Tsangarides, C., 2004, “A Bayesian Approach to Model Uncertainty,” IMF Working Paper No. 04/68 (Washington: International Monetary Fund).
- Wooldridge, J.M., 2002, Econometric Analysis of Cross Section and Panel Data, 1st edition (Cambridge, MA: MIT Press).

### Tables — key reported statistics (selected entries preserved exactly)
- Table 1. Robustness of Growth Determinants — Marginal Evidence of Importance
  - 5-year averages / 8-year averages — Posterior inclusion probabilities, posterior means, and posterior standard errors reported for variables including:
    - 1 Log(initial income): 0.87, -0.211, 0.204 (first panel); 0.97, -0.202, 0.066 (second panel)
    - 2 Log(investment): 0.84, 0.147, 0.184; 0.73, 0.212, 0.077
    - 3 Log(population growth): 0.66, -0.322, 0.497; 0.61, -0.208, 0.522
    - 5 Life expectancy: 0.70, 0.007, 0.017; 0.47, 0.006, 0.001
    - 9 Openness: 0.85, 0.103, 0.217; 0.29, -0.011, 0.039
    - Panel year entries (examples): Panel 1990: 0.70, 0.023, 0.056 (first panel); Panel 1984: 0.75, -0.045, 0.012 (second panel)
  - Note: "Boxed areas indicate inclusion probabilities above 0.50."

- Table 2. LIBMA Comparison with BACE and FLS — Marginal Evidence of Importance
  - Columns for FLS prior 1, FLS prior 9, BACE with 5-year and 8-year averages.
  - Selected entries:
    - 1 Log(initial income): 1.00, -0.705, 0.056; 1.00, -0.708, 0.003; 1.00, -0.712, 0.003
    - 5 Life expectancy: 1.00, 0.042, 0.008; 1.00, 0.043, 0.000; 1.00, 0.042, 0.000
    - 7 Debt: 1.00, -0.315, 0.039; 1.00, -0.315, 0.001; 1.00, -0.320, 0.002
  - Notes:
    - "Boxed areas indicate inclusion probabilities above 0.50."
    - "Areas in yellow are common to the results in Table 1; areas in green (blue) have P(inclusion)>0.50 in Table 1 but not in Table 2 (have P(inclusion)>0.50 in Table 2 but not in Table 1)."

- Table 3. LIBMA Comparison with BACE and FLS (Panel Data Format) — Marginal Evidence of Importance
  - Multiple columns for FLS prior 1, FLS prior 9, and BACE across 5-year and 8-year averages.
  - Selected entries:
    - 1 Log(initial income): consistently 1.00 with posterior means and standard errors such as -0.286, 0.253 / -0.287, 0.064 / -0.286, 0.064 in one block; 1.00, -0.411, 0.129 in another.
    - 6 Log(inflation): entries such as 1.00, -0.086, 0.213; 1.00, -0.085, 0.045.
    - 10 Terms of Trade: 0.96, 0.260, 0.860; 0.96, 0.261, 0.742; 0.96, 0.260, 0.739.
  - Notes:
    - "Boxed areas indicate inclusion probabilities above 0.50."
    - Same color-coding note as Table 2.

### Appendix I — Monte Carlo experiment results (summary of methodology and findings)
- Objective:
  - Assess performance of LIBMA when instrument count is reduced.
  - Compute posterior model probabilities, inclusion probabilities, and parameter statistics.
- Data generating setup:
  - Universe contains 12 variables: 5 exogenous (2 time invariant), 6 endogenous, and the lagged dependent variable (predetermined).
  - Number of periods T = 6.
  - Number of individuals N varied: 50; 75; 90; and 100.
  - Instrument sets examined:
    - (i) all lags of instruments,
    - (ii) 2 lags of the instruments,
    - (iii) 1 lag of the instruments.
  - Both collapsed and non-collapsed forms of instruments used.
  - For full (collapsed) forms, moment conditions:
    - all lags: 119 (51) moment conditions;
    - 2 lags: 95 (36) moment conditions;
    - 1 lag: 73 (29) moment conditions.
- Monte Carlo specifics:
  - 500 instances of the data generating process generated.
  - Random components drawn from Normal distributions:
    - v_it ~ N(0, ^2_v) with ^2_v = 0:10;
    - eta_i ~ N(0, ^2_eta) with ^2_eta = 0:10.
  - Prior probability of inclusion for each variable = 0:50.
- Main findings:
  - Collapsing instruments improves inclusion probabilities dramatically for both included and non-included variables, especially for smaller N (improvement often by factor of 1.5 or more).
  - As sample size increases, posterior inclusion probabilities approach 1 for relevant variables; median posterior inclusion probabilities for non-relevant variables decrease with sample size.
  - Among collapsed forms, using 1 or 2 lags rather than the full set of lags generally yields higher inclusion probabilities, particularly for endogenous variables and for lower values of N; no clear distinction between 1 or 2 lags.
  - For higher N, selecting fewer lags among collapsed instruments does not dramatically improve results.
  - Collapsed instruments reduce the ratio of instruments to sample size, G/N, improving results; further reducing lags yields diminishing returns as G/N becomes smaller.
  - For the investigation in the paper, collapsed instruments using 2 lags are used.
- Parameter estimation and model selection:
  - Collapsing instruments reduces bias and variance of parameter estimates; improvements increase with sample size.
  - Stacking (uncollapsed) instruments gives better results for recovery of the true model in some cases, but even when model selection performance is poorer, BMA differentiates relevant from non-relevant variables.
  - Table A2 reports median estimated parameters versus true parameter values; Table A3 reports posterior probability statistics for the true model and recovery rates.

### Table A. Instruments for various options (T = 6; m = 5; q = 6)
- Instrument option rows with G/N reported for N = 50, 75, 90, 100:
  - Uncollapsed full: 119 instruments — G/N = 2.38, 1.59, 1.32, 1.19
  - Uncollapsed 2 lags: 95 instruments — G/N = 1.90, 1.27, 1.06, 0.95
  - Uncollapsed 1 lag: 73 instruments — G/N = 1.46, 0.97, 0.81, 0.73
  - Collapsed full: 51 instruments — G/N = 1.02, 0.68, 0.57, 0.51
  - Collapsed 2 lags: 36 instruments — G/N = 0.72, 0.48, 0.40, 0.36
  - Collapsed 1 lag: 29 instruments — G/N = 0.58, 0.39, 0.32, 0.29

_Italic: Content derived solely from _wp09268 - References (PDF)._

### Appendix II: Monte Carlo data generating process

### _wp09268 - Appendix II: Monte Carlo data generating process

### Data generating process (DGP) specification
- Universe of potential explanatory variables: 12 variables (5 exogenous variables, of which 2 are time invariant; 6 endogenous variables; and the lagged dependent variable).
- Time invariant exogenous variables (for individual i and period t):
  - (1_it  2_it)' = r_i, with Pr[r_i = (m p^{3/2} / n p^{3/2})] = 1/9; for i = 1, : : : , N; m = 0,1,2; n = 0,1,2.
  - r_i is a vector random variable with two independent and uniformly distributed elements with discrete support {0, p^{3/2}, 2 p^{3/2}}. Support size selected so variance of the resultant random variable is 1.
- Three exogenous variables (x1_it, x2_it, x3_it):
  - (x1_it x2_it x3_it)' = r_x_t with r_x_t ~ N(0, I_3); for t = 0,1, : : : , T; i = 1, : : : , N. I_3 is the three dimensional identity matrix.
- Endogenous variables (w1_it ... w6_it):
  - (w1_it ... w6_it)' = 0.4055 (w1_it ... w6_it)_{t-1} - 0.2454 v_it 1 + r_w_it ; for t = 1, : : : , T.
  - (w1_i0 ... w6_i0)' = -0.2454 v_it 1 + r_w_i0 .
  - v_it ~ N(0, σ_v^2); r_w_it ~ N(0, I_6). Here 1 denotes a vector of 1's with appropriate dimension.
  - Overall error term v_it is assumed normally distributed.
- Dependent variable generation:
  - For t = 0:
    - y_i0 = (1/(1-β)) ((i0 x_i0 w_i0)′ m + ξ_i + v_i0)
    - v_i0 ~ N(0, σ_v^2) and ξ_i ~ N(0, σ_ξ^2).
    - κ = 0.23, i0 = (1_i0  2_i0)', x_i0 = (x1_i0 x2_i0 x3_i0)', w_i0 = (w1_i0 ... w6_i0)'.
    - m = (1 0 1 1 0 1 1 1 0 0 0)' is the model selection vector: true model includes 1 time invariant variable, 2 regular exogenous variables, and 3 endogenous variables.
  - For t = 1,2, ..., T:
    - y_it = β y_{i,t-1} + (it x_it w_it)′ m + ξ_i + v_it ;
    - v_it ~ N(0, σ_v^2); ξ_i ~ N(0, σ_ξ^2).
- The theoretical R^2 of the generated models varies between 0.50 and 0.60.

### Simulation design and summary diagnostics
- Simulation settings reported: comparisons for sample sizes N = 50, 75, 90, 100; α = 0.50; and σ_ν^2 = 0.1.
- Tables reported in the appendix:
  - Table A1: "Model Recovery: Medians and Variances of Posterior Inclusion Probability for Each Variable" — reports median and variance of posterior inclusion probabilities for each variable (y_{t-1}, x1, x2, x3, x4, x5, w1...w6) under configurations: Not stacked vs Stacked; All lags used, 1 lag used, 2 lags used; for each N = 50, 75, 90, 100.
  - Table A2: "Model Recovery: Medians and Variances of Estimated Parameter Values" — reports true values, medians, and variances of estimated parameter values (BMA coefficients) for the same variable set and sample configurations as in Table A1.
  - Table A3: "Posterior Probabilities" — reports:
    - Probability of retrieving the true model (% Correct) for each N and configuration (Not stacked / Stacked; All lags used, 1 lag used, 2 lags used).
    - Posterior probability of the true model: summary statistics by N (Mean, Variance, Q1, Median, Q3) for each configuration.
    - Posterior probability ratio of true model / best among the other models (reported across the same sample and lag configurations).
- Selected simulation outcome highlights (exact values preserved from reported tables):
  - Theoretical R^2 range: 0.50 and 0.60.
  - Table A3: Probability of retrieving the true model (% Correct):
    - N=50: % Correct = 0 2 16 13 (reported across configurations).
    - N=75: % Correct = 11 15 13 11 3 (reported across configurations).
    - N=90: % Correct = 3 16 11 20 6 21 (reported across configurations).
    - N=100: % Correct = 7 20 12 27 8 23 (reported across configurations).
  - Table A3: Posterior probability of the true model — selected statistics (N=100, one configuration example reported verbatim):
    - Mean 0.017 0.169 0.064 0.282 0.029 0.218
    - Variance 0.341 0.548 0.464 0.681 0.340 0.573
    - Median 0.017 0.169 0.064 0.282 0.029 0.218
    - Q3 0.111 0.730 0.340 1.061 0.197 0.915
  - Table A1 and Table A2 report numerous medians and variances for posterior inclusion probabilities and coefficient estimates respectively; entries include variable-specific values such as:
    - For N=50, variable y_{t-1} inclusion medians and variances include values like 0.34830 .0039, 0.55600 .0229, 0.44660 .0128, 0.71920 .0394, 0.36630 .0055, 0.67040 .0288 (across different configurations).
    - For N=100, variable x3 inclusion medians/variances include 0.82910 .03020 .99850 .00490 .99270 .00920 .99900 .01110 .94150 .02160 .99930 .0041 (as reported across configurations).
  - Note: The appendix presents extensive tabulated medians and variances for each variable across sample sizes and lag/instrument configurations; the above bullets preserve representative reported numeric formats from those tables.

### Variables, definitions, and data sources (Table B1)
- Dependent variable:
  - DIFFY — Penn World Table 6.2 — Growth of real GDP per capita (2000 US dollars at PPP).
- Examples of explanatory variables (variable name — Type — Source — Definition as reported):
  - LNY0 — Endogenous/Predet — Penn World Table 6.2 — Logarithm of initial real GDP per capita (2000 US dollars at PPP).
  - LNI — Endogenous — Penn World Table 6.2 — Logarithm of real investment as ratio to GDP (2000 US dollars at PPP).
  - DIFFI — Penn World Table 6.2 — Growth of real investment to GDP (2000 US dollars at PPP).
  - LTOTED2 — Endogenous — Barro and Lee dataset — Logarithm of total average stock of years of primary and secondary education.
  - LFEXP2 — Endogenous — World Development Indicators (World Bank) — Life expectancy at birth (total) with filled in years.
  - LNINFLE — Endogenous — International Financial Statistics (IMF) — Logarithm of one plus the inflation rate.
  - BALGDP — Endogenous — World Economic Outlook (IMF) — Government balance as share of GDP, current LCU.
  - TOTGR — Exogenous — World Economic Outlook (IMF) — Terms of trade (goods and services) growth.
  - G3VLNONROL — Exogenous — World Economic Outlook (IMF) — G-3 (=US, Euro, Japan) real exchange rate volatility faced by country.
  - CENLAT — Invariant — Gallup, Mellinger, Sachs (CID datasets) — Latitude of country centroid.
  - Regional dummies: EAP, ECA, MENA, LAC, SA, SSA, OTHER — Invariant — World Bank — regional dummy indicators.
- Table B1 provides a comprehensive listing of all variables, their classification (endogenous/exogenous/invariant/predetermined), source datasets (e.g., Penn World Table 6.2, Barro and Lee dataset, World Development Indicators, International Financial Statistics, World Economic Outlook, Sambanis, Doyle and Sambanis, Gallup/Mellinger/Sachs), and short definitions as reported.

*Appendix II: Monte Carlo data generating process — source material as provided in the content unit.*

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