Panel Regression

MacroEconometricModels.jl provides a comprehensive panel regression module following Stata's xtreg/xtiverg/xtlogit/xtprobit conventions. The module covers linear panel models, panel instrumental variables, panel discrete choice, and six specification tests with four covariance estimators.

• Linear panel (estimate_xtreg): Fixed Effects, Random Effects (Swamy-Arora), First-Difference, Between, Correlated Random Effects (Mundlak 1978), Arellano-Bond, Blundell-Bond
• Panel IV (estimate_xtiv): FE-IV, RE-IV/EC2SLS (Baltagi 1981), FD-IV, Hausman-Taylor (1981)
• Panel logit (estimate_xtlogit): Pooled, FE conditional (Chamberlain 1980), RE (Gauss-Hermite quadrature), CRE
• Panel probit (estimate_xtprobit): Pooled, RE, CRE (no FE — incidental parameters problem)
• Panel marginal effects: AME with delta-method SEs for panel logit/probit
• Specification tests: Hausman, Breusch-Pagan LM, F-test for FE, Pesaran CD, Wooldridge AR, Modified Wald
• Covariance estimators: Entity-cluster (Arellano 1987), time-cluster, two-way cluster (Cameron-Gelbach-Miller 2011), Driscoll-Kraay (1998) HAC

using MacroEconometricModels, Random, DataFrames
Random.seed!(42)

# ---- PWT: growth regression panel ----
pwt = load_example(:pwt)
df_pwt = DataFrame(pwt.data, pwt.varnames)
df_pwt.country = pwt.group_names[pwt.group_id]
df_pwt.year = pwt.time_id
# Filter valid observations and create log variables
valid = .!isnan.(df_pwt.rgdpna) .& .!isnan.(df_pwt.rkna) .& .!isnan.(df_pwt.emp) .&
        .!isnan.(df_pwt.pop) .& .!isnan.(df_pwt.hc) .& .!isnan.(df_pwt.labsh) .&
        .!isnan.(df_pwt.csh_i) .&
        (df_pwt.emp .> 0) .& (df_pwt.pop .> 0) .& (df_pwt.rgdpna .> 0) .& (df_pwt.rkna .> 0)
df_pwt = df_pwt[valid, :]
df_pwt.lngdppc = log.(df_pwt.rgdpna ./ df_pwt.pop)  # log GDP per capita
df_pwt.lnk = log.(df_pwt.rkna ./ df_pwt.emp)         # log capital per worker
pd_pwt = xtset(df_pwt, :country, :year)

# ---- DDCG: democracy and growth panel ----
ddcg = load_example(:ddcg)
df_ddcg = DataFrame(ddcg.data, ddcg.varnames)
df_ddcg.country = ddcg.group_names[ddcg.group_id]
df_ddcg.year = ddcg.time_id
valid_ddcg = .!isnan.(df_ddcg.y) .& .!isnan.(df_ddcg.dem)
df_ddcg = df_ddcg[valid_ddcg, :]
pd_ddcg = xtset(df_ddcg, :country, :year)

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Quick Start

Recipe 1: Fixed effects — growth regression

# PWT: log GDP per capita on human capital and log capital per worker
m_fe = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk])
report(m_fe)

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Recipe 2: Random effects

m_re = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:re)
report(m_re)

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Recipe 3: Hausman test (FE vs RE)

ht = hausman_test(m_fe, m_re)
report(ht)

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Recipe 4: Panel IV (FE-IV with simulated endogeneity)

# Synthetic panel with endogenous regressor and instrument
N, T_p = 50, 20
n = N * T_p
df_iv = DataFrame(id=repeat(1:N, inner=T_p), t=repeat(1:T_p, N),
                  x=randn(n), z=randn(n))
alpha_i = repeat(randn(N), inner=T_p)
df_iv.x_endog = 0.5 .* df_iv.z .+ randn(n)
df_iv.wage = alpha_i .+ 1.5 .* df_iv.x .+ 2.0 .* df_iv.x_endog .+ randn(n)
pd_iv = xtset(df_iv, :id, :t)
m_iv = estimate_xtiv(pd_iv, :wage, [:x], [:x_endog]; instruments=[:z])
report(m_iv)

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Recipe 5: Panel logit — democracy and development

# DDCG: democracy (0/1) on log GDP — Lipset modernization hypothesis
m_logit = estimate_xtlogit(pd_ddcg, :dem, [:y]; model=:re)
report(m_logit)

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Recipe 6: Specification test battery

bp = breusch_pagan_test(m_re)
report(bp)

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Linear Panel Models

Fixed Effects (Within Estimator)

The within estimator eliminates time-invariant unobserved heterogeneity by demeaning within each panel unit (Baltagi 2021):

\[\tilde{y}_{it} = \tilde{x}_{it}' \beta + \tilde{e}_{it}\]

where:

• $\tilde{y}_{it} = y_{it} - \bar{y}_i$ is the within-demeaned outcome
• $\tilde{x}_{it} = x_{it} - \bar{x}_i$ is the within-demeaned regressor
• $\beta$ is estimated by OLS on demeaned data
• Entity effects $\hat{\alpha}_i = \bar{y}_i - \bar{x}_i' \hat{\beta}$ are recovered after estimation

The model reports three R-squared variants: within (variation within entities), between (variation of entity means), and overall (total variation).

# PWT: within-country variation in GDP per capita
m_fe = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; cov_type=:cluster)
report(m_fe)

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The within R-squared measures how well human capital and capital deepening explain within-country GDP growth after removing country fixed effects. A high rho (fraction of variance due to $\alpha_i$) indicates persistent cross-country differences dominate total variation.

Two-way fixed effects absorb both entity and time effects:

m_twoway = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; twoway=true)
report(m_twoway)

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Random Effects (GLS)

The random effects estimator treats entity effects as random draws from a distribution and uses GLS with the Swamy-Arora (1972) variance component estimator:

\[y_{it} - \hat{\theta} \bar{y}_i = (x_{it} - \hat{\theta} \bar{x}_i)' \beta + (1 - \hat{\theta}) \mu + \text{error}\]

where:

• $\hat{\theta} = 1 - \sqrt{\hat{\sigma}_e^2 / (\hat{\sigma}_e^2 + T_i \hat{\sigma}_u^2)}$ is the quasi-demeaning parameter
• $\hat{\sigma}_e^2$ is from the within regression
• $\hat{\sigma}_u^2$ is from the between regression (Swamy-Arora 1972)

m_re = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:re)
report(m_re)

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RE is efficient under the assumption $E[\alpha_i \mid x_{it}] = 0$. The Hausman test evaluates this assumption.

First-Difference

The first-difference estimator removes entity effects by differencing consecutive observations:

\[\Delta y_{it} = \Delta x_{it}' \beta + \Delta e_{it}\]

m_fd = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:fd)
report(m_fd)

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FD is consistent under the same assumptions as FE but may be more efficient when errors follow a random walk.

Between Estimator

The between estimator regresses entity means on entity-mean regressors:

\[\bar{y}_i = \bar{x}_i' \beta + \bar{\alpha}_i + \bar{e}_i\]

m_be = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:between)
report(m_be)

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The between estimator exploits cross-country variation. Countries with higher average human capital and capital per worker have higher average GDP per capita.

Correlated Random Effects (Mundlak)

The CRE approach (Mundlak 1978) relaxes the RE exogeneity assumption by augmenting the RE model with group means of time-varying regressors:

\[y_{it} = x_{it}' \beta + \bar{x}_i' \gamma + \alpha_i + e_{it}\]

m_cre = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:cre)
report(m_cre)

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The slope estimates $\hat{\beta}$ from CRE are numerically equivalent to FE. The $\hat{\gamma}$ coefficients on the group means test whether the RE assumption holds — significant $\gamma$ indicates correlation between regressors and entity effects.

Arellano-Bond and Blundell-Bond

Dynamic panel estimators handle lagged dependent variables using GMM with internal instruments.

Arellano-Bond (1991) differences the equation and uses lagged levels as instruments:

# Synthetic dynamic panel (lagged DV requires careful construction)
N_d, T_d = 50, 20
n_d = N_d * T_d
df_dyn = DataFrame(id=repeat(1:N_d, inner=T_d), t=repeat(1:T_d, N_d),
                   invest=randn(n_d), output=randn(n_d))
alpha_d = repeat(randn(N_d), inner=T_d)
df_dyn.growth = alpha_d .+ 0.8 .* df_dyn.invest .- 0.3 .* df_dyn.output .+ randn(n_d)
# Create lagged variable
df_dyn.growth_lag = vcat(missing, df_dyn.growth[1:end-1])
for i in 1:N_d; df_dyn.growth_lag[(i-1)*T_d + 1] = missing; end
df_dyn = dropmissing(df_dyn)
pd_dyn = xtset(df_dyn, :id, :t)
m_ab = estimate_xtreg(pd_dyn, :growth, [:invest, :output]; model=:ab)
report(m_ab)

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Blundell-Bond (1998) adds level equations with lagged differences as instruments, improving efficiency when the autoregressive parameter is close to unity:

m_bb = estimate_xtreg(pd_dyn, :growth, [:invest, :output]; model=:bb)
report(m_bb)

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Keywords

Return Values

Panel Instrumental Variables

The estimate_xtiv function handles endogeneity in panel data through four IV strategies.

FE-IV

Within-transform all variables, then apply 2SLS on the demeaned data:

m_feiv = estimate_xtiv(pd_iv, :wage, [:x], [:x_endog]; instruments=[:z], model=:fe)
report(m_feiv)

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RE-IV (EC2SLS)

The Baltagi (1981) EC2SLS estimator quasi-demeans all variables, then uses $[\tilde{Z}, \bar{Z}_i]$ as instruments:

m_reiv = estimate_xtiv(pd_iv, :wage, [:x], [:x_endog]; instruments=[:z], model=:re)
report(m_reiv)

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FD-IV

First-difference all variables, then apply 2SLS:

m_fdiv = estimate_xtiv(pd_iv, :wage, [:x], [:x_endog]; instruments=[:z], model=:fd)
report(m_fdiv)

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Hausman-Taylor

The Hausman-Taylor (1981) estimator handles endogenous time-invariant regressors by using within-deviations of time-varying exogenous variables as instruments:

# Time-invariant variables (e.g., education level, geographic endowment)
N_ht, T_ht = 50, 20
n_ht = N_ht * T_ht
df_ht = DataFrame(id=repeat(1:N_ht, inner=T_ht), t=repeat(1:T_ht, N_ht),
                  experience=randn(n_ht))
df_ht.region = repeat(randn(N_ht), inner=T_ht)        # time-invariant exogenous
df_ht.education = repeat(randn(N_ht), inner=T_ht)     # time-invariant endogenous
alpha_ht = repeat(randn(N_ht) .+ 0.5 .* randn(N_ht), inner=T_ht)
df_ht.earnings = alpha_ht .+ 1.5 .* df_ht.experience .+ 0.3 .* df_ht.region .+ 0.8 .* df_ht.education .+ randn(n_ht)
pd_ht = xtset(df_ht, :id, :t)
m_ht = estimate_xtiv(pd_ht, :earnings, [:experience], Symbol[];
                      model=:hausman_taylor,
                      time_invariant_exog=[:region],
                      time_invariant_endog=[:education])
report(m_ht)

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Keywords

Return Values

Panel Discrete Choice

Panel Logit

The estimate_xtlogit function estimates panel logistic regression with four methods:

• Pooled: Standard logit ignoring panel structure (with optional cluster-robust SEs)
• FE (conditional): Chamberlain (1980) conditional likelihood — eliminates entity effects by conditioning on sufficient statistics. Only within-entity variation identifies coefficients.
• RE: Gauss-Hermite quadrature integration over the random effect distribution
• CRE: Mundlak-style augmentation of the RE model with group means

# DDCG: democracy transition on log GDP (Lipset hypothesis)
m_pooled = estimate_xtlogit(pd_ddcg, :dem, [:y])
report(m_pooled)

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# FE conditional logit — within-country variation only
m_fe_logit = estimate_xtlogit(pd_ddcg, :dem, [:y]; model=:fe)
report(m_fe_logit)

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# RE logit — integrates over country-level heterogeneity
m_re_logit = estimate_xtlogit(pd_ddcg, :dem, [:y]; model=:re)
report(m_re_logit)

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The RE logit coefficient on log GDP captures the effect of economic development on the probability of democracy, integrating over unobserved country-level heterogeneity. Positive and significant coefficients support the Lipset (1959) modernization hypothesis.

Panel Probit

The estimate_xtprobit function supports pooled, RE, and CRE models. No FE probit is available because there is no conditioning trick analogous to the logit case — the incidental parameters problem biases FE probit coefficients (Wooldridge 2010, §15.8).

m_probit = estimate_xtprobit(pd_ddcg, :dem, [:y]; model=:re)
report(m_probit)

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Panel Marginal Effects

marginal_effects computes AMEs with delta-method standard errors for panel logit and probit models. For RE and CRE models, the marginal effects integrate over the random effect distribution using Gauss-Hermite quadrature.

me = marginal_effects(m_re_logit)
report(me)

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Keywords (Logit/Probit)

Specification Tests

Six specification tests help choose between estimators and diagnose violations of model assumptions.

Hausman Test (FE vs RE)

Tests whether the RE assumption $E[\alpha_i \mid x_{it}] = 0$ holds (Hausman 1978). Rejection favors FE.

m_fe2 = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk])
m_re2 = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:re)
ht = hausman_test(m_fe2, m_re2)
report(ht)

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Breusch-Pagan LM Test

Tests for the presence of random effects: $H_0: \sigma_u^2 = 0$ (Breusch & Pagan 1980). Rejection suggests pooled OLS is inefficient and RE or FE is preferred.

bp = breusch_pagan_test(m_re2)
report(bp)

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F-Test for Fixed Effects

Tests joint significance of all entity fixed effects: $H_0: \alpha_1 = \alpha_2 = \cdots = \alpha_N$.

ft = f_test_fe(m_fe2)
report(ft)

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Pesaran CD Test

Tests for cross-sectional dependence in panel residuals (Pesaran 2004). Under $H_0$, residuals are uncorrelated across entities.

cd = pesaran_cd_test(m_fe2)
report(cd)

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Wooldridge AR Test

Tests for first-order serial correlation in first-differenced residuals (Wooldridge 2010). Under $H_0$, no serial correlation.

ar = wooldridge_ar_test(m_fe2)
report(ar)

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Modified Wald Test

Tests for groupwise heteroskedasticity: $H_0: \sigma_i^2 = \sigma^2$ for all $i$ (Greene 2012). Rejection suggests entity-specific error variances.

mw = modified_wald_test(m_fe2)
report(mw)

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Recommended Workflow

Covariance Estimators

All linear panel models support four covariance estimators via the cov_type keyword:

# Driscoll-Kraay standard errors for PWT growth regression
m_dk = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; cov_type=:driscoll_kraay)
report(m_dk)

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Complete Example

A full panel analysis workflow using the Penn World Table:

# Growth regression: log GDP per capita on human capital and capital deepening
m_fe_full = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk])
m_re_full = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:re)

# Hausman test: FE vs RE
ht = hausman_test(m_fe_full, m_re_full)
report(ht)

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# Breusch-Pagan: test for random effects
bp = breusch_pagan_test(m_re_full)
report(bp)

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# Diagnostics on FE model
cd = pesaran_cd_test(m_fe_full)
report(cd)

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ar = wooldridge_ar_test(m_fe_full)
report(ar)

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# CRE as robustness check
m_cre_full = estimate_xtreg(pd_pwt, :lngdppc, [:hc, :lnk]; model=:cre)
report(m_cre_full)

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The CRE group-mean coefficients test the RE exogeneity assumption. Significant group means indicate that countries with higher average human capital or capital intensity differ systematically in ways correlated with the entity effects — favoring FE over RE.

Common Pitfalls

1. Forgetting xtset. All panel estimators require PanelData created via xtset(df, :group, :time). Passing a raw DataFrame throws an error.

2. Including time-invariant regressors in FE. The within-transformation eliminates all time-invariant variables. Use RE, CRE, or Hausman-Taylor to estimate their effects.

3. Using FE probit. There is no conditioning trick for probit (unlike logit). The package correctly excludes :fe from estimate_xtprobit — use :re or :cre instead.

4. Weak instruments in panel IV. Check first_stage_f in the PanelIVModel output. Values below 10 indicate weak instruments (Staiger & Stock 1997).

5. Ignoring cross-sectional dependence. Standard cluster-robust SEs assume independence across entities. Run pesaran_cd_test and switch to cov_type=:driscoll_kraay if rejected.

6. Unbalanced panels with AB/BB. Arellano-Bond and Blundell-Bond GMM require sufficient time periods per entity for the instrument matrix. Very short panels may produce singular moment conditions.

References

• Arellano, M. (1987). Computing Robust Standard Errors for Within-Groups Estimators. Oxford Bulletin of Economics and Statistics 49(4), 431-434. DOI
• Arellano, M. & Bond, S. (1991). Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations. Review of Economic Studies 58(2), 277-297. DOI
• Baltagi, B. H. (1981). Simultaneous Equations with Error Components. Journal of Econometrics 17(2), 189-200. DOI
• Baltagi, B. H. (2021). Econometric Analysis of Panel Data. 6th ed. Springer. ISBN 978-3-030-53952-8.
• Blundell, R. & Bond, S. (1998). Initial Conditions and Moment Restrictions in Dynamic Panel Data Models. Journal of Econometrics 87(1), 115-143. DOI
• Breusch, T. S. & Pagan, A. R. (1980). The Lagrange Multiplier Test and Its Applications to Model Specification in Econometrics. Review of Economic Studies 47(1), 239-253. DOI
• Cameron, A. C., Gelbach, J. B. & Miller, D. L. (2011). Robust Inference with Multiway Clustering. Journal of Business & Economic Statistics 29(2), 238-249. DOI
• Cameron, A. C. & Miller, D. L. (2015). A Practitioner's Guide to Cluster-Robust Inference. Journal of Human Resources 50(2), 317-372. DOI
• Chamberlain, G. (1980). Analysis of Covariance with Qualitative Data. Review of Economic Studies 47(1), 225-238. DOI
• Driscoll, J. C. & Kraay, A. C. (1998). Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data. Review of Economics and Statistics 80(4), 549-560. DOI
• Feenstra, R. C., Inklaar, R. & Timmer, M. P. (2015). The Next Generation of the Penn World Table. American Economic Review 105(10), 3150-3182. DOI
• Greene, W. H. (2012). Econometric Analysis. 7th ed. Prentice Hall. ISBN 978-0-131-39538-1.
• Hausman, J. A. (1978). Specification Tests in Econometrics. Econometrica 46(6), 1251-1271. DOI
• Hausman, J. A. & Taylor, W. E. (1981). Panel Data and Unobservable Individual Effects. Econometrica 49(6), 1377-1398. DOI
• Lipset, S. M. (1959). Some Social Requisites of Democracy: Economic Development and Political Legitimacy. American Political Science Review 53(1), 69-105. DOI
• Mundlak, Y. (1978). On the Pooling of Time Series and Cross Section Data. Econometrica 46(1), 69-85. DOI
• Pesaran, M. H. (2004). General Diagnostic Tests for Cross Section Dependence in Panels. CESifo Working Paper No. 1229. DOI
• Staiger, D. & Stock, J. H. (1997). Instrumental Variables Regression with Weak Instruments. Econometrica 65(3), 557-586. DOI
• Swamy, P. A. V. B. & Arora, S. S. (1972). The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models. Econometrica 40(2), 261-275. DOI
• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. 2nd ed. MIT Press. ISBN 978-0-262-23258-6.