Pooled Regression Model

• Models for Panel Data: Pooled Regression Model
• Naceur Khraief, Tunis Business School - 2025

Pooled Regression Model

• This is a basic approach to analyzing panel data, which includes both cross-sectional and time-series dimensions.
• Assumes a constant intercept and constant slopes across all cross-sectional units like countries or firms.
• It ignores heterogeneity among entities and treats panel data as a large pooled dataset.
• Error variance is constant across time and entities, known as homoscedasticity.
• Errors are not correlated across time, indicating no autocorrelation.
• It assumes all differences between individuals are captured by regressors and that there are no entity-specific effects.
• Uses standard OLS regression, making it simple to estimate.
• Functions effectively if there is no significant heterogeneity across entities.
• Suitable for short panels, such as those with a few time periods.
• Disadvantages include ignoring individual and time heterogeneity, potentially leading to biased estimates if entity-specific effects are present.
• There is a risk of omitted variable bias if unobserved individual effects correlate with regressors.
• May violate key assumptions such as heteroskedasticity and autocorrelation in time-series data.
• Best used when entities (cross-sections) don't show significant differences.
• Fixed effects (FE) or random effects (RE) models are unnecessary based on statistical tests like the F-test for FE or the Breusch-Pagan LM test for RE.

Model Assumptions and Equations

• Analysis begins by assuming the simplest model version, without heterogeneity among groups.
• α₁ = α₂ = ... = αN, expressing equality of alpha across all groups
• yit = xitβ + α + εit, with i = 1, ..., N and t = 1, ..., T, representing the pooled model equation
• OLS is the efficient estimator under classical assumptions: zero conditional mean of εit, homoscedasticity, uncorrelatedness across observations i, and strict exogeneity of xit.

Stacked Observations and Estimators

• Stack T observations for individual i in a single equation: yi = Xiγ + εi.
• γ = [α, β']' now includes the constant term.
• If εit is well-behaved, then E(εit|Xi) = 0, t = 1, 2, ..., T, E(εiεi'|Xi) = σ²IT, and E(εiεj'|Xi) = 0, if i ≠ j.
• The most efficient estimator of γ is γ = (X'X)⁻¹X'y.
• The estimated variance is Var(γ) = s²(X'X)⁻¹, with s² = (e'e) / (NT - 1), where e = y - Xŷ.

Robust Covariance Matrix Estimation & Bootstrapping

• It is likely that there is correlation across observations such that E(εiεi'|Xi) = Ωi.
• Asymptotic covariance matrix estimated as Asy. Var(γ) = (X'X)⁻¹ (Σᵢ X'εiεi'Xi) (X'X)⁻¹.
• With heteroscedasticity across individuals, yi = Xiδ + ωi.
• The disturbance vector ωi consists of εit plus omitted components.
• Var(ωi|Xi) = σ²IT + Σi = Ωi.
• The ordinary least squares estimator of δ changes in this setting.

Equations for Estimators and Covariance Matrices

• δ = (X'X)⁻¹X'y = [Σ Xi'Xi]⁻¹ [Σ Xi'yi].
• The true asymptotic covariance matrix is expressed in terms of the probability limit:
• Asy. Var[δ] = (1/n) plim [Σ Xi'Xi]⁻¹ plim [Σ Xi'ωiωi'Xi] plim [Σ Xi'Xi]⁻¹.
• Alternative matrix estimates are introduced using within-group vectors of T residuals.

Robust Estimation Using Group Means

• The pooled regression model is estimated using the sample means of the data.
• Implied regression model obtained by pre-multiplying each group by 1/Ti', where i' is a row vector of ones.
• Equations for group means:
  • (1/T)i'yi = (1/T)i'Xiβ + (1/T)i'ωi
  • yi = xiβ + wi
  • E[wi] = 0 and Var(wi|xi) = σ²i = (1/T²)i'Σii, with Σi unspecified.
• Heteroscedastic regression relies on the White estimator for appropriate inference.
  • White estimator uses OLS but calculates an alternative standard error for heteroskedasticity
  • Variance can be calculated with a heteroscedasticity-robust standard error.

Estimation with First Differences

• First differencing is a method of estimation.
• The process transforms latent heterogeneity out of the model with: yiₜ = cᵢ + xiₜβ + εiₜ
• This includes first differences equation with ∆yiₜ = ∆ci + (∆xiₜ)'β + ∆εiₜ
• ∆yiₜ = (∆xiₜ)'β + εiₜ - εi,ₜ₋₁
• ∆yiₜ = (∆xiₜ)'β + uiₜ

Advantages, Disadvantages, and Technical Issues

• Advantage: Removes latent heterogeneity whether fixed or random effects apply.
• Disadvantage: Differencing removes time-invariant variables; applications can be unhelpful like the "impact of Education on the Wage"
• Technical issues:
  • Trade-off between cross-observation correlation ci and disturbance uit = εit − εit-1.
  • The new disturbance uit is autocorrelated where ∆xi and ∆εi could be correlated using feasible GLS.

Within- and Between- Groups Estimators

• A pooled regression model has three possible formulations:
  • Original: yiₜ = α + xiₜβ + εiₜ where i = 1...N and t = 1...T
  • Term of deviations from group means: yiₜ - ȳi = (xiₜ − x̄i)'β + (εiₜ − ε̄i) where i = 1...N and t = 1...T
  • In terms of the group means (between deviation): ȳi = α + x̄'i + ε̄i

Pooled OLS and Sums of Squares

• Where ẞ is estimated by pooled OLS using total sum of squares and cross products:
  • Sₓₓᵀ = ΣᵢΣₜ (xiₜ − x̄)(xiₜ − x̄)'
  • Sₓᵧᵀ = ΣᵢΣₜ (xiₜ − x̄)(yiₜ − ȳ)
  • x̄ = 1/NT ΣᵢΣₜ xiₜ and ȳ = 1/NT ΣᵢΣₜ yiₜ

(b) Within-Group's sums

• The moments matrices for (b) are within-group's (I.E., deviations from the group means) sums of squares and cross products:
  • Sₓₓᵂ = ΣᵢΣₜ (xiₜ − x̄i)(xiₜ − x̄i)'
  • Sₓᵧᵂ = ΣᵢΣₜ (xiₜ − x̄i)(yiₜ − ȳi)

Means over groups

• For (c), the means of group mean are the overall mean, where moment matrices are between-groups sums of squares and cross products:
  • Sₓₓᴮ = Σᵢ T(x̄i − x̄)(x̄i − x̄)'
  • Sₓᵧᴮ = Σᵢ T(x̄i − x̄)(ȳi − ȳ)
  • The following are easy to verify that:
    • Sₓₓᵀ = Sₓₓᵂ + Sₓₓᴮ
    • Sₓᵧᵀ = Sₓᵧᵂ + Sₓᵧᴮ

Least Squares Estimator

• Three possible least squares estimator of ẞ correspond to these decompositions:
  • The least squares estimator(a) in the pooling regression is: β̂ᵀ = (Sₓₓᵀ)⁻¹(Sₓᵧᵀ) = (Sₓₓᵂ + Sₓₓᴮ)⁻¹(Sₓᵧᵂ + Sₓᵧᴮ)
  • The within-groups estimator(b) from is: β̂ᵂ = (Sₓₓᵂ)⁻¹(Sₓᵧᵂ)= (ΣᵢΣₜ (xiₜ − x̄i)(xiₜ − x̄i)')⁻¹ (ΣᵢΣₜ (xiₜ − x̄i)(yiₜ − ȳi)), is the least squares dummy variable (LSDV) estimator.
  • The alternative estimator(c) would be the between-groups estimator β̂ᴮ =(Sₓₓᴮ)⁻¹(Sₓᵧᴮ), this is the least squares estimator based on the N sets of group means.