Panel Data Analysis

Panel Data

• Cross-sectional data involves observing 'n' units at a single point in time.
• Panel data involves observing units repeatedly over a period of time.
• Panel data provides the ability to analyze behavioral responses over time, such as those resulting from interventions.
• Panel data allows controlling for unobserved, time-invariant characteristics of units, which results in addressing omitted variables bias more convincingly.

Types of Panel Data

• Observations for all variables across all units in all time periods describe a balanced panel.
• Data is called an unbalanced panel if variables are not observed for every unit in every time period.
• When 'n' (number of units) is much larger than 'T' (number of time periods), the data is referred to as a micro panel.
• When 'T' is much larger than 'n' the data is referred to as a macro panel.
• Tools discussed work for unbalanced micro panels if missing data is random and not subject to sample selection.
• Special tools covered in time series analysis are needed for the econometric analysis of macro panels where T ≫ n.

Traffic Fatalities and Alcohol Taxes

• The dataset is a balanced micro panel of 48 US continental states from 1982–1988 with 336 observations.
• Focus lies on calculating the average causal effect of beer tax (measured in 1988 dollars) on traffic fatality rate.
• FatalRate is the number of traffic deaths per 10,000 inhabitants annually.
• A core question is whether higher alcohol taxes reduce traffic fatalities and serve as an effective policy to enhance road safety.

OLS Regression Interpretation

• An OLS regression suggests 0.15 more traffic fatalities per 10,000 inhabitants for each $1 increase in beer tax.
• A one standard deviation increase in beer tax (SD = 0.48) is correlated with a 0.13 standard deviation rise in fatality rate (SD = 0.57).
• The effect was statistically insignificant, with |t| = 1.15 < 1.64 and p = 0.269 > 0.1.
• OLS regression is expected to have omitted variables bias from factors influencing traffic fatalities: automobile quality, road quality, cultural attitudes on drinking/driving, and car density.

Before and After Comparison

• Adding suitable controls can help avoiding omitted variables bias.
• Confounders necessitate observation for meaningful control.
• An alternative approach compares panel data before and after to control for time-invariant unobserved heterogeneity across units.
• The core panel data model is FatalRateit = βo + β₁ BeerTaxit + Ziβ2 + Uit, where Zi represents time-invariant unobserved factors.
• Changes in FatalRate over time cannot be driven by Zi, because Zi does not vary over time.
• Examining fatality rates at the start and end of the period (1982 and 1988) results in FatalRatei,1982 = β0 + β1BeerTaxi,1982 + Z′iβ2 + ui,1982 and FatalRatei,1988 = β0 + β1BeerTaxi,1988 + Z′iβ2 + ui,1988.
• The alteration between 1988 and 1982 removes the effect of Zi which are constant omitted variables, expressed as FatalRatei,1988 - FatalRatei,1982 = β1(BeerTaxi,1988 – BeerTaxi,1982) + ui,1988 – ui,1982.
• Before-and-after comparisons account for omitted variable bias from time-invariant confounders unlikely to change significantly between 1982 and 1988, like cultural attitudes toward drinking and driving.

OLS in Differences and Interpretation

• A regression constant adding to difference models shows time trend in traffic fatalities across states due to progress in automobile safety and medicine
• Comparison covering 1982–1988 found the predicted negative effect of beer taxes on traffic fatalities.
• A $1 increase in beer tax in a state corresponds to a one-death decrease per 10,000 residents on average.
• This effect has economic meaning, one standard deviation higher tax sees 0.88 fall in fatality
• Significance lies at 1% with specific stats.
• Validity of causal effect depends on absence of time-varying factors affecting fatality and correlating with beer tax.

Linear Regression Model with Unit Fixed Effects

• Before-and-after comparisons exclude data from 1983–1987.
• Using every observation, the linear regression model reads Yit = β0 + X′itβ1 + Z′iβ2 + uit.
• In this model, Xit represents time-varying regressors.
• Key regression model definition is αi ≡ β0 + Z′iβ2 is Yit = αi + X′itβ1 + uit.
• Here, αi are unit i's fixed effect.
• Unit fixed effects in the application are state fixed effects.

Least-Squares Dummy Variables Estimation

• In the linear regression model Yit = αi + X′itβ1 + uit, unit fixed effect αi is a specific regression constant.
• To estimate the model, one can run an OLS regression that includes dummies D2,...,Dn, remembering the dummy variable trap.
• Formula is Yit = β0 + X′itβ1 + γ2D2i +...+ γnDni + uit, where Dji = 1 for j = i and Dji = 0 otherwise.
• Estimation includes k coefficients of time-varying regressors and (n – 1) dummy coefficients.

Unit Demeaning (Within Transformation)

• The linear regression model with fixed effects is again considered, Yit = αi + X′itβ1 + uit.
• The equation's sides are averaged over time, giving Ȳi = αi + X̄′iβ1 + ūi, with variable definitions.
• Unit demeaning involves taking the difference to eliminate the unit fixed effect: (Yit − Ȳi) = (Xit − X̄i)′β1 + (uit − ūi).
• With unit-demeaned variables, this condenses to Y eit = X e′β + ueit.

OLS Regression after Unit Demeaning

• Analogous to the before-and-after comparison,the unit fixed effect disappears: Y eit = Xe′β1 + ueit, controls unobserved variables invariant over time.
• All variables that are time-invariant disappear after unit demeaning, which makes estimating their effect impossible.
• Dummy model holds true because time-invariant variables have perfect multicollinearity with dummies.
• Any Xit = Xi follows Xi = X1D1i + X2D2i +...+ XnDni resulting perfect multicollinearity with unit dummies D1, D2,...,Dn.

Comparison of Regression Types

• Estimating equation yields an OLS regression from the model of least-squares dummy variables.
• Two observation data provides two co-efficient vectors after and before the comparison.
• To control effects, experts can run an OLS regression over the unit model.
• Goal to estimate time dummies for the fixed effect.
• Ordinary least squares helps to estimate effects in least squares dummy variables where â₁ = βo and â¡ = βo + y¡ for each i ≠ 1.

OLS Regression of the Unit-Demeaned Model

• The unit-demeaned model uses OLS estimator regressing the demeaned 'Y' on the demeaned 'X', is the fixed-effects or FE estimator
• Vectors and matrices of variables should be defined.

FE Estimator

• The unit-demeaned regression model in matrix form is: Y = X’β₁ + U
• FE estimator minimizes the sum of the residuals with: β̂₁FE = (X'X)⁻¹X'Y
• Desirable sample properties will be apparent if X has full rank and X'X is thus invertible.
• FE assumption results in in unbiased, consistent, asymptotically normal outcomes with causal inference

FE Assumption 1

• Exogenous regressors are in the unit-demeaned regression model means assuming a conditional mean-zero unit-demeaned error.
• Represented as E(u eit |Xeit) = 0 for each i = 1,...,n and t = 1,...,T
• Exogeneity is violated whenever time-varying determinants of a regressand correlated with the regressors are ommitted.
• When excluded those act as confounders introducing omitted variables bias.
• Omission of any time-invariant determinants does not introduce this bias.

Exogeneity in FE vs OLS

• Unit-demeaned error and regressors contain past, present, and future values.
• This is why first FE assumption, the error correlating with the past and present, will be violated.
• Exogeneity assumption is vital for FE regression with panel data given it being stronger than for cross-sectional data.
• The beer tax would be violated as well with a strong fatality relationship.
• All regressors must have changes with no change to the regressand over time.

FE Assumption 2

• Second FE says variables must be iid through all units.
• This means joint distribution must be of the same and independent units.
• Autocorrelation measurement allows for same variable unit usage.
• Panel and time series data have autocorrelation since a state beer tax only changes with time.
• Sampling units becomes automatic if following random sampling from a population over time.

FE Assumptions 3 & 4

• FE follows the OLS assumption with assuming the rareness in outliers.
• Finite fourth variable moment is assumed for each j = 1,..., k, and i = 1,...n, and t = 1,...T.
• The fourth FE assumption eliminates multicollinearity.
• The matrix becomes linearly independent if assuming full rank and then is invertible.

FE Assumptions

• Assumes exogeneity of the regressors for each observation.
• It includes distribution across units which means random sampling.
• Raress of Outliers and finite moments across multiple units
• Multicollinearity absence where columns become linear.
• An iid and exogeneity unit implies strong exogeneity.

FE and OLS Estimators Relationship

• FE estimators have standard sampling like that of the OLS.

Unbiasedness of the FE Estimator

• Consider the FE estimator
• FE1 and FE2 imply strong exogeneity.

Consistency of FE Estimator

• Define X as the T x k matrix and U as the T-dimensional vector for unit i
• LLN yields equations
• On the other hand, the LLN combined with gives:

Sampling Distribution of the FE Estimator

• Under FE assumptions, the estimator is not biased.
• FE becomes a consistent model.
• Asymptotically becomes a normal model.

Cluster-Robust Standard Errors

• Different periods create autocorrelated errors.
• Errors to both create error estimations

Regression Analysis

• Pooled OLS regressions show statistically significant amounts.
• Fixed analysis effect leads to statistically significant results shown by theory.
• Rising beers rates affects traffic in states.
• Time invariant rates are mostly different across states.

FE regression

• Includes state fixed effects, controlling variables between states but remaining constant by time, quality, landscape, etc.
• Time fixed effects control variables across time but affect states equally.
• Alcohol taxes are the time-varying factors that change state alcohol taxes.

Model specification

• Identify regressor
• Think of missing factors
• control variables to diminish variation
• Models with additional variable testing

Regression Analysis: Cont'd

• If the F test is accurate, time factors are most signicant
• Regressors and traffic laws end up insignificant.
• policy doubles then becomes accurate of alcohol.

Threats to Validity

• FE Regression can still suffer.
• Confounding factors effect beer tax
• Alcohol tax have great impacts on effects
• Alcohol abuse is plausible at tax being higher.