12. Panel Data, II Flashcards
What does persistence mean and how is persistence modelled?
Persistence refers to the idea that past outcomes or behaviors have a continuing influence on current outcomes or behaviors. In other words, persistence means that the effects of an event or decision do not vanish immediately but continue to affect the system over time.
In dynamic panel data models, persistence is typically modeled by including lagged dependent variables as explanatory variables. These lagged variables represent past outcomes and are included to capture the ongoing influence of previous periods on the current period’s outcomes
A lagged dependent variable is the value of the dependent variable from a previous time period (Yt−1). Yit=ρYit−1+δDit+ui+ϵit
* ρ: Captures how much the past outcome (Yit−1) influences the current outcome (Yit).
What are dynamic panel data models?
Dynamic panel data models include as part of their specification both lagged dependent variables and unobserved individual specific effects. These models allow for empirical modelling of dynmaics while accounting for individual-level heterogenity –> They enable os to figure out whether past behavoir directly afffects current behavoir or whether individuals are simply predisposed to behavore in one way or another.
The general dynamic panel data model is:
yi,t=γyi,t−1+Xi,tβ+ai+ui,t, Where:
* yi,t: Dependent variable for unit i at time t.
* yi,t−1: Lagged dependent variable (previous value of y).
* Xi,t: Vector of exogenous explanatory variables.
* γ,β: Coefficients to estimate.
* ai: unobservable individual-specific effects (time-invariant).
* ui,t: Random disturbance term (error).
Why is OLS biased and incosistent when used to estimate dynamic panel data models?
- The general dynamic panel data model is:
yi,t=γyi,t−1+Xi,tβ+ai+ui,t - For the disturbance term, we assume that:
E[ui,t |yi,t-1,…,yi,1, xi,t, xi,t-1,…,xi,1]=0. Mean zero conditional assumption aka the distibance term, ui,t, has an expected value of zero, conditional on all available explanatory variables (including the lagged dependent variable and other explanatory variables). This means that the disturbance term should not be correlated with the explanatory variables, ensuring that the model’s estimates are unbiased. - A lot of studies estimate a variation of the general dynamic panel data model, yet they ignore the individual-spefic effect ai.
yi,t=γyi,t−1+Xi,tβ+ui,t, where ui,t=ai+ui,t
Thus the lagged dependent variables becomes correlated with the disturbance term, since it is correlated with ai by construction - violated the zero mean conditional assumption. Therefore we should be cautios when drawing inferences based on paramter estimates produces by cross-sectional regression models. - Even standard panel data estimators are not approproiate for estimating the general dunamic model, even though the typical fixed effects estimator removes individual-specific effects, it still leads to biased and inconsistent estimates. The problem arises because after the transformation, there remains a correlation between the transformed lagged dependent variable and the transformed error term. Specifically, the lagged dependent variable is correlated with the error term, and this correlation causes bias. The bias is of order 1/T and so remains a problem in typical panel data sets where T is small.
Why are instrumental variables used in dynamic panel data models, and how do they address the issues with individual effects and lagged dependent variables?
Instrumental variables are used in dynamic panel data models to address the issue of correlation between the lagged dependent variable and the disturbance term after removing individual-specific effects. Transforming the equation eliminates correlation with individual effects but creates new endogeneity problems: a correlation between the lagged dependent variable and the disturbance in the transformed equation. To address this, instrumental variables (IV) are used.
What is the anderson-Hsiao (instrumental variable( estimator)?
Yet, even after differencing, Δyi,t−1 is correlated with Δui,t because yi,t−1 is a function of ui,t−1. Therefore the AH estimator uses lagged variables as instruments to address this correlation:
The Anderson-Hsiao estimator uses first-differencing to eliminate the problem of correlation between the lagged
yit−yit−1=γ(yit−1−yit−2)+β(xit−xit−1)+(uit−uit−1)
This is rewritten as:
Δyit=γΔyit−1+βΔxit+Δuit
While first-differencing removes individual effects, the lagged dependent variable still correlates with the disturbance term. To address this, instrumental variables, such as:
1. Levels instrument: yi,t−2, assuming no serial correlation in ui,t
2. Differences instrument: Δyi,t−2, the differenced lag.
While both approaches solve the problem of endogeneity, they come with limitations.
Problems
The Anderson-Hsiao estimator can suffer from issues like large variances and biases, especially with smaller samples and when the lagged dependent variable is close to 1. The first-differencing estimator can also have singularities. Improvements have been made using the GMM framework, which uses additional instruments provided by the panel structure, leading to more efficient estimators than the original Anderson-Hsiao estimators
What is the first difference GMM estimator and how does it work for dynamic panel data?
The first-difference GMM Estimator is a method that removes individual effects from the model to address the correlation between lagged dependent variables and unobserved individual effects. It works as follows:
- The dynamic panel model is first differenced to eliminate individual-specific effects ai. For the model:
yit=γyi,t−1+Xitβ+ai+uit.
The first-differenced version is:
Δyit=γΔyi,t−1+ΔXitβ+Δuit - Instrumental Variables (IV):
After differencing, there remains correlation between the lagged dependent variable and the disturbance term. The solution is to use instruments, which are lagged values of the endogenous variables:
E[zitΔuit]=0, where zit are valid instruments like the lagged dependent variable yi,t−2 and explanatory variables Xit - Moment Conditions:
These instruments provide moment conditions for GMM estimation:
E[zit(yit−Xitβ)]=0
The optimal GMM estimator is then derived using these moment conditions.
What are the challenges of the First-Difference GMM Estimator, and how do subsequent improvements address them?
What is the Beck-Katz standard for panel data and how does it address common econometric problems?
The Beck-Katz standard is a widely used method for analyzing pooled time-series cross-section data (panel data). Pooling helps deal with the “few cases, many variables” problem by increasing observations and degrees of freedom + make it possible to control fro exogenous schoks (time fixed effects) and reduce omitted variables bias (unit fixed effects).
There are four four common panel data issues that Beck and Katz try to adress with their method:
1. Autocorrelation (serial correlation): This occurs when errors are correlated across time periods, meaning the error at time t is related to the error at time t−1. The Beck-Katz method accounts for this by including lagged dependent variables to eliminate the correlation.
2. Heteroscedasticity: This refers to the issue where the variance of the errors is not constant across units or time periods. Some units (e.g., countries with higher GDP) may exhibit higher error variance than others. Beck-Katz handles this by adopting robust standard errors to account for such differences in error variance.
3. Cross-sectional correlation: Errors can be correlated across different units (e.g., countries), often due to common external shocks (such as a global economic crisis). The Beck-Katz approach controls for this by including unit fixed effects, thus isolating the effect of each unit.
4. Non-spherical errors: This problem occurs when the errors are both autocorrelated and heteroscedastic simultaneously. The Beck-Katz method addresses this by using fixed effects models and panel-corrected standard errors, which correct for both serial and cross-sectional correlations in the errors.
What is 1/4 potential sources of problems in panel data anlysis with a lagged dependent variable and period and unit dummies that Plümper et. al. adresses?
Absorption of cross-sectional variance by unit dummies
Estimation with unit fixed effects can
1. Lead to the absorption of most of the theoretical interesting cross-sectional varianace in the data: When you include unit fixed effects, you are effectively controlling for all differences between the units. These fixed effects account for cross-sectional variance, meaning they absorb (eliminate) all the variations between the units that do not change over time. The result is that you are no longer accounting for between-unit differences, and the analysis focuses only on the within-unit changes over time. This can be problematic, especially when the cross-sectional variation is theoretically important.
2. Make it impossible to estimate the effect of time invariant exogenous variables and serverely biases the estimate of party time invariant variables: Unit fixed effects make it impossible to estimate the effects of time-invariant exogenous variables—variables that do not change over time. Additionally, partly time-invariant variables, like some institutional features, might get severely biased estimates because the fixed effects model tries to isolate the within-country variation and ignores between-country differences.
Conclusion: If your theory predicts an influence of a level-measure X (time-invariant?) on a change-measure (time-varying?) Y, you must not include unit fixed effects. This is because unit fixed effects will absorb the variability in the level-measure X between countries, effectively eliminating the ability to estimate its effect on the change-measure Y.
What is 2/4 potential sources of problems in panel data anlysis with a lagged dependent variable and period dummies that Plümper et. al. adresses?
Absorption of time-series variance by the lagged dependent variable and period dummies
Estimation with lagged dependent variables and time fixed effects can
1. Absorb a large portion of the time trend in the dependent variable (like government spending), leaving less variance for the explanatory variables to explain.**
2. Make it difficult to identify the true effects of other independent variables because much of the trend is attributed to the lagged dependent variable and period dummies.
Conclusion: If you include a lagged measure of your dependent variable or period dummies, you may absorb time trends in the dependent variable
What is 3/4 potential sources of problems in panel data anlysis with a lagged dependent variable and period and unit dummies that Plümper et. al. adresses?
Misspecification of the lag structure
In panel data analysis, lag structure refers to the time delay between changes in the independent variable(s) (X) and their subsequent impact on the dependent variable (Y). In the context of lagged dependent variables (Yit-1), the assumption is that the previous period’s value of the dependent variable influences the current value.
The issue is that it is a common to assume uniform lags ( the same time delay across all units/countries). This assumes that the effect of a given change in an independent variable (X) on the dependent variable (Y) occurs in the same timeframe across all units. However, in reality, the time it takes for changes in X to affect Y may differ by country or unit. If researchers use a uniform lag structure, this could distort the findings because it overlooks the varying response times across different countries or units.
Conclusion: The magnitude of the effect of lagged DVs may differ between units (→ it may be problematic to assume uniform lags).
What is 4/4 potential sources of problems in panel data anlysis with a lagged dependent variable and period and unit dummies that Plümper et. al. adresses?
Neglect of parameter slope heterogenity
In panel data analysis the assumption that the relationship between an independent variable and the dependent variable stays the same across all periods can be unrealistic, as the impact of such variables may change over time. Political dynamics can evolve, and not accounting for this can lead to biased results.
Researchers should allow for parameter heterogeneity, meaning the coefficients of key variables can vary across time periods. This can be done by including period dummies or interacting the variable of interest with time periods to capture shifts in its effect over time.
Conclusion: For longer observation periods, slopes of key variables and error term variances may change overtime (time dependence of coefficients)
What is the difference between fixed effects models and random effects models?
- FE models: Focus on explaining within-unit variation by controlling for time-invariant characteristics of the units (countries, individuals). They remove differences across units (such as unobserved characteristics) by including unit-specific intercepts, meaning they cannot account for variation that doesn’t change over time within the units.
- RE Models: Assume that there is variability across units but treat this variability as random. RE models include random coefficients that allow for variation between units, providing more flexibility in modeling contextual and heterogeneity effects. RE models estimate both within and between unit effects, making them more generalizable.
What is the problem of hierarchical data in panel analysis, and how does the Random Effects (RE) model address it?
Hierarchical data structures, such as individuals nested in countries or outcomes over time nested in individuals, lead to correlation between observations within the same higher-level entity. Standard regression models assume that residuals are independent, which is incorrect for hierarchical data. This can cause biased estimates and incorrect standard errors.
The RE model solves this problem by partitioning the residual variance into two components:
1. Higher-level variance (between units).
2. Lower-level variance (within units).
By separating these two sources of variance, the RE model allows for a more accurate estimation of both individual-level and group-level effects, and it corrects the standard errors to account for the hierarchical structure.
Model formulation:
Yij = Bo + B1X1ij + B2zj + (uj+eij), where
* Yij = The dependent variable (outcome) for observation i within higher-level unit j.
* B0 = Intercept
* X1ij = Covariate at the lower level (time-varying)
* Zj = Covariate at the higher level (time-invariant)
* Uj = Random effect for the higher-level unit. This represents the unobserved characteristics of the higher-level unit that influence the outcome. The random effect uj is allowed to vary across higher-level units, reflecting the fact that each higher-level unit might have its own unique influence on the outcome.
* Eij = Residual at the lower level reflecting the variation in outcomes that is not explained by the covariates.
What is the problem of omitted variable bias and endogeneity in RE models, and how does it arise?
The problem of omitted variable bias and endogeneity in RE models stems from the exogeneity assumption made in the model. Specifically, RE models assume that the residuals are independent of the covariates, both at the higher level (between) and lower level (within), which is represented by the following assumption:
1. E(uj|xij, zj) = 0
2. E(eij|xij, zj) = 0
In practical terms, this assumption means that the residuals at both levels (higher level uj and lower level
eij) are assumed to be uncorrelated with the covariates
xij (time-varying) and zj (time-invariant).
However, this assumption often doesn’t always hold in practice. The main issue is that covariates like xij (which are time-varying) can have both within-group effects (how the covariate changes over time within a unit, like an individual) and between-group effects (how the covariate varies between higher-level units, like countries). When both these effects are present, it leads to a situation where the residuals are correlated with the covariates, violating the assumptions of the RE model.
