Question

Consider the unobserved effects panel data model for a random draw \(i\), where \(a_{t}\) denotes different year intercepts: $$\begin{aligned}y_{i t}=& \alpha_{2} d_{2}+\ldots+\alpha_{T} d T+\beta_{1} x_{i t}+\beta_{2} x_{i t 2}^{2}+\gamma_{1} z_{i 1}+\gamma_{2} z_{i 2}+\gamma_{3} z_{i 2}^{2} \\\&+\gamma_{4} x_{i t} z_{i 1}+a_{i}+u_{i t}, t=1,2, \ldots, T\end{aligned}$$ Assuming a balanced panel, write down, for a given \(i\), the CRE equation that you would estimate using the RE estimator. Which parameter estimates should be identifical to the fixed effects estimates?

Short answer

The CRE equation should incorporate means of time-varying variables. Coefficients \(\beta_1\) and \(\beta_2\) match the fixed effects estimates.

Step-by-step solution

Understanding the Panel Data Model

The given model is a panel data model where the dependent variable, \(y_{it}\), is explained by several types of regressors: \(d\) are year intercepts, \(x_{it}\) and \(x_{it2}^2\) are time-varying regressors, \(z_{i1}, z_{i2}\), and \(z_{i2}^2\) are time-invariant regressors. \(x_{it} z_{i1}\) represents interaction terms, while \(a_i\) captures unobserved individual effects and \(u_{it}\) is the error term.

Constructing the CRE Model

The Correlated Random Effects (CRE) model is designed to handle potential correlation between \(a_i\) and the regressors. The CRE equation that would be estimated using the Random Effects (RE) estimator can be formulated by adding the individual means of time-variant variables: \(\bar{x}_{i} = \frac{1}{T}\sum_{t=1}^{T} x_{it}\) and similarly for other time-varying variables like \(\bar{x}_{i2}^2\). The CRE equation is:\[ y_{it} = \alpha_2 d_2 + \ldots + \alpha_T d_T + \beta_1 x_{it} + \beta_2 x_{it}^2 + \gamma_1 z_{i1} + \gamma_2 z_{i2} + \gamma_3 z_{i2}^2 + \gamma_4 x_{it}z_{i1} + \delta_1 \bar{x}_{i} + \delta_2 \bar{x}^2_{i} + a_i + u_{it} \]

Identifying Parameters Consistent with Fixed Effects

In CRE modeling, \(\beta_1\) and \(\beta_2\) for time-varying variables \(x_{it}\) and \(x_{it}^2\) should be identical to those obtained from a fixed effects (FE) model. This is because the FE model effectively differences out the \(a_i\) term, focusing solely on within-individual variation, and these coefficients are primarily estimated based on that variation.

Key concepts

unobserved effects model

In panel data econometrics, the unobserved effects model plays a critical role in capturing individual-specific traits that do not change over time. For instance, when studying economic outcomes, each individual may have unique characteristics or preferences that cannot be directly observed but still influence the dependent variable.

This model can be mathematically represented as: \[ y_{it} = ext{{observable factors}} + a_i + u_{it} \]Here, \(a_i\) stands for the unobserved effect, which is constant across time periods for each individual, and \(u_{it}\) is the idiosyncratic error term that varies over time and individuals.

Capturing these unobserved effects correctly is essential, as ignoring them may lead to biased or inconsistent coefficient estimates and misinterpretation of the model's results in panel data analysis.

fixed effects estimator

The fixed effects estimator is a technique that is commonly used to control for unobserved heterogeneity in panel data models. It does so by removing time-invariant characteristics from the data, effectively focusing on within-individual variation over time. One might imagine it as using **differences** within each subject over time rather than differences across subjects.

This approach allows researchers to obtain unbiased estimates of the parameters of interest even when the unobserved effects \(a_i\) are correlated with the regressors. The transformation typically applied, called "demeaning" or "within transformation," subtracts the individual-specific mean from each observation. This eliminates the constant unobserved effect component, as shown in this demeaning transformation:\[ y_{it} - ar{y}_i = (x_{it} - ar{x}_i) \beta + (u_{it} - ar{u}_i) \]In this equation, \(\bar{y}_i\) represents the average outcome for individual \(i\), and similarly for other terms. Using this method provides a clear view of how changes over time affect the dependent variable, independent of the time-invariant characteristics.

correlated random effects (CRE)

The correlated random effects (CRE) model offers a middle ground between fixed effects (FE) and random effects (RE) models by allowing correlation between the unobserved effects \(a_i\) and the regressors, within a framework that's estimated using the RE approach.

The CRE model incorporates individual means of the time-varying regressors, denoted as \(\bar{x}_{i}\), into the model. This adjustment accounts for potential correlation by incorporating individual-specific averages in the regressors:\[ y_{it} = ext{{year effects}} + eta_1 x_{it} + eta_2 x_{it}^{2} + ext{{other terms}} + ext{individual means} + a_i + u_{it} \]This specification bridges the gap by allowing researchers to estimate coefficients on time-varying variables (like \(\beta_1\) and \(\beta_2\)) similarly to what is achieved using an FE estimator.

The strength of the CRE model lies in its capability to flexibly adjust for the potential endogeneity arising from correlations between the unobserved effect and the regressors, making it a robust choice in situations where traditional random effects models may fail.

time-varying regressors

Time-varying regressors are central to analyzing panel data as they change across time periods for the same individuals or entities. This variability allows researchers to capture the dynamics and causal relationships affecting the outcome variable in more nuanced ways. In the context of panel data econometrics, these regressors provide valuable insights into how changes in the explanatory variables influence the dependent variable over time.

When estimating panel data models, it's crucial to distinguish between time-varying and time-invariant regressors. Unlike time-invariant variables, which remain constant for each entity, time-varying variables require special attention since they contribute to the longitudinal aspect of the study. For example, in the exercise considered, \(x_{it}\) and \(x_{it}^{2}\) are such regressors that provide dynamic insights across time periods.By capturing these changes, panel data models can effectively be used to assess the temporal impacts, such as policy changes or economic shocks, on individual units. Moreover, the correct handling of these variables is pivotal for identifying consistent and meaningful parameter estimates, particularly in models that employ fixed or correlated random effects techniques.