Question

Consider the following linear panel data model: 10.125 \(^{y_{i t}}=x_{1, i t}^{\prime} \beta_{1}+x_{2, i t}^{\prime} \beta_{2}+w_{1, i}^{\prime} \gamma_{1}+w_{2, i}^{\prime} \gamma_{2}+\alpha_{i}+u_{i t}\), where \(w_{k, i}\) are time-invariant and \(x_{k, i t}\) are time-varying explanatory variables. The variables with index \(1\left(x_{1, i t}\right.\) and \(\left.w_{1, i}\right)\) are strictly exogenous in the sense that \(E\left\\{x_{1, i t} \alpha_{i}\right\\}=0, E\left\\{x_{1, i s} u_{i t}\right\\}=0\) for all \(s, t\), \(E\left\\{w_{1, i} \alpha_{i}\right\\}=0\) and \(E\left\\{w_{1, i} u_{i t}\right\\}=0\). It is also assumed that \(E\left\\{w_{2, i} u_{i t}\right\\}=0\) and that the usual regularity conditions (for consistency and asymptotic normality) are met. (a) Under which additional assumptions would OLS applied to ( \(\underline{10.125}\) ) provide a consistent estimator for \(\beta=\left(\beta_{1}, \beta_{2}\right)^{\prime}\) and \(\gamma=\left(\gamma_{1}, \gamma_{2}\right)^{\prime}\) ? (b) Consider the fixed effects (within) estimator. Under which additional assumption(s) would it provide a consistent estimator for \(\beta\) ? (c) Consider the OLS estimator for \(\beta\) based upon a regression in first- differences. Under which additional assumption(s) will this provide a consistent estimator for \(\beta\) ? (d) Discuss one or more alternative consistent estimators for \(\beta\) and \(\gamma\) if it can be assumed that \(E\left\\{x_{2, i s} u_{i t}\right\\}=0\) (for all \(s, t\) ), and \(E\left\\{w_{2, i} u_{i t}\right\\}=0\). What are the restrictions, in this case, on the number of variables in each of the categories? (e) Discuss estimation of \(\beta\) if \(x_{2, \text { it }}\) equals \(y_{i, t-1}\), (f) Discuss estimation of \(\beta\) if \(x_{2, i t}\) includes \(y_{i, t-1}\). (g) Would it be possible to estimate both \(\beta\) and \(\gamma\) consistently if \(x_{2, i t}\) includes \(y_{i, t-1}\) ? If so, how? If not, why not? (Make additional assumptions, if necessary.)

Short answer

Consistency of OLS requires exogeneity. Fixed effects and first-difference estimators need strict exogeneity of \(x_{1,it}\), \(x_{2,it}\). Using GMM or 2SLS methods can address endogeneity due to lagged dependent variables.

Step-by-step solution

Understanding OLS Consistency

The Ordinary Least Squares (OLS) estimator is consistent if the regressors are exogenous and the error term has zero mean conditional on the regressors. For OLS to be consistent in this model, assume that the regressors are not correlated with the error term, that is, for some type of exogeneity.

OLS Assumptions for Consistency

To ensure OLS provides consistent estimates of \beta and \gamma, we need: 1) Strict exogeneity of all regressors with respect to the error term: \(E[x_{1,it} u_{it}] = 0\), \(E[x_{2,it} u_{it}] = 0\), \(E[w_{1,i} u_{it}] = 0\), \(E[w_{2,i} u_{it}] = 0\). 2) No perfect multicollinearity. 3) Homoscedasticity and no autocorrelation of the error term.

Assumptions for Fixed Effects Estimator

The fixed effects (within) estimator removes time-invariant effects by differencing. It will provide consistent estimates for \beta if: 1) Time-varying regressors are strictly exogenous conditional on \alpha_i: \(E[x_{1,it} u_{it}] = 0\), \(E[x_{2,it} u_{it}] = 0\). 2) \(w_{k,i}\) are not correlated with the error term: \(E[w_{1,i} u_{it}] = 0\), \(E[w_{2,i} u_{it}] = 0\).

Assumptions for First-Differences OLS

The first-differences estimator removes individual effects by differencing. It will be consistent for \beta if: 1) The error term follows a random walk or doesn’t exhibit serial correlation: \(E[(u_{it} - u_{it-1}) | x_{1,it}, x_{1,it-1}, x_{2,it}, x_{2,it-1}] = 0\). This implies \(x_{2,it}\) is strictly exogenous: \(E[x_{2,it} u_{it-1}] = 0\).

Alternative Consistent Estimators

If \(E[x_{2,is} u_{it}] = 0\) and \(E[w_{2,i} u_{it}] = 0\), instrument variable techniques like Two-Stage Least Squares (2SLS) can be applied. Restrictions on number of variables: Instruments must be correlated with \(x_{2,it}\) but uncorrelated with \(u_{it}\). The model must satisfy rank and order conditions for identification.

Estimating \beta when \(x_{2,it} = y_{i,t-1}\)

If \(x_{2,it}\) equals \(y_{i,t-1}\), \(x_{2,it}\) becomes endogenous due to potential correlation with \(u_{it}\). Techniques like Generalized Method of Moments (GMM) using lagged values of exogenous variables as instruments can be used for consistent estimation.

Estimating \beta when \(x_{2,it}\) includes \(y_{i,t-1}\)

If \(x_{2,it}\) includes \(y_{i,t-1}\), endogeneity issues arise. Lagged dependent variables as regressors necessitate using instrumental variables or dynamic panel data methods like Arellano-Bond estimator for consistent results.

Consistency of \(\beta\) and \(\gamma\) when \(x_{2,it}\) includes \(y_{i,t-1}\)

Consistently estimating both \beta and \gamma is challenging if \(x_{2,it}\) includes \(y_{i,t-1}\) due to endogeneity. Using GMM or 2SLS with appropriately lagged instruments could help, but may not fully eliminate bias, especially if \gamma terms are involved.

Key concepts

Ordinary Least Squares (OLS) consistency

To ensure the Ordinary Least Squares (OLS) estimator is consistent, certain conditions must be satisfied. First, the regressors need to be exogenous, meaning they should not be correlated with the error term. This condition can be written as: \( E[x_{1,it} u_{it}] = 0 \), \( E[x_{2,it} u_{it}] = 0 \), as well as \( E[w_{1,i} u_{it}] = 0 \), \( E[w_{2,i} u_{it}] = 0 \).
Additionally, there should be no perfect multicollinearity among the regressors, and the error term must be homoscedastic (having constant variance) and not exhibit autocorrelation. If all these conditions are met, OLS will yield consistent estimates of the parameters \( \beta \) and \( \boldsymbol{\beta}_{1} \).

fixed effects estimator

The fixed effects (within) estimator is designed to remove time-invariant individual effects \( \boldsymbol{\beta}_{1} \) by differencing. This approach is especially effective when dealing with panel data. For the fixed effects estimator to provide consistent estimates for \( \beta \), it is necessary for the time-varying regressors to be strictly exogenous with respect to \( \boldsymbol{\beta}_{1} \):
\( E[x_{1,it} u_{it}] = 0 \), \( E[x_{2,it} u_{it}] = 0 \). It is also crucial that the time-invariant variables \( \boldsymbol{\beta}_{1} \) are not correlated with the error term: \( E[w_{1,i} u_{it}] = 0 \), \( E[w_{2,i} u_{it}] = 0 \).
When these conditions are met, the fixed effects estimator will consistently estimate \( \beta \).

first-differences estimator

The first-differences estimator removes individual specific effects by differencing sequential observations. This method is particularly useful to control for any unobserved heterogeneity that is time-invariant. For the first-differences estimator to provide consistent estimates of \( \beta \), the error term should not exhibit serial correlation.
This can be written as: \( E[(u_{it} - u_{it-1}) | x_{1,it}, x_{1,it-1}, x_{2,it}, x_{2,it-1}] = 0 \).
This implies that the regressors \( x_{2,it} \) must be strictly exogenous: \( E[x_{2,it} u_{it-1}] = 0 \). With these conditions in place, this estimator helps to account for potential endogeneity related to omitted variables.

exogeneity

Exogeneity is a key concept for ensuring the consistency of various estimators in econometrics. A variable is considered exogenous if it is not correlated with the error term in the model. This requirement ensures that the regressors do not carry any hidden variations that could be attributed to the error term. Strict exogeneity, which is a stronger form, means: \( E[x_{1,it} u_{it}] = 0 \), \( E[x_{2,it} u_{it}] = 0 \) holds for all time periods.
This implies that past, present, or future values of the regressors should not influence the error term. Achieving exogeneity ensures unbiased and consistent parameter estimates.

instrumental variables

Instrumental variables (IV) are used when there is endogeneity within the explanatory variables. An instrument is a variable that is correlated with the endogenous regressor but uncorrelated with the error term. Using IV, consistent estimates can be obtained even when traditional methods like OLS fail.
The method typically used is Two-Stage Least Squares (2SLS). Here, the first stage involves regressing the endogenous variable on the instruments to get predicted values. In the second stage, these predicted values are used in place of the actual endogenous variables in the regression model.
This process ensures that the estimates are free from the bias caused by endogeneity.

Generalized Method of Moments (GMM)

The Generalized Method of Moments (GMM) is a powerful estimation technique used when dealing with panel data models, particularly in the presence of endogeneity. GMM constructs optimal estimates by using moment conditions derived from the data.
For instance, in a model where \( x_{2,it} \) equals \( y_{i,t-1} \), GMM can utilize lagged values of the dependent variable as instruments. This is advantageous since it accounts for potential endogeneity and heteroskedasticity.
Using moment conditions effectively allows GMM to provide consistent and efficient parameter estimates, even in the presence of serial correlation and heteroskedasticity.

Arellano-Bond estimator

The Arellano-Bond estimator is a GMM-based method tailored for dynamic panel data models, where lagged dependent variables are included as regressors. This estimator addresses the endogeneity problem arising from the correlation between lagged dependent variables and the error term.
It achieves this by using lagged values of the dependent variable as instruments, ensuring consistency. This estimator is particularly effective in managing small sample biases and offers a robust way to handle dynamic panels. The Arellano-Bond technique also deals with serial correlation and heteroskedasticity in the error term, assuring reliable parameter estimates.

endogeneity

Endogeneity occurs when an explanatory variable is correlated with the error term, leading to biased and inconsistent parameter estimates. This issue can arise due to omitted variables, measurement errors, or simultaneity.
To address endogeneity, techniques like Instrumental Variables (IV) or the Generalized Method of Moments (GMM) can be employed. These methods use instruments - variables that are correlated with the endogenous regressor but uncorrelated with the error term - to achieve consistent estimates.
Recognizing and addressing endogeneity is crucial for obtaining reliable and valid results in econometric models, ensuring the robustness of the statistical inferences made from the analysis.