### Introduction

While structural breaks are a widely examined topic in pure time series, their impacts on panel data models have garnished less attention.

However, in their forthcoming paper Chowdhury and Russell (2018) demonstrate that structural breaks can cause bias in the instrumental variable panel estimation framework.

This work highlights that structural breaks shouldn't be limited to pure time series models and warrant equal attention in panel data models.

## The Model

For simplicity, consider the same AR(1) dynamic panel data model used by Chowdhury and Russell

$$y_{it} = \alpha y_{it-1} + \eta_i + \nu_{it}$$

In this model, $\eta_i$ represents the individual fixed effects and $\nu_{it}$ represents the random error terms.

Now consider an additive break in the fixed effects at time $T_B$ such that

$$y_{it} = \alpha y_{it-1} + (\eta_i + \delta_{iT_B}) + \nu_{it}, t \ge T_B$$

where

$$ E[\delta_{iT_B} \nu_{it}] \ne 0, t\lt T_B\\ E[\delta_{iT_B} \nu_{it}] = 0, t\ge T_B\\ E[\delta_{tT_B} \eta_i] \ne 0 $$

Note that just as the fixed effects, $\eta_i$, are different across each individual the impact of the structural break, $\delta_{iT_B}$, on the fixed effects is different across the individuals.

### Model Summary

- Dynamic panel data model.
- Individual specific structural break in fixed effects ($\delta_{tT_B}$).
- $E[\delta_{tT_B} \eta_i] \ne 0$.

## The Arellano-Bond Estimation Method

In static panel data models, like the one-way fixed effects model, demeaning or differencing is used to address heterogeneity. However, Nickell (1981) showed that in dynamic panel data models this process creates a bias in the coefficient estimates.

To address this issue, lagged levels or differences of the dependent variable are used as instruments. The Dynamic Panel Data approach, popularized by Arellano Bond (1991), uses a system of equations, one for each time period, and different instruments in each equation.

This is done to allow the use of newly available lagged variables as instruments as we move forward through the time series. From these new instruments, the Arellano-Bond moment conditions are formed.

### Arellano-Bond Moment Conditions

- Difference Estimator $$E[y_{it-s}(\delta \nu_{it})] = 0\ \text{for}\ t = 3, 4, \ldots, T\ \text{and}\ 2 \le S \le t-1$$
- Level Estimator $$E [\Delta y_{it-s}(\nu_{it} + \eta_i)] = 0\ \text{for}\ t = 3, 4, \ldots, T\ \text{and}\ 2 \le S \le t-1$$

## The Bias

### The Arellano-Bond Difference Estimator

When structural breaks are present the Arellano-Bond moment conditions are no longer valid. To demonstrate, consider the difference equation moments when there is a structural break at $T_B = 3$, $t = 4$, and $s = 2$:

$$E[y_{it-s}(\Delta \nu_{it})]\\ = E[y_{i2}({y_{i4} - \alpha y_{i3} - \eta_i - \delta_{iT_B}} - {y_{i3} - \alpha y_{i2} - \eta_i})] \\ = E[y_{i2}\Delta y_{i4}] - \alpha E[y_{i2} \Delta y_{i2}] - \boxed{E[y_{i2} \delta_{iT_B}] }$$

Structural breaks introduce bias into the GMM difference estimates through the boxed term in the equation above, $\boxed{E[y_{i2} \delta_{iT_B}] \ne 0}$.

### The Arellano-Bond Level Estimator

Now consider the level equation moments when there is a structural break at $T_B = 3$, $t = 4$, and $s = 2$:

$$E[\Delta y_{it-s}(\nu_{it} + \eta_i)] = [(y_{i3} - y_{i2})(\nu_{i4} + \eta_i)]\\ = E[(\alpha y_{it-2} + \delta_{iT_B} + \nu_{i3} + \eta_i - y_{it-2})(\nu_{i4} + \eta_i)]\\ = E[\big((\alpha - 1)y_{it-2} + \eta_i\big)\eta_i] + \boxed{E[\delta_{iT_B}\eta_i] }$$

In this case structural breaks introduce bias into the GMM level estimates through the boxed term, $\boxed{E[\delta_{iT_B}\eta_i] \ne 0}$.

## The Double-D GMM Estimator

Chowdhury and Russell (2018) propose the use of a new *Double-D* GMM estimator. The *Double-D* estimator uses lagged differences as instruments but correlates them with the lagged *differences* of the fixed effects such that the moments are given by $E[\Delta y_{i,t-s} \Delta \nu_{it}]$ where $S \ge 2$.

To see how this eliminates the bias consider the case where $t = 5$ and $S = 2$:

$$E[\Delta y_{it-2}(\Delta \nu_{it} + \Delta \eta_i)] = E[ \Delta y_{it-2} (\Delta \nu_{it})]\\ = E[(\alpha y_{it-2} + \delta_{iT_B} + \nu_{i3} + \eta_i - y_{it-2})(\Delta \nu_{i5})]\\ = E[\big((\alpha - 1)y_{i3} + \eta_i\big)(\Delta \nu_{i5})] + \boxed{E[(\delta_{iT_B} \Delta \nu_{i5})]}$$

Note that in the *Double-D* moment equation the boxed term $\boxed{E[(\delta_{iT_B} \Delta \nu_{i5})]}$ is equal to zero and the moments are valid.

## Conclusion

Structural breaks cannot be ignored, whether working with pure time series models or panel data models. When introduced into dynamic panel data models, structural breaks bias the Arellano-Bond moments, in turn biasing the coefficient estimates. Chowdhury and Russell (2018) propose a promising solution to this bias, the *Double-D* estimator.

## References

Chowdhury, R. A., & Russell, B. (2018). The difference, system and ‘Double‐D’GMM panel estimators in the presence of structural breaks. *Scottish Journal of Political Economy, 65*(3), 271-292.

Nickell, S. (1981). Biases in Dynamic Models with Fixed Effects. *Econometrica, 49*(6), 1417-1426. doi:10.2307/1911408