A simple dynamic panel data approach for macro policy assessment

ABSTRACT

We propose a simple-to-implement dynamic panel difference-in-difference method to assess the impact of (place-based) macro policies. The idea is to exploit both serial correlation and cross-sectional dependence within a panel and implement a semi- difference-in-difference decomposition. Different from existing panel data methods, the proposed approach is easy to implement and conduct statistical inference and can incorporate multiple treated groups at different time thresholds. Monte Carlo simulations show the proposed estimator performs reasonably well with finite samples. An application to assess the general impact of a free trade zone policy is also provided.

KEWORDS:

• Dynamic panel model
• Difference-in-Difference
• Free Trade Zone
• GDP growth
• Treatment Effect

Acknowledgments

The author(s) would like to thank Jonathan Presler, David Schwegman, Alfonso Flores-Lagunes for their kind help and discussion.

A Understanding the Dynamic Panel Data Difference-in-Difference Model

Model (1) can be understood in at least three ways.

Firstly, it is a dynamic panel data model with a difference-in-difference decomposition. Comparing with a traditional DID regression, we only add the post-treatment period indicator $T_t$ and its interaction term of the location indicator $T_t\ast D_l$ since a location-specific (fixed) effect $d_i$ is already included in the original dynamic panel data model in Equation (1)(1) $y_{it+1}=\gamma y_{it}+\beta x_{it}+\alpha T_t+\eta T_t{D}_i+d_i+{\epsilon }_{it+1}i=1,..,N;t=1,\dots ,T$(1) . Consequently, $\eta$ is the parameter of interest which captures the effect of a macro policy. Explicitly, Equation (1)(1) $y_{it+1}=\gamma y_{it}+\beta x_{it}+\alpha T_t+\eta T_t{D}_i+d_i+{\epsilon }_{it+1}i=1,..,N;t=1,\dots ,T$(1) can be rewritten as

(3) $y_{it+1}=\gamma y_{it}+\beta x_{it}+u_{it+1}$(3)

(4) $u_{it+1}=\alpha T_t+\eta T_t\ast D_i+d_i+{\epsilon }_{it+1}$(4)

Clearly, Equation (3)(3) $y_{it+1}=\gamma y_{it}+\beta x_{it}+u_{it+1}$(3) is a first-order dynamic panel data model (without fixed effects); Equation (4)(4) $u_{it+1}=\alpha T_t+\eta T_t\ast D_i+d_i+{\epsilon }_{it+1}$(4) is a difference-in-difference decomposition for the residuals from the first equation. We have two remarks on the model in Equations (3)(3) $y_{it+1}=\gamma y_{it}+\beta x_{it}+u_{it+1}$(3) and (4(4) $u_{it+1}=\alpha T_t+\eta T_t\ast D_i+d_i+{\epsilon }_{it+1}$(4) ): First, the lag dependent variable $y_{it-1}$ here not only captures the serial correlation but also could act as a proxy for the unobserved characteristics of each individual (e.g., state or province). Second, Equation (4)(4) $u_{it+1}=\alpha T_t+\eta T_t\ast D_i+d_i+{\epsilon }_{it+1}$(4) is slightly different from the traditional DID model since we include the fixed effect ‘$d_i$’ rather than the dummy variable for a specific location $l$ ‘$D_i$’. For instance, if we only have two individuals (e.g., one is treated and the other is in the control group): $i=l,other$, then the above model is equivalent to

(5) $y_{it+1}=\gamma y_{it}+\beta x_{it}+u_{it+1}$(5)

(6) $u_{it+1}=\alpha T_t+\eta T_t\ast D_i+\theta \ast D_i+{\epsilon }_{it+1}$(6)

which is a standard dynamic panel data model with a DID decomposition on the residuals. This follows the same logic of LSDV estimation for the fixed effects model (Baltagi 2008).

Secondly, it is a panel DID model. Under mild assumptions such as unconfoundedness given lagged outcomes, the proposed dynamic panel data model collapses to the unconfoundedness DID model with panel data mentioned in Imbens and Wooldridge (2009) as the following model shows¹⁰

(7) $y_{it+1}-y_{it}=\alpha +\eta \ast G_{it}+(\gamma -1)y_{it}+{\epsilon }_{it+1}$(7)

where $t$ is the treatment cutoff time threshold and $G_{it}:=(T_{t+1}-T_t)D_i=D_i$. This is a within-group transformation. Defining the pre-treatment period, $t=0$, we can obtain an identical equation as in section 6.5.4 of Imbens and Wooldridge (2009):

(8) $y_{i1}-y_{i0}=\alpha +\eta \ast G_i+(\gamma -1)y_{i0}+{\epsilon }_{i1}$(8)

in which they demonstrate that this unconfoundedness-based approach in the context of panel data is more attractive than the traditional DID with repeated cross-sectional data. As a consequence, the causal interpretation of $\eta$ on the dependent variable is more credible.

Finally, the proposed model (1) focuses on a short-run causal analysis. Using a first-difference transformation (as is to be done in the GMM procedure), the proposed model essentially estimates a one-period impact at the treatment cutoff time. This can easily be seen in the estimation procedure. The first-difference transformation of model (1) is:

(9) $y_{it+1}-y_{it}=\gamma (y_{it}-y_{it-1})+\beta (x_{it}-x_{it-1})+\alpha (T_t-T_{t-1})+\eta (T_t-T_{t-1})D_i+{\epsilon }_{it+1}-{\epsilon }_{it}$(9)

For each treated city or individual $i\in l$, the variable $(T_t-T_{t-1})D_i$ is equal to zero except in the first post-period of the policy. Moreover, one can also use an $s$-difference transformation of model (1) as follows:

(10) $y_{it+1}-y_{it-s}=\gamma (y_{it}-y_{it-s-1})+\beta (x_{it}-x_{it-s-1})+\alpha (T_t-T_{t-s-1})+\eta (T_t-T_{t-s-1})D_i+{\epsilon }_{it+1}-{\epsilon }_{it-s}$(10)

Then, the proposed model essentially estimates an $s$-period impact around the treatment cutoff time. Here $s$ cannot be too large as we require enough exogenous lag dependent variables as instruments to identify the transformed model. For the instrument variables (IV), GMM procedure on dynamic panel models refers to Anderson and Hsiao (1981), Arellano and Bond (1991), Ahn and Schmidt (1995), Blundell and Bond (1998), Baltagi (2013) and bias-correction methods in Kiviet (1995), Bun and Kiviet (2003), Bruno (2005a, 2005b), Dhaene and Jochmans (2015) and references therein. This short-period analysis facilitates the causal inference of the proposed model as Assumption 3 is more likely to be satisfied in short post-treatment periods.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

¹ DID can be regarded as a special case of a two-way fixed effects model.

² Assumption 3.1 in Abadie (2005) states that conditional on the covariates, the average outcomes for treated and controls would have followed parallel paths in absence of the treatment.

³ One referee raised the jackknife method to our attention and we really appreciate it.

⁴ We also tried ${\sigma }_s^2$=2 and obtained similar results. The results are available upon request.

⁵ One possible explanation is that for a linear dynamic panel model, the jackknife estimator is a weighted combination of the typical GMM estimator while the bias-correction LSDV estimator also use the GMM estimator as initial estimates.

⁶ We excluded the data of three provinces who were blamed for GDP data manipulation. The nominal GDP per capita was deflated into real comparable GDP per capita with a base year of 1978 and the growth rate of GDP per capita was calculated based on that.

⁷ By Arellano and Bond (1991), the Sargan test in the two-step GMM procedure has been shown to have a better power than that in the one-step GMM while Hausman test in both procedures show a tendency to over-reject.

⁸ Applying the tvdiff command in Stata, we found that the test for parallel trend passed using the ‘leads’ but did not when using the ‘time trend’ For details, please see the Stata file for tvdiff.

⁹ Both bias-correction estimation methods yield marginal significant results.

¹⁰ We eliminate the exogenous covariates $x_{it}$ here for comparison.:

Additional information

Funding

Zhou's research is supported by National Natural Science Foundation of China [Grant No. 71833004 and Grant No. 714711080].