Model specification for long panel dataset with heteroskedasticity, autocorrelation, cross-sectional dependence, and non-stationarity

[Nuno Fernandes]

Dear Statalist,

For my master's thesis I'm working with a balanced panel dataset (N=10, T=72) spanning 2018-2023 on a monthly basis. My research examines how certain characteristics affect the competitiveness of platforms over time. I have one dependent variable (market share) and three independent variables (measures of complement availability, diversity, and quality). My goal is to analyse linear and non-linear (quadratic) effects as well as interactions between the independent variables and to conduct hypothesis tests on them.

After running various diagnostics, I've identified the following:

1. Heteroskedasticity: Modified Wald test (xttest3) rejected the null hypothesis of homoskedasticity (p < 0.001).

2. Autocorrelation: Wooldridge test for autocorrelation (xtserial) indicated the presence of first-order autocorrelation (F = 756.180, p < 0.001).

3. Cross-sectional dependence: Pesaran CD test (xtcdf) showed significant cross-sectional dependence for all variables (p < 0.001).

4. Time fixed effects: Joint test of time dummies (testparm i.mdate) was significant (p < 0.001), indicating relevant time effects.

5. Non-stationarity: Fisher-type unit-root test (xtunitroot fisher) showed:
- Dependent variable (market share) is non-stationary (p = 0.1123)
- One independent variable (complement availability) is non-stationary (p = 0.9657)
- Two independent variables (diversity and quality) are stationary (p < 0.05)

6. Cointegration: Westerlund test (xtcointtest westerlund) rejected the null hypothesis of no cointegration (p = 0.0169), indicating all panels are cointegrated.

7. No endogeneity: Durbin-Wu-Hausman tests for each variable did not indicate endogeneity issues.

8. Distribution: Market share, complement availability, and diversity are right-skewed, while quality appears more normally distributed. However, I read that normality is not a prerequisite for panel data regression.

9. Residuals: Given the sample size (720 observations), I didn't test for normality of residuals, assuming the central limit theorem would apply.

Initially, I planned to use xtscc with time (i.mdate) and entity fixed effects, as suggested by a robust Hausman test using xtoverid (p = 0.0176). Before I decided in favour of xtscc, I also researched alternative commands such as xtpcse, xtgls and read that xtreg was not suitable. However, given the non-stationarity and cointegration, I'm reconsidering this approach and I've now looked into xtpmg, xtdcce2, and xtdolshm for dealing with non-stationarity/cointegration.

My question: What model specification would be most appropriate given these characteristics of my panel dataset? I'm using StataNow/BE 18.5.

I would greatly appreciate advice on:
1. The most suitable model to use
2. A specific Stata command for this model that addresses the issues mentioned above

Thank you in advance for your valuable insights.
Nuno

[Jeff Wooldridge]

The main issue is the unit roots and whether you have cointegration in the equation. Even if you do, the inference is nonstandard, but at least you know the equation is meaningful.

[Nuno Fernandes]

The main issue is the unit roots and whether you have cointegration in the equation. Even if you do, the inference is nonstandard, but at least you know the equation is meaningful.

Hi Jeff, thank you for your helpful response. I now better understand the importance of accounting for the unit roots and cointegration in my model.

Since the Westerlund test confirmed cointegration, I’ve reconsidered using xtscc and have started looking into xtdcce2 instead, as it explicitly handles cross-sectional dependence, which is a significant issue in my data. Additionally, I saw that xtdcce2 estimates both short- and long-run coefficients while including error correction. However, I’ve read that xtdcce2 is typically recommended for large N and large T panels, while my dataset has N = 10 and T = 72. Given this, do you think xtdcce2 is still appropriate for my dataset, or would a command like xtpmg—despite not directly addressing cross-sectional dependence—be more suitable?