According to my understanding, when a dependent variable like Y is added as the first/2nd lag of itself as the independent variable in the model, it will arise the issue of endogeneity. In panel data studies, we use this kind of model for the dynamic panel when a number of cross-sections increases the number of time dimensions. The appropriate measure we use is GMM but as your data is purely cross-sectional you should consult youtube videos on it. I hope you will find the best estimation technique to tackle your study there.

A lagged dependent variable added to the regression will become a Dynamic regression on panel data. If the time period is less than the number of cross section variables, use GMM. Otherwise any time series regression can be used - OLS,, DOLS and ARDL method of estimation can be used, depending on your requirements and tests.

Dear Zahid Bashir and Chung Tin Fah Many thanks for your suggestion, that really helps. Still, I am wondering because, the X variables, i.e. all the explanatory variables (of course, except the lagged Y) are time-invariant; they vary only across cross-sectional units. Could you refer me to any specific model that includes the lagged value of the dependent variable as an explanatory variable along with other variables that are time-invariant? If I enforce a traditional cross-section model, how can I handle the endogeneity issue?

Stay safe.

Mohammad Hossain

welcome dear

If you have cross-sectional data without a time dimension you will not be able to include a lagged dependent variable. To do this you will need to add a time dimension and use panel data. GMM estimators are available for estimating such models (which will be subject to endogeneity as mentioned in the answers above). I recommend that you read the following article and download the xtabond2 program for use in Stata. The article discusses how to estimate such models and provides examples with Stata computer output.

Roodman, D. (2009). How to do xtabond2: An introduction to "Difference" and "System" GMM in Stata. The Stata Journal, 9(1), 86-136.

A paper that provides an application of the GMM methods with a discussion of many of the various choices is as follows:

Matousek, R., and Nguyen, T. N. and Stewart, C. (2017) ‘Note on a non-structural model using the disequilibrium approach: Evidence from Vietnamese banks’, Research in International Business and Finance, 41, pp. 125 – 135.

Note that these GMM procedures are appropriate for large N and small T. If you have moderate or small N and moderate T it may be more appropriate to use a bias corrected dynamic panel fixed-effects estimator that can be used when N is not regarded as large. Such estimators can be implemented in Stata too. See the following papers (these only allow one lagged dependent variable).

Bruno, G.S.F. (2005a) ‘Approximating the bias of the LSDV estimator for dynamic unbalanced panel data models’, Economics Letters, 87, 361-366.

Bruno, G.S.F. (2005b) ‘Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals’, The Stata Journal, 5, 473-500.

An example paper that uses Bruno's estimator and panel unit root tests is given in:

Goda, T., Stewart, C. and Torres García, A. (2019) ‘Absolute Income Inequality and Rising House Prices’, forthcoming in Socio-Economic Review.

Heartiest thanks, Chris Stewart for your wonderful direction. I think I should gather more series to make it a panel data analysis. The fact is only the Y variable is accumulating in time and all the X variables are time-invariant. I have made some experiments and find that regression of Y on Xs and y(-1) produce results of high R-squared, many significant and few insignificant coefficients and violation of underlying assumptions of normality, homoscedasticity, and no multicollinearity. If I can get a few relevant time series I will make it a panel data model. Stay safe.

Dear Mohammad, if data are possibly nonstationary once a time-series dimension is added you may need to consider panel integration and cointegration tests. See the paper by Goda et al (2019) referenced in my previous answer for some panel integration and cointegration tests that account for any cross-sectional dependence that exists in the data. Best wishes, Chris.