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Donggyu Sul Publications

Publish Date
Discussion Paper

This paper points out some pitfalls in the use of two-way fixed effects (TWFE) regressions when outcome variables contain nonlinear or stochastic trend components.  When a policy change shifts trend paths of outcome variables conventional TWFE estimation can distort results and invalidate inference. A robust solution is proposed by identifying determinants of dynamic club membership based on the idea of relative convergence, which can be assessed empirically by the so-called ‘logt’ test (Phillips & Sul, 2007a). Club membership in each time period is estimated by recursive regression, transforming outcome variables to statistically stable, stationary status. Time varying club membership can then be used to identify the determinants of club memberships by running a panel logit or ordered logit regression. This approach is applied to study COVID-19 vaccination data across 50 states and the District of Columbia (DC). A new weekly database is created to track individual state and DC vaccination policies and mandates over the period from March 2021 to February 2022. Initially two convergent clubs are identified. Later evidence of the vaccination rates across states reveals a single convergent club. The primary determinant of this merger of sub-clubs is found to be federal-level vaccine mandates.


The concept of relative convergence, which requires the ratio of two time series to converge to unity in the long run, explains convergent behavior when series share commonly divergent stochastic or deterministic trend components. Relative convergence of this type does not necessarily hold when series share common time decay patterns measured by evaporating rather than divergent trend behavior. To capture convergent behavior in panel data that do not involve stochastic or divergent deterministic trends, we introduce the notion of weak σ-convergence, whereby cross section variation in the panel decreases over time. The paper formalizes this concept and proposes a simple-to-implement linear trend regression test of the null of no σ-convergence. Asymptotic properties for the test are developed under general regularity conditions and various data generating processes. Simulations show that the test has good size control and discriminatory power. The method is applied to examine whether the idiosyncratic components of 90 disaggregate personal consumption expenditure (PCE) price index items σ-converge over time. We find strong evidence of weak σ-convergence in the period after 1992, which implies that cross sectional dependence has strenthened over the last two decades. In a second application, the method is used to test whether experimental data in ultimatum games converge over successive rounds, again finding evidence in favor of weak σ-convergence. A third application studies convergence and divergence in US States unemployment data over the period 2001-2016.


This paper introduces a new estimation method for dynamic panel models with fixed effects and AR(p) idiosyncratic errors. The proposed estimator uses a novel form of systematic differencing, called X-differencing, that eliminates fixed effects and retains information and signal strength in cases where there is a root at or near unity. The resulting “panel fully aggregated” estimator (PFAE) is obtained by pooled least squares on the system of X-differenced equations. The method is simple to implement, free from bias for all parameter values, including unit root cases, and has strong asymptotic and finite sample performance characteristics that dominate other procedures, such as bias corrected least squares, GMM and system GMM methods. The asymptotic theory holds as long as the cross section (n) or time series (T) sample size is large, regardless of the n/T ratio, which makes the approach appealing for practical work. In the time series AR(1) case (n = 1), the FAE estimator has a limit distribution with smaller bias and variance than the maximum likelihood estimator (MLE) when the autoregressive coefficient is at or near unity and the same limit distribution as the MLE in the stationary case, so the advantages of the approach continue to hold for fixed and even small n. For panel data modeling purposes, a general-to-specific selection rule is suggested for choosing the lag parameter p and the procedure works in a standard manner, aiding practical implementation. The PFAE estimation method is also applicable to dynamic panel models with exogenous regressors. Some simulation results are reported giving comparisons with other dynamic panel estimation methods.


While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution which holds uniformly in the autoregressive coefficient rho including stationary and unit root cases. The rate of convergence is root of n when |τ| < 1 and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when τ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity.


Some extensions of neoclassical growth models are discussed that allow for cross section heterogeneity among economies and evolution in rates of technological progress over time. The models offer a spectrum of transitional behavior among economies that includes convergence to a common steady state path as well as various forms of transitional divergence and convergence. Mechanisms for modeling such transitions and measuring them econometrically are developed in the paper. A new regression test of convergence is proposed, its asymptotic properties are derived and some simulations of its finite sample properties are reported. Transition curves for individual economies and subgroups of economies are estimated in a series of empirical applications of the methods to regional US data, OECD data and Penn World Table data.

Keywords: Economic growth, Growth convergence, Heterogeneity, Neoclassical growth, Relative transition, Transition curve, Transitional divergence

JEL Classification: 030, 040, C33


Two groups of applied econometricians have figured prominently in empirical studies of growth convergence. In terms of a popular caricature, one group believes it has found a black hat of convergence (evidence for growth convergence) in the dark room of economic growth, even though the hat may not exist (the task may be futile). A second group believes it has found a black coat of divergence (evidence against growth convergence) even though this object also may not exist (empirical reality, including the nature of growth divergence, is ever more complex than the models used to characterize it). The present paper seeks to light a candle to see whether there is a hat, a coat or another object of identifiable clothing in the room of regional and multi-country economic growth. After our examination, we find that the candle power of applied econometrics is too low to clearly distinguish a black hat in the huge dark room of economic growth. However, in our theory model, we find an important new role for heterogeneity over time and across economies in the transitional dynamics of economic growth; and, in our empirical work, these transitional dynamics reveal an elusive shadow of the conditional convergence hat in both US regional and inter-country OECD growth patterns.

Keywords: Convergence Parameter, Conditional Convergence, Economic Growth, Growth Convergence, Heterogeneity, Neoclassical Economics, Transition measures

JEL Classification Numbers: 040, C32, C33

Econometrics Journal

This paper deals with cross section dependence, homogeneity restrictions and small sample bias issues in dynamic panel regressions. To address the bias problem we develop a panel approach to median unbiased estimation that takes account of cross section dependence. The new estimators given here considerably reduce the effects of bias and gain precision from estimating cross section error correlation. The paper also develops an asymptotic theory for tests of coefficient homogeneity under cross section dependence, and proposes a modified Hausman test to test for the presence of homogeneous unit roots. An orthogonalization procedure is developed to remove cross section dependence and permit the use of conventional and meta unit root tests with panel data. Some simulations investigating the finite sample performance of the estimation and test procedures are reported.

Keywords: Autoregression, Bias, Cross section dependence, Dynamic factors, Dynamic panel estimation, GLS estimation, Homogeneity tests, Median unbiased estimation, Modified Hausman tests, Median unbiased SUR estimation, Orthogonalization procedure, Panel unit root test

JEL Classification: C32, C33