Econometrics

# Measuring Convergence

Is regional variation in healthcare spending converging or diverging over time?  How would you know?   How would you measure convergence?

A paper by Panopoulou and Pantelidis (2013) answers just such a question.  They define 3 types of convergence.  These include beta convergence, sigma convergence, and stochastic convergence.   I explain each one below.

### Beta Convergence

Beta convergence occurs when high-cost regions are the ones most likel to have the lowest growth rates. One can test for beta convergence using the following regression:

• yi = α + βω0,i + ei

where yi is the average spending growth rate in region i and ω0,i is the initial spending level in region i. If the coefficient β is negative, then one can state that the spending distribution is experiencing beta convergence.

### Sigma Convergence

Sigma convergence describes whether the spending distribution shrinks over time regardless of the relative position of each region. This concept was introduced by Barro and Sala-i-Martin (1990). Specifically, sigma convergence refers to :a decrease over time in the cross-sectional variation of the natural logarithm of the variable of interest. Typically, the sample standard deviation and/or the coefﬁcient of variation (cross-sectional mean over cross-sectional deviation) is used to measure variation.”

### Stochastic Convergence

Under this type of convergence, the persistence of shocks on the variable of interest is examined. Speciﬁcally, when stochastic convergence exists, shocks in the variable of interest relative to the panel average are temporary. Typically, testing for stochastic convergence is performed via standard panel unit root tests, where the existence of a unit root indicates that the effect of a shock is permanent causing divergence of the series from the sample mean.

### Sigma Convergence in Cross-State Health Expenditures

Panopoulou and Pantelidis find the following:

We examine convergence in HCEs using the PS methodology, employing an example following HCEs for US states from 1980 to 2004. Our empirical ﬁndings do not support the existence of full convergence among the US states. However, there is evidence that the US states form two clubs (consisting of 18 and 32 states, respectively) that converge to different equilibria. Moreover, HCEs seem to be determined by the geographical position of each state. Speciﬁcally, states located in the south and west part of the US are members of the low-spending convergence club, while the remaining states located in the north and east part of the US form the high-spending convergence club.

We also extend our analysis to examine the cross-state disparities of nine major components of HCEs. Our results suggest full convergence among the US states for only two components of HCEs, namely, ‘physician and other professional services’ and ‘home health care’. A similar result holds for ‘durable medical products’, where 47 states converge, while Nebraska, New Hampshire, and New Jersey form a separate convergent club. Furthermore, there is evidence of club convergence related to geographical characteristics for two components (i.e., ‘dental services’ and ‘nursing’).

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