Steven Durlauf, Paul Johnson and Jonathan Temple 1105trend breaks raises the general question of whether convergence dynamics obey a
nonlinear process. Chonget al.(2008) employ a smooth transition autoregressive
process to model output and conclude that OECD member countries are converg-
ing to the US level of output per capita. Their analysis is again difficult to interpret
in economic terms, as their functional form assumptions do not correspond to any
particular economic model and therefore invite questions about the appropriate
choice of nonlinear structure.
A second criticism of time series tests has been made by Michelacci and Zaffa-
roni (2004). They argue that evidence of unit roots in per capita output may be
spurious because of a lack of attention to the possibility that dependence in these
individual series exhibits long memory, that is, dependence decays at a hyperbolic
rather than a geometric rate. If it is the case that the individual series are stationary
in levels, the differences between them cannot contain unit roots, so that con-
vergence is occurring. They further argue that long memory can explain the 2%
convergence rate found in cross-section regressions. This is an intriguing argument,
although Durlaufet al.(2005) question the strength of the empirical evidence for
long memory as well as its theoretical plausibility. Michelacci and Zaffaroni also
do not directly study the behavior of per capita output differences, so it is unclear
how to match their analysis with other studies.
The time series andσapproaches to convergence are merged in Evans (1996).
He studies the time series properties ofσt^2 , the cross-section variance of logyi,t.He
shows that, when there is no cointegration among the series logyi,t,σt^2 may be
represented as a unit root process with a quadratic time trend, and this suggests
a time series test of convergence based on unit root tests applied to a time series
forσt^2. He uses this test to conclude that there is convergence to a common trend
among 13 industrial countries over the period 1870–1989 and among a group of
51 countries over the period 1950–92, although the evidence in the latter case is
less conclusive.
Evans (1997) provides a time series approach to estimating rates of convergence.
For the contiguous US states over the period 1929–91, he finds that about one-
third of the point estimates are negative and about two-thirds of the confidence
intervals contain zero. For a sample of 48 countries over the period 1950–90, about
half of the point estimates are negative and all but two of the confidence intervals
contain zero.
23.6.1 Transitions versus steady-state dynamics
There are important differences between the time series approach to convergence
and theβand distribution shape approaches. As argued in Bernard and Durlauf
(1996), time series tests assume that the underlying stochastic processes are time
invariant, so that countries have transited to an invariant output process. In con-
trast, cross-section approaches, such asβ-convergence andσ-convergence, are
motivated by the assumption that countries are in transition to a steady-state,
so that the data for a given country at timetare drawn from a different stochastic
process than the data at some future time.^29 Bernard and Durlauf further indicate