1104 The Econometrics of Convergence
Bernard and Durlauf (1995) define convergence for a set of countriesI as
occurring if:
lim
T→∞Proj(logyi,t+T−logyj,t+T∣
∣&t)= 0 ∀i,j∈I, (23.15)where Proj(a|b)denotes the projection ofaonband&tdenotes some information
set, which will generally include functions of time as well as current and lagged
values of logyi,tand logyj,t. Such a&twould imply a type of unconditional con-
vergence, whereas inclusion of control variables such as investment rates would
admit conditional convergence, but this has not been explored in the literature.
Most implementations of this definition have generally focused on the detection
of deterministic or stochastic trends in logyi,t−logyj,t, as the presence of either
implies a violation of (23.15). Consequently, time series tests of convergence have
typically been implemented by testing for the presence of a unit root in the logyi,t−
logyj,tprocess.
Using an approach based on unit root tests, Bernard and Durlauf (1995) find
little evidence of convergence in a group of 15 advanced industrialized economies
between 1900 and 1989 based on data from Maddison (1982, 1989). Hobijn
and Franses (2000) similarly find little evidence of convergence across a group of
112 Penn World Table countries over the period 1960–89.^28 Their work is based on
a clustering algorithm to identify groups of converging countries. They find many
small clusters, which they view as having distinct steady-states; but their multi-
plicity, and the absence of controls for structural characteristics, means that these
clusters could simply reflect differences in those characteristics, rather than differ-
ences in long-run outcomes due to differences in initial conditions. The breadth
of the sample used also suggests that the Bernard and Durlauf (1996) argument,
about the need for consideration of the substantive economic assumptions that
underlie time series methods for studying convergence, is applicable in this case.
One criticism of these tests focuses on the validity of unit root tests in the pres-
ence of structural breaks. Perron (1989) has argued that the failure to allow for
structural breaks can lead to spurious evidence in support of the presence of a unit
root (or, more precisely, can diminish the ability to reject the null of a unit root).
Greasley and Oxley (1997) impose breaks exogenously and find convergence for
Denmark and Sweden, in contrast to Bernard and Durlauf (1995), who did not
allow for breaks. Li and Papell (1999) allow for endogenous trend breaks and find
that this reduces, relative to Bernard and Durlauf (1995), the number of country
pairs that fail to exhibit convergence.
Carlino and Mills (1993) study US regions and reject convergence except with
specifications that allow for a trend break in 1946. However, a trend break vio-
lates (23.15), as it implies that some component of logyi,t−logyj,tis predictable
in the long run. Thus claims that allowing for data breaks produces evidence in
favor of convergence invites the question of what is meant by convergence. Note
that the sort of violation of (23.15) implied by a trend break is different from the
type implied by a unit root, as a break associated with the level of output means
that the output difference between two countries is always bounded. The issue of