Paul Johnson, Steven Durlauf and Jonathan Temple 1147time series analysis raised above continue to apply. Important variables are either
measured at infrequent intervals, or show little year-to-year variation. Moreover,
variation in annual growth rates may give misleading answers about the longer-
term growth process. For this reason, growth research using panel data has typically
averaged data over five- or ten-year periods. Given the lack of data before 1960, this
implies that growth panels not only have relatively few cross-sectional units, but
also low values ofT, often just five or six once lagged values have been included
as explanatory variables or instruments.^18
Most of the estimated models have been based on the hypothesis of conditional
convergence, namely that countries converge to parallel equilibrium growth paths,
the levels of which are a function of a few variables. As we saw in section 24.3,
a corollary is that an equation for growth (essentially the first difference of log
output) should contain some dynamics in lagged output. In this case, the growth
equation can be rewritten as a dynamic panel data model in which current output
is regressed on controls and lagged output, as in Islam (1995). In statistical terms
this rewritten model is identical in all respects, except that the coefficient on initial
output (originallyβ)is now 1+β:
logyi,t=( 1 +β)logyi,t− 1 +ψXi,t+πZi,t+αi+μt+εi,t. (24.31)This regression is a panel analogue to the cross-section regression (24.11), but now
includes a country-specific effectαiand a time-specific effectμt. The inclusion of
time effects is important in the growth context, not least because the means of
the log output series will typically increase over time, given productivity growth at
the world level. Inclusion of a country-specific effect allows permanent differences
in the level of income between countries that are not captured byXi,torZi,t.In
principle, one can also allow the parameters 1+β,ψ, andπto differ across countries
or regions.
Standard random effects estimators require that the individual effectsαiare dis-
tributed independently of the explanatory variables, and this requirement is clearly
violated for a dynamic panel such as (24.31) by construction, given the depen-
dence of logyi,tonαi. Hence the vast majority of panel data growth studies use
a fixed effects (within-group) estimator. Given their popularity, it is important to
understand how these estimators work. In a fixed effects regression there is a full
set of country-specific intercepts, one for each country, and inference proceeds
conditional on the particular countries observed, a natural choice in this context.
Identification of the slope parameters, usually constrained to be the same across
countries, relies on variation over time within each country. The “between” varia-
tion, namely the variation across countries in the long-run averages of the variables,
is not used.
The key strength of this method, familiar from the microeconometric litera-
ture, is the ability to address one form of unobserved heterogeneity: any omitted
variables that are constant over time will not bias the estimates, even if these
omitted variables are correlated with the explanatory variables. Intuitively, the
country-specific intercepts can be seen as picking up the combined effects of all