Palgrave Handbook of Econometrics: Applied Econometrics

1278 Spatial Analysis of Economic Convergence

Other types of kernel visualizations are shown in Figure 27.6. The conditional
densities (Figure 27.6(a)) have been estimated for a 71-year transition period from
1929 to 2000 for the US. By stacking the conditional densities, the figure shows
how the regional income distribution from 1929 evolves into that in 2000. The
highest-density region (HDR) plot in Figure 27.6(b) identifies the smallest region in
the sample space that covers a given probability. For each conditional distribution,
the darker shaded area is the 50% HDR, while the lighter shade represents the 99%
HDR. The HDR plot reflects strong convergence over the 71-year period, as each of
the conditional modes (dots) falls away from the diagonal, indicating that states
tend to move towards the overall mean of the distribution.^8
Alternatively, estimates of the marginal densities for the two periods could be
examined. Bianchi (1997) has suggested that the multimodality of a density could
serve as an indication of polarization. In this way, movement to a single mode
would be reflective of convergence. It should be noted that this marginal approach
risks loss of information on the transitions and can mask internal mixing.
While kernels provide novel visualizations of growth dynamics, their use with
spatially referenced data is not without problems. To gain some understanding of
the potential implications for spatial dependence in the estimation, rewrite the
kernel density estimator for a given time period as:

fˆ(x;h)=n−^1

∑n

i= 1

Kh(x−Xi), (27.27)

where the incomesX 1 ,...,Xnare no longer independent but are identically dis-
tributed with a common density, so thatCov(Xj,Xj+k)depends only onk. In such

a setting the bias offˆ(x;h)is robust to the dependence, but the variance becomes:

V

[

fˆ(x;h)

]

=n−^1 V

[

Kh(x−X 1 )

]

+ 2 n−^1

n∑− 1

j= 1

( 1 −j/n)COV

[

Kh(x−Xj),Kh(x−Xj+ 1 )

]

. (27.28)

Applications of kernel density estimators to regional income datasets have implic-
itly assumed that the observations were pairwise independent. With spatially
referenced data, however, such an assumption can be questionable. The impli-
cations for the properties of kernel density estimators and the related methods
of stochastic dominance (Carrington, 2006) and relative distribution approaches
(Janikas, 2007) have been largely ignored to date in the growth literature.

27.3.3.2 Regional conditioning and spatial filtering

Canova and Marcet (1995) suggest that income cross-correlations among countries
need to be treated prior to constructing kernels. Their approach is to base the
kernel on:

x∗i,t=xi,t/

∑

i

xi,t, (27.29)