Palgrave Handbook of Econometrics: Applied Econometrics

Sergio J. Rey and Julie Le Gallo 1277

the directionality between the two paths is distinct, with New Jersey’s evolution
moving downwards towards the mean of the distribution, while for Virginia the
dynamics are upward.^7 Rey and Ye (2008) develop approaches for summarizing the
properties of these time paths and their use in comparative analysis.

27.3.3 Stochastic kernels

A very active area of exploratory analysis has been the application of stochastic
kernels to regional growth series (Magrini, 1999; López-Bazoet al., 1999; Bulli,
2001; Basile, 2006). Here the focus is on the evolution of the distribution itself
and a number of novel approaches towards the estimation of densities and their
interpretation have been suggested.

27.3.3.1 Estimation

Lettingfx(t)represent the regional income density forneconomies in periodt,
the evolution of the cross-sectional distributions is modeled through the use of a
stochastic kernel:

fx(t+s)=

∫∞

−∞

Mt,sfx(t)dx, (27.24)

whereMt,sis the stochastic kernel which traces where points infx(t)move to in
fx(t+s). The kernel can be viewed as a continuous analog of the Markov transition
matrix that we examined in the previous section. As such, the stochastic kernel con-
tains important information regarding the distributional dynamics and thus the
question of its estimation becomes important. One approach relies on an estimate
of the conditional distribution:

Mˆt,s=fˆx(t+s)|x(t)=

fˆx(t+s),x(t)

fˆx(t)

. (27.25)

Estimates of the joint or marginal densities themselves rely on, somewhat confus-
ingly, kernel density estimates. For example:

fˆx(t)=fˆ(x,t;h)=(nh)−^1

∑n

i= 1

K{(x−Xi,t)/h}, (27.26)

whereKis a function such that

∫
K(x)dx=1, referred to as the kernel,his the
bandwidth andXi,tis the income for economyifor a given time period.
Based on the estimate of the stochastic kernel, alternative visualizations can
be generated to explore the implied transitional dynamics. These include three-
dimensional representations and the analogous two-dimensional contour plot.
Evidence of polarization in the income distributions would be reflected in peaks in
the 3D kernel or concentrated values in the contour. The sphericality of either of
the graphs would be indicative of heightened income mobility and leap-frogging,
while elongated ellipses along the 45-degree line would suggest a lack of mobility
and convergence.