8.2 Counterfactual Income Dynamics
8.2.1 Sources of the Growth Differential With Respect to a Hypothetical Average Economy
Following De la Fuente (1995), we are now able to quantify the immediate
determinants of growth and convergence during the period. The sources of the
growth differential with respect to a hypothetical representative economy,
basically the average country over the period, are computed using the above
parameter estimates.
The following code decomposes each country's growth rate differential with
respect to the sample average into five factors: the contribution of
physical capital accumulation, the impact of the working-age population
growth, the contribution of human capital accumulation, the Neoclassical
convergence effect, and the impact of a fixed effect reflecting differences
in efficiency.
pkap=((x[,1]-mean(x[,1]))*b[2,])+((x[,2]-mean(x[,2]))*b[3,])
wagrowth=(x[,3]-mean(x[,3]))*b[4,]
hkap=(x[,4]-mean(x[,4]))*b[5,]
convergence=(x[,5]-mean(x[,5]))*b[6,]
fixed=((x[,6]-mean(x[,6]))*b[7,])+((x[,7]-mean(x[,7]))*b[8,])
+((x[,8]-mean(x[,8]))*b[9])+((x[,9]-mean(x[,9]))*b[10,])
8.2.2 Univariate Kernel Density Estimation and Bandwidth Selection
- {hcrit, crit} =
denbwsel(x{, h, K, d})
- starts an interactive tool for kernel density bandwidth selection
using the WARPing method
- fh =
denest(x{, h, K, d})
- computes the kernel density estimate on a grid using the WARPing
method
- {fh, fhl, fhu} =
denci(x{, h, alpha, K, d})
- computes the kernel density estimate and pointwise confidence intervals
on a grid using the WARPing method
|
Suppose we are given a sample of independent, identically distributed
realizations of a random variable
. Now,
if a smooth kernel function
is
centered around each observation
and if we average over these
functions in the observations, we obtain the kernel density estimate defined
as follows
 |
(8.1) |
where the kernel function is a symmetric probability density function.
Practical application of kernel density estimation is crucially dependent on
the choice of the smoothing parameter
. A measure of accuracy in order to
assess how closely
estimates
is the Integrated
Squared Error,
.
Stone (1984) shows that a data-driven bandwidth
that
asymptotically minimizes
is given by
 |
(8.2) |
with
the cross validation
function, and where
.
Park and Turlach (1992) provide an overview over the existing bandwidth
selection methods. We choose here to perform the Least Squares Cross
Validation criterion instead of, for instance, the Biased CV or the Smoothed
CV criteria that need either a very large sample size or pay with a large
variance. Still, note that it remains difficult to recommend once and for
all a particular bandwidth selector. One should therefore compare the
resulting density estimates determined by different selection methods.
Our goal, here, is first to select an optimal bandwidth and second to
estimate kernel densities of the world income distribution. We first load
the necessary libraries. The smoother
quantlib automatically loads
the xplore
and the kernel
quantlibs. The plot
quantlib is used for graphing the resulting cross validation and density
functions.
library("smoother")
library("plot")
Second, we call the quantlet
denbwsel
that needs the univariate data
vector as input and that will open a selection box which offers you the
choice between different bandwidth selectors, as well as the possibility to
change parameters such as the kernel, the search grid, etc. Among them, the
LSCV criteria.
I60=x[,5]./max(x[,5])
I85=(x[,5].+y)./max(x[,5].+y)
{hcrit1,crit1}=denbwsel(I60)
{hcrit2,crit2}=denbwsel(I85)
Obviously, the CV function is not minimized within the automatically
selected range of bandwidth. The bandwidth that minimizes the CV criterion
is below the selected lower bound. We must increase the search grid for
.
If one manually selects a lower bound for
equal to
, the following
graphics are displayed that show the LSCV function in the upper left, the
selected optimal bandwidth in the upper right, the resulting kernel density
estimate in the lower left, and some information about the search grid and
the kernel in the lower right. The graphical display is shown in Figure 8.1.
Figure:
LSCV for the worldwide income per working-age person in 1960
normalized relative to the maximum.
growdist.xpl
|
The optimal bandwidth corresponding to the world per working-age person
income distribution in 1960 is therefore
. It is stored in
hcrit1. Note that the Sheather and Jones (1991)' selector chosen by Di Nardo, Fortin, and Lemieux (1996)
finds a bandwidth equal to
. We open a second selection box in order to
compute the optimal bandwidth corresponding to the world per working-age person
output in 1985. The lower
bound of the search grid is now set to
and the corresponding optimal
bandwidth obtained by least squares cross validation is now equal to
. There is apparently more structure in the final distribution as compared
to the initial distribution.
Confidence intervals can be derived under some restrictive assumptions
(see Härdle; 1991) and written as
![$ \left[ \widehat{f}_{h}(x)-z_{1-\frac{\alpha }{2}}\sqrt{\frac{\widehat{f}_{h}(x...
...{2}}\sqrt{\frac{\widehat{f}_{h}(x)\left\Vert K\right\Vert _{2}^{2}}{nh}}\right]$](xploreapplichtmlimg1025.gif) |
(8.3) |
where
is the
quantile of
the standard normal distribution.
In XploRe, confidence intervals are computed using
denci. The
following quantlet code computes the confidence intervals for the optimal
bandwidth previously selected by least squares cross validation, selecting a
Gaussian kernel, a discretization binwidth
, and significance
level
.
d=(max(I60)-min(I60))./200
{fh60,clo60,cup60}=denci(I60,hcrit1,0.10,"gau",d)
We propose now to decompose changes in the world income distribution on the
basis of simple counterfactual densities. More specifically, and as proposed
by De la Fuente (1995), what would the density of income have been in 1985
in a hypothetical world where the relative income of each country changed
only due to factor accumulation, with all economies displaying average
behavior in terms of all other variables? We simulate three such
counterfactual densities. One is the density as defined above. Another is
the density that one would have observed in 1985 if the relative income of
each country changed only due to the Neoclassical convergence effect. The
last one is the density that the empirical model is able to predict.
In a first step, we compute the relative per working-age person income under
the above assumptions and the observed density in 1985, and then estimate
the corresponding counterfactual density. Comparing these densities with the
density estimates corresponding to the real world in 1960 and 1985 gives a
clear visual insight of the sources of the world income dynamics. Univariate
density estimates are computed using
denest
This quantlet only approximates the kernel density by the WARPing method.
This method has the statistical efficiency of kernel methods while being
computationally comparable to histogram methods as it performs smoothing
operations on the bin counts rather than the raw data as in traditional
kernel density estimation (see Härdle and Scott; 1992).
It is also
possible to evaluate the density estimate at all observations by using
denxest
instead of
denest.
w1=(x[,5].+pkap.+hkap.+wagr.+conv.+fix.+mean(y))
./max(x[,5].+pkap.+hkap.+wagr.+conv.+fix.+mean(y))
fhpred=denest(w1,hcrit2,"gau",d)
w2=(x[,5].+pkap.+hkap.+wagr.+mean(y))
./max(x[,5].+pkap.+hkap.+wagr.+mean(y))
fhfac=denest(w2,hcrit2,"gau",d)
w3=(x[,5].+conv.+mean(y))./max(x[,5].+conv.+mean(y))
fhconv=denest(w3,hcrit2,"gau",d)
fh85=denest(I85,hcrit2,"gau",d)
Figure:
Univariate Density Estimates
and Confidence Intervals. Upper left: Per working-age person income in 1960
(solid blue line) with pointwise confidence intervals (dashed blue lines)
and in 1985 (red solid line). Upper right: Real (red line) and predicted
(magenta line) per working-age person income densities in 1985. Lower left
and right: Per working-age person income in 1960 (solid blue line) and
counterfactual income densities in 1985 if countries would have differ only
in factor accumulation (left) or in the Neoclassical convergence effect
(right).
growdist.xpl
|
The above density estimates are displayed in Figure 8.2. To distinguish the
densities, we choose to color them with the quantlet
setmask.
Technically,
setmask
handles mask vectors that contain numerical
information to control the graphical display of the data points. This
explains the name of the function. Density estimates are drawn as solid
lines and confidence intervals as dashed lines.
fh60=setmask(fh60,"line","blue")
clo60=setmask(clo60,"line","blue","thin","dashed")
cup60=setmask(cup60,"line","blue","thin","dashed")
fh85=setmask(fh85,"line","red")
fhfac=setmask(fhfac,"line","yellow")
fhconv=setmask(fhconv,"line","green")
fhpred=setmask(fhpred,"line","magenta")
To display Figure 8.2, we need to create a display which consists of four
windows. This is achieved through the command
createdisplay. The
command
show
allows us to specify the data sets that will be
plotted in each plot of the display. After
show
has been called,
one controls the layout of the display by
setgopt.
disp1=createdisplay(2,2)
show(disp1,1,1,fh60,clo60,cup60,fh85)
show(disp1,1,2,fh85,fhpred)
show(disp1,2,1,fh60,fhfac)
show(disp1,2,2,fh60,fhconv)
setgopt(disp1,1,1,"title","Density Confidence Intervals",
"xlabel","Income in 1960 and 1985","ylabel",
"density estimates")
setgopt(disp1,1,2,"title","Predicted vs 1985","xlabel",
"Income (Predicted & 85)")
setgopt(disp1,2,1,"title","1960 plus factor accumulation
effect","xlabel","Income (60 and factor accumulation)")}
setgopt(disp1,2,2,"title","1960 plus Neoclassical convergence
effect",}
"xlabel","Income (60 and convergence)")}
The upper left of Figure 8.2 displays both the density estimate of the per
working-age person income in 1960 together with the corresponding confidence
intervals (solid and dashed blue lines), and the per working-age person
income density estimate in 1985 (red line). On the one hand, the
distribution of income at the beginning of the period appears to be
unimodal, most of the economies clustering in what one might call a
middle-income class. On the other hand, the underlying density in 1985 seems
to be consistent with a multimodal distribution suggesting that countries
follow different development paths and that they tend to cluster into
different income classes. The population of economies in 1985 seems to have
at least three modes. The initial middle income class vanished: some
countries caught up and joined a club of rich countries and others felt into
a poverty trap. This is the ``Twin Peaks'' scenario illustrated among others
by Quah (1996). At least, the structure of the worldwide income
distribution in 1985 does not fit anymore within the computed confidence
intervals corresponding to the income density estimate in 1960. There is a
very systematic shift over times. This suggests a great amount of mobility
within the system and over the period under study.
Where does this mobility exactly come from? What are the most important
factors in determining the worldwide income distribution dynamics? The
upper right display of Figure 8.2 shows a counterfactual income density
estimate that the empirical model estimated above has been able to predict
together with the income density estimate in 1985. Although, the model
appears to be able to predict the formation of the two modes for the highest
income classes, and therefore to capture, at least partially, the
convergence phenomenon, it is unable to fit the poverty trap that arose
during the period. The lower left display suggests that differences in
factor accumulation together with the differences in efficiency as proxied
by the continental dummies, may be partially responsible for this wealth
trap. But this cannot explain the all story. Something else is going on, and
I leave here this issue for future exploration. Finally, the lower right
display illustrates a collapsing over time of the world income distribution
to a degenerate point limit. If all economies were displaying average
behavior in terms of factor's accumulation and efficiency, poor countries
would catch up with rich ones.
8.2.3 Multivariate Kernel Density Estimation
- fh =
denestp(x{, h, K, d})
- computes a multivariate density estimate on a grid using the WARPing
method
- gs =
grcontour2(x, c{, col})
- generates a contour plot from a 3-dimensional data set x
|
All above formulas can be easily generalized to multivariate observations
, and
. The
kernel function
has to be replaced by a multivariate kernel
. One
takes a product kernel
 |
(8.4) |
where
.
The following quantlet computes two-dimensional density estimates for
different data sets via the function
denestp, where the kernel is
Gaussian and the bandwidth chosen arbitrarily to
. The surface of each
bivariate density estimate is then illustrated via contour plots with
contour lines
. The function
grcontour2
allows us to
generate contours corresponding to a bivariate density estimate.
library("graphic")
bi1=I60~I85
r=rows(bi1)
d=(max(bi1)-min(bi1))./20
fh1=denestp(bi1,0.05,"gau",d)
c1=(1:5).*max(fh1[,3])./10
gs1=grcontour2(fh1,c1)
bi2=I60~w1
d=(max(bi2)-min(bi2))./20
fhpred=denestp(bi2,0.05,"gau",d)
c2=(1:5).*max(fhpred[,3])./10
gspred=grcontour2(fhpred,c2)
bi3=I60~w2
d=(max(bi3)-min(bi2))./20
fhfac=denestp(bi3,0.05,"gau",d)
c3=(1:5).*max(fhfac[,3])./10
gsfac=grcontour2(fhfac,c3)
bi4=I60~w3
d=(max(bi4)-min(bi4))./20
fhconv=denestp(bi4,0.05,"gau",d)
c4=(1:5).*max(fhconv[,3])./10
gsconv=grcontour2(fhconv,c4)
disp2=createdisplay(2,2)
z=setmask(I60~I60,"line","red")
show(disp2,1,1,gs1,z)
show(disp2,1,2,gspred,z)
show(disp2,2,1,gsfac,z)
show(disp2,2,2,gsconv,z)
setgopt(disp2,1,1,"title","Bivariate Density Estimate",
"xlabel","Income (1960)","ylabel","Income (1985)")
setgopt(disp2,1,2,"title","2D Density Estimate",
"xlabel","Income (1960)",
"ylabel","Predicted Income in 1985")
setgopt(disp2,2,1,"title","2D Density Estimate",
"xlabel","Income (1960)","ylabel","Factor Accumulation")
setgopt(disp2,2,2,"title","2D Density Estimate",
"xlabel","Income (1960)","ylabel",
"Neoclassical Convergence")
Figure:
Contours of Bivariate Density
Estimates. The x-axis is the per working-age person income in 1960. The
y-axis is respectively: the per working-age person income in 1985 (upper
left), the predicted income in 1985 (upper right), and the relative income
of each country changed only due to factor accumulation (lower left) and to
the Neoclassical convergence effect (lower right) with all economies
displaying average behavior.
growdist.xpl
|
The above quantlet leads to Figure 8.3.
The display in the upper left box is
the bivariate density estimate of per working-age person incomes in 1960 and
1985. If most observations concentrate along the 45
-line, then
countries in the distribution remain where they started. In reality, poor
(rich) countries do concentrate under (above) the 45
-line. Note
also that whatever the class of income from which a country starts displays
both catching up and lagging behind especially when a country started in the
middle income class. This corroborates the emergent ``twin peaks'' in the
cross-country distribution documented, for instance, by Quah (1996). This
density estimate also corroborates the economic historian's notion of
convergence clubs; that is of countries catching up with one another but
only within particular subgroups. If one isolates the Neoclassical
convergence effect, then we obtain the density estimate displayed in the
lower right box. Note how much the graph rotates counter-clockwise. This
illustrates a potential for poor countries to overtake through the
Neoclassical convergence effect. In fact, the twin peaks scenario arises
mainly because of differences in the accumulation of reproducible factors
(see the lower left display). However, the model as it is specified does not
provide a perfect fit of the distribution dynamics at work under the period
under study. In particular, it does not allow us to recover and to explain
the formation of the poverty trap in the real distribution.
Still, counterfactual income dynamics as computed and analyzed above allow
us to provide explanations to the regularities characterizing the evolution
of the world income distribution. Although this is a new step in
understanding cross country patterns of growth, much remains to be done. At
least, this article provides an exercise which allows us to study the role
of specific explanatory factors in explaining observed patterns of
cross-country income distribution dynamics.