The package enables to estimate the degree of heterogeneity across
cross-sectional units in panel data. The methods are developed by:
Setup
We have balanced panel data \(\{ y_{it}
\}\) for \(i = 1, \dots, N\) and
\(t = 1, \dots, T\), where
- \(y_{it}\): A scalar random
variable for unit \(i\) in period \(t\).
- \(N\): The number of
cross-sectional units.
- \(T\): The length of time
series.
Assume that the individual time series \(\{
y_{it} \}\) for \(t = 1, \dots,
T\) is strictly stationary across time but heterogeneous across
units.
To examine the degree of heterogeneity, we focus on estimating the
cumulative distribution function (CDF), density function, and moments
(i.e., the means, variances, and correlations) of the heterogeneous mean
\(\mu_i\), \(k\)-th autocovariance \(\gamma_{k,i}\), and \(k\)-th autocorrelation \(\rho_{k,i}\):
- \(\mu_i = E(y_{it} \mid i)\),
- \(\gamma_{k,i} = E[(y_{it} -
\mu_i)(y_{i,t-k} - \mu_i) \mid i]\),
- \(\rho_{k,i} = \gamma_{k,i} /
\gamma_{0,i}\),
where \(E(\cdot \mid i)\) denotes
the population mean for the individual time series. In words, \(\mu_i\), \(\gamma_{k,i}\), and \(\rho_{k,i}\) are the population mean, \(k\)-th autocovariance, and \(k\)-th autocorrelation, respectively, for
the individual time series. Note that \(\gamma_{0, i}\) corresponds to the
heterogeneous variance. These variables are treated as i.i.d. random
variables across units.
The estimators are the corresponding sample analogues:
- \(\hat \mu_i = \frac{1}{T} \sum_{t=1}^T
y_{it}\).
- \(\hat \gamma_{k,i} = \frac{1}{T-k}
\sum_{t=k+1}^T (y_{it} - \hat \mu_i)(y_{i,t-k} - \hat
\mu_i)\).
- \(\hat \rho_{k,i} = \hat \gamma_{k,i} /
\hat \gamma_{0,i}\).
Denoting \(\xi_i = \mu_i\), \(\gamma_{k,i}\), or \(\rho_{k,i}\), the parameters of interest
are:
- The CDF \(F_{\xi}(\cdot) = \Pr(\xi_i \le
\cdot)\).
- The density function \(f_{\xi}(\cdot)\).
- The mean \(E(\xi_i)\).
- The variance \(var(\xi_i)\).
- The correlations, such as \(cor(\mu_i,
\gamma_{k,i})\).
Okui and Yanagi (2019) propose the empirical distribution estimation
of the CDF and the moments of the true \(\mu_i\), \(\gamma_{k,i}\), and \(\rho_{k,i}\) using the estimated \(\hat \mu_i\), \(\hat \gamma_{k,i}\), and \(\hat \rho_{k,i}\). Okui and Yanagi (2020)
propose the nonparametric kernel smoothing estimation for the density of
the true \(\mu_i\), \(\gamma_{k,i}\), and \(\rho_{k,i}\) using the estimated \(\hat \mu_i\), \(\hat \gamma_{k,i}\), and \(\hat \rho_{k,i}\). Since these estimators
have asymptotic biases, they propose the half-panel jaccknife (HPJ) bias
correction and the third-order jackknife (TOJ) bias correction. In
addition, they also consider the cross-sectional bootstrap procedures
for performing statistical inference.
The HPJ and TOJ bias correction requires \(T \ge 4\) and \(T
\ge 6\), respectively.
Functions
The panelhetero package provides the following
functions:
nemoment(): The naive estimation of the moments of the
heterogeneous mean, the heterogeneous autocovariance, and the
heterogeneous autocorrelation without bias correctionhpjmoment(): The HPJ bias-corrected estimation of the
moments of the heterogeneous mean, the heterogeneous autocovariance, and
the heterogeneous autocorrelationtojmoment(): The TOJ bias-corrected estimation of the
moments of the heterogeneous mean, the heterogeneous autocovariance, and
the heterogeneous autocorrelationneecdf(): The naive empirical CDF estimation without
bias correctionhpjecdf(): The HPJ bias-corrected empirical CDF
estimationtojecdf(): The TOJ bias-corrected empirical CDF
estimationnekd(): The naive kernel density estimation without
bias correctionhpjkd(): The HPJ bias-corrected kernel density
estimationtojkd(): The TOJ bias-corrected kernel density
estimation
Below we discuss the arguments and the returned values of each
function.
nemoment()
The nemoment() function enables to implement the naive
estimation of the moments of the heterogeneous quantities without bias
correction. The parameters of interest are the means, variances, and
correlations:
- \(E(\mu_i)\)
- \(E(\gamma_{k,i})\)
- \(E(\rho_{k,i})\)
- \(var(\mu_i)\)
- \(var(\gamma_{k,i})\)
- \(var(\rho_{k,i})\)
- \(cor(\mu_i, \gamma_{k,i})\)
- \(cor(\mu_i, \rho_{k,i})\)
- \(cor(\gamma_{k,i},
\rho_{k,i})\)
The usage is
nemoment(data, acov_order = 0, acor_order = 1, R = 1000)
with:
data: An \(N \times
T\) matrix of panel data. Each row corresponds to individual time
series.acov_order: A non-negative integer \(k\) of the order of autocovariance. Default
is 0.acor_order: A positive integer \(k\) of the order of autocorrelation.
Default is 1.R: A positive integer of the number of bootstrap
repetitions. Default is 1000.
The function returns a list that contains the following elements:
estimate: A vector of the parameter estimatesse: A vector of the standard errors computed by the
cross-sectional bootstrapci: A matrix of the 95% confidence intervals computed
by the cross-sectional bootstrapquantity: A matrix of the estimated heterogeneous
quantitiesacov_order: The order of autocovarianceacor_order: The order of autocorrelationN: The number of cross-sectional unitsS: The length of time seriesR: The number of bootstrap repetitions
Note: The bootstrap results depend on random number generation. It is
highly recommended to use set.seed() before
nemoment(). The same comment also applies to the other
functions.
hpjmoment()
The hpjmoment() function enables to implement the HPJ
bias-corrected estimation of the moments of the heterogeneous quantities
The parameters of interest are the same as nemoment(). The
usage is
hpjmoment(data, acov_order = 0, acor_order = 1, R = 1000)
with the same arguments as nemoment(). The function returns
a list that contains the same elements as nemoment().
tojmoment()
The tpjmoment() function enables to implement the TOJ
bias-corrected estimation of the moments of the heterogeneous quantities
The parameters of interest are the same as nemoment(). The
usage is
tojmoment(data, acov_order = 0, acor_order = 1, R = 1000)
with the same arguments as nemoment(). The function returns
a list that contains the same elements as nemoment().
neecdf()
The neecdf() function enables to implement the naive
empirical CDF estimation without bias correction. The parameters of
interest are the CDFs of \(\mu_i\),
\(\gamma_{k,i}\), and \(\rho_{k,i}\).
The usage is
neecdf(data, acov_order = 0, acor_order = 1, R = 1000, ci = TRUE)
with:
data: An \(N \times
T\) matrix of panel data. Each row corresponds to individual time
series.acov_order: A non-negative integer \(k\) of the order of autocovariance. Default
is 0.acor_order: A positive integer \(k\) of the order of autocorrelation.
Default is 1.R: A positive integer of the number of bootstrap
repetitions. Default is 1000.ci: A logical whether to compute the 95% confidence
interval for the CDF. Default is TRUE.
The function returns a list that contains the following elements:
mean: A plot of the estimated distribution of the
heterogeneous mean made by ggplot2::stat_function()acov: A plot of the estimated distribution of the
heterogeneous autocovariance made by
ggplot2::stat_function()acor: A plot of the estimated distribution of the
heterogeneous autocorrelation made by
ggplot2::stat_function()mean_func(): A function that returns the value of the
estimated distribution of the heterogeneous meanacov_func(): A function that returns the value of the
estimated distribution of the heterogeneous autocovarianceacor_func(): A function that returns the value of the
estimated distribution of the heterogeneous autocorrelationmean_ci_func(): A function that returns the 95%
confidence interval for the estimated distribution of the heterogeneous
meanacov_ci_func(): A function that returns the 95%
confidence interval for the estimated distribution of the heterogeneous
autocovarianceacor_ci_func(): A function that returns the 95%
confidence interval for the estimated distribution of the heterogeneous
autocorrelationquantity: A matrix of the estimated means,
autocovariances, and autocorrelationsacov_order: The order of autocovarianceacor_order: The order of autocorrelationN: The number of cross-sectional unitsS: The length of time seriesR: The number of bootstrap repetitions
Note: In each plot, x-axis limits are set to the minimum and maximum
of the estimated quantity.
hpjecdf()
The hpjecdf() function enables to implement the HPJ
bias-corrected empirical CDF estimation. The parameters of interest are
the same as neecdf(). The usage is
hpjecdf(data, acov_order = 0, acor_order = 1, R = 1000, ci = TRUE)
with the same arguments as neecdf(). The function returns a
list that contains the same elements as neecdf().
Note 1: Due to bias correction, the HPJ bias-corrected empirical
distribution estimator may be a non-monotonic function. To address this
issue, the HPJ bias-corrected estimator is converted into a monotonic
function using the Rearrangement package.
Note 2: The mean_func(), acov_func(), and
acor_func() functions returned by hpjecdf()
are not subject to rearrangement. If desired, you can manually use the
Rearrangement package to get a rearrangement
estimate.
Note 3: Due to bias correction, the HPJ or TOJ bias-corrected
empirical distribution estimate may be less than 0 or greater than 1. We
adjust the estimate to be between 0 and 1.
tojecdf()
The hpjecdf() function enables to implement the HPJ
bias-corrected empirical CDF estimation. The parameters of interest are
the same as neecdf(). The usage is
hpjecdf(data, acov_order = 0, acor_order = 1, R = 1000, ci = TRUE)
with the same arguments as neecdf(). The function returns a
list that contains the same elements as neecdf().
Note: The same comments about hpjecdf() also apply to
tojecdf().
nekd()
The nekd() function enables to implement the naive
kernel density estimation without bias correction using the Gaussian
kernel. The parameters of the interest are the density function of \(\mu_i\), that of \(\gamma_{k,i}\), and that of \(\rho_{k,i}\).
The usage is
nekd(data, acov_order = 0, acor_order = 1, mean_bw = NULL, acov_bw = NULL, acor_bw = NULL)
with:
data: An \(N \times
T\) matrix of panel data. Each row corresponds to individual time
series.acov_order: A non-negative integer \(k\) of the order of autocovariance. Default
is 0.acor_order: A positive integer \(k\) of the order of autocorrelation.
Default is 1.mean_bw: The bandwidth used for estimating the density
function of the heterogeneous mean. Default is NULL, and the plug-in
bandwidth is obtained from KernSmooth::dpik().acov_bw: The bandwidth used for estimating the density
function of the heterogeneous autocovariance. Default is NULL, and the
plug-in bandwidth is obtained from KernSmooth::dpik().acor_bw: The bandwidth used for estimating the density
function of the heterogeneous autocorrelation. Default is NULL, and the
plug-in bandwidth is obtained from KernSmooth::dpik().
The function returns a list that contains the following elements:
mean: A plot of the estimated density of the
heterogeneous mean made by ggplot2::stat_function()acov: A plot of the estimated density of the
heterogeneous autocovariance made by
ggplot2::stat_function()acor: A plot of the estimated density of the
heterogeneous autocorrelation made by
ggplot2::stat_function()mean_func(): A function that returns the value of the
estimated density of the heterogeneous meanacov_func(): A function that returns the value of the
estimated density of the heterogeneous autocovarianceacor_func(): A function that returns the value of the
estimated density of the heterogeneous autocorrelationbandwidth: A vector of the bandwidthsquantity: A matrix of the estimated means,
autocovariances, and autocorrelationsacov_order: The order of autocovarianceacor_order: The order of autocorrelationN: The number of cross-sectional unitsS: The length of time series
Note: In each plot, x-axis limits are set to the minimum and maximum
of the estimated quantity.
hpjkd()
The hpjkd() function enables to implement the HPJ
bias-corrected kernel density estimation. The parameters of interest are
the same as nekd(). The usage is
hpjkd(data, acov_order = 0, acor_order = 1, mean_bw = NULL, acov_bw = NULL, acor_bw = NULL)
with the same arguments as nekd(). The function returns a
list that contains the same elements as nekd().
Note: Due to bias correction, the HPJ bias-corrected density estimate
may be less than 0. The estimate is adjusted so that it is not less than
0.
tojkd()
The tojkd() function enables to implement the TOJ
bias-corrected kernel density estimation. The parameters of interest are
the same as nekd(). The usage is
tojkd(data, acov_order = 0, acor_order = 1, mean_bw = NULL, acov_bw = NULL, acor_bw = NULL)
with the same arguments as nekd(). The function returns a
list that contains the same elements as nekd().
Note: The same comment about hpjkd() also applies to
tojkd().
Examples
The following simple example illustrates the use of the
panelhetero package.
To begin with, install the package with:
install.packages("panelhetero")
library(panelhetero)
Or
# install.packages("devtools") # if necessary
# install.packages("ggplot2") # if necessary
devtools::install_github("tkhdyanagi/panelhetero", build_vignettes = TRUE)
library(panelhetero)
Then, using the simulation() function, generate
artificial data from an AR(1) model with random coefficients as
follows.
set.seed(1)
y <- panelhetero::simulation(N = 300, S = 8)
The nemoment(), hpjmoment(),
tojmoment() functions enable to estimate the moments of the
heterogeneous quantities. For example, you can implement the HPJ
bias-corrected estimation as follows.
result1 <- hpjmoment(data = y, acov_order = 0, acor_order = 1)
result1$estimate
#> E(mean) E(acov) E(acor) var(mean) var(acov)
#> 0.045292150 0.389074482 0.359539625 0.900338806 0.073253948
#> var(acor) cor(mean, acov) cor(mean, acor) cor(acov, acor)
#> 0.072506331 -0.004269697 -0.004974914 0.004231555
result1$se
#> E(mean) E(acov) E(acor) var(mean) var(acov)
#> 0.05726649 0.02105856 0.03394039 0.08499369 0.01637006
#> var(acor) cor(mean, acov) cor(mean, acor) cor(acov, acor)
#> 0.01602074 0.09375047 0.11572946 0.08964072
result1$ci
#> 95% CI lower 95% CI upper
#> E(mean) -0.06258859 0.1569463
#> E(acov) 0.34940286 0.4290926
#> E(acor) 0.29244305 0.4238191
#> var(mean) 0.74091639 1.0741703
#> var(acov) 0.04360967 0.1067161
#> var(acor) 0.04252445 0.1048688
#> cor(mean, acov) -0.18701211 0.1735921
#> cor(mean, acor) -0.22024162 0.2249527
#> cor(acov, acor) -0.17275447 0.1712729
The neecdf(), hpjecdf(),
tojecdf() functions enable to implement the empirical CDF
estimation. For example, you can implement the HPJ bias-corrected
empirical CDF estimation with:
result2 <- hpjecdf(data = y,
acov_order = 0,
acor_order = 1,
R = 100,
ci = FALSE)
result2$mean
The nekd(), hpjkd(), tojkd()
functions enable to implement the kernel density estimation. For
example, you can implement the HPJ bias-corrected kernel density
estimation with:
result3 <- hpjkd(data = y, acov_order = 0, acor_order = 1)
result3$mean
Each function makes this type of figure using the
ggplot2 package. You can customize it via standard
commands for the ggplot2 package. For example, we can
customize the title and the theme as follows.
result3$mean +
ggtitle("") +
theme_classic()
The neecdf(), hpjecdf(),
tojecdf(), nekd(), hpjkd(), and
tojkd() functions return the mean_func(),
acov_func(), and acor_func() functions that
return the estimated CDFs and the estimated density functions. Using
these functions, you can make figures by yourself. For example, you can
make a figure of the HPJ bias-corrected density for the heterogeneous
mean as follows.
ggplot(data = data.frame(x = c(-3.5, 3.5)), aes(x = x)) +
stat_function(fun = result3$mean_func) +
ggtitle("")
