This paper proposes nonparametric kernel-smoothing estimation for panel data to examine the degree of heterogeneity across cross-sectional units. Our procedure is model-free and easy to implement, and provides useful visual information, which enables us to understand intuitively the properties of heterogeneity. We rst estimate the sample mean, autocovariances, and auto- correlations for each unit and then apply kernel smoothing to compute estimates of their density and cumulative distribution functions. The kernel estimators are consistent and asymptotically normal under double asymptotics, i.e., when both cross-sectional and time series sample sizes tend to in nity. However, as these exhibit biases given the incidental parameter problem and the nonlinearity of the kernel function, we propose jackknife methods to alleviate any bias . We also develop bandwidth selection methods and bootstrap inferences based on the asymptotic properties. Lastly, we illustrate the success of our procedure using an empirical application of the dynamics of US prices and Monte Carlo simulation .
This paper proposes nonparametric kernel-smoothing estimation for panel data to examine the degree of heterogeneity across cross-sectional units. We first estimate the sample mean, autocovariances, and autocorrelations for each unit and then apply kernel smoothing to compute their density functions. The dependence of the kernel estimator on bandwidth makes asymptotic bias of very high order affect the required condition on the relative magnitudes of the cross-sectional sample size (N) and the time-series length (T). In particular, it makes the condition on N and T stronger and more complicated than those typically observed in the long-panel literature . We also consider a split-panel jackknife method to correct bias, construction of confidence intervals, and estimation of cumulative distribution functions based on the density estimation. An empirical application and Monte Carlo simulations illustrate our procedure in finite samples .
R
Our procedure is model-free and easy to implement, and provides useful visual information, which enables us to understand intuitively the properties of heterogeneity. We rst
We first
R
auto- correlations
autocorrelations
R
estimates of their density and cumulative distribution
their density
R
kernel estimators are consistent and asymptotically normal under double asymptotics, i.e., when both
dependence of the kernel estimator on bandwidth makes asymptotic bias of very high order affect the required condition on the relative magnitudes of the
R
and time series sample sizes tend to in nity. However, as these exhibit biases given the incidental parameter problem and the nonlinearity of the kernel function, we propose jackknife methods to alleviate any bias
sample size (N) and the time-series length (T). In particular, it makes the condition on N and T stronger and more complicated than those typically observed in the long-panel literature
R
develop bandwidth selection methods and bootstrap inferences
consider a split-panel jackknife method to correct bias, construction of confidence intervals, and estimation of cumulative distribution functions
R
asymptotic properties. Lastly, we illustrate the success of our procedure using an empirical application of the dynamics of US prices
density estimation. An empirical application
R
simulation
simulations illustrate our procedure in finite samples