代做Nonparametric Econometrics Problem Set 1 2024帮做R程序

Nonparametric Econometrics

Problem Set 1

Due March 12, 2024

Exercises

1. The Kernel density estimator is defined as

where X1, X2, . . . , Xn are iid observations. If the Kernel function K(·) satisfies that ( i ) ∫ K(u)du = 1; ( ii ) K(u) = K (−u); ( iii ) ∫ u2K(u)du = κ2 < ∞, prove that

2. In our lectures, we define a CDF estimator called ”Empirical Distribution”, i.e.

where X1, X2, . . . , Xn are iid observations with CDF F(·).

(a) Find the bias, variance and MSE of this empirical distribution estimator.

(b) Show that F(x) − F(x) = Op(n −1/2).

3. Suppose that you have a sample of wages for people working in Beijing, Shanghai and Shenzhen. For each city, you collect 500 observations. You estimate the density functions fbj (x), fsh(x) and fsz(x) for each group using the same bandwidth, respectively. Can you use (fbj(x) + fsh(x) + fsz(x)) / 3 to estimate the density function of the combined sample?

4. (Simulation) ( i ) Generate an iid sample from standard normal distribution N(0, 1) with sample size n = 300. ( ii ) Estimate the density function using Epanechnikov kernel K(u) = 0.75(1 − u 2 )1{|u| ≤ 1}. The bandwidth is h = 0.2, 0.5 and 1. ( iii ) Plot standard normal density function and your three estimated density functions in a single figure. Discuss the importance of bandwidth.

5. Exercise 1.17 in Li and Racine (2007).

Hint: The data Italy wages is available in R-package "np" . You can install this package by

install . packages (’np ’)

Then data Italy can be loaded by

library (’np ’)

data ( Italy )

summary ( Italy )