NONPARAMETRIC ESTIMATION OF A CONDITIONAL MEAN FUNCTION
Let Y and X be random variables. These notes are concerned with estimating the
conditional mean function g ( x) = E (Y | X = x) without making any assumptions about its shape.
The data consist of n o
Econometrics 381: Section 4
January 28, 2011
1. Practice problem 1:
Let Y = 0 + 1 X + U and E(Ui |X) = c, E(Ui2 |X) = 2 (Xi X)2 + c2 , where c is some
constant dierent from 0. Suppose you have a random sample of size n. Let 0 and 1 be
OLS estimates of 0 a
NOTES ON THE MULTIVARIATE NORMAL AND CHI-SQUARE DISTRIBUTION AND ON
THE POWER OF HYPOTHESIS TESTS
I.
The Multivariate Normal and Chi-Square Distributions
A. A random vector X with dim( X ) = d has the multivariate normal distribution with mean vector
and
THE RESET TEST
This is a test of the hypothesis that a regression model has the right functional form. Let
the model be
(1)
Y = 0 + 1 X 1 + . + K X K + U ; E (U | X 1 ,., X K ) = 0 .
In this model, Y may be the logarithm of another variable, and some or a
THE PE TEST
Consider a linear and log-linear model. ssume for simplicity that there is only one righthand side variable. Additional right-hand side variables can be included in the usual way.
The linear model is
Y = 0 + 1 X + U .
The log-linear model is
l
NOTES ON KERNEL NONPARAMETRIC REGRESSION
The problem is to estimate the function g in the model
Y = g ( X ) + U ; E (U | X ) = 0 .
In other words, we want to estimate the conditional mean function g ( x) = E (Y | X = x) . The data
are a random sample cfw_
Applied Econometrics 281, Handout 2
1
Joint distributions
1. Consider the joint distribution of income X and education Y .
Y =1 Y =2 Y =3
X=5
0.22
0.15
0.08
X = 10
0.11
0.14
0.10
X = 20
0.05
0.07
0.08
Calculate E(X|Y ) for dierent values of Y . Calculate