1
Partitioned Matrix
Consider the following matrix
a11
a21
A =
.
(pq )
ap1
It can be "partitioned" as
A=
a12
a22
.
ap2
. a1q
. a2q
. .
. apq
A11
A12
(rs)
(r (q s)
(pr)s)
(pr )(q s)
A21
A22
125
For example, A = 3 1 1 can be partitioned as A =
076
3
1
w
Extensions to Tobit and Heckit regression
models
1
Multiple censoring points
A useful extension to the traditional Tobit model
yi
=
yi
=
0
+
yi
0
1 xi
+ ui
if yi > 0
if yi
0
is when data are subject to two forms of censoring. For example, yi could be
full
Marginal eects
Suppose our model for the latent variable y is:
y=
+ x+u
We have studied a number of cases of interest:
A. The econometrician observes y = y . Then
@E (y jx)
=
@x
B. Probit model: y = 1 (y > 0). The marginal eect of interest is:
@E (y jx)
@
Other Discrete Dependent Variable Models
Our model is still
yi =
+ xi + ui
where yi is a latent dependent variable (for example, net utility associated to
making a certain choice), and the only thing we observe is the index:
1
0
yi =
1
if yi
0
if yi < 0
L
Discrete Dependent Variables
We are now going to deal with discrete dependent variables. An example is
a binary variable
yi =
1
0
if person i works
otherwise
The objective of regression models involving discrete dependent variables is
to understand what v
Economics 102C - Stanford University
Prof. Luigi Pistaferri
Program evaluation
1. Introduction
Consider an individual who is faced with the problem of choosing whether to take or not
a given "treatment". The type of interventions or treatments one might c
HYPOTHESIS TESTING
1
Linear Hypotheses
Suppose you want to test a set of linear hypotheses on the vector of parameters
from the usual linear model
y = X + u
under the assumption that OLS is unbiased and consistent. The way this is
written in matrix algeb
Panel data
1
Panel data
Cross-sectional data typically collect information on N units (individuals, rms,
countries, etc.) at a given moment in time. Time-series data collect information
on a single unit (typically, a country) over many (T ) time periods.
Problems with the IV estimator
So far, we have only discussed the merits of the IV estimator. We will now also discuss its
weaknesses.
Problems with the IV estimator
1
1.1
High standard errors
One of the principal weaknesses of the IV estimator is that it
Law of iterated expectations (LIE)
This law states that
E(y) = Ex E(y | x)
where Ex means that the expectation is calculated with respect to the distribution of x. In
other words
b
E ( y ) = E ( y | x ) f (x )dx
a
where a and b are the lower and upper sup
MATRIX ALGEBRA
Why is it useful?
- simple k-variables OLS expressions (and estimators other than OLS)
- any data set you deal with is a big matrix
- useful at the programming stage (i.e., for Gauss, Matlab not for Stata)
- its elegant!
A matrix is a recta