# Endogeneity test

Discussion in 'Scientific Statistics Math' started by minimus, Dec 14, 2010.

1. ### minimusGuest

Hello,

I asked it before but could not get a response to my question that I state
below. I would like to once again ask if someone would be willing to discuss
with me my question...

Thanks and kind regards.

Let us consider the following equations:

y = betazero + betaone x + u (1)
x = alfaone z + e (2)

Suppose x is endogenous in (1). Suppose z is an instrument for x in (2).
If I understand it correctly, the logic of the endogeneity test goes as
follows.

"x" has two components: "z" and "e".

"z" is uncorrelated with "u". Hence, the only way "x" is correlated with "u"
is if "e" is correlated with "u".

Hence we cast the following equation, where n is an error term, and test if
tetaone is 0:

u = tetaone e + n (3)

We do not observe u and e but could get the residuals from (1) and (2) and
test this.
But ok this does not work because residuals for u are by construction
uncorrelated with x.

Then we consider the following equation, where we predict e from equation
(2) and test if tetaone is 0:

y = betazero + betaone x + tetaone e + n (4)

If it turns out that tetaone is not zero then we have endogeneity.

Question 1:

This last sentence is my problem. Why do I conclude that there is
endogeneity if tetaone is not zero? The sort of logic I can think of is the
following but it does not convince me:

In equation (1) I replace "u" with "tetaone e + n". If it turns out that
"tetaone" is significant, this will mean that we leave "e" in "u" in
equation (1), which then means that "e" is part of "u"? But this does not
tell me that e is a determinant of u. Ok I am confused here probably and
don't quite know how I should look at the equations below.

y = betazero + betaone x + u (1)
y = betazero + betaone x + tetaone e + n (4)

Question 2:

From (2) we know that

e = x - alfaone z

I consider predicted e in (4). But (4) has the "x" variable and "e" is also
defined by "x". Then x and predicted e will be perfectly correlated by
construction right? Why is this not a problem in (4)?

minimus, Dec 14, 2010

2. ### Art KendallGuest

From your use of "endogenous" and "instrument" I speculated that this
is an econometrics concept. perhaps you should look for an econometrics
newsgroups or discussion list.
I found many hits when I Googled "Hausman endogeneity test" e.g.,
http://en.wikipedia.org/wiki/Hausman_test

Art

Art Kendall, Dec 14, 2010

3. ### minimusGuest

Yes, it is an econometrics concept but I could not find any other group
whose name suggests to be more suitable for the question I am asking. For
googling the term: No, that does not and will not help. I am asking a

minimus, Dec 14, 2010
4. ### Bruce WeaverGuest

Here are a couple possibilities.

http://www.forumjar.com/forums/Econometrics

I'm not an econometrician, so do not recognize most of the mailing
lists from that second link. I do recognize the Stata list though,
and believe that Stata is pretty popular with folks who do
econometrics. So you might try there.

Bruce Weaver, Dec 14, 2010
5. ### Richard StartzGuest

You might check the textbook by Davidson and MacKinnon.

Richard Startz, Dec 15, 2010
6. ### minimusGuest

Thanks. I will have a look at it.

minimus, Dec 15, 2010