Two stage least squares

lindah · 2011-08-27 18:31:26

HI guys,

May I ask if anyone here is a bit of a regression expert?

You are given a dependent variable y, and independent variable x such that the regression would be:

There is the possibility that both x and y have measurement errors in them.
You are also given the lagged (by one period) dataset of x and y (x_1 and y_1 respectively).

How would you go about fixing this error?
______________________________________________________________________________________________________________________
May I ask if my approach is correct?
I would use the lagged data to estimate the true values of x & y, so that the errors are no longer correlated so that the model would look as follows

Then this would be the equivalent of running a two way least squares regression with x_{1} and y_{1} as instrumental variables?

Thank you in advance for any feedback

Regards,
Linda

Last edited by lindah (2011-08-27 18:50:00)

bobbym · 2011-08-27 19:05:51

Hi lindah;

Pertaining to measurement error, if the data is a measurement then
of course there is a measurement error. If it is a floating point number
then there is an error. The least squares regression was designed for that.

lindah · 2011-08-27 19:15:07

Hi bobbym,

Thank you for looking

I was shown that if there was a measurement error in for e.g. X then we could write the error as follows:

where is the 'true" figure of X
If this were substituted into the original regression we would obtain

So the error term of this regression is affected by , so there is correlation between X and the error term which isn't ideal for a regression

Does it make it different in this context?

bobbym · 2011-08-27 19:23:56

Hi lindah;

For one thing this notation implies 2 sources of error.

So that notation is a little strange.
Also beta is an error magnifier but that
does not mean that the error term is a function of beta. Gauss invented least
squares to minimize the error in data that was a measurement.
Back then it was astronomical data for the orbit of Ceres.

lindah · 2011-08-27 19:33:19

Hi bobbym;

What I thought was any measurement errors were captured in

.

I'll take the excerpt as follows from the question:

Discuss the possibility that the estimated coefficients from the model are inconsistent due to measurement error bias. What strategies are available to reduce such bias?

I also found this:
http://support.sas.com/documentation/cd … ect003.htm

The question was in the context of Instrumental Variables Estimation, so it seems like my lecturer is pushing us to use this to correct for measuring errors.

Or am I confusing everyone in the process even more?

Thanks,
Linda

Last edited by lindah (2011-08-27 19:39:01)

bobbym · 2011-08-27 19:43:21

Hi lindah;

Yes, like a constant of integration it all gets combined up in one
error term.

I have not looked at the html yet, so please hold on.

Yikes, I think all that proves is that the guys who wrote SAS are a heck of a lot smarter than I am.

In their examples they have one thing you do not.
They know the distribution that the errors come from!
You do not.
You are forced to assume they are random!
I also can not assume a SNC for them.

lindah · 2011-08-27 20:15:34

Hi bobbym,

What's SNC in this context?

bobbym · 2011-08-27 20:18:41

Hi;

SNC is the standard
normal curve.

Sorry, I meant to say we do not even have the luxury
of saying they are normally distributed. We do not
even know what physical process the data models.

We have generic equations and no
real data, no process. Nothing to give
a clue as to how the errors might be distributed.

lindah · 2011-08-27 21:07:52

Hi bobbym;

Actually y = Treasury bill rate and x = inflation rates. For this question we were told to assume errors are normally distributed with E[e] = 0

It's such a weird question!!

bobbym · 2011-08-27 21:15:49

Hi lindah;

It is not weird now it is just tough. Is that saying they are
normally distributed with a mean of 0? But what is the standard
deviation?

I am going to get a little sleep, sorry but I am out of energy.
Will be back after a nap.

lindah · 2011-08-27 23:26:13

Hi bobbym,

Hope you have a good nap, wish I could have one!!!

The variance of the errors are just assumed to be

, with a mean of 0 - so i.i.d

I appreciate the feedback given!

bobbym · 2011-08-28 01:16:23

Hi lindah;

Do you know how to use the standard error for regression?

This might clear up one or two points.

http://en.wikipedia.org/wiki/Regression_analysis

http://en.wikipedia.org/wiki/Errors-in-variables_model

lindah · 2011-08-29 19:23:00

Hi bobbym;

Thanks for the links. The errors-in-variables methods were what I had to address - so finding instrument variables for the independent variable in question
In my case there was no need to account for measurement errors in the dependent variable.

Thanks for posting the links!!

bobbym · 2011-08-29 19:26:38

Hi lindah;

Your welcome. Glad the links helped they helped me alot.

Incidentally the whole subject of least squares starts with both Legendre and Gauss. Both got the idea but Legendre published first. Like Newton, Leibniz and calculus they both argued over the rights to it.

Math Is Fun Forum

#1 2011-08-27 18:31:26

Two stage least squares

#2 2011-08-27 19:05:51

Re: Two stage least squares

#3 2011-08-27 19:15:07

Re: Two stage least squares

#4 2011-08-27 19:23:56

Re: Two stage least squares

#5 2011-08-27 19:33:19

Re: Two stage least squares

#6 2011-08-27 19:43:21

Re: Two stage least squares

#7 2011-08-27 20:15:34

Re: Two stage least squares

#8 2011-08-27 20:18:41

Re: Two stage least squares

#9 2011-08-27 21:07:52

Re: Two stage least squares

#10 2011-08-27 21:15:49

Re: Two stage least squares

#11 2011-08-27 23:26:13

Re: Two stage least squares

#12 2011-08-28 01:16:23

Re: Two stage least squares

#13 2011-08-29 19:23:00

Re: Two stage least squares

#14 2011-08-29 19:26:38

Re: Two stage least squares

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