Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.22...
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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.22 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: http:// learningresources.lse.ac.uk/141/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/
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Transcript of Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.22...
Christopher Dougherty
EC220 - Introduction to econometrics (review chapter)Slideshow: exercise r.22
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/141/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
EXERCISE R.22
R.22 A random variable X has unknown population mean X and population variance X .
A sample of n observations {X1, ..., Xn} is generated.Show that
is an unbiased estimator of X.
Show that the variance of Z does not tend to zero as n tendsto infinity and that therefore Z is an inconsistent estimator,despite being unbiased.
1
nnnn XXXXXZ 111321 21
21
...81
41
21
2
EXERCISE R.22
2
nnnn
nnnn
nnnn
XEXEXEXEXE
XEXE
XEXEXE
XXXXXEZE
111321
111
321
111321
21
21
...81
41
21
21
21
...
81
41
21
21
21
...81
41
21
To demonstrate unbiasedness, we start by using the first expected value rule to decompose the expression and then the second to take the factors of 2 out of each expectation.
EXERCISE R.22
3
X
Xnn
XnXnXXX
nnnn XEXEXEXEXEZE
11
11
111321
21
21
...81
41
21
21
21
...81
41
21
21
21
...81
41
21
Each expectation is equal to X. (We are thinking about the sample at the planning stage, before the sample is actually generated.) The coefficients of X add to 1, so the estimator is unbiased.
EXERCISE R.22
4
nnnn
nnnn
nnnn
XX
XXX
XX
XXX
XXXXXZ
var21
var21
...
var641
var161
var41
21
var21
var...
81
var41
var21
var
21
21
...81
41
21
varvar
22122
321
111
321
111321
Now consider the variance. Using variance rule 1, we can decompose it into the sum of the variances plus twice the covariance of each observation with every other observation.
EXERCISE R.22
5
nnnn
nnnn
nnnn
XX
XXX
XX
XXX
XXXXXZ
var21
var21
...
var641
var161
var41
21
var21
var...
81
var41
var21
var
21
21
...81
41
21
varvar
22122
321
111
321
111321
We are assuming that the observations are generated independently and hence that the covariances are all zero.
EXERCISE R.22
6
nnnn
nnnn
nnnn
XX
XXX
XX
XXX
XXXXXZ
var21
var21
...
var641
var161
var41
21
var21
var...
81
var41
var21
var
21
21
...81
41
21
varvar
22122
321
111
321
111321
We use variance rule 2 to take the coefficients out of the variances. Remember that we have to square the coefficients when we do this.
EXERCISE R.22
7
22
22222
222
222
222
22122
321
31
121
131
21
21
...641
161
41
21
21
...641
161
41
var21
var21
...
var641
var161
var41
var
XX
Xnn
XnXnXXX
nnnn
n
XX
XXXZ
Each variance is equal to X. (Again, we are thinking about the sample at the planning stage, before the sample is actually generated.)
2
EXERCISE R.22
8
22
22222
222
222
222
22122
321
31
121
131
21
21
...641
161
41
21
21
...641
161
41
var21
var21
...
var641
var161
var41
var
XX
Xnn
XnXnXXX
nnnn
n
XX
XXXZ
Hence we obtain an expression for the variance. It decreases as n increases.
EXERCISE R.22
9
22
22222
222
222
222
22122
321
31
121
131
21
21
...641
161
41
21
21
...641
161
41
var21
var21
...
var641
var161
var41
var
XX
Xnn
XnXnXXX
nnnn
n
XX
XXXZ
But it does not tend to zero. The distribution of the estimator does not collapse to a spike and so it is inconsistent.
EXERCISE R.22
10
22
22222
222
222
222
22122
321
31
121
131
21
21
...641
161
41
21
21
...641
161
41
var21
var21
...
var641
var161
var41
var
XX
Xnn
XnXnXXX
nnnn
n
XX
XXXZ
The reason is that, as n increases, the additional observations are being given sharply decreasing weights, and hence the estimator is not benefiting from the increase in the sample size.
nnnn XXXXXZ 111321 21
21
...81
41
21
Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use.
27.08.06
