Consider the linear regression model y = XBo + u (1) where y is T x 1 with tth element yt, X is Tx k with tth row x, u is T x 1 with tth element ut, Bo is a k x 1 vector of unknown parameters. Assume that (1) is the true model for y, X is fixed in repeated samples, rank(X) k, E[u] = 0 and Var[u] = o²IT for some unknown scalar constant o. Let T = (X'X)−¹X'y that is, T is the Ordinary Least Squares (OLS) estimator of 30 based on (1). Let BT = Dy be an estimator of 30 based on (1) where D = (X'X)-¹X' + C and C is ak x T matrix of constants. =

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
icon
Related questions
Question
Consider the linear regression model
Y XBo + u
(1)
where y is T x 1 with tth element yt, X is T x k with tth row x, u is T × 1 with
tth element ut, Bo is a k x 1 vector of unknown parameters. Assume that (1) is
the true model for y, X is fixed in repeated samples, rank(X) k, E[u] = 0
and Var[u] = IT for some unknown scalar constant o. Let T (X'X)-¹X'y
that is, T is the Ordinary Least Squares (OLS) estimator of 30 based on (1). Let
BT = Dy be an estimator of 30 based on (1) where D = (X'X)-¹X' + C and C is
a kx T matrix of constants.
(a) Show that is an unbiased estimator of 30 if and only if CX = 0.
(b) If CX=0 then it follows that
=
Var[Br] - Var[T] =CC".
=
-
Show that CC' is a positive semi-definite matrix by construction.
(c) What do the results in parts (a)-(b) imply about the statistical properties of
the OLS estimator, ÔT?
Transcribed Image Text:Consider the linear regression model Y XBo + u (1) where y is T x 1 with tth element yt, X is T x k with tth row x, u is T × 1 with tth element ut, Bo is a k x 1 vector of unknown parameters. Assume that (1) is the true model for y, X is fixed in repeated samples, rank(X) k, E[u] = 0 and Var[u] = IT for some unknown scalar constant o. Let T (X'X)-¹X'y that is, T is the Ordinary Least Squares (OLS) estimator of 30 based on (1). Let BT = Dy be an estimator of 30 based on (1) where D = (X'X)-¹X' + C and C is a kx T matrix of constants. (a) Show that is an unbiased estimator of 30 if and only if CX = 0. (b) If CX=0 then it follows that = Var[Br] - Var[T] =CC". = - Show that CC' is a positive semi-definite matrix by construction. (c) What do the results in parts (a)-(b) imply about the statistical properties of the OLS estimator, ÔT?
Expert Solution
steps

Step by step

Solved in 5 steps with 19 images

Blurred answer
Similar questions
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman