- (Serial Correlation) Consider the following time series regression model Yt = Bo+ B1x, +, t = 1,,T Et = Ut + pult-1, PE (0, 1), where rt and ut are i.i.d. over t and are independent of each other. Assume Eu be the OLS estimator of B1 from regressing yt on at (with an intercept). For simplicity, in the following = assume {r : 1
- (Serial Correlation) Consider the following time series regression model Yt = Bo+ B1x, +, t = 1,,T Et = Ut + pult-1, PE (0, 1), where rt and ut are i.i.d. over t and are independent of each other. Assume Eu be the OLS estimator of B1 from regressing yt on at (with an intercept). For simplicity, in the following = assume {r : 1
MATLAB: An Introduction with Applications
6th Edition
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Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![2. (Serial Correlation) Consider the following time series regression model
Yt = Bo + B1t +,
t = 1,..,T
Et = Ut + pUt-1,
PE (0, 1),
where x and ut are i.i.d. over t and are independent of each other. Assume E[u] = 0, E[u?] = o². Let B1
be the OLS estimator of B1 from regressing yt on xt (with an intercept). For simplicity, in the following we
assume {rt :1<t<T} are fixed (non-random).
1
(a) Show that
V[e] = (1+ p²)o², Cov[et, &t–1] = po², Covlet, et-j] = 0 for j > 2.
(b) Let i = -2=1 Tt. Use your answer to part (a) to show that
T.
T.
vE(a; – 1)6 = (1+ p²)o² E(# – #)² + 2po² (a - )(x1-1 – ).
V
t=1
t=1
t=2
(Hint: use the fact that for generic random variables z1, VIE-1 21] = E V[zı]+2E-2 E Cou[zt, 21-e).)
t%3D2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe960b7b-8746-4256-b957-606edabfae78%2F6c504469-c5ce-4932-95d4-369ffdc942a2%2Fx8xtmg9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. (Serial Correlation) Consider the following time series regression model
Yt = Bo + B1t +,
t = 1,..,T
Et = Ut + pUt-1,
PE (0, 1),
where x and ut are i.i.d. over t and are independent of each other. Assume E[u] = 0, E[u?] = o². Let B1
be the OLS estimator of B1 from regressing yt on xt (with an intercept). For simplicity, in the following we
assume {rt :1<t<T} are fixed (non-random).
1
(a) Show that
V[e] = (1+ p²)o², Cov[et, &t–1] = po², Covlet, et-j] = 0 for j > 2.
(b) Let i = -2=1 Tt. Use your answer to part (a) to show that
T.
T.
vE(a; – 1)6 = (1+ p²)o² E(# – #)² + 2po² (a - )(x1-1 – ).
V
t=1
t=1
t=2
(Hint: use the fact that for generic random variables z1, VIE-1 21] = E V[zı]+2E-2 E Cou[zt, 21-e).)
t%3D2
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