ECMT3150: The Econometrics of Financial Markets (Semester 1, 2023) Tutorial 1c
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ECMT3150: The Econometrics of Financial Markets
(Semester 1, 2023)
Tutorial 1c
1. Consider the random walk model: for t > 1;
yt = yt — 1 + "t ;
where {"t } ~ wn(0;a).
(a) Show that y^t (`) = E[yt+`|Ft] = yt for all ` > 0.
(b) Show that the forecast error e^t (`) := yt+` - y^t (`) has variance `a , which diverges as ` 二 o (i.e., hopeless to forecast RW in the distant future).
(c) DeÖne the ACF as pt;j = . Show that pt;j 二 1 as t 二 o for all Öxed lag j (long memory).
2. [RT Ex: 2.7] Consider the daily simple returns of IBM stock, CRSP value-weighted index, CRSP equal-weighted index, and the S&P composite index from January 1980 to December
2008. The index returns include dividend distributions.
The data Öle is d-ibm3dxwkdays8008 .txt, which has 12 columns. The columns are (year, month, day, IBM, VW, EW, SP, M, T, W, H, F), where M, T, W, R, and F denotes indicator variables for Monday to Friday, respectively.
Use a regression model to study the e§ects of trading days on the equal weighted index returns. What is the Ötted model? Are the weekday e§ects signiÖcant in the returns at the 5% level? Use the HAC estimator of the covariance matrix to obtain the t ratio of regression estimates. Does the HAC estimator change the conclusion of weekday e§ects? Are there serial correlations in the regression residuals? If yes, build a regression model with time series error to study weekday e§ects.
3. [Mid-sem exam 2018s1]
(a) (10 marks) You are given with the following Öve data generating processes (DGPs). They all
share the same model on the level:
yt = 8yt — 1 + ut ; I8I < 1; (1)
but with di§erent error dynamics as follows:
DGP 1 : ut = 2"t with {"t } ~ wn(0; 1).
DGP 2 : ut = 2"t with {"t } ~ mds and Var("t I"t — 1 ; "t —2 ; : : :) = 2t: DGP 3 : ut = 0:8ut — 1 + "t , with {"t } ~ wn(0; 1).
DGP 4 : ut = "t + 0:3"t — 1 , with {"t } ~ wn(0; 1).
DGP 5 : ut = 7t "t , {"t } ~ iid N (0; 1), 7t(2) = 0:01 + 0:3ut(2)— 1 .
Carol, after taking ECMT2150, attempts to make inference on 8 under each of the Öve DGPs, by running a simple linear regression of yt on yt — 1 . The ordinary least squares estimator OLS and
its asymptotic variance estimator V一ar ( OLS ) are given by
OLS = —2t V一ar ( OLS ) = 一 yt(2)— 1 ;
where s2 is the sample variance of the regression residuals.
Copy the following table to your answer booklet and complete with the correct answers. Each correct answer is worth one mark. You do not need to show intermediate steps.
DGP |
OLS consistent? (yes/no) |
V一ar (OLS ) consistent? (yes/no) |
1 |
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2 |
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3 |
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4 |
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5 |
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(b) (3 marks) Suppose the DGP is given by (1) with {ut } ~ wn(0; 1), except that 8 = 1. Carol
wants to run a t-test, and decides to reject the null hypothesis of 8 = 1 if the t statistic
t = s=
exceeds the critical value of the student t distribution with n - 1 degrees of freedom. Comment on the validity of Carolís statistical inference procedure.
2023-04-05