Assignments/announcements:
NOTE: no office hours on Monday March 1
due to HDSI event; see
https://datascience.ucsd.edu/newsandevents/events/3rdanniversary
Exercise E moved to the Final Exam; so
HW9 (due March 7):
From Ch. 6: 32, 34, 35 + Exercise C.
From Ch. 9: 12, 14, 44 + Exercises D and F.
DO NOT COLLABORATE WITH ANYONE ON THE PROBLEMS OF THE FINAL EXAM!
TAKEHOME FINAL EXAM (due by noon of Wednesday March 17NOTE:
Final Exam deadline POSTPONED to March 18, Thursday by 11:59pm):
Do the following exercises:
Exercise E and from Ch. 9: Ex. 45, 46, 47;
Exercise G, H and from Ch. 10: Ex. 6 (do not use ex. 10.2 for this 
just use the Empirical rule of Paradigm 10.1.4);
Exercise I and from Ch. 11: ex. 12, 19, 21;
additionally, do exercises J and K.
Exercise G:
Suppose we want to fit an AR(p) model to the Wolfer sunspots data as in Exercise D.
(i) Use the empirical rule of p. 335 to identify the order p.
(ii) Use the R function ar (choose option YuleWalker)
to fit AR(r) models to the data for r=1,..., 12. For each r, record the
estimated variance of the white noise driving the AR model; denote it by sigma^2_r.
(iii) For each r, compute the AIC = n log (sigma^2_r) + 2r (see Eq. (10.9.2).
(iv) Plot AIC as a function of r=1,..., 12. Find the r that minimizes the AIC.
Is this different from your answer to part (i)?
Exercise H:
(i) Use the R function arma to fit all ARMA(p,q) models for p=0,1,2 and q=0,1,2,
i.e., 9 models, to the Wolfer sunspots data. Create a table that shows the
estimated variance of the white noise driving the ARMA model for all combinations of
p and q.
(ii) Use AIC to select the best ARMA model in this case.
Exercise I:
Let X_t be the GARCH(p,q) model of Definition 11.4.1.
Let p=1=q and show that X_t satisfies an ARCH(infinity) equation.
Identify the ARCH(infinity) coefficients.
Exercise J:
Let X_t be the ARCH(p) process of Example 11.1.6. Let G_t denote the information set:
(X_s for s < t).
(i) Show that X_t has mean zero, i.e., EX_t=0.
[Hint: first show E(X_tG_t) =0 and use the law of iterated expectations.]
(ii) Show that X_t
is a weakly stationary white noise, and compute its variance.
(iii) Define Y_t=X_t^2, i.e., the square of X_t.
Find the optimal predictor of Y_t
given G_t, denote it by \hat Y_t, and show that it is linear in
the variables (Y_s for s < t).
(iv) Let U_t= Y_t  \hat Y_t. Show that U_t is a mean zero, white noise.
Use this to claim that that Y_t satisfies an AR model with U_t as input.
(v) Do the above suffice to prove that Y_t is a linear time series?
[Hint: the answer is NObut need to argue for it!]
Exercise K:
Let X_t denote the Wolfer sunspots data. Denote \mu=EX_t and
f(w) the spectral density at point w.
(i) Estimate f(w) using the best fitted AR model from Exercice G.
(ii) Use the above, together with a CLT for the sample mean, to write down
a 95% confidence interval for \mu.
(iii) Let Y_t = log (X_t + 1). Use the R function ar.yw
(with aic = TRUE) to fit an AR model to the Y_t data. Does the chosen AR order for
Y_t coincide with that of X_t (as done in Exercice G)?
(iv) What is the (nonlinear) model for X_t that is implied by the fitted
AR model to the Y_t data?
WEEK 10 READING: Paradigm 9.8.1, Paradigm 9.8.3, Example 9.8.6, Paradigm 9.9.1; Remark 10.1.1, Paradigm 10.1.4, Paradigm 10.3.6 [recall the definition of YuleWalker estimators from Remark 5.8.6], Example 10.3.7, Remark 10.3.8, Definition 10.9.6 (Note: in Eq. (10.9.2) we should have used \sigma ^2_r instead of \sigma ^2); Ch. 11.1 plus Definition 11.4.1.
DISCUSSION/PRESENTATION MATERIAL BELOW:
Monday class: Read Paradigm 9.8.1, Paradigm 9.8.3, Example 9.8.6, Paradigm 9.9.1. Do
Exercise E and from Ch. 9: Ex. 45, 46, 47.
Wednesday class : Read Remark 10.1.1, Paradigm 10.1.4, Paradigm 10.3.6 [recall the definition of YuleWalker
estimators from Remark 5.8.6], Example 10.3.7, Remark 10.3.8, Definition 10.9.6. Do
Exercise G, H and from Ch. 10: Ex. 6 (do not use ex. 10.2 for this 
just use the Empirical rule of Paradigm 10.1.4).
Friday class: Read Ch. 11.1 plus Definition 11.4.1. Do
Exercise I and from Ch. 11: ex. 12, 19, 21.
Instructor:  Prof. Dimitris Politis Email: dpolitis@ucsd.edu Tel.: (858) 5345861 Office: SDSC217E Office hours: Monday 11am to 11:30, Wednesday 9:30am11am, or by appointment; students can also chat with the instructor after class on Mondays and Wednesdays 


TA: 


Textbook 
TIME SERIES: A FIRST COURSE WITH BOOTSTRAP
STARTER,
by Tucker McElroy and Dimitris Politis,
Chapman and Hall/CRC Press, 2020.


Structure of the course 


Course Grade  