Name:______Stu ID:______Oct. 4, 2012
MIDTERM EXAMINATION Statistics 153 D. R. Brillinger
The exam is closed book. There are 2 questions, and a bonus question.. Answer both. Answer in the space provided. Show your work. Gfor your answers.
A calculator may be used: but please use only the following analytical keys: add, subtract, multiply, divide, square root.
You have exactly 50 minutes for the exam.
Please no questions to the proctor. If you are not sure of the meaning of a question, set down an interpretation, and provide your answer.
Question 1. (Chapter 4) Consider the series
Yt = et + q et-4 + q et-8 + ... (*)
for t=... with {et} a zero mean white noise.
a) What general time series class is this an example of?
Pages 1, 2
b) Derive and discuss its autocovariance function.
c) Can this series be stationary? If yes, when?
d) Can it be represented as a stationary AR? If yes which one?
e) What is an invertible series/process?
f) Is (^) ever one? When?
g) Define the autocorrelation function, rho_h.
h) What are rho_0, rho_1, rho_2, rho_3, rho_4 for the series (*)?
i) What can you say about the partial correlation function of (*)?
Pages 3,4
Question 2. Consider the time series
Y_t = a cos wt + b sin w + X_t (**)
with X_t ... and a, b, w fixed and known.
a) Show Y_t = a cos wt + b sin w satisfies
Y_t - 2 cos wt Y_{t-1} + Y_{t-2} = 0
[Reminder. 2 cos phi cos psi = ]
b) Consider the time series
Y_t = a cos wt + b sin w + X_t (**)
with X_t ... and w fixed and known.
How would you use the "operator" PSI of
PSI Y_t = Y_t - 2 cos wt Y_{t-1} + Y_{t-2 (***)
to create a stationary series from (**)
Pages 5,6
Question 3.
Consider the series {u_t}, {v_t} defined by
u_t = phi u_{t-1} + e_t
v_t = phi v_{t-1{ + f_t
with |phi| < 1,where E and f are white noises with means 0 and variances sigma_e and sigma_f respectively, further all the e_ are independent of all the f_s for s,r = ...
Consider the series Y_t = u_t +V_t, and Z_t = U_t - V_t
a) Derive the time series distributions of Y_t and Z_t.
Question 4. {Seasonal]
How can the model be written as a linear model?
Buyes-Balot
How would you compute estimates of the seasonal effects?
ARIMA
Question ? a). Derive the acf of the process defined by
=
Sketch it.
Comment on the result.
[Seasonal] Monthly daya. What does the acf look like? Reason for your answer?
Question ? a). Derive the acf of the process defined by
Sketch it.
Comment on the result.
Chapter 3. Trends
Chapter 5. Models for nonstationary series
Chapter 6 Model specification
Check Chatfield, Harvey, Brockwell-Davis, Shumway-Stoffer