MSDS 596 Homework

MSDS 596 Homework 10 Due November 28 2017
Notes. The lowest grade among all eleven homework will be dropped, so NO late submission will be
accepted. All homework assignment must be written on standard 8.5 by 11 paper and stapled together.
Computer generated output without detailed explanations and remarks will not receive any credit. You may
type out your answers, but make sure to use different fonts to distinguish your own words with computer
output. Only hard copies are accepted, except under special circumstances. For the simulation and data
analysis problems, keep the code you develop as you may be asked to present your work later.
1 (80 pts). Use scan("lt.txt") to read in the data from the file lt.txt, which is a vector of length 500.
Consider a local trend model for the data
yt = st + et, et ∼ N(0, 0.25);
st+1 = st + ηt, ηt ∼ N(0, 0.01), s0 ∼ N(0.2, 2.25).
Use your own program for part (a), (b), and (c).
(a) Write your own program to implement the Kalman filter. Calculate the exact log likelihood of the
data. [Hint. Use s1|0 = 0.2, and Σ1|0 = 2.26 as the initial values.]
(b) Plot the predicted state variables st|t−1 for 1 ≤ t ≤ T. Also plot the 95% confidence intervals
st|t−1 ± 2pΣt|t−1. Your plot should look similar to Figure 11.4 of the textbook.
(c) Plot the filtered state variables st|
for 1 ≤ t ≤ T. Also plot the 95% confidence intervals st|t±2pΣt|t.
(d) Pretend that you do not know σ2e and σ2η, but you do have the prior knowledge that s0 ∼ N(0.2, 2.25),
use the dlm package to find the MLE of σ2e and σ2η.
(e) Reproduce the plots of part (b) and (c), using the MLE you obtained from part (d). Also, plot the
smoothed state variable st|T , together with the 95% confidence intervals st|T ± 2
p
Σt|T , for every t.
(f) Repeat part (d), using the R function StructTS() with type="level".
2 (20 pts). Rewrite the model (1−.3B)(1−.5B)(1−.6B +.4B2
)(rt −1.3) = at in the original form without
using the back-shift operator B.
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