Stat 551 Exam 2
December 12, 2012
I have neither given nor received unauthorized assistance on this examination.
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Signature
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Name Printed
There are 12 parts on this exam. I will score every part out of 10 points and take your best
10 of 12 scores. (Budget your time accordingly and find things you can do.)
1
10 pts 1. A mean 0 bivariate stationary time series
{( y
1t
, y2t )}t =−∞ has cross-covariance function
∞
 1 γc
Γ ( 0) = 
, Γ (1) = θ Γ ( 0 ) , and Γ ( h ) = 0 for h > 1
2 
γ c c 
for some c > 0, −1 < γ < 1, and − 1 < θ < 1 .
a) Write in explicit matrix form (you need not simplify, but you must plug in everything you can)
the variance of the variable y11 − 2 y21 + y12 .
10 pts b) Suppose that ( y11 , y21 ) = ( 3, 7 ) and y12 = 4 . Write out in explicit matrix form (again you must
plug in everything you can but need not simplify) the best linear prediction of y23 .
2
10 pts c) Suppose that, in fact, the process is Gaussian and γ = 0 . What is the large n approximate
distribution of ( n times) the bivariate variable giving empirical cross correlations at lags
h and k ,
n ( ρˆ1,2 ( h ) , ρˆ1,2 ( k ) )
(Be as explicit as possible.)
10 pts d) Identify a time invariant linear filter L , that when (at least in theory) applied to the { y1t }
sequence (say, Y1 ) will produce a white noise sequence, "whitening" the 1st coordinate series. For
Y2 the { y2t } sequence, what is the large n approximate distribution of the empirical cross
correlation between L Y1 and Y2 at lag h = 3 ?
3
10 pts 2. Suppose that I decide that the sample autocorrelation function for the first n values of a mean 0
stationary series { yt } can be described roughly as "oscillating at some not completely obvious
period inside a roughly exponentially declining 'wrapper' whose rate of decline is not exactly clear."
A cartoon might look like:
Propose a sensible parametric form for an autocorrelation function that might be fit to the observed
series (provide for both a parameter governing the period of oscillation and a parameter governing
the rate of decay) and write the matrix form for a function you would propose to optimize in order
the estimate the correlation parameters and the process variance.
Proposed autocorrelation function:
Function to be optimized:
4
10 pts
3. Below are some values of a time series { yt } through time n = 12 . Compute Holt-Winters
seasonal (with d = 4 ) forecasts for times t = 13,14,15,16 using smoothing constants α = β = γ =
1
.
2
(In doing these computations, for purposes of this exam, you may round even intermediate
calculations to one decimal place.)
t
1
2
3
4
5
6
7
8
yt
2
3
1
2
4
5
4
5
t
9
10
11
12
13
14
15
16
yt
7
6
6
7
--
--
--
--_
aˆt
bˆt
cˆt
yˆt
aˆt
bˆt
cˆt
yˆt
5
10 pts 4. Argue carefully that an ARMA ( 2, 2 ) process { yt } satisfying
yt = .2 yt −1 − .2 yt − 2 + ε t + .2ε t −1 + .2ε t − 2
for {ε t } mean 0 white noise with variance σ 2 is both causal and invertible.
10 pts 5. Consider an ARMA (1,1) process { yt } satisfying
yt = −.5 yt −1 + ε t + .2ε t −1
for {ε t } mean 0 white noise with variance σ 2 and an ARIMA(1,1,1) process
satisfying
{ut }
( ut − ut −1 ) = −.5 ( ut −1 − ut −2 ) + ε t + .2ε t −1
for {ε t } mean 0 white noise with variance σ 2 . Find spectral densities for { yt } and {ut } .
6
10 pts 6. Below are plots of power transfer functions for the differencing operators D, D4 , and D* = DD4 .
There are time series plots and spectral densities below for realizations of 5 different moving
average models and a differenced version of each. For each set of plots, identify the correct MA
model and the differencing operator that was used to make the second pair of plots for that model.
The models were
Model A:
Model B:
Model C:
Model D:
Model E:
yt
yt
yt
yt
yt
= ε t + .8ε t −1
= ε t + .2ε t −1
= ε t − .8ε t −1
= ε t + .8ε t −1 + .2ε t − 4
= ε t + .2ε t −1 + .8ε t − 4
Case 1:
Series
Spectral Density
Differenced Series
Spectral Density
Model is ________________
Differencing Operator is ____________________
7
Case 2:
Series
Spectral Density
Differenced Series
Spectral Density
Model is ________________
Case 3:
Series
Differenced Series
Model is ________________
Differencing Operator is ____________________
Spectral Density
Spectral Density
Differencing Operator is ____________________
8
Case 4:
Series
Spectral Density
Differenced Series
Spectral Density
Model is ________________
Case 5:
Series
Differenced Series
Model is ________________
Differencing Operator is ____________________
Spectral Density
Spectral Density
Differencing Operator is ____________________
9
7. BDM considers a data set consisting of the numbers of goals scored by England against Scotland in
a series of n = 52 annual soccer matches. We'll here consider a generalized state space analysis of
those data. We'll suppose that conditional on Poisson means λt for t = 1, 2, ,53 the numbers of goals
scored ( y1 , y2 , , y53 ) are independent and yt  Poisson ( λt ) .
10 pts a) Give a joint probability mass function for Y53 conditioned on the means. That is, specify
f ( y1 , , y53 | λ1 , , λ53 ) . Then provide a joint pdf for the logarithms of the means lt = ln ( λt )
supposing that these follow a mean 0 normal random walk model with variance σ 2 under the
assumption that l1  N ( 0,100 ) (100 is the variance). (You don't need to rewrite anything, but note
then that λt = exp ( lt ) and we thus have a fully specified model for all of the yt 's and lt 's .)
10 pts b) On the next page is some BUGS code and output for analysis of the goal data. Two analyses are
included. The first uses an assumption that σ 2 = 1 and the second employs a U ( 0,10 ) prior
distribution on σ . For both analyses say when (at which t ) the English goal scoring capacity seemed
at its lowest and what is a sensible prediction interval for y53 .
Analysis 1
|
|
|
|
|
|
|
|
|
|
|
|
|
Analysis 2
10
10 pts c) What, overall, seem to be the effects of computing under the assumption that σ = 1 rather than
giving σ a prior distribution? Argue that the differences you see "make sense."
Analysis 1
model {
l[1]~dnorm(0,.01)
for (i in 2:53) {
epsilon[i]~dnorm(0,tau)
l[i]<-l[i-1]+epsilon[i]
}
for (i in 1:53) {
lambda[i]<-exp(l[i])
y[i]~dpois(lambda[i])
}
}
list(y=c(0,1,0,2,4,1,0,1,5,1,4,2,1,3,1,1,1,1,0,1,1,0,2,0,2,0,0,1,0,1,2,2,1,2,4,1,4,1,0,0,4,1,0,1,0,
1,1,2,1,1,0,0,NA),tau=1)
11
lambda[1]
lambda[2]
lambda[3]
lambda[4]
lambda[5]
lambda[6]
lambda[7]
lambda[8]
lambda[9]
lambda[10]
lambda[11]
lambda[12]
lambda[13]
lambda[14]
lambda[15]
lambda[16]
lambda[17]
lambda[18]
lambda[19]
lambda[20]
lambda[21]
lambda[22]
lambda[23]
lambda[24]
lambda[25]
lambda[26]
lambda[27]
lambda[28]
lambda[29]
lambda[30]
lambda[31]
lambda[32]
lambda[33]
lambda[34]
lambda[35]
lambda[36]
lambda[37]
lambda[38]
lambda[39]
lambda[40]
lambda[41]
lambda[42]
lambda[43]
lambda[44]
lambda[45]
lambda[46]
lambda[47]
lambda[48]
lambda[49]
lambda[50]
lambda[51]
lambda[52]
lambda[53]
y[53]
mean
0.5112
0.6439
0.7125
1.633
2.721
1.376
0.8744
1.497
3.465
2.036
3.199
2.136
1.607
2.112
1.302
1.088
0.986
0.8642
0.5921
0.7459
0.8581
0.6745
1.061
0.7148
0.9747
0.4887
0.445
0.6485
0.5889
1.022
1.567
1.773
1.57
2.048
3.009
1.909
2.716
1.275
0.7521
0.8749
2.226
1.151
0.6579
0.7251
0.5999
0.8449
1.038
1.356
0.9557
0.6917
0.3842
0.3274
0.5541
0.5533
sd
0.5018
0.5569
0.5845
1.009
1.332
0.831
0.5413
0.9134
1.602
1.033
1.438
1.076
0.8887
1.157
0.7678
0.6978
0.6855
0.6441
0.4613
0.5601
0.6542
0.5285
0.7443
0.5511
0.6815
0.4033
0.3839
0.5254
0.439
0.7082
0.9388
0.9956
0.9464
1.139
1.434
1.054
1.344
0.7959
0.551
0.6033
1.203
0.7495
0.495
0.5536
0.4774
0.6238
0.7096
0.8668
0.6773
0.5513
0.3763
0.3835
1.372
1.563
MC_error
0.03194
0.03305
0.03291
0.05155
0.05704
0.04009
0.02565
0.04273
0.06386
0.04197
0.05409
0.04151
0.0368
0.0447
0.0307
0.02867
0.02793
0.02725
0.01895
0.0231
0.028
0.02218
0.02826
0.0229
0.0252
0.01602
0.01521
0.02076
0.016
0.02728
0.032
0.0324
0.03213
0.03595
0.03702
0.02907
0.02851
0.01928
0.01522
0.01461
0.02018
0.01344
0.009646
0.009893
0.008122
0.009785
0.008914
0.007996
0.006009
0.004563
0.00281
0.002337
0.006582
0.007518
val2.5pc
0.05587
0.08563
0.1138
0.3376
0.7822
0.2858
0.1622
0.3994
1.184
0.5778
1.125
0.629
0.4714
0.5932
0.2824
0.2235
0.1714
0.1273
0.08747
0.1202
0.1405
0.1034
0.1916
0.1085
0.1766
0.07101
0.05724
0.08672
0.09961
0.1954
0.3703
0.4402
0.3763
0.5307
0.9561
0.5296
0.8177
0.2831
0.1296
0.1726
0.596
0.2391
0.1081
0.1139
0.08884
0.1406
0.1943
0.2761
0.1621
0.09311
0.03201
0.01447
0.007843
0.0
median
0.3448
0.478
0.5577
1.404
2.494
1.205
0.761
1.283
3.201
1.847
2.954
1.953
1.408
1.866
1.142
0.9301
0.8198
0.6978
0.4719
0.5956
0.6808
0.5321
0.8861
0.5674
0.8024
0.3765
0.3356
0.509
0.478
0.8485
1.361
1.568
1.355
1.81
2.769
1.688
2.47
1.092
0.6103
0.7234
1.993
0.9778
0.5259
0.5778
0.4684
0.6819
0.8605
1.157
0.7851
0.5407
0.2688
0.1994
0.192
0.0
val97.5pcstart
1.917
10000
2.174
10000
2.375
10000
4.131
10000
5.912
10000
3.505
10000
2.269
10000
3.816
10000
7.347
10000
4.554
10000
6.63
10000
4.7
10000
3.849
10000
4.973
10000
3.221
10000
2.859
10000
2.807
10000
2.567
10000
1.797
10000
2.214
10000
2.608
10000
2.068
10000
2.999
10000
2.192
10000
2.755
10000
1.565
10000
1.443
10000
2.067
10000
1.739
10000
2.815
10000
3.936
10000
4.23
10000
3.964
10000
4.913
10000
6.442
10000
4.569
10000
5.987
10000
3.305
10000
2.191
10000
2.476
10000
5.222
10000
3.059
10000
1.974
10000
2.188
10000
1.871
10000
2.471
10000
2.888
10000
3.552
10000
2.71
10000
2.14
10000
1.395
10000
1.385
10000
3.306
10000
4.0
10000
sample
52001
52001
52001
52001
52001
5200
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
52001
Analysis 2
model {
sigma~dunif(0,10)
tau<-1/(sigma*sigma)
12
l[1]~dnorm(0,.01)
for (i in 2:53) {
epsilon[i]~dnorm(0,tau)
l[i]<-l[i-1]+epsilon[i]
}
for (i in 1:53) {
lambda[i]<-exp(l[i])
y[i]~dpois(lambda[i])
}
}
list(y=c(0,1,0,2,4,1,0,1,5,1,4,2,1,3,1,1,1,1,0,1,1,0,2,0,2,0,0,1,0,1,2,2,1,2,4,1,4,1,0,0,4,1,0,1,
0,1,1,2,1,1,0,0,NA))
list(sigma=1)
lambda[1]
lambda[2]
lambda[3]
lambda[4]
lambda[5]
lambda[6]
lambda[7]
lambda[8]
lambda[9]
lambda[10]
lambda[11]
lambda[12]
lambda[13]
lambda[14]
lambda[15]
lambda[16]
lambda[17]
lambda[18]
lambda[19]
lambda[20]
lambda[21]
lambda[22]
lambda[23]
lambda[24]
lambda[25]
lambda[26]
lambda[27]
lambda[28]
lambda[29]
lambda[30]
mean
1.218
1.238
1.273
1.34
1.401
1.382
1.378
1.437
1.537
1.511
1.533
1.477
1.415
1.389
1.323
1.276
1.235
1.207
1.182
1.174
1.169
1.157
1.168
1.153
1.16
1.145
1.144
1.162
1.182
1.228
sd
0.3318
0.3042
0.2903
0.326
0.3848
0.3511
0.3398
0.4094
0.5415
0.4988
0.5419
0.464
0.388
0.3678
0.3055
0.2828
0.2745
0.2737
0.2766
0.277
0.2802
0.2835
0.2802
0.2809
0.2816
0.2874
0.2876
0.2822
0.2699
0.2716
MC_error
0.01347
0.01069
0.008317
0.008372
0.01285
0.01076
0.01018
0.01536
0.0247
0.02194
0.02455
0.01908
0.01342
0.0114
0.005829
0.003788
0.005301
0.007195
0.008829
0.009265
0.00968
0.01073
0.009498
0.01071
0.01011
0.01143
0.01156
0.009881
0.008226
0.004637
val2.5pc
0.5093
0.588
0.6998
0.8391
0.9185
0.9067
0.9118
0.9566
0.9917
0.9861
0.9898
0.9754
0.948
0.9278
0.8321
0.7381
0.6447
0.5871
0.5334
0.5287
0.5181
0.4935
0.527
0.5017
0.5
0.4696
0.4668
0.5019
0.5535
0.6653
median
1.231
1.243
1.259
1.289
1.314
1.309
1.309
1.332
1.359
1.353
1.357
1.343
1.321
1.309
1.281
1.258
1.239
1.224
1.212
1.206
1.203
1.198
1.201
1.194
1.198
1.19
1.191
1.199
1.208
1.233
val97.5pcstart
1.903
10001
1.882
10001
1.934
10001
2.189
10001
2.471
10001
2.344
10001
2.286
10001
2.58
10001
3.066
10001
2.933
10001
3.058
10001
2.81
10001
2.485
10001
2.407
10001
2.1
10001
1.932
10001
1.793
10001
1.71
10001
1.666
10001
1.653
10001
1.653
10001
1.633
10001
1.649
10001
1.623
10001
1.632
10001
1.616
10001
1.613
10001
1.64
10001
1.66
10001
1.769
10001
sample
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
13
lambda[31]
lambda[32]
lambda[33]
lambda[34]
lambda[35]
lambda[36]
lambda[37]
lambda[38]
lambda[39]
lambda[40]
lambda[41]
lambda[42]
lambda[43]
lambda[44]
lambda[45]
lambda[46]
lambda[47]
lambda[48]
lambda[49]
lambda[50]
lambda[51]
lambda[52]
lambda[53]
sigma
y[53]
1.282
1.319
1.348
1.404
1.453
1.41
1.405
1.32
1.268
1.255
1.278
1.222
1.183
1.166
1.149
1.152
1.151
1.151
1.13
1.109
1.088
1.08
1.087
0.09086
1.096
0.2913
0.3226
0.3465
0.415
0.481
0.4129
0.4174
0.3172
0.2825
0.2778
0.3019
0.2738
0.2708
0.2782
0.2824
0.2833
0.2867
0.2929
0.3026
0.3178
0.3377
0.3549
0.376
0.1244
1.118
0.003429
0.006162
0.008884
0.01422
0.01923
0.01476
0.01461
0.006537
0.002939
0.002966
0.004068
0.004366
0.007353
0.008718
0.01019
0.009916
0.009806
0.009795
0.01165
0.01362
0.01552
0.01626
0.01559
0.00783
0.01646
0.7754
0.8344
0.8809
0.9311
0.952
0.9388
0.9335
0.8452
0.7511
0.7233
0.765
0.6603
0.5824
0.538
0.5162
0.5165
0.5111
0.5007
0.4487
0.3904
0.3325
0.2841
0.2616
0.001356
0.0
1.256
1.271
1.284
1.302
1.319
1.306
1.303
1.27
1.249
1.242
1.25
1.225
1.205
1.195
1.187
1.188
1.189
1.19
1.18
1.174
1.166
1.165
1.171
0.01111
1.0
1.983
2.162
2.294
2.593
2.824
2.593
2.594
2.168
1.936
1.891
2.005
1.781
1.676
1.652
1.632
1.636
1.637
1.649
1.629
1.609
1.602
1.613
1.652
0.392
4.0
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
10001
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
14