STAT 153 - Introduction to Time Series Spring 2023 Practice Final
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STAT 153 - Introduction to Time Series
Spring 2023
Practice Final
April 28, 2023
1. Figure 1 shows the periodogram for a SOI dataset (SOI measures fluctuations in the surface air pressure difference between Tahiti and Darwin). This dataset has length n = 480. I decided to model the dataset as coming from a stationary process {Xt }
0.0
0.1
0.2
0.3
Frequency (j/n)
0.4
0.5
Figure 1: Periodogram of the SOI dataset
with spectral density f .
(a) The value of the periodogram at j = 40 i.e., I(40/480) = I(1/12) equals 11.04.
Using this, construct a 95% confidence interval for f(1/12) (you may use the fact that P{χ2(2) ≤ 0.05} = 0.025 and P{χ2(2) ≤ 7.38} = 0.975). Is this a good confidence interval? (2 + 1 = 3 points).
(b) The value of the periodogram at j = 10 i.e., I(10/480) = I(1/48) equals 0.52. Us-
ing this, construct a 95% confidence interval for f(1/48). Is this a good confidence interval? (2 + 1 = 3 points).
(c) Instead of just using I(1/12) as an estimator for f(1/12), I decided to smooth the periodogram around 1/12. Specifically, I chose m = 4 and constructed the
estimate
fˆ(1/12) := 4 I ( ) .
This estimate turned out to be 1.35. Using this estimate, construct a 95% confi- dence interval for f(1/12) (you may use the fact that P{χ18(2) ≤ 8.23} = 0.025 and P{χ18(2) ≤ 31.53} = 0.975). How does this interval compare to your interval in (a)? Do you need any assumptions on the spectral density f for this interval to be a valid confidence interval for f(1/12)? (2 + 1 + 1 = 4 points).
2. (a) Provide a sequence {aj } such that the filter which takes input {Xt } and outputs Yt = 对−∞ aj Xt −j has a power transfer function that is strictly increasing in the range [0, 1/2]. Provide reasoning for your answer (2 points).
(b) Consider a linear time-invariant filter whose power transfer function is plotted in Figure 2. I have sent an input process {Xt } through this filter to obtain an output process {Yt }. Of the two plots given in Figure 3, one corresponds to realizations of the input {Xt } while the other corresponds to realizations of the output {Yt }. Identify the plot which corresponds to the input and the plot which corresponds to the output. Provide reasoning for your answer. (2 points).
Power Transfer Function of a filter
0.3
Lambda
Figure 2: Power Transfer Function of a Filter
(c) Provide a sequence {aj } such that the filter which takes input {Xt } and outputs Yt = 对−∞ aj Xt −j has a power transfer function that is strictly decreasing in the range [0, 1/2]. Provide reasoning for your answer. (2 points).
(d) Consider a linear time-invariant filter whose power transfer function is plotted in Figure 4. I have sent an input process {Xt } through this filter to obtain an output process {Yt }. Of the two plots given in Figure 5, one corresponds to realizations of the input {Xt } while the other corresponds to realizations of the output {Yt }. Identify the plot which corresponds to the input and the plot which corresponds to the output. Provide reasoning for your answer. (2 points).
Power Transfer Function of a filter
0.5
Lambda
Figure 4: Power Transfer Function of a Filter
0 50 100 150 200
Time
Plot Two
200
Time
Figure 5: Two Plots
3. Consider the time series model Yt = st +Zt where st is a seasonal function with period d and {Zt } is white noise.
(a) Show that Xt = Yt − Yt −d is a stationary process. (2 points).
(b) Show that the best linear predictor for Xnd+1 in terms of X1 ,X2 , . . . ,Xnd −1,Xnd
equals
−X1 − 2X1+d − 3X1+2d − · · · − X1+(n− 1)d
n + 1 .
(2 points)
(c) Find the partial autocorrelation function of {Xt }. (2 points).
4. Determine whether each of the following statements is true or false. Provide reasons
in each case. You will not be awarded points if no reason is provided. (6 points). (a) An ARIMA(p,d,q) model with non-zero d is always non-stationary.
(b) The computation of the Conditional Least Squares (or conditional sum of squares)
estimates of the parameters of an ARMA process does not require any iterative optimization routine.
(c) Every stationary process must be an ARMA(p,q) process for some finite p and q .
(d) Given every sequence of real numbers {γ(h),h ≥ 0}, there exists a stationary process {Xt } whose autocovariance function equals γ(h) for every h ≥ 0.
(e) If b0 , . . . ,bn −1 denotes the DFT of a dataset x0 , . . . ,xn −1, then bj = n −j for every
1 ≤ j ≤ (n − 1).
(f) The periodogram of the dataset xt = cos(2π(0.065)t),t = 0, . . . , 99 will have only
one spike.
2023-05-12