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PSTAT 174/274: Time Series 2023
发布时间:2023-02-25
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Examination Paper for PSTAT174/274
PSTAT 174/274:
Time Series
2023
The relative weights attached to each question are as follows:
[174] A1 (18), A2 (11), A3 (35).
[274] A1 (30), A2 (30), A3 (40).
The numbers in square brackets indicate the relative weight attached to each sub-part of the question.
• Please start new sub-questions on a new page.
• All working should be provided for an answer to receive complete marks.
• Any numerical answers should be stated either as a fraction or as a numerical answer rounded to two decimal places.
• PAY ATTENTION TO THE QUESTIONS SPECIFIED FOR 174, 274 and joint questions for both 174 & 274.
1 Formula Page
Notation:
Unless otherwise indicated, in all questions:
• {ϵt } denotes a sequence of uncorrelated zero-mean random variables with constant variance σ 2 . I.e. {ϵt } ∼ WN(0,σ2 ).
• denote AR polynomial by ϕ(·) and the MA polynomial by θ(·).
• δi,j is the Kronecker-delta function
δij = {1(0)
if i j,
if i = j.
• γ(·) denotes the Autocovariance function and ρ(·) denotes the Auto- correlation function.
• B denotes the backshift operator and ∇ denotes the time series differ- ence operator
has
Var(Y) = |
[Wold Decomposition Theorem] Any zero-mean nondeterministic covariance- stationary process {Yt } can be decomposed as ∞ Yt =工 βj ϵt −j + κt j=0 where
• β0 = 1 and 之 • ϵt ∼ WN(0,σ2 ), • {βj } and {ϵt } are unique, • {κt } is deterministic, • ϵt is the limit of linear combinations of Ys , s ≤ t, • E[κt ϵs] = 0, ∀t,s. |
[Wold Decomposition: ACVF] γ(k) = σ 2 |
∞ 工 βj βj+k j=0 |
a + aw + aw2 + aw3 + ··· + awn = [Quadratic Equation] ax2 + bx + c = 0 has roots
|
A1 (174 & 274 attempt) - attempt all sub-questions unless a specific sub-
question states otherwise.
Let
Xt := λ + ηt
Yt := µ + ϵt −1
Zt := ∇ϵt
with λ,µ ∈ R and where {ϵt } ⊥ {ηt } are independent white noise processes with {ϵt } ∼ WN(0,σ2 ), and {ηt } ∼ WN(0,ση(2)). Find ex- pressions for the following in terms of model parameters and where possible use notation of Kronecker-Delta functions:
(b) Autocovariance function of (Yt − ϵt ). [3]
(c) cov(Zt ,Yt −k ), for all k . [3]
(d) E(Xt(2)) in terms of λ and ση(2) . [3]
(e) E(Zt −k Yt ), for all k . [3]
(f) E ((Yt − ϵt )2 ). [3]
(g) (ONLY 274 attempt)
Show that |γX (k)| ≤ γX (0), |γY (k)| ≤ γY (0) and for all lags k ∈ N
(h) (ONLY 274 attempt)
Show that |ρX (k)| ≤ 1, |ρY (k)| ≤ 1 and |ρZ (k)| k ∈ N
|γZ (k)| ≤ γZ (0)
[3]
≤ 1 for all lags
[9]
A2 (174 & 274 attempt) - attempt all sub-questions unless a specific sub-
question states otherwise.
In this question you will consider ergodicity and time series averages. Consider the following MA(2) time series model:
Yt = ϵt − θ 1 ϵt −1 − θ2 ϵt −2
(a) Explain with a diagram and no more than 4-5 sentences the dif- ference between an ensemble average and a time series average. [6]
(b) State a sufficient condition for the ensemble average to match the time series average. [3]
(c) (ONLY 274 attempt)
Show that the variance of the T sample mean estimator for this model as a function of σ 2 , θ 1 and θ2 is given by
VaT(Y) =
+
( 1 −
)
+ ( 1 −
)
(1 θ21 θ2(2))
[16]
(d) You are told that σ 2 = 1, θ 1 = 0.2 and θ2 = −0.1 is the variance
than the T = 20 sample size ensemble average i.i.d. case? Justify your answer with a relevant calculation and explanation using the formula in Equation (1) provided above in part (c) of
this question. [5]
A3 (174 & 274 attempt) - attempt all sub-questions unless a specific
sub-question states otherwise.
In this question you will consider properties of sample estimators in time series context and perform inference and testing. You may use R to assist you in calculations, but you should state the formula of any estimator you are calculating as well as the value you obtain for the estimator, and if R is used, the R command utilised. You can either obtain the solution by calculator or by R.
(a) You are given T = 15 samples from a time series:
{y1 , . . . ,y15 }
= {−0.92, 1.08, 0.20, −0.63, 1.36, 1.33, − 1.06,
Calculate the Sample mean, the Sample ACVF (for lags k = 0,1,2,3,4) and the Sample ACF (for lags k = 0,1,2,3,4) and plot the correlogram. Comment on whether this time series data is pure white noise or not based on the Correlogram you construct. Remark on the effect of such a small sample size on this conclusion. [13]
(b) Apply a relevant hypothesis test based on the sample ACF to determine if the time series is statistically different from White Noise based on this sample observation. Describe the
Null and Alternative hypothesis, state the relevant test statis- tic and the distribution of this test statistic under the Null (asymptotic distribution), write an expression for the p-value or the decision region and finally state the outcome of your inference testing procedure. Remark on the effect of such a small sample size on this conclusion. [12]
(c) Calculate a 95% confidence interval for estimator r(2) and determine if the estimator is statistically different from 0. You may assume that the Central Limit Theorem applies and therefore the estimator has an asymptotic Gaussian distribu- tion and you may use Bartlett’s formula for the evaluation of the finite sample standard error. Finally state the outcome of your findings. Remark on the effect of such a small sample size on this conclusion. [10]
(d) (ONLY 274 attempt)
Apply the Box-Pierce portmanteau test to determine whether you can reject the following null statement based on the data. You may consider maximum lag k = 3 for this test analysis.
— H0 : The data are independently distributed (i.e. the
correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process).
— H1 : The data are not independently distributed; they
exhibit serial correlation.
state the relevant test statistic and the distribution of this test statistic under the Null (asymptotic distribution), write an expression for the p-value or the decision region and finally state the outcome of your inference testing procedure. [5]