STAT7017 Final Project Page 1 of 3
Big Data Statistics – Final Project (Deferred)
Total of 45 Marks
Let Z denote the set of integers. A sequence of random vector observations (𝕏t : t = 1, . . . , T ) with values in Rp is called a p-dimensional (vector) time series. We denote the sample mean and sample covariance matrix by
𝕏:= T
t=1
1 T
1 T
𝕊0 := T −1 (𝕏t −𝕏t)(𝕏t −𝕏t)′.
t=1
𝕏t,
The lag-τ sample cross-covariance (aka. autocovariance) matrix is defined as
1 T
𝕊τ := T −1 (𝕏t −𝕏t)(𝕏t−τ −𝕏t)′.
t=τ+1
The lag-τ cross-correlation is given by
where D = diag(1/√s11, 1/√s22, . . . , 1/√spp) and the values come from 𝕊0 =
[1] [1]
[1]
Question 1 [12 marks]
Simulation is a helpful way to learn about vector time series. Define the matrices
0.8 0.4 2.0 0.5 A= −0.3 0.6 , Σ= 0.5 1.0 .
Generate 300 observations from the “vector autoregressive” VAR(1) model
𝕏t =A𝕏t−1+εt (1)
where εt ∼ N2(0, Σ), i.e., they are i.i.d. bivariate normal random variables with mean zero and covariance Σ. Note that when simulating is it customary omit the first 100 or more observations and you can start with 𝕏0 = (0, 0)′.
Also generate 300 observations from the “vector moving average” VMA(1) model
𝕏t =εt +Aεt−1. (2)
(a) Plot the time series 𝕏t for the VAR(1) model given by (1)
(b) Obtain the first five lags of sample cross-correlations of 𝕏t for the VAR(1) model, i.e.,
ρ1,…,ρ5.
(c) Plot the time series 𝕏t for the MA(1) model given by (2).
Dale Roberts – Australian National University
ρτ =D𝕊τD
𝔼[𝕏t] = 0, some authors (e.g., [C]) omit 𝕏t and consider the symmetrised lag-τ sample cross-
covariance given by
Cτ :=
1 T−τ
(𝕏t𝕏′t+τ +𝕏t+τ𝕏′t). t=1
[sij ].
Assuming
2T
Last updated: February 23, 2020
[1] [5]
[3]
STAT7017 Final Project Page 2 of 3 (d) Obtain the first two lags of sample cross-correlations of 𝕏t for the MA(1) model.
(e) Implement the test from [A] and reproduce the simulation experiment given in Section 5. This means you need to generate Table 1 from [A].
(f) The file q-fdebt.txt contains the U.S. quarterly federal debts held by (i) foreign and international investors, (ii) federal reserve banks, and (iii) the public. The data are from the Federal Reserve Bank of St. Louis, from 1970 to 2012 for 171 observations, and not seasonally adjusted. The debts are in billions of dollars. Take the log transformation and the first difference for each time series. Let (𝕏t ) be the differenced log series.
Test H0 : ρ1 = . . . = ρ10 = 0 vs Ha : ρτ ̸= 0 for some τ ∈ {1, . . . , 10} using the test from [A]. Draw the conclusion using the 5% significance level.
Question 2 [13 marks]
More generally, a p-dimensional time series 𝕏t follows a VAR model of order l, VAR(l), if
l
𝕏t =a0+Ai𝕏t−i +εt (3)
i=1
where a0 is a p-dimensional constant vector and Ai are p × p (non-zero) matrices for i > 0, and
i.i.d. εt ∼ Np (0, Σ) for all t with p × p covariance matrix Σ.
One day you might want to “build a model” using the VAR(l) framework. One of the first things you need to do is to determine the optimal order l. Tiao and Box (1981) suggest using sequential likelihood ratio tests; see Section 4 in [B]. Their approach is to compare a VAR(l) model with a VAR(l − 1) model and amounts to considering the hypothesis testing problem
H0 :Al =0 vs. H1 :Al ̸=0.
We can do this by determining model parameters using a least-squares approach. We rewrite (3)
as
𝕏′t =Xt′𝔸+ε′t
whereXt =(1,𝕏′t−1,…,𝕏′t−l)′ isa(pl+1)-dimensionalvectorand𝔸=[a0,A1,…,Al]isa p×1+l×(p×p)=p×(pl+1)matrix. Withobservationsattimest=l+1,…,T,wewrite the data as
X = X𝔸 + E (4) where X is a (T −l)×p matrix with the ith row being 𝕏′l+i, X is a (T −l)×(pl+1) design
matrix with the ith row being X′ , and E is a (T −l)×p matrix with the ith row being ε′ . l+i l+i
The matrix 𝔸 contains the coefficient parameters of the VAR(l) model and let Σε,l be the corresponding innovation covariance matrix. Under a normality assumption, the likelihood ratio for the testing problem is
|Σˆε,l| (T−l)/2 λ1= |Σˆε,l−1| .
The likelihood ratio test of H0 is equivalent to rejecting H0 for large values of |Σˆε,l|
−2log(λ1)=−(T−l)log |Σˆε,l−1| .
Dale Roberts – Australian National University
Last updated: February 23, 2020
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STAT7017 Final Project Page 3 of 3 A commonly used statistic is Bartlett’s approximation given by
|Σˆε,l| M(l)=−(T −l−1.5−pl)log |Σˆε,l−1| ,
which follows asymptotically (as n → ∞ and p fixed) a χ2 distribution with p2 degrees of freedom. The following methodology is suggested for selecting the order l:
1. Select a positive integer P , which is the maximum VAR order that we would like to consider. 2. Setup the regression framework (4) for the VAR(P ) model. That is, there are T − P
observations (i.e., rows) in the X matrix.
3. For l = 0, . . . , P compute the least-squares estimate of the AR coefficient matrix 𝔸. For
l = 0, we have 𝔸 = a0. Then compute the ML estimate for Σε, l given by Σˆε,l :=(1/T −P)R′lRl
where Rl = 𝕏 − X𝔸 is the residual matrix of the fitted VAR(l) model.
4. For l = 1,…,P, compute test statistic M(l) and its p-value, which is based on the
asymptotic χ2 distribution. p2
5. Examine the test statistics sequentially starting with l = 1. If all the p-values of the M(l) test statistics are greater than the specified type I error for l > m, then a VAR(m) model is specified. This is so because the test rejects the null hypothesis Al = 0, but fails to reject Al = 0 for l > m.
Consider a bivariate time series 𝕏 = (𝕩TB, 𝕩CPI) where 𝕩TB is the change in monthly US treasury tttt
bills with maturity 3 months and 𝕩CPI is the inflation rate, in percentage, of the U.S. monthly t
consumer price index (CPI). This data from the Federal Reserve Bank of St. Louis. The CPI rate is 100 times the difference of the log CPI index. The sample period is from January 1947 to December 2012. The data are in the file m-cpib3m.txt.
(a) Plot the time series 𝕏t.
(b) Select a VAR order for 𝕏t using the methodology (described above).
(c) Drawing on your results obtained in this project and the theory discussed in class, explain and demonstrate (e.g., simulation study) what might happen with this methodology if the dimensionality p of the time series becomes large.
Question 3 [20 marks]
The recent paper [C] is concerned with extensions of the classical Marchenko-Pastur to the time series case. Reproduce their simulation study which is found in Section 5 and Figure 1.
References
[A] Li, McLeod (1981). Distribution of the Residual Autocorrelations in Multivariate ARMA Time Series Models, J.R. Stat. Soc. B 43, No. 2, 231–239.
[B] Tiao and Box (1981). Modelling multiple time series with applications. Journal of the American Statistical Association, 76. 802 – 816.
[C] Liu, Aue, Paul (2015). On the Marchenko-Pastur Law for Linear Time Series. Annals of Statistics Vol. 43, No. 2, 675–712.
[D] Wang, Aue, Paul (2017). Spectral analysis of sample autocovariance matrices of a class of linear time series in moderately high dimensions. Bernouilli 23(4A), 2181–2209.
Dale Roberts – Australian National University
Last updated: February 23, 2020