ST414 – Spectral Analysis of
Time Series Data
Lecture 7
20 February 2014
Last Time
• Asymptotics for the spectral density matrix
• Statistical inference for coherence
analyses
• Estimating VAR parameters
• Causality
• The VAR spectral density matrix
2
Today’s Objectives
• VAR spectral density matrix example
• Linear filters
• Tapering
3
The VAR Spectrum
Recall the spectral density for an AR(p) is
𝜎2
𝑓 𝜔 =
,
2
|Φ(exp −𝑖2𝜋𝜔 )|
where
𝑝
𝜙𝑗 𝑧 𝑗
Φ 𝑧 =1−
𝑗=1
4
The VAR Spectrum
The spectral density matrix for a VAR(p) is
𝑓 𝜔 = Φ(exp −𝑖2𝜋𝜔 )−1 Σ[Φ(exp −𝑖2𝜋𝜔 )∗ ]−1 ,
where
𝑝
Φ𝑗 𝑧 𝑗
Φ 𝑧 = 𝐼𝑑 −
𝑗=1
5
Sales Example
6
Sales Example
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Sales Example
VAR(2) for Sales/Lead data:
0.280 − 0.730
Φ1 =
0.028 − 0.516
0.205 − 2.177
Φ2 =
−0.011 − 0.153
1.431 − 0.022
Σ=
−0.022
0.077
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Sales Example
Causality tests:
𝐻0 : Sales does not Granger-cause Lead
p = 0.267
𝐻0 : Lead does not Granger-cause Sales
p = 1.806 x 10-8
𝐻0 : No instantaneous causality
p = 0.422
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Sales Example
10
Sales Example
11
Sales Example
12
Seasonality Example
Suppose X(t) is weakly stationary, sampled
monthly. Consider the differenced series Y(t)
= X(t)-X(t-12). What is the spectrum for Y(t)?
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Seasonality Example
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Seasonality Example
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Seasonality Example
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Seasonality Example
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Seasonality Example
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Differencing Example
This time, consider the first difference:
Y(t) = X(t) – X(t-1)
What is the spectrum for Y(t)?
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Differencing Example
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Differencing Example
21
Moving Average Example
Consider now a moving average X(t):
1
𝑌 𝑡 =
𝑋 𝑡−6 +𝑋 𝑡+6
24
1
+
12
5
𝑋(𝑡 − 𝑟)
𝑟=−5
What is the spectrum for Y(t)?
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Moving Average Example
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Moving Average Example
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Moving Average Example
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Moving Average Example
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Linear Filters
Let X(t) be a weakly stationary process with
spectrum 𝑓𝑋 𝜔 , and let 𝜓𝑗
be a
𝑗𝜖𝒁
sequence with
𝑗𝜖𝒁 |𝜓𝑗 |
𝑌 𝑡 =
< ∞. Let
𝑗𝜖𝒁 𝜓𝑗 𝑋
𝑡−𝑗 .
Then the spectrum of 𝑌 𝑡 is
𝑓𝑌 𝜔 = |A 𝜔 |2 𝑓𝑋 𝜔 ,
where A 𝜔 = 𝑗𝜖𝒁 𝜓𝑗 exp(−𝑖2𝜋𝜔𝑗).
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Linear Filters
Let X(t) be a weakly stationary P-variate process
with spectral density matrix 𝑓𝑋 𝜔 , and let Ψ𝑗
𝑗𝜖𝒁
be a sequence of QxP matrices with 𝑗𝜖𝒁 ||Ψ𝑗 || <
∞, where || ∙ || is any matrix norm. Let
𝑌 𝑡 = 𝑗𝜖𝒁 Ψ𝑗 𝑋 𝑡 − 𝑗 .
Then the QxQ spectral density matrix of 𝑌 𝑡 is
𝑓𝑌 𝜔 = A 𝜔 𝑓𝑋 𝜔 A 𝜔 ∗ ,
where A 𝜔 = 𝑗𝜖𝒁 Ψ𝑗 exp(−𝑖2𝜋𝜔𝑗).
28
Tapering
If X(t) is zero-mean weakly stationary with
spectral density 𝑓𝑋 𝜔 , one can show that
0.5
𝐸 𝐼 𝜔𝑗
=
𝑊𝑇 (𝜔𝑗 − 𝜔) 𝑓𝑋 𝜔 𝑑𝜔,
−0.5
where 𝑊𝑇 (𝜔) is the Fejer kernel.
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Tapering
Let h(t) be a data taper, with DFT
𝑇
𝐻𝑇 𝜔 = 𝑇 −1/2
ℎ 𝑡 exp(−𝑖2𝜋𝜔𝑡) .
𝑡=1
Define 𝑌 𝑡 = ℎ 𝑡 𝑋(𝑡). Then
0.5
𝐸 𝐼𝑌 𝜔𝑗
=
𝑊𝑇 (𝜔𝑗 − 𝜔) 𝑓𝑋 𝜔 𝑑𝜔,
−0.5
where 𝑊𝑇 𝜔 = |𝐻𝑇 𝜔 |2 .
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Example
31
Shumway & Stoffer, 2004
Example
32
Example
33
Example
34