ST414 – Spectral Analysis of
Time Series Data
Lecture 6
18 February 2014
Last Time
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Introduction to bivariate processes
The cross-covariance function
The cross-spectrum
Coherence analysis
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Today’s Objectives
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Asymptotics for the spectral density matrix
Estimating VAR parameters
Causality
The VAR spectral density matrix
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Nonparametric Estimation
Given a bivariate time series (𝑋1 t , 𝑋2 t )′,
construct the bivariate DFTs 𝑑 𝜔𝑗 =
(𝑑𝑋1 𝜔 , 𝑑𝑋2 𝜔 )′, where
𝑇
𝑑𝑋𝑗 𝜔𝑗 = 𝑇 −1/2
𝑋𝑗 𝑡 exp(−𝑖2𝜋𝜔𝑗 𝑡)
𝑡=1
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Nonparametric Estimation
The periodogram matrix is
𝑰 𝜔𝑗 = 𝑑 𝜔𝑗 𝑑 𝜔𝑗
∗
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Nonparametric Estimation
The smoothed periodogram matrix is
𝒇 𝜔𝑗
1
=
2𝑀 + 1
𝑀
𝑘=−𝑀
𝑘
𝑰(𝜔𝑗 + )
𝑇
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Nonparametric Estimation
More generally,
𝑀
𝒇 𝜔𝑗 =
𝑘=−𝑀
where
ℎ𝑘 ≥ 0,
|𝑘|≤𝑀 ℎ𝑘
𝑘
ℎ𝑘 𝑰(𝜔𝑗 + )
𝑇
= 1, and ℎ𝑘 = ℎ−𝑘
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Nonparametric Estimation
From here, an estimate of the squared
coherence is
2
𝜌𝑋𝑌
𝜔 =
|𝑓𝑋𝑌 𝜔 |2
𝑓𝑋 𝜔 𝑓𝑌 𝜔
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Nonparametric Estimation
Let Z(t) be bivariate Gaussian white noise
(0, Σ), Σ non-singular, and let 𝑰(𝜔) be the
periodogram matrix of Z(t). Then as 𝑇 → ∞,
𝑰(𝜔) converges in distribution to a random
matrix 𝑌𝑌 ∗ , where 𝑌~𝑁 𝐶 (0, Σ).
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Nonparametric Estimation
𝐸 𝒇 𝜔
𝐶𝑜𝑣 𝑓𝑝𝑞 𝜔 , 𝑓𝑟𝑠 𝜆
2
𝑀
ℎ
𝑘=−𝑀 𝑘
→ 𝑓
𝑝𝑟
→𝒇 𝜔
0 if 𝜔 ≠ 𝜆
𝜔 𝑓𝑠𝑞 𝜔 if 𝜔 = 𝜆 ≠ 0, 0.5
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Nonparametric Estimation
|𝜌𝑋𝑌 𝜔 |
2
𝑀
ℎ
𝑘=−𝑀 𝑘
~𝐴𝑁( 𝜌𝑋𝑌 𝜔 , 1 − 𝜌𝑋𝑌 𝜔
𝑡𝑎𝑛ℎ−1 ( 𝜌𝑋𝑌 𝜔 )
2
𝑀
ℎ
𝑘=−𝑀 𝑘
2
/2)
~𝐴𝑁(𝑡𝑎𝑛ℎ−1 ( 𝜌𝑋𝑌 𝜔 ), 1/2)
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Nonparametric Estimation
Alternatively, if we use a uniform kernel for
2
smoothing, under 𝐻0 : 𝜌𝑋𝑌
𝜔 = 0, we have
2
𝜌𝑋𝑌
𝜔
𝐹=
2𝑀
2
1 − 𝜌𝑋𝑌
𝜔
𝐹~𝐹(2, 2 2𝑀 + 1 − 2)
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SOI Example
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SOI Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
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Sales Example
−2.682 ∙ 2𝜋
−4.173 ∙ 2𝜋
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Least Squares Estimation
Consider the K-variate VAR(p) model:
𝑋 𝑡 = Φ1 𝑋 𝑡 − 1 + ⋯ Φ𝑝 𝑋 𝑡 − 𝑝 + 𝑍 𝑡
Assume “presamples”
𝑋 0 , 𝑋 −1 , … , 𝑋 −𝑝 + 1
are available.
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Least Squares Estimation
Rewrite as
𝑋 = 𝐵𝑌 + 𝑍.
The least squares estimator is
𝐵 = 𝑋𝑌′ 𝑌𝑌 ′ −1
Moreover,
𝛴 = 𝑇 −1 (𝑋 − 𝐵𝑌)(𝑋 − 𝐵𝑌)′
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Causality
If Y(t) can be predicted more efficiently if the
past values of the process X(t) is taken into
account, then X(t) Granger-causes Y(t).
If Y(t+1) can be predicted more efficiently if
X(t+1) is taken into account, then X(t)
instantaneously-causes Y(t).
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Causality
𝑋(𝑡)
=
𝑌(𝑡)
𝑝
𝑘=1
𝜙11,𝑘 𝜙12,𝑘
𝜙21,𝑘 𝜙22,𝑘
𝑋(𝑡 − 𝑘)
𝑍1 (𝑡)
+
𝑌(𝑡 − 𝑘)
𝑍2 (𝑡)
X(t) does not Granger-cause Y(t) if 𝜙21,𝑘 = 0 for
all k.
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Causality
𝜎11 𝜎12
𝛴= 𝜎 𝜎
21 22
Y(t) does not instantaneously-cause X(t) if
𝜎21 = 𝜎12 = 0.
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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
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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
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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 = 0.2669
𝐻0 : Lead does not Granger-cause Sales
p = 1.806 x 10-8
𝐻0 : No instantaneous causality
p = 0.4221
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Sales Example
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Sales Example
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Sales Example
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