Time Series

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  • Topic: Autoregressive moving average model, Time series, Time series analysis
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TIME SERIES ANALYSIS

Chapter Three
Univariate Time Series Models

Chapter Three

Univariate time series models c WISE

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3.1

Preliminaries

We denote the univariate time series of interest as yt.
• yt is observed for t = 1, 2, . . . , T ;
• y0, y−1, . . . , y1−p are available;
• Ωt−1 the history or information set at time t − 1.
Call such a sequence of random variables a time series.

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Martingales
Let {yt} denote a sequence of random variables and let It =
{yt, yt−1, . . .} denote a set of conditioning information or information set based on the past history of yt. The sequence {yt, It} is called a martingale if
• It−1 ⊂ It (It is a filtration)
• E [|yt|] < ∞
• E [yt|It−1] = yt−1 (martingale property)

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Random walk model
The most common example of a martingale is the random walk model yt = yt−1 + εt,

εt ∼ W N (0, σ 2)

where y0 is a fixed initial value.
Letting It = {yt, . . . , y0} implies E [yt|It−1] = yt−1 since E [εt|It−1] = 0.

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Law of Iterated Expectations
Definition 1. In general, for information sets It and Jt such that It ⊂ Jt (Jt is the bigger info set). The Law of Iterated Expectations says E [Y |It] = E [E [Y |Jt]|It].

Let {yt, It} be a martingale. Then
E [yt|It−2] = E [E [yt|It−1]|It−2] = E [yt−1|It−2] = yt−2. It follows that
E [yt|It−k ] = yt−k .

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Martingale difference sequence
Definition 2. Let {εt} be a sequence of random variables with an associated information set It. The sequence {εt, It} is called a martingale difference sequence (MDS) if
• It−1 ⊂ It
• E [εt|It−1] = 0 (MDS property)

If {yt, It} is a martingale, a MDS {εt, It} may be constructed by defining εt = yt − E [yt|It−1].

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Linear time series models
Wold’s decomposition theorem (c.f. Fuller (1996) pg. 96) states that any covariance stationary time series {yt} has a linear process or infinite order moving average representation of the form


yt = µ +



ψk εt−k ,
k=0

2
ψk < ∞

ψ0 = 1,
k=0

εt = W N (0, σ 2)
where εt is followed by the white noise (W N ) process.

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In the Wold form, it can be shown that
E [yt] = µ


γ0 = Var(yt) = σ 2

2
ψk
k=0


γj

= Cov(yt, yt−j ) = σ 2

ψk ψk+j
k=0

ρj

Chapter Three

=


k=0 ψk ψk+j

2
k=0 ψk

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The moving average weights in the Wold form are also called impulse responses since
∂yt+s
= ψs ,
∂εt

s = 1, 2, . . .

For a stationary and ergodic time series
lim ψs = 0

s→∞

and the long-run cumulative impulse response


ψs < ∞.
s=0

A plot of ψs against s is called the impulse response function (IRF).

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A very rich and practically useful class of stationary and ergodic processes is the autoregressive-moving average (ARMA) class of models made popular by Box and Jenkins (1976).
ARMA(p, q ) models take the form of a pth order stochastic difference equation
yt − µ = φ1(yt−1 − µ) + · · · + φp(yt−p − µ) +εt + θ1εt−1 + · · · + θq εt−q
εt ∼ W N (0, σ 2).

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The ARMA(p, q ) model may be compactly expressed using lag
polynomials.
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Lag Operator Notation
The lag operator L is defined such that for any time series {yt}, Lyt = yt−1. It has the following properties: L2yt = L · Lyt = yt−2, L0 = 1 and L−1yt = yt+1.
The operator ∆ = 1 − L creates the first difference of a time series: ∆yt = (1 − L)yt = yt − yt−1.
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Define φ(L) = 1 − φ1L − · · · − φpLp and θ(L) = 1 + θ1L + · · · + θq Lq , the ARMA model may then be expressed as
φ(L)(yt − µ) = θ(L)εt....
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