Time Series

TIME SERIES ANALYSIS

Chapter Three
Univariate Time Series Models

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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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