Time Series - STAT - EPFL

Continuous/discrete time models
Two main situations:
□ available data are part of a random sequence {Y t }, for which time t takes only integer values,
i.e. t ∈ Z. Thus Y t does not exist at (say) t = 0.5;
□ available data are values of a random function {Y (t)} that exists for all t ∈ R (or R + ) but is
only observed at a limited number of times.
In some cases (e.g. rainfall, with the time unit being hours) the observed data are
∫ t
t−1
Y (t)dt.
In this case, and particularly if we will want to use different time scales, there is a good case for
building a continuous-time model but estimating its parameters etc. using the cumulated/discrete
time data. Otherwise conclusions made for different time scales may be incoherent.
Time Series Autumn 2008 – slide 22
Measures of dependence
Definition 2 Let {Y t } t∈T be a stochastic process. Then
(a) if E(|Y t |) < ∞, then we define the mean (or expectation) of the process to be µ t = E(Y t ). If
non-constant µ t is sometimes called the trend;
(b) if var(Y t ) < ∞ for all t ∈ T , then we define the (auto)covariance function of the process to be
γ(s,t) = cov(Y s ,Y t ) = E {(Y s − µ s )(Y t − µ t )} , s,t ∈ T ,
and the (auto)correlation function of the process to be
ρ(s,t) =
γ(s,t)
{γ(s,s)γ(t,t)}
1/2,
s,t ∈ T .
□ Note that var(Y t ) = cov(Y t ,Y t ) = γ(t,t).
□ The Cauchy–Schwarz inequality gives |ρ(s,t)| ≤ 1 for all s,t ∈ T , with ρ(t,t) = 1 for all t.
□ The function γ(s,t) is semi-definite positive: ∑ a i a j γ(t t ,t j ) ≥ 0 for any a 1 ,... ,a k and any
{t 1 ,... ,t k } ⊂ T .
Time Series Autumn 2008 – slide 23
11