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Universit` di Pavia a

Introduction to Stochastic processes
Eduardo Rossi

Stochastic Process

Stochastic Process: A stochastic process is an ordered sequence of random variables defined on a probability space (Ω, F, P ). {Yt (ω), ω ∈ Ω, t ∈ T }, such that for each t ∈ T , yt (ω) is a random variable on the sample space Ω, and for each ω ∈ Ω, yt (ω) is a realization of the stochasticprocess on the index set T (that is an ordered set of values, each corresponds to a value of the index set). Time Series: A time series is a set of observations {yt , t ∈ T0 }, each one recorded at a specified time t. The time series is a part of a realization of a stochastic process, {Yt , t ∈ T } where T ⊇ T0 . An infinite series {yt }∞ t=−∞

Eduardo Rossi c

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Time series econometrics 2011

2Mean, Variance and Autocovariance The unconditional mean µt = E[Yt ] = Yt f (Yt )dYt (1)

Autocovariance function: The joint distribution of (Yt , Yt−1 , . . . , Yt−h ) is usually characterized by the autocovariance function: γt (h) = = = Cov(Yt , Yt−h ) E[(Yt − µt )(Yt−h − µt−h )] ... (Yt − µt )(Yt−h − µt−h )f (Yt , . . . , Yt−h )dYt . . . dYt−h

The autocorrelation function ρt (h) = γt(h) γt (0)γt−h (0)

Eduardo Rossi c

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Time series econometrics 2011

3

Stationarity

Weak (Covariance) Stationarity: The process Yt is said to be weakly stationary or covariance stationary if the second moments of the process are time invariant: E[Yt ] E[(Yt − µ)(Yt−h − µ)] = = µ < ∞ ∀t γ(h) < ∞ ∀t, h

Stationarity implies γt (h) = γt (−h) = γ(h).

Eduardo Rossi c

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Timeseries econometrics 2011

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Stationarity

Strict Stationarity The process is said to be strictly stationary if for any values of h1 , h2 , . . . , hn the joint distribution of (Yt , Yt+h1 , . . . , Yt+hn ) depends only on the intervals h1 , h2 , . . . , hn but not on the date t itself: f (Yt , Yt+h1 , . . . , Yt+hn ) = f (Yτ , Yτ +h1 , . . . , Yτ +hn ) ∀t, h Strict stationarity impliesthat all existing moments are time invariant. Gaussian Process The process Yt is said to be Gaussian if the joint density of (Yt , Yt+h1 , . . . , Yt+hn ), f (Yt , Yt+h1 , . . . , Yt+hn ), is Gaussian for any h1 , h2 , . . . , hn . (2)

Eduardo Rossi c

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Time series econometrics 2011

5

Ergodicity

The statistical ergodicity theorem concerns what information can be derived from anaverage over time about the common average at each point of time. Note that the WLLN does not apply as the observed time series represents just one realization of the stochastic process. Ergodic for the mean. Let {Yt (ω), ω ∈ Ω, t ∈ T } be a weakly stationary process, such that E[Yt (ω)] = µ < ∞ and T E[(Yt (ω) − µ)2 ] = σ 2 < ∞ ∀t. Let y T = T −1 t=1 Yt be the time average. If y T converges inprobability to µ as T → ∞, Yt is said to be ergodic for the mean.

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Time series econometrics 2011

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Ergodicity

To be ergodic the memory of a stochastic process should fade in the sense that the covariance between increasingly distant observations converges to zero sufficiently rapidly. For stationary process it can be shown that absolutely summable autocovariances, i.e.∞ |γ(h)| < ∞, are sufficient to ensure h=0 ergodicity. Ergodic for the second moments
T

γ(h) = (T − h)−1

(Yt − µ)(Yt−h − µ) → γ(h)
t=h+1

P

(3)

Ergodicity focus on asymptotic independence, while stationarity on the time-invariance of the process.

Eduardo Rossi c

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Time series econometrics 2011

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Example

Then {Yt } is strictly stationary but not ergodic. ProofObviously we have that Yt = u0 for all t ≥ 0. Stationarity follows from: E[Yt ] E[Yt2 ] E[Yt Yt−1 ] = = = E[u0 ] = 0 E[u2 ] = σ 2 0 E[u2 ] = σ 2 0 (5)

Consider the stochastic process {Yt } defined by   u t = 0 with u0 ∼ N (0, σ 2 ) 0 Yt =  Yt−1 t > 0

(4)

Eduardo Rossi c

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Time series econometrics 2011

8

Example

Thus we have µ = 0, γ(h) = σ 2 ρ(h) = 1 are time invariant....