ECON 3350/7350 Univariate Time Series – I
Eric Eisenstat
The University of Queensland
Tutorial 2
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 2 1 / 8
Stationarity
When a single realisation of observations is available, the aggregation of observations over time is important.
Definition
A stochastic process is stationary if the data generating process is such that the mean, variance and covariances are independent of time.
E(yt) = μ
V ar(yt) = E[(yt − μ)2] = σy2 = γ0 Cov(yt, yt−k) = E[(yt − μ)((yt−k − μ)] = γk k = 1,2,…
These conditions must be satisfied for all values of t.
At this initial stage we will only consider stationary processes and we will relax this assumption later in the course.
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Autoregressive (AR) Models
To model the dependence in yt upon its own past behaviour. AR(1)
AR(2)
AR(p)
yt = a0 + ayt−1 + εt
yt =a0 +ayt−1 +a2yt−2 +εt
yt =a0 +ayt−1 +…+apyt−p +εt p
yt =a0 +aiyt−i +εt i=1
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 2 3 / 8
Moving Average (MA) Models
MA(1)
MA(2)
MA(q)
yt = μ + εt + β1εt−1
yt =μ+εt +β1εt−1 +β2εt−2
yt =μ+εt +β1εt−1 +…+βqεt−q
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The Autoregressive Moving Average Model (ARMA)
ARMA(1, 1)
yt =a0 +a1yt−1 +εt +β1εt−1
ARMA(2, 2)
yt = a0 + a1yt−1 + a2yt−2 + εt + β1εt−1 + β2εt−2
ARMA(3, 1)
yt = a0 + a1yt−1 + a2yt−2 + a3yt−3 + εt + β1εt−1
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Autocovariance and Autocorrelation
Definitions
V ar(yt) = E[(yt − μ)2] = γ0
Cov(yt,yt−k)=E[(yt −μ)(yt−k −μ)]=γk k=1,2,…
Definitions
Autocovariance Function: γk , k = 1, 2, …. If the process is stationary γk =γ−k.
Autocorrelation Function (ACF): ρk = γk , k = 1, 2, … γ0
Correlogram or SACF: Plot of the sample autocorrelation function, rk, against k.
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Partial Autocorrelation Function (PACF)
Definition
The partial autocorrelation function (φkk) is given by the kth coefficients in the corresponding AR(k) system of autoregressions. AR(1) yt = a0 + a1yt−1; φ11 = a1
AR(2) yt = a0 + a1yt−1 + a2yt−2; φ22 = a2
.
AR(k)yt =a0+a1yt−1+a2yt−2+…+akyt−k; φkk =ak
We identify the DGP by plotting the sample ACF (SACF) and sample PACF (SPACF) together.
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PACF (cont.)
Using the Yule-Walker equations, we can work out the PACF from the ACF,
φ11 = ρ1
φ22 = (ρ2 − ρ21)/(1 − ρ21)
.
ρk − k−1 φk−1,j ρk−j j=1
,k=3,4,5,… where,φk,j =φk−1,j −φkkφk−1,k−j,j=1,2,3,…,k−1
φkk = 1−k−1φ ρ j=1 k−1,j j
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