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Lecture 3: ARIMA(p,d,q) models
Florian Pelgrin
University of Lausanne, Ecole des HECDepartment of mathematics (IMEA-Nice)
Sept. 2011 - Jan. 2012
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 1 / 17
Introduction
Motivation
Characterize the main properties of ARIMA(p) models.
Discuss asymptotic equivalence with ARMA(p,q) models
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 2 / 17
Introduction
Road map
1 Introduction
2 ARIMA(p,d,q) modelDefinitionFundamental representationsEquivalence with ARMA(p,q) modelsAutoregressive approximationMoving average approximation
3 Application
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 3 / 17
ARIMA(p,d,q) model Definition
1. Definition
To some extent, ARIMA(p,d,q) models are a generalization ofARMA(p,q) models : the d-differenced process ∆dXt is(asymptotically) an ARMA(p,q) process :
On the other hand, the statistical properties of the two models aredifferent, especially in terms of forecasting.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 4 / 17
ARIMA(p,d,q) model Definition
Definition
A stochastic process (Xt)t≥−p−d is said to be an ARIMA(p, d , q)—anintegrated mixture autoregressive moving average model—if it satisfies thefollowing equation :
Φ(L)(1− L)dXt = µ+ Θ(L)εt ∀t ≥ 0
where εt is a (weak) white noise process with variance σ2ε , the lag
polynomials are given by :
Φ(L) = 1− φ1L− · · · − φpLp with φp 6= 0
Θ(L) = 1 + θ1L + · · ·+ θqLq with θq 6= 0,
and the initial conditions :
Z−1 = {X−1, · · · ,X−p−d , ε−1, · · · , ε−q}
are such that :
Cov(εt ,Z−1) = 0 ∀t ≥ 0.Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 5 / 17
ARIMA(p,d,q) model Definition
Remarks :
1. The stochastic process Xt also writes :
Φ(L)∆dXt = µ+ Θ(L)εt or Φ(L)Yt = µ+ Θ(L)εt
where Yt = ∆dXt = (1− L)dXt .
2. Initial conditions are fundamental.Example : Consider the two stochastic processes :
Xt = ρXt−1 + εt
Yt ≡ Xt − Xt−1 = ηt
where ρ = 0.9, εt and ηt are (weak) white noises.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 6 / 17
ARIMA(p,d,q) model Definition
AR(1) model Random walk (without drift)
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Note: The black, blue, and red solid lines correspond respectively to X_0 = 0, X_0 = 2, and X_0 = -2.
Figure: Initial conditions and stationarity
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 7 / 17
ARIMA(p,d,q) model Fundamental representations
1.2. Fundamental representations
Definition (Fundamental representation)
Let (Xt)t≥−p−d denote the following ARIMA(p,d,q) stochastic process :
Φ(L)∆dXt = µ+ Θ(L)εt ∀t ≥ 0
where Φ(L) = 1− φ1L− · · · − φpLp and Θ(L) = 1 + θ1L + · · ·+ θqLq,θq 6= 0,φp 6= 0, µ is a constant term, (εt) is a weak white noise, and theinitial conditions are uncorrelated with εt (t ≥ 0). This representation issaid to be causal or fundamental if and only if :
(i) All of the roots of the (inverse) characteristic equation associated toΦ are of modulus less (larger) than one.
(ii) All of the roots of the (inverse) characteristic equation associated toΘ are of modulus less (larger) than one.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 8 / 17
ARIMA(p,d,q) model Fundamental representations
Definition (Fundamental minimal representation)
Let (Xt)t≥−p−d denote the following ARIMA(p,d,q) stochastic process :
Φ(L)∆dXt = µ+ Θ(L)εt ∀t ≥ 0
where Φ(L) = 1− φ1L− · · · − φpLp and Θ(L) = 1 + θ1L + · · ·+ θqLq,θq 6= 0,φp 6= 0, µ is a constant term, (εt) is a weak white noise, and theinitial conditions are uncorrelated with εt (t ≥ 0). This representation issaid to be causal or fundamental if and only if :
(i) All of the roots of the (inverse) characteristic equation associated toΦ are of modulus less (larger) than one.
(ii) All of the roots of the (inverse) characteristic equation associated toΘ are of modulus less (larger) than one.
(iii) The (inverse) characteristic polynomials have no common roots.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 9 / 17
ARIMA(p,d,q) model Equivalence with ARMA(p,q) models
1.3. Equivalence with ARMA(p,q) models
Proposition
Let (Xt)t≥−p−d denote a minimal and causal ARIMA(p, d , q) stochasticprocess :
Φ(L)(1− L)dXt = µ+ Θ(L)εt .
The stochastic process defined by :
Yt = ∆dXt = (1− L)dXt
is asymptotically equivalent to an ARMA(p, q) process.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 10 / 17
ARIMA(p,d,q) model Autoregressive approximation
1.4. Autoregressive approximation
Definition
The autoregressive approximation (and not the AR(∞) representation) ofa causal and minimal ARIMA(p,d,q) stochastic process is given by :
At(L)Xt = µ0 + εt + h(t)′Z−1.
where At(L) =t∑
j=0ajL
j , a0 = 1, and the aj terms are the coefficients of the
division (by increasing powers) of Φ(u) by Θ(u), µ0 is a constant term,and h(t) ∈ Rp+d+q (with lim
t→+∞h(t) = 0).
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 11 / 17
ARIMA(p,d,q) model Moving average approximation
1.5. Moving average approximation
Definition
The moving average approximation (and not the MA(∞) representation)of a causal and minimal ARIMA(p,d,q) stochastic process is given by :
Xt = µ1 + Bt(L)εt + h(t)′Z−1.
where Bt(L) =t∑
j=0bjL
j , b0 = 1, and the bj terms are the coefficients of the
division (by increasing powers) of Θ(u) by Φ(u), µ1 is a constant term,and h(t) ∈ Rp+d+q (with lim
t→+∞h(t) = 0).
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 12 / 17
ARIMA(p,d,q) model Moving average approximation
Example
Starting from :
(1− φL)(1− L)Xt = εt − θεt−1 ∀t ≥ 0
with the initial conditions ε−1, X−1 and X−2 ;One gets :
∆Xt =1− θL
1− φLεt − φtθε−1 + φt+1(X−1 − X−2)
=1− θL
1− φLεt + h′tZ−1.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 13 / 17
Application
2. Application : modelling of the US stock marketdividends
Visual inspection of the autocorrelation function may indicate thatthe (log) US dividend is not stationary. The autocorrelogram of thefirst-differenced variable decreases quickly to zero—the first-differencevariable may be stationary.
Unit root tests tends to favor the assumption of non-stationarity (seelater on).
The chosen specification is then an ARIMA(p,1,q) model :
Φ(L)(1− L)dt = Θ(L)εt
where εt is a white noise process, and :
Φ(L) = 1− φ1L− · · · − φpLp
Θ(L) = 1 + θ1L + · · ·+ θqLq.
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 14 / 17
Application
Figure 34: (Partial) autocorrelogram of (log) US stock market dividents Level First-difference
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 15 / 17
Application
Estimation of an ARIMA(2,1,1) model
Parameter Estimate Std. Error t-stat. p-value
c 0.0007 0.0003 1.8777 0.0611φ1 0.7342 0.1096 6.6982 0.0000φ2 0.1206 0.0619 1.9477 0.0521θ1 -0.6442 0.1038 -6.2008 0.0000
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 16 / 17
Application
Figure 35: Adjusted error terms
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1975 1980 1985 1990 1995 2000 2005
Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 17 / 17
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