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Time Series Analysis STA457H1

创建日期：2024-05-16 14:19:56

所属分类：统计学statistics

标签： STA457H1

Time Series Analysis

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Time Series Analysis

STA457H1
ARIMA Models
Lecture 5

Learning Objectives
By the end of this chapter, you should be able to do the following:
Understand the sufficient and necessarily conditions for AR,MA, and ARMA
models.
Evaluate stationary, causality, and invertibility in time series models.
Use R with real time series data and interpret outcomes of analyses.

Linear Processes
The Box-Jenkins methodology requires that the model used in describing and forecasting a time
series be both stationary and invertible.
Definition
The process {xt} is a stationary if it remains in statistical equilibrium with probabilistic
properties that do not change over time, in particular varying about a fixed constant mean
level and with constant variance.
The process {xt} is an invertible if its weights decline (does not depend on time), and hence
the Box-Jenkins model can be used to express xt as a function of past x observations (that
is, xt−1, xt−2, . . . ).
For a stationary time series without trends or seasonal effects. That is; if necessary, any
trends, seasonal or cyclical effects have already been removed from the series, we might
construct a linear model for a time series with autocorrelation.
The most important special cases of linear processes are:
Autoregressive (AR) Model.
Moving-Average (MA) Model.
Autoregressive Moving Average (ARMA) Model.
Adding nonstationary models to the mix ARMA models leads to the autoregressive
integrated moving average (ARIMA) model.
Autoregressive (AR) Models with Zero Mean
Definition
A time series {xt}, with zero mean, is called autoregressive process of order p
and denoted by AR(p), if it can be written in the form:
xt = ϕ1xt−1 + ϕ2xt−2 + · · ·+ ϕpxt−p + wt (1)
where wt is a random shock, usually assumed to be white noise, i.e.,
wt ∼ wn(0, σ2w ), and ϕ1, · · · , ϕp(ϕp ̸= 0) are parameters.
With backshift operator, the AR process can be written as
Φp(B)xt = wt , (2)
where Φp(B) = 1− ϕ1B − ϕ2B2 − · · · − ϕpBp is a polynomial of degree p.

Autoregressive (AR) Models with Mean µ
If the mean, µ, of xt is not zero, replace xt by xt − µ to get:
xt − µ = ϕ1(xt−1 − µ) + ϕ2(xt−2 − µ) + · · ·+ ϕp(xt−p − µ) + wt (3)
or write
xt = δ + ϕ1xt−1 + ϕ2xt−2 + · · ·+ ϕpxt−p + wt , (4)
where
δ = µ(1− ϕ1 − ϕ2 − · · · − ϕp)
For example, the AR(2) model
xt = 1.5 + 1.2xt−1 − 0.5xt−2 + wt
is
xt − µ = 1.2(xt−1 − µ)− 0.5(xt−2 − µ) + wt
where 1.5 = µ(1− 1.2− (−0.5))⇒ 1.5 = 0.3µ.
Thus, the model has mean µ = 5 and the model can be written as
xt − 5 = 1.2(xt−1 − 5)− 0.5(xt−2 − 5) + wt

A Simulated AR(1) Series: xt = 0.8xt−1 + wt
set.seed(12345)
ar1.sim<-arima.sim(n=100,list(order=c(1,0,0),ar=0.8))
par(mfrow=c(1,3))
plot(ar1.sim, ylab="x(t)", main="Simulated AR(1)",lwd=2)
acf(ar1.sim, 50, main="Autocorrelation Function",lwd=2)
acf(ar1.sim, type="partial", main="Partial ACF",lwd=2)
Simulated AR(1)
Time
x(t
)
0 20 40 60 80 100
−
2
0
2
4
0 10 20 30 40 50
−
0.
2
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
2
0.
0
0.
2
0.
4
0.
6
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated AR(1) process: (1− 0.8B)xt = wt

A Simulated AR(1) Series: xt = −0.8xt−1 + wt
set.seed(12345)
ar11.sim<-arima.sim(n=100,list(order=c(1,0,0),ar=-0.8))
par(mfrow=c(1,3))
plot(ar11.sim, ylab="x(t)", main="Simulated AR(1)",lwd=2)
acf(ar11.sim, main="Autocorrelation Function",lwd=2)
acf(ar11.sim, type="partial", main="Partial ACF",lwd=2)
Simulated AR(1)
Time
x(t
)
0 20 40 60 80 100
−
4
−
2
0
2
0 5 10 15 20
−
0.
5
0.
0
0.
5
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
6
−
0.
4
−
0.
2
0.
0
0.
2
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated AR(1) process: (1 + 0.8B)xt = wt

A Simulated AR(2) Series: xt = 1.1xt−1 − 0.3xt−2 + wt
set.seed(1965)
ar2<-arima.sim(n=200,list(order=c(2,0,0),ar=c(1.1,-0.3)))
par(mfrow=c(1,3))
plot(ar2, ylab="x(t)", main="Simulated AR(2)",lwd=2)
acf(ar2, main="Autocorrelation Function",lwd=2)
acf(ar2, type="partial", main="Partial ACF",lwd=2)
Simulated AR(2)
Time
x(t
)
0 50 100 150 200
−
4
−
2
0
2
4
0 5 10 15 20
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
2
0.
0
0.
2
0.
4
0.
6
0.
8
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated AR(2) process: (1− 1.1B + 0.3B2)xt = wt

Causal Conditions for an of AR(1) Process
The autoregressive process of order 1, AR(1), xt = ϕxt−1 + wt (equivalently
(1− ϕB)xt = wt ) is a causal process if it is stationary with values that are not
depending on the future. In this case, the absolute value of the root of (1− ϕz) = 0
must lie outside the unit circle.
That is, the AR(1) is causal process if
|z| = |1
ϕ
| > 1 or equivalently if |ϕ| < 1 (5)
Note that causality implies stationarity but not the other way around.
If the process is not stationary then it will not causal.
Examples
1.) (1− 0.4B)xt = wt is a causal process because the absolute value of the root of
(1− 0.4z) = 0 is |z| = |1/0.4| = 2.5 > 1.
2.) (1 + 1.8B)xt = wt is not stationary process (not casual) because |root| of
(1 + 1.8z) = 0 is |z| = | 1−1.8 | = 0.56 < 1.
3.) xt = 0.5xt−1 + wt is a causal process because |0.5| < 1.

Causal Conditions for an of AR(2) Process
The AR(2) model,
xt = ϕ1xt−1 + ϕ2xt−2 + wt ,
is causal when the two roots of 1− ϕ1z − ϕ2z2 lie outside of the unit circle. Using the quadratic
formula, this requirement can be written as
∣∣∣∣ϕ1 ±
√
ϕ21 + 4ϕ2
−2ϕ2
∣∣∣∣ > 1.
In this case, the necessary and sufficient conditions for causality are:
|ϕ2| < 1, ϕ1 + ϕ2 < 1, and ϕ2 − ϕ1 < 1
Recall the solution for quadratic equation p(z) = a+ bz + cz2 is
z =
−b ±
√
b2 − 4ac
2a
Examples
1.) xt = 1.1xt−1 − 0.4xt−2 + wt is causal.
2.) xt = 0.6xt−1 − 1.3xt−2 + wt is not stationary (|ϕ2| ≮ 1).
3.) xt = 0.6xt−1 + 0.8xt−2 + wt is not stationary (ϕ1 + ϕ2 ≮ 1).
4.) xt = −0.4xt−1 + 0.7xt−2 + wt is not stationary (ϕ2 − ϕ1 ≮ 1).

Causal Conditions for an of AR(p) Process
The Autoregressive process of order p, AR(p)
xt = ϕ1xt−1 + · · ·+ ϕpxt−p + wt ,
is said to be casual if the all absolute values of the roots of
1− ϕ1z − ϕ2z2 − · · · − ϕpzp = 0 lie outside the unit circle.
Examples
1.) xt = −0.4xt−1 + 0.21xt−2 + wt is a casual process as |roots| of
1 + 0.4z − 0.21z2 = (1− 0.3z)(1 + 0.7z) = 0 are | 10.3 | = 3.3 > 1 and
| 1−0.7 | = 1.4 > 1.
2.) xt = 1.7xt−1 − 0.42xt−2 + wt is not a stationary process because the
absolute value of one of the two roots of
1− 1.7z + 0.42z2 = (1− 0.3z)(1− 1.4z) = 0 lies inside the unit circle,
which is | 11.4 | = 0.71 < 1.
3.) xt = −0.25xt−2 + wt is a casual process because |roots| of
1 + 0.25z2 = 0 are | ± 2i | = 2 > 1, where i = √−1.

Finding the Roots in R
In R, the function polyroot(a), where a is the vector of polynomial coefficients
in increasing order can be used to find the zeros of a real or complex polynomial
of degree n − 1
p(x) = a1 + a2x + · · ·+ anxn−1.
The function Mod() can be used to check whether the roots of p(x) have
modulus > 1 or not.
Example
The AR(4) process xt = −0.3xt−1 + 0.7xt−2 − 1.2xt−3 + 0.1xt−4 + wt is not stationary
as
roots <- polyroot(c(1,-0.3, 0.7,-1.2,0.1))
Mod(roots)
## [1] 0.8829242 0.8829242 1.1250090 11.4024243

Moving Average (MA) Models
Definition
A time series {xt}, with zero mean, is called moving Average process of order
q and denoted by MA(q), if it can be written in the form:
xt = wt + θ1wt−1 + θ2wt−2 + · · ·+ θqwt−q (6)
where wt is a white noise, wt ∼ wn(0, σ2w ), and θ1, · · · , θq(θq ̸= 0) are
parameters.
With backshift operator, the MA process can be written as
xt = Θq(B)wt , (7)
where Θq(B) = 1 + θ1B + θ2B2 + · · ·+ θqBq is polynomial of degree q.
The MA(q), ∀q ≥ 1, process is always stationary (regardless of the
values of θj , j = 1,2, · · · ,q).
If the |roots| of 1 + θ1z + θ2z2 + · · ·+ θqzq = 0 > 1, the MA(q) process is
said to be invertible.

A Simulated MA(1) Series: xt = wt + 0.8wt−1
set.seed(6436)
ma1.sim<-arima.sim(n=200,list(order=c(0,0,1),ma=0.8))
par(mfrow=c(1,3))
plot(ma1.sim, ylab="x(t)", main="Simulated MA(1)",lwd=2)
acf(ma1.sim, main="Autocorrelation Function",lwd=2)
acf(ma1.sim, type="partial", main="Partial ACF",lwd=2)
Simulated MA(1)
Time
x(t
)
0 50 100 150 200
−
3
−
2
−
1
0
1
2
3
4
0 5 10 15 20
−
0.
2
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
2
0.
0
0.
2
0.
4
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated MA(1) process: xt = (1 + 0.8B)wt

A Simulated MA(1) Series: xt = wt − 0.8wt−1
set.seed(6436)
ma11.sim<-arima.sim(n=200,list(order=c(0,0,1),ma=-0.8))
par(mfrow=c(1,3))
plot(ma11.sim, ylab="x(t)", main="Simulated MA(1)",lwd=2)
acf(ma11.sim, main="Autocorrelation Function",lwd=2)
acf(ma11.sim, type="partial", main="Partial ACF",lwd=2)
Simulated MA(1)
Time
x(t
)
0 50 100 150 200
−
4
−
2
0
2
0 5 10 15 20
−
0.
5
0.
0
0.
5
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
4
−
0.
3
−
0.
2
−
0.
1
0.
0
0.
1
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated MA(1) process: xt = (1− 0.8B)wt

A Simulated MA(2) Series: xt = wt +0.8wt−1 +0.6wt−2
set.seed(14762)
ma2<-arima.sim(n=200,list(order=c(0,0,2),ma=c(0.8,0.6)))
par(mfrow=c(1,3))
plot(ma2, ylab="x(t)", main="Simulated MA(2)",lwd=2)
acf(ma2, main="Autocorrelation Function",lwd=2)
acf(ma2, type="partial", main="Partial ACF",lwd=2)
Simulated MA(2)
Time
x(t
)
0 50 100 150 200
−
4
−
2
0
2
4
0 5 10 15 20
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Lag
AC
F
Autocorrelation Function
5 10 15 20
−
0.
2
0.
0
0.
2
0.
4
0.
6
Lag
Pa
rti
al
A
CF
Partial ACF
Figure: A simulated MA(2) process: xt = (1 + 0.8B + 0.6B2)wt

Non-uniqueness of MA Models and Invertibility
If we replace θ by 1/θ, the autocorrelation function for MA(1), ρ(h) = θ1+θ2
(studied later), will not be changed. There are thus two processes,
xt = wt + θwt−1
and
xt = wt +
1
θ
wt−1,
show identical autocorrelation pattern, and hence the MA(1) coefficient is
not uniquely identified.
For the general moving average process, MA(q), there is a similar
identifiability problem.
The problem can be resolved by assuming that the following operator is
invertible
1 + θ1B + θ2B2 + · · ·+ θqBq
i.e., all absolute roots of 1 + θ1z + θ2z2 + · · ·+ θqzq = 0 lie outside the
unit circle.

Examples - Invertibility of MA Models
Example
1.) The MA(1) process xt = wt + 0.4wt−1 is an invertible process because
|θ| = |0.4| < 1 (or equivalently the root of 1 + 0.4z = 0 in absolute value
is z = 1/0.4 = 2.5 > 1).
2.) The MA(1) process xt = w1 + 1.8wt−1, is not invertible process because
the root of 1 + 1.8z = 0 in absolute value is z = 1/1.8 = 0.56 < 1.
3.) The MA(3) process xt = wt − 0.3wt−1 + 0.7wt−2 − 1.2wt−3 is not
invertible as
roots <- polyroot(c(1,-0.3, 0.7,-1.2))
Mod(roots)
## [1] 0.8810509 0.8810509 1.0735363

Mixed Autoregressive-Moving Average (ARMA)
Models
Definition
A time series {xt}, with zero mean, is called Autoregressive-Moving Average
process of order (p,q) and denoted by ARMA(p,q), if it can be written as
xt =
p∑
i=1
ϕixt−i︸︷︷︸
Autoregressive Part
+
q∑
j=0
θjwt−j︸︷︷︸
Moving Average Part
, (8)
where wt ∼ wn(0, σ2w ) is white noise, ϕi & θj for i = 1,2, · · · ,p, j = 0,1, · · · ,q,
are the parameters of the Autoregressive & Moving Average parts
respectively, and θ0 = 1.

Mixed Autoregressive-Moving Average (ARMA)
Models
With backshift operator, the process can be rewritten as:
Φp(B)xt︸︷︷︸
Autoregressive Part
= Θq(B)wt︸︷︷︸
Moving Average Part
, (9)
where Φp(B) = 1−∑pi=1 ϕiB i and Θq(B) = 1 +∑qj=1 θjB j .
If the mean, µ, of xt is not zero, replace xt by xt − µ to get:
Φp(B)(xt − µ) = Θq(B)wt .
can also be written as:
xt = δ + ϕ1xt−1 + · · ·+ ϕpxt−p︸︷︷︸
Autoregressive Part
+wt + θ1wt−1 + · · ·+ θqwt−q︸︷︷︸
Moving Average Part
, (10)
where δ = µ(1− ϕ1 − ϕ2 − · · · − ϕp) and µ is called the intercept obtained from
the output of the R function arima().

Three Conditions for ARMA Models
The ARMA model is assumed to be stationary, invertible and identifiable:
The condition for the stationarity is the same for the pure AR(p) process:
i.e., the roots of 1− ϕ1z − ϕ2z2 − · · · − ϕpzp lie outside the unit circle.
The condition for the invertibility is the same for the pure MA(q) process:
i.e., the roots of 1 + θ1z + θ2z2 + · · ·+ θqzq lie outside the unit circle.
The identifiability condition means that the model is not redundant: i.e.,
Φp(B) = 0 and Θq(B) = 0 have no common roots.

Stationary and Invertibility Conditions for ARMA
Models
Model Stationary conditions Invertibility conditions
MA(1) None |θ1| < 1
xt = δ + at + θ1at−1
MA(2): None θ1 + θ2 < 1
xt = δ + at + θ1at−1 + θ2at−2 θ2 − θ1 < 1
|θ2| < 1
AR(1)
xt = δ + ϕ1xt−1 + at |ϕ1| < 1 None
AR(2) ϕ1 + ϕ2 < 1 None
xt = δ + ϕ1xt−1 + ϕ2xt−2 + at ϕ2 − ϕ1 < 1
|ϕ2| < 1
ARMA(1, 1)
xt = δ + ϕ1xt−1 + at + θ1at−1 |ϕ1| < 1 |θ1| < 1

Example: Model Redundancy (Shared Common
Roots in ARMA Models)
Consider the ARMA(1,2):
xt = 0.2xt−1 + wt − 1.1wt−1 + 0.18wt−2,
this model can be written as
(1− 0.2B)xt = (1− 1.1B + 0.18B2)wt
or equivalently
(1− 0.2B)xt = (1− 0.2B)(1− 0.9B)wt
cancelling (1− 0.2B) from both sides to get:
xt = (1− 0.9B)wt
Thus, the process is not really an ARMA(1,2), but it is a MA(1) ≡ ARMA(0,1).

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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