ARMA models
Financial time series – lecture four
Financial time series – lecture four
ARMA models
Outline
ACF and PACF. Detailed accounts for ACF and PACF of
ARMA models. Need to understand basic concepts and
definitions of ACF and PACF. Know how to compute ACF for
ARMA models.
MLE and LS estimations for ARMA models. Understand the
standard MLE and LS approach. Able to distinguish exact and
conditional estimates. Able to write out the optimization form
of these two approach for ARMA models.
ARMA model prediction. Understand the general form of
prediction for linear time series models. Able to conduct
multiple step ahead forecast for AR, MA and simple ARMA
models. Able to compute the prediction error and it’s
variance. (2.4.4., 2.5.4 and 2.6.4 of Tsay book)
Financial time series – lecture four
ARMA models
The autoregressive model AR(p)
First order AR
X
t
=
φ
0
+
φ
1
X
t

1
+
w
t
where
w
t
∼
WN
(0
, σ
2
).
Properties of AR(1) process:
E
(
X
t
)
=
φ
0
Var
(
X
t
)
=
σ
2
1

φ
2
1
if

φ
1

<
1
ρ
(
h
)
=
φ
h
1
ACF examples of AR(1) processes.
ts.sim = arima.sim(list(order = c(1,0,0), ar = 0.8),
n = 200);
acf(ts.sim,lag.max=12);
Financial time series – lecture four
ARMA models
AR(1) with positive
φ
1
.
0
2
4
6
8
10
12
0.0
0.2
0.4
0.6
0.8
1.0
Lag
ACF
Series
ts.sim
Figure:
ACF when
φ
1
=0.8
Financial time series – lecture four
ARMA models
AR(1) with negative
φ
1
.
ts.sim = arima.sim(list(order = c(1,0,0), ar = 0.8),
n = 200);
acf(ts.sim,lag.max=12);
0
2
4
6
8
10
12
ï
0.5
0.0
0.5
1.0
Lag
ACF
Series
ts.sim
Financial time series – lecture four
ARMA models
AR(1) with small
φ
1
.
Time
ts.sim
0
50
100
150
200
ï
3
ï
2
ï
1
0
1
2
3
Figure:
ACF when
φ
1
=0.2
Financial time series – lecture four
ARMA models
AR(1) with large
φ
1
. What are the differences?
Time
ts.sim
0
50
100
150
200
ï
15
ï
10
ï
5
0
5
10
Figure:
ACF when
φ
1
=0.99
Financial time series – lecture four
ARMA models
Properties of AR(1)
The stationary condition for AR(1) process is

φ
1

<
1.
X
t
=
φ
0
+
φ
1
X
t

1
+
w
t
Var
(
X
t
) =
φ
2
1
Var
(
X
t

1
) +
σ
2
If
X
t
is stationary,
(1

φ
2
1
)
Var
(
X
t
) =
σ
2
,
thus

φ
1

<
1.
Condition on the past, we have
E
(
X
t

X
t

1
) =
φ
0
+
φ
1
X
t

1
,
Var
(
X
t

X
t

1
) =
Var
(
w
t
) =
σ
2
.
Financial time series – lecture four
ARMA models
MA representation of AR(1)
For simplicity, consider the mean zero series
X
t
=
φ
1
X
t

1
+
w
t
=
φ
1
(
φ
1
X
t

2
+
w
t

1
) +
w
t
=
φ
2
1
X
t

2
+
φ
1
w
t

1
+
w
t
=
· · ·
=
w
t
+
φ
1
w
t

1
+
φ
2
1
w
t

2
+
· · ·
The shock at
t

k
has an exponentially decaying influence on
X
t
. The associated coefficients are called the
ψ
weights.
Financial time series – lecture four
ARMA models
Higher order AR models
The zero mean AR(p) model is
X
t
=
φ
1
X
t

1
+
φ
2
X
t

2
+
· · ·
+
φ
p
X
t

p
+
w
t
,
where
w
t
is uncorrelated with
X
t

j
for any
j
. Note that mean
is zero here.
Include the mean, the model is
X
t
=
φ
0
+
φ
1
X
t

1
+
φ
2
X
t

2
+
· · ·
+
φ
p
X
t

p
+
w
t
,
where
φ
0
=
μ
(1

φ
1
 · · · 
φ
p
), and
μ
=
E
(
X
t
).
Stationarity condition: all roots of
1

φ
1
B
 · · · 
φ
p
B
p
= 0
are outside the unit disc.
B
is the backshift operator.
Financial time series – lecture four
ARMA models
AR(2) with complex roots
Consider an AR(2)
X
t
= 1
.
5
X
t

1

0
.
75
X
t

2
+
w
t
, the roots
of 1

1
.
5
B
+ 0
.
75
B
2
are outside of unit circle. This will
ensure a stationary AR(2) process.
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 Fall '19
 Autoregressive moving average model, ARMA