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Simulations with AR(1) Processes
• To simulate 500 values from Yt = 0.6Yt−1 + et with et ∼ NID(0, 1), use the R
command:
arima.sim(list(ar=0.6), n=500)
• To simulate 600 values from Yt = −0.4Yt−1 + et with et ∼ NID(0, 1), use the R
command:
arima.sim(list(ar=-0.4), n=600)
Note: We usuall discard some values at the beginning of each simulated series for
further analysis.
Examples: Obtain and plot the last 300 values together with their acf for each
simulated series.
Solution:
• d1=arima.sim(list(ar=0.6), n=500)[201:500]
ts.plot(d1)
acf(d1)
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Time
d1
0 50 100 150 200 250 300
−4 0 2
Simulated AR(1) with phi=0.6
0 5 10 15 20
0.0 0.6
Lag
ACF
Series d1
• d2=arima.sim(list(ar=-0.4), n=600)[301:600]
ts.plot(d2)
acf(d2)
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Time
d2
0 50 100 150 200 250 300
−3 0 2
Simulated AR(1) with phi=−0.4
0 5 10 15 20
−0.5 0.5
Lag
ACF
Series d2
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AR(1) with lag operator L
Consider the stationary AR(1) process given by
Yt = φYt−1 + et, where |φ| < 1 and {et} ∼ W N(0, σ2e).
Using the lag operator L, this can be written as
Yt = φLYt + et, where Yt−1 = LYt.
This is equivalent to (1 − φL)Yt = et
. Since |φ| < 1, it is clear that the solution to
1 − φL = 0, L =1φ > 1.
Note: The equation 1 − φL = 0 associate with this AR(1) process is known as
its charateristic equation. (see P71 of the textbook). Therefore, |L| > 1 for
stationarity. This is known as the root is outside the unit circle.
Diagram:
Example: Consider the stationary AR(1) process given by Yt = 0.7Yt−1 + et
. Write
down this using the lag operator L and the corresponding charateristic equation.
Hence find the root of this equation and show that the root satisfies |L| > 1 or the
root is outside the unit circle.
Solution:
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Exercise: Consider the AR(1) process given by Yt = −0.3Yt−1 + et
. Write down this
using the lag operator L and the corresponding charateristic equation. Hence find
the root of this equation and determine whether this process is stationary.
Solution:
Remark: In AR(1) processes, the current Yt depends only on the immediate past
STAT485/685作业代做
value Yt−1 or the conditional probability distribution of Yt given the history dependent
only on the immediate past Yt−1. That is,
P(Yt
|Yt−1, Yt−2, · · ·) = P(Yt
|Yt−1)
and is known as the Markov Property. Therefore, a stationary AR(1) is also known
as a Markov Process.
Now we look at the second order AR(2) process in detail.
4.5 AR(2) Processes - P71 of the textbook
Consider an AR(2) process generated by the recursive equation
Yt = φ1 Yt−1 + φ2 Yt−2 + et, {et} ∼ W N(0, σ2e).
Using the lag operator, this can be written as
(1 − φ1 L − φ2 L2) Yt = et.
The associate charateristic equation for this AR(2) process is 1 − φ1 L − φ2 L
2 = 0.
For convenience, let φ(L) = 1 − φ1 L − φ2 L
2
. Then we can write the above AR(2)
process as
φ(L) Yt = et,
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Example: Assuming {Yt} given in Yt = φ1 Yt−1 + φ2 Yt−2 + et
, {et} ∼ W N(0, σ2e) is
stationary, find
(i) E(Yt).
(ii) By evaluating the both sides of V ar(Yt) = V ar(φ1 Yt−1 + φ2 Yt−2 + et), find an
expression relating γ0, γ1 and σ
2
e
. Recall that under the stationary conditions,
Cov(Yt−k, et) = E(Yt−ket) = 0, k ≥ 1.
(ii) By evaluating the both sides of V ar(Yt − φ1 Yt−1 − φ2 Yt−2) = V ar(et), find an
expression relating γ0, γ1, γ2 and σ
2
e
.
Solution:
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.
Solution to this charateristic equation is useful in many applications. Therefore, we
look the methods of solving quadratic equations.
A Review of Solving Quadratic Equations
Consider the charateristic equation φ(L) = 1 − φ1 L − φ2 L
2 = 0 for
Yt = φ1 Yt−1 + φ2 Yt−2 + et
.
Since this charateristic equation is a quadratic, it is easy to solve either by factorisation
or from the fomula.
• Factorise: Let 1 − φ1 x − φ2 x
2 = (1 − δ1x)(1 − δ2x). Then the solutions are
• Formula: Let ax2 + bx + c be a quadratic equation. Then the solutions are.
Example: Consider the AR(2) process given by Yt = 1.3Yt−1 − 0.4Yt−2 + et
.
(i) Write down this AR(2) process using the lag operator L.
(ii) What is the corresponding charateristic polynomial φ(L)?
(iii) Solve the equation φ(L) and show that both roots are outside the unit circle.
Solution:
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.
Example: For each of the following AR(2) process:
(i) Write down this AR(2) process using the lag operator L.
(ii) What is the corresponding charateristic polynomial φ(L)?
(iii) Solve the equation φ(L) and show that both roots are outside the unit circle.
(a) Yt = −0.3Yt−1 + 0.1Yt−2 + et
.
(b) Yt = 1.5Yt−1 − 0.4Yt−2 + et
.
(c) Yt = 1.3Yt−1 + 0.5Yt−2 + et
.
(d) Yt = −Yt−1 + 6Yt−2 + et
.
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Solution:
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.
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Stationarity of AR(2) Processes
An Important Result:
A Stationary solution to the AR(2) process
Yt = φ1Yt−1 + φ2Yt−2 + et
exists if and only if the roots of the associtaed charateristic equation
1 − φ1L − φ2L
2 = 0
are outside the unit circle. That is, both roots larger than 1 in absolute value.
Example: Identify the stationary AR(2) processes in the previous example. Give
reasons.
Solution
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Note: It can be shown that the quadratic equation 1 − φ1L − φ2L
2 = 0 has roots
outside the unit cicle if the following three inequalities hold:
|φ2| < 1
φ2 + φ1 < 1
φ2 − φ1 < 1.
Example: For each of the AR(2) process (a), (b), (c) and (d) above
(i) Write down φ1 and φ2.
(ii) Write down all the three inequalities as given above.
(iii) Using (ii), determine the stationarity of each process.
(iv) Are the results in (iii) consistent with stationarity/nonstationarity from the
previous example?
Solution:
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