Moving-average (MA) model Model with finite time lags of memory! Some daily stock returns have minor serial corre- lations. Can be modeled as MA or AR models. MA(1) model 1. Form: r t = µ + a t − θa t−1 2. Stationarity: always stationary. 3. Mean (or expectation): E (r t )= µ. 4. Variance: V ar (r t ) = (1 + θ 2 )σ 2 . 5. Autocovariance: (a). Lag 1: Cov (r t ,r t−1 )= −θσ 2 . (b). Lag l: Cov (r t ,r t−l ) = 0 for l> 1. Thus, r t is not related to r t−2 , r t−3 , ··· . 1
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Moving-average (MA) model
Model with finite time lags of memory!
Some daily stock returns have minor serial corre-
lations. Can be modeled as MA or AR models.
MA(1) model
1. Form: rt = µ+ at − θat−1
2. Stationarity: always stationary.
3. Mean (or expectation): E(rt) = µ.
4. Variance: V ar(rt) = (1+ θ2)σ2.
5. Autocovariance:
(a). Lag 1: Cov(rt, rt−1) = −θσ2.
(b). Lag l: Cov(rt, rt−l) = 0 for l > 1.
Thus, rt is not related to rt−2, rt−3, · · · .
1
ACF: ρ1 = −θ1+θ2
, ρl = 0 for l > 1.
Finite memory! MA(1) models do not remember
what happen two time periods ago.
0 5 10 15 20
0.25
0.50
0.75
1.00ACF of MA(1) with θ = (0.5)
0 5 10 15 20
0.0
0.2
0.4PACF of MA(1) with θ = (0.5)
0 5 10 15 20
0.0
0.5
1.0ACF of MA(1) with θ = (-0.5)
0 5 10 15 20
-0.3
-0.2
-0.1
0.0PACF of MA(1) with θ = (-0.5)
ACF and PACF for MA(1) model
6. Forecast (at origin t = n):
(a). 1-step ahead: rn(1) = µ − θan. Why? Be-
cause at time n, an is known, but an+1 is not.
(b). 1-step ahead forecast error: en(1) = an+1
with variance σ2a .
(c). Multi-step ahead: rn(l) = µ for l > 1.
Thus, for an MA(1) model, the multi-step ahead
forecasts are just the mean of the series. Why?
Because the model has memory of 1 time period.
(d). Multi-step ahead forecast error:
en(l) = an+l − θan+l−1.
(e). Variance of multi-step ahead forecast error:
(1 + θ2)σ2a = variance of rt.
7. Invertibility:
Concept: rt is a proper linear combination of atand the past observations {rt−1, rt−2, · · · }.
Why is it important? It provides a simple way to
obtain the shock at.
For an invertible model, the dependence of rt on
rt−l converges to zero as l increases.
MA(1) model with condition |θ| < 1:
at = (rt − µ) +∞∑i=1
θi(rt−i − µ)
or AR(∞) model
rt =µ
1− θ+ at −
∞∑i=1
θirt−i.
Invertibility of MA models is the dual property of
stationarity for AR models.
MA(2) model
1. Form: rt = µ+ at − θ1at−1 − θ2at−2, or
rt = µ+ (1− θ1B − θ2B2)at.
2. Stationary with E(rt) = µ.
3. Invertibility:
all the roots of θ(z) = 1−θ1z−θ2z2 = 0 lie outsidethe unit circle.
Decompose 1− θ1z − θ2z2 = (1− α1z)(1− α2z).
Then |α1| < 1 and |α2| < 1.
rt = µ+ (1− α1B)(1− α2B)at.
Let ut = (1− α2B)at. Then
rt = µ+ ut − α1ut−1 and ut = at − α2at−1.
Given {rt}, we can invert {ut} and then invert {at}.
{0 if j = 1,2, · · · , l.an+j if j = 0,−1,−2, · · · .
D. Forecast error:
en(l) = pn+l − pn(l) =l−1∑j=0
ψjan+l−j ,
where ψi can be calculated, recursively:
ψj =j−1∑i=0
πj−iψi , j = 1,2, · · · , l − 1.
πj is the coefficients of the expansion:
π(B) =ϕp(B)(1−B)d
θq(B)= 1−
∞∑j=1
πjBj.
pt+l =∞∑j=1
πjpt+l−j + at+l.
E. Forecast variance:
Var[en(l)] = σ2a
l−1∑j=0
ψ2j .
F. Forecast interval (limit) (FI):pn(l)−Nα2σa
√√√√√ l−1∑j=0
ψ2j , pn(l) +Nα
2σa
√√√√√ l−1∑j=0
ψ2j
where Nα
2is the α/2-quantile of the standard nor-
mal distribution,
In SAS, the output gives the forecasting intervalsfor AR(p), MA(q), and ARMA(p,q) models.
Note that rt = pt − pt−1, where pt = lnPt. SASoutputs give the forecasting value of pn+l, i.e.,pn(l), and the forecasting intervals of pn+l, i.e.,[Lα(l), Uα(l)].