Byron Gangnes Econ 427 lecture 11 slides Moving Average Models.
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Transcript of Byron Gangnes Econ 427 lecture 11 slides Moving Average Models.
![Page 1: Byron Gangnes Econ 427 lecture 11 slides Moving Average Models.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649f0a5503460f94c1e505/html5/thumbnails/1.jpg)
Byron Gangnes
Econ 427 lecture 11 slidesEcon 427 lecture 11 slides
Moving Average Models
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Byron Gangnes
Review: Review: Wold’s TheoremWold’s Theorem• Review: Wold’s Theorem says that we
can represent any covariance-stationary time series process as an infinite distributed lag of white noise:
0
2
( ) ,
(0, ),
t t i t ii
t
y B L b
WN
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Byron Gangnes
Finite approximationsFinite approximations• The problem is that we do not have infinitely
many data points to use in estimating infinitely many coefficients!
• We want to find a good way to approximate the unknown (and unknowable) true model that drives real world data.
– The key to this is to find a parsimonious, yet accurate, approximation.
• We will look at two building blocks of such models now: the MA and AR models, as well as their combination.
• We start by examining the dynamic patterns generated by the basic types of models.
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Byron Gangnes
Moving Average (MA) modelsMoving Average (MA) models• MA(1)
• Intuition? – Past shocks (innovations) in the series feed
into the succeeding period.
1
2
(1 )
(0, )
t t t
t
t
y
L
where WN
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Byron Gangnes
Properties of an MA(1) seriesProperties of an MA(1) series• Uncond. Mean =
• Uncond. Variance=
1
1( ) ( ) 0t t t
t t
Ey E
E E
1
21
2 2 2
2 2
var( ) var
var( ) var( )
(1 )
t t t
t t
y
This uses 2 statistics results:
1) for 2 random vars x and y, E(x+y)=E(x)+E(y)
2) If a is a constant:
E(ax) = a E(x)
This uses 2 statistics results:
1) var(x+y)= var(x) +var(y),
if they are uncorrelated
2) var(ax) = a2var(x)
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Byron Gangnes
Properties of an MA(1) seriesProperties of an MA(1) series• Conditional mean (the mean given that we
are a particular time period, t; i.e. given an “information set” ):
1 1 1
1 1 1
1
( | ) ( ) |
( | ) ( | )
0
t t t t t
t t t t
t
E y E
E E
1 1 2 3, , ,...t t t t
We read this as “the expected value of yt-1 conditional on the information set .”
That just means we want the expected value taking into account that we know the values of the past epsilons.
t−1
t−1
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Byron Gangnes
Properties of an MA(1) seriesProperties of an MA(1) series• Conditional variance =
2
1 1 1
21
2 2
var( | ) ( | ) |
( | )
( )
t t t t t t
t t
t
y E y E y
E
E
To see this, plug in the MA(1) expression for yt and the expression for the expected value of yt conditional on the information set (which we just calculated). Notice they differ only by , the innovation in y that we have not yet observed
t
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Byron Gangnes
Properties of an MA(1) seriesProperties of an MA(1) series• Autocovariance function:
• Let’s look at how that is derived…
1 1( ) ( ) ( )( )
, 1
0,
t t t t t tE y y E
otherwise
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Byron Gangnes
Autocovariance functionAutocovariance function• Here is the derivation of that:
E (t +t−1)(t− +t−−1)( ) =
=E tt−( ) + E tt−−1( ) + E t−1t−( ) + E t−1t−−1( )
when =1:
=E tt−1( ) +E tt−( ) +E t−1t−1( ) + E t−1t−( )
=0 + 0 + + 0
When , all terms are zero. Check it! >1
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Byron Gangnes
Properties of an MA(1) seriesProperties of an MA(1) series• Autocorrelation function is just this divided
by the variance:
2, 1( )
( ) 1(0)
0, otherwise
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Byron Gangnes
Can yCan ytt be written another way? be written another way?• Under certain circumstances, we can write an
MA process in an autoregressive representation– The series can be written as a function of past lags of
itself (plus a current innovation)
• Why does this form have intuitive appeal?– Today’s value of the variable is related explicitly to
past values of the variable
yt=t +φyt−1 +φ
yt− + ...,
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Byron Gangnes
InvertibilityInvertibility• In this case, we say that the MA process is
invertible
• This will occur when
• See the book for a demonstration of how this works and why the key criterion.
<1
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Byron Gangnes
What does an MA(1) look like?What does an MA(1) look like?
ma1seriesa = whitenoise + .7*whitenoise(-1)
-3
-2
-1
0
1
2
3
4
80 82 84 86 88 90 92 94 96 98 00 02 04
MA1SERIESE
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Byron Gangnes
Correlogram from MA(1)Correlogram from MA(1)