Byron Gangnes

Econ 427 lecture 11 slidesEcon 427 lecture 11 slides

Moving Average Models

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

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.

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

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)

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

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

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

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

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

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− + ...,

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

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

Byron Gangnes

Correlogram from MA(1)Correlogram from MA(1)