Time Series Basics (2)

Fin250f: Lecture 3.2

Fall 2005

Reading: Taylor, chapter 3.5-3.7, 3.9(skip 3.6.1)

Outline

Linear stochastic processes Autoregressive process Moving average process Lag operator Forecasting AR and MA’s The ARMA(1,1) Trend plus noise models Bubble simulations

Linear Stochastic Processes

Linear modelsTime series dependenceCommon econometric frameworksEngineering background

AR(1)Autoregressive Process,

Order 1

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Xt = a+φXt−1 +et(Xt −μ ) =φ(Xt−1 −μ ) +et|φ |<1

var(et ) =σ e2

AR(1) Properties

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E(Xt ) = μ

E(Xt ) =a

1−φ

Et (Xt+1) = φ(Xt − μ ) +μ

Et−1(Xt ) = a +φXt−1

var(Xt ) =σ e

2

(1−φ2 )

ρ j = cor(Xt ,Xt− j ) = φ j

AR(m)

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(Xt −μ ) = φ j (Xt− j −μ )j=1

m

∑ +et

Moving Average Process of Order 1, MA(1)

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Xt = μ +θet−1 + et

MA(1) Properties

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E(Xt ) = μ

Et (Xt+1) = μ +θetvar(Xt ) = (1+θ 2 )σ e

2

cor(Xt ,Xt−1) =θ

(1+θ 2 )

cor(Xt ,Xt− j ) = 0 j ≥ 2

MA(m)

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Xt = μ + θ jet− j + etj=1

m

∑

AR->MA

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Xt =φXt−1 +etXt−1 =φXt−2 +et−1

Xt =φ(φXt−2 +et−1) +etXt =φ2Xt−2 +φet−1 +et

Xt =φmXt−m + φ jet− jj=0

m

∑ , |φ |<1

Xt = φ jet− jj=0

∞

∑

Lag Operator (L)

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LXt = Xt−1

LkXt = Xt−kLkμ = μ

Using the Lag Operator

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Xt − μ = φ(Xt−1 − μ ) + etXt − μ = φL(Xt − μ ) + et(1−φL)(Xt − μ ) = et

An important feature for L

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Xt = φXt−1 + et(1−φL)Xt = et

Xt =1

(1−φL)et

Xt = φ jLjetj=0

∞

∑ = (φL) j etj=0

∞

∑

1(1−φL)

= (φL) jj=0

∞

∑

MA -> AR

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Xt = μ +θet−1 +etXt −μ = (1+θL)et

1(1+θL)

(Xt −μ ) = et

(−θL) j (Xt −μ )j=0

∞

∑ = et

MA->AR

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Xt −μ = −(−θ ) jj=1

∞

∑ (Xt− j −μ ) +et

Xt −μ = (−1) j−1θ jj=1

∞

∑ (Xt− j −μ ) +et

Xt −μ =θ (−θ ) j−1

j=1

∞

∑ (Xt− j −μ ) +et

|θ |<1

Forecasting the AR(1)

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(Xt+1 − μ ) = φ(Xt − μ ) + et+1

Et (Xt+1 − μ ) = φ(Xt − μ ) +Et (et+1)

Et (et+1) = 0

ft ,1 = Et (Xt+1) = μ +φ(Xt − μ )

Forecasting the AR(1): Multiperiods

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(Xt+1 − μ ) = φ(Xt − μ ) + et+1

(Xt+2 − μ ) = φ(Xt+1 − μ ) + et+2

(Xt+2 − μ ) = φ(φ(Xt − μ ) + et+1) + et+2

(Xt+2 − μ ) = φ2 (Xt − μ ) +φet+1 + et+2

Et (Xt+2 − μ ) = φ2 (Xt − μ ) +φEtet+1 +Etet+2

ft ,2 = Et (Xt+2 ) = μ +φ2 (Xt − μ )

ft ,N = Et (Xt+N ) = μ +φN (Xt − μ )

Forecasting an MA(1)

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Xt = μ +θet−1 + etXt+1 = μ +θet + et+1

Et (Xt+1) = μ +θEt (et )

The ARMA(1,1): AR and MA parts

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Xt − μ = φ(Xt−1 − μ ) +θet−1 + et

var(Xt ) =1+ 2φθ +θ 2

1−φ2σ e

2

ρ j = cor(Xt ,Xt− j ) = Aφ j

A =(1+φθ )(φ +θ )φ(1+ 2φθ +θ 2 )

ARMA(1,1) with L

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(1−φL)(Xt −μ ) = (1+θ )et

et =(1−φL)(1+θL)

(Xt −μ )

et = (1−φL) (−θL) jj=0

∞

∑ (Xt −μ )

ARMA(1,1) with L

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(Xt −μ ) =θ (j=1

∞

∑ −θ ) j−1(Xt− j −μ ) +

φ (−θ ) j−1(Xt− j −μ )j=1

∞

∑ +et

(Xt −μ ) = (φ +θ ) (j=1

∞

∑ −θ ) j−1(Xt− j −μ ) +et

Forecasting 1 Period

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(Xt −μ ) = (φ +θ ) (j=1

∞

∑ −θ ) j−1(Xt− j −μ ) +et

ft ,1 = μ +(φ +θ ) (j=1

∞

∑ −θ ) j−1(Xt− j −μ )

ARMA(p,q)

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Xt −μ = φi (Xt−i −μ )i=1

p

∑ + θ jet− jj=1

q

∑ +et

Why ARMA(1,1)?

Small, but persistent ACF’sComparing the AR(1) and ARMA(1,1)

AR(1) ACF’s

ARMA(1,1) ACF’s

Adding an AR(1) to an MA(0)(Trend plus noise)

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Zt = Xt +Yt(Xt − μ X ) = φ(Xt − μ X ) + et(1−φL)(Xt − μ X ) = etYt = μY +utZt is ARMA(1,1)

Why Is This Useful?(Taylor 3.6.2)

Returns follow a combination processSum of:

Small, but very persistent trend Independent noise term

Trend Plus Noise

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rt = ut +ε tut = φut−1 + etcov(rt ,rt−1) = cov(φut−1 + et +ε t ,ut−1 +ε t−1)

Trend Plus Noise

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cov(rt ,rt−1) = cov(φut−1,ut−1)

cov(rt ,rt−1) = φσ u

2

cov(rt ,rt− j ) = φ jσ u

2

cor(rt ,rt−1) =cov(rt ,rt−1)σ r

2=

φσ u

2

(σ u

2 +σ ε

2 )

A =σ u

2

(σ u

2 +σ ε

2 )

cor(rt ,rt− j ) = Aφ j

Parameter Example

A small bigA = 0.02,

Trend Plus Noise ACF

Temporary Pricing ErrorsBubbles(3.6.1)

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log(Pt ) = log(Pt*) +ut

log(Pt*) = log(Pt−1

* ) +ε tut =φut−1 +etRt = log(Pt ) − log(Pt−1)

Rt = log(Pt*) − log(Pt−1

* ) +ut −ut−1

Rt = ε t +ut −ut−1

AR(1) Difference

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ut = φut−1 + etut−1 = φut−2 + et−1

ut − ut−1 = φ(ut−1 − ut−2 ) + et − et−1

ARMA(1,1)

Returns = ARMA + noise

Variance Ratio

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Rt = ε t +(ut −ut−1)

ut =φut−1 +et

B =var(ut −ut−1)

var(Rt )=

2 var(ut ) − 2φvar(ut−1)var(Rt )

B =2(1−φ)var(ut )

σ ε2 + 2(1−φ)var(ut )

Return Autocorrelations

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Rt = ε t +ut −ut−1

cor(Rt ,Rt−1) = Bcor(ut −ut−1,ut−1 −ut−2 )

cor(Rt ,Rt− j ) = BAφ j

A =(1−φ)(φ −1)

2φ(1−φ)=

−(1−φ)2φ

< 0

An Example

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var(ε t ) = 0.001,var(et ) = 0.001

φ = 0.99,B = 0.49

Bubble Price Simulation

Return ACF

Outline

Linear stochastic processes Autoregressive process Moving average process Lag operator Forecasting AR and MA’s The ARMA(1,1) Trend plus noise models Bubble simulations