Financial Econometrics Econ 40357 ARIMA Part 2: Autoregressive - - PowerPoint PPT Presentation

Financial Econometrics Econ 40357 ARIMA Part 2: Autoregressive Models

N.C. Mark

University of Notre Dame and NBER

August 30, 2020

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Autoregressive (AR) models.

These are models with more durable, persistent dependence over time. Let ǫt

iid

∼ (0, σ2

ǫ ), and |ρ| < 1. Then the AR(1) model is

yt = a + ρyt−1 + ǫt where E (yt) = µ = a (1 − ρ), Var (yt) = σ2

ǫ

(1 − ρ2), ρ (yt, yt−k) = ρk Note: a = µ (1 − ρ) , which means we can also write it as yt = µ (1 − ρ) + ρyt−1 + ǫt

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The MA representation of the AR(1)

The AR(1) can also be represented as an MA(∞) . yt = a + ρ(a + ρyt−2 + ǫt−1)

• yt−1

+ ǫt = a + ρa + ρ2(a + ρyt−3 + ǫt−2)

• yt−2

+ ρǫt−1 + ǫt = a + ρa + ρ2a + ǫt + ρǫt−1 + ρ2ǫt−2 + ρ2yt−3 . . . = a

• 1 + ρ + ρ2 + ρ3 + · · ·
• a/(1−ρ)

+ ǫt + ρǫt−1 + ρ2ǫt−2 + ρ3ǫt−3 + · · · = a 1 − ρ + ǫt + ρǫt−1 + ρ2ǫt−2 + ρ3ǫt−3 + · · ·

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What is the mean E(yt)? E (yt) = E

• a

1 − ρ + ǫt + ρǫt−1 + ρ2ǫt−2 + ρ3ǫt−3 + · · ·

• =
• a

1 − ρ + Eǫt + ρEǫt−1 + ρ2Eǫt−2 + ρ3Eǫt−3 + · · ·

• =

a 1 − ρ

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What is the Variance Var(yt)? σ2

y

= Var (yt) = E

• ǫt + ρǫt−1 + ρ2ǫt−2 + ρ3ǫt−3 + · · ·

2 = E

• ǫ2

t + ρ2ǫ2 t−1 + ρ4ǫ2 t−2 + · · · + 2ρǫtǫt−1 + 2ρ2ǫtǫt−2 + · · ·

• =

 Eǫ2

t + ρ2Eǫ2 t−1 + ρ4Eǫ2 t−2 + · · · + 2ρEǫtǫt−1 + 2ρ2Eǫtǫt−2 + · · ·

• 

 = σ2

ǫ

• 1 + ρ2 + ρ4 + · · ·
• =

σ2

ǫ

1 − ρ2

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What is the autocorrelation function? First, write the AR(1) in deviations from the mean form, yt = µ (1 − ρ) + ρyt−1 + ǫt yt − µ = ρ (yt−1 − µ) + ǫt Then, γ1 = Cov (yt, yt−1) = E (yt − µ) (yt−1 − µ) = E (ρ (yt−1 − µ) + ǫt) (yt−1 − µ) = ρE (yt−1 − µ)2

• Var(yt−1)

+ E (ǫt (yt−1 − µ))

• =

ρVar (yt) = ρσyσy Hence, ρ (yt, yt−1) = ρ

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γ2 = Cov (yt, yt−2) = E (yt − µ) (yt−2 − µ) = E (ρ (yt−1 − µ) + ǫt) (yt−2 − µ) = ρE (yt−1 − µ) (yt−2 − µ)

• γ1

+ E (ǫt (yt−2 − µ))

• =

ργ1 ρ (yt, yt−2) = ργ1 σyσy = ρρσyσy σyσy = ρ2 We can infer that ρ (yt, yt−k) = ρk

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AR(1) forecasts

Et(˜ yt+1) = ρ˜ yt Et(˜ yt+2) = ρEt(˜ yt+1) = ρ2 ˜ yt Hence, Et(˜ yt+k) = ρk ˜ yt Try it out on daily stock returns.

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Realization of an AR(1) with ρ = 0.96

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How to generate in Eviews

’ Generate white noise process series e = nrnd ’Generate persistent AR(1) smpl @first @first series sto = 0 ’ Initial conditions smpl @first+1 @last series sto = .96*sto(-1)+.5*e ’ Recursion series y = sto delete sto (To get impulse response: Quick, estimate VAR) (arima models.wf1 and pgm)

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Impulse Response Function

The impulse response function traces the effect of a one time,

• ne-standard deviation shock today ǫt = σǫ, on the current and all

future values yt, yt+1, yt+2, .... Stationary processes will revert to their mean values. Let’s analyze as deviations from the mean (set µ = 0). AR(1): yt = ρyt−1 + ǫt, 0 < ρ < 1. yt = ǫt yt+1 = ρyt = ρǫt yt+2 = ρyt+1 = ρ2ǫt yt+k = ρkǫt

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Another representation of impulse response. MA representation (mean suppressed µ = 0), yt = ǫt + ρǫt−1 + ρ2ǫt−2 + ρ3ǫt−3 + · · · One time shock ǫt, with all other shocks shut down, ǫk = 0, k = t yt = ǫt yt+1 = ρǫt yt+2 = ρ2ǫt and so on. Later, I will show you how to generate implulse responses in Eviews.

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Impulse response of AR(1)

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AR(1) with negative ρ

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Unit Root Nonstationarity

Why |ρ| < 1 is necessary for stationarity? It is usually the case that 0 < ρ < 1 in economics and finance (persistence). What happens to the mean and the variance of yt when ρ = 1? What happens to the impulse response function when ρ = 1? (permanent effect).

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Realization of a driftless Random Walk

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Random walk with drift

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The AR(2) model. Back to Stationary Models.

Let ǫt

iid

∼

• 0, σ2

ǫ

• . The second-order autoregressive model (AR(2)) is

yt = a + ρ1yt−1 + ρ2yt−2 + ǫt and is stationary if |ρ1 + ρ2| < 1. Assume stationarity, take expectations µy = a + ρ1µy + ρ2µy a = µy 1 − ρ1 − ρ2

Computing variance and autocovariances by hand is too complicated. It involves taking variance and first-order covariance σ2

y

= ρ2

1σ2 y + ρ2 2σ2 y + 2ρ1ρ2γ1 + σ2 ǫ

γ1 = ρ1σ2

y + ρ2γ1 → γ1 =

ρ1σ2

y

1 − ρ2 Then you must to solve these two equations for σ2

y and γ1.

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AR(2) Impulse Response Function

AR(2) with µ = 0 (or in deviation from mean form). yt = ρ1yt−1 + ρ2yt−2 + ǫt Let y0 = y−1 = 0, One-time shock at time 1, ǫ1, with all other shocks shut down. Trace effect recursively y1 = ǫ1 y2 = ρ1y1 = ρ1ǫ1 y3 = ρ1y2 + ρ2y1 =

• ρ2

1 + ρ2

• ǫt

y4 = ρ1y3 + ρ2y2 = ρ1

• ρ2

1 + 2ρ2

• ǫ1

and so on Is possible to get cyclical impulse responses.

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Realization and Impulse Response AR(2) ρ1 = 0.8, ρ2 = −0.8

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AR(2) forecasts

Form the forecasts and input recursively. Et (˜ yt+1) = ρ1 ˜ yt + ρ2 ˜ yt−1 Et (˜ yt+2) = ρ1 (Et (˜ yt+1)) + ρ2 ˜ yt Et (˜ yt+3) = ρ1 (Et (˜ yt+2)) + ρ2Et (˜ yt+1)

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Extensions

1

No need to stop at AR(2). Can add more and more lags.

2

In MA model, can add more and more lagged shocks.

3

Difference between MA and AR. AR is dependence across time of observations. MA is dependence across time of shocks.

4

MA memory is finite

5

AR memory is infinite (but diminishes exponentially)

6

Can combine MA and AR. Here’s ARMA(1,1) yt = a + ρyt−1 + ǫt + θ1ǫt−1

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