Regression With Autocorrelated...

OLS Regression Auto-correlated Models Regression with Autocorrelated Errors

Regression With Autocorrelated ErrorsEDU 7309 Project

Xiaowen Hu & Wenkai Bao

Southern Methodist University

Apr. 7th, 2010

Xiaowen Hu & Wenkai Bao Regression With Autocorrelated Errors


Linear Regression Model

Settings and Assumptions


yi = β0 + β1xi1 + . . . + βkxik + εi ,

where yi , xi1, . . . , xik are observations of k + 1 variables, andεi

iid∼ N(0, σ2).

E(εi) = 0 for i = 1, . . . , nVar(εi) = σ2 for i = 1, . . . , ncov(εi , εj) = E(εiεj) = 0 for i 6= j




Least Square Regression

β0, . . . , βk unknownResidual ei = yi − β̂0 − β̂1xi1 − . . .− β̂kxik

β̂0, . . . , β̂k minimizes∑n

i=1 e2i



Autocorrelated Errors

Relaxing The Assumptions

What if cov(εi , εj) 6= 0?More specific, autocorrelation among errorsThis may occur when

Missing true explanatory variablesMisspecification of models (linear vs. quadratic)Pure correlated errors (true autocorrelation)

β̂0, . . . , β̂k are still unbiasedthat is, expectations of β̂’s are β’s.




Relaxing The Assumptions

However...Variance of errors may be underestimatedVariance of β̂’s may be underestimatedConfidence intervals may not be applicableSpurious regressione.g. two uncorrelated variables may appear related.




Remedial Options

Add predictorsHigher order predictorsTransform variablesCochrane-Orcutt (1949)



ARMA Models

Basic Definitions

Time series processA collection of random variables Xt ’s. i.e. {Xt}, where t istime index.Autocovariance

γ(t1, t2) = cov(Xt1 , Xt2)

Autocorrelation

ρ(t1, t2) =γ(t1, t2)

σ(Xt1)σ(Xt2)

White noiseA type of time series that

Xt ’s are identically distributedγ(t1, t2) = 0γ(t , t) = σ2



ARMA Models

Autoregressive Models

AR(p) ModelAn autoregressive model of order p is

Xt − µ− φ1(Xt−1 − µ)− . . .− φp(Xt−p − µ) = wt ,

where µ is mean, and wt is white noise.



ARMA Models

Operator Form of AR Models

Introduce backward shift operator0.8BXt = 0.8Xt−1, 1.2B2Xt = 1.2Xt−2Similar properties as algebraic counterparts(0.3B + 1.6B)Xt = 1.9BXt = 1.9Xt−1

Rewrite AR(p) model as

(1− φ1B − . . .− φpBp)(Xt − µ) = wt .

Denote it simply as φ(B)(Xt − µ) = wt



ARMA Models

AR(1) Examples

Xt − 0.9Xt−1 = wt , Xt + 0.9Xt−1 = wt

realization of AR(1) with positive phi

Time

0 20 40 60 80 100

−6

−4

−2

02

realization of AR(1) with negative phi

Time

0 20 40 60 80 100

−4

−2

02

4



ARMA Models

Moving Average Models

MA(q) ModelA moving average model of order q is

Xt − µ = wt − θ1wt−1 − . . .− θqwt−q,

where µ is mean, and wt is white noise.



ARMA Models

Operator Form of MA Models

Rewrite MA(q) model as

Xt − µ = (1− θ1B − . . .− θqBq)wt .

Denote it simply as Xt − µ = θ(B)wt



ARMA Models

MA(1) Examples

Xt = wt − 0.9wt−1, Xt = wt + 0.9wt−1

realization of MA(1) with positive theta

Time

0 20 40 60 80 100

−3

−1

12

3

realization of MA(1) with negative theta

Time

0 20 40 60 80 100

−4

−2

02



ARMA Models

Stationarity

Stationary Processes

A process {Xt}, t = 0,±1,±2, . . . is stationary ifµt is constantσ2

t is constantcov(Xt1 , Xt2) only depends on |t1 − t2|

All MA(q) models are stationaryNot all AR(p) models are stationaryStationary AR(p) models can be written as MA(∞) form



ARMA Models

Invertibility

Invertible MA ProcessesAn MA process is invertible if it can be written as AR(∞) form

(1− φ1B − φ2B2 − . . .)(Xt − µ) = wt

All AR(p) models are invertibleNot all MA(q) models are invertible



ARMA Models

ARMA(p, q) Models

ARMA(p, q) ModelA time series is ARMA model of order p, q if it is stationary,invertible, and

Xt−µ−φ1(Xt−1−µ)−. . .−φp(Xt−p−µ) = wt−θ1wt−1−. . .−θqwt−q

A combination of AR model and MA modelOperator formφ(B)(Xt − µ) = θ(B)wt



ARMA Models

ARMA(1,1) Examples

Xt − 0.9Xt−1 = wt − 0.7wt−1, Xt + 0.9Xt−1 = wt − 0.7wt−1

realization of ARMA(1,1) with +phi & +theta

Time

0 20 40 60 80 100

−3

−1

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realization of ARMA(1,1) with −phi & +theta

Time

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ARMA Models

Models Considered

More general: ARUMA, ARIMAe.g. random walkInfinite varianceOnly considering stationary and invertible modelsρ(t − k , t) = ρ(k) for any t



ARMA Models

ACF Plots: MA(1)

Xt = wt − 0.9wt−1, Xt = wt + 0.9wt−1

realization of MA(1) with positive theta

Time

0 20 40 60 80 100

−3

−1

12

3

0 20 40 60 80 100

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true autocorrelation of MA(1) with positive theta

lag

au

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sample autocorrelation of MA(1) with positive theta

lag

au

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realization of MA(1) with negative theta

Time

0 20 40 60 80 100

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true autocorrelation of MA(1) with negative theta

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sample autocorrelation of MA(1) with negative theta

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ARMA Models

ACF Plots: AR(1)

Xt − 0.9Xt−1 = wt , Xt + 0.9Xt−1 = wt

realization of AR(1) with positive phi

Time

0 20 40 60 80 100

−6

−4

−2

02

0 20 40 60 80 100

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true autocorrelation of AR(1) with positive phi

lag

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sample autocorrelation of AR(1) with positive phi

lag

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realization of AR(1) with negative phi

Time

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ARMA Models

ACF Plots: ARMA(1,1)

Xt − 0.9Xt−1 = wt − 0.7wt−1, Xt + 0.9Xt−1 = wt − 0.7wt−1

realization of ARMA(1,1) with +phi & +theta

Time

0 20 40 60 80 100

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−1

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true autocorrelation of ARMA(1,1) with +phi & +theta

lag

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sample autocorrelation of ARMA(1,1) with +phi & +theta

lag

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realization of ARMA(1,1) with −phi & +theta

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ARMA Models

PACF Plots

ACF plots identify the order of MA modelTo identify orders of AR(MA) modelsUse partial ACF plots

PACF ρ∗(k)

is the correlation coefficient between Xt and Xt−k with the lineareffect of {Xt−1, . . . , Xt−(k−1)}, on each, removed.



ARMA Models

PACF Plots

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partial sample autocorrelation of AR(1) with positive phi

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partial sample autocorrelation of AR(1) with negative phi

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partial sample autocorrelation of MA(1) with positive phi

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partial sample autocorrelation of MA(1) with negative phi

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partial sample autocorrelation of ARMA(1,1) with +phi & +theta

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partial sample autocorrelation of ARMA(1,1) with −phi & +theta



Detecting Autocorrelated Errors

Graphical Check

Residual plotACF/PACF plots

AR(p) MA(q) ARMA(p, q)ACF Tails off Cuts off Tails off

after lag qPACF Cuts off Tails off Tails off

after lag p



Detecting Autocorrelated Errors

Objective Tests

Durbin-Watson d test

Durbin-Watson statistics

Dk =

∑nt=k+1(et − et−k )2∑n

t=1 e2t

Dk ≈ 2(1− ρk )

Limited to AR(1) models

The runs testRun: an uninterrupted sequence of + or - signs of theresiduals



Theories

OLS Regression

Modelyt = β′xt + εt , t = 1, 2, . . . , n

Matrix formy = Xβ + ε,

where X = [x1, . . . , xn]′, ε= (ε1, . . . , εn)′ with

variance-covariance matrix Γ= {γ(s, t)}



Theories

ARMA Transformation

Suppose {εt} follows ARMA model

φ(B)εt = θ(B)wt ,

where {wt} is white noiseMatrix form

φ(B)

θ(B)ε = w ,

where ε= (ε1, . . . , εn)′, w= (w1, . . . , wn)

′

TransformationMultiply φ(B)

θ(B) on both sides of y = Xβ + ε,

φ(B)

θ(B)y =

φ(B)

θ(B)Xβ + w

Independent error term assumption is satisfied



Theories

Cochrane and Orcutt Algorithm

Obtain residuals via OLS routine

yt = β̂′xt + et

Fit an ARMA model to et

φ̂(B)et = θ̂(B)wt

Apply ARMA transformation to linear model

φ̂(B)

θ̂(B)yt = β̂′ φ̂(B)

θ̂(B)xt + wt ,

denoted as ut =β̂′vt+wt

Run OLS regression again on transformed model



Illustrative Example

Pollution, Temperature, Mortality Example

SettingsWeekly data 1970 through 1979 in Los AngelesCardiovascular Mortality (Mt ), Particulates (Pt ),Temperature (Tt )





DataCardiovascular Mortality

Time

mor

t

0 100 200 300 400 500

7090

110

130

Temperature

Time

tem

p

0 100 200 300 400 500

−20

010

20

Particulates

Time

part

0 100 200 300 400 500

2040

6080

100





Correlations

mort

50 60 70 80 90 100

7080

9010

011

012

013

0

5060

7080

9010

0

temp

70 80 90 100 110 120 130 20 40 60 80 100

2040

6080

100

part




Obtain OLS Residuals

Models consideredMt = β0 + β1t + εt

Mt = β0 + β1t + β2(Tt − T.) + εt

Mt = β0 + β1t + β2(Tt − T.) + β3(Tt − T.)2 + εt

Mt = β0 + β1t + β2(Tt − T.) + β3(Tt − T.)2 + β4Pt + εt





> fit <‐ lm(mort~ trend + temp + temp2 + part, na.action=NULL) > summary(fit) Call: lm(formula = mort ~ trend + temp + temp2 + part, na.action = NULL) Residuals: Min 1Q Median 3Q Max ‐19.0760 ‐4.2153 ‐0.4878 3.7435 29.2448 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 81.592238 1.102148 74.03 < 2e‐16 *** trend ‐0.026844 0.001942 ‐13.82 < 2e‐16 *** temp ‐0.472469 0.031622 ‐14.94 < 2e‐16 *** temp2 0.022588 0.002827 7.99 9.26e‐15 *** part 0.255350 0.018857 13.54 < 2e‐16 *** ‐‐‐ Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.385 on 503 degrees of freedom Multiple R‐squared: 0.5954, Adjusted R‐squared: 0.5922 F‐statistic: 185 on 4 and 503 DF, p‐value: < 2.2e‐16





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−3 −2 −1 0 1 2 3

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PACF of residuals




Fit an AR(2) model to et

> (fit2<‐ar.ols(fit$resid, aic=F,order=2)) Call: ar.ols(x = fit$resid, aic = F, order.max = 2) Coefficients: 1 2 0.2205 0.3625 Intercept: ‐0.002895 (0.2472) Order selected 2 sigma^2 estimated as 30.92




Apply ARMA transformation to linear model

> Mort<‐filter(mort, c(1,‐.2205,‐.3625),sides=1)[3:508] > Trend<‐filter(trend, c(1,‐.2205,‐.3625),sides=1)[3:508] > Temp<‐filter(temp, c(1,‐.2205,‐.3625),sides=1)[3:508] > Temp2<‐filter(temp2, c(1,‐.2205,‐.3625),sides=1)[3:508] > Part<‐filter(part, c(1,‐.2205,‐.3625),sides=1)[3:508]




Fit OLS Regression on Transformed Model

> fit3 = lm(Mort~ Trend + Temp + Temp2 + Part) > summary(fit3) Call: lm(formula = Mort ~ Trend + Temp + Temp2 + Part) Residuals: Min 1Q Median 3Q Max ‐17.4256 ‐3.4915 ‐0.3200 3.0912 17.9067 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 34.835498 0.672217 51.822 < 2e‐16 *** Trend ‐0.027775 0.003861 ‐7.193 2.32e‐12 *** Temp ‐0.196162 0.038710 ‐5.067 5.68e‐07 *** Temp2 0.016758 0.002210 7.582 1.66e‐13 *** Part 0.229008 0.022589 10.138 < 2e‐16 *** ‐‐‐ Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.281 on 501 degrees of freedom Multiple R‐squared: 0.3068, Adjusted R‐squared: 0.3012 F‐statistic: 55.43 on 4 and 501 DF, p‐value: < 2.2e‐16




Fit OLS Regression on Transformed Model

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−3 −2 −1 0 1 2 3

−2

02

4

Theoretical Quantiles

Sta

nd

ard

ize

d r

esid

ua

ls

Normal Q−Q

149

89

148

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

ACF of residuals after transformation

0 5 10 15 20 25

−0

.10

.00

.10

.20

.3

Lag

Pa

rtia

l A

CF

PACF of residuals after transformation





##compare parameter estimates > rbind(summary(fit)$coefficients[,1], summary(fit3)$coefficients[,2]) (Intercept) trend temp temp2 part [1,] 81.59224 ‐0.02684 ‐0.47247 0.02259 0.25535 [2,] 0.67222 0.00386 0.03871 0.00221 0.02259 ##compare parameter standard errors > rbind(summary(fit)$coefficients[,2], summary(fit3)$coefficients[,2]) (Intercept) trend temp temp2 part [1,] 1.10215 0.00194 0.03162 0.00283 0.01886 [2,] 0.67222 0.00386 0.03871 0.00221 0.02259


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