Regression with Autocorrelated Errors

U.S. Wine Consumption and Adult Population – 1934-2002

Data Description

• Y=U.S. Annual Wine Consumption (Millions of Gallons)

• X=U.S. Adult Population (Millions of People)• Years – 1934-2002 (Post Prohibition)• Model:

2

11

10

,0~

...

iid

XY

t

qtqttt

ttt

Ordinary Least Squares Regression

64.4828.236631.497.347^^

ett PW

Regression StatisticsMultiple R 0.965612383R Square 0.932407274Adjusted R Square 0.931398427Standard Error 48.64438813Observations 69

ANOVAdf SS MS F Significance F

Regression 1 2186985.91 2186985.91 924.23 0.0000Residual 67 158540.53 2366.28Total 68 2345526.43

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -347.9736 21.9895 -15.8245 0.0000 -391.8649 -304.0824apop_m 4.3092 0.1417 30.4012 0.0000 4.0263 4.5921

present isation Autocorrel

64.170,158.170,1

0.11989871158540.53

19008.8044 :TestWatson -Durbin

1

2

2

21

U

UL

n

tt

n

ttt

dDW

npdnpd

e

eeDW

Covariance Structure (q=1)

122

21

2

1

2

22

1

12

2

2

2

2

2

112121

2

2

2

2222

2

222

12

222

2

2

11

2110

,1

1

1

1

1

,

1,

10

:GeneralIn

1,,

11

1

10

10 :Assuming

11,0~

tttt

nn

n

n

n

t

ttk

ktttt

ttttttt

COVVV

VV

V

COVCOVVE

VCOVCOV

VVVE

VE

iidXY

ε

Generalized Least Squares (q=1)

εTXβTYTY*IVTTYTI

VTT

VT

TεYV

1111'11

1'1

1

1

Transform

1000

0100

0010

0001

1

1000

100

010

001

1000

1100

1110

1111

1

1000

1100

1110

1111

1

1

1

1

1

1

100

010

001

0001

:4 whereCase heConsider t

1000

01000

0010

0001

00001

:Define

1

1

1

1

22

2

2

2

2

2

2

2

2

22

2222

232222

2

2

2

22

2222

232222

2

2

23

2

2

32

2

2

2

2

21

2

1

2

2

V

n

VV

nn

n

n

Estimated GLS (q=1)

i

iii

iEGLS

iEGLSie

e

n

ttt

n

tt

SEtsSE

pns

SEtV

pn

een

en

^

^^

2^

^^^

2

,

^

,

^1^1^2^^

1^1^

2^1^1^1^1^

^

^

^

^

^

2^

1^

^^^2^

^

^^

21

^

1

2^

)0(/1'

)1()0(

'

1'

''''

10000

01000

00010

00001

00000)0(

)1()0()0(

)1(1)1(

1)0(

1

EGLS

^

EGLS

^

EGLS

^

1

EGLS

^

XTTX'β

βXYTTβXYYTTX'XTTX'β

T

Estimated GLS (q=1) – Wine/Population Data

38.2104372.0

9348.004372.)0(/39.4

358.90.4546

4.2540

2066.0

4.2540

0.206631.515-

31.515-5482.225

5479.8024.2540

347.23-

10.9348-0000

010.9348-000

00010.9348-0

000010.9348-

000003552.0

82.8920.93482147.89)1(2297.69)0(

^

^^

2^

2

1

^

2^

1^

2^^^^

i

iii

e

SEtsSEs

tV

EGLS

^

EGLS

^

β

β

T

present ision tocorrelatconcludeAunot Do

64.170,158.170,167.1

U

UL

dDW

npdnpdDW

Estimated GLS (General q) - I

)()(

1000000

0100000

0000010

0000001

)()(

:))1(2 (assuming :matrixation transformObtain the 5)

: ofion DecompositCholesky Obtain the 4)

')0(

: and vector 1 - errors lagged theof tsCoefficien theEstimate 3)

)(

)2(

)1(

)0()2()1(

)2()0()1(

)1()1()0(

,...,1,0)(

: vector1 andmatrix following theand lag toancesautocovari theEstimate 2)

:Residuals OLSObtain and Regression SquaresLeast Ordinary Fit 1)

1

^^1

^^

1

^^1

^^

2^

11

^2^1

2

^

^

^

^^^

^^^

^^^

1^

nnnqn

nqqnqqq

qn

Vqx

qqq

q

q

qhn

eeh

qxqxqq

q

q

q

q

t

n

hthtt

2

1^

2

1211112q

^

11

1/2^^

q

^

q

^

q

^

q

^

q

^^

q

^

q

^^

q

^

q

^

OLS

^

OLS1

OLS

^

T

TTT

TTT0TPT

VT

P'PΓΓ

γργΓρ

γΓ

βXYeYX'XX'β

Estimated GLS (General q) - II

qpndfSE

t

isSEqpn

s

SEt

V

qpn

i

ii

thiiiii

iEGLS

iEGLSi

e

e

'on with distributi- t with thecompare and

: for the statistics-Obtain t 11)

ofelement diagonal '

')0(

:errors standard theand parameters siveautoregres of estimatesfor MS residualObtain 10)

:tscoefficien regressionfor statistics- testimatedObtain 9)

': ofmatrix covariance- varianceestimated Obtain the 8)

'

''

:model ed transform thefrom anceerror vari Estimated Obtain the 7)

'':estimate squaresleast dgeneralize estimated Obtain the 6)

^

^

i

1^^2

^

^

2

,

^

,

^

2^^

2^

q

^q

^^

1^^

EGLS

^

EGLS

^

EGLS

^^^

EGLS

^

^^1^^

EGLS

^

Γγρ

XTTX'ββ

βXYTTβXY

YTTX'XTTX'β

SAS Proc Autoreg Output The AUTOREG Procedure Dependent Variable wine Ordinary Least Squares Estimates SSE 158540.525 DFE 67 MSE 2366 Root MSE 48.64439 SBC 738.318203 AIC 733.84999 Regress R-Square 0.9324 Total R-Square 0.9324 Durbin-Watson 0.1199

Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 -347.9736 21.9895 -15.82 <.0001 adpop 1 4.3092 0.1417 30.40 <.0001

Estimates of Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0 2297.7 1.000000 | |********************| 1 2147.9 0.934807 | |******************* |

SAS Proc Autoreg Output

Preliminary MSE 289.8

Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value 1 -0.934807 0.043717 -21.38

Yule-Walker Estimates SSE 18516.1612 DFE 66 MSE 280.54790 Root MSE 16.74956 SBC 596.454422 AIC 589.752103 Regress R-Square 0.5702 Total R-Square 0.9921 Durbin-Watson 1.6728

Standard Approx Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -347.2297 74.0420 -4.69 <.0001 adpop 1 4.2540 0.4546 9.36 <.0001