9bWJ4riXFBGGECh12 Autocorrelation

8/3/2019 9bWJ4riXFBGGECh12 Autocorrelation

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Gujarati(2003): Chapter 12


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Definition of Autocorrelation

We assumed of the CLRMs errors that Cov (ui , uj) = 0 forij, i.e. Errors are not serially correlated, or, no autocorrelation

This is essentially the same as saying there is no pattern in theerrors.

Obviously we never have the actual us, so we use theirsample counterpart, the residuals (the s).

If there are patterns in the residuals from a model, we say that

they are autocorrelated.

Some stereotypical patterns we may find in the residuals aregiven on the next 3 slides.


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Positive Autocorrelation

+

-

-

t

+

1

t

+

-

Time

tu


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Negative Autocorrelation

+

-

-

t

+

1

tu

+

-

tu

Time


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No pattern in residuals

No autocorrelation

+

tu

-

-

+

1t

+

tu


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Definition: First-order of Autocorrelation, AR(1)

If Cov (ut, us) = E (ut us) 0 where t sYt = 1 + 2 X2t + ut t = 1,,T

and ifut = ut-1+ vt

where -1 < < 1 ( : RHO)and vt ~ iid (0, v

2) (white noise)

This scheme is called first-order autocorrelation and denotes as AR(1)

Autoregressive : The regression ofut can be explained byitself lagged one period.

(RHO) : the first-order autocorrelation coefficientor coefficient of autocovariance

0),cov(

)var(

0)(

2

st

vt

t

vv

v

vE


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Autocorrelation AR(1) :

Cov (ut u t-1) > 0 => 0 -1


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1973 230 320 u1973 .

... .

1995 558 714 u19951996 699 822 u19961997 881 907 u19971998 925 1003 u19981999 984 1174 u19992000 1072 1246 u2000

Year Consumptiont = 1 + 2 Incomet + errortExample of serial correlation:

TaxRate1999

TaxRate2000

Error term

represents

other factors

that affect

consumption

t

~ iid(0, v

2)

TaxRate2000 = TaxRate1999 + v2000 u

t=

u

t-1+ v

t

The current year Tax Rate may be determined by previous year rate


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The consequences of serial correlation:

(are same as those of heteroscedasticity)

1. The estimated coefficients are still unbiased.

E(k) = k^

3. The standard error of the estimated coefficient,Se(k)will be biased. Therefore, t- and F tests are not valid.

^

^2. The variances of the kis no longer the smallest.

So, OLS estimators are not BLUE

Therefore, when the regression has AR(1) errors,

The estimators are not BLUE. t and F tests are invalid.


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Detecting Autocorrelation:

The Durbin-Watson Test

The Durbin-Watson (DW) is a test for first order autocorrelation -i.e. it assumes that the relationship is between an error and the

previous one

ut= ut-1 + vt (1)

where vt N(0, v2

). The DW test statistic actually tests

H0: =0andH1: 0

The test statistic is calculated by

DW

u u

u

t tt

T

tt

T

12

2

2

2


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Detecting Autocorrelation:


We can also write

(2)where is the estimated correlation coefficient. Since isa correlation, it implies that .

Rearranging forDWfrom (2) would give 0DW4.

If = 0, DW= 2. So roughly speaking, do not reject thenull hypothesis ifDW is near 2 i.e. there is littleevidence of autocorrelation

Unfortunately, DWhas 2 critical values, an upper criticalvalue (du) and a lower critical value (dL), and there is alsoan intermediate region where we can neither reject nor notreject H0.

11 p

DW 2 1( )


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Durbin-Watson test(Cont.)

d = 2 (1- )==> = 1 -

==> = 1-

^

^d

2

d2

^

Since -1 1^implies 0 d 4

DW =

2 (1 - ) (ut ut-1)

2t=2

T^ ^

ut2

t=1

T ^

^

(d)


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Conditions which Must be Fulfilled for DW to be a Valid Test 1. Constant term in regression 2. Regressors are non-stochastic 3. No lags of dependent variable


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Ex: How to detect autocorrelation ?Gujarati(2003) Table12.4, pp.460


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Run OLS: and check the t-value of the coefficientttt vuu 1

9385.02

122904.01

21914245.0

DW


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From OLS regression result: where dor DW* = 0.1229

Check DW Statistic Table(At 5% level of significance, k = 1, n=40)

dL = 1.246

du = 1.344

0 1.246 1.344 2

dL du

DW*

0.1229

Durbin-Watson Autocorrelation test

RejectH0

region

H0 : no autocorrelation

= 0

H1 : yes, autocorrelation exists.

or > 0

positive autocorrelation


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Example 2:

UMt = 23.1 - 0.078 CAPt - 0.146 CAPt-1 + 0.043Tt^

(15.6) (2.0) (3.7) (10.3)

R2 = 0.78 F = 78.9 u = 0.677 RSS = 29.3 DW = 0.23 n = 68_

^

(i) K = 3 (number of independent variables)

(ii) n = 68 , = 0.01 significance level

0.05

(iii) dL = 1.525 , du = 1.703 0.05

dL = 1.372 , du = 1.546 0.01

Reject H0, positive autocorrelation exists

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9bWJ4riXFBGGECh12 Autocorrelation