9.1 Lecture #9 Studenmund (2006) Chapter 9 Objectives The nature of autocorrelation The consequences...

Lecture #9Studenmund (2006) Chapter 9

Objectives • The nature of autocorrelation• The consequences of autocorrelation• Testing the existence of autocorrelation• Correcting autocorrelation

Time Series Data

Time series process of economic variables

e.g., GDP, M1, interest rate, exchange rate,

imports, exports, inflation rate, etc.

Realization

An observed time series data set generated from a time series process

Remark: Age is not a realization of time series process.Time trend is not a time series process too.

Decomposition of time series

random

Cyclical orseasonal

Xt = Trend + seasonal + random

Static ModelsStatic Models

Ct = 0 + 1Ydt + t

Subscript “t” indicates time. The regression is a contemporaneous relationship, i.e., how does current consumption (C) be affected by current Yd?

Example: Static Phillips curve model

inflatt = 0 + 1unemployt + t

inflat: inflation rateunemploy: unemployment rate

Finite Distributed LagLag Models

Forward Distributed Lag Effect (with order q)

Effect at time t+2

Economic actionat time t

Effect at time t

Ct =0+0Ydt+t

Effect at time t+1

Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t

Effect at time t+q ….

….Ct+q=0+1Ydt+q+…+1Ydt+tCt=0+1Ydt+…+1Ydt-q+t

Economic actionat time t

Effect at time t-1

Backward Distributed Lag Effect

Yt= 0+0Zt+1Zt-1+2Zt-2+…+2Zt-q+t

Initial state: zt = zt-1 = zt-2 = c

Effect at time t-q ….Effect

at time t-3 Effect

at time t-2

C = 0 + 00Ydt + 11Ydt-1 + 22Ydt-2 + t

Long-run propensity (LRP) = (00 + + 11 + + 22)

Permanent unit change in C for 1 unit permapermanentnent (long-run) change in Yd.

Distributed Lag model in general:

Ct = 0 + 0Ydt + 1Ydt-1 +…+ qYdt-q

+ other factors + t

LRP (or long run multiplier) = 0 + 1 +..+ q

Time Trends

Linear time trend

Yt = 0 + 1t + t Constant absolute change

Exponential time trend

ln(Yt) = 0 + 1t + t Constant growth rate

Quadratic time trend

Yt = 0 + 1t + 2t2 + t Accelerate change

For advances on time series analysis and modeling , welcome to take ECON 3670ECON 3670

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

If Cov (t, s) = E (t s) 0 where t s

Yt = 0 + 1 X1t + t t = 1,……,T

and if t = t-1 + ut

where -1 < < 1 ( : RHORHO)

and ut ~ iid (0, u2) (white noise)

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

Autoregressive : The regression of t can be explained by itself lagged one period.

(RHORHO) : the first-order autocorrelation coefficient or ‘coefficient of autocovariance’

1990 230 320 u1990

… … … …. ... … … ….2002 558 714 u2002

2003 699 822 u2003

2004 881 907 u2004

2005 925 1003 u2005

2006 984 1174 u2006

2007 1072 1246 u2007

Year Consumptiont = 0 + 1 Incomet + errort

Example of serial correlation:

TaxPay2006

TaxPay2007

Error termrepresents

other factorsthat affect

consumption

uutt ~ iid(0, ~ iid(0, uu22))

TaxPay2007 = TaxPay2006 + uu2007

t = t-1 + uut

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

If t = 1 t-1 + ut

it is AR(1), first-order autoregressive

If t = 1 t-1 + 2 t-2 + ut

it is AR(2), second-order autoregressive

If t = 1 t-1 + 2 t-2 + …… + n t-n + ut

it is AR(n), nth-order autoregressive

……………………………………………….

High orderautocorrelation

Autocorrelation AR(1) :

Cov (t t-1) > 0 => 0 < < 1 positive AR(1)

Cov (t t-1) < 0 => -1 < < 0 negative AR(1)

-1 < < 1

If t = 1 t-1 + 2 t-2 + 3 t-3 + ut

it is AR(2), third-order autoregressive

Positive autocorrelation

i Cyclical: Positive autocorrelation

xx x x

xThe current error term tends to have the same sign as the previous one.

Negative autocorrelation

No autocorrelation

xxx0 timex

The current error term tends to have the opposite sign from the previous.

The current error term tends to be randomly appeared from the previous.

The meaning ofThe meaning of : The error term t at time t is a linear combination of the current and past disturbance.

0 < < 1

-1 < < 0

The further the period is in the past, the smaller is the weight of that error term (t-1) in determining t

= 1 The past is equal importance to the current.

> 1 The past is more importance than the current.

The consequences of serial correlation:

3. The standard error of the estimated coefficient, Se(k) becomes large

^2. The variances of the k is no longer the smallestno longer the smallest

Therefore, when AR(1) is existing in the regression,The estimation will not be “BLUE”

BLUEBLUE1. The estimated coefficients are still unbiasedstill unbiased.

E(k) = k^

If E(tt-1) 0, and t = t-1 + ut , then

If = 0, zero autocorrelation, than Var(1)AR1 = Var(1)^ ^

If 0, autocorrelation, than Var(1)AR1 >> Var(1)^ ^

Two variable regression model: Yt = 0 + 1X1t + t

The OLS estimator of 1,

If E(t t-1) = 0 then Var (1) = ^ 2

===> 1 =

Var (1)AR1= + + 2^ 2 22 xt xt+1 xt xt+2

xt2 xt

2 xt2 xt

-1 < < 1

+ ….

The AR(1) variance is not the smallest

Example:

t = t-1 + ut ==> t = [ t-2 + ut-1] + ut

==> t-2 = t-3 + ut-2 => t = 2 [ t-3 + ut-2] + ut-1 + ut

==> t-1 = t-2 + ut-1 t = 2 t-2 + ut-1 + ut

t = 3 t-3 + 2 ut-2 + ut-1 + ut

E(t t-1) =2

E(t t-3) = 2 2

E(t t-2) = 2

…………….E(t t-k) = k-1 2

Autoregressive scheme:

It means the more periods in the past,the less effect on current period

k-1 becomes smaller and smaller

How to detect autocorrelation ?

DW* or d*

5% level of significance,

k = 1k = 1, n=24n=24

DW* = 0.9107

dL = 1.27

du = 1.45

kk is the number of independent variables (excluding the intercept)

DW* << ddLL

From OLS regression result: where d or DW* = 0.9107

Check DW Statistic Table (At 5% level of significance, k’ = 1, n=24)

dL = 1.27du = 1.45

0 1.27 1.45 2

0.91070.9107

Durbin-Watson Autocorrelation test

Reject H0

region

H0 : no autocorrelation = 0H1 : yes, autocorrelation exists. or > 0 positive autocorrelation

Durbin-Watson test

OLS : Y = 0 + 1 X2 + …… + k Xk + t

obtain t , DW-statistic(d) ^

Assuming AR(1) process: t = t-1 + ut

I. H0 : ≤ 0 no positive autocorrelation

H1 : > 0 yes, positive autocorrelation-1 < < 1

Compare dd* and ddLL, dduu (critical values)DW*

if dd* < ddLL ==> reject H0

if d* > dduu ==> not reject H0

if ddLL dd* dduu ==> this test is inconclusive

Durbin-Watson test(Cont.)

Since -1 -1 1^

implies 0 dd 4

DW = 2 (1 - ) (t - t-1)2

(dd)dd 2(1)^

dd ≈ 2 (1- )

==> ≈ 1 -

==> ≈ 1-

0 1.27 1.45 22

(4-d(4-dLL))(4-d(4-dUU))

2.732.732.552.55

Durbin-Watson test(Cont.)II. H0 : ≥0 no negative autocorrelation

H1 : < 0 yes, negative autocorrelation

we use (4-d) (when dd is greater than 2)

if (4 - d) < dL

or 4 - dL < d < 4 ==> reject H0

if dL (4 - d) du

or 4 - du > d > 2 ==> not reject H0

if dL (4 - d) du

or 4 - du d 4 - dL ==> inconclusive

0 1.27 1.45 22

(4 - dL)(4-d(4-dUU))

2.732.732.552.55

Durbin-Watson test(Cont.)II. H0 : =0 No autocorrelation

H1 : 0 two-tailed test for auto correlation either positive or negative AR(1)

If d < dL

or d > 4 - dL

==> reject H0

If du < d < 4 - du ==> not reject H0

If dL d du

or 4 - du d 4 - dL

==> inconclusive

For example :

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 = 0.677 SSR = 29.3 DW = 0.23DW = 0.23 n = 68_

(i) K = 3 (number of independent variable)Observed

(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

(excluding intercept)

H0 : = 0 positive autocorrelation H1 : > 0

0 dL du 2

reject H0 not

reject

inconclusive

DW (d)

4-du 4-dL 4

inconclusive

reject H0

H0 : = 0negative autocorrelation

H1 : < 0

not reject

2.45 2.297

2.63 2.475

1.3721.525

1.5461.703

1% & 5%Critical values

The assumptions underlying the d(DW) statistics :

1. Intercept term must be included.2. X’s are nonstochastic

3. Only test AR(1) : t = t-1 + ut where ut ~ iid (0, u2)

4. Not include the lagged dependent variable,

Yt = 0+ 1 Xt1 + 2 Xt

2 + …… + kXtk + Yt-1 + t

(autoregressive model)

5. No missing observation 1970 100 15

1980 235 2081 N.A. N.A.82 N.A. N.A.93 253 3794 281 4195

... ... ...

... ...

missing

Lagrange MultiplierLagrange Multiplier ( (LMLM)) Test Test or called Durbin’s m testOr Breusch-Godfrey (BG) test of higher-order autocorrelation

^Test Procedures:(1) Run OLS and obtain the residuals t.

(3) compute the BG-statistic: ((nn--pp)R)R22

(4) compare the BG-statistic to the 2p (pp is # of degree-order)

(5) If BG > 2p, reject Ho,

it means there is a higher-order autocorrelation If BG < 2

p, not reject Ho,

it means there is a no higher-order autocorrelation

^ ^ ^ ^^ ^ ^ ^

(2) Run t against all the regressors in the model

plus the additional regressors, t-1, t-2, t-3,…, t-p.

t = 0 + 1 Xt + t-1 + t-2 + t-3 + … + t-pp + u

Obtain the RR22 value from this regression.

Remedy: 1. First-difference transformation

Yt = 0 + 1 Xt + t

Yt-1 = 0 + 1 Xt-1 + t-1 assume = 1

==> Yt - Yt-1 = 0 - 0 + 1 (Xt - Xt-1) + (t - t-1)

==> Yt = 1 Xt + t

no intercept

2. Add a trend (T)Yt = 0 + 1 Xt + 2 T + t

Yt-1 = 0 + 1 Xt-1 + 2 (T -1) + t-1

==> (Yt - Yt-1) = (0 - 0) + 1 (Xt - Xt-1) + 2 [T- (T -1)] + (t - t-1)

==> Yt = 1 Xt + 2*1 + ’t

==> Yt = 2* + 1 Xt + ’t

If 1* > 0 => an upward trend in Y^

(2 > 0)^

3. Cochrane-Orcutt Two-step procedureCochrane-Orcutt Two-step procedure (CORC)(1). Run OLS on Yt = 0 + 1 Xt + t

and obtains t ^

(3). Use the to transform the variables :^

Yt* = Yt - Yt-1

^Xt* = Xt - Xt-1

-) Yt-1 = 0 + 1 Xt-1 + t-1 ^ ^ ^^

Yt = 0 + 1 Xt + t

(4). Run OLS on Yt* = 0

* + 1* Xt

* + ut

(2). Run OLS on t = t-1 + ut^

and obtains ^

Where u~(0, )

GeneralizedGeneralized Least SquaresLeast Squares

(GLS)(GLS)methodmethod

(Yt - Yt-1)= 0(1-) +1(Xt - Xt-1) + (t -t-1) ^ ^^^

4. Cochrane-OrcuttCochrane-Orcutt Iterative Procedure

(5). If DW test shows that the autocorrelation still existing, than it needs to iterate the procedures from (4). Obtains the t

(6). Run OLS

t* = t-1

* + ut’^ ^

(1 - )DW2

and obtains which is the second-round estimated ^

Xt** = Xt - Xt-1 Yt-1 = 0 + 1 Xt-1 + t-1

(7). Use the to transform the variable^

Yt** = Yt - Yt-1 Yt = 0 + 1 Xt + t

^ ^ ^ ^^

Cochrane-Orcutt Iterative procedure(Cont.)

(8). Run OLS onYt

** = 0** + 1

** Xt** + t

Where is ^^(Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + (t - t-1)^ ^ ^^ ^ ^

(9). Check on the DW3 -statistic, if the autocorrelation is still existing, than go into third-round procedures and so on.

Until the estimated ’s differs a little ^^( - < 0.01).

Example: Studenmund (2006) Exercise 14 and Table 9.1, pp.342-344

Low DW statistic

Obtain the Residuals

(Usually after you run regression, the residuals will be immediately stored in this icon

Give a new name for the residual series

Run regression of the current residual on the lagged residual

Obtain the estimated ρρ(“rho”)

ttt 1ˆˆ

(3) Transform the Y* and X*

New series are created,

but each first observation

is lost.

Obtain the estimated result

which is improved

Run the

transformed

regression

The Cochrane-Orcutt Iterative procedure in the EVIEWS

The is the EVIEWS’ Command to run the iterative procedure

(5)~(9)

The result of the Iterative procedure

The DW

is improved

This is the

estimated ρρEach

variable is

transformed

Generalized least Squares (GLS)

Yt = 0 + 1 Xt + t t = 1,……,T (1)

Assume AR(1) : t = t-1 + ut -1 < < 1

Yt-1 = 0 + 1 Xt-1 + t-1 (2)

(1) - (2) => (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + (t - t-1)

GLS => Yt* = 0

* + 1* Xt

* + ut

5. Prais-Winsten transformation

To avoid the loss of the first observation, the first observation of Y1

* and X1* should be transformed as :

X1* = 1 - 2 (X1)

* = 1 - 2 (Y1)^

Edit the figure hereTo restorethe first observation

but Y2* = Y2 - Y1 ; X2

* = X2 - X1^ ^

Y3* = Y3 - Y2 ; X3

* = X3 - X2^^

. …..

* = Yt - Yt-1 ; Xt* = Xt - Xt-1

6. Durbin’s Two-step method : Since (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + ut

=> Yt = 0* + 1 Xt - 1 Xt-1 + Yt-1 + ut

Yt = 0 + 1 Xt + t

III. Run OLS on model : Yt* = 0 + 1 Xt

* + ’t

and 1 = 1^ ^where 0 = 0 (1 - )^ ^

I. Run OLS => this specification

Yt = 0* + 1

* Xt - 2* Xt-1 + 3

* Yt-1 + ut

Obtain 3* as an estimated (RHO)^ ^

II. Transforming the variables :

Yt* = Yt - 3

* Yt-1 as Yt* = Yt - Yt-1

and Xt* = Xt - 3

* Xt-1 as Xt* = Xt - Xt-1

Including this

lagged term of Y

Obtain the estimated

ρρ(“rho”)^

Lagged Dependent VariableLagged Dependent Variable and Autocorrelation

Compare h* to Z where Zc ~ N (0,1) normal distribution

If |h*| > Zc => reject H0 : = 0 (no autocorrelation)

Yt = 0 + 1 X1t + 2 X2t

+ …… + k Xk.t + 1 Yt-1 +t

DW statistic will often be closed to 2 or

DW does not converge to 2 (1 - )^DW is not reliable

Durbin-h Test: Compute h* =

^ 1 - n*Var (1)

Limitation of Durbin-Watson Test:

Durbin-h Test: Compute h* =

^ 1 - n*Var (1)

Therefore reject

H0 : = 0 (no autocorrelation)

2)10617.0(*241

24*7772.0*

h* = 4.458 > Z

9.1 Lecture #9 Studenmund (2006) Chapter 9 Objectives The nature of autocorrelation The consequences...

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