lecture_3_-_2014

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Applied Econometrics

Introduction to Time Series

Roman Horvth

Lecture 3

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Contents

StationarityWhat it is and what it is for

Some basic time series models

Autoregressive (AR) Moving average (MA)

Consequences of non-stationarity (spuriousregression)

Testing for (non)-stationarity Dickey-Fuller test

Augmented Dickey-Fuller test

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Xtis stationary if:the series fluctuates around a constant long runmean

Xt has finite variance which is not dependent upon timeCovariance between two values of X

tdepends only on the

difference apart in time (e.g. covariance between Xtand Xt-1isthe same as for Xt-8and Xt-9)

E(Xt

) = (mean is constant in t)Var(Xt) =

2 (variance is constant in t)Cov(Xt,Xt+k) = (k) (covariance is constant in t)

If data not stationary, spurious regression problem

(Weak) Stationarity

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Examples of Times Series Models

ARautoregressive models Xt= + *Xt-1+ ut is called AR(1) process Xt= + 1*Xt-1+ 2*Xt-2+.+k*Xt-k+ ut is AR(k) process

MAmoving average models Xt= + ut + 2*ut-1 is called MA(1) process

Xt= + ut + 2*ut-1+.+k*ut-k .is called MA(k) process

If you combine AR and MA process, you getARMA process E.g. ARMA (1,1) is Xt= +*Xt-1+ut+ 2ut-1

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Is MA and AR process stationary?

Compute mean, variance and covariance andcheck if it depends on time

For AR process, you may easily derive that theprocess is stationary if < 1

For MA (1) process, mean is , variance is ut2

*(1+2

2) and covariance cov(Xt, Xt-k) is either 0if k>1 or ut

2*2 ,so it does not depend on time

(MA(k) is stationary process)

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White noise process:

Xt = ut ut ~ IID(0, 2)

Example of Stationary Time Series

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

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Another Example of Stationary Time Series

Xt= 0.5*Xt-1+ ut ut ~ IID(0, 2)

Stationary without drift

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

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Example of Non-stationary Time Series

Yt= + *t + ut,wheret is time trend

Take the expected value E(Yt) = + *t ,clearly the mean depends on time and the series

is non-stationary

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In contrast a non-stationary time series has at leastone of the following characteristics:Does not have a long run mean which the series

returnsVariance is dependent upon time and goes toinfinity as the sample period approaches infinity

Correlogram does not die out - long memory

Non-stationary time series

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Example of Non-stationary Time Series

The level of GDP is not constant; the mean increases over

time.

UK GDP Level

0

20

40

60

80

100

120

1992Q

3

1993Q

2

1994Q

1

1994Q

4

1995Q3

1996Q2

1997Q1

1997Q4

1998Q3

1999Q2

2000Q1

2000Q4

2001Q

3

2002Q

2

2003Q

1

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Non-stationary time seriescorrelogram

UK GDP (Yt)

For non-stationary series the Autocorrelation Function (ACF)declines towards zero at a slow rate as kincreases.

0 1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ACF-Y

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Some transformation = first differenece, logarithm, second

difference ...First difference of UK GDP (Yt = Yt - Yt-1) is stationary:- growth rate is reasonably constant through time- variance is also reasonably constant through time

Possible solutions of non-stationarity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1992Q3

1993Q2

1994Q1

1994Q4

1995Q3

1996Q2

1997Q1

1997Q4

1998Q3

1999Q2

2000Q1

2000Q4

2001Q3

2002Q2

2003Q1

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Stationary time series - correlogram

UK GDP Growth ( Yt)

ACF decline towards zero as kincreasesDecline of ACF is rapid for stationary series

0 1 2 3 4 5 6 7

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

ACF-DY

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Non-stationary Time Series Continued

Random Walk

Xt= Xt-1+ ut, where ut ~ IID(0, 2)

Mean is constant in t: E(Xt) = E(Xt-1)

X1=X0+ u1 (take initial valueX0)X2=X1 + u2= (X0 + u1) + u2

Xt=X0 + u1+ u2 ++ ut (take expectations)

E(Xt) = E(X0 + u1+ u2 ++ ut) = E(X0)= constant

Variance is not constant in t:

Var(Xt) = Var(X0) + Var(u1) ++ Var(ut)

= 0+ 2

++ 2

=t 2

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Random walk

0

0.5

1

1.5

2

2.5

Xt

= Xt-1

+ ut

ut

~ IID(0, 2)

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AR(1) process: Xt= + Xt-1 + ut ;(ut ~ IID(0, 2))

< 1stationary process - process forgetspast

= 1non-stationary process - process doesnot forget past

= 0 without drift

0 with drift

MA processis always stationary

Relationship between stationary and non-

stationary process

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Summary on basic time series processes

AR (k) process

MA(k) process

ARMA (p,l) process If you k-th difference the data, then you have

ARIMA (p,k,l)estimation of ARIMA models

is a subject of next lecture

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Spurious Regression

(spurious correlation)

Problem that time-series data usually includes trend

Result:

Spurious correlation (variables with similar trends

are correlated)Spurious regression (independent variable with

similar trend looks as dependent = strong statisticalrelationship)

coefficient significant (high adjusted-R2, larget-statistics) ... even if unrelated in economicterms

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How to avoid spurious regression:

3 approaches to non-stationarity

1. Include a time trendas an independent variable (old-fashioned) yt= c + 1xt+ 2t + ut .....(t = 1,2, ..., T)

2. 1st differencethe data if variables I(1); 2nd difference ifI(2)

= converts non-stationary variables into stationaryvariables

Problems:

theory often about levels

detrending loss of information

3. Cointegration+ ECM

= Long-run relationship +short-run adjustment

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(A) Informal methods:- Plot time series- Correlogram

(B) Formal methods:

- Statistical test for stationarity- Dickey-Fuller tests.

How do we identify non-stationary

processes?

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Informal Procedures to identify non-stationary

processes(a) Constant mean?

(b) Constant variance?

0 50 100 150 200 250 300 350 400 450 500

-200

-150

-100

-50

0

50

100

150

200 var

0 50 100 150 200 250 300 350 400 450 500

0

2

4

6

8

10

12

RW2

f d d f

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processes

Diagnostic testCorrelogram for stationary process (dies outrapidly, series has no memory)

0 50 100 150 200 250 300 350 400 450 500

-0.25

0.00

0.25

0.50whitenoise

0 5 10

-0.5

0.0

0.5

1.0ACF-whitenoise

f l d id if i

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processes

Diagnostic testCorrelogram for a random walk (does not die out,high autocorrelation for large values of k)

0 50 100 150 200 250 300 350 400 450 500

0.0

2.5

5.0

7.5

10.0

12.5randomwalk

0 5 10

0.25

0.50

0.75

1.00ACF-randomwalk

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Dickey-Fuller Test

Test based on Yt

= Yt-1

+ ut

- DF test to determine whether =1

Yes unit root non-stationary

No no unit root

Dynamic model:

Subtract Yt-1 ...Yt- Yt-1= (-1)Yt-1+ ut

Reparameterise: Yt= Yt-1+ ut

where = (-1)

Test =0 equivalent to test =1

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Augmented Dickey-Fuller Test

Augment dynamic modelYt= Yt-1+ ut:

1) Constant or drift term (0)

Yt= 0+ Yt-1+ ut

2) Time trend (T)

Yt= 0+ T + Yt-1+ ut

3) Lagged values of the dependent variable

Yt= 0+ T + Yt-1+ 1Yt-1+ 2Yt-2+ ... + ut

Find the right specification, trade-off parsimony vs. whitenoise in residual

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Critical values

DF/ADF use t- and F-statistics but critical values arenot standard

Problems:

distributions of these statistics are non-standard special tables of critical values (derived from

numerical simulations)

Usual t- and F-tests not valid in presence of unit

roots

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