Applied Econometrics

Seminar 2Introduction to Time Series

(Stationarity and Unit Root Testing)

Please note that for interactive manipulation you need Mathematica 6 version of this .pdf. Mathematica 6 will be available soon at all Lab's Computers at IES

http://staff.utia.cas.cz/barunikJozef Barunik ( barunik @ utia. cas . cz )

|

A brief revision

To warm you up, please tell me what are the main ideas and why we use these concepts(from lecture):

StationarityAR processMA processAutocorrelation function (ACF)

there is interactive study material on ARIMA at my web pagehttp://staff.utia.cas.cz/barunik

|

2 Seminar2.nb

Stationarity

A time series 8rt< is said to be weakly stationary if both mean of rtand covariancebetween rt and rt-l are time-invariant - EHrtL = m and CovHrt , r-l L = gl .

In other words, stationarity requires distribution of time series to be constant under timeshift, weak stationarity, which is assumed more often requires only fluctuation with con-stant variation around constant level.

|

Seminar2.nb 3

Why do we need to care about stationarity ?

]:[ nonstationary series can strongly influence its behavior and properties (persistence ofshocks might be infinite) ]:[ spurious regressions - 2 trending variables over time which are totally unrelated willhave high R2

]:[ assumptions for assymptotic analysis is not valid for nonstationary series (we can nottest hypotheses validly)

|

4 Seminar2.nb

Tests of Stationarity - Visually

Simple Plot of series is always first stepWhich series is stationary?

|

Seminar2.nb 5

Tests of Stationarity - Visually

Let us load the data from previous pictures

CZK/EUR 2007-2008

Run JMulti

load file CZK_EUR_2008.txt

|

6 Seminar2.nb

Tests of Stationarity - Visually: CZK/EUR

Can we see constant mean and variance at exchange rate?

|

Seminar2.nb 7

Tests of Stationarity - Visually: CZK/EUR cont.

What about ACF/PACF - for stationary series, it should decay exponentially, or be non-significant if there is no pattern

|

8 Seminar2.nb

Tests of Stationarity - Visually: CZK/EUR cont.

What we can do is to difference the exchange rate, so we plot "returns", as rt =Pt

Pt-1- 1 or

continuously rt = lnI PtPt-1

M

|

Seminar2.nb 9

Tests of Stationarity - Visually: CZK/EUR cont.

Let's look at ACF/PACF of differenced series

|

10 Seminar2.nb

Tests of Stationarity - Formally (Unit Root Tests)

Exact tests can be found in the lecture, or in textbooks

H0: unit rootHA: no unit root (stationarity)

Thus we need to reject the null hypothesis to be sure we have stationary time series. Ifseries has unit root, it is not stationary

usual tests: Dickey-Fuller, Augmented Dickey-Fuller (ADF), Phillip-Perron test, KPSS test|

Seminar2.nb 11

Unit Root Tests in JMulti - ADF

ADF test - Dyt = f yt-1 + ut

H0 : f = 0 versus H1 : f < 0(or process yt has unit root and is nonstationary, against alternative that it is stationary

Options: constant (intercept in regression), time trend (trend stationarity), seasonal dum-mies (mondays...)

|

12 Seminar2.nb

What does ADF tells us about our CZK/EUR series?

let's run on levels and differences, try using constant (consistency), residuals should bewhite noise, also significant autocorrelation should not be present among remaining lags.

for exchange rate (level):statistics is -1.6090 (10% level critical value is -1.62 )we can not reject the null of unit root even at 10% - series are not stationary !!!

for differenced exchange rate:statistics is -9.5549 (1% level critical value is -2.56)we can strongly reject the null of unit root - series are stationary !!!

|

Seminar2.nb 13

Unit Root Tests in JMulti - KPSS

We test

H0 : yt~IH0L againstH1 : yt~IH1L

thus null hypothesis is, that series are stationary ! (different from ADF, take care), againsta unit root

we can again choose level or trend stationarity

|

14 Seminar2.nb

Next Example

load PX50.txt data

PSE index for 2007-2008

Let's repeat the analysis with different series|

Seminar2.nb 15