module 5_ tests for stationarity part 2

8/10/2019 Module 5_ Tests for Stationarity Part 2

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Subject Business Economics

Paper No and Title Introduction to time series regression: unit root tests and cointegration

Module No and Title 5 Tests for Stationarity (Part 2): Unit Root Tests and Transformation of Time Series

Module Tag BSE_P_M

1. Learning Outcomes

After studying this module, you ill !e a!le to understand in detail: The significance of unit root test in testing the stationarity of a series" The Augmented #ic$ey %uller Test and its significance in analysing the stationarity of

a series" Philli&s'Perron Unit Root Test The &rocess of transforming the non'stationary time series into a stationary time

series"

2. Unit Root TestsUnit root test has !ecome one of the most idely used method for testing the stationarity ofthe series" et s first discuss the unit root test !riefly, then e ill &roceed to its a&&lica!ilityin testing the stationarity of the series"

Any series is generated !y random al$ if,

X t = X t-1 + u t

u t is inde&endently, identically distri!uted ith mean *ero and constant +ariance, that is, theerror term is a hite noise"

u t ~ iid(0 , 2

here:t is the +aria!le s +alues ta$en o+er the time t

u t is the error term

In general,X t = !X t-1 + u t

If - . /, e consider the model to !e a random al$ &rocess hich has an inherent feature ofnon'stationarity in it"

2.1 Random "a#$ %it& 'ri t

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0n adding the drift +aria!le in the random al$ model, e get random al$ ith drift modelhich is re&resented as !elo :

X t = ) + X t-1 + u t

here:1 is the drift +aria!le

2.2 Random "a#$ %it& 'ri t and Trend

A model hich contains !oth the drift and trend +aria!le in the model, e term it as randomal$ ith drift and trend"

X t = ) + X t-1 + ! + u t

here:- is the trend +aria!le

Random al$ is a unit root test if the coefficient of t'/ is e ual to one"

2.* Unit Root Test 'ic$e - u##er Test

To test the stationarity of a series, e assume our model to !e:

X t = !X t-1 + t

3e ta$e a null hy&othesis hich ill further im&ly hether the model is a random al$

&rocess or not" As a result, our hy&othesis ill !e as under:

/ 0 /=

/ 1

/

3e test the model using a t'test and a&&ly the follo ing formula:

t.

""

/

E S

4o e+er,

does not follo e actly t'distri!ution" Rather it follo s a Tau distri ution "

Therefore, to test the significance e loo$ at the ta!le +alues of Tau ta!le"

If the calculated +alue is less than the ta!le tau +alue, then the null hy&othesis is acce&ted andthe series is a unit root random al$ and hence is non'stationary"

If calculated +alue is greater than the ta!le tau +alue, then the null hy&othesis is re6ectedim&lying that the series is stationary"

4o e+er, the &rocess can !e tested using another method as ell" 3e ta$e our original model,that is,

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X t = !X t-1 + t

3e su!tract t'/ from !oth the sides" 3e, therefore, get:X t X t-1 = !X t-1 X t-1 + t

0n sim&lifying the a!o+e e uation, e get,

3X t = (!-1 X t-1 + t

3e can rite the a!o+e e uation as:3X t = X t-1 + t

here: . - 7 / . first difference o&erator

8efore e &roceed onto the testing &rocedure, it must !e noted that if . 9 or - . /, then thisould im&ly that our model is a random al$ &rocess and hence is non'stationary" ee&ing

the a!o+e mentioned +ie s into consideration, e ma$e the follo ing hy&othesis:

/ 0 ! = 1 or ! 1 = 0 or = 0/ 1 4 0

3e again test the model using the follo ing formula, assuming that our - follo s a taudistri!ution"

t=

;"" E S

~ Tau distri ution

Summing u&,

If calculated +alue is less than the ta!le +alue, then e acce&t the null hy&othesis andreach a conclusion that the series is non'stationary"

If calculated +alue is more than the ta!le +alue, then e re6ect the null hy&othesis andreach a conclusion that the series is stationary"

2.5 6ugmented 'ic$e u##er Test

Another +ersion of this test is that hen < t is not a hite noise term or they are correlatedith each other" Under these circumstances e use, Augmented #ic$ey %uller Test" It is

conducted !y augmenting the follo ing e uations:

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Y t = Y t-1 + u t

Pure RandomWal

Y t = 1 + Y t-1 + u t

(RandomWalk with

Drift)

Y t = 1 + 2 t + Y t-1 + u t

RandomWalk withDrift and

Trend

Then e add lagged +alue of the de&endent +aria!le (

=

m

i

it Y

/

) hich in true sense isaugmentation" It ma$es the error term < t a hite noise term"

So, hen the error term is con+erted into hite noise, then our e uation !ecomes:

37 t = ! 1 + ! 2t + 7 t-1 +

=

m

i

it i Y

/

+ u t

here: u t is the &ure hite noise term

=> t . > t 7 > t'/=> t'/ . > t'/ 7 > t'2 and so on"

The num!er of lagged +aria!les are so included that they are enough to ma$e the error termserially uncorrelated"

Under the Augmented #ic$ey %uller Test, e assume the same hy&othesis, that is,

4 9: . 94 / : ? 9

3e use the follo ing formula to estimate the calculated +alue"

t=

;

"" E S

~ 6ugmented 'ic$e u##er Tau distri ution

If the calculated +alue is less than the ta!le +alue, e conclude that the error term is auto'correlated and hence the series is non'stationary" 0n the other hand, if calculated +alue ismore than the ta!le +alue, e re6ect the null hy&othesis stating that the series is stationary innature"

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2.8 T&e 9&i##i:s-9erron Unit Root Test

The test, as the name suggested, as in+ented !y 9.;.ue o t&e Unit Root Test

The significance of the unit root test de&ends on the si*e and &o er of the test" 8y si*e, emean the strength of the le+el of significance, or in other terms, the &ro!a!ility of committingthe Ty&e I error" 0n the other hand, !y &o er of the test e estimate the &ro!a!ility of re6ecting the null hy&othesis hen it is false"

8efore continuing further, let s ta$e a loo$ !ac$ at the ty&e of errors committed in the testing &rocedures"

?rror / 0 "&en@T :e A Re6ected 4 9 is trueT :e AA (more &arm u# Acce&ted 4 9 is false

The &o er of a test can !e calculated !y su!tracting the &ro!a!ility of Ty&e II error from Ty&eI" The ma imum &ossi!le &o er can !e e ual to one"

2. .1 BiCe o t&e Test

The #ic$ey'%uller test is sensiti+e to its &rocedure of conducting it" If for e am&le, our truemodel is a random al$ model, ho e+er, e estimate random al$ ith drift model" Thenthe conclusions deri+ed from it may !e rong at say 5 &er cent le+el of significance" This is

!ecause in the case of the latter, the true le+el of significance is much larger than 5 &er cent"Boreo+er, if e also e clude the mo+ing a+erage com&onent from the model, this couldfurther distort the si*e"

2. .2 9o%er o t&e Test

It is an inherent characteristic in the #ic$ey %uller Test that they ha+e a lo &o er and tend toacce&t the null hy&othesis more fre uently, and there!y increasing the &ro!a!ility of committing Ty&e II error" The reason can !e attri!uted to the follo ing:

a" The &o er of a test de&ends more on the time s&an of the data as com&ared to the num!er of o!ser+ations" %or e am&le, the &o er of a test ould !e more in case of C9o!ser+ations o+er C9 years as com&ared to /99 o!ser+ations o+er /99 days"

!" If in a model, say,t . t'/ D ut E / and not e actly e ual to one, then the unit root test ould declare the model to !e

non'stationary"

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Economic TimeSeries

#i$erenceStationar%

Process&'as a

stoc(astictrend)

Trend Stationar%Process

&'as adeterministic

trend)

c" The a!o+e mentioned models assume a single unit root, that is, they are integrated of theorder /" If the model is integrated of order higher than /, then it may result in more thanone unit root" In the latter case, e use 'ic$e 9antu#a Test "

d" The unit root tests are inca&a!le of catching any structural !rea$s in the model"

*. Trans ormation o Don-Btationar Time Beries

0n regressing the time series hich is non'stationary, e may incur a &ro!lem of s&uriousregression" Therefore, to a+oid them, it is im&ortant to con+ert the non'stationary time seriesinto stationary time series" 4o e+er, the con+ersion de&ends on hether the time series isdifference stationary or trend stationary"

*.1 'i erence Btationar 9rocess

If a time series contains a unit root, then the first difference of such time series ill !estationary" Similarly, if the time series is integrated of higher order than /, say I(2), then

e ha+e to difference it t ice" et s estimate it using the follo ing &rocedure,

3e ta$e a +aria!le > and o!ser+e its +alues o+er the &eriod of time" The model ill !e

re&resented as:

7 t = 7 t-1 + u t

here: > is the +aria!le ta$en into considerationt is the length of time &eriodu t is the error term

Sim&lifying the e uation, e get> t ' > t'/ . u t

=> t . u t

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Any series is generated !y &ure random al$ is e ual to hite noise, if it is a firstdifference e uation"

Using the #ic$ey %uller Test,

t=

;"" E S

~ Tau distri ution

3e conclude saying,

If calculated +alue is less than the ta!le +alue, then e acce&t the null hy&othesis andreach a conclusion that the series is non'stationary"If calculated +alue is more than the ta!le +alue, then e re6ect the null hy&othesis andreach a conclusion that the series is stationary"

*.2 Trend Btationar 9rocess

The &rocess states that any series hich has a trend is a non'stationary series"3e ta$e the follo ing model

7 t = ) + !t + u t

7 t ) !t = u t

The model states that if from a series e su!tract the trend, then the series !ecomes astationary series" 4ere u t is a linearly detrended time series"

It can also !e &ossi!le that the trend may !e non'linear in nature" %or e am&le,

7 t = ) + ! 1t + ! 2t 2 + u t

Under this, the residuals ill no !e uadratically detrended time series"

4o e+er, it should !e noted that if a series is difference stationary &rocess !ut e treat itas trend stationary &rocess, then it is called under-di erencing " 0n the other hand, if theseries is a trend stationary &rocess !ut e treat it as difference stationary &rocess, then itis called oEer-di erencing "

The conse uences of such mis'inter&retation can result into s&ecification errors hich can

!e serious, de&ending u&on ho one handles the &ro&erties of serial correlation of theresulting error terms"

In general, most of the macroeconomic time series are difference stationary &rocess rather than trend stationary &rocess"

5. Bummar

The to&ic can !e summarised as follo s:

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A random al$ model has an inherent characteristic of non'stationarity"

A random al$ model, random al$ ith drift model, random al$ ith drift andtrend modelF they all follo Tau distri!ution"

Stationarity can !e chec$ed on the !asis of hether the model has a unit root inherentin it or not" The other methods li$e #ic$ey'%uller Test and Augmented #ic$ey %uller test can also !e used for testing the stationarity of the series"

Philli&s'Perron test uses the non'&arametric statistical methods to a+oid any serialcorrelation in the error term ithout adding the lagged difference terms as in the caseof Augmented #ic$ey %uller Test"

If the model is integrated of order higher than /, then it may result to more than oneunit root and in that case e use 'ic$e 9antu#a Test "

An economic time series is of t o ty&es: %irstly, difference stationary hich follo s astochastic trend, Secondly, trend stationary hich follo s a deterministic trend"

If a series is difference stationary &rocess !ut e treat it as trend stationary &rocess,then it is called under'differencing" 0n the other hand, if the series is a trendstationary &rocess !ut e treat it as difference stationary &rocess, then it is calledo+er'differencing" Bost of the macroeconomic time series are difference stationary

&rocess rather than trend stationary &rocess"

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module 5_ tests for stationarity part 2

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