Stationarity and Unit Roots

8/13/2019 Stationarity and Unit Roots

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Lecture 5: Stationarity and Unit Roots

Reading Asteriou P229-239 and Chapter 16

(or Enders Chapter 4)

1



Background

Up to now we have been mostly looking at

cross-section methods, today we will begin to

move towards time-series econometrics and

we will focus on this for the remainder of thecourse.

Sometimes the methods we have looked at

are applicable to time-series however, inmany other cases they are not applicable (or

not directly at least).

2



Background

In a cross section model such as:

Yi = α + β1X1i + εi

Recall that E(εi) was 0.

Thus if we have a sample then on average Y will be known if we

know X1. We can write this as: E(Yi|X1i) = α + β1X1i

E(Yi|X1i) is pronounced expectation of Yi given X1i

Implication: we don‟t need to know the errors to know what Y will beon average as long as we know X1. [since on average errors are 0]

Another way to say this is, if we know X then we know the averagevalue that Y will take.

The important point is for a given value of X, the average for Y is

constant [since no random ε in E(Yi|X1i) = α + β1X1i].

3



This is not always the case for time-series and

this causes us some problems

(Aside: recall that the average for Y was usedto find our estimates for α and β in OLS)

4



0 50 100 150 200 250 300 350 400 450

.25

.5

.75

1

1.25

1.5

1.75

2

2.25

2.5

Many Economic Series Do not Conform to the

Assumptions of Classical Econometric Theory

Share Prices

0 50 100 150 200 250

.35

.4

.45

.5

.55

.6

.65

.7 Exchange Rate

1960 1965 1970 1975

8.7

8.8

8.9

9

9.1

9.2

9.3

Income

5



Before considering this though we

will talk a little about time series….

6



Time series data

We view realisations of economic time series as beinggenerated by a stochastic process;

the particular realisation of a variable at one point in time is just one

possible outcome from an inherently random variable

So we can think of this as

Yt = (……) + εt Where (….) is some relationship to observed variables

When we are dealing with time series data, a common

starting point is to ask:

Well what can the past values of the series itself tell us

about the likely future path?

For the next few lectures we will be focusing on this

question

7





Some observations regarding time-series

When we look at a time-series, such as the one

above, we often notice that: The stochastic process generating economic data show a

distinct tendency to sustain movements in one direction

Mean of the series is often not constant

0 50 100 150 200 250

.35

.4

.45

.5

.55

.6

.65

.7

Exchange Rate

9



We distinguish between two types

of series:

Stationary and Non-stationary

[Very important to understand which you are

dealing with as OLS is wrong when the seriesis non-stationary!!!]

10





Stationary Series

A stationary series has: Constant mean

E(Yt) = u

Constant variance

Var(Yt) = E (Yt – u)

2

= σ2

The covariance between two time periods depends only on thelag between the time periods and not on the point in time at whichthe covariance is computed γ

k = Cov(Yt,Yt-k) = Constant for all t and k≠0.

E.g. Cov(Y2008, Y2008-s)=0.5 then Cov(Y2004, Y2004-s) should also be

0.5 since the gap is 4 years here too! This should be the same forany set of observations s years apart

Basically this is saying that there shouldn‟t be periods when thepast doesn‟t matter much and periods when it matters a lot – theeffect of the past should be the same in all periods

12



Example in Stata

13



Note, there are other types of

stationarity. However we won’t deal

with them in this course!

14



Non-Stationary Series

There may be no long-run mean to which the

series returns [e.g. there may be a trend in the

data]

The variance may be time dependent and go toinfinity as time approaches to infinity

Theoretical autocorrelations may not decay overtime but, in finite samples, the sample

correlogram dies out slowly

[we‟ll talk about this later today]

15



Conditions for Stationarity in a simple model

Consider the model:Yt= ФYt-1 + εt

Stationarity requires that │Ф│ < 1

So that the impact of a disturbance dies out over time!

To see why consider the alternatives:

If │Ф│> 1 we have an explosive series (because shocks compoundover time rather than dying out)

To see this: Remember we assume that the same process is at work in

each period: so Yt-1= ФYt-2 + εt-1

Yt= ФYt-1 + ε t becomes Yt= Ф[ФYt-2 + ε t-1] + ε t

Yt= Ф2Yt-2 + Ф ε t-1 + ε t

Note that the shock from the previous period is having a bigger impact in thisperiod, if we kept substituting we would see the shock from n periods ago has an

effect of Фn now so the shock keeps having a bigger and bigger impact!

If Ф= 1 then the series has a unit root => a shock has the

same effect every period i.e. never dies out

16



In Economics, sustained explosive seriesare less common (output is an example),

however unit roots are common in many

series so we will spend a little timeconsidering them now….

17 Thanks to Tom Pierse for the pointing out the output example!



18

Using OLS.

Recall that the OLS estimator is:

Notice this includes the mean, the variance and the

covariance

This is why we need the three of these to be finite

constants.

Otherwise we would not get a stable unique

estimate for the coefficient using OLS!!

)(

),(ˆ

1

2

1

X Var

Y X Cov

X X

Y Y X X

N

j

j

N

j

j j





Suppose we were to plot the series

Y t - what would it look like?

Well each period, we’d start at the value from

last year then we’d add a bit on (which may benegative) but the change is random so may

look something like:

20



Non-stationary time series: Random Walk

Random Walk

0

0.5

1

1.5

2

2.5

y t = y t-1 + ε t εt ~ n (0, σ 2 )

21



A key point is, if I’d told you the value at the

start was close to zero, you would still not

have been able to accurately predict the actual

outcome at the end, since there was just aseries of random movements…..

…. Like watching a drunk guy (or girl) stumblingalong, you’re never sure which way they’ll walk next!

[Our best guess is our current information]

22



Why is a random walk nonstationary? Y t = Y t-1 + ε t

E(ε t) = 0,Denote the initial value of Y = y0

Mean:

y1 = y0+ ε 1

y2 = (y0+ ε 1) + ε 2

y3

= (y0

+ ε1

+ ε2

) + ε3yt = yo + Σ ε t

E (Yt) = y0 = Constant

[Satisfies 1st requirement for stationarity]

Variance:

Var(Y t ) = Var(Y0 ) + Var (ε 1) +…+ Var (ε t ) = 0 + σ 2 +…+ σ 2

= t σ 2

(variance is not constant through time)

Hence as t → ∞ the variance of Yt approaches infinity

[Thus violates the 2nd requirement for stationarity => non-stationary] 23



E.g. Exchange rates may have a unit root:

0 50 100 150 200 250

.35

.4

.45

.5

.55

.6

.65

.7

Exchange Rate

24



Return to the Random Walk model

yt=

y

t-1 +

εty1= y0 + ε0

y2 = y1+ ε1

= (y0 + ε0) + ε1

yt = y0 + εi

Thus the general solution if y0 is a given initial condition is

yt = y0 +

i.e. our best prediction of the value of the series t periods

from now is the value now (since E( )=0)

t

i 1

t

i 1 t

25

t

i 1 t





Non-stationary time series

UK GDP (Y t )

The level of GDP (Y) is not constant and the mean increases over time. Hence the level of

GDP is an example of a non-stationary time series.

Here though the non-stationarity looks to be caused by a trend rather than a random walk!!

GDP Level

0

20

40

60

80

100

120

1 9 9 2

Q 3

1 9 9 3

Q 2

1 9 9 4

Q 1

1 9 9 4

Q 4

1 9 9 5

Q 3

1 9 9 6

Q 2

1 9 9 7

Q 1

1 9 9 7

Q 4

1 9 9 8

Q 3

1 9 9 9

Q 2

2 0 0 0

Q 1

2 0 0 0

Q 4

2 0 0 1

Q 3

2 0 0 2

Q 2

2 0 0 3

Q 1

27



When we have a series with a unit

root or a trend, differencing it may

lead to a stationary series…..

28



First difference of GDP is stationary

ΔY t = Y t - Y t-1since it fluctuates around about 0.6, and the variance is stable….

Stationary time series

GDP Growth

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 9 9 2 Q 3

1 9 9 3 Q 2

1 9 9 4 Q 1

1 9 9 4 Q 4

1 9 9 5 Q 3

1 9 9 6 Q 2

1 9 9 7 Q 1

1 9 9 7 Q 4

1 9 9 8 Q 3

1 9 9 9 Q 2

2 0 0 0 Q 1

2 0 0 0 Q 4

2 0 0 1 Q 3

2 0 0 2 Q 2

2 0 0 3 Q 1

29





Testing For non-stationarity

Constant covariance - use of correlogram

Covariance between two values of Y t depends only on thedifference apart in time for stat ionary series.

Cov(Y t ,Y t -k ) = (k )

i.e. covariance is constant in t , and depends only on the lengthapart of the observations.

(A) Covariance for 1980 and 1985 is the same as for 1990 and1995. (i.e. t = 1980 and 1990, k = 5)

(B) Covariance for 1980 and 1987 is the same as for 1990 and

1997. (i.e. t = 1980 and 1990, k = 7)But:

(C) Covariance for 1980 and 1985 may differ from that between1990 and 1997, since gap is different (i.e. k differs between thetwo periods)

31



Testing For non-stationarity

Constant covariance - use of correlogram

One simple test is based on the autocorrelation

function (ACF)

The ACF at lag k, denoted by k is defined as:

k = k / 0 = Covariance at lag k / Variance

Note: if k=0 then ρo = 1

As with any correlation coefficient k lies between -1

and 1

The quickerkgoes to zero as k increases, the quicker

shocks are dying out

A plot of k against time is known as the sample

correlogram

ˆ

ˆ

32

̂ ̂

ˆ

ˆ

ˆ



Note:

When dealing with a single variable Yt. It

turns out that the autocorrelation equals

the coefficient (Ф)

E.g. if we have: Yt= ФYt-1 + ε t

Then: ρ1= Ф

Since: Yt-1= ФYt-2 + ε t-1

Yt= Ф(ФYt-2 + ε t-1) + ε t

Yt= Ф2Yt-2 + Фε t-1 + ε t

ρ2= Ф2

33





Non-stationary time series

UK GDP (Y t ) - correlogram

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

35





WHITE NOISE PROCESS

y t = ε t ε t ~ n (0, σ 2 )

•Correlogram:

Stationary time series

White Noise

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

37





Other Stationary processes If 0<Ф<1 e.g. y t = 0.2 y t-1 + εt the correlations decrease to 0

quickly (if Ф is close to 0 the correlations fall quickly)

If -1<Ф<0 e.g. y t = -0.2 y t-1 + εt the correlations decrease to 0

quickly but oscillate between positive and negative

correlations. (Again, if Ф is close to 0 the correlations fall

quickly)

39



We will use ACF plots quite a lot so its

important to get used to understanding

them (later in the course we’ll look at

PACF plots too)!

40



Statistical Tests of Stationarity

But first we’ll take a break!

41



Box Pierce Test

The Box-Pierce statistic tests the joint hypothesis

that all k are simultaneously equal to zero. (i.e.

series is white noise process (and thus stationary)

The test statistic is approx. distributed as a χ2

distribution with m df.

n = sample size

m = lag length

If B.P. > χ2m (α) then reject H0: k = 0

2

1

ˆ.. k

M

k

n P B

ˆ

42

ˆ



Ljung-Box Q-Statistic

Box-Pierce statistic performs poorly in small sample so E-views uses the Ljung-Box (Q) statistic instead.

n is the sample size, k is the sample autocorrelation at lag k ,and m is the number of lags being tested.

Null hypothesis is still that all k are simultaneously equal to

zero [more of a test whether the process is white noise whenused on the ACF]

Process may be stationary but not white noise!!

ˆ

ˆ

43





A Second Test of Stationarity

The Dickey Fuller Test We can test H0: Ф = 1 [or H0: =0 ] by a t statistic.

However the „t statistic‟ does not follow a students t-distribution.Under the null hypothesis of nonstationarity, Dickey and Fuller(1979) have tabulated the critical values of the conventionallycalculated t- statistic. These critical values have been further

extended and improved by MacKinnon (1991) If by the critical values in MacKinnon(1991):

We cannot reject H0: =0,

We cannot reject H0: Ф = 1,

We cannot reject the unit root hypothesis,

We cannot reject nonstationarity If we can reject H0: =0,

We can reject H0: Ф = 1,

We conclude the series is stationary.

45



The Augmented Dickey Fuller Test

The error term in the DF test is unlikely to be white noise i.e. there is usually autocorrelation present!

To „whiten‟ the error terms, empirical studies oftenspecify lags of the dependent variable as follows:

∆Yt = Yt-1 + + εt

This is known as the Augmented Dickey Fuller statistic(ADF)

The appropriate lag length can be determined by AIC,SBC or through an LM test

The H0: =0 is that there is a unit root (as with the DFtest)

it

p

i i y

1

46





Trend Stationary

- Stochastic trend

- Deterministic trend

48



Stochastic trend:In this case the trend is due to

shocks not dying out so we see atrend

49

St h ti t d d lk d l



Stochastic trend: e.g. random walk model From earlier remember E(yt) = yo

If yt is the most recent observation/realisation of thedata generating process of y then yt is the unbiased

estimator of all future values of yt+s

ut is a random shock in the random walk model, but it has a

permanent effect on the conditional mean

E(yt+s│t) = yt , but E(yt+s│t+1) = yt+1

Because ut , ut+1 … are random we have no reason tobelieve that the series mean will revert to yt from yt+1.

The random shock has permanent effects on the mean.

Such a sequence is said to have a stochastic trend

The series may seem to increase/decrease due to a

sequence of positive (negative) shocks

50



Stochastic Trend

We have seen that a series with a unit root

will wander aimlessly around. However if wehave an intercept in the model:

y t = α + y t -1 + εt

Then each period, Yt is changing by α + εt sowe will observe a „trend‟ in the data. Thistype of trend is known as a stochastic

trend A random walk model with an intercept, is

known as a ‟random walk with drift‟ 51



Non-stationary time series: Random Walk with Drift

y t = α + y t-1 + u t u t ~ n (0, σ 2 ) (α >0)

Errors still persist, but α ensures there is a pattern – the larger α is the clearer the

pattern since the shock each period contributes less in relative terms…

Random Walk with Drift

0

2

4

6

8

10

12

52



Why we care about a stochastic trend:

If we have a stochastic trend the series is not

stationary => classical inference (OLS) is

wrong for Yt

However by taking the first difference we get

a series which is stationary – so we can use

OLS on the first difference, ∆yt.

We would call Yt “difference stationary” 53



Difference Stationary

54





Deterministic trend: In this casethere is a real tendency for the

series to increase over time (i.e. itsnot just due to errors not dying out)

56

d



Deterministic trend

Suppose the series increases by b1 each

period (or decreases if b1 <0). We could

express this as:

We can remove the trend by subtracting β1tfrom both sides:

We call this ‘detrending’ the series.

Yt* = Yt - β1t

Yt* = α + β2Yt-1 + εt

Classical inference is valid on the detrended series provided |β2|<1

57

Yt = α + β1(t) + β2Yt-1 + εt

Yt - β1(t) = α + β1(t) - β1(t) + β2Yt-1 + εt







T i f i i ( i )!



Testing for stationarity (again)!

We would carry out all 3 forms of the DF (or

ADF) test:

No intercept and no trend

intercept (drift) but no trend Both intercept and trend

The we use AIC or SBC to choose betweenthe models – and trust the result of the test

for that model

61

P bl i T i f U i R



Problems in Testing for Unit Roots

We need to determine the appropriate lag length in the ADFequation

The ADF only tests for a single unit root. Although we can applythe ADF test procedure to differences of the yt in order to testhigher unit root hypothesis

The ADF has low power to distinguish between a unit root and anear unit root process. In other words it has low power to reject afalse null hypothesis.

Incorrectly including or omitting a drift or trend term lowers thepower of the test to reject the null hypothesis of a unit root.

The tests fail to account for structural breaks in the time series. A

structural break may make a stationary series appear nonstationary.

62

Trend Stationary and Difference Stationary Time Series



Trend Stationary and Difference Stationary Time Series

When the true data generating process is

unknown it is sensible to plot and observe the

data and to start statistical tests with the most

general model Also AIC and SBC criteria can be used to see

which specification best suits the data….

See A&H figure 16.5 on Page 298

63

O i f i d f



Over view of remainder of course:

TIME-SERIES

STATIONARY

One variable

Homoskedastic

ARMA

Heteroskedasticity

GARCH

More than 1Variable

VAR

NONSTATIONARY

One Variable

ARIMA

More than 1variable

Cointegration

(VECM)

Documents

Stationarity and Unit Roots