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Lecture 3
Stephen G Hall
Stationarity
NON-STATIONARY TIME SERIES
OVER THE LAST DECADE OR SO WE HAVE BEGUN TO
UNDERSTAND THAT ECONOMETRIC ANALYSIS OF TIME
SERIES DATA CAN BE SERIOUSLY MISLEADING WHEN WE
ARE DEALING WITH NON STATIONARY DATA.
EARLIER STUDIES POINTED TO THE PROBLEM
GRANGER AND NEWBOLD(1974) OR YULE(1926)
eg
BUT IT IS ONLY RECENTLY THAT WE HAVE BEGUN TO
DEVELOP A BODY OF TECHNIQUES WHICH ALLOW US TO
DEAL WITH THESE PROBLEMS
THESE THREE LECTURES ARE AIMED AT GIVING YOU A
WORKING GRASP OF THIS BODY OF TECHNIQUES.
REFERENCES
CUTHBERTSON K., HALL S.G. and TAYLOR M.P. APPLIED
ECONOMETRIC TECHNIQUES, SIMON AND SCHUSTER 1991,
(OVERVIEW AND INTUITION)
ENGLE R.F. and GRANGER C.W.J. LONG RUN ECONOMIC
RELATIONSHIPS, OXFORD UNIVERSITY PRESS 1991
(REPRINTS OF KEY READINGS)
BANERJEE A. DOLADO J. GALBRAITH J.W. AND HENDRY
D.F. COINTEGRATION, ERROR CORRECTION AND THE
ECONOMETRIC ANALYSIS OF NON-STATIONARY DATA,
OXFORD 1993. MORE ON ASYMPTOTIC THEORY AND
DETAILED PROOFS.
SOME DEFINITIONS
STATIONARITY
A STOCHASTIC PROCESS IS STRICTLY (STRONGLY)
STATIONARY IF ITS PROBABILITY LAW IS NOT TIME
DEPENDENT. THAT IS TO SAY IF WE TAKE ANY
CONSECUTIVE SUBSET OF THE TIME SERIES ITS JOINT
DISTRIBUTION FUNCTION IS IDENTICAL TO ANY OTHER
SIMILAR SUBSET.
WEAK (SECOND ORDER, COVARIANCE) STATIONARITY IS
IMPLIED BY
E( X t ) = E( X t+h ) =  < 
E( X t2 ) = E( X t2+h ) =  < 
E( X t X t - j ) = E( X t+h X t+h- j ) =  ij < 
AN INTEGRATED PROCESS IS ONE WHICH MAY BE
REDUCED TO STATIONARITY BY DIFERENCING, DENOTED
I(J) WHERE J IS THE NUMBER OF DIFFERENCES. I(0) IS
STATIONARY
ORDERS OF MAGNITUDE AND CONVERGENCE
LET Xt BE A SEQUENCE OF REAL NUMBERS AND Yt BE A
SEQUENCE OF POSITIVE REAL NUMBERS.
THEN X IS OF SMALLER ORDER IN MAGNITUDE THAN Y IF;
lim T  X T / Y T = 0 denoted X T = o( Y T )
X IS AT MOST OF ORDER IN MAGNITUDE Y IF;
| X t | / Y t  M all t, denoted X T = O( Y T )
THESE TWO DEFINITIONS ARE ABOUT HOW FAST X AND Y
GROW RELATIVE TO EACH OTHER, IMPLICITLY WHICH OF
THE TWO WILL COME TO DOMINATE THE OTHER OVER
TIME.
IF Wt IS A SEQUENCE
VARIABLES THEN;
OF
RANDOM
STOCHASTIC
Wt CONVERGES IN PROBABILITY TO W IF
lim t  Pr(| W t - W |>  ) = 0,  > 0, denoted plim W t is W
Wt IS OF SMALLER ORDER IN PROBABILITY THAN Yt IF;
plim W T / Y T = 0, denoted W T = o p ( Y T )
Wt IS AT MOST OF ORDER IN PROBABILITY Yt IF THERE
EXISTS A POSITIVE REAL NUMBER M SUCH THAT;
Pr(| W T | M  Y T )   ,  > 0, denoted W T = O p ( Y T )
THESE THREE ARE ABOUT WHERE W IS GOING IN THE
LONG RUN AND IF WE EXPECT W OR Y TO BECOME
DOMINANT, AND HOW DOMINANT.
ERGODIC AND MIXING PROCESSES
AN ERGODIC PROCESS IS A SLIGHTLY STRONGER FORM
OF WEAK STATIONARITY WHERE IN ADDITION WE
REQUIRE
 -1

lim t   t  COV( X t , X t+i )  = 0
 i=1

t
MIXING IS A PARTICULAR
INDEPENDENCE.
FORM
OF
ASYMPTOTIC
UNIFORM MIXING MEANS THAT ASYMPTOTICALLY THE
CONDITIONAL PROBABILITY OF X GIVEN Y IS EQUAL TO
THE UNCONDITIONAL PROBABILITY OF X. (ie Y TELLS US
NOTHING ABOUT X)
STRONG MIXING MEANS THAT ASYMPTOTICALLY THE
JOINT PROBABILITY OF X AND Y IS EQUAL TO THE
PRODUCT OF THE INDIVIDUAL PROBABILITIES OF X AND
Y. SO AGAIN THEY ARE BASICALLY UNRELATED.
WIENER OR BROWNIAN PROCESSES
NORMAL ASYMPTOTICS WORKS OVER TIME AS TIME
GOES FROM ZERO TO INFINITY. THIS MEANS THAT
VARIANCES OF NON-STATIONARY PROCESSES GENERALLY
BECOME UNBOUNDED. THE TRICK USED HERE IS TO MAP
THE ZERO TO INFINITY INTERVAL OF DISCRETE TIME
INTO A CONTINUOUS INTERVAL OVER 0-1.
A WIENER PROCESS IS LIKE A RANDOM WALK BUT IN
CONTINUOUS TIME MAPPED OVER THE UNIT INTERVAL.
THE WIENER PROCESS IS DENOTED W(r) FOR r BETWEEN
ZERO AND ONE. W(r) IS DISTRIBUTED AS NORMAL WITH
ZERO MEAN AND VARIANCE r.
THE TRICK IS TO LEARN HOW TO MAP NONSTATIONARY
VARIABLES INTO THE UNIT INTERVAL AND RELATE THEM
TO KNOWN DISTRIBUTIONS.
CONSTRUCTING A WIENER PROCESS
LET
S t = S t -1 + vt S 0 = 0
vt ~ IN(0,1)
THEN E( S | 0 ) = t
2
t
THE IDEA IS TO FIND A MAPPING WHICH TAKES THIS
SERIES WHICH SPREADS FROM ZERO TO INFINITY AND
MAPS IT ONTO THE UNIT INTERVAL.
WE CONSTRUCT A NEW SERIES AS FOLLOWS;
RT (r) = S [Rt] / T
WHERE [rT] IS THE INTEGER PART OF rT AND r IS BETWEEN
0 AND 1.
THIS CREATES A `STEP' FUNCTION WHICH GETS FINER AS r
GETS SMALLER. IN THE LIMIT RT(r) TENDS TO W(r) AS T
BECOMES LARGE.
The following slides illustrate this
Step representation of a random walk over 10 points
Step representation of a random walk over 100 points
Step representation of a random walk over 1000 points
SPURIOUS REGRESSION
THE STANDARD OLS ESTIMATOR IS
 = (X X ) (X Y)
-1
THIS RESTS ON THE ASSUMPTION THAT (1/T)(X'X) AND
(1/T)(X'Y) CONVERGES ON SOME CONSTANT. MANY OF THE
PROOFS OF CONSISTENCY ETC. ALSO USE THIS
ASSUMPTION.
UNDER NON-STATIONARITY NEITHER OF THESE TWO
TERMS MAY EXIST, AS T GROWS BOTH MAY EXPLODE IN
DIFFERENT WAYS AND THE RESULTING ESTIMATOR CAN
BE HIGHLY MISLEADING.
CONSIDER A MONTE CARLO EXPERIMENT
2
=

+
~
IID(0,
1 )
Yt
Y t -1 u t u t
2
X t =  X t -1 + vt vt ~ IID(0,  2 )
  1
NOW CONSIDER THE REGRESSION
Yt = 0 + 1 X t + t
AS X AND Y ARE UNRELATED WE WOULD HOPE THAT THE
COEFFICIENT ON X WOULD CONVERGE ON ZERO.
THIS IS NOT THE CASE. OLS MAXIMISES CORRELATIONS
AND IN NON STATIONARY DATA ENTIRELY SPURIOUS
CORRELATIONS MAY EXIST WHICH DO NOT DISAPPEAR NO
MATTER HOW LARGE THE SAMPLE.
YULE(1926) FIRST POINTED THIS OUT IN A PRAGMATIC WAY
AND GRANGER AND NEWBOLD(1974) COINED THE TERM
SPURIOUS REGRESSION AND WARNED OF AN R2>DW.
PHILLIPS(1986) DEMONSTRATES FORMALLY THAT OLS
ESTIMATORS OF THE COEFFICIENTS DO NOT HAVE ANY
WELL DEFINED LIMITING DISTRIBUTION AND AS T GROWS
THE PROBABILITY OF FINDING A `SIGNIFICANT'
RELATIONSHIP RISES.
THE FOLLOWING SLIDES SHOW THE ACTUAL
DISTRIBUTIONS WHICH EXIST UNDER VARIOUS
ASSUMPTIONS REGARDING THE PROPERTIES OF X AND Y
Frequency distribution for the correlation between X and Y when
they are both I(0)
Frequency distribution for the correlation between X and Y when
they are both I(1)
Frequency distribution for the correlation between X and Y when
they are both I(2)
Frequency distribution for the correlation between X and Y when X
is I(1) and Y is (2)
DETERMINISTIC AND STOCHASTIC TRENDS
ONE SOLUTION CONSIDERED TO THE PROBLEM OF NONSTATIONARITY WAS DETRENDING THE DATA. THIS IS NOT
NOW REGARDED AS SATISFACTORY FOR TWO REASONS.
FIRST. DETERMINISTIC TRENDS CAN NOT REMOVE A
STOCHASTIC UNIT ROOT.
X t =  + X t - 1 + vt
X0 = 0
t
X t =  t +  vi
i=0
ONLY THE DRIFT IS REMOVED NOT THE UNIT ROOT.
SECOND. THE DISTRIBUTIONAL PROBLEMS ASSOCIATED
WITH SPURIOUS REGRESSIONS APPLY EQUALLY TO THE
ESTIMATION OF DETERMINISTIC EFFECTS. WE TEND TO
ACCEPT THERE PRESENCE TOO EASILY.
SO WE NEED TO KNOW ABOUT THE STATIONARITY
PROPERTIES OF OUR DATA TO MAKE SENSE OF ANY
ESTIMATION RESULTS.
TESTING FOR STATIONARITY
THIS IS USUALLY UNDERTAKEN IN TERMS OF TESTING FOR
A UNIT ROOT, CONSIDER
Y t =  Y t -1 + vt
THEN THE NULL HYPOTHESIS OF A UNIT ROOT IMPLIES
THAT
=1
IF THIS IS SIGNIFICANTLY LESS THAN 1 THEN WE CAN
REJECT THE UNIT ROOT HYPOTHESIS IN FAVOUR OF A
STATIONARY ALTERNATIVE.
UNDER THE NULL HOWEVER Y IS NONSTATIONARY AND SO
THE DISTRIBUTION OF THE TEST STATISTIC IS NONNORMAL AND SO WE CAN NOT USE STANDARD `t' TABLES.
CORRECT CRITICAL VALUES WERE FIRST TABULATED BY
DICKEY(1976) AND USED IN DICKEY AND FULLER (1979,
1981)
THE DICKEY-FULLER TEST MODELS
THREE BASIC MODELS ARE CONSIDERED, IN A SLIGHTLY
DIFFERENT PARAMETERISATION TO THE LAST SLIDE.
a  Y t =  a Y t - 1 + vt
b  Y t =  +  b Y t - 1 + vt
c  Y t =  +  c Y t - 1 + t + vt
H0 :  = 0 H1 :  < 0
TWO TEST PROCEDURES ARE PROPOSED
i) Tˆ
ii) ˆ/SE( ˆ )
Dickey-Fuller
5% critical
values for both
tests
TEST i)
TEST ii)
-7.7
-1.95
T=100
-7.9
T=INFINITY -8.1
MODEL b
-1.95
-1.95
T=50
-13.3
T=100
-13.7
T=INFINITY -14.1
MODEL c
-2.93
-2.89
-2.86
T=50
-19.8
-3.50
T=100
-20.7
T=INFINITY -21.8
-3.45
-3.41
MODEL a
T=50
TESTING IN THE PRESENCE OF A GENERAL DYNAMIC
ERROR PROCESS.
THE TESTS OUTLINED ABOVE ASSUME THAT THE ERROR
PROCESS OF THE MODEL IS WELL BEHAVED.
IN GENERAL WE WOULD EXPECT A SIMPLE MODEL OF THIS
TYPE TO HAVE A RICH DYNAMIC STRUCTURE.
FULLER(1976) DEMONSTRATES THAT A HIGH ORDER AR
MODEL CAN BE DEALT WITH USING THE FOLLOWING
MODEL.
SAID AND DICKEY(1984) EXTEND THE PROOF TO INCLUDE
ARMA MODELS.
THE AUGMENTED DICKEY-FULLER TEST (ADF)
k
a  Y t =  a Y t - 1 +   Y t - i + vt
i +1
k
b  Y t =  +  b Y t - 1 +   Y t - i + vt
i=1
k
c  Y t =  +  c Y t - 1 + t +   Y t - i + vt
i=1
H0 :  = 0 H1 :  < 0
THESE HAVE THE SAME ASYMPTOTIC DISTRIBUTION AS
THE DF TESTS.
NON-PARAMETRIC TESTS OF A UNIT ROOT
TO AVOID USING MORE SPECIAL TABLES IT IS USEFUL TO
HAVE TESTS WHICH RELY ON LESS STRINGENT
DISTRIBUTIONAL ASSUMPTIONS. THE MAIN TESTS USED
ARE THE PHILLIPS(1987) AND PHILLIPS AND PERON(1988)
TESTS;
SAME THREE BASIC MODELS AS THE DF TEST
a  Y t =  a Y t - 1 + vt
b  Y t =  +  b Y t - 1 + vt
c  Y t =  +  c Y t - 1 + t + vt
H0 :  = 0 H1 :  < 0
BUT THE ERRORS NEED NOT BE WHITE NOISE and normal.
THE FOLLOWING ASSUMPTIONS ARE MADE.
E( vt ) = 0 t

sup t E | vt | <  FOR SOME  > 2
 2 = lim t  E( t -1 Y t2 ) EXISTS, AND  2 > 0
vt IS STRONGLY MIXING

 2 = E( v12 ) +  E( v1 ,v j )
j=2
THE TEST
DEFINE
T
2
-1
S v = T  vˆt2
t =1
l
T
j=1
t = j +1
~2
2
-1
S Tl = S v + 2 T   l (j)  v~t v~t - j
-1
(j)
=
1
j(l
+
1
)
l
THEN THE CORRESPONDING TESTS FOR THE
CASE WITHOUT TREND ARE
T
2 -1
2
2
-2
=
T

0.5(
)[
(
)
S Tl S v T  Y t - 1 Y - 1 ]
Z
t= 2
Z  = ( S v / S Tl )( ˆ/SE( ˆ ))
Standatd
DF
distrbution
T
- 0.5( S tl2 - S v2 )[ S Tl ( T - 2  ( Y t - 1 - Y - 1 )2 )0.5 ] - 1
t= 2
MULTIPLE UNIT ROOTS
WE OFTEN WANT TO ESTABLISH THE ORDER OF
INTEGRATION OF A SERIES.
DICKEY AND PANTULA(1987) POINT OUT THAT TESTING
FOR I(1) THEN I(2) THEN I(3) ETC, IS NOT A VALID TEST
SEQUENCE AS THE ALTERNATIVE IS STATIONARITY.
THE CORRECT PROCEDURE IS TO START FROM THE
HIGHEST PROBABLE ORDER OF INTEGRATION, SAY I(3),
THEN TEST I(3) AGAINST I(2), IF WE REJECT I(3) THEN TEST
I(2) AGAINST I(1) AND SO ON.
SEASONAL INTEGRATION
UNIT ROOTS MAY EXIST AT ANY FREQUENCY (NOT JUST
ZERO) AND SOME LITERATURE EXISTS ON SEASONAL
INTEGRATION. IF
X t - X t -4 = vt
THEN GENERALLY THE FIRST DIFFERENCE OF X WILL NOT
BE STATIONARY AND X MUST BE SEASONALLY
DIFFERENCED. WE CAN WRITE
X t - X t - 4 = ( X t - X t -1 ) + ( X t -1 - X t - 2 ) + ( X t - 2 - X t - 3 ) + ( X t - 3 - X t - 4 )
(1 - L4 )X = (1 - L)(1  L  L2  L3 )X = S(L)
X IS SEASONALLY INTEGRATED OF ORDER d,D (SI(d,D) IF
D
 S(L ) X
d
IS STATIONARY
FURTHER
(1 - L4 ) = (1 - L)(1 + L)(1 + L2 )
= (1 - L)(1 + L)(1 - iL)(1 + iL)
SO THERE ARE 4 ROOTS; 0, HALF YEARLY AND A PAIR OF
COMPLEX CONJUGATES AT 4 QUARTERS
TESTING SEASONAL UNIT ROOTS
LETS ASSUME
 (L) X t = (1 -  1 L)(1+ 2 L)(1+ 3 L2 ) X t + vt
THEN AFTER SOME MANIPULATION WE CAN WRITE
(1 - L4 ) X t =  1 Z 1t - 1 +  2 Z 2t - 1 +  4 Z 3t - 1 +  3 Z 3T - 2 +  + vt
2
3
=
(1
+
L
+
+
Z 1t
L L ) Xt
2
3
=
-(1
L
+
Z 2t
L L ) Xt
2
=
-(1
Z 3t
L ) Xt
THE HEGY (HYLLEBERG ENGLE GRANGER AND YOO(1990))
TESTS FOR SEASONAL UNIT ROOTS ARE THEN
H 0 : UNIT ROOT 0 FREQUENCY =>  1 = 0
H 0 : UNIT ROOT HALF SEASONAL F =>  2 = 0
H 0 : UNIT ROOT AT SEASONAL F =>  3 =  4 = 0
CRITICAL VALUES TABULATED IN HEGY
Example
The Great Crash, The Oil Price
Shock and the Unit Root Hypothesis
by P Perron, Econometrica 1989, 57
no.6, pp 1361-1401
Perron notes that the implications for economic analysis of a
unit root and a deterministic trend are completely different.
xt    xt 1  ut
a shock should persist for ever, the variance into the future
grows exponentially, policy will last forever.
xt   T  ut
Here a shock will not last beyond the current period, the
variance of x does not grow into the future, policy can have
no lasting effect
Many studies have found unit roots and Peron questioned
this.
He proposed 2 possible ways of viewing the world
3 possible models all based around a random walk non-stationary
process. Break in level, break in trend and both together.
3 alternatives based around deterministic trends, Break in level,
trend or both.
His point is it may be very difficult to tell the two apart
Log of nominal USA wages
Log of real USA GNP
Log of USA common stock price
Splitting into sub samples often implies the sub samples are
stationary
Correlogram often dies away quickly suggesting stationarity
He undertook a monte carlo study,
data was generated by
yt   1  (  2   1 ) DU t   1t  et
and
yt   1   1t  (  2   1 ) DT *  et
and then he estimated a standard DF test for a unit root.
This process was replicated 10,000 times
Cumulative density function of the DF coefficient as the size of the
break in the mean increases
Same thing for the breaking trend model
Coefficient results
Monti carlo produces new critical values which vary with the
break size and he can then retest the data based on this more
general model.
Surprisingly although the critical value changes with the
break size it doesn’t change much
He then re-tests the original data assuming the presence of a
break with these new critical values and finds that most of the
data can be treated as stationary
The message: don’t use the tests mechanically, make sensible
judgements about when we need to worry about non-stationarity.