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Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version.
The Land of the Unit RootsThe Land of the Unit Roots
Gloria González-RiveraUniversity of California, RiversideandJesús Gonzalo U. Carlos III de Madrid
Why should we care about the existence of unit roots?Why should we care about the existence of unit roots?
• Growth
• Forecast
• The effect of a shock
• Spurious Regression
•Asymptotic Results
• Testing for unit roots
• Problems of testing for unit roots
• Structural Breaks
Some plots: InflationSome plots: Inflation
Some Plots: Production
Some Plots: Dutch stock market indexSome Plots: Dutch stock market index
How do we model Growth?How do we model Growth?
Most of the macro-economic variables: GNP, Consumption, Investment, ...etc show a growing pattern through time. This pattern is impossible to be captured with our stationary ARMA models:
)(ˆlim)|()(ˆ
)()(
)(
0
22
lZZElZ
ZVarZE
aLZ
nl
nlnn
iiatt
tt
tZ
tt
How do we describe trends as the following?
How do we model Growth? (cont)How do we model Growth? (cont)
Two options:
• An ARMA model with a deterministic trend component
(TS=Trend Stationary)
• A Unit Root process with a drift term (DS=difference stationary)
t
tcZEaLtcZ ttt )(
ttt aZZ 1
1. Deterministic time trend
2. Stochastic trend. Unit root processes
tcZEaLtcZ ttt )(
ttt aZZ 1
21
0
1
0
21
0
2
1
0
121
)(
))()()((),cov(
)(var)(
)(
.....2
:onsubstituti backward with; :conditions initial
a
Nt
jjt
Nt
jjt
tttt
a
Nt
jjttt
Nt
jjtt
tttttt
N
NtaaE
ANtZANtZEZZ
NtaNtA
aNtAZ
aaZaZZ
AZNt
Random walk with drift
1: tocompared large for
)()(
)(22
2
Nt
Nt
Nt
NtNt
Nt
aa
a
tt
How can we relate processes 1. and 2. ?
ttt
tt
aZZ
aLtcZ
1.2
)(.1
Consider the process:
)1)....(1)(1()......1(
:roots in polynomial AR thedecomposed
)....1()......1(
),(where.3
112
111
11
LrLrLrLL
aLLuLL
qpARMAuutcZ
pp
p
tq
qtp
p
ttt
)1(Z2.) (model)()1()1(
)1(
difference and 3. modelGet
)()1)....(1(
)......1()1(
then321and1 Suppose
1.) (model)(
1 then,stationary is If
)()1)....(1)(1(
)......1(
t
11
112
1
111
1
112
11
1
IaLuLZL
utcZ
utcZ
aLaLrLr
LLuL
...p ,irr
aLtcZ
iru
aLaLrLrLr
LLu
ttt
tt
tt
tt
p
t
i
tt
it
tt
p
t
Differencing this type of non-stationary process makes it stationary
Warning
tt
tt
aLLZL
aLtcZ
)()1()1(
)(
Non-invertible MA
)2()()1(
...311 that Suppose
2
112
11
IZaLZL
pirrr
ttt
i
Two differences are needed to achieved stationarity
* In general, ),,( qdpARIMAZt
d-differences are necessary for stationarity
tqtd
p aLZLL )()1)((
ForecastsForecasts
Deterministic time trend:
t
tal
alnln
lnlnnlnln
nlnln
lnln
tt
Z
aLMSE
lZZEMSE
aaaale
aalnclZ
aLlncZ
aLtcZ
ofpart stationary
)(var....)1(lim
)...1())(ˆ(
....)(
....)()(ˆ
)()(
)(
222
21
221
22
21
2
112211
11
Forecasts (cont)Forecasts (cont)Unit Root:
........
....)|(
.........
......)|(
)|(......)|()|()|(
......
).....()()(
...)(ˆ
)()1(
121
11
121
1111
11
11
1211
11
nl
nllnnln
nnn
nlnlnlnlnln
nnnnlnnlnnln
nnlnlnln
nnnlnlnlnlnln
nlnln
t
Z
t
a
aZlZE
Zaa
aaaaZE
ZZEZEZEZE
ZZZZZ
ZZZZZZZZ
aalZ
aLZL
t
Forecasts: ExamplesForecasts: Examples
nnnln
ttt
nnln
tt
aZlZE
aaZL
ZlZE
aZL
)|(
0...;)1(.2
)|(
0....)1(.1
3211
21
Forecast error
2
2221
21
2
112122111
1122111111
11
)(ˆlim
.....)1()1(1)(ˆ
)...1(......)1()1(
.....)....()....(
)())1(ˆ(....))1(ˆ())(ˆ(
)(ˆ)(
lZZE
lZZE
aaaa
aaaaaaa
ZZZZlZZlZZ
lZZle
nlnl
anln
nllnlnln
nnllnlnnllnln
nnnnnlnnln
nlnn
(l elements in sum )
Forecasts: Examples (cont)Forecasts: Examples (cont)Example
ARIMA(0,1,1)
22
221
21
2
11
)1)(1(1
)1(....)1(1)(ˆ
)1(
a
anln
ttt
l
lZZE
aaZL
The Effect of a ShockThe Effect of a Shock t ah t Z
Transitory shock:
Permanent shock: 0tahtZ
h
0tahtZ
h
Examples:
(1)
h as 0h
tahtZ
....tah...2hta2
1htahtahtZ
...2ta21tatatZ
1|| ta1tZtZ
The effect of a shock (cont)The effect of a shock (cont)
0
0,2
0,4
0,6
0,8
1
1,2
T-1 TT+1
T+2T+3
T+4T+5
T+6T+7
T+8T+9
T+10T+11
T+12
e
0
0,2
0,4
0,6
0,8
1
1,2
T-1 TT+1
T+2T+3
T+4T+5
T+6T+7
T+8T+9
T+10T+11
T+12
y
The unit impulse response function for the AR(1) process yt = 0.8 yt-1 + et
The effect of a shock (cont)The effect of a shock (cont)
The Effect of a Shock (cont)The Effect of a Shock (cont)
t ah t Z
(2)
h as 0h1tahtZ
....ta...2hta1htahtahtZ...2ta1tatatZ
| ta1tZtZ
(3)
0)1(tahtZ
hta)L(~
L1hta
)1(htZ
ta)L(~
L1ta
)1(tZ
ta)L(~
)L1(ta)1(ta)L(tZ)L1(
The Effect of a Shock (cont)The Effect of a Shock (cont)
t ah t Z
Q1: Calculate the effect of a perturbation on at in the following TS model:
ta)L(tu wheretutctZ
Spurious RegresionsSpurious Regresions
Consider two independent random walks:
0,01
1
kttkttstttt
ttt
vEvuEustvEuvxx
uyy
By construction, there is no relation between x and y.
Consider the regression
ttt xy e
Q2: Which values do you expect the estimates of and will take?What about the R2? You will find the right answer in two more lectures.
Some Asymptotic ResultsSome Asymptotic Results
Consider the stationary case
)21(T
12
0T
12
1
t
1t2yˆvarˆE
t
1t2y
t
1tyty
ˆ:OLS
e
e
11 e ttt yy
Asymptotically (CLT) from the previous lecture:
)1,0()ˆ( 2 NT
Some asymptotic results (cont)Some asymptotic results (cont)
When the asymptotic result is not valid to perform inferences because
1
ondistributi ddegeneratea has ˆ
0)1ˆ(0)ˆvar(
T
What to do when ?1
ondistributiFuller -Dickey: ondistributi standard-nona has
e
e
t
1t2y
t
t1ty
1ˆ:1under ;
t
1t2y
t
t1ty
ˆ
tt
ttt
yT
yT
T2
12
1
1
1
)1ˆ(e
t
ttyT
?1
1e)1,0(),0(
0.....
2
011
Nt
ytNy
yy
tt
ttt
eee
12
11
1
2
1
2
11
2
1
2
11
2
1)(
2
1)(
2
1
)(2
1
2)(
2112
(by LLN)
2
2
2
12
221
220
221
221
21
221
211
221
2
221
e
e
e
ee
eee
ee
eee
D
ttt
tt
T
ttt
ttT
ttt
ttT
tt
t
t
tt
ttt
ttttt
ttttttt
yT
TT
yy
T
Ty
Ty
T
yyyyy
yyy
yyyy
?1
12
2
t
tyT
zero.not islimit in the variance thebecause variable
random a toeconvergencget that wedoingBy . 2Tby sum thisdivide tohave weso
2T Order2)1t2t)(1t(t)3/1()
t
1t2y(Var
t t2
T)1T(2)1t(21t
2Ey
t
1t2yE
2)1t()1t2yvar()2)1t(,0(N1ty
In summary, the statistic
t
t
ttt
yT
yT
T1
22
1
1
1
)1ˆ(e
has a non-standard distribution known as the Dickey-Fuller distributionthat is dominated by the chi-squared behavior of the numerator.
We can construct a pseudo-t test as
1
1ˆ
ˆ1ˆ
2
2
21
22ˆ
ˆ
T
sy
st tt
tt
e
ee
This pseudo-t test does not have the usual limiting Gaussian distribution
-1.95
5%
%5)65.1(
%595.1ˆ
1ˆ
ZP
P
Dickey-Fuller distribution
Normal distributionReject the unit root too often
Some Asymptotic Results (cont)Some Asymptotic Results (cont)
The asymptotic distributions can be written in a more compact way
e
1
0
2W
)12)1(W)(2/1(1
0
2W
1
0
WdW
t
21ty
2T
1t
t1tyT
1
)1ˆ(T
2)dr2)r(W(
)12)1(W)(2/1(
2)dr2)r(W(
WdW
ˆˆ1ˆ
t
Some Asymptotic Results (cont)Some Asymptotic Results (cont)
where W(r) is a Brownian Motion (see the applets from this lecture).A Brownian Motion is defined by the following properties:•W(0)=0•W(t) has stationary and independent increments and for all t and ssuch that for t>s we have W(t)-W(s) is N(0, (t-s))•W(t) is N(0,t) for every t•W(t) is sample path continuous.
Testing for Unit Roots (DF test)Testing for Unit Roots (DF test)
Problem: Tests for unit root are conditional on the presence of deterministic regressors; and tests for the presence of deterministic regressors are
conditional on the presence of unit root.
Reparametrization of the model
ity)(stationar0:
root)(unit 0:
)1(
1
0
11
111
eee
H
H
yyy
yyyy
ttttt
ttttt
Dickey-Fuller consider three models with deterministic regressors:
ttt
ttt
ttt
yty
yy
yy
ee
e
11
11
11
)3(
)2(
)1(
Pseudo-t statistics for :
(3) model
(2) model
(1) model
0:
0:
1
0
H
H
Pseudo-F statistic)/(
/)(
kTRSS
rRSSRSSF
U
UR
for :
(3) model 0:
(3) model 0:
(2) model 0:
30
20
10
FH
FH
FH
ttt
ttt
ttt
yty
yy
yy
ee
e
11
11
11
)3(
)2(
)1(
05.0)41.3t(P
05.0)86.2(P
05.0)95.1(P
05.0)64.1e(P
Testing for Unit Roots: A procedure for the DF testTesting for Unit Roots: A procedure for the DF test
1. Start with a general model
2.) to(go
root)unit (no
0:
0:
1
0
accept
reject
H
H
2. Test for trend
3.) to(go
Normal usemay 0)(0: 30 accept
rejectFH
3. Estimate
4.) to(go
root)unit (no
0:
0:
1
0
1
accept
reject
H
H
yy ttt
e
4. Test for drift
5.) to(go
Normal usemay 0)(0: 10 accept
rejectFH
5. Estimate
(stop)
root)unit (no
0:
0:
1
0
1
accept
reject
H
H
yy ttt
e
ttt yty e 1
Augmented Dickey-Fuller testAugmented Dickey-Fuller test
The previous results are only valid when the error term et is iid.
If this is not the case, for instance if et follows a linear process:
then it can be proved that it can be proved that we can re-write the DF regression
by adding lags of the increments of yt-1 until the error term becomes
iid. This solves the problem. The strategy is the same and the A.D
are the same as before.
ta)L(t e
ta
p
1i
ityi1tytty
Q3: Think on two different ways to choose the right order “p”.
Q4: Discuss briefly two reasons why we deal with the null of unit root instead of the null of stationarity.
Q5: I am sure you have read and heard many many many times that unit root tests do not have power. What about other tests? Any comments?
Structural Breaks versus unit RootsStructural Breaks versus unit Roots
See notes and discussion in class.
See Part IV of “Unit Roots, Cointegration and Unit Roots, Cointegration and Structural ChangeStructural Change” by Maddala and Kim. Cambridge University Press 1998.
Applications in FinanceApplications in Finance
Asset price levels (logs) US marketAsset price levels (logs) US market
US stock market returns US stock market returns 1980-20001980-2000
Looking at information in past prices: Looking at information in past prices: returns and past returnsreturns and past returns
Histogarm US returnsHistogarm US returns
Interest rate USAInterest rate USA
Histogram of Interest rates in the USAHistogram of Interest rates in the USA
Scatter plot: unveiling information of past interest Scatter plot: unveiling information of past interest rate levelrate level
ACF and PAC for interest ratesACF and PAC for interest rates
AC and PAC of interest rate changes
Regression for the DF testRegression for the DF test
Regression for the DF testRegression for the DF test
ADF test for interest ratesADF test for interest rates
Model fitModel fit
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