Unit Root Tests: Methods and Problems. Roger Perman Applied Econometrics Lecture 12. Unit Root Tests. How do you find out if a series is stationary or not?. Order of Integration of a Series. A series which is stationary after being differenced once
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How do you find out if a series is stationary or not?
Order of Integration of a Series
A series which is stationary after being differenced once
is said to be integrated of order 1 and is denoted by I(1).
In general a series which is stationary after being
differenced d times is said to be integrated of order d,
denoted I(d). A series, which is stationary without
differencing, is said to be I(0)
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Informal Procedures to identify nonstationary processes
Informal Procedures to identify nonstationary processes
Informal Procedures to identify nonstationary processes
Statistical Tests for stationarity: Simple ttest
Set up AR(1) process with drift (b0)
Yt = b0 + b1Yt1 + t t ~ iid(0,σ2) (1)
Simple approach is to estimate eqn (1) using OLS and examine estimated b1
Use a ttest with null Ho: b1 = 1 (nonstationary)
against alternative Ha: b1 < 1 (stationary).
Test Statistic: TS = (b1 – 1) / (Std. Err.(b1))
reject null hypothesis when test statistic is large negative
 5% critical value is 1.65
Statistical Tests for stationarity: Simple ttest
Dickey Fuller (DF) approach to non stationarity testing
Three different regression can be used to test the presence of a unit root
The difference between the three regressions concerns the
presence of deterministic elements b0 and b2t.
1 – For testing if Y is a pure Random Walk
2 – For testing if Y is a Random Walk with Drift
3 – For testing if Y is a Random walk with Drift and Deterministic Trend
The simplest model (appropriate only if you think there are no
other terms present in the ‘true’ regression model)
Use the t statistic and compare it with the the table of critical values computed by Dickey and Fuller. If your t value is outside the confidence interval, the null hypothesis of unit root is rejected
Statistic
A more general model (allowing for ‘drift’)
Statistic  Use the F statistic to check if = b0 = 0 using the
non standard tables
Statistic  use the t statistic to check if =0 , again using
nonstandard tables
Sample size of n = 25 at 5% level of significance for eqn. (2)
τμcritical value = 3.00 ttest critical value = 1.65
Δpt1 = 0.007  0.190pt1
(1.05) (1.49)
= 0.190 τμ = 1.49 > 3.00
hence cannot reject H0 and so unit root.
Incorporating time trends in DF test for unit root
Different DF tests – Summaryttype test
ττΔYt = b0 + βYt1 + b2trend+ t
(a) Ho:β = 0Ha:β < 0
τμΔYt = b0 + βYt1+ t
(b) Ho: β = 0 Ha: β < 0
τΔYt = βYt1 + t
(c)Ho: β = 0 Ha: β < 0
Critical values from Fuller (1976)
Different DF tests – Summary Ftype test
Φ3ΔYt = b0 + Yt1 + b2 trend+ t
(a) Ho:β = b2=0Ha: 0 and/or b20
Φ1 ΔYt = b0 + Yt1+ t
(b) Ho:= b0=0Ha: 0 and/or b0 0
Critical values from Dickey and Fuller (1981)
Summary of DickeyFuller Tests
(Critical values for n = 100)
Augmented Dickey Fuller (ADF) test for unit root
Dickey Fuller tests assume that the residuals t in the DF regression are non autocorrelated.
Solution: incorporate lagged dependent variables.
For quarterly data add up to four lags.
ΔYt = b0 + Yt1 + θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + θ4ΔYt4 + t (3)
Problem arises of differentiating between models.
Use a general to specific approach to eliminate insignificant variables
Check final parsimonious model for autocorrelation.
Check Ftest for significant variables
Use Information Criteria. Tradeoff parsimony vs. residual variance.
Consider The Following Series and Its Correlogram
This variable Y is clearly trended and you have to determine if this trend is stochastic
or deterministic. After having created the difference variable Y estimate the
following model, with as many lags of Y as you think appropriate.
(in the example I choose 4 lags of the variable Y)
Choose Between Alternative Models  The ModelProgress Results
Both the FTest and the Schwarz Information Criteria indicates
that MODEL 4 is the one to be preferred
After having estimated, according to the previous analysis, the following equation
the relevant hypotheses to examine are (in this particular case)
b
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To do this perform an FTest and use the statistic
Wald test for linear restrictions: Subset
LinRes F( 2,493) = 5.0781 [0.0066] **
Be careful here. The value 5.0781 is not significant at the 5% critical value, although PcGive marks it as significant (it is using the conventional F distribution).
Therefore we cannot reject the null hypothesis, and so infer that we do not have a deterministic time trend in the equation. Hence, we can continue the analysis using
and then use
statistic  use the F statistic to check if = b0 = 0 using the
non standard tables
statistic  use the t statistic to check if =0 , again using
nonstandard tables
The tstat cannot reject the null hypothesis of Unit Root while the Fstat
rejects the null hypothesis that the drift is equal to zero. Therefore we can
conclude that the model most likely to describe the true DGP is
Look at the Series – Is there a Trend?
Yes
No
Estimate
Estimate
Use
to test
Use
to test
b
=
H
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b
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b
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b
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Accept
Reject
Reject
Accept
Pure Random Walk
test =0 using the
tstat. from step 1
using
test =0 using the
tstat. from step 1
using
Reject
Accept
Reject
Accept
Unit Root +Trend
No Unit Root
Stable Series,
use normal test
to check the drift
Random
Walk + Drift
Use
Normal Test procedure
to determine the
presence of
Time trend or Drift
To determine if there
is a drift as well
Alternative statistical test for stationarity
One further approach is the Sargan and Bhargava (1983) test which uses the DurbinWatson statistic.
If Yt is regressed on a constant alone, we then examine the residuals for serial correlation.
Serial correlation in the residuals (long memory) will fail the DW test and result in a low value for this test.
This test has not proven so popular.
Testing Strategy for Unit Roots
Testing Strategy for Unit Roots
Example Real GDP (2000 Prices) Seasonally Adjusted
GDP
Time
r
k
Time
r
k
(2) Incorporate Linear Trend since data is trending upwards
(3) Determine Lag length of ADF test
Model
EQ ( 1) ΔYt = b0+b2 trend+ Yt1 + θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + θ4ΔYt4 + t
EQ ( 2) ΔYt = b0+b2 trend+ Yt1 + θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + t
EQ ( 3) ΔYt = b0+b2 trend+ Yt1 + θ1ΔYt1 + θ2ΔYt2 + t
EQ ( 4) ΔYt = b0+b2 trend+ Yt1 + θ1ΔYt1 + t
EQ ( 5) ΔYt = b0+b2 trend+ Yt1 + t
Use both the Ftest and the Schwarz information Criteria (SC).
Reduce number of lags where Ftest accepts.
Choose equation where SC is the lowest
i.e. minimise residual variance and number of estimated parameters.
(3) Determine Lag length of ADF test
Progress to date
Model T p loglikelihood Schwarz Criteria
EQ( 1) 190 7 OLS 156.91128 1.8450
EQ( 2) 190 6 OLS 157.12068 1.8196
EQ( 3) 190 5 OLS 160.37203 1.8262
EQ( 4) 190 4 OLS 162.16872 1.8175
EQ( 5) 190 3 OLS 162.17130 1.7899
Tests of model reduction
EQ( 1) > EQ( 2): F(1,183) = 0.40382 [0.5259]Accept model reduction
EQ( 1) > EQ( 3): F(2,183) = 3.3947 [0.0357]*Reject model reduction
EQ( 1) > EQ( 4): F(3,183) = 3.4710 [0.0173]*
EQ( 1) > EQ( 5): F(4,183) = 2.6046 [0.0374]*
Some conflict in results. Ftests suggest equation (2) is preferred to equation (1) and equation (3) is not preferred to equation (2).
Additionally, the relative performance of these three equations is confirmed by information criteria.
Therefore adopt equation (2).
(B) Conduct Formal Tests
EQ( 2) Modelling DY by OLS (using Lab2.in7)
The estimation sample is: 1956 (2) to 2003 (3)
Coefficient Std.Error tvalue tprob Part.R^2
Constant 0.505231 0.3552 1.42 0.157 0.0109
Trend 0.00655304 0.004772 1.37 0.171 0.0101
Y_1 0.0141798 0.01307 1.08 0.279 0.0064
DY_1 0.0119522 0.07297 0.164 0.870 0.0001
DY_2 0.142437 0.07241 1.97 0.051 0.0206
DY_3 0.185573 0.07332 2.53 0.012 0.0336
AR 15 test: F(5,179) = 0.68451 [0.6357]
Main issue is serial correlation assumption for this test.
Can we accept the null hypothesis of no serial correlation? Yes!
Apply Ftype test Include time trend in specification
Φ3: ΔYt = b0 + b2 trend + Yt1 + θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + t(a) Ho:= b2=0Ha:β 0 and/or b20
PcGive Output : Test/Exclusion Restrictions.
Test for excluding: [0] = Trend [1] = Y_1
F(2,184) = 2.29 < 6.39 = 5% C.V. (by interpolation).
Hence accept joint null hypothesis of unit root and no time trend
(next test whether drift term is required).
NB Critical Values (C.V.) from Dickey and Fuller (1981) for Φ3
Sample Size (n) 25 50 100 250 500
C.V. at 5% 7.24 6.73 6.49 6.34 6.30
Apply Ftype test Exclude time trend from specification
Φ1: ΔYt = b0 + Yt1+ θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + t
(b) Ho: = b0=0Ha: 0 and/or b0 0
PcGive Output : Test/Exclusion Restrictions.
Test for excluding: [0] = Constant[1] = Y_1
F(2,185) = 10.27 > 4.65 = 5% C.V.
Hence reject joint null hypothesis of unit root and no drift.
NB Critical Values (C.V.) from Dickey and Fuller (1981) for Φ1
Sample Size (n) 25 50 100 250 500
C.V. at 5% 5.18 4.86 4.71 4.63 4.61
Apply ttype test(τμ)
τμΔYt = b0 + Yt1+ θ1ΔYt1 + θ2ΔYt2 + θ3ΔYt3 + t
(b) Ho: = 0 Ha: < 0
τμ = 1.64 > 2.88 = 5% C.V.
Hence accept null of unit root.
N.B. Critical Values (C.V.) from Fuller (1976) for τμ
Sample Size (n) 25 50 100 250 500
C.V. at 5% 3.00 2.93 2.89 2.88 2.87
EQ(2a) Modelling DY by OLS (using Lab2.in7)
The estimation sample is: 1956 (2) to 2003 (3)
Coefficient Std.Error tvalue tprob Part.R^2
Constant 0.0535255 0.1343 0.399 0.691 0.0009
Y_1 0.00352407 0.002150 1.64 0.103 0.0143
DY_1 0.0218516 0.07279 0.300 0.764 0.0005
DY_2 0.131601 0.07215 1.82 0.070 0.0177
DY_3 0.172115 0.07283 2.36 0.019 0.0293
AR 15 test: F(5,180) = 0.50464 [0.7725]
τμ= 1.64 > 2.88 (5% C.V.) hence we can not reject the null of unit root.
Problem Number 1: Structural Breaks
Perron (1989)  He argues that most macroeconomic variables
are not unit root processes. They are Trend Stationary with
Structural Breaks
All these events have changed the mean of a process like GDP
If you do not recognize the structural break, you’ll find unit root
where there is not
With Structural Change All Unit Root Tests Are Biased
Towards the Non Rejection of a Unit Root
DickeyFuller test for Y; DY on
Variable Coefficient Std.Error tvalue
Y_1 0.065307 0.037352 1.748
\sigma = 1.16902 DW = 2.35 DW(Y) = 0.2242 DF(Y) = 1.748
Critical values used in DF test: 5%=1.943 1%=2.587
RSS = 132.5618324 for 1 variables and 98 observations
Information Criteria:
SC = 0.348867 HQ = 0.333159 FPE=1.38056 AIC = 0.32249
The Unit Root Hypothesis is not rejected.
Perron proposed a method to overcome this problem  But you
need to know when the structural break happened
Problem Number Two : Low Power
The Power of a test is the probability of rejecting a false
Null Hypothesis 
Unit Root Tests
Y is a unit root series;
Z is a nearunit root series
Is = 0 in ΔYt = b0 + Yt1 + t