1. Unit Root Tests: Methods and Problems
Roger Perman
Applied Econometrics Lecture 12

2. How do you find out if a series is stationary or not?
Unit Root Tests
How do you find out if a series is stationary or not?

3. Y = b0 + Y + e ® I (1) D Y = Y - Y = b0 + e ® I (0)
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) Y = b0 + Y + e I
(1) t t - 1 t D Y = Y - Y = b0 + e I
(0) t t t - 1 t

4. Informal Procedures to identify non-stationary processes
(1) Eye ball the data (a) Constant mean?
(b) Constant variance?

5. Informal Procedures to identify non-stationary processes
(2) Diagnostic test - Correlogram
Correlation between 1980 and k.
For stationary process correlogram dies out rapidly.
Series has no memory is not related to 1985.

6. Informal Procedures to identify non-stationary processes
(2) Diagnostic test - Correlogram
For a random walk the correlogram does not die out.
High autocorrelation for large values of k

7. Statistical Tests for stationarity: Simple t-test
Set up AR(1) process with drift (b0)
Yt = b0 + b1Yt-1 + t t ~ iid(0,σ2) (1)
Simple approach is to estimate eqn (1) using OLS and examine estimated b1
Use a t-test with null Ho: b1 = 1 (non-stationary)
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

8. Statistical Tests for stationarity: Simple t-test
Simple t-test based on AR(1) process with drift (b0)
Yt = b0 + b1Yt-1 + t t ~ iid(0,σ2) (1)
Problem with simple t-test approach
(1) lagged dependent variables => b1 biased downwards in small samples (i.e. dynamic bias)
(2) When b1 =1, we have non-stationary process and standard regression analysis is invalid
(i.e. non-standard distribution)

9. Dickey Fuller (DF) approach to non- stationarity testing
Dickey and Fuller (1979) suggest we subtract Yt-1 from both sides of eqn. (1)
Yt - Yt-1 = b0 + b1Yt-1 - Yt-1 +  t  t ~ iid(0,σ2)
ΔYt = b0 + Yt-1 +  t  = b1 – (2)
Use a t-test with: null Ho:  = 0 (non-stationary or Unit Root)
against alternative Ha:  < 0 (stationary).
- Large negative test statistics reject non- stationarity
- This is known as unit root test since in eqn. (1) Ho: b1 =1.

10. The difference between the three regressions concerns the
Variants of DF test
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

11. 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

12. 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
non-standard tables

13. Example
Sample size of n = 25 at 5% level of significance for eqn. (2)
τμ-critical value = t-test critical value = -1.65
Δpt-1 = pt-1
(-1.05) (-1.49)
 = τμ = > -3.00
hence cannot reject H0 and so unit root.

14. Incorporating time trends in DF test for unit root
Some time series clearly display an upward or downward trend (non-stationary mean).
Should therefore incorporate trend in the regression used for the DF test.
ΔYt = b0 +  Yt-1 + b2 trend +  t (4)
It may be the case that Yt will be stationary around a trend. Although if a trend is not included series is non-stationary.

15. ττ ΔYt = b0 + βYt-1 + b2 trend +  t
Different DF tests – Summary t-type test
ττ ΔYt = b0 + βYt-1 + b2 trend +  t
(a) Ho: β = 0 Ha: β < 0
τμ ΔYt = b0 + βYt-1 +  t
(b) Ho: β = 0 Ha: β < 0
τ ΔYt = βYt-1 +  t
(c) Ho: β = 0 Ha: β < 0
Critical values from Fuller (1976)

16. Φ3 ΔYt = b0 +  Yt-1 + b2 trend + t
Different DF tests – Summary F-type test
Φ3 ΔYt = b0 +  Yt-1 + b2 trend + t
(a) Ho: β = b2 = 0 Ha:   0 and/or b2 0
Φ1 ΔYt = b0 +  Yt-1 + t
(b) Ho:  = b0 = 0 Ha:   0 and/or b0  0
Critical values from Dickey and Fuller (1981)

17. Summary of Dickey-Fuller Tests
(Critical values for n = 100)

18. 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 +  Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 +  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 F-test for significant variables
Use Information Criteria. Trade-off parsimony vs. residual variance.

19. 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)

23. Choose Between Alternative Models - The Model-Progress Results
Both the F-Test and the Schwarz Information Criteria indicates
that MODEL 4 is the one to be preferred

24. To do this perform an F-Test and use the statistic
Unit Root Testing
After having estimated, according to the previous analysis, the following equation
the relevant hypotheses to examine are (in this particular case) H : b , b , b = b , , 2 v H : b , b , b b , , 1 2
To do this perform an F-Test and use the statistic

25. PcGive output of test result:
Wald test for linear restrictions: Subset
LinRes F( 2,493) = [0.0066] **
Be careful here. The value 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
non-standard tables

26. The t-stat cannot reject the null hypothesis of Unit Root while the F-stat
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

27. Reject Accept Reject Accept Reject Accept Reject Accept
Look at the Series – Is there a Trend?
Yes No
Estimate
Estimate
Use
Use
to test
to test H : b , b = , H : b , b , b = b , , 2 v v H : b , b , b 1 H : b , b , b , , 1 2
Reject
Accept
Reject
Accept
test =0 using the
t-stat. from step 1
using
Pure Random Walk
test =0 using the
t-stat. from step 1
using
Reject
Accept
Reject
Accept
No Unit Root
Unit Root +Trend
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

28. Alternative statistical test for stationarity
One further approach is the Sargan and Bhargava (1983) test which uses the Durbin-Watson 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.

29. Testing Strategy for Unit Roots
Three main aspects of Unit root testing
- Deterministic components (constant, time trend).
- ADF Augmented Dickey Fuller test - lag length
- use F-test or Schwarz Information Criteria
- In what sequence should we test?
- Phi and tau tests

30. Testing Strategy for Unit Roots
Formal Strategy
(A) Set up Model
(1) Use informal tests – eye ball data and correlogram
(2) Incorporate Time trend if data is upwards trending
(3) Specification of ADF test
– how many lags should we incorporate to avoid serial correlation?

31. Example- Real GDP (2000 Prices) Seasonally Adjusted
(1) Plot Time Series - Non-Stationary
(i.e. time varying mean and correlogram non-zero)
GDP
Time r k

32. (1) Plot First Difference of Time Series - Stationary
Unit Root Testing
(1) Plot First Difference of Time Series - Stationary
(i.e. constant mean and correlogram zero)
Time r k

33. Unit Root Testing
(2) Incorporate Linear Trend since data is trending upwards

34. (3) Determine Lag length of ADF test
Unit Root Testing
(3) Determine Lag length of ADF test
Estimate general model and test for serial correlation
EQ ( 1) ΔYt = b0 +b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 + t
EQ( 1) Modelling DY by OLS (using Lab2.in7)
The estimation sample is: 1956 (2) to 2003 (3) n = 190
Coefficient Std.Error t-value t-prob Part.R^2
Constant
Trend Y_
DY_
DY_
DY_
DY_
AR 1-5 test: F(5,178) = [0.1308]
Test accepts null of no serial correlation.
Nevertheless we use F-test and Schwarz Criteria to check model.

35. (3) Determine Lag length of ADF test
Unit Root Testing
(3) Determine Lag length of ADF test
Model
EQ ( 1) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 + t
EQ ( 2) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t
EQ ( 3) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + t
EQ ( 4) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + t
EQ ( 5) ΔYt = b0+b2 trend+ Yt-1 + t
Use both the F-test and the Schwarz information Criteria (SC).
Reduce number of lags where F-test accepts.
Choose equation where SC is the lowest
i.e. minimise residual variance and number of estimated parameters.

36. (3) Determine Lag length of ADF test
Unit Root Testing
(3) Determine Lag length of ADF test
Progress to date
Model T p log-likelihood Schwarz Criteria
EQ( 1) OLS
EQ( 2) OLS
EQ( 3) OLS
EQ( 4) OLS
EQ( 5) OLS
Tests of model reduction
EQ( 1) --> EQ( 2): F(1,183) = [0.5259] Accept model reduction
EQ( 1) --> EQ( 3): F(2,183) = [0.0357]* Reject model reduction
EQ( 1) --> EQ( 4): F(3,183) = [0.0173]*
EQ( 1) --> EQ( 5): F(4,183) = [0.0374]*
Some conflict in results. F-tests 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).

37. (B) Conduct Formal Tests
Unit Root Testing
(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 t-value t-prob Part.R^2
Constant
Trend Y_
DY_
DY_
DY_
AR 1-5 test: F(5,179) = [0.6357]
Main issue is serial correlation assumption for this test.
Can we accept the null hypothesis of no serial correlation? Yes!

38. Unit Root Testing
Apply F-type test - Include time trend in specification
Φ3: ΔYt = b0 + b2 trend + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t (a) Ho:  = b2 = 0 Ha: β  0 and/or b2 0
PcGive Output : Test/Exclusion Restrictions.
Test for excluding: [0] = Trend [1] = Y_1
F(2,184) = < 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)
C.V. at 5%

39. Φ1: ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t
Unit Root Testing
Apply F-type test - Exclude time trend from specification
Φ1: ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t
(b) Ho:  = b0 = 0 Ha:   0 and/or b0  0
PcGive Output : Test/Exclusion Restrictions.
Test for excluding: [0] = Constant[1] = Y_1
F(2,185) = > 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)
C.V. at 5%

40. τμ ΔYt = b0 +  Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t
Unit Root Testing
Apply t-type test (τμ)
τμ ΔYt = b0 +  Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t
(b) Ho:  = 0 Ha:  < 0
τμ = > = 5% C.V.
Hence accept null of unit root.
N.B. Critical Values (C.V.) from Fuller (1976) for τμ
Sample Size (n)
C.V. at 5%

41. Unit Root Testing
EQ(2a) Modelling DY by OLS (using Lab2.in7)
The estimation sample is: 1956 (2) to 2003 (3)
Coefficient Std.Error t-value t-prob Part.R^2
Constant Y_
DY_
DY_
DY_
AR 1-5 test: F(5,180) = [0.7725]
τμ = > (5% C.V.) hence we can not reject the null of unit root.

42. Problems With Unit Root Tests

43. 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
For example
1929 Depression
Oil Shocks
Technological Change
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

44. The Unit Root Hypothesis is not rejected.
Unit Root Test For Y
Dickey-Fuller test for Y; DY on
Variable Coefficient Std.Error t-value Y_
\sigma = DW = DW(Y) = DF(Y) =
Critical values used in DF test: 5%= %=-2.587
RSS = for 1 variables and 98 observations
Information Criteria:
SC = HQ = FPE= AIC =
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

45. Consider The Following VariableY = . 5 Y + D + e t t - 1 t
for t = . . 49 D = t 2
for t = 50 . .
100

46. The Power of a test is the probability of rejecting a false
Problem Number Two : Low Power
The Power of a test is the probability of rejecting a false
Null Hypothesis -
Unit Root Tests
Low Power to Distinguish Between Unit and near Unit Root
Low Power to Distinguish Between Trend and Drift
Y is a unit root series;
Z is a near-unit root series

47. Test result is based on the standard error of 
Is  = 0 in ΔYt = b0 + Yt-1 + t
Test result is based on the standard error of 
- Measure of how accurate is our estimated coefficient
- with increasing observations we become more certain.
In this case, power of the test is the ability to reject the null of non-stationarity when it is false (equivalently, the ability to accept alternative hypothesis of stationarity).
Low power implies a series may be stationary but Dickey-Fuller test suggests unit root.
Low power is especially a problem when series is stationary but close to being unit root.
One solution to low power is to increase the number of observations by increasing the span of data. However, there may be differences in economic structure or policy which should be modelled differently. Alternative solution to low power is a number of joint ADF tests.
- Take information from a number of countries.
- And pool coefficients. (i.e. combine information).