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

1

Unit Root Tests: Methods and Problems

Roger Perman

Applied Econometrics Lecture 12

2

Unit Root Tests

How do you find out if a series is stationary or not?

3

Y b0 Y I

Y Y Y b0 I

t t t

t t t t

1

1

(1)

(0)

Order of Integration of a Series

A series which is stationary after being differenced onceis 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)

4

Informal Procedures to identify non-stationary processes

(1) Eye ball the data (a) Constant mean?

(b) Constant variance?

0 50 100 150 200 250 300 350 400 450 500

-200

-150

-100

-50

0

50

100

150

200var

0 50 100 150 200 250 300 350 400 450 500

0

2

4

6

8

10

12RW2

5

Informal Procedures to identify non-stationary processes

(2) Diagnostic test - Correlogram Correlation between 1980 and 1980 + k.

For stationary process correlogram dies out rapidly.Series has no memory. 1980 is not related to 1985.

0 50 100 150 200 250 300 350 400 450 500

-0.25

0.00

0.25

0.50whitenoise

0 5 10

-0.5

0.0

0.5

1.0ACF-whitenoise

6

Informal Procedures to identify non-stationary processes

(2) Diagnostic test - CorrelogramFor a random walk the correlogram does not die out.

High autocorrelation for large values of k

0 50 100 150 200 250 300 350 400 450 500

0.0

2.5

5.0

7.5

10.0

12.5randomwalk

0 5 10

0.25

0.50

0.75

1.00ACF-randomwalk

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) processwith 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 –1 (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

Variants of DF test

Three different regression can be used to test the presence of a unit root

Y Y

Y b Y

Y b Y b t

t t

t t

t t

1

0 1

0 1 2

The difference between the three regressions concerns the presence of deterministic elements b0 and b2t.

1 – For testing if Y is a pure Random Walk2 – For testing if Y is a Random Walk with Drift3 – For testing if Y is a Random walk with Drift and Deterministic Trend

11

Y Yt t 1

The simplest model (appropriate only if you think there are noother 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

Y b Yt t 0 1

A more general model (allowing for ‘drift’)

1

Statistic - Use the F statistic to check if = b0 = 0 using thenon 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 = -3.00 t-test critical value = -1.65

Δpt-1 = -0.007 - 0.190pt-1

(-1.05) (-1.49)

= -0.190 τμ = -1.49 > -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

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

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

Model Hypothesis TestStatistic

Critical Values for95% and 99%

Confidence Interval

Y b Y btt t 0 1 2 = 0 -3.45 and -4.04

b0 = 0 given = 0 3.11 and 3.78

b2 = 0 given = 0 2.79 and 3.53

= b2 = 0 36.49 and 8.73

= b0 = b2 = 0 24.88 and 6.50

Y b Yt t 0 1 =0 -2.89 and -3.51

b0 = 0 given = 0 2.54 and 3.22

= b0 = 0 14.71 and 6.70

Y Yt t 1 =0 -1.95 and -2.60

(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 variablesCheck final parsimonious model for autocorrelation.

Check F-test for significant variablesUse Information Criteria. Trade-off parsimony vs. residual variance.

19

Consider The Following Series and Its Correlogram

0 50 100 150 200 250 300 350 400 450 500

100

200

Y

0 5 10 15 20 25 30

.25

.5

.75

1ACF-Y

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)

20

21

22

23

Choose Between Alternative Models - The Model-Progress Results

Model 1:

Model 2:

Model 3:

Model 4:

Model 5:

Y b b t Y Y Y Y Y

Y b b t Y Y Y Y

Y b b t Y Y Y

Y b b t Y Y

Y b b t Y

t t t t t t

t t t t t

t t t t

t t t

t t

0 1 1 1 1 2 2 3 3 4 4

0 2 1 1 1 2 2 3 3

0 2 1 1 1 2 2

0 2 1 1 1

0 2 1

Both the F-Test and the Schwarz Information Criteria indicates that MODEL 4 is the one to be preferred

24

Unit Root Testing

Y b b t Y Yt t t 0 2 1 1 1

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

v

H b b b

0 0 2 0

1 0 2 0

0 0

0 0

: , , , ,

: , , , ,

To do this perform an F-Test and use the statistic

3

25

PcGive output of test result:Wald test for linear restrictions: SubsetLinRes 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

Y b Y Yt t t 0 1 1 1

and then use

1

statistic - use the F statistic to check if = b0 = 0 using thenon 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-statrejects 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

Y b Y Yt t t 0 1 1 1

27

Look at the Series – Is there a Trend?

Yes No

Estimate Y b b t Y Yt t j t j 0 2 1 Y b Y Yt t j t j 0 1

Estimate

H b b bv

H b b b

0 0 2 0

1 0 2 0

0 0

0 0

: , , , ,

: , , , ,

3Use to test Use 1 to testH b

vH b

0 0

1 0

0 0

0 0

: , ,

: , ,

Accept

Reject Reject Accept

test =0 using the t-stat. from step 1

using

Reject

No Unit Root

Normal Test procedure to determine the presence of Time trend or Drift

Accept

Unit Root +Trend

Use 2

To determine if there is a drift as well

Pure Random Walk

test =0 using the t-stat. from step 1

using

Accept

RandomWalk + Drift

Reject

Stable Series,use normal testto check the drift

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

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?

Testing Strategy for Unit Roots

31

Example- Real GDP (2000 Prices) Seasonally Adjusted

(1) Plot Time Series - Non-Stationary (i.e. time varying mean and correlogram non-zero)

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

50

75

100Y

0 5 10

0.25

0.50

0.75

1.00ACF-Y

k

Time

GDP

r

32

Unit Root Testing

(1) Plot First Difference of Time Series - Stationary (i.e. constant mean and correlogram zero)

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

-1

0

1

2

3DY

0 5 10

-0.5

0.0

0.5

1.0ACF-DY

Time

k

r

33

(2) Incorporate Linear Trend since data is trending upwards

Unit Root Testing

34

Unit Root Testing

(3) Determine Lag length of ADF testEstimate 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 0.538887 0.3597 1.50 0.136 0.0121Trend 0.00701814 0.004836 1.45 0.148 0.0114Y_1 -0.0156708 0.01330 -1.18 0.240 0.0075DY_1 -0.0191048 0.07395 -0.258 0.796 0.0004DY_2 0.137352 0.07297 1.88 0.061 0.0190DY_3 0.188071 0.07354 2.56 0.011 0.0345DY_4 0.0474897 0.07473 0.635 0.526 0.0022

AR 1-5 test: F(5,178) = 1.7263 [0.1308] Test accepts null of no serial correlation. Nevertheless we use F-test and Schwarz Criteria to check model.

35

Unit Root Testing

(3) Determine Lag length of ADF test

ModelEQ ( 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 + tEQ ( 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 lowesti.e. minimise residual variance and number of estimated parameters.

36

(3) Determine Lag length of ADF testProgress to dateModel T p log-likelihood Schwarz Criteria EQ( 1) 190 7 OLS -156.91128 1.8450EQ( 2) 190 6 OLS -157.12068 1.8196EQ( 3) 190 5 OLS -160.37203 1.8262EQ( 4) 190 4 OLS -162.16872 1.8175EQ( 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 reductionEQ( 1) --> EQ( 3): F(2,183) = 3.3947 [0.0357]* Reject model reductionEQ( 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. 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).

Unit Root Testing

37

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 0.505231 0.3552 1.42 0.157 0.0109Trend 0.00655304 0.004772 1.37 0.171 0.0101Y_1 -0.0141798 0.01307 -1.08 0.279 0.0064DY_1 -0.0119522 0.07297 -0.164 0.870 0.0001DY_2 0.142437 0.07241 1.97 0.051 0.0206DY_3 0.185573 0.07332 2.53 0.012 0.0336

AR 1-5 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!

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_1F(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 500C.V. at 5% 7.24 6.73 6.49 6.34 6.30

39

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_1F(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 500C.V. at 5% 5.18 4.86 4.71 4.63 4.61

40

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 = 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 500C.V. at 5% -3.00 -2.93 -2.89 -2.88 -2.87

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 0.0535255 0.1343 0.399 0.691 0.0009Y_1 0.00352407 0.002150 1.64 0.103 0.0143DY_1 -0.0218516 0.07279 -0.300 0.764 0.0005DY_2 0.131601 0.07215 1.82 0.070 0.0177DY_3 0.172115 0.07283 2.36 0.019 0.0293

AR 1-5 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.

42

Problems With Unit Root Tests

43

Perron (1989) - He argues that most macroeconomic variables are not unit root processes. They are Trend Stationary withStructural 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 rootwhere there is not

With Structural Change All Unit Root Tests Are Biased Towards the Non Rejection of a Unit Root

Problem Number 1: Structural Breaks

44

Dickey-Fuller test for Y; DY onVariable Coefficient Std.Error t-valueY_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%=-2.587RSS = 132.5618324 for 1 variables and 98 observationsInformation Criteria:SC = 0.348867 HQ = 0.333159 FPE=1.38056 AIC = 0.32249

Unit Root Test For Y

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 Variable

Y Y D

Dt

t

t t t

t

05

0 0 49

2 50 100

1.

..

..

for

for

0 10 20 30 40 50 60 70 80 90 100

-2.5

0

2.5

5

Y

0 1 2 3 4 5 6 7 8 9 10 11 12

.25

.5

.75

1ACF-Y

46

Problem Number Two : Low Power

The Power of a test is the probability of rejecting a false Null Hypothesis -

• Low Power to Distinguish Between Unit and near Unit Root• Low Power to Distinguish Between Trend and Drift

Unit Root Tests

0 10 20 30 40 50 60 70 80 90 100

-8

-6

-4

-2

0

2

4

Y1 Z1

Y is a unit root series;

Z is a near-unit root series

47

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).