We examine the behaviour of Dickey Fuller test (DF) in the case of noisy data using Monte Carlo simulation. The findings show clearly that the size distortion of DF test becomes larger as the noise increases in the data.Hypothesis testing; Unit root test; Monte Carlo Analysis.

Sahar S. Qaqeesh

Stephen Hall

English

Research Papers in Economics

 
 
 
 
 
DEPARTMENT OF ECONOMICS 
   
 
 
 
 
The behaviour of Dickey Fuller test in the 
case of noisy data: to what extent we can trust 
the outcome 
 
 
 
 
 
 
Stephen G. Hall, University of Leicester, UK 
Sahar S. Qaqeesh, Univerersity of Leicester, UK 
 
 
 
 
 
Working Paper No. 09/18 
September 2009 
Updated 18/09/2009 
Updated 21/09/2009 
 
   1 
The Behaviour of Dickey Fuller Test in the Case of Noisy Data: To What 
Extent We Can Trust the Outcome 
 
Stephen G. Hall 
Department of Economics 
University of Leicester 
 
Sahar S. Qaqeesh 
Department of Economics 
University of Leicester 
 
September 18, 2009 
 
ABSTRACT 
We examine the behaviour of Dickey Fuller test (DF) in the case of noisy data using 
Monte Carlo simulation. The findings show clearly that the size distortion of DF test 
becomes larger as the noise increases in the data. 
Keywords: Hypothesis testing; Unit root test, Monte Carlo Analysis. 
JEL classification: C01, C12, C15. 
 
1. INTRODUCTION 
 Various unit root tests have been employed in empirical work to identify the order of 
integration of economic variables. So far Dickey Fuller (DF) test remains the most 
famous one. A substantial body of research examines the main characteristics of DF 
test and particularly its main shortcomings such as low power, large size distortion 
and sensitivity to the true data generating process (DGP). Diebold and Rudebusch 
(1989a,  1991)  and  DeJong,  Nankervis,  Savin,  and Whiteman  (1992)  examine  the 
power of DF test when the process has short memory with a unit root close to unity 
and  provide  strong  evidence  that  DF  test  has  low  power  when  the  process  is 
fractionally integrated.  
The Size distortion of DF test is also examined heavily in the literature. Perron (1989), 
Hamori and Tokihisa (1997), Montañés and Reyes (1998), Leybourne and Newbold 
(2000), Sen (2001, 2003, 2008) and Kim, Lybourne and Newbold (2004) examine the 
behaviour of DF test in the case of structural breaks. Cheung and Lai (1998), and 
Cook and Manning (2004) examine the influence of the lag selection process using   2 
standard information criteria on the size distortion of ADF test. Granger and Hallman 
(1991), Kramer and Davies (2002) examine the robustness of DF test in the case of 
improper  transformations  of  the  data.  Phillips  and  Perron  (1988),  Schwert  (1989), 
Agiakloglou  and  Newbold  (1992)  analyze  the  performance  of  DF  test  when  the 
process  that  generates  the  time  series  contains  moving  average  term.  The  main 
findings for all these studies show that the distribution of the unit root test statistics is 
different from the distribution proposed by Dickey-Fuller. Accordingly a severe size 
distortion occurs and the power of DF test becomes questionable. 
In  this  paper,  we  examine  the  performance  of  DF  test  when  the  process  that 
generates the time series contains noise. The main focus will be on the size distortion 
of DF test as the noise increases in the data. This paper is organized as follows: 
section 2 the motivation, section 3 the full design of Monte Carlo experiment that is 
employed to illustrate the behaviour of DF test and the size distortion that occurs as a 
result of the  measurement errors in  the  data, Section  4 the empirical results  and 
section 5 the conclusion. 
 
2.  THE MOTIVATION 
The  motivation  for  this  study  comes  from  some  simple  tests  on  the  order  of 
integration on Jordanian inflation. Figure 1 shows monthly inflation in Jordan from 
1997 to 2007, if we consider the spectrum for this series it suggests a non-stationary 
process,  but  as  table  1  shows  the  DF  test  strongly  suggests  that  the  series  is 
stationary. In order to check this we then defined the inflation rate over a 12 month 
period, shown in figure 2 and this clearly looks non-stationary and indeed, as table 1 
shows, the DF test strongly suggests non stationarity 
Table 1: Dickey-Fuller tests for stationarity of Jordanian Inflation 
DF Test using (AIC), Exogenous: Constant 
  t-statistics, no. of lags 
INFLATION RATE (H=1 MONTH)  **-9.368042(1) 
INFLATION RATE (H=12 MONTH)  *-0.899845(12) 
*   The null hypothesis of non-stationarity can't be rejected at (1%,5%,10%) significance  levels. 
**  The null hypothesis of non-stationarity is rejected at (1%, 5%,10%)  significance  levels. 
Of course these two tests are in clear contradiction of each other as, 
11
1
1
1 1
12 11 2 1 1 12
12 ... ) ( ... ) ( ) ( t t t t t t t t t t t t P P P P P P P P  
Where t
j  is the rate of inflation at time t over j  periods. So if monthly inflation is 
stationary then annual inflation which is simply the sum of monthly inflation must also   3 
be  stationary.  We  believe  that  the  explanation  for  this  contradiction  lies  in  the 
presence of measurement error. This is evident in figure 1 as we can clearly see here 
that many months exhibited negative inflation, this is not a phenomenon which is 
observed by Jordanians and hence it would seem to be a problem with the actual 
measurement of the price level. 
If this is the case then we can also show why taking a longer period for the inflation 
calculation would give a more meaningful test statistic. We begin by assuming that 
the true price level in logs is a random walk 
t t t P P 1
* *  
But the observed log of the price level is subject to measurement error 
t t t v P P
*  
Where both  t t v and  are IID noise processes and observed inflation is given by 
j t t t
j P P  
Now 
1
1
t t t t v v   and hence v t
2 2 1 2 ) var(  
And 
12 11 2 1
12 ... t t t t t t t v v  and hence  v t
2 2 12 2 12 ) var(  
It is therefore evident that by measuring inflation over a longer period the size of the 
random walk variance grows relative to the size of the measurement error and hence 
measurement error has less effect on measured inflation. 
To examine the influence of the noisy data on the size distortion and the empirical 
power of DF test more completely, we now turn to a formal Monte Carlo Simulation. 
 
3. Monte Carlo Analysis/ Experiment Design 
The Monte Carlo experiment begins with Data Generating Process. The steps of data 
generating are as follows:- 
Step one:  Generate a data set using the simplest model of time series which is a 
non-stationary  normal  random  walk.  So  the  first  data  set  ( )  is  a  random  walk 
process without a drift and it is generated by an AR (1) model of the form:- 
   4 
   =   +  ,   t= 1, 2, ……., T                      Eq.   (1) 
Where,   
  1 
0 x  = 0 (the initial value)  
    = Random disturbance is generated from normal distribution with zero mean and                
          constant variance (
2) equals one, i.e.   ~ N(0,1). 
 
Step two: Create noises in the data set ( ) by adding random disturbances with zero 
mean and fixed variance and create a measured variable y. 
y t=   +  t,     t= 1, 2, ……., T                             Eq.   (2) 
Where we may vary the variance of this error to investigate the effects of different 
levels of measurement error relative to the random walk component, 
       t~ N(0,0), N(0,0.5), N(0,1), N(0,1.5), N(0,2), N(0,3), N(0,4), N(0,5), N(0,6), and  N(0,7).    
 We consider the following samples sizes T= 25, 50, 100, 150, and 200 observations 
and we perform 50,000 replications for each sample size and for each variance.  We 
chose 50,000 replications on the grounds that for each sample size using variance 
zero ( t~ N(0,0))   we needed 50,000 replications to  exactly replicate the standard 
Dickey Fuller critical values. 
In order to show the size distortion  of DF test as noise increases in the data , we 
calculate   the    percentage   of rejection   of   the   null   hypothesis at 5% level of 
significance using the normal critical values of DF test with constant model
1.  
 
4. Empirical Results 
In table 2 we show clearly that the percentage of rejection of the null   hypothesis at 
5% level of significance increases dramatically as the noise increases in the data. 
The null hypothesis of non stationrity is rejected more often in favour of the alternative 
(the stationarity).  
                                                           
1 DF test asymptotic critical values at 5% level of significance under sample sizes  (25, 50, 100, 150, and 
200) are as follows: -2.99%, -2.92%, -2.89%, -2.88% and -2.88%.   5 
The benchmark case in this experiment is the one where the variance equals zero (no 
noise  embedded)  and  both  generated  data  sets  ( )  and  (y t)  are  equal.  The 
percentage of rejection of the null hypothesis for all samples' sizes is exactly 5% (see 
table 2) which means that DF test is able to identify the truth about the unit root using 
the normal critical values (95%) of the time. When the noise increases, from variance 
0.5 till variance 7, the size distortion becomes larger and the percentage of rejection 
increases dramatically which means that DF test provide misleading results using the 
same normal critical values. 
The results also show that the size distort ion becomes larger when the sample size 
increases  even  at  lower  variances  which  implies  that  even  very  large  samples 
containing measurement error will give incorrect inference . Figure 3  demonstrates 
that under sample size 50, 100, 150 and 200 the percentage of the rejection of the 
null hypothesis increases faster than the case of sample size 25. The influence of the 
noise appears more quickly when the sample size is big, for example the percentage 
of rejection reaches 100% at variance six for both sample si ze 150 and 200 and at 
variance seven for sample size 100 while in the case of sample 25 we need to add 
more noises to reach 100%.   
It is crystal clear that the distribution of the t -statistic  when the data set contains 
noises is different from the distribution proposed by Dickey Fuller where the process 
is a pure random walk. In this paper we propose a new set of critical values that can 
be used as an indication to identify the unit root in noisy data. The proposed critical 
values in table 3   are derived from  the  distribution of t-statistic  values  across the 
replications. The critical values at 1%, 5% and 10% are calculated as the first and fifth 
and tenth percentile of the t-statistic distribution.  As a benchmark, the critical values 
at variance zero equal exactly the asymptomatic critical values under the DF test. It is 
obvious that the critical values  become bigger  (in absolute values) when the noise 
increases  in the data and this mean   that  the  new  t-statistic  distribution  will have 
heavier and fatter tails than normal fat tails. 
 
5. Conclusion 
The  main  objective  of  this  experiment  is  to  prove  that  the  rejection  of  the  null 
hypothesis of unit root under DF test  in some cases should not be taken without 
further investigation.  
We prove by Monte Carlo simulation that the size distortion of DF test becomes larger 
as the noise increases in the data and faster as the sample size becomes bigger. 
We believe that DF normal critical values can be misleading and implausible when 
the data set contains noise. Instead the proposed critical values (table 3) can be more 
reliable in identifying the truth about unit root properties.   6 
References 
Agiakloglou, C. and Newbold, P. (1992). "Empirical Evidence on Dickey-Fuller-Type Tests", Journal of 
time Series Analysis, Vol. 13, No. 6, pp. 471-483. 
 
Cheung Yin-Wong, Lai Kon S., (1998). " Power of the augmented dickey-fuller test with information-
based lag selection", Journal of Statistical Computation and Simulation, Vol. 60, pp. 57-65. 
 
Cook S., Manning N., (2004),  "Lag optimisation and finite-sample size distortion of unit root tests", 
Economics Letters, Vol. 84, pp. 267-274. 
 
DeJong, D. N., Nankervis, J. C., Savin, N. E. and Whiteman, C. H. (1992), "The Power Problems of 
Unit Root in Time Series with Autoregressive Errors", Journal of Econometrics, Vol. 53, pp. 323-343. 
 
DeJong, D. N., Nankervis, J. C., Savin, N. E. and Whiteman, C. H. (1992), "Integration Versus Trend 
Stationary in Time Series", Econometrica, Vol. 60, No. 2, pp. 423-433.. 
 
Dickey,D.A., and Fuller, W.A., (1979). "Distribution of the estimators for autoregressive time series with 
a unit root". Journal of the American Statistical Association, Vol.  74, No. 366, pp. 427-431. 
 
Dickey,D.A., and Fuller, W.A., (1981). " Likelihood ratio statistics for autoregressive time series with a 
unit root'. Econometrica, Vol. 49, No. 4, pp. 1057-1072. 
 
Diebold, F. X., and Rudebusch, G. D., (1989), "Long memory and persistence in aggregate output", 
Journal of Monetary Economics, Vol. 24,  pp. 189-209. 
 
Diebold, F. X., and Rudebusch, G. D., (1991), "On the Power of Dickey-Fuller tests against fractional 
alternatives", Economics Letters, Vol. 35, pp. 155-160. 
 
Granger,  C.W.J.,  and  Hallman,  J.,  (1991),  "Nonlinear  transformations  of  integrated  time  series", 
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Economics Letters, Vol. 57, pp. 245-253. 
 
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Trend Misspecification, Journal of Time Series Analysis, Vol. 25, pp.755-764. 
 
Kramer, W., and Davies, L.,(2002), "Testing for unit roots in the context of misspecified logarithmic 
random walks", Economics letters, Vol. 74, pp. 313-319. 
 
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under the alternative hypothesis", Econometric Theory. Vol. 16, pp. 779-789. 
 
Montañés, A., and Reyes M., (1998), "Effect of a Shift in the Trend Function on Dickey-Fuller Unit Root 
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Econometrica, Vol. 57, No. 6,  pp. 1361-1401. 
 
Phillips,  P.  C.  B.,  and  Perron  P.,  (1988),  "Testing  for  a  Unit  Root  in  Time  Series  Regression", 
Biomètrika, Vol. 75, No. 2, pp. 335-346. 
 
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International Economic Review, Vol. 33, No. 3, pp. 661-680. 
 
Sen,  A.  (2001),  "  Behaviour  of  Dickey-Fuller  F-tests  under  the  trend-break  stationary  alternative", 
Statistics and Probability Letters, Vol. 55, pp. 257-268. 
 
Sen, A. (2003), " On unit-root tests when the alternative is a trend-break stationary process", Journal of 
Business and Economic Statistics, Vol. 21, No.1, pp. 174-184. 
 
Sen,  A.  (2008),  "  Behaviour  of  Dickey-Fuller  tests  when  there  is  a  break  under  the  unit  root  null 
hypothesis",  Statistics and probability Letters,  Vol. 78, pp.  622-628. 
 
Schwert, G. W., (1989)."Tests for unit roots :a Monte Carlo Investigation", Journal of Business and 
Economics Statistics, Vol. 7, No. 2,  pp. 147-159. 
 
 
 
   7 
Figure (1) 
 
 
 
 
Figure (2) 
 
 
 
 
 
 
   8 
Figure (3) 
 
0.00
20.00
40.00
60.00
80.00
100.00
120.00
0 0.5 1 1.5 2 3 4 5 6 7
VARIANCES
P
E
R
C
E
N
T
A
G
E
 
O
F
 
R
E
J
E
C
T
I
O
N
 
SAMPLE = 25
SAMPLE = 50
SAMPLE = 100
SAMPLE = 150
SAMPLE = 200
 
 
Table 2 
REPLICATIONS 
=50,000 
THE PERCENTAGE OF REJECTION OF THE NULL HYPOTHESIS OF UNIT ROOT 
AT 5% LEVEL OF SIGNIFICANCE 
SAMPLE SIZES = 25, 50, 100, 150, AND 200 
25 
t-statistic<-2.99 
50 
t-statistic<-2.92 
100 
t-statistic<-2.89 
150 
t-statistic<-2.88 
200 
t-statistic<-2.88 
VARIANCE = 0  5.04%  4.79%  5.01%  5.02%  4.96% 
VARIANCE = 0.5  9.90%  10.89%  11.73%  11.82%  11.85% 
VARIANCE = 1  25.10%  31.77%  34.92%  36.59%  36.67% 
VARIANCE = 1.5  44.00%  55.92%  61.39%  63.48%  63.86% 
VARIANCE = 2  60.00%  74.08%  79.38%  81.23%  81.96% 
VARIANCE = 3  79.47%  92.35%  95.27%  96.10%  96.43% 
VARIANCE = 4  88.78%  97.90%  99.11%  99.46%  99.54% 
VARIANCE = 5  93.18%  99.44%  99.87%  99.94%  99.96% 
VARIANCE = 6  95.23%  99.86%  99.98%  100.00%  100.00% 
VARIANCE = 7  96.45%  99.96%  100%  100.00%  100.00% 
   9 
Table 3 
REPLICATIONS 
=50,000 
THE CRITICAL VALUES FOR 
ALL SAMPLES’ SIZES 
25  50  100  150  200 
CRITICAL VALUES  1%  5%  10%  1%  5%  10%  1%  5%  10%  1%  5%  10%  1%  5%  10% 
VARIANCE = 0  -3.77  -2.99  -2.63  -3.56  -2.90  -2.59  -3.49  -2.89  -2.58  -3.48  -2.88  -2.57  -3.46  -2.88  -2.57 
VARIANCE = 0.5  -4.21  -3.36  -2.98  -4.02  -3.33  -2.97  -4.00  -3.34  -2.98  -4.02  -3.32  -2.97  -4.00  -3.32  -2.97 
VARIANCE = 1  -5.08  -4.15  -3.69  -5.12  -4.27  -3.83  -5.29  -4.41  -3.94  -5.36  -4.44  -3.97  -5.40  -4.45  -4.00 
VARIANCE = 1.5  -5.77  -4.80  -4.33  -6.12  -5.18  -4.70  -6.58  -5.57  -5.02  -6.78  -5.71  -5.13  -6.94  -5.79  -5.21 
VARIANCE = 2  -6.29  -5.26  -4.80  -6.87  -5.91  -5.41  -7.63  -6.57  -5.99  -8.05  -6.87  -6.24  -8.35  -7.06  -6.40 
VARIANCE = 3  -6.87  -5.84  -5.36  -7.80  -6.84  -6.36  -9.09  -8.04  -7.45  -9.87  -8.68  -8.03  -10.47  -9.13  -8.42 
VARIANCE = 4  -7.18  -6.16  -5.67  -8.34  -7.40  -6.91  -9.98  -8.97  -8.41  -11.03  -9.92  -9.29  -11.88  -10.62  -9.92 
VARIANCE = 5  -7.36  -6.36  -5.86  -8.70  -7.75  -7.28  -10.52  -9.56  -9.05  -11.80  -10.76  -10.18  -12.83  -11.66  -11.01 
VARIANCE = 6  -7.50  -6.49  -5.99  -8.93  -7.98  -7.52  -10.91  -9.99  -9.49  -12.32  -11.35  -10.81  -13.51  -12.41  -11.82 
VARIANCE = 7  -7.59  -6.57  -6.07  -9.11  -8.15  -7.70  -11.19  -10.30  -9.81  -12.72  -11.77  -11.26  -13.98  -12.95  -12.41 
 

Behavior of Dickey-Fuller t-tests when there is a break under the alternative hypothesis&quot;,

Behaviour of Dickey-Fuller F-tests under the trend-break stationary alternative&quot;,

Behaviour of Dickey-Fuller tests when there is a break under the unit root null hypothesis&quot;,

Behaviour of Dickey-Fuller Unit-Root Tests Under Trend Misspecification,

Distribution of the estimators for autoregressive time series with a unit root&quot;.

Effect of a Shift in the Trend Function on Dickey-Fuller Unit Root Tests&quot;.

Empirical Evidence on Dickey-Fuller-Type Tests&quot;,

for unit roots :a Monte Carlo Investigation&quot;,

Integration Versus Trend Stationary in Time Series&quot;,

Lag optimisation and finite-sample size distortion of unit root tests&quot;,

Likelihood ratio statistics for autoregressive time series with a unit root'.

Long memory and persistence in aggregate output&quot;,

Nonlinear transformations of integrated time series&quot;,

On the Power of Dickey-Fuller tests against fractional alternatives&quot;,

On unit-root tests when the alternative is a trend-break stationary process&quot;,

Power of the augmented dickey-fuller test with informationbased lag selection&quot;,

Testing for a unit root in the presence of a variance shift&quot;

Testing for a Unit Root in Time Series Regression&quot;,

Testing for unit roots in the context of misspecified logarithmic random walks&quot;,

The Great Crash, the Oil Price Shock and the Unit Root Hypothesis&quot;,

The Power Problems of Unit Root in Time Series with Autoregressive Errors&quot;,

Trends and random walks in macroeconomic time series: A re-examination&quot;,

The Behaviour of Dickey Fuller test in the case of noisy data: to what extent we can trust the outcome.

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The Behaviour of Dickey Fuller test in the case of noisy data: to what extent we can trust the outcome.

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