This document discusses time series analysis and stationarity testing. It explains that time series data is common in finance and can be observed at different intervals. When using time series data, variables may influence each other with lags and non-stationary variables can cause spurious regressions. The Dickey-Fuller test and Augmented Dickey-Fuller test are introduced to test for a unit root and determine if a time series is stationary or non-stationary. The tests compare a test statistic to critical values, and a failure to reject the null hypothesis of a unit root means the series is non-stationary.

1. Advanced econometrics and Stata
L9-10 Time series
Dr. Chunxia Jiang
Business School, University of Aberdeen, UK
Beijing , 17-26 Nov 2019

2. Finance models & time series data
 Many applications in finance are concerned with the
analysis of time series data
 Monthly observations of a company’s share prices
 Quarterly observations on GDP
 Data can be observed at different but regular intervals
(yearly, quarterly, monthly, daily, seconds – high
frequency data)

3. Things to be careful about when
using time series data
 One time series variable can influence another
with a lag
 If variables are non stationary, they cause a
problem of spurious regression
 Non-stationary variables should not be included in
the regression model
 We can transform variables to make them
stationary
 We can check whether they are cointegrated

4. Stochastic processes
 A random or stochastic process is a collection of random
variables ordered in time
 i.e. GDP for a particular year could be any number,
depending on economic and political environment. The
figure in the sample for that year is a particular
realization of all such possibilities
 GDP is a stochastic process and actual values we
observed are particular realization of that process
 In time series, we see the realization to draw inferences
about the underlying stochastic process

5. Stochastic processes
 A stochastic process is said to be stationary if its mean and
variance are constant over time and the value of the covariance
between two time periods depends only on the distance or gap
or lag between the two periods and not the actual time at which
the covariance is computed
 If a time series is stationary, its mean, variance, and autocovariance
(at various lags) remain the same no mater at what point we
measure them: they are time invariant
 If a time series is not stationary, it is called a nonstationary time
series
 A nonstationary time series will have a time-varying mean or a time-
varying variance or both

6. Random walk
 Random walk without drift (no constant or intercept
term)
 Random walk with drift (with constant)
t
t
t u
Y
Y 
 1

7. Unit root stochastic process
 If rho=1, we face the unit root problem: a situation of
nonstationarity
 We say that the series contains a unit root.
 If the absolute value of rho is less than 1, the time
series is stationary
1
1
1 



  
 t
t
t u
Y
Y

8. A stationary AR(1) series and a random walk
-10
0
10
20
30
40
50
60
70
1 16 31 46 61 76 91 106 121 136 151 166 181 196 211 226 241 256 271 286 301 316 331 346 361 376 391 406 421 436 451 466 481 496
yt ynst
Stationary
Non-Stationary

9. Why are stationarity so important
 If a time series is nonsationary, we can study its
behaviour only for the time period under
consideration.
 It is not possible to generalize it to other time periods.
 For the purpose of forecasting, such a time series may
be of little practical value

10. Tests for stationarity
 Autocorrelation function
 Unit root test

11. Autocorrelation
 When we consider a time series variable we want to look at
how its value today is correlated to its value yesterday, or
two days ago, three days ago…
 These correlations are called autocorrelations
 For a stationary series the first autocorrelation tends to be
quite high but the effect dies out quickly
 For a non-stationary series, the correlations are very high
even when considering several lags

12. Univariate time series analysis
 Autocorrelations are useful but not enough to
determine whether a series is stationary
 We need to carry out some regression analysis
 Let’s look at the simple representation of a time series
called Autoregressive Process of order 1 or AR(1):
t
t
t e
y
y 

 1


1


1


1

 Stationary
Non Stationary
Explosive

13. Univariate regression analysis
 Let’s take the previous AR(1) process:
 Given that a value of θ equals to 1 indicates that
the series is non-stationary, to find out about
stationarity we could simply use a t test for the null
hypothesis that θ=1, against the alternative that is
less than 1.
 Unfortunately this is not the correct procedure.
t
t
t e
y
y 

 1



14. Dickey-Fuller equation
Let’s subtract yt-1 from both sides of the equation:
This can be rewritten as:
t
t
t e
y
y 

 1


t
t
t
t
t e
y
y
y
y 



 

 1
1
1 

  t
t
t e
y
y 



 1
1


t
t
t e
y
y 


 1



15. Dickey-Fuller test
(or unit root test)
 From this equation:
 We test if rho is equal to zero, against the alternative that it is less
than zero.
 Rejection of the null implies that the series is stationary.
 If you fail to reject the null, the series is non stationary.
 Use the Dickey-Fuller tables – the Dickey-Fuller (DF) test.
 We then compare the DF test statistics with the critical values in the
DF tables.
 To reject the null, DF test statistics has to be a large negative number
or we can say that the DF test statistic>DF critical values in absolute
value.
t
t
t e
y
y 


 1



16. Dickey-Fuller test
(or unit root test)
 t value of the estimated coefficient doesnot follow the t
distribution even in large samples. It follows the tau statistic
 Example



 where values in parentheses are DF values (similar to t-values).
DF on Yt-1 is –1.49 which in absolute term, is smaller than the
critical value of -2.93 . Hence, we cannot reject the null
hypothesis and we conclude that Yt is non-stationary.
50
n
0.09
R
(-1.49)
(-1.05)
(9)
0.190Y
0.0066
ΔY
2
1
t
t




 

17. 1% and 5% critical values for the
Dickey-Fuller test
Without trend With trend
Sample size 1% 5% 1% 5%
T=25 -3.75 -3.00 -4.38 -3.60
T=50 -3.58 -2.93 -4.15 -3.50
T=100 -3.51 -2.89 -4.04 -3.45
T=250 -3.46 -2.88 -3.99 -3.43
T=500 -3.44 -2.87 -3.98 -3.42
T= -3.43 -2.86 -3.96 -3.41
There are different types of critical values depending on how you specify your model:

18. Augmented DF test
 In DF test, it is assumed that error term is uncorrelated.
 If the error term is correlated, use ADF –adding the
lagged values of the dependent variable ∆Yt
 The number of lagged difference terms to include is
often determined empirically, being enough so that the
error term is serially uncorrelated.
 The rest is the same as DF test
(12)
v
ΔY
β
δY
α
ΔY t
p
1
j
j
t
j
1
t
t 








19. Limitations of unit root tests
 DF test is sensitive to the way it is conducted given
different versions of DF test: random walk, random
walk with drift, random walk with drift and trend
 DF tests tend to accept the null of unit root more
frequently

20.  Augmented Dickey and Fuller (1979) (ADF) test
 dfuller lM2Growth, trend lags(1) regress
 dfuller d.lM2Growth, trend lags(1) regress
 The Kwiatkowski–Phillips–Schmidt–Shin Kwiatkowski, et
al. (1992) test
 kpss lM2Growth, auto
 kpss d.lM2Growth, auto
DF/ADF Test