Introduction to Time Series



Is it Time Series Data?
Is it Cross Sectional Data?
Is it Panel (Pooled Data)?
About Data

Qualitative Data


Qualitative Surveys
Quantitative Data


Primary Data: Questionnaire Research
Secondary Data: Existing Data from
records
On Data More!



Time Series Data
 Monthly, Annually, Quarterly, Weekly,
Daily
Cross Sectional Data
 Questionnaire Data
 Data for variables based on a single
period
Panel Data
 As a combination of time series and
cross-sectional data
On the types of Data More !
Years
GDP (m.$)
M2 /GDP
1996
115,2
0,55
1997
116,4
0,56
1998
117,8
0,55
1999
115,4
0,60
2000
119,5
0,70
2001
120,0
0,71
2002
124,1
0,70
2003
125,9
0,72
2004
132,4
0,74
Time Series Data
for a particular
country
On the types of Data More
Country
GDP (m.$)
(Year: 2004)
M2 /GDP
(Year: 2004)
Albania
215,2
0,55
Belgium
206,4
0,56
Denmark
217,8
0,55
Estonia
115,4
0,60
Finland
119,5
0,70
Germany
920,0
0,71
Holland
224,1
0,70
Ireland
150,9
0,72
Litvania
132,4
0,74
Crosssectional
Data for GDP and
M2 by country for
2004
On the types of Data More
Country
Albania
Belgium
Denmark
Years
GDP (m.$)
(Year: 2004)
M2 /GDP
(Year: 2004)
2002
215,2
0,55
2003
206,4
0,56
2004
217,8
0,55
2002
215,4
0,60
2003
219,5
0,70
2004
220,0
0,71
2002
224,1
0,70
2003
250,9
0,72
2004
232,4
0,74
Panel
Data
What is Econometrics?
Econometrics  Economic
Measurement
Mathematical Economics
Economics
Mathematics
Econometrics
Mathematical
Statistics
Statistical Economics
Statistics
Methodology of Econometrics








1. Statement of theory or hypothesis
2. Specification of the mathematical model of the
theory
3. Specification of the econometric model of the
theory
4. Obtaining the data
5. Estimation of the parameters of the
econometric model
6. Hypothesis Testing
7. Forecasting or Prediction
8.Using the model for control or policy purposes
Stationarity



Empirical work based on time series
data assumes that the underlying
time series is stationary.
Otherwise, are econometric results
spurious or non-sense?
Then, what is Stationarity?
Definition of “Stationarity”


A time series might be equal to its
value plus a purely random shock
(or error term). Thus, this means a
random walk phenomenon
(Especially in financial time series).
Thus, the relationship between two
series need to be the one with
“stationarity” charactetistic.
A Stationary Stochastic Process


A random or stochastic process is a
collection of random variables
ordered in time.
Thus, a type stochastic process is
the so-called stationary stochastic
process.
Specification of Stationarity
Mean
: E (Yt) = µ
Variance : var (Yt) = E (Yt - µ)2 = 2
Covariance: k = E[(Yt - µ)(Yt+k - µ)]
In short, if a time series is stationary, its mean, variance and
autocovariance (at various lags) remain the same no matter at
what point we measure them; they are time invariant.
Such a time series will tend to return to its mean (mean
reversion) and fluctuations (its variance) around this mean will
have a broadly constant amplitude.
Tests of Stationarity

1. Graphical Analysis

2. Autocorrelation Function (ACF) and Correlogram

3. The Unit Root Tests (The main Focus of “all the
tests of Stationarity”)
1. Graphical Analysis of Stationarity
8.0
7.5
7.0
6.5
6.0
5.5
60
65
70
75
80
85
90
95
00
05
LGDP
Here GDP is increasing, that is, showing an upward trend, suggesting
perhaps that the mean value of the GDP has been changing. This
suggests that may be GDP is non-stationary. This ituitive feel is the
starting point of more formal tests of stationarity.
2. Autocorrelation Function (ACF) and Correlogram
3. The Unit Root Test
This is a test of stationarity (or nonstationarity) that has become
widely popular over the past several years (Gujarati, 2003).
The starting point is the unit root (stochastic) process with:
Yt  Yt 1  ut
A white noise error
term
-1 ≤ p ≤ 1
When p = 1, the above equation becomes a random walk model without drift, which is a nonstationary stochastic process.
Contiuned with manipulation:
Yt  Yt 1  ut
Yt  Yt 1  Yt 1  Yt 1  ut
  - 1Yt 1  ut
Yt 1  Yt 1  ut
This shows that since ut is a white noise error term, it is
stationary, which means that the first differences of a random
walk time series are stationary.
Continued
Yt 1  Yt 1  ut
In practice, we estimate the above equation and test the
null hypothesis that  = 0. If  = 0, then p = 1, that is
we have a unit root in Y variable. This means it is NOT
stationary.
When  < 0, then, p < 1 and Y is STATIONARY. Since  =
(p-1), for stationarity p must be less than one. For this to
happen  must be negative.
Question:
Which test to use for Unit Root?


The estimated t value of Yt-1 does
follow the t distribution even in
large samples: It does not have an
symptotic normal distribution.
So? What is the alternative?
a. Dickey-Fuller (DF) Test (1979)



DF has shown that under the null hypothesis that
=0, the estimated t value of the coefficient of Yt-1
follows the  (tau) statistic.
DF have computed the critical values of 
statistic on the basis of Monte Carlo simulations.
In the literature tau test is known as the DF test.
DF Test

A random walk process may have
no drift, or it may have drift or it
may have both deterministic and
stochastic trends. To allow for the
various possibilities, the DF test is
estimated in three different forms
(under three different null
hypotheses).
DF Test
Yt is a random walk:
Yt 1  Yt 1  ut
Yt is a random walk with drift:
Yt 1  1  Yt 1  ut
Yt is a random walk with drift
around a stochastic trend:
Yt 1  1   2t  Yt 1  ut
In each case, the null hypothesis is that =0; that is, there is a unit root – the time series
is non-stationary. It is extremely important to note that the critical values of the tau test
to test the hypothesis that =0, are different for each of the preceeding three
specifications of the DF test.
b. The Augmented Dickey-Fuller (ADF) Test
(1981)



In conducting the DF test, it was assumed that
the error term ut was uncorrelated.
But in case the ut is correlated, DF have
developed a test, known as ADF test.
The test is conducted by “augmenting” the
preceeding three equations by adding the lagged
values of the dependent variable Yt .
The ADF Test

The ADF test consists of estimating
the following regression:
m
Yt  1   2t  Yt 1    i Yt i  t
i 1
The Phillips-Perron (PP) Unit Root Tests
(1988)

Phillips and Perron (1988)
procedures, which compute a
residual variance that is robust to
auto-correlation, are applied to test
for unit roots as an alternative to
ADF unit root test.
Walter Enders’ (1995) suggestion for
Unit Root Tests:
Walter Enders’ (1995) suggestion for
Unit Root Tests:


Start to test unit roots from the most general
model (with drift+trend) to the most specific
(without drift and trend) model.
Then, compare results. If in any model, the null
hypothesis is accepted, =0, then, there is a unit
root. The series is non-stationary.
A Professional Table for Unit Roots:
Developed by Salih KATIRCIOGLU
Table 1. ADF and PP Tests for Unit Root
Statistics (Levels)
ln y
Lag
ln T
lag
ln Tour
lag
ln X
lag
ln M
lag
-3.20***
-0.62
2.94
-2.72
-1.03
5.38
(1)
(2)
(2)
(4)
(9)
(8)
-1.10
-1.91
2.57
-1.74
-1.93
2.60
(2)
(2)
(0)
(3)
(3)
(8)
-3.61**
-1.03
-1.91***
-3.61***
-1.43
-1.86
(0)
(2)
(0)
(0)
(3)
(2)
-0.31
-2.38
1.42
-0.30
-2.22
1.14
(2)
(2)
(2)
(6)
(6)
(3)
-2.54
-1.56
2.62
-2.36
-1.72
2.41
(0)
(2)
(2)
(3)
(3)
(2)
Statistics
(First
Differences)
∆ln y
Lag
∆ln T
lag
∆lnTour
lag
∆ln X
lag
∆ln M
lag
T (ADF)
 (ADF)
 (ADF)
T (PP)
 (PP)
 (PP)
-4.69*
-4.75*
-4.23*
-5.39*
-5.46*
-4.19*
(1)
(1)
(0)
(4)
(4)
(1)
-5.26*
-6.55*
-5.96*
-7.47*
-6.77*
-5.96*
(3)
(1)
(0)
(4)
(4)
(0)
-6.05*
-6.08*
-7.84*
-8.89*
-9.20*
-7.87*
(1)
(1)
(0)
(3)
(4)
(1)
-5.10*
-5.96*
-5.61*
-6.67*
-5.88*
-5.77*
(3)
(1)
(1)
(3)
(4)
(1)
-6.60*
-6.44*
-6.07*
-6.94*
-7.10*
-6.07*
(3)
(1)
(0)
(2)
(4)
(0)
T (ADF)
 (ADF)
 (ADF)
T (PP)
 (PP)
 (PP)
Note:
y represents real gross domestic product; T is real trade volume; Tour is total tourist arrivals to Cyprus; X is
total real exports; and finally, M is total real imports. All of the series are at their natural logarithms.
T represents the most general model with a drift and trend;  is the model with a drift and without trend; 
is the most restricted model without a drift and trend.
Numbers in brackets are lag lengths used in ADF test (as determined by AIC set to maximum 3) to remove
serial correlation in the residuals. When using PP test, numbers in brackets represent Newey-West
Bandwith (as determined by Bartlett-Kernel).
* **
, and *** denote rejection of the null hypothesis at the 1%, 5% and 10% levels respectively.
Tests for unit roots have been carried out in E-VIEWS 4.1.
The Question is:




There must be a lag level for testing unit roots.
So, which lag level to select in running unit root
tests?
Two popular tests: Akaike Information Criterion
(AIC) and Schwartz Information Criterion (SIC).
These lag tests will be introduce in the application
session.
EVIEWS 5.1 does these tests automatically.
Power of Test


Most tests of the DF type have low power;
that is, they tend to accept the null of
unit root more frequently than is
warranted. That is, these tests may find a
unit root even when none exists.
There are some reasons for low power,
please refer to Gujarati (2003): p. 819.
c. KPSS Test for Unit Root


To confirm the test results obtained from the ADF and
PP tests, Kwiatkowski Phillips, Schmidt and Shin’s test
(1992) (KPSS) is suggested to eliminate a possible low
power against stationary near unit root processes
which occurs in the ADF and PP tests.
The KPSS test complements the ADF and PP tests in
which the null hypothesis of KPSS test is that a series
is stationary. This means that a stationary series is
likely to have insignificant KPSS statistics and
significant ADF and PP statistics.
Now is the Application Time of Unit
Root Tests