Time Series
Econometrics:
Asst. Prof. Dr. Mete Feridun
Department of Banking and Finance
Faculty of Business and Economics
Eastern Mediterranean University
What is a time series?
A time series is any series of data that
varies over time. For example
• Monthly Tourist Arrivals from Korea
• Quarterly GDP of Laos
• Hourly price of stocks and shares
• Weekly quantity of beer sold in a pub
Because of widespread availability of
time series databases most empirical
studies use time series data.
Caveats in Using Time Series
Data in Applied Econometric
Modeling
• Data Should be Stationary
Data Should be Stationary
• Presence of Autocorrelation
• Guard Against Spurious Regressions
• Establish Cointegration
• Reconcile SR with LR Behavior via ECM
• Implications to Forecasting
• Possibility of Volatility Clustering
What is a Stationary Time
Series?
• A Stationary Series is a Variable with
constant Mean across time
• A Stationary Series is a Variable with
constant Variance across time
These are Examples of
Non-Stationary Time Series
16000 12000 10000 9000
14000 8000
10000
8000
7000
12000
8000
6000 6000
10000
6000 5000
8000
4000 4000
4000
6000
3000
2000
4000 2000
2000
2000 0 0 1000
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
AUSTRALIA CANADA CHINA GERMANY
24000 45000 50000 7000
20000 40000 6000
40000
35000 5000
16000
30000
30000 4000
12000
25000 3000
20000
8000
20000 2000
10000
4000 15000 1000
0 10000 0 0
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
HONGKONG JAPAN KOREA MALAYSIA
7000 30000 12000 60000
6000 25000 10000
50000
5000
20000 8000
40000
4000
15000 6000
3000
30000
10000 4000
2000
20000
1000 5000 2000
0 0 0 10000
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
SINGAPORE TAIWAN UK USA
These are Examples of
Stationary Time Series
8000 6000 4000 6000
6000 4000 3000 4000
4000
2000 2000 2000
2000
0 1000 0
0
-2000 0 -2000
-2000
-4000 -4000 -1000 -4000
-6000 -6000 -2000 -6000
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
AUST CAN CHI GERM
12000 12000 12000 2000
8000 8000 8000
1000
4000 4000 4000
0
0 0 0
-1000
-4000 -4000 -4000
-2000
-8000 -8000 -8000
-12000 -12000 -12000 -3000
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
HONG JAP KOR MAL
3000 12000 5000 30000
4000
2000
8000 3000 20000
1000 2000
4000 10000
1000
0
0
0 0
-1000 -1000
-4000 -2000 -10000
-2000
-3000
-3000 -8000 -4000 -20000
92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04 92 94 96 98 00 02 04
SING TWN UKK US
What is a “Unit Root”?
If a Non-Stationary Time
Series Yt has to be
“differenced” d times to make
it stationary, then Yt is said
to contain d “Unit Roots”. It is
customary to denote Yt ~ I(d)
which reads “Yt is integrated
of order d”
Establishment of Stationarity
Using Differencing of
Integrated Series
• If Yt ~ I(1), then Zt = Yt – Yt-1 is Stationary
• If Yt ~ I(2), then Zt = Yt – Yt-1 – (Yt – Yt-2 )is
Stationary
Unit Root Testing: Formal Tests
to Establish Stationarity of
Time Series
• Dickey-Fuller (DF) Test
• Augmented Dickey-
Fuller (ADF) Test
• Phillips-Perron (PP) Unit
Root Test
• Dickey-Pantula Unit
Root Test
• GLS Transformed
Dickey-Fuller Test
• ERS Point Optimal Test
• KPSS Unit Root Test
• Ng and Perron Test
What is a Spurious
Regression?
A Spurious or Nonsensical relationship
may result when one Non-stationary
time series is regressed against one
or more Non-stationary time series
The best way to guard against
Spurious Regressions is to check for
“Cointegration” of the variables used
in time series modeling
Symptoms of Likely Presence
of Spurious Regression
• If the R2 of the regression is greater
than the Durbin-Watson Statistic
• If the residual series of the regression
has a Unit Root
Cointegration
• Is the existence of a long run
equilibrium relationship among time
series variables
• Is a property of two or more variables
moving together through time, and
despite following their own individual
trends will not drift too far apart since
they are linked together in some sense
Two Cointegrated Time
Series
55
X Y
50
45
40
35
30
25
20
15
10
0 10 20 30 40 50 60 70 80 90 100
Cointegration Analysis:
Formal Tests
• Cointegrating Regression Durbin-
Watson (CRDW) Test
• Augmented Engle-Granger (AEG) Test
• Johansen Multivariate Cointegration
Tests or the Johansen Method
Error Correction Mechanism
(ECM)
• Reconciles the Static LR Equilibrium
relationship of Cointegrated Time
Series with its Dynamic SR
disequilibrium
• Based on the Granger Representation
Theorem which states that “If
variables are cointegrated, the
relationship among them can be
expressed as ECM”.
Forecasting: Main
Motivation
• Judicious planning
requires reliable
forecasts of decision
variables
• How can effective
forecasting be
undertaken in the light
of non-stationary nature
of most economic
variables?
• Featured techniques:
Box-Jenkins Approach
and Vector Auto
regression (VAR)
Approaches to Economic
Forecasting
The Box-Jenkins Approach
• One of most widely used methodologies for
the analysis of time-series data
• Forecasts based on a statistical analysis of
the past data. Differs from conventional
regression methods in that the mutual
dependence of the observations is of primary
interest
• Also known as the autoregressive integrated
moving average (ARIMA) model
Approaches to Economic
Forecasting
The Box-Jenkins Approach
Advantages
• Derived from solid mathematical statistics foundations
• ARIMA models are a family of models and the BJ approach is
a strategy of choosing the best model out of this family
• It can be shown that an appropriate ARIMA model can
produce optimal univariate forecasts
Disadvantages
• Requires large number of observations for model
identification
• Hard to explain and interpret to unsophisticated users
• Estimation and selection an art form
Approaches to Economic
The Box-Jenkins Approach
Forecasting
Differencing the series Identify model to be
to achieve stationarity tentatively entertained
Estimate the parameters
of the tentative model
No
Diagnostic checking. Is
Use the model for the model adequate?
Yes
forecasting and
control
Approaches to Economic
The Box-Jenkins Approach-Identification Tools
Forecasting
• Correlogram – graph showing the ACF and the PACF at
different lags.
• Autocorrelation function (ACF)- ratio of sample
covariance (at lag k) to sample variance
• Partial autocorrelation function (PACF) – measures
correlation between (time series) observations that are k
time periods apart after controlling for correlations at
intermediate lags (i.e., lags less than k). In other words, it is
the correlation between Yt and Yt-k after removing the effects
of intermediate Y’s.
Approaches to Economic
The Box-Jenkins Approach-Identification
Forecasting
Theoretical Patterns of ACF and PACF
Type of Typical Pattern Typical
Model of ACF Pattern of
PACF
AR (p) Decays Significant
exponentially or spikes through
with damped sine lags p
wave pattern or
both
MA (q) Significant spikes Declines
through lags q exponentially
ARMA Exponential decay Exponential
(p,q) decay
Approaches to Economic
The Box-Jenkins Approach-Diagnostic Checking
Forecasting Checkin
How do we know that the model we estimated is a reasonable
fit to the data?
Check
residuals
Rule of thumb: None of the ACF and the PACF are
individually statistically significant
Formal test: m
Box-Pierce Q Q N rk2
k 1
ˆm2
2
Ljung-Box LB LB n(n 2)
k 1
n k m
k
Approaches to Economic
Some issues in the Box-Jenkins modeling
Forecasting
Judgmental decisions
• on the choice of degree of order
• on the choice of lags
Data mining
• can be avoided if we confine to AR processes only
• fit versus parsimony
Seasonality
• observations, for example, in any month are often affected by
some seasonal tendencies peculiar to that month.
• the differencing operation – considered as main limitation for
a series that exhibit moving seasonal and moving trend.
Vector Autoregression (VAR)
Introduction
• VAR resembles a SEM modeling – we consider several
endogenous variables together. Each endogenous variables is
explained by its lagged values and the lagged values of all
other endogenous variables in the model.
• In the SEM model, some variables are treated as endogenous and
some are exogenous (predetermined). In estimating SEM, we
have to make sure that the equation in the system are identified –
this is achieved by assuming that some of the predetermined
variables are present only in some equation (which is very
subjective) – and criticized by Christopher Sims.
• If there is simultaneity among set of variables, they should all
be treated on equal footing, i.e., there should not be any a priori
distinction between endogenous and exogenous variables.
Vector Autoregression (VAR)
Its Uses
Forecasting
VAR forecasts extrapolate expected values of current and future
values of each of the variables using observed lagged values of
all variables, assuming no further shocks
Impulse Response Function (IRFs)
IRFs trace out the expected responses of current and future
values of each of the variables to a shock in one of the VAR
equations
Vector Autoregression (VAR)
Its Uses
Forecast Error Decomposition of Variance (FEDVs)
FEDVs provide the percentage of the variance of the error
made in forecasting a variable at a given horizon due to specific
shock. Thus, the FEDV is like a (partial) R2 for the forecast
error
Granger Causality Tests
Granger-causality requires that lagged values of variable A are
related to subsequent values in variable B, keeping constant the
lagged values of variable B and any other explanatory variables
Vector Autoregression (VAR)
Mathematical Definition
[Y]t = [A][Y]t-1 + … + [A’][Y]t-k + [e]t or
Yt1 A11 A12 A13 ... A1 p t 1
Y 1
A '
A '
A '
... A t k e1t
' Y 1
2
A 2
11 12 13 1p
2
Yt 21 A22 A23 ... A2 p Yt 1 A' 21 A' 22 A' 23 ... A' 2 p Yt k e2t
Y3 3 3
t A31 A32 A33 ... A3 p Yt 1 ... A'31 A'32 A'33 ... A'3 p Yt k e3t
... ... ... ... ... ... ... ... ... ... ... ... ... ...
'
p
Yt
Ap1
Ap 2 Ap 3 ... App Yt p1
A' p1 A' p 2 A' p 3 ... A pp Yt k e pt
p
where:
p = the number of variables be considered in the system
k = the number of lags be considered in the system
[Y]t, [Y]t-1, …[Y]t-k = the 1x p vector of variables
[A], … and [A'] = the p x p matrices of coefficients to be estimated
[e]t = a 1 x p vector of innovations that may be contemporaneously
correlated but are uncorrelated with their own lagged values and
uncorrelated with all of the right-hand side variables.
Vector Autoregression (VAR)
Example
Consider a case in which the number of variables n is 2, the
number of lags p is 1 and the constant term is suppressed. For
concreteness, let the two variables be called money, mt and
output, yt .
The structural equation will be:
mt 1 yt 11mt 1 12 yt 1 mt
yt 2 yt 21mt 1 22 yt 1 yt
Vector Autoregression (VAR)
Example
Then, the reduced form is
11 1 21 12 1 22 1 1
mt mt 1 yt 1 mt yt
1 1 2 1 1 2 1 1 2 1 1 2
11mt 1 12 yt 1 1t
21 2 11 22 2 12 2 1
yt mt 1 yt 1 mt yt
1 1 2 1 1 2 1 1 2 1 1 2
21mt 1 22 yt 1 2t
Vector Autoregression (VAR)
Example
Among the statistics computed from VARs are:
Granger causality tests – which have been interpreted as
testing, for example, the validity of the monetarist proposition
that autonomous variations in the money supply have been a
cause of output fluctuations.
Variance decomposition – which have been interpreted as
indicating, for example, the fraction of the variance of output
that is due to monetary versus that due to real factors.
Impulse response functions – which have been interpreted as
tracing, for example, how output responds to shocks to money
(is the return fast or slow?).
Vector Autoregression (VAR)
Granger Causality
In a regression analysis, we deal with the dependence of one
variable on other variables, but it does not necessarily imply
causation. In other words, the existence of a relationship
between variables does not prove causality or direction of
influence.
In our GDP and M example, the often asked question is whether
GDP M or M GDP. Since we have two variables, we are
dealing with bilateral causality.
Given the previous GDP and M VAR equations:
mt 1 yt 11mt 1 12 yt 1 mt
yt 2 mt 21mt 1 22 yt 1 yt
Vector Autoregression (VAR)
Granger Causality
We can distinguish four cases:
Unidirectional causality from M to GDP
Unidirectional causality from GDP to M
Feedback or bilateral causality
Independence
Assumptions:
Stationary variables for GDP and M
Number of lag terms
Error terms are uncorrelated – if it is, appropriate
transformation is necessary
Vector Autoregression (VAR)
Granger Causality – Estimation (t-test)
mt 11mt 1 12 yt 1 1t
yt 21mt 1 22 yt 1 2 t
A variable, say mt is said to fail to Granger cause another variable,
say yt, relative to an information set consisting of past m’s and y’s
if: E[ yt | yt-1, mt-1, yt-2, mt-2, …] = E [yt | yt-1, yt-2, …].
mt does not Granger cause yt relative to an information set
consisting of past m’s and y’s iff 21 = 0.
yt does not Granger cause mt relative to an information set
consisting of past m’s and y’s iff 12 = 0.
In a bivariate case, as in our example, a t-test can be used to test
the null hypothesis that one variable does not Granger cause
another variable. In higher order systems, an F-test is used.
Vector Autoregression (VAR)
Granger Causality – Estimation (F-test)
1. Regress current GDP on all lagged GDP terms but do not
include the lagged M variable (restricted regression). From this,
obtain the restricted residual sum of squares, RSSR.
2. Run the regression including the lagged M terms (unrestricted
regression). Also get the residual sum of squares, RSSUR.
3. The null hypothesis is Ho: i = 0, that is, the lagged M terms do
not belong in the regression.
( RSS R RSSUR ) / m
F
RSSUR /( n k )
5. If the computed F > critical F value at a chosen level of
significance, we reject the null, in which case the lagged m
belong in the regression. This is another way of saying that m
causes y.
Vector Autoregression (VAR)
Variance Decomposition
Our aim here is to decompose the variance of each element of
[Yt] into components due to each of the elements of the error
term and to do so for various horizon. We wish to see how
much of the variance of each element of [Yt] is due to the first
error term, the second error term and so on.
Again, in our example:
mt 1 yt 11mt 1 12 yt 1 mt
yt 2 mt 21mt 1 22 yt 1 yt
The conditional variance of, say mt+j, can be broken down into
a fraction due to monetary shock, mt and a fraction due to the
output shock, yt .
Vector Autoregression (VAR)
Impulse Response Functions
Here, our aim is to trace out the dynamic response of each
element of the [Yt] to a shock to each of the elements of the
error term. Since there are n elements of the [Yt], there are n2
responses in all.
From our GDP and money supply example:
mt 1 yt 11mt 1 12 yt 1 mt
yt 2 mt 21mt 1 22 yt 1 yt
We have four impulse response functions:
mt j / mt mt j / yt
yt j / mt yt j / yt
Vector Autoregression (VAR)
Pros and Cons
Advantages
The method is simple; one does not have to worry about
determining which variables are endogenous and which
ones exogenous. All variables in VAR are endogenous
Estimation is simple; the usual OLS method can be applied
to each equation separately
The forecasts obtained by this method are in many cases
better than those obtained from the more complex
simultaneous-equation models.
Vector Autoregression (VAR)
Pros and Cons
Some Problems with VAR modeling
• A VAR model is a-theoretic because it uses less prior
information. Recall that in simultaneous equation models
exclusion or inclusion of certain variables plays a crucial role
in the identification of the model.
• Because of its emphasis on forecasting, VAR models are
less suited for policy analysis.
• Suppose you have a three-variable VAR model and you decide
to include eight lags of each variable in each equation. You will
have 24 lagged parameters in each equation plus the constant
term, for a total of 25 parameters. Unless the sample size is
large, estimating that many parameters will consume a lot of
degree of freedom with all the problems associated with that.
Vector Autoregression (VAR)
Pros and Cons
• Strictly speaking, in an m-variable VAR model, all the m
variables should be (joint) stationary. If they are not stationary,
we have to transform (e.g., by first-differencing) the data
appropriately. If some of the variables are non-stationary, and
the model contains a mix of I(0) and I(1), then the transforming
of data will not be easy.
• Since the individual coefficients in the estimated VAR models
are often difficult to interpret, the practitioners of this technique
often estimate the so-called impulse response function. The
impulse response function traces out the response of the
dependent variable in the VAR system to shocks in the error
terms, and traces out the impact of such shocks for several
periods in the future.
