Download Time Series Econometrics - Eastern Mediterranean University

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
• 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
14000
10000
12000
10000
9000
8000
8000
7000
8000
6000
6000
10000
6000
5000
8000
4000
4000
4000
6000
2000
4000
2000
2000
0
92
94
96
98
00
02
04
3000
2000
0
92
94
96
AUSTRALIA
98
00
02
04
1000
92
94
96
CANADA
24000
45000
20000
40000
98
00
02
04
92
94
96
CHINA
98
00
02
04
02
04
02
04
GERMANY
50000
7000
6000
40000
35000
16000
5000
30000
30000
4000
12000
25000
8000
3000
20000
20000
2000
10000
4000
15000
0
1000
10000
92
94
96
98
00
02
04
0
92
94
96
HONGKONG
98
00
02
04
0
92
94
96
JAPAN
00
02
04
92
94
96
KOREA
7000
30000
12000
6000
25000
10000
20000
8000
15000
6000
10000
4000
5000
2000
5000
98
98
00
MALAYSIA
60000
50000
40000
4000
3000
30000
2000
1000
0
0
92
94
96
98
00
SINGAPORE
02
04
20000
0
92
94
96
98
TAIWAN
00
02
04
10000
92
94
96
98
UK
00
02
04
92
94
96
98
USA
00
These are Examples of
Stationary Time Series
8000
6000
4000
6000
6000
4000
3000
4000
2000
2000
2000
4000
2000
0
1000
0
0
-2000
-2000
0
-4000
-4000
-1000
-6000
-6000
-2000
92
94
96
98
00
02
04
92
94
96
AUST
98
00
02
04
-2000
-4000
-6000
92
94
96
CAN
98
00
02
04
92
94
96
CHI
12000
12000
12000
8000
8000
8000
4000
4000
4000
98
00
02
04
00
02
04
00
02
04
GERM
2000
1000
0
0
0
0
-1000
-4000
-4000
-4000
-8000
-8000
-8000
-12000
-12000
92
94
96
98
00
02
04
-2000
-12000
92
94
96
HONG
98
00
02
04
-3000
92
94
96
JAP
3000
98
00
02
04
92
94
96
KOR
12000
98
MAL
5000
30000
4000
2000
8000
20000
3000
1000
2000
4000
10000
1000
0
0
0
-1000
0
-1000
-2000
-4000
-2000
-10000
-3000
-3000
-8000
92
94
96
98
SING
00
02
04
-4000
92
94
96
98
TWN
00
02
04
-20000
92
94
96
98
UKK
00
02
04
92
94
96
98
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 DickeyFuller 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 Forecasting
The Box-Jenkins Approach
Differencing the series
to achieve stationarity
Identify model to be
tentatively entertained
Estimate the parameters
of the tentative model
No
Use the model for
forecasting and
control
Yes
Diagnostic checking. Is
the model adequate?
Approaches to Economic Forecasting
The Box-Jenkins Approach-Identification Tools
• 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 Forecasting
The Box-Jenkins Approach-Identification
Theoretical Patterns of ACF and PACF
Type of
Model
AR (p)
MA (q)
ARMA (p,q)
Typical Pattern of
ACF
Typical
Pattern of
PACF
Decays exponentially Significant spikes
or with damped sine
through lags p
wave pattern or both
Significant spikes
Declines
through lags q
exponentially
Exponential decay
Exponential
decay
Approaches to Economic Forecasting
The Box-Jenkins Approach-Diagnostic Checking
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:
 Box-Pierce Q
m
Q  N  rk2
k 1
2
ˆ


 Ljung-Box LB LB  n(n  2)  k  2 m

nk 
k 1 

m
Approaches to Economic Forecasting
Some issues in the Box-Jenkins modeling
 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   A
 2   11
Yt   A21
Y 3    A
 t   31
...   ...
 p 
Yt   Ap1
A12
A22
A32
...
Ap 2
A13
A23
A33
...
Ap 3
...
...
...
...
...
1

Y
A1 p  t 1 
 A'11
 2
 '

A2 p  Yt 1 
 A 21
A3 p  Yt 31   ...   A'31

 
...  ... 
 ...
 
 A' p1
App  Yt p1 

 
'
12
A
A' 22
A'32
...
A' p 2
'
13
A
A' 23
A'33
...
A' p 3
...
...
...
...
...
1

Y
A   t  k  e1t 
 
 2
A' 2 p  Yt  k  e2t 
 3 
A'3 p  Yt  k   e3t 
 



...  ...
... 
'  p 
A pp  Yt  k  e pt 
 
'
1p
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   2t
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  RSS UR ) / m
F
RSS UR /( 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.