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Time Series
Econometrics
Didar Erdinc, Ph.D.
Associate Professor of Economics
American University in Bulgaria
+
Vector Autoregression (VAR)
Introduction
+ 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 = b1 yt + g11mt-1 + g12 yt-1 + emt
yt = b2 mt + g 21mt-1 + g 22 yt-1 + e 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
yt  j /  mt
¶mt+ j / ¶e yt
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.