Financial Econometrics
Introduction to Systems Approach
Introduction
• Complete the description of Error Correction
Models and assess an example of cointegration
using PPP.
• Explain the problems associated with
simultaneous equation estimation.
• Describe the order condition and why this is
important when estimating a system of
equations
• Introduce the Vector Autoregressive (VAR)
approach to estimating simultaneous equation
models.
Error Correction Term
• The error correction term tells us the speed with which
our model returns to equilibrium following an exogenous
shock.
• It should be negatively signed, indicating a move back
towards equilibrium, a positive sign indicates movement
away from equilibrium
• The coefficient should lie between 0 and 1, 0 suggesting
no adjustment one time period later, 1 indicates full
adjustment
• The error correction term can be either the difference
between the dependent and explanatory variable (lagged
once) or the error term (lagged once), they are in effect
the same thing.
Example of ECM
• The following ECM was formed, using 60
observations:
yˆ t  0.78  0.24xt  0.32(ut 1 )
(0.56) (0.12) (0.08)
(SE in parenthese s, u is the residual from
a cointegrat ing relationsh ip)
Example of an ECM
• The error correction term has a t-statistic
of 4, which is highly significant supporting
the cointegration result.
• The coefficient on the error correction term
is negative, so the model is stable.
• The coefficient of -0.32, suggests 32%
movement back towards equilibrium
following a shock to the model, one time
period later.
Potential Problems with
Cointegration
• The ADF test often indicates acceptance of the null
hypothesis (no cointegration), when in fact cointegration
is present.
• The ADF test is best when we have a long time span of
data, rather than large amounts of observations over a
short time span. This can be a problem with financial
data which tends to cover a couple of years, but with
high frequency data (i.e. daily data)
• It is only really used for bi-variate cointegration tests,
although it can be used for multivariate models, a
different set of critical values is required.
Multivariate Approach to
Cointegration
• A different approach to testing for cointegration
is generally required when we have more then 2
endogenous variables in the model.
• If we assume all the variables are endogenous,
we can construct a VAR and then test for
cointegration.
• One of the most common approaches to
multivariate cointegration is the Johansen
Maximum Likelihood (ML) test.
• This test involves testing the characteristic roots
or eigenvalues of the π matrix (coefficients on
the lagged dependent variable).
Steps in Testing for Cointegration
1)
2)
3)
4)
5)
6)
Test all the variables to determine if they are I(0), I(1)
or I(2) using the ADF test.
If both variables are I(1), then carry out the test for
cointegration
If there is evidence of cointegration, use the residual to
form the error correction term in the corresponding
ECM
Add in a number of lags of both explanatory and
dependent variables to the ECM
Omit those lags that are insignificant to form a
parsimonious model
Use the ECM for dynamic forecasting of the dependent
variable and assess the accuracy of the forecasts.
Simultaneous Equations
• When we run a regression using OLS, we
assume the explanatory variables are
exogenous
• If the explanatory variables are endogenous, the
estimates produced by OLS are biased
• This means are estimator is not BLUE, therefore
are t and F statistics are invalid.
• This is known as simultaneous equation bias
• However a potential problem occurs with
deciding which variables are exogenous.
Measures to Overcome the Bias
• One way to overcome the bias is to form
‘reduced form’ equations, in which we rearrange
our model, by a process of substitution, until all
the explanatory variables are exogenous.
• Another method involves using instrumental
variable techniques and involves finding
exogenous variables to act as instruments for
the endogenous variables.
• A popular way to overcome the problem in the
finance literature is to estimate a system of
equations, termed a vector autoregressive (VAR)
approach.
Reduced-Form Equations
• When producing a reduced form equation, we have to
ensure our equation is identified.
• This means that we can form the coefficients in our
structural equation from the estimated reduced form
coefficients. There are three forms of identification:
- Exactly Identified – We can form unique
values for the structural coefficients
- Under Identified – It is not possible to form
the structural coefficients
- Over Identified – We produce more then one
value for the structural coefficients
Order Condition of Identification
• To determine if an equation is identified, underidentified or over-identified, we need to apply the
order condition
• We also need to test the rank condition in theory,
however the order condition is usually adequate
to ensure identifiability
• The easiest way of applying the order condition,
is to use the following: In a model of M
simultaneous equations in order for an equation
to be identified, it must exclude at least M-1
variables (endogenous and exogenous)
Example of order condition
• The following equations have (1) as
exactly identified, (2) as overidentified
yt   0  1st   2Gt  ut
(1)
st   0  1st 1  vt
(2)
Gt  exogenous
st 1  pre det er min ed (exogenous)
Indirect Least Squares (ILS)
• This consists of three stages:
1) Obtain the reduced form equations.
2) Apply OLS to the reduced formequations.
3) Obtain estimates of the original structural
coefficients from the estimated reducedform coefficients.
• This approach assumes the original
equations are exactly identified.
Instrumental Variables (IV)
• An instrumental variable can be interpreted as a
‘proxy’ for a variable which suffers from
simultaneity.
• In effect the instrumental variable is highly
correlated with the endogenous variable, but is
uncorrelated with the error term.
• If such a variable exists, OLS can be used to
estimate the model with the IV instead of the
endogenous variable.
• One way to incorporate an IV into a model is to
use two-stage-least squares.
Two-Stage-Least Squares
• Given the following model, where equation
(1) is overidentified:
yt   0  1st  ut
(1)
st   0  1 yt   2 xt   3 st 1  vt (2)
xt  exogenous
st 1  pre det er min ed
Two-Stage Least Squares- Stage 1
• To get rid of the correlation between the
endogenous variable s(t) and the residual u(t) in
equation (1), regress s(t) on all the predetermined (exogenous) variables in the system.
• In this case it is just x(t) and s(t-1) as the
explanatory variables.
• Obtain the fitted value of the variable s(t), which
is uncorrelated with the residual. This can now
be used as an instrument in the original
equation.
Stage 2
• The final step involves regressing the initial
equation (1), but using the fitted value for s(t) as
an instrument for s(t)
• The fitted value is a good instrument because it
is similar to s(t), but uncorrelated with the
residual.
• The standard errors will need to be adjusted,
however this is a fairly standard routine, that
most computer software automatically
completes.
Features of 2SLS
• It can be directly applied to an equation in a
system, without needing to take into account any
other equations.
• It can be used for both exactly identified and
overidentified equations.
• Easy to use, the only information required is
what the exogenous variables are.
• Given that the S.E can be calculated, t-statistics
can also be used.
• It is a large-sample technique.
The Vector Autoregressive
Approach (VAR)
• To overcome the problems of endogenous variables, one
way around the problem is to estimate a system of
equations,
• All the equations are identified, as we exclude all the
level variables as explanatory variables
• In this case there are as many equations as there are
variables in the system, with each variable acting as a
dependent variable.
• This dependent variable is then regressed against lags
of the other variables in the system and itself.
• The lagged variables are pre-determined (exogenous)
• When specifying the VAR, it is important to decide on the
optimal number of lags to include.
Model (Equities (s) and interest
rates (r))
VAR Assumptions
OLS estimation of a single equation in the Unrestricted VAR
******************************************************************************
Dependent variable is DR3
123 observations used for estimation from 1961Q2 to 1991Q4
******************************************************************************
Regressor
Coefficient
Standard Error
T-Ratio[Prob]
DR3(-1)
-.40309
.46681
-.86351[.390]
DR3(-2)
-1.1887
.47229
-2.5168[.013]
DR3(-3)
-.48694
.46430
-1.0488[.297
DR3(-4)
-.19538
.45873
-.42592[.671]
DR10(-1)
.49745
.46501
1.0698[.287]
DR10(-2)
.86432
.47243
1.8295[.070]
DR10(-3)
.48701
.46394
1.0497[.296]
DR10(-4)
-.0057767
.45559
-.012680[.990]
DTBILL(-1)
.26585
.17750
1.4978[.137
DTBILL(-2)
.25603
.18008
1.4218[.158]
DTBILL(-3)
.30794
.18041
1.7069[.091]
DTBILL(-4)
.10721
.17680
.60636[.546]
K
-.5659E-3
.063335
-.0089349[.993
******************************************************************************
As well as the usual diagnostic tests……
Main Uses of VARs
•
•
•
•
This model is often used to link different
markets, such as the bond and stock markets.
Testing for causality between variables
The main use is forecasting, due to the
dynamic nature of this model, it can produce
reasonable dynamic forecasts of the variables
in the system
When the VAR is adapted slightly, it can be
turned into a Vector Error Correction Model
(VECM), this can then be used for assessing
long as well as short run relationships.
Causality Tests
• Regression does not imply causality, to test for
causality a lagged model needs to be used.
• ‘Granger Causality’ is the main approach to
testing for causality, in general the term causality
is not used, instead it is said A ‘Granger causes’
B.
• Granger Causality tests can also be used to
determine if a variable is exogenous or not.
Granger Causality
• To test for ‘Granger causality’ between two
variables, simply run the following VAR:
Granger Causality
• To determine if there is any evidence of causality from s
to r, we conduct a F-test for joint significance of the
lagged explanatory variables. If jointly significantly
different to 0, then causality run from s to r.
• If the same process is carried out on the other equation
and this time the lagged explanatory variables are
insignificant, we say ‘s causes r’.
• If we carry out the tests on both equations and it appears
s causes r and r causes s then we have bi-causality, i.e.
causality runs in both directions, many argue this
suggests an invalid relationship between the variables.
Conclusion
• Simultaneous equation bias is a serious
problem in econometric modelling
• We can to an extent overcome the
problem by using a reduced-form equation
approach, assuming the equation is
identified exactly
• A further way around the problem is to use
a VAR, where all the explanatory variables
are lagged, therefore pre-determined.