Practice questions:
1. Consider the following vector autoregressive model:
where yt is a p × 1 vector of variables determined by k lags of all p variables in
the system, ut is a p× 1 vector of error terms, β0 is a p× 1 vector of constant
term coefficients and βi are p × p matrices of coefficients on the i th lag of y.
(a) If p = 2, and k = 3, write out all the equations of the VAR in full,
carefully defining any new notation you use that is not given in the
question.
Answer
P=2 qnd k=3 implies that there are two variables in the system, and
that both equations have three lags of two variables. The VAR can be
written in long-hand for as;
Where
, and β coefficiencies on the lags of yt are
defined as follows. βijk refers to the kth lag of the ith variable in the jth
equation. This seems like a natural notation to use, although of course any
sensitive would also be correct.
(b) Why have VARs become popular for application in economics and finance,
relative to structural models derived from some underlying theory?
answer
This question can be answered by basically discussing the advantages
of VARs compared with structural models. The structural models
require the researcher to specify some variables as being exogenous (if
all variables were endogenous, then none of the equations would be
identified, and therefore estimation of the structural equations would be
impossible). This can be viewed as a restriction (a restriction that the
exogenous variables do not have any simultaneous equations feedback),
often called an “identifying restriction”. Determining what are the
identifying restrictions is supposed to be based on economic or financial
theory, but Sims, who first proposed the VAR methodology, argued that
such restrictions were “incredible”. He thought that they were too
loosely based on theory and were often specified by researchers on the
basis of giving the restrictions that the models required to make the
equations identified. Under a VAR, all the variables have equations, and
so in a sense, every variable is endogenous, which takes the ability to
misrepresent (either deliberately or inadvertently) or to mis-specify the
model in this way, out of the hands of the researcher. Another possible
reason why VARs are popular in the academic literature is that standard
form VARs can be estimated using OLS since all of the lags on the RHS
are counted as pre-determined variables. Further, a glance at the
literature which has sought to compare the forecasting accuracies of
structural models with VARs, reveals that VARs seem to be rather
better at forecasting (perhaps because the identifying restrictions are not
valid). Thus, from a purely pragmatic point of view, researchers may
prefer VARs if the purpose of the modelling exercise is to produce
precise point forecasts
(a) Discuss any weaknesses you perceive in the VAR approach to
econometric modelling.
Answer
VARs have, of course, also been subject to criticisms. The most important of these criticisms
is that VARs are theoretical. In other words, they use very little information form economic
or financial theory to guide the model specification process. The result is that the models
often have little or no theoretical interpretation, so that they are of limited use for testing and
evaluating theories. Second, VARs can often contain a lot of parameters. The resulting loss
indegrees of freedom if the VAR is unrestricted and contains a lot of lags, could lead to a loss
of efficiency and the inclusion of lots of irrelevant or marginally relevant terms. Third, it is
not clear how the VAR lag lengths should be chosen. Different methods are available (see
below), but they could lead to widely differing answers. Finally, the very tools that have been
proposed to help to obtain useful information from VARs, i.e. impulse responses and
variance decompositions, are themselves difficult to interpret.
(b) Two researchers, using the same set of data but working independently,
arrive at different lag lengths for the VAR equation (6.99). Describe and
evaluate two methods for determining which of the lag lengths is more
appropriate.
Answer
The two methods that we have examined are model restrictions and
information criteria. The model restrictions approach involves starting with
the larger of the two models and testing whether it can be restricted down
to the smaller one using the likelihood ratio test based on the determinants
of the variance-covariance matrices of residuals in each case. The
alternative approach would be to examine the value of various
information criteria and to select the model that minimizes the criteria.
Since there are only two models to compare, either technique could be
used. The restriction approach assumes normality for the VAR error
terms, while use of the information criteria does not. On the other hand,
the information criteria can lead to quite different answers depending on
which criterion is used and the severity of its penalty term. A completely
different approach would be to put the VARs in the situation that they
were intended for (e.g. forecasting, making trading profits, determining
a hedge ratio etc.), and see which one does best in practice.
(c) Explain how fixed effects models are equivalent to an ordinary least square’s
regression with dummy variables.
Answer
Fixed effects models are equivalent to an ordinary least square regression
with dummy variables because they both account for the unobserved
heterogeneity across units by including unit-specific intercepts. The dummy
variables capture the fixed effects of each unit by taking a value of one for
that unit and zero otherwise.
(d) How does the random effects model capture cross-sectional heterogeneity in
the intercept term?
The random effects model captures cross-sectional heterogeneity in the intercept
term by allowing the intercept to vary across individuals or groups. This means
that each individual or group has its own baseline level of the outcome variable
that is influenced by some unobserved factors. The random effects model
assumes that these unobserved factors are independent of the explanatory
variables.
1. (a) What stylised features of financial data cannot be explained using linear
time series models?
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Frequency: Stock market prices are measured every time there is a trade or
somebody posts a new quote, so often the frequency of the data is very high -Nonstationarity: Financial data (asset prices) are covariance non-stationary; but if we
assume that we are talking about returns from here on, then we can validly
consider them to be stationary.
-Linear Independence: They typically have little evidence of linear (autoregressive)
dependence, especially at low frequency.
-Non-normality: They are not normally distributed – they are fat-tailed.
-Volatility pooling and asymmetries in volatility: The returns exhibit volatility
clustering and leverage effects. Of these, we can allow for the non-stationarity
within the linear (ARIMA) framework, and we can use whatever frequency of data
we like to form the models, but we cannot hope to capture the other features using
a linear model with Gaussian disturbances.
(b) Which of these features could be modelled using a GARCH(1,1)
process?
GARCH models are designed to capture the volatility
clustering effects in the returns (GARCH(1,1) can model the
dependence in the squared returns, or squared residuals), and they
can also capture some of the unconditional leptokurtosis, so that
even if the residuals of a linear model of the form given by the first
part of the equation in part (e), the tuˆ’s, are leptokurtic, the
standardized residuals from the GARCH estimation are likely
to be less leptokurtic. Standard GARCH models cannot,
however, account for leverage effects.
(c) Why, in recent empirical research, have researchers preferred GARCH
(1,1) models to pure ARCH(p)?
(d) Describe two extensions to the original GARCH model. What additional
characteristics of financial data might they be able to capture?
2. (a) Discuss briefly the principles behind maximum likelihood.
(b) Describe briefly the three hypothesis testing procedures that are available
under maximum likelihood estimation. Which is likely to be the easiest to
calculate in practice, and why?
(c) OLS and maximum likelihood are used to estimate the parameters of a
standard linear regression model. Will they give the same estimates? Explain
your answer.
3. (a) Distinguish between the terms ‘conditional variance’ and
‘unconditional variance’. Which of the two is more likely to be relevant
for producing:
i. 1-step-ahead volatility forecasts
ii. 20-step-ahead volatility forecasts.
(a) If ut follows a GARCH(1,1) process, what would be the likely result if a
regression of the form (8.110) were estimated using OLS and assuming a
constant conditional variance?
(b) Compare and contrast the following models for volatility, noting their
strengths and weaknesses:
i. Historical volatility
ii. EWMA
iii. GARCH(1,1)
iv. Implied volatility.
4. (a) Briefly outline Johansen’s methodology for testing for cointegration between
a set of variables in the context of a VAR.
(b) A researcher uses the Johansen procedure and obtains the following test
statistics (and critical values):
r λmax
0 38.962
1 29.148
2 16.304
3 8.861
4 1.994
95% critical value
33.178
27.169
20.278
14.036
3.962
(c) Determine the number of cointegrating vectors.
(c) ‘If two series are cointegrated, it is not possible to make inferences regarding
the cointegrating relationship using the Engle–Granger technique since the
residuals from the cointegrating regression are likely to be autocorrelated.’ How
does Johansen circumvent this problem to test hypotheses about the
cointegrating relationship?
(d) Give one or more examples from the academic finance literature of where the
Johansen systems technique has been employed. What were the main results
and conclusions of this research?
(f). Compare the Johansen maximal eigenvalue test with the test based on the
trace statistic. State clearly the null and alternative hypotheses in each case.
4. (a) What are the advantages of constructing a panel of data, if one is available,
rather than using pooled data?
(b) What is meant by the term ‘seemingly unrelated regression’? Give examples
from finance of where such an approach may be used.
(c) Distinguish between balanced and unbalanced panels, giving examples of
each.
(d) Explain how fixed effects models are equivalent to an ordinary least squares
regression with dummy variables.
(e) How does the random effects model capture cross-sectional heterogeneity in
the intercept term?
(f) What are the relative advantages and disadvantages of the fixed versus
random effects specifications and how would you choose between them for
application to a particular problem?