Solutions to the Review Questions at the End of Chapter 9
1. (a) This was a rather silly question since the answer is largely given away by
the question in part (b)! Nonetheless, although there are several methods that
could be used to determine whether there is evidence of daily seasonality in
stock returns, a simple method would be to obtain a sample of daily stock
returns and regress them on 5 day-of-the-week dummy variables. The
coefficient estimates would then be interpreted as the average return on each
day of the week, and if some of these were statistically significant but with
differing signs, this could be taken as evidence of daily seasonalities.
(b) The problem is one of perfect multicollinearity between the five daily
dummy variables and the constant term known as the dummy variable trap.
The sum of the five daily dummy variables will be one in every time period,
and this will be identical to the column of ones used for the constant. The
result is that the implicit assumption of the columns of the matrix of
explanatory variables being independent of one another has been violated, and
hence there is not enough separate information in the sample to be able to
calculate the values of all of the coefficients. The (XX) matrix will be singular
and therefore its inverse will not exist. The solution is simple: either use all 5
daily dummy variables but no intercept term, or drop one of the dummy
variables and still include the intercept. These two methods of dealing with the
problem are equivalent with identical RSS, and only the interpretation of the
coefficient estimates will change.
(c) The first step is to calculate the t-ratios. These are 0.232, -2.691, 0.673, 0.039, and –0.141 for the intercept, D1, D2, D3 and D4 respectively. The
interpretation of the intercept coefficient is the value of the return when all of
the variables (including the daily dummies) are zero, which in this case is the
average Friday return. The coefficients on the daily dummies can be
interpreted as the average deviation of that day’s return from the average
return for all days of the week. Only one of these dummy variables is
significant – the dummy for Monday, and we would thus conclude that the
average return on Monday was significantly lower than the average return for
the whole week, but there is no statistically significant evidence of any other
daily seasonalities given these results.
(d) Intercept dummy variables work by changing the regression intercept
estimate if a certain set of conditions hold, while slope dummies work by
changing the slope(s). For example, suppose that the regression model under
study for a sample of daily returns is
yt = 1 + 2x2t + 3x3t + ut .
A model containing these variables but also including intercept dummy
variables would be
yt = 1 + 2x2t + 3x3t + 4D1t + 5D2t + 6D3t + 7D4t + ut .
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As before, if we include the intercept in the regression, we only want 4 dummy
variables, and any 4 of the 5 daily dummies (defined as above) could be
included. A model containing the explanatory variables and slope dummy
variables would be
yt = 1 + 2x2t + 3x3t + 4D1tx2t + 5D2tx2t + 6D3x2t + 7D4tx2t + 8D1tx3t +
9D2tx3t + 10D3tx3t + 11D4tx3t + ut .
The dummy variables are defined identically as in the intercept dummy case,
and again one less dummy is needed than the total number of days in the
week. The dummies are now multiplied by explanatory variables so that each
of the slopes on x2 and x3 are permitted to vary from one day to the next.
(e) The financial year ends at approximately the end of March in the UK. So
one way to test the hypothesis that stock returns are different at the end of the
tax year compared with other times of the year would be to obtain a long
sample of monthly returns and to regress the returns on a dummy variable
taking the value one in March and zero otherwise. If, everything else equal,
investors were selling to realise losses in March, we would expect the
coefficient on this dummy to be negative and statistically significant due to
excess selling pressure. Thus average March returns would be significantly
lower than average returns over the whole year.
2. (a) A switching model is simply one where the behaviour of the series is
permitted to change from one type to another under the model. For example,
any regression containing seasonal dummy variables would be a simple kind
of switching model, since the behaviour of the series will be different at
different times.
Threshold autoregressive (TAR) models are those where the variable under
study is assumed to follow one autoregressive process in a given regime and
other autoregressive processes in other regimes. Movements from one regime
to another occur when a variable (not necessarily the variable under study)
rises above or falls below a particular value. Markov switching models, in their
simplest forms, assume that a variable can be drawn from one of several
regimes, each regime having its own mean and variance. The key distinction
between the two classes of models is that TAR models assume that the
threshold variable governing the regime is known, and under the model, once
this threshold is set, the variable is in one of the regimes alone. The Markov
switching model, on the other hand, assumes that the state-determining or
forcing variable is unobserved. The variable under study is thus never
completely in one regime or another, but rather is in each regime with some
probability at each point in time.
The decision on which of the two model classes is more appropriate for a
particular application would be made on the grounds of whether the statedetermining variable was observable or not, and what type of dynamics were
of interest in the model. For example, if the financial theory does not suggest
the forcing variable, then the Markov switching approach may be preferable.
On the other hand, if theory suggests an obvious choice of switching variable,
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or if it is of interest to use an AR-type model, then the TAR would be more
appropriate.
There have been very few comparisons of the two approaches that I know of –
authors seem to just adopt a particular approach and use it without discussing
the alternatives available.
(b) (i) The Markov property applies if a process is “path independent” – that
is, it is only the current value of a series or the current set of probabilities that
determine where the process will be during the next time period, and none of
the values of the series or probabilities during previous time periods. Thus, a
series that followed an AR(1) model would possess the Markov property. An
algebraic expression for the Markov property is given in equation (9.10) on
page 464. The implication of a process having the Markov property is that its
development can be described using only a vector of current probabilities and
a single transition matrix.
(ii) A transition matrix, in the context of Markov switching models, is a matrix
that maps a set of current probabilities to a set of future probabilities. Thus it
will describe the probabilities of the process being in a particular state in the
next period, conditioned upon it being in a given state during this period.
(c) A SETAR model is a TAR model where the state-determining variable is the
dependent variable used in the regression. The use of SETAR models rather
than a more general TAR removes one item to decide on (the statedetermining variable). But there are many others – the number of regimes, the
number of lags in each regime, the value(s) of the threshold(s), and the lag
with which the variable will switch. The major difficulty with SETAR (and
indeed all TAR) models is that it is impossible to easily and validly estimate all
of these quantities at the same time. They depend on eachother, and also, the
threshold causes a discontinuity in the function that would be maximised (if
ML is used) or minimised (if NLS) is used.
Therefore, the usually easiest way to estimate such models is to use as much
knowledge as possible from financial theory and to assume values for other
parameters and then to estimate as little as possible. For example, it may be
the case that the number of regimes, the delay value and the threshold values
can be assumed from theory. This would leave only the number of lags in each
regime together with the coefficients to be estimated. This could be validly
done using information criteria to determine the lag lengths for each regime
and ML or NLS to estimate the coefficients.
(d) Standard information criteria of the form described in Chapter 5 could be
employed to determine the appropriate length of the lags in each regime.
There would be one value of the criterion for each model order, and the model
that order that minimised the value of the criterion would be the one selected.
The problem with this approach is that, if the series under study resides in one
of the regimes for a considerably shorter time than it resides in the others, a
very short lag length will typically be selected for that regime. The reason is
that the reduction in the overall residual sum of squares is unlikely to be big if
it covers only a small number of observations. The upshot is that the use of
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standard information criteria applied globally to the whole model would
typically be to lead to long lag lengths for all regimes that the series spends a
high proportion of time in and short lag lengths for regimes that it did not
enter very often. A solution is to define an information criterion that does not
penalise the whole model for additional parameters in one state, i.e. a criterion
that is a function of the separately calculated residual sums of squares for both
of the regimes and of the number of lags and of the number of observations in
each of those regimes. An algebraic example of such a criterion was given in
equation (9.26) on page 476.
(e) If there are transactions costs that are non-negligible, this can lead PPP not
to hold since there would be deviations from PPP, which may appear to
represent profitable trading opportunities since the law of one price is
violated, but that in practice are unprofitable once these costs are taken into
account. Thus, a threshold model may be useful for this, since it would allow
PPP not to hold if the deviations from PPP were sufficiently small that
transactions costs would imply that this situation could persist indefinitely,
while the PPP relationship would be restored if the deviations from it became
sufficiently large to warrant cross-border trading, which would restore
equilibrium. In the linear case with no thresholds, the PPP relationship is can
be estimated for the current example using France and Germany (before the
advent of the EURO currency!) by:
ln( fx F / G ,t )   0   1 ln( p F ,t )   2 ln( p G ,t )  u t
where ln( fx F / G ,t ) is the log of the exchange rate, expressed in French francs per
German mark, and ln( p F ,t ) and ln( p G ,t ) are the logs of the French and German
consumer
price
series
respectively.
We
could
define
ln( fxF / G ,t )  ln( p F ,t )  ln( p G ,t ) as the deviation from PPP (see Chapter 7). This
could be generalised to allow for a different relationship between the three
variables according to whether the deviation from PPP is larger than some
upper threshold value r, or smaller (more negative) than a lower threshold s:
  0   1 ln( p F ,t )   2 ln( p G ,t )  u1t if ln( fx F / G ,t 1 )  ln( p F ,t 1 )  ln( p G ,t 1 )  r

ln( fx F / G ,t )    0   1 ln( p F ,t )   2 ln( p G ,t )  u 2t if s  ln( fx F / G ,t 1 )  ln( p F ,t 1 )  ln( p G ,t 1 )  r
    ln( p )   ln( p )  u
if ln( fx F / G ,t 1 )  ln( p F ,t 1 )  ln( p G ,t 1 )  s
0
1
F ,t
2
G ,t
3t

We could then consider the different values of the parameter estimates in each
of the 3 regimes (although we could not validly conduct hypothesis tests in the
usual way since it is very likely that the variables in the model are nonstationary!). The values of the thresholds (r and s) could be imposed on the
basis of some assumed size of transactions costs, or they could be estimated.
(f) The problem is essentially that the threshold no longer exists under the null
hypothesis that the SETAR model collapses to a linear model with the same
lag lengths as were in each part of the SETAR. This fact means that the usual
basis of asymptotic theory for testing hypotheses is not applicable, so that the
test statistics would not follow the distributions that we would have assumed
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of them. There are procedures available for testing hypotheses such as this in
the context of TAR models, but these are quite complex – see Hansen (1996),
for example.
(g) It is tempting to think that more complex models are bound to produce
more accurate forecasts than simpler models since the former should be able
to capture more of the relevant features of the data. However, this is certainly
not the case, for complex models may have a tendency to fit to sample-specific
features of the data that are not replicated during future (out of sample)
periods, and therefore lead to less accurate forecasts. This issue was discussed
in Chapter 5. However, the use of SETAR models for producing out of sample
prediction brings with it an additional problem, namely the possibility that the
regime that the variable will reside in during the forecasted observations will
be incorrectly predicted. If the SETAR model fits the data well, it is likely that
the behaviour will be quite different between the two regimes, and therefore
that the forecasts from each regime would also be different. This being the
case, forecasting the regime wrongly could cause a big source of forecast
inaccuracy for the series, and in practice it is often very difficult to forecast the
regime that the series will be in with any reasonable accuracy. Thus, any
improvement in forecast accuracy from accurate prediction of the variable
conditional upon a correct forecast of the regime that it will be in, is likely to
be more than outweighed by incorrectly forecasting the regime. Overall,
therefore, regime switching models have produced surprisingly poor forecasts,
even when they appear to fit the data very well – see Dacco and Satchell
(1999).
3. Both of these questions concern volatility dynamics rather than dynamics in
the conditional mean – therefore, in both cases the appropriate answer would
be to use a model for volatility dynamics but which also allowed for varying
behaviour over time. For (i), a plausible model would be a GARCH with some
daily dummy variables included in the conditional variance equation, e.g.
yt =  + yt-1 + ut , ut  N(0,t2)
t2 = 0 + 1 ut21 +t-12 + 1D1t + 2D2t + 3D3t + 4D4t
where D1, .., D4 are Monday, .., Thursday dummy variables. A more
sophisticated model could also allow the coefficients on the lagged squared
error or lagged conditional variance terms to also vary across the days of the
week. If Monday volatility dynamics are different from other days of the week,
we would expect to see 1 significant.
For (ii), some sort of threshold model is required, and the question suggests
that the threshold variable is observable (and is the value of the previous day’s
volatility). Thus, an appropriate model would be a GARCH model with a
threshold in the conditional variance that switches, e.g.
yt =  + yt-1 + ut , ut  N(0,t2)
t2 = 0 + 1 ut21 +t-12 + 1It
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where It = 1 if t-12 > 0.1, and zero otherwise. Note that the question does not
specify what is “volatility”, so it is assumed in this answer that it is equated
with “conditional variance”. Again, this dummy variable would only allow the
intercept in the conditional variance (i.e. the unconditional variance) to vary
according to the previous day’s volatility. A similar dummy variable could be
applied to the lagged squared error or lagged conditional variance terms to
allow them to vary with the size of the previous day’s volatility.
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