Solutions to the Review Questions at the End of Chapter 10

1. (a) 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 holds, 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
that 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 state-determining 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, 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.

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(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 determines
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 (10.10). 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 ones removes one item to decide on (the state-determining
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 each other 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 6 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 of 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 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 (RSSs) 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 (10.26).

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(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

where is the log of the exchange rate, expressed in French francs per
German mark, and and are the logs of the French and German
consumer price series, respectively. We could define
as the deviation from PPP (see Chapter 8). 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

We could then consider the different values of the parameter estimates in each of the
three 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 non-stationary!). 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 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 6. 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.

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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 +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 +t-12 + 1It
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.

4. (a) State space models are a way of formulating a time-series model that allows for
time-varying parameters involving a transition equation and a measurement equation.
In the way that state space models are used in economics and finance, the
measurement equation describes the relationship between the variables of interest
(akin to a standard regression model), while the transition equation describes how the
‘parameters’ from the measurement equation vary over time. The Kalman filter
provides a recursive method for calculating the appropriate estimates of the state
vector (let us call this ty1:t) – it is adaptive in the sense that it makes use of
information available to time t and then modifies the estimates of t as t grows and
more data points become known.

(b) Both of these techniques provide methods to estimate the state vector (e.g., t) in a
time-varying parameters model. The key difference is that the Kalman filter only uses

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observations up to time t to produce estimates of the state vector at that time ( ty1:t),
while the smoother uses all T observations (ty1:T). The filter moves forwards through
the data while the smoother uses all information in the sample. So which of the two
methods to use will depend upon the objective: the filter can provide real time
estimates of the state vector to use in prediction, for example, while the smoother
provides a retrospective best fit of all of the data.

(c) This is actually fairly straightforward: in essence we estimate the state space
model assuming that the state variables (e.g., t) do vary over time and then look at
the variance of the error term in the state equation. Since this error term (referred to
as 2 in the book) is the only driver in the transition equation, if this is ‘small’ then t
will not be varying much over time and can be assumed to be constant. In order to
assess whether 2 is ‘small’ or not, we construct the ratio of 2 to u2, i.e., the ratio
of these hyperparameters: the variance of the error term in the state equation to the
variance of the error term in the measurement equation. The larger is this ratio, the
more evidence there is that the state variable is time-varying and we would test for
the extent to which this ratio is statistically significantly different from zero.

(d) Such a model would be a combination of the two simple time-varying parameter
specifications that were described in the book where we allow both the intercept and
the slope in the model to vary

yt = t + txt + ut
t+1 = t + 1t
t+1 = t + 2t

where yt is the return on the fund at time t in excess of the risk-free rate, xt is the
return on a proxy for the market portfolio at time t, and 1t and 2t are the two sets of
error terms in the transition equations.

(e) The Kalman filter constructs estimates of the state vector, while maximum
likelihood is used to provide estimates of the hyperparameters (i.e., 2 to u2) in the
state space formulation.

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