BUS 10123 Chapter Notes - Chapter 10: The Dummies, Multicollinearity, Dependent And Independent VariablesExam
SchoolKent State University
DepartmentBusiness Administration Interdisciplinary
Course CodeBUS 10123
ProfessorEric Von Hendrix
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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 (XX) 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:
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3x3t + ut .
A model containing these variables but also including intercept dummy
variables would be:
7D4t + ut .
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:
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.
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