# BUS 10123 Chapter Notes - Chapter 10: The Dummies, Multicollinearity, Dependent And Independent VariablesExam

by OC2511532

School

Kent State UniversityDepartment

Business Administration InterdisciplinaryCourse Code

BUS 10123Professor

Eric Von HendrixStudy Guide

FinalThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**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:

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

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 .

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.

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