In re: High-Tech Employee Antitrust Litigation

Filing 716

Omnibus Declaration of Christina J. Brown in Support of #715 Reply re Joint Motion to Exclude the Expert Testimony of Edward E. Leamer, Ph.D. , #714 Reply to Joint Motion to Strike the Improper Rebuttal Testimony in Dr. Leamer's Reply Expert Report or, in the Alternative, MOTION for Leave to Submit a Reply Report of Dr. Stiroh filed by Apple Inc.. (Attachments: #1 Exhibit A, #2 Exhibit B, #3 Exhibit C, #4 Exhibit D, #5 Exhibit E, #6 Exhibit F, #7 Exhibit G, #8 Exhibit H, #9 Exhibit I, #10 Exhibit J, #11 Exhibit K, #12 Exhibit L, #13 Exhibit M, #14 Exhibit N, #15 Exhibit O, #16 Exhibit P, #17 Exhibit Q)(Related document(s) #715 , #714 ) (Brown, Christina) (Filed on 2/27/2014) Modified text on 2/28/2014 (dhmS, COURT STAFF).

EXHIBIT K

OMNIBUS BROWN DECLARATION

Copyright © 2004

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Library of Congress Cataloging-in-Publication Data
Verbeek, Marno.
A guide to modern econometrics / Marno Verbeek. – 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-470-85773-0 (pbk. : alk. paper)
1. Econometrics. 2. Regression analysis. I. Title.
HB139.V465 2004
330 .01 5195 – dc22
2004004222
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-470-85773-0
Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India
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AN INTRODUCTION TO LINEAR REGRESSION

24

is a specific value chosen by the researcher. If this hypothesis is true we know that
the statistic
0
b − βk
tk = k
(2.49)
se(bk )
has a t distribution with N − K degrees of freedom. If the null hypothesis is not true,
0
the alternative hypothesis H1 : βk = βk holds. As there are no unknown values in tk ,
it becomes a test statistic that can be computed from the estimate bk and its standard
error se(bk ). The usual testing strategy is to reject the null hypothesis if tk realizes
a value that is very unlikely if the null hypothesis is true. In this case this means
very large absolute values for tk . To be precise, one rejects the null hypothesis if the
probability of observing a value of |tk | or larger is smaller than a given significance
level α, often 5%. From this, one can define the critical values tN−K;α/2 using
P {|tk | > tN−K;α/2 } = α.
For N − K not too small, these critical values are only slightly larger than those of
the standard normal distribution, for which the two-tailed critical value for α = 0.05
is 1.96. Consequently, at the 5% level the null hypothesis will be rejected if
|tk | > 1.96.
The above test is referred to as a two-sided test because the alternative hypothesis
0
allows for values of βk on both sides of βk . Occasionally, the alternative hypothesis is
one-sided, for example: the expected wage for a man is larger than that for a woman.
0
0
Formally, we define the null hypothesis as H0 : βk ≤ βk with alternative H1 : βk > βk .
Next we consider the distribution of the test statistic tk at the boundary of the null
0
hypothesis (i.e. under βk = βk , as before) and we reject the null hypothesis if tk is too
large (note that large values for bk lead to large values for tk ). Large negative values
for tk are compatible with the null hypothesis and do not lead to its rejection. Thus for
this one-sided test, the critical value is determined from
P {tk > tN−K;α } = α.
Using the standard normal approximation again, we reject the null hypothesis at the
5% level if
tk > 1.64.
Regression packages typically report the following t-value,
tk =

bk
,
se(bk )

sometimes referred to as the t-ratio, which is the point estimate divided by its standard
error. The t-ratio is the t-statistic one would compute to test the null hypothesis that
βk = 0, which may be a hypothesis that is of economic interest as well. If it is rejected,
it is said that ‘bk differs significantly from zero’, or that the corresponding variable

HYPOTHESIS TESTING

31

2.5.7 Size, Power and p-Values

When an hypothesis is statistically tested two types of errors can be made. The first
one is that we reject the null hypothesis while it is actually true, and is referred to as a
type I error. The second one, a type II error, is that the null hypothesis is not rejected
while the alternative is true. The probability of a type I error is directly controlled by
the researcher through his choice of the significance level α. When a test is performed
at the 5% level, the probability of rejecting the null hypothesis while it is true is 5%.
This probability (significance level) is often referred to as the size of the test. The
probability of a type II error depends upon the true parameter values. Intuitively, if the
truth deviates much from the stated null hypothesis, the probability of such an error
will be relatively small, while it will be quite large if the null hypothesis is close to the
truth. The reverse probability, that is, the probability of rejecting the null hypothesis
when it is false, is known as the power of the test. It indicates how ‘powerful’ a test
is in finding deviations from the null hypothesis (depending upon the true parameter
value). In general, reducing the size of a test will decrease its power, so that there is
a trade-off between type I and type II errors.
Suppose that we are testing the hypothesis that β2 = 0, while its true value is in fact
0.1. It is clear that the probability that we reject the null hypothesis depends upon the
standard error of our OLS estimator b2 and thus, among other things, upon the sample
size. The larger the sample the smaller the standard error and the more likely we are
to reject. This implies that type II errors become increasingly unlikely if we have large
samples. To compensate for this, researchers typically reduce the probability of type
I errors (that is of incorrectly rejecting the null hypothesis) by lowering the size α of
their tests. This explains why in large samples it is more appropriate to choose a size
of 1% or less rather than the ‘traditional’ 5%. Similarly, in very small samples we may
prefer to work with a significance level of 10%.
Commonly, the null hypothesis that is chosen is assumed to be true unless there
is convincing evidence of the contrary. This suggests that if a test does not reject,
for whatever reason, we stick to the null hypothesis. This view is not completely
appropriate. A range of alternative hypotheses could be tested (for example β2 = 0,
β2 = 0.1 and β2 = 0.5), with the result that none of them is rejected. Obviously,
concluding that these three null hypotheses are simultaneously true would be ridiculous.
The only appropriate conclusion is that we cannot reject that β2 is 0, nor that it is
0.1 or 0.5. Sometimes, econometric tests are simply not very powerful and very large
sample sizes are needed to reject a given hypothesis.
A final probability that plays a role in statistical tests is usually referred to as the
p-value. This p or probability value denotes the minimum size for which the null
hypothesis would still be rejected. It is defined as the probability, under the null, to
find a test statistic that (in absolute value) exceeds the value of the statistic that is
computed from the sample. If the p-value is smaller than the significance level α, the
null hypothesis is rejected. Many modern software packages supply such p-values and
in this way allow researchers to draw their conclusions without consulting or computing
the appropriate critical values. It also shows the sensitivity of the decision to reject
the null hypothesis, with respect to the choice of significance level. For example, a
p-value of 0.08 indicates that the null hypothesis is rejected at the 10% significance
level, but not at the 5% level.