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shows a z-distribution and a t-distribution. The t-distribution is very sim-
ilar to the normal (Z) distribution, with t being slightly more spread out.
The equation for the t-distribution is:
b
t (2-6)
x2
s/ i
i
where the denominator is the standard error of b, commonly denoted as
sb (the standard error of a is sa).
Since is unobservable, we have to make an assumption about it in
order to calculate a t-distribution for it. The usual procedure is to test for
the probability that, regardless of the regression™s estimate of ”which
is our b”the true is really zero. In statistics, this is known as the ˜˜null
hypothesis.™™ The magnitude of the t-statistic is indicative of our ability
to reject the null hypothesis for an individual variable in the regression
equation. When we reject the null hypothesis, we are saying that our
regression estimate of is statistically signi¬cant.
We can construct 95% con¬dence intervals around our estimate, b, of
the unknown . This means that we are 95% sure the correct value of
is in the interval described in equation (2-7).
b t0.025 sb (2-7)

Formula for 95% confidence interval for the slope
Figure 2-2 shows a graph of the con¬dence interval. The graph is a
t-distribution, with its center at b, our regression estimate of . The mark-
ings on the x-axis are the number of standard errors below or above b.
As mentioned before, we denote the standard error of b as sb. The lower
boundary of the 95% con¬dence interval is b t0.025 sb, and the upper
boundary of boundary of the 95% con¬dence interval is b t0.025 sb. The




CHAPTER 2 Using Regression Analysis 33
34




F I G U R E 2-1

Z-distribution vs t-distribution


0.4




0.35




0.3




0.25
probability density




0.2




0.15




0.1



t t
0.05

Z Z

0
-6 -4 -2 0 2 4 6
for Z, standard deviations from mean
For t, standard errors from mean
F I G U R E 2-2

t-distribution of B around the Estimate b


0.4




0.35




0.3
probability density


0.25




0.2




0.15

area = 2.5%
area =2.5%
0.1




0.05




0
-6 -4 -2 0 2 4 6
B =b+t0.025 sb
B =b
B= b“t 0.025sb B measured in standard
errors away from b
35
area under the curve for any given interval is the probability that will
be in that interval.
The t-distribution values are found in standard tables in most statis-
tics books. It is very important to use the 0.025 probability column in the
tables for a 95% con¬dence interval, not the 0.05 column. The 0.025 col-
umn tells us that for the given degrees of freedom there is a 21„2% prob-
ability that the true and unobservable is higher than the upper end of
the 95% con¬dence interval and a 21„2% probability that the true and
unobservable is lower than the lower end of the 95% con¬dence interval
(see Figure 2-2). The degrees of freedom is equal to n k 1, where n
is the number of observations and k is the number of independent vari-
ables.
Table 2-3 is an excerpt from a t-distribution table. We use the 0.025
column for a 95% con¬dence interval. To select the appropriate row in
the table, we need to know the number of degrees of freedom. Assuming
n 10 observations and k one independent variable, there are eight
degrees of freedom (10 1 1). The t-statistic in Table 2-3 is 2.306 (C7).
That means that we must go 2.306 standard errors below and above our
regression estimate to achieve a 95% con¬dence interval for . The re-
gression itself will provide us with the standard error of . As n, the
number of observations, goes to in¬nity, the t-distribution becomes a z-
distribution. When n is large”over 100”the t-distribution is very close
to a standardized normal distribution. You can see this in Table 2-3 in
that the standard errors in Row 9 are very close to those in Row 10, the
latter of which is equivalent to a standardized normal distribution.
The t-statistics for our regression in Table 2-1B are 3.82 (D33) and
56.94 (D34). The P-value, also known as the probability (or prob) value,
represents the level at which we can reject the null hypothesis. One minus
the P-value is the level of statistical signi¬cance of the y-intercept and
independent variable(s). The P-values of 0.005 (E33) and 10 11 (E34) mean
that the y-intercept and slope coef¬cients are signi¬cant at the 99.5% and
99.9% levels, respectively, which means we are 99.5% sure that the true
y-intercept is not zero and 99.9% sure that the true slope is not zero.10

T A B L E 2-3

Abbreviated Table of T-Statistics


A B C D

4 Selected t Statistics

5 d.f.\Pr. 0.050 0.025 0.010
6 3 2.353 3.182 4.541
7 8 1.860 2.306 2.896
8 12 1.782 2.179 2.681
9 120 1.658 1.980 2.358
10 In¬nity 1.645 1.960 2.326




10. For spreadsheets that do not provide P-values, another way of calculating the statistical
signi¬cance is to look up the t-statistics in a Student™s t-distribution table and ¬nd the level
of statistical signi¬cance that corresponds to the t-statistic obtained in the regression.


PART 1 Forecasting Cash Flows
36
The F test is another method of testing the null hypothesis. In mul-
tivariable regressions, the F-statistic measures whether the independent
variables as a group explain a statistically signi¬cant portion of the var-
iation in Y.
We interpret the con¬dence intervals as follows: there is a 95% prob-
ability that true ¬xed costs (the y-intercept) fall between $22,496 (F33) and
$91,045 (G33); similarly, there is a 95% probability that the true variable
cost (the slope coef¬cient) falls between $0.77 (F34) and $0.84 (G34).
The denominator of equation (2-6) is called the standard error of b,
or sb. The standard error of the Y-estimate, which is de¬ned as
n
1 ˆ
Yi)2
s (Yi
n 2 i1



is $16,014 (B23). The larger the amount of scatter of the points around
the regression line, the greater the standard error.11

Precise Con¬dence Intervals12
Earlier in the chapter, we estimated 95% con¬dence intervals by subtract-
ing and adding two standard errors of the y-estimate around the regres-
sion estimate. In this section, we demonstrate how to calculate precise
95% con¬dence intervals around the regression estimate using the equa-
tions:

x2
1 o
t0.25s (2-8)
x2
n i

95% confidence interval for the mean forecast

x2
1 o
t0.025s 1 (2-9)
x2
n i

95% confidence interval for a specific year™s forecast
In the context of forecasting adjusted costs as a function of sales,
equation (2-8) is the formula for the 95% con¬dence interval for the mean
adjusted cost, while equation (2-9) is the 95% con¬dence interval for the
costs in a particular year. We will explain what that means at the end of
this section, after we present some material that illustrates this in Table
2-1B, page 2.
Note that these con¬dence intervals are different than those in equa-
tion (2-7), which was around the forecast slope only, i.e., b. In this section,


11. This standard error of the Y-estimate applies to the mean of our estimate of costs, i.e., the
average error if we estimate adjusted costs and expenses many times. This is appropriate in
valuation, as a valuation is a forecast of net income and / or cash ¬‚ows for an in¬nite
number of years. The standard error”and hence 95% con¬dence interval”for a single
year™s costs is higher.
12. This section is optional, as the material is somewhat advanced, and it is not necessary to
understand this in order to be able to use regression analysis in business valuation.
Nevertheless, it will enhance your understanding should you choose to read it.


CHAPTER 2 Using Regression Analysis 37
we are calculating con¬dence intervals around the entire regression fore-
cast.
The ¬rst 15 rows of Table 2-1B, page 2, are identical to the ¬rst page
and require no explanation. The $989,032 in B16 is the average of the 10
years of sales in B6“B15.
Column D is the deviation of each observation from the mean, which
is the sales in Column B minus the mean sales in B16. For example, D6

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