Chapter 7
Regression Models and Hypothesis
Testing
Abstract This chapter covers the basics of adapting regression models, also known
as linear fits in physics, to find the parameters that best explain data in a model and
then estimate the error bars of the parameters. The analysis of the model’s reliability
stimulates a discussion of χ
2 and t-distributions and their role in testing hypotheses
regarding the parameters; for example, whether a parameter can be omitted from the
fit. A more elaborate method, based on the F-test, follows. The chapter closes with
a discussion of parsimony as a guiding principle when building models.
Both in physical and social sciences as well as in economics we often try to
describe the behavior of observable parameters—the measurements—in terms of
model parameters. Frequently the dependence of the measurements on the parameters of a model—the fit parameters—is linear; the archetypical problem of this type
is the fit of data points to a straight line. Let us therefore assume that we are given
a list of n data points (s i , y i ) for i = 1, . . . , n that come from varying a variable s
and observing another quantity y. Figure 7.1 shows two examples. Visual inspection
indicates that the data points cluster around a straight line that we can describe by
two parameters a and b, given by y i = as i + b. Our task is thus to find the slope
a and the intercept b. This is most easily accomplished by writing one copy of the
equation for each of the n measurements, which defines one row in the following
matrix-valued equation
⎛
⎜
⎜
⎝
. . .
y i
. . .
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
. . .
. . .
s i 1
. . .
. . .
⎞
⎟
⎟
⎠
a
b
.
(7.1)
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_7) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_7
69
Regression Models and Hypothesis
Testing
Abstract This chapter covers the basics of adapting regression models, also known
as linear fits in physics, to find the parameters that best explain data in a model and
then estimate the error bars of the parameters. The analysis of the model’s reliability
stimulates a discussion of χ
2 and t-distributions and their role in testing hypotheses
regarding the parameters; for example, whether a parameter can be omitted from the
fit. A more elaborate method, based on the F-test, follows. The chapter closes with
a discussion of parsimony as a guiding principle when building models.
Both in physical and social sciences as well as in economics we often try to
describe the behavior of observable parameters—the measurements—in terms of
model parameters. Frequently the dependence of the measurements on the parameters of a model—the fit parameters—is linear; the archetypical problem of this type
is the fit of data points to a straight line. Let us therefore assume that we are given
a list of n data points (s i , y i ) for i = 1, . . . , n that come from varying a variable s
and observing another quantity y. Figure 7.1 shows two examples. Visual inspection
indicates that the data points cluster around a straight line that we can describe by
two parameters a and b, given by y i = as i + b. Our task is thus to find the slope
a and the intercept b. This is most easily accomplished by writing one copy of the
equation for each of the n measurements, which defines one row in the following
matrix-valued equation
⎛
⎜
⎜
⎝
. . .
y i
. . .
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
. . .
. . .
s i 1
. . .
. . .
⎞
⎟
⎟
⎠
a
b
.
(7.1)
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_7) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_7
69