data set. In fact, it has often been argued that the simplicity of the regression
process is probably its most powerful feature. Then why do we even attempt
to discuss it at length? Well, it is also true that although the general idea of
regression is relatively simple, the simplicity of the process lends itself to
hasty application without much thought. Thus, ironically, the pervasiveness
and simplicity of the regression approach is the reason to discuss aspects of
the regression approach in some detail.
Proper use of regression is possible only if we thoroughly understand the
standard regression scenario and the deviations of our situation from the
standard. Let us therefore examine the standard regression scenario as ap-
plied to physical systems. Central to physical systems is the study of cause
and effect; that is, the response of the system to a particular stimulus. If the
expectation is that the response is proportional to the stimulus, the process
of linear regression comes in very handy in measuring this constant of pro-
portionality. The typical experiment involves subjecting the system to a se-
ries of inputs or a stimulus in a specified range and measuring the response
of the system to those inputs. The input–output pairs then form the data set
on which the regression is run. The independent variable in this case is the
input or stimulus, and the slope of the regression is the constant of propor-
tionality connecting the stimulus to the response.
Let us now highlight some aspects of the experimental process. Note
that in this case, since the experimenter is the one administering the stimu-
lus to the system, he or she can design it to be accurate with a very small
magnitude of error. We can therefore assume that there is no error in the
input data. However, the output data come from the response of the system
and may not be known accurately due to imperfect experimental conditions.
Also, if the experimental conditions do not change dramatically during the
course of the trials, we may assume that the error manifest in each observa-
tion of the response is drawn from a common probability distribution. Thus,
here is a situation where the input is known relatively accurately, and the
source of error is only in the output with an error standard deviation that is
constant across all observations.
Now let us see how stock price data relates to the scenario described
above. First, it may be argued that the price of a stock is known exactly.
Then where is the source of error? Note that we use just one representative
value for the price in a given time period, while in actuality the price changes
constantly within the period itself. Therefore, a reasonable case may be
made for the existence of uncertainty or error associated with our choice for
the price in the time period. Next, in our situation, there is no distinct sep-
aration of cause and effect. The prices of the two stocks may very well be
feeding off each other, as discussed in the error correction model for cointe-
gration in Chapter 5. Since it is not possible to easily separate cause from ef-
fect in this scenario, and both of the price values are read as outputs, we are
Testing for Tradability 109