which is the ordinary least squares procedure. The denominator term, in
fact, weights each data point in the cost function to be inversely propor-
tional to the variance of the individual error terms. Thus, it may also be
thought of as the sum of squared errors each normalized by its variance.
This approach to regression using the chi-squared merit function is some-
times also called the weighted least squares approach.
However, typical applications adopting a weighted least squares ap-
proach assume the error only in the response variable and not in both.
Specifically, those applications do not have the term with g^2 in the denomi-
nator. The presence of this term in the denominator complicates the mini-
mization of the chi-squared function, in that the derivative of the
chi-squared function with respect to gis now nonlinear, and so we may need
to resort to numerical methods to solve this. In summary, the regression
process can get fairly involved if we are to account for the varying error
probability distributions in the measurement with errors in measurement for
both the variables.
Nevertheless, if there is a way to construct a price series such that the er-
rors associated with the observation in each time period may be assumed to
be the same, then we can do away with these complications, work with just
ordinary least squares, and arrive at a reasonable answer. Let us see if we
might be able to do that.
Note that in the previous paragraph we mentioned that we choose only
one representative price for a given time period from the range of stock
price movement for that period. In a typical scenario where the length of a
period is one trading day, the standard convention in the construction of
daily time series is to use the closing price at the end of the trading day; that
is, the latest price in the process of serial price adjustment. Let us call this
method for recording the price time series the close-closemethod. Time se-
ries of this type are usually constructed to perform a mark to market of stock
inventory. Given the nature of the auction process and the price discovery
mechanism, this is definitely a reasonable choice. But is this approach to
data construction appropriate for our purposes? It is seductive by habit to
use the same construction regardless of purpose even though such an exer-
cise may be ill-advised. Care must be taken to ensure that the process of data
construction is a reflection of the specific purpose at hand.
Our purpose is to examine the price relationship between two stocks. In
this quest, to examine price relationships, more important than the closing
price in a given time period is the answer to the question, “At what price in
the time period was the liquidity a maximum?” That would be the consen-
sus price in the time chunk at which the most buyers and sellers agreed that
the price was right and a lot of shares changed hands. Therefore, conclusions
drawn on the maximum liquidity price series of two stocks would be more
reliable than using the close-close approach. A reasonable proxy for the
Testing for Tradability 111