Linear regression is the most
widely used of all statistical techniques:
it is the study of *linear* (i.e., straight-line) relationships
between variables, usually
under an assumption of *normally distributed errors*.

The first thing you ought to
know about linear regression is how
the strange term *regression* came to be applied to
the subject of linear statistical models. This type of predictive
model was first studied in depth by a 19th-Century scientist,
Sir Francis Galton. Galton was a self-taught naturalist,
anthropologist,
astronomer, and statistician--and a real-life Indiana Jones
character. He was famous for his explorations, and he wrote a
best-selling book on how to survive in the wilderness entitled
"Shifts and Contrivances Available in Wild Places." (The book is still
in print and still considered a useful resource--you
can find a copy in Perkins Library. Among other handy hints for
staying alive--such as how to treat spear-wounds or extract your
horse from quicksand--it introduced the concept of the sleeping
bag to the Western World.)

Galton was a pioneer in the
application of statistical methods
to human measurements, and in studying data on relative heights
of fathers and their sons, he observed the following phenomenon: a
taller-than-average father tends to produce a taller-than-average
son, but the son is likely to be *less* tall than the father
in terms of his relative position within his own population. Thus, for
example, if the father's height is** x** standard
deviations from the mean within his own population, then you should
predict that the son's height will be** rx** (r times x) standard
deviations from the mean within his own population, where r is
a number *less than 1 in magnitude*. (**r** is what will
be defined below as the *correlation *between the height
of the father and the height of the son.) The same is true of
virtually *any* physical measurement than can be performed
on parents and their offspring. This seems at first glance like
evidence of some genetic or sociocultural mechanism for damping
out extreme physical traits, and Galton therefore termed it a
"*regression toward mediocrity*," which in modern
terms is a "*regression to the mean*." But the
phenomenon discovered by Galton is a mathematical inevitability:
unless every son is *exactly* as tall as his father in a
relative sense (i.e., unless the correlation is exactly equal
to 1), the predictions *must* regress to the mean regardless
of the underlying mechanisms of inheritance or culture.

Regression to the mean is an
inescapable fact of life. Your children
can be *expected* to be less exceptional (for better or worse)
than you are. Your score on a final exam in a course can be *expected*
to be less good (or bad) than your score on the midterm exam.
A baseball player's batting average in the second half of the
season can be *expected* to be closer to the mean (for all
players) than his batting average in the first half of the season. And
so on. The key word here is "expected." This does
not mean it's *certain* that regression to the mean will
occur, but that's the way to bet! (More precisely, that's the
way to bet if you wish to minimize squared error.)

We have already seen a
suggestion of regression-to-the-mean in
some of the time series forecasting models we have studied already:
plots of forecasts tend to be* smoother*--i.e., they exhibit
less variability--than the plots of the original data. (This
is not true of random walk models, but it is generally true of
smoothing models and other models in which coefficients other
than a constant term are estimated.) The intuitive explanation
for the regression effect is simple: the thing we are trying
to predict usually consists of a predictable component ("signal")
and a statistically independent *unpredictable* component
("noise"). The best we can hope to do is to predict
the value of the signal, and then let the noise fall where it may.
Hence our forecasts will tend to exhibit less variability than
the actual values, which implies a regression to the mean.

Another way to think of the
regression effect is in terms of sampling
bias. Suppose we select a sample of baseball players whose batting
averages were much higher than the mean (or students whose grades
were much higher than the mean) in the first half of the year.
Presumably their averages were unusually high in part because
they were "good" and in part because they were "lucky." The fact that
they did so well in the first half of the year
makes it probable that *both* their ability and their luck
were better than average during that period. In the second half
of the year they may be just as good, but they probably will not
be as lucky. So we should predict that in the second half their
performance will be closer to the mean.

Now, why do we often assume
that relationships beween variables
are *linear*?

- Because linear relationships are the
*simplest non-trivial relationships*that can be imagined (hence the easiest to work with), and..... - Because the "true" relationships between our variables
are often at least
*approximately*linear over the range of values that are of interest to us, and... - Even if they're not, we can often
*transform*the variables in such a way as to linearize the relationships.

And why do we often assume the
errors of linear models are *normally
distributed*?

- This assumption is often justified by
appeal to the
*central limit theorem*of statistics, which states that the*sum*of a sufficiently large number of independent random variables--whatever their individual distributions--approaches a normal distribution. Much data in business and economics is obtained by*aggregation*--e.g., adding up the effects of all transactions occurring in a month, quarter, year, or whatever. Insofar as individual transactions may occur randomly and somewhat independently, we might expect the variations in the aggregate data to be approximately normal in distribution. - It is (again) mathematically convenient: it implies that
the
optimal coefficient estimates for a linear model are those that
minimize the
*mean squared error*(which are easily calculated), and it justifies the use of a host of statistical tests based on the normal family of distributions. (This family includes the t distribution, the F distribution, and the Chi-square distribution.) - Even if the "true" error process is not normal in terms of the original units of the data, it may be possible to transform the data so that your model's prediction errors are approximately normal.

A **variable** is, by
definition, *a quantity that varies*. (If
it didn't vary, it would be a constant, not a variable.) In fitting
statistical models in which some variables are used to predict
others, what we hope to find is that the different variables do
*not* vary *independently* (in a statistical sense),
but that they tend to vary *together*.

In particular, when fitting *linear*
models, we hope to find
that one variable (say, Y) is varying as a *straight-line*
function of another variable (say, X). In other words, if all
other possibly-relevant variables could be held fixed, we would
hope to find the *graph* of Y versus X to be a straight
line (apart from the inevitable random errors or "noise").

A measure of the absolute
amount of "variability" in
a variable is (naturally) its **variance**, which is defined
as its *average squared deviation from its own mean*.
Equivalently,
we can measure variability in terms of the **standard deviation**,
which is defined as the square root of the variance. The standard
deviation has the advantage that it is measured in the same units
as the original variable, rather than squared units.

Our task in predicting Y might
be described as that of "explaining"
some or all of its variance--i.e., *why*, or under what
conditions,
does it deviate from its mean? Why is it not constant? That is,
we would like to be able to improve on the "naive" predictive
model: Ý(t) = CONSTANT, in which the best value for the
constant is presumably the historical mean of Y. More precisely,
*we hope to find a model whose prediction errors are smaller,
in a mean square sense, than the deviations of the original variable
from its mean*.

In using *linear* models
for prediction, it turns out very
conveniently that the *only* statistics of interest (at least
for purposes of estimating coefficients) are the mean and variance
of each variable and the **correlation coefficient*** *
between each pair of variables. The coefficient of correlation
between X and Y is commonly denoted by the letter **r**.

The correlation coefficient
between two variables is a statistic
that measures the *strength of the linear relationship* between
them, on a relative (i.e., unitless) scale of -1 to +1. That is,
it measures the extent to which a linear model can be used to
predict the deviation of one variable from its mean given knowledge
of the other's deviation from its mean at the same point in time.

The correlation coefficient is
most easily computed if we first **standardize**
each of the variables--i.e., express it in units of standard deviations
from its own mean. The standardized value of X is commonly denoted
by X*, and the value of X* at period t is defined as:

**X*(t) = (X(t) -
AVERAGE(X))/STDEV(X)**

where AVERAGE(X) and STDEV(X) are the sample mean (average) and standard deviation of X, in Excel notation. For example, suppose that AVERAGE(X) = 20 and STDEV(X) = 5. If X(t) = 25, then X*(t) = 1, if X(t) = 10, then X*(t) = -2, and so on.

Now, **the correlation
coefficient is equal to the average product
of the standardized values** of the two variables. That is,
if we let X* and Y* denote the standardized values of X and Y,
we have:

**r = AVERAGE(X*Y*) =
(X*(1)Y*(1) + X*(2)Y*(2) + ... + X*(n)Y*(n))/n**

... where n is the sample size. Thus, for example, if X and Y are stored in columns on a spreadsheet, you can use the AVERAGE and STDEV functions to compute their averages and standard deviations, then you can create two new columns in which the values of X* and Y* in each period are computed according to the formula above. Then create a third new column in which X* is multiplied by Y* in every period. The average of the values in the last column is the correlation between X and Y. (Of course, in Excel, you can just use the =CORREL(x,y) function to calculate a correlation coefficient!)

If the two variables tend to
vary on the *same sides* of
their respective means at the same time, then the average product
of their deviations (and hence the correlation between them) will
be *positive*, since the product of two numbers with the
same sign is positive. Conversely, if they tend to vary on *opposite*
sides of their respective means at the same time, their correlation
will be *negative*. If they vary *independently* with
respect to their means--that is, if one is equally likely to be
above or below its mean regardless of what the other is doing--then
the correlation will be *zero*.

The correlation coefficient is
not only the average product of
the standardized values, but also: **the correlation coefficient
is the "best" coefficient for multiplying the standardized
value of one variable in order to predict the standardized value
of the other**. That is, the "best" linear model (in
a minimum-squared-error sense) for predicting Y* from X* turns
out to be:

**Ý*(t) = rX* (t)**

Thus, if X is observed to be 1 standard deviation above its own mean, then we should predict that Y will be r standard deviations above its own mean; if X is 2 standard deviations below its own mean, then we should be predict that Y will be 2r standard deviations below its own mean, and so on.

In graphical terms, this means
that, on a scatter plot of Y* versus
X*, the best-fit line for predicting Y* from X* is *the line
that passes through the origin and has slope* r. (Note: this
fact is not supposed to be obvious, but it is easily proved by
elementary differential calculus of several variables.)

If we want to obtain a formula
for predicting Y from X in *unstandardized
terms*, we just need to substitute the formulas for the standardized
values in the prededing equation, which then becomes:

**(Ý(t) -
AVERAGE(Y))/STDEV(Y)** = **r (X(t) - AVERAGE(X))/STDEV(X)**

If we now rearrange this equation and collect constant terms, we obtain:

**Ý(t) = a + bX(t)**

where:

**b = r(STDEV(Y)/STDEV(X))**

**a = AVERAGE(Y) -
b(AVERAGE(X))**

Notice that, as we claimed earlier, the coefficients in the linear equation for predicting Y from X depend only on the means and standard deviations of X and Y and on their coefficient of correlation. (By the way, these formulas for a and b are mathematically equivalent to the formulas for "b0" and "b", respectively, given on page 206 of your text. Personally, I find it much easier to remember the formulas in terms of the means and standard deviations and the correlation coefficient, and I find it easier to write the formula for the correlation coefficient in terms of the standardized values X* and Y* than in terms of the original values X and Y, as given on page 204 of your text.)

*Perfect* positive
correlation (r = +1) or perfect negative
correlation correlation (r = -1) is only obtained if one variable
is an *exact* linear function of the other, without error.
In such a case, one variable is merely a linear transformation
of the other--they aren't really "different" variables
at all!

In general, therefore, we find
less-than-perfect correlation,
which is to say, we find that r is less than 1 in absolute value.
Therefore our prediction for Y* will typically be *smaller*
in absolute value than our observed value for X*. That is, we
will always predict Y to be closer to its own mean, in units of
its own standard deviation, than X* *was observed to be,
which is Galton's phenomenon of regression to the mean.

So, the technical explanation of the regression-to-the-mean effect hinges on two mathematical facts: (i) the correlation coefficient, calculated in the manner described above, happens to be the coefficient that minimizes the squared error in predicting Y* from X*, and (ii) the correlation coefficient is never larger than 1 in absolute value, and it is only equal to 1 when Y* is an exact (noiseless) linear function of X*.

The term "regression" has stuck and has even mutated from an intransitive verb into a transitive one since Galton's time. We don't merely say that the predictions for Y "regress to the mean"--we now say that we are "regressing Y on X" when we estimate a linear equation for predicting Y from X, and we refer to X as a "regressor" in this case.

When we have fitted a linear
model, we can compute its mean squared
prediction error and compare this to the variance of the original
variable. As noted above, we hope to find that the MSE is *less*
than the original variance. The relative amount by which the mean
squared error is less than the variance of the original variable
is referred to as the *fraction* of the variance that was
*explained* by the model. For example, if the MSE is 20%
less than the original variance, we say we have "explained
20% of the variance."

It turns out that **in a
simple regression model **(a linear
model with only one *"*X" variable), **the fraction
of variance explained is precisely the square of the correlation
coefficient**--i.e., the square of r. Hence, the
fraction-of-variance-explained
has come to be known as "r-squared."

In a *multiple* regression model (a linear model with
two
or more "regressors"), there are many correlation coefficients
that must be computed, in addition to all the means and variances.
For example, we must consider the correlation between *each*
X variable and the Y variable, and also the correlation between
each *pair* of X variables. In this case, it still
turns
out that the model coefficients and the fraction-of-variance-explained
statistic can be computed entirely from knowledge of the means,
standard deviations, and correlation coefficients among the
variables--but
the computations are no longer easy. We will leave the details
to the computer...