Notes on linear
regression analysis (pdf file)

Introduction
to linear regression analysis

Mathematics
of simple regression

Regression examples

·
Beer sales vs. price, part 1: descriptive
analysis

·
Beer sales vs. price, part 2: fitting a simple
model

·
Beer sales vs. price, part 3: transformations
of variables

·
Beer sales vs.
price, part 4: additional predictors

What to look for in regression
output

What’s a good
value for R-squared?

What's the bottom line? How to compare models

Testing the assumptions of linear regression

Additional notes on regression
analysis

Stepwise and all-possible-regressions

Excel file with
simple regression formulas

Excel file with regression formulas
in matrix form

If you are a PC Excel user, you *must* check this out:

RegressIt: free Excel add-in for
linear regression and multivariate data analysis

Percent
of variance explained vs. percent of standard deviation explained

An example in
which R-squared is a poor guide to analysis

Guidelines for
interpreting R-squared

The
question is often asked: "what's a good value for R-squared?" or
“how big does R-squared need to be for the regression model to be
valid?” Sometimes the claim
is even made: "a model is not useful unless its R-squared is at least
x", where x may be some fraction greater than 50%. The correct response to this question is
polite laughter followed by: "That depends!" A former student of mine landed a job at
a top consulting firm by being the only candidate who gave that answer during
his interview.

R-squared
is the “percent of variance explained” by the model. That is,** R-squared is the fraction by which the variance of the errors is less
than the variance of the dependent variable. **(The latter number would be the
error variance for a constant-only model, which merely predicts that every
observation will equal the sample mean.)
It is called R-squared because in a simple regression model it is just *the square of the correlation* between
the dependent and independent variables, which is commonly denoted by
“r”. In a *multiple* regression model R-squared is
determined by pairwise correlations among *all*
the variables, including correlations of the independent variables with each
other as well as with the dependent variable. In the latter setting, the square root of
R-squared is known as “multiple R”, and it is equal to the
correlation between the dependent variable and the regression model’s
predictions for it. (Note: if the
model does not include a constant, which is a so-called “regression
through the origin”, then R-squared has a different definition. See this page for more
details. *You cannot compare R-squared
between a model that includes a constant and one that does not*.)

Generally
it is better to look at ** adjusted R-squared** rather than
R-squared and to look at the

Now, **what is the relevant variance that requires
explanation, and how much or how little explanation is necessary or useful?** There is a huge range of
applications for linear regression analysis in science, medicine, engineering,
economics, finance, marketing, manufacturing, sports, etc.. In some situations the variables under
consideration have very strong and intuitively obvious relationships, while in
other situations you may be looking for very weak signals in very noisy
data. The decisions that depend on
the analysis could have either narrow or wide margins for prediction error, and
the stakes could be small or large.
For example, in medical research, a new drug treatment might have highly
variable effects on individual patients, in comparison to alternative
treatments, and yet have statistically significant benefits in an experimental
study of thousands of subjects.
That is to say, the amount of variance explained when predicting
individual outcomes could be small, and yet the estimates of the coefficients
that measure the drug’s effects could be significantly different from
zero (as measured by low P-values) in a large sample. A result like this could save many lives
over the long run and be worth millions of dollars in profits if it results in
the drug’s approval for widespread use.

Even in
the context of a single statistical decision problem, there may be many ways to
frame the analysis, resulting in different standards and expectations for the
amount of variance to be explained in the linear regression stage. We have seen by now that there are many *transformations*
that may be applied to a variable before it is used as a dependent variable in
a regression model: deflation, logging, seasonal adjustment, differencing. All
of these transformations will change the variance and may also change the *units*
in which variance is measured. Logging completely changes the the units of measurement
(roughly speaking, the error measures become percentages rather than absolute
amounts, as explained here),
while deflation and seasonal adjustment and differencing generally reduce the
variance significantly when properly applied. Therefore, if the dependent
variable in the regression model has already been transformed in some way, it
is possible that much of the variance has already been "explained"
merely by that process. With
respect to which variance should improvement be measured in such cases: that of
the original series, the deflated series, the seasonally adjusted series, the
differenced series, or the logged series?
*You cannot meaningfully compare
R-squared between models that have used different transformations of the
dependent variable,* as the example below will illustrate.

Moreover,
variance is a hard quantity to think about because it is measured in *squared units* (dollars squared, beer
cans squared….). It is easier
to think in terms of *standard deviations*,
because they are measured in the same units as the variables and they directly
determine the widths of confidence intervals. So, it is instructive to also consider
the **“percent of standard deviation
explained,”** i.e., the percent by which the standard deviation of the
errors is less than the standard deviation of the dependent variable. This is equal to **one minus the square root of 1-minus-R-squared**. Here is a table that shows the
conversion:

For
example, if the model’s R-squared is 90%, the variance of its errors is
90% less than the variance of the dependent variable and the standard deviation
of its errors is 68% less than the standard deviation of the dependent
variable. That is, the standard
deviation of the regression model’s errors is about 1/3 the size of the
standard deviation of the errors that you would get with a constant-only model. That’s very good, but it
doesn’t sound quite as impressive as “NINETY PERCENT
EXPLAINED!”.

If the
model’s R-squared is 75%, the standard deviation of the errors is exactly
one-half of the standard deviation of the dependent variable. Now, suppose that the addition of
another variable or two to this model increases R-squared to 76%. That’s better, right? Well, by the
formula above, this increases the percent of standard deviation explained from
50% to 51%, which means the standard deviation of the errors is reduced from
50% of that of the constant-only model to 49%, a shrinkage of 2% in relative
terms. Confidence intervals for
forecasts produced by the second model would therefore be about 2% narrower
than those of the first model, on average, not enough to notice on a
graph. You should ask
yourself: is that worth the
increase in model complexity?

An
increase in R-squared from 75% to 80% would reduce the error standard deviation
by about 10% in relative terms.
That begins to rise to the level of a perceptible reduction in the
widths of confidence intervals. But don’t forget, confidence intervals
are realistic guides to the accuracy of predictions *only if the model’s assumptions are correct*. When adding more variables to a model, you need to
think about the cause-and-effect assumptions that implicitly go with them, and
you should also look at how their addition changes the estimated coefficients
of other variables. Do they become
easier to explain, or harder? And
do the residual stats and plots indicate that the model’s assumptions are
OK? If they aren’t, then you
shouldn’t be obsessing over small improvements in R-squared anyway. Your problems lie elsewhere.

Another
handy rule of thumb: *for small values
(R-squared less than 25%), the percent of standard deviation explained is
roughly one-half of the percent of variance explained.* So, for example, a
model with an R-squared of 10% yields errors that are 5% smaller than those of
a constant-only model, on average.

**How big an R-squared is “big
enough”, or cause for celebration or despair? ** That depends on the decision-making
situation, and it depends on your objectives or needs, and it depends on how
the dependent variable is defined.
In some situations it might be reasonable to hope and expect to explain
99% of the variance, or equivalently 90% of the standard deviation of the
dependent variable. In other cases,
you might consider yourself to be doing very well if you explained 10% of the
variance, or equivalently 5% of the standard deviation, or perhaps even less. The following section gives an example
that highlights these issues. If
you want to skip the example and go straight to the concluding comments, click here.

**An example in which
R-squared is a poor guide to analysis:**
Consider the U.S. monthly auto sales series that was used for
illustration in the first chapter of these notes, whose graph is reproduced
here:

The units
are $billions and the date range shown here is from January 1970 to February
1996. Suppose that the objective of
the analysis is to predict monthly auto sales from monthly total personal
income. I am using these variables
(and this antiquated date range) for two reasons: (i) this very (silly) example was used
to illustrate the benefits of regression analysis in a textbook that I was
using in that era, and (ii) I have seen many students undertake self-designed
forecasting projects in which they have blindly fitted regression models using
macroeconomic indicators such as personal income, gross domestic product,
unemployment, and stock prices as predictors of nearly everything, the logic being that they reflect the
general state of the economy and therefore have implications for every kind of
business activity. Perhaps so, but
the question is whether they do it in a *linear,
additive* fashion that stands out against the background noise in the
variable that is to be predicted, and whether they adequately explain *time patterns* in the data, and whether they
yield *useful* predictions and
inferences in comparison to other ways in which you might choose to spend your
time. Return to top
of page.

The
corresponding graph of personal income (also in $billions) looks like this:

There is no
seasonality in the income data. In
fact, there is almost no pattern in it at all except for a trend that increased
slightly in the earlier years. (This
is not a good sign if we hope to get forecasts that have any specificity.) By comparison, the seasonal
pattern is the most striking feature in the auto sales, so the first thing that
needs to be done is to *seasonally adjust*
the latter. Seasonally adjusted
auto sales (independently obtained from the same government source) and
personal income line up like this when plotted on the same graph:

The strong and
generally similar-looking trends suggest that we will get a very high value of
R-squared if we regress sales on income, and indeed we do. Here is the summary table for that
regression:

Adjusted R-squared
is almost 97%! However, a result
like this is to be expected when regressing a strongly trended series on *any other strongly trended series*,
regardless of whether they are logically related. Here are the line fit plot and
residuals-vs-time plot for the model:

The
residual-vs-time plot indicates that the model has some terrible problems.
First, there is very strong *positive
autocorrelation in the errors,* i.e., a tendency to make the same error many
times in a row. In fact, the lag-1
autocorrelation is 0.77 for this model.
It is clear why this happens:
the two curves do not have exactly the same shape. The trend in the auto sales series tends
to vary over time while the trend in income is much more consistent, so the two
variales get out-of-synch with each other.
This is typical of nonstationary time series data. Second, *the model’s largest errors have
occurred in the more recent years and especially in the last few months* (at
the “business end” of the data, as I like to say), which means that
we should expect the next few errors to be huge too, given the strong positive
correlation between consecutive errors.
And finally, *the local variance of
the errors increases steadily over time*. The reason for this is that random
variations in auto sales (like most other measures of macroeconomic activity)
tend to be consistent over time in *percentage*
terms rather than absolute terms, and the absolute level of the series has
risen dramatically due to a combination of inflationary growth and real
growth. As the level as grown, the
variance of the random fluctuations has grown with it. Confidence intervals for forecasts in the
near future will therefore be way too narrow, being based on average error
sizes over the whole history of the series. So, despite the high value of
R-squared, this is a *very* bad
model. Return to top
of page.

One way to try to
improve the model would be to *deflate *both
series first. This would at least
eliminate the inflationary component of growth, which hopefully will make the
variance of the errors more consistent over time. Here is a time series plot showing auto
sales and personal income after they have been deflated by dividing them by the
U.S. all-product consumer price index (CPI) at each point in time, with the CPI
normalized to a value of 1.0 in February 1996 (the last row of the data). This does indeed flatten out the trend
somewhat, and it also brings out some fine detail in the month-to-month
variations that was not so apparent on the original plot. In particular, we begin to see some
small bumps and wiggles in the income data that roughly line up with larger bumps
and wiggles in the auto sales data.

If we fit a simple
regression model to these two variables, the following results are obtained:

Adjusted R-squared
is only 0.788 for this model, which is worse, right? Well, no. We “explained” some of the
variance in the original data by deflating it prior to fitting this model. Because the dependent variables are not
the same, it is not appropriate to do a head-to-head comparison of
R-squared. Arguably this is a better
model, because it separates out the real growth in sales from the inflationary
growth, and also because the errors have a more consistent variance over
time. (The latter issue is not the
bottom line, but it is a step in the direction of fixing the model
assumptions.) Most interestingly,
the deflated income data shows some fine detail that matches up with similar
patterns in the sales data.
However, the error variance is still a long way from being constant over
the full two-and-a-half decades, and the problems of badly autocorrelated
errors and a particularly bad fit to the most recent data have not been
solved.

Another
statistic that we might be tempted to compare between these two models is the
standard error of the regression, which normally is the best bottom-line
statistic to focus on. The second
model’s standard error is much larger: 3.253 vs. 2.218 for the first model. But wait… *these two numbers cannot be directly compared, either, because they are
not measured in the same units.*
The standard error of the first model is measured in units of *current dollar*s, while the standard
error of the second model is measured in units of *1996 dollar*s. Those
were decades of high inflation, and 1996 dollars were not worth nearly as much
as dollars were worth in the earlier years. (In fact, a 1996 dollar was only worth
about one-quarter of a 1970 dollar.)
Return to top of page.

The slope
coefficients in the two models are also of interest. Because the units of the dependent
and independent variables are the same in each model (current dollars in the
first model, 1996 dollars in the second model), *the slope coefficient can be interpreted as the predicted increase in
dollars spent on autos per dollar of increase in income.* The slope coefficients in the two models
are nearly identical: 0.086 and
0.087, implying that on the margin, 8.6% to 8.7% of additional income is spent
on autos.

Let’s now try
something totally different: fitting a simple time series model to the deflated
data. In particular, let’s
fit a **random-walk-with-drift model,**
which is logically equivalent to fitting a constant-only model to the* first difference *(period to period
change) in the original series. Let
the differenced series be called AUTOSALES_SADJ_1996_DOLLARS_DIFF1 (which is
the name that would be automatically assigned in RegressIt). Notice that we are now 3 levels deep in
data transformations: seasonal
adjustment, deflation, and differencing!
This sort of situation is very common in time series analysis. Here are the results of fitting this
model, in which AUTOSALES_SADJ_1996_DOLLARS_DIFF1 is the dependent variables
and there are no independent variables, just the constant. This model merely predicts that each
monthly difference will be the same, i.e., it predicts constant growth relative
to the previous month’s value.

Adjusted R-squared
has dropped to zero! This is not a
problem: a constant-only regression always has an R-squared of zero, but that
doesn’t necessarily imply that it is not a good model for the particular
dependent variable that has been used. We should look instead at the
standard error of the regression.
The units and sample of the dependent variable are the same for this
model as for the previous one, so their regression standard errors can be
legitimately compared. (The
sample size for the second model is actually 1 less than that of the first
model due to the lack of period-zero value for computing a period-1 difference,
but this is insignificant in such a large data set.) The regression standard error of this
model is only 2.111, compared to 3.253 for the previous one, a reduction of
roughly one-third, which is a very significant improvement. (The residual-vs-time plot for this
model and the previous one have the same vertical scaling: look at them both and compare the size
of the errors, particularly those that have occurred recently.) The reason why this model’s
forecasts are so much more accurate is that it looks at* last month’s actual sales values*, whereas the previous model
only looked at personal income data.
*It is often the case that the best
information about where a time series is going to go next is where it has been
lately.*

There is no line
fit plot for this model, because there is no independent variable, but here is
the residual-versus-time plot:

These residuals
look quite random to the naked eye, but they actually exhibit *negative autocorrelation*, i.e., a
tendency to alternate between overprediction and underprediction from one month
to the next. (The lag-1
autocorrelation here is -0.356.)
This often happens when differenced data is used, but overall the errors
of this model are much closer to being independently and identically
distributed than those of the previous two, so we can have a good deal more
confidence in any confidence intervals for forecasts that may be computed from
it. Of course, this model does not shed
light on the relationship between personal income and auto sales.

So, what is the
relationship between auto sales and personal income? That is a complex question and it will
not be further pursued here except to note that there some other simple things
we could do besides fitting a regression model. For example, we could compute the *percentage of income spent on automobiles
over time*, i.e., just divide the auto sales series by the personal income
series and see what the pattern looks like. Here is the resulting picture:

This chart nicely
illustrates cyclical variations in the fraction of income spent on autos, which
would be interesting to try to match up with other explanatory variables. The range is from about 7% to about 10%,
which is generally consistent with the slope coefficients that were obtained in
the two regression models (8.6% and 8.7%).
However, this chart re-emphasizes what was seen in the residual-vs-time
charts for the simple regression models:
the fraction of income spent on autos is not consistent over time. In particular, notice that the fraction
was increasing toward the end of the sample, exceeding 10% in the last month.

The bottom line
here is that **R-squared was not of any
use in guiding us through this particular analysis toward better and better
models**. In fact, among the
models considered above, the worst one had an R-squared of 97% and the best one
had an R-squared of zero. At
various stages of the analysis, data transformations were suggested: seasonal adjustment, deflating,
differencing. (Logging was not
tried here, but would have been an alternative to deflation.) And every time the dependent variable is
transformed, it becomes impossible to make meaningful before-and-after
comparisons of R-squared.
Furthermore, regression was probably not even the best tool to use here
in order to study the relation between the two variables. It is not a “universal wrench”
that should be used on every problem.
Return to top of page.

**So,
what IS a good value for R-squared?
**It
depends on the variable with respect to which you measure it, it depends on the
units in which that variable is measured and whether any data transformations have
been applied, and it depends on the decision-making context. If the dependent variable is a
nonstationary (e.g., trending or random-walking) time series, an R-squared
value very close to 1 (such as the 97% figure obtained in the first model
above) may not be very impressive.
In fact, if R-squared is very close to 1, and the data consists of time
series, this is usually a bad sign rather than a good one: there will often be significant time
patterns in the errors, as in the example above. On the other hand, if the dependent
variable is a properly *stationarized* series (e.g., differences or
percentage differences rather than levels), then an R-squared of 25% may be
quite good. In fact, an R-squared of 10% or even less could have some information
value when you are looking for a weak signal in the presence of a lot of noise
in a setting where even a *very *weak
one would be of general interest. Sometimes there is a lot of value in
explaining only a very small fraction of the variance, and sometimes there
isn't. Data transformations such as logging or deflating also change the
interpretation and standards for R-squared, inasmuch as they change the
variance you start out with.

However, **be very careful when evaluating a model
with a low value of R-squared**.
In such a situation: (i) it
is better if the set of variables in the model is determined a priori (as in
the case of a designed experiment or a test of a well-posed hypothesis) rather
by searching among a lineup of randomly selected suspects; (ii) the data should
be clean (not contaminated by outliers, inconsistent measurements, or
ambiguities in what is being measured, as in the case of poorly worded surveys
given to unmotivated subjects); (iii) the coefficient estimates should be individually
or at least jointly significantly different from zero (as measured by their
P-values and/or the P-value of the F statistic), which may require a large
sample to achieve in the presence of low correlations; and (iv) it is a good
idea to do **cross-validation**
(out-of-sample testing) to see if the model performs about equally well on data
that was not used to identify or estimate it, particularly when the structure
of the model was not known a priori.
It is easy to find spurious (accidental) correlations if you go on a
fishing expedition in a large pool of candidate independent variables while
using low standards for acceptance.
I have often had students use this approach to try to predict stock
returns using regression models--which I do not recommend--and it is not
uncommon for them to find models that yield R-squared values in the range of 5%
to 10%, but they virtually never survive out-of-sample testing. (You should buy index funds instead.)

There are a variety
of ways in which to cross-validate a model. A discussion of some of them can be
found here. If your software doesn’t offer
such options, there are simple tests you can conduct on your own. One is to split the data set in half and
fit the model separately to both halves to see if you get similar results in
terms of coefficient estimates and adjusted R-squared.

When working with **time series data**, if you compare the
standard deviation of the errors of a regression model which uses exogenous
predictors against that of a simple time series model (say, an autoregressive
or exponential smoothing or random walk model), you may be disappointed by what
you find. If the variable to be
predicted is a time series, it will often be the case that most of the
predictive power is derived from its own history via lags, differences, and/or
seasonal adjustment. This is the reason why we spent some time studying the
properties of time series models before tackling regression models.

**A rule of thumb for small values of R-squared**: If R-squared is small (say 25% or less),
then the fraction by which the standard deviation of the errors is less than
the standard deviation of the dependent variable is *approximately* one-half of R-squared, as shown in the table above. So,
for example, if your model has an R-squared of 10%, then its errors are only
about 5% smaller on average than those of a constant-only model, which merely
predicts that everything will equal the mean. Is that enough to be useful, or
not? Another handy reference
point: if the model has an
R-squared of 75%, its errors are 50% smaller on average than those of a
constant-only model. (This is not an approximation: it follows directly from the fact that
reducing the error standard deviation to ½ of its former value is
equivalent to reducing its variance to ¼ of its former value.)

In general you
should **look at adjusted R-squared rather
than R-squared**. Adjusted
R-squared is an *unbiased* estimate of
the fraction of variance explained, taking into account the sample size and
number of variables. Usually
adjusted R-squared is only slightly smaller than R-squared, but it is possible
for adjusted R-squared to be zero or negative if a model with insufficiently
informative variables is fitted to too small a sample of data.

**What measure of
your model's explanatory power should you report to your boss or client or
instructor? **If
you used regression analysis, then to be perfectly candid you should of course
include the adjusted R-squared for the regression model that was actually
fitted (whether to the original data or some transformation thereof), along
with other details of the output, somewhere in your report. You should more strongly emphasize the **standard error of the regression**,
though, because that measures the predictive accuracy of the model in real
terms, and it scales the width of all confidence intervals calculated from the
model. You may also want to report
other practical measures of error size such as the **mean absolute error** or **mean
absolute percentage error** and/or **mean absolute scaled error**.

**What should never
happen to you: **Don't
ever let yourself fall into the trap of fitting (and then promoting!) a
regression model that has a respectable-looking R-squared but is actually very
much inferior to a simple time series model. If the dependent variable in your
model is a nonstationary time series, be sure that you do a comparison of error
measures against an appropriate time series model. Remember that what R-squared measures is
the proportional reduction in error variance that the regression model achieves* in comparison to a constant-only model
(i.e., mean model) fitted to the same dependent variable*, but the
constant-only model may not be the most appropriate reference point, and the
dependent variable you end up using may not be the one you started with if data
transformations turn out to be important.

**And finally: R-squared is not the bottom line**

Go on to next topic: How to compare models