Notes on linear
regression analysis (pdf file)

Introduction
to linear regression analysis

Regression
example, part 1: descriptive analysis

Regression
example, part 2: fitting a simple model

Regression
example, part 3: transformations of variables

Regression
example, 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

Spreadsheet with
regression formulas (new version including RegressIt output)

Stepwise and all-possible-regressions

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 an intercept-only model.) It is called R-squared because in 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.

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, 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 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 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 R-squared be measured--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.

Morever, 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 an intercept-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 intercept-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
an intercept-only model, on average.

Generally
it is better to look at *adjusted*
R-squared rather than R-squared and to look at the *standard error of the regression* rather than the standard deviation
of the errors. However, the
difference between them is usually very small unless you are trying to estimate
too many coefficients from too small a sample in the presence of too much
noise. (Specifically, adjusted R-squared is equal to 1 minus (n-1)/(n-k-1) times 1-minus-R-squared, where n is the sample size
and k is the number of independent variables.) Adjusted R-squared bears the same
relation to the standard error of the regression that R-squared bears to the
standard deviation of the errors, so the same table applies to the former pair
as well as the latter pair.

**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
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 which yields *useful* predictions
and inferences in the process, 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. Second, *the model’s largest errors have occurred in the last few periods*
(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 variance of the errors increases steadily over time, both locally
and globally*. 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.

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 an intercept-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 intercept. 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: an intercept-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.
The model does not shed light on the relationship between personal
income and auto sales, but neither did the previous two models, to be perfectly
honest.

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 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, 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 and the situation is such that even a weak signal would be
informative or valuable.

However, **caution is needed when evaluating a model
with a very 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 large lineup of randomly
selected suspects; (ii) 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 (iii) 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 an intercept-only model, which merely
predicts that everything will equal the mean. Another handy reference point: if the model has an R-squared of 75%,
its errors are 50% smaller on average than those of an intercept-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.)

**What value of
R-squared should you report to your boss or client? **If you used
regression analysis, then to be perfectly candid you should of course include
the R-squared for the regression model that was actually fitted, 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**.

**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 an
intercept-only model (i.e., mean model) fitted to the same dependent variable*,
but the intercept-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**