(Linear) Regression: The workhorse of empirical research in the social sciences

All example files discussed below can be loaded from the Data >
Manage page. Click the `examples`

radio button and press
`Load`

.

Start by selecting a response variable and one or more explanatory variables. If two or more explanatory variables are included in the model we may want to investigate if any interactions are present. An interaction exists when the effect of an explanatory variable on the response variable is determined, at least partially, by the level of another explanatory variable. For example, the increase in price for a 1 versus a 2 carrot diamond may depend on the clarity level of the diamond.

In the *Summary* tab we can test if two or more variables
together add significantly to the fit of a model by selecting variables
in the `Variables to test`

dropdown. This functionality can
be very useful to test if the overall influence of a variable of type
`factor`

is significant.

Additional output that requires re-estimation:

- Standardize: Coefficients can be hard to compare if the explanatory variables are measured on different scales. By standardizing the response variable and the explanatory variables before estimation we can see which variables move-the-needle most. Radiant standardizes data by replacing the response variable \(Y\) by \((Y - mean(Y))/(2 \times sd(Y))\) and replacing all explanatory variables \(X\) by \((X - mean(X))/(2 \times sd(X))\). See Gelman 2008 for discussion
- Center: Replace the response variable Y by Y - mean(Y) and replace all explanatory variables X by X - mean(X). This can be useful when trying to interpret interaction effects
- Stepwise: A data-mining approach to select the best fitting model. Use with caution!
- Robust standard errors: When
`robust`

is selected the coefficient estimates are the same as OLS. However, standard errors are adjusted to account for (minor) heterogeneity and non-normality concerns.

Additional output that does not require re-estimation:

- RMSE: Root Mean Squared Error and Residual Standard Deviation
- Sum of Squares: The total variance in the response variable split into the variance explained by the Regression and the variance that is left unexplained (i.e., Error)
- VIF: Variance Inflation Factors and Rsq. These are measures of multi-collinearity among the explanatory variables
- Confidence intervals: Coefficient confidence intervals

The *Predict* tab allows you calculate predicted values from a
regression model. You can choose to predict responses for a dataset
(i.e., select `Data`

from the `Prediction input`

dropdown), based on a command (i.e., select `Command`

from
the `Prediction input`

dropdown), or a combination of the two
(i.e., select `Data & Command`

from the
`Prediction input`

dropdown).

If you choose `Command`

you must specify at least one
variable and value to get a prediction. If you do not specify a value
for each variable in the model, either the mean value or the most
frequent level will be used. It is only possible to predict outcomes
based on variables in the model (e.g., `carat`

must be one of
the selected explanatory variables to predict the `price`

of
a 2-carat diamond).

- To predict the price of a 1-carat diamond type
`carat = 1`

and press return - To predict the price of diamonds ranging from .5 to 1 carat at steps
of size .05 type
`carat = seq(.5,.1,.05)`

and press return - To predict the price of 1,2, or 3 carat diamonds with an ideal cut
type
`carat = 1:3, cut = "Ideal"`

and press return

Once the desired predictions have been generated they can be saved to
a CSV file by clicking the download icon on the top right of the screen.
To add predictions to the dataset used for estimation, click the
`Store`

button.

The *Plot* tab is used to provide basic visualizations of the
data as well as diagnostic plots to validate the regression model.

We have access to data from a company selling men’s and women’s
apparel through mail-order catalogs (dataset `catalog`

). The
company maintains a database on past and current customers’ value and
characteristics. Value is determined as the total $ sales to the
customer in the last year. The data are a random sample of 200 customers
from the company’s database and has the following 4 variables

- Sales = Total sales (in $) to a household in the past year
- Income = Household income ($1000)
- HH.size = Size of the household (# of people)
- Age = Age of the head of the household

The catalog company is interested in redesigning their Customer Relationship Management (CRM) strategy. We will proceed in two steps:

- Estimate a regression model using last year’s sales total. Response
variable: sales total for each of the 200 households; Explanatory
variables: household income (measured in thousands of dollars), size of
household, and age of the household head. To access this dataset go to
*Data > Manage*, select`examples`

from the`Load data of type`

dropdown, and press the`Load`

button. Then select the`catalog`

dataset. - Interpret each of the estimated coefficients. Also provide a statistical evaluation of the model as a whole.
- Which explanatory variables are significant predictors of customer value (use a 95% confidence level)?

**Answer:**

Select the relevant variables mentioned above and press the
`Estimate model`

button or press `CTRL-enter`

(`CMD-enter`

on mac). Output from *Model > Linear
regression (OLS)* is provided below:

The null and alternate hypothesis for the F-test can be formulated as follows:

- \(H_0\): All regression coefficients are equal to 0
- \(H_a\): At least one regression coefficient is not equal to zero

The F-statistic suggests that the regression model as a whole
explains a significant amount of variance in `Sales`

. The
calculated F-statistic is equal to 32.33 and has a very small p.value
(< 0.001). The amount of variance in sales explained by the model is
equal to 33.1% (see R-squared).

We can replicate the standard F-test that is reported as part of all
regression output by selecting `income`

,
`HH.size`

, and `Age`

in the
`Variables to test`

box. The relevant output is shown
below.

Note that in this example, “model 1” is a regression without
explanatory variables. As you might expect, the explained variance for
model 1 is equal to zero. The F-test compares the *fit* of model
1 and model 2, adjusted for the difference in the number of coefficients
estimated in each model. The test statistic to use is described below.
\(R^2_2\) is the explained variance for
model 2 and \(R^2_1\) is the explained
variance for model 1. \(n\) is equal to
the number of rows in the data, and \(k_2\) (\(k_1\)) is equal to the number of estimated
coefficients in model 2 (model 1).

\[ \begin{eqnarray} F & = & \frac{(R^2_2 - R^2_1)/(k_2 - k_1)}{(1 - R^2_2)/(n - k_2 - 1)} \\\\ & = & \frac{(0.331 - 0)/(3 - 0)}{(1 - 0.331)/(200 - 3 - 1)} \\\\ & = & 32.325 \end{eqnarray} \]

We can use the provided p.value associated with an F-statistic of 32.325 to evaluate the null hypothesis. We can also calculate the critical F-statistic using the probability calculator. As we can see from the output below that value is 2.651. Because the provided p.value is < 0.001 and the calculated F-statistic is larger than the critical value (32.325 > 2.651) we reject null hypothesis that all coefficient are equal to zero.

The coefficients from the regression can be interpreted as follows:

- For an increase in income of $1000 we expect, on average, to see an increase in sales of $1.7754, keeping all other variables in the model constant.
- For an increase in household size of 1 person we expect, on average, to see an increase in sales of $22.1218, keeping all other variables in the model constant.
- For an increase in the age of the head of the household of 1 year we expect, on average, to see an increase in sales of $0.45, keeping all other variables in the model constant.

For each of the explanatory variables the following null and alternate hypotheses can be formulated:

- H0: The coefficient associated with explanatory variable x is equal to 0
- Ha: The coefficient associated with explanatory variable x is not equal to 0

The coefficients for `Income`

and `HH.size`

are
both significant (p.values < 0.05), i.e., we can reject H0 for each
of these coefficients. The coefficient for `Age HH`

is not
significant (p.value > 0.05), i.e., we cannot reject H0 for
`Age HH`

. We conclude that a change in Age of the household
head does not lead to a significant change in sales.

We can also use the t.values to evaluate the null and alternative
hypotheses for the coefficients. Because the calculated t.value for
`Income`

and `HH.size`

is **larger**
than the *critical* t.value we reject the null hypothesis for
both effects. We can obtain the critical t.value by using the
probability calculator in the *Basics* menu. For a t-distribution
with 196 degrees of freedom (see the `Errors`

line in the
`Sum of Squares`

table shown above) the critical t.value is
1.972. We have to enter 0.025 and 0.975 as the lower and upper
probability bounds in the probability calculator because the alternative
hypothesis is `two.sided`

.

The data `ideal`

contains simulated data that is very
useful to demonstrate what data for, and residuals from, a regression
should ideally look like. The data has 1,000 observations on 4
variables. `y`

is the response variable and `x1`

,
`x2`

, and `x3`

are explanatory variables. The
plots shown below can be used as a bench mark for regressions on real
world data. We will use *Model > Linear regression (OLS)* to
conduct the analysis. First, go the the *Plots* tab and select
`y`

as the response variable and `x1`

,
`x2`

, and `x3`

as the explanatory variables.

`y`

, `x2`

, and `x3`

appear (roughly)
normally distributed whereas `x1`

appears (roughly) uniformly
distributed. There are no indication of outliers or severely skewed
distributions.

In the plot of correlations there are clear associations among the response and explanatory variables as well as among the explanatory variables themselves. Note that in an experiment the x’s of interest would have a zero correlation. This is very unlikely in historical data however. The scatter plots in the lower-diagonal part of the plot show that the relationships between the variables are (approximately) linear.

The scatter plots of `y`

(the response variable) against
each of the explanatory variables confirm the insight from the
correlation plot. The line fitted through the scatter plots is
sufficiently flexible that it would pickup any non-linearities. The
lines are, however, very straight, suggesting that a linear model will
likely be appropriate.

The dashboard of six residual plots looks excellent, as we might expect for these data. True values and predicted values from the regression form a straight line with random scatter, i.e., as the actual values of the response variable go up, so do the predicted values from the model. The residuals (i.e., the differences between the values of the response variable data and the values predicted by the regression) show no pattern and are randomly scattered around a horizontal line. Any pattern would suggest that the model is better (or worse) at predicting some parts of the data compared to others. If a pattern were visible in the Residual vs Row order plot we might be concerned about auto-correlation. Again, the residuals are nicely scattered about a horizontal axis. Note that auto-correlation is a problem we are really only concerned about when we have time-series data. The Q-Q plot shows a nice straight and diagonal line, evidence that the residuals are normally distributed. This conclusion is confirmed by the histogram of the residuals and the density plot of the residuals (green) versus the theoretical density of a normally distributed variable (blue line).

The final diagnostic we will discuss is a set of plots of the residuals versus the explanatory variables (or predictors). There is no indication of any trends or heteroscedasticity. Any patterns in these plots would be cause for concern. There are also no outliers, i.e., points that are far from the main cloud of data points.

Since the diagnostics look good, we can draw inferences from the
regression. First, the model is significant as a whole: the p.value on
the F-statistic is less than 0.05 therefore we reject the null
hypothesis that all three variables in the regression have slope equal
to zero. Second, each variable is statistically significant. For
example, the p.value on the t-statistic for `x1`

is less than
0.05 therefore we reject the null hypothesis that `x1`

has a
slope equal to zero when `x2`

and `x3`

are also in
the model (i.e., ‘holding all other variables in the model
constant’).

Increases in `x1`

and `x3`

are associated with
increases in `y`

whereas increases in `x2`

are
associated with decreases in `y`

. Since these are simulated
data the exact interpretation of the coefficient is not very
interesting. However, in the scatterplot it looks like increases in
`x3`

are associated with decreases in `y`

. What
explains the difference? Hint: consider the correlation plots.

Both linear and log-log regressions are commonly applied to business data. In this example we will look for evidence in the data and residuals that may suggest which model specification is appropriate for the available data.

The data `diamonds`

contains information on prices of
3,000 diamonds. A more complete description of the data and variables is
available from the *Data > Manage* page. Select the variable
`price`

as the response variable and `carat`

and
`clarity`

as the explanatory variables. Before looking at the
parameter estimates from the regression go to the *Plots* tab to
take a look at the data and residuals. Below are the histograms for the
variables in the model. `Price`

and `carat`

seem
skewed to the right. Note that the direction of skew is determined by
where the *tail* is.

In the plot of correlations there are clear associations among the
response and explanatory variables. The correlation between
`price`

and `carat`

is very large (i.e., 0.93).
The correlation between `carat`

and `clarity`

of
the diamond is significant and negative.

The scatter plots of `price`

(the response variable)
against the explanatory variables are not as clean as for the
`ideal`

data in Example 2. The line fitted through the
scatter plots is sufficiently flexible to pickup non-linearities. The
line for `carat`

seems to have some curvature and the points
do not look randomly scattered around that line. In fact the points seem
to fan-out for higher prices and number of carats. There does not seem
to be very much movement in `price`

for different levels of
`clarity`

. If anything, the price of the diamond seems to go
down as clarity increase. A surprising result we will discuss in more
detail below.

The dashboard of six residual plots looks less than stellar. The true
values and predicted values from the regression form an S-shaped curve.
At higher actual and predicted values the spread of points around the
line is wider, consistent with what we saw in the scatter plot of
`price`

versus `carat`

. The residuals (i.e., the
differences between the actual data and the values predicted by the
regression) show an even more distinct pattern as they are clearly not
randomly scattered around a horizontal axis. The Residual vs Row order
plot looks perfectly straight indicating that auto-correlation is not a
concern. Finally, while for the `ideal`

data in Example 2 the
Q-Q plot showed a nice straight diagonal line, here dots clearly
separate from the line at the right extreme. Evidence that the residuals
are not normally distributed. This conclusions is confirmed by the
histogram and density plots of the residuals that show a more spiked
appearance than a normally distributed variable would.

The final diagnostic we will discuss is a set of plots of the
residuals versus the explanatory variables (or predictors). The
residuals fan-out from left to right in the plot of residuals vs carats.
The scatter plot of `clarity`

versus residuals shows outliers
with strong negative values for lower levels of `clarity`

and
outliers with strong positive values for diamonds with higher levels of
`clarity`

.

Since the diagnostics do not look good, we should
**not** draw inferences from this regression. A log-log
specification may be preferable. A quick way to check the validity of a
log-log model change is available through the *Data >
Visualize* tab. Select `price`

as the Y-variable and
`carat`

as the X-variable in a `Scatter`

plot.
Check the `log X`

and `log Y`

boxes to produce the
plot below. The relationship between log-price and log-carat looks close
to linear. Exactly what we are looking for!

We will apply a (natural) log (or *ln*) transformation to both
`price`

and `carat`

and rerun the analysis to see
if the log-log specification is more appropriate for the available data.
This transformation can be done in *Data > Transform*. Select
the variables `price`

and `carat`

. Choose
`Transform`

from the `Transformation type`

drop-down and choose `Ln (natural log)`

from the
`Apply function`

drop-down. Make sure to click the
`Store`

button so the new variables are added to the dataset.
Note that we cannot apply a log transformation to `clarity`

because it is a
categorical
variable.

In *Model > Linear regression (OLS)* select the variable
`price_ln`

as the response variable and `carat_ln`

and `clarity`

as the explanatory variables. Before looking at
the parameter estimates from the regression go to the *Plots* tab
to take a look at the data and residuals. Below are the histograms for
the variables in the model. Note that `price_ln`

and
`carat_ln`

are not right skewed, a good sign.

In the plot of correlations there are still clear associations among
the response and explanatory variables. The correlation between
`price_ln`

and `carat_ln`

is extremely large
(i.e., .93). The correlation between `carat_ln`

and
`clarity`

of the diamond is significant and negative.

The scatter plots of `price_ln`

(the response variable)
against the explanatory variables are now much cleaner. The line through
the scatter plot of `price_ln`

versus `carat_ln`

is (mostly) straight. Although the points do have a bit of a blocked
shape around the line, the scattering seems mostly random. We no longer
see the points fan-out for higher values of `price_ln`

and
`carat_ln`

. There seems to be a bit more movement in
`price_ln`

for different levels of `clarity`

.
However, the `price_ln`

of the diamond still goes down as
`clarity`

increases which is unexpected. We will discuss this
result below.

The dashboard of six residual plots looks much better than for the
linear model. The true values and predicted values from the regression
(almost) form a straight line. Although at higher and lower actual and
predicted values the line is perhaps still very slightly curved. The
residuals are much closer to a random scatter around a horizontal line.
The Residual vs Row order plot still looks perfectly straight indicating
that auto-correlation is not a concern. Finally, the Q-Q plot shows a
nice straight and diagonal line, just like we saw for the
`ideal`

data in Example 2. Evidence that the residuals are
now normally distributed. This conclusion is confirmed by the histogram
and density plot of the residuals.

The final diagnostic we will discuss is a set of plots of the
residuals versus the explanatory variables (or predictors). The
residuals look much closer to random scatter around a horizontal line
compared to the linear model. Although for low (high) values of
`carat_ln`

the residuals may be a bit higher (lower).

Since the diagnostics now look much better, we can feel more
confident about drawing inferences from this regression. The regression
results are available in the *Summary* tab. Note that we get 7
coefficients for the variable clarity compared to only one for
`carat_ln`

. How come? If you look at the data description
(*Data > Manage*) you will see that clarity is a categorical
variables with levels that go from IF (worst clarity) to I1 (best
clarity). Categorical variables must be converted to a set of dummy (or
indicator) variables before we can apply numerical analysis tools like
regression. Each dummy indicates if a particular diamond has a
particular clarity level (=1) or not (=0). Interestingly, to capture all
information in the 8-level clarity variable we only need 7 dummy
variables. Note there is no dummy variable for the clarity level I1
because we don’t actually need it in the regression. When a diamond is
**not** of clarity SI2, SI1, VS2, VS1, VVS2, VVS1 or IF we
know that in our data it must be of clarity I1.

The F-statistic suggests that the regression model as a whole
explains a significant amount of variance in `price_ln`

. The
F-statistic is very large and has a very small p.value (< 0.001) so
we can reject the null hypothesis that all regression coefficients are
equal to zero. The amount of variance in `price_ln`

explained
by the model is equal to 96.6. It seems likely that prices of diamonds
will be much easier to predict than demand for diamonds.

The null and alternate hypothesis for the F-test can be formulated as follows: H0: All regression coefficients are equal to 0 Ha: At least one regression coefficient is not equal to zero

The coefficients from the regression can be interpreted as follows:

- For a 1% increase in carats we expect, on average, to see a 1.809% increase in the price of a diamond of, keeping all other variables in the model constant
- Compared to a diamond of clarity I1 we expect, on average, to pay 100x(exp(.444)-1) = 55.89% more for a diamond of clarity SI2, keeping all other variables in the model constant
- Compared to a diamond of clarity I1 we expect, on average, to pay 100x(exp(.591)-1) = 80.58% more for a diamond of clarity SI1, keeping all other variables in the model constant
- Compared to a diamond of clarity I1 we expect, on average, to pay 100x(exp(1.080)-1) = 194.47% more for a diamond of clarity IF, keeping all other variables in the model constant

The coefficients for each of the levels of clarity imply that an
increase in `clarity`

will increase the price of diamond. Why
then did the scatter plot of clarity versus (log) price show price
decreasing with clarity? The difference is that in a regression we can
determine the effect of a change in one variable (e.g., clarity) keeping
all other variables in the model constant (i.e., carat). Bigger,
heavier, diamonds are more likely to have flaws compared to small
diamonds so when we look at the scatter plot we are really seeing the
effect of not only improving clarity on price but also the effect of
carats which are negatively correlated with clarity. In a regression, we
can compare the effects of different levels of clarity on (log) price
for a diamond of **the same size** (i.e., keeping carat
constant). Without (log) carat in the model the estimated effect of
clarity would be incorrect due to
omitted
variable bias. In fact, from a regression of `price_ln`

on `clarity`

we would conclude that a diamond of the highest
clarity in the data (IF) would cost 59.22% less compared to a diamond of
the lowest clarity (I1). Clearly this is not a sensible conclusion.

For each of the explanatory variables the following null and alternate hypotheses can be formulated: H0: The coefficient associated with explanatory variable X is equal to 0 Ha: The coefficient associated with explanatory variable X is not equal to 0

All coefficients in this regression are highly significant.

Add code to
*Report
> Rmd* to (re)create the analysis by clicking the
icon on the bottom
left of your screen or by pressing `ALT-enter`

on your
keyboard.

If a plot was created it can be customized using `ggplot2`

commands or with `patchwork`

. See example below and
*Data
> Visualize* for details.

Copy-and-paste the full command below into the RStudio console (i.e., the bottom-left window) and press return to gain access to all materials used in the linear regression module of the Radiant Tutorial Series:

usethis::use_course("https://www.dropbox.com/sh/s70cb6i0fin7qq4/AACje2BAivEKDx7WrLrPr5m9a?dl=1")

Data Exploration and Pre-check of Regression (#1)

- This video shows how to use Radiant to explore and visualize data before running a linear regression
- Topics List:
- View data
- Visualize data

Interpretation of Regression Results and Prediction (#2)

- This video explains how to interpret the regression results and calculate the predicted value from a linear regression model
- Topics List:
- Interpret coefficients (numeric and categorical variables)
- Interpret R-squared and adjusted R-squared
- Interpret F-test result
- Predict from a regression model

Dealing with Categorical Variables (#3)

- This video shows how to deal with categorical variables in a linear regression model
- Topics List:
- Check the baseline category in Radiant
- Change the baseline category

Adding New Variables into a Regression Model (#4)

- This video demonstrates how to test if adding new variables will lead to a better model with significantly higher explanatory power
- Topics List:
- Set up a hypothesis test for adding new variables in Radiant
- Interpret the F-test results
- Compare this F-test to the default F-test in regression summary

Linear Regression Validation (#5)

- This video demonstrates how to validate a linear regression model
- Topics List:
- Linearity (scatter plots, same as the one in the pre-check)
- Normality Check (Normal Q-Q plot)
- Multicollinearity (VIF)
- Heteroscedasticity

- This video demonstrates when and how to run a log-log regression
- Topics List:
- Transform data with skewed distributions by natural log function
- Interpret the coefficients in a log-log regression

To illustrate the interpretation of coefficients in a regression model we start with the following equation:

\[ S_t = a + b P_t + c D_t + \epsilon_t \]

where \(S_t\) is sales in units at time \(t\), \(P_t\) is the price in $ at time \(t\), \(D_t\) is a dummy variable that indicates if a product is on display in a given week, and \(\epsilon_t\) is the error term.

For a continuous variable such as price we can determine the effect of a $1 change, while keeping all other variables in the model constant, by taking the partial derivative of the sales equation with respect to \(P\).

\[ \frac{ \partial S_t }{ \partial P_t } = b \]

So \(b\) is the marginal effect on sales of a $1 change in price. Because a dummy variable such as \(D\) is not continuous we cannot use differentiation and the approach needed to determine the marginal effect is a little different. If we compare sales levels when \(D = 1\) to sales levels when \(D = 0\) we see that

\[ a + b P_t + c \times 1 - a + b P_t + c \times 0 = c \]

For a linear model \(c\) is the marginal effect on sales when the product is on display.

To illustrate the interpretation of coefficients in a semi-log regression model we start with the following equation:

\[ ln S_t = a + b P_t + c D_t + \epsilon_t \]

where \(ln S_t\) is the (natural) log of sales at time \(t\). For a continuous variable such as price we can again determine the effect of a small change (e.g., $1 for a $100 dollar product), while keeping all other variables in the model constant, by taking the partial derivative of the sales equation with respect to \(P\). For the left-hand side of the equation we can use the chain-rule to get

\[ \frac {\partial ln S_t}{\partial P_t} = \frac{1}{S_t} \frac{\partial S_t}{\partial P_t} \]

In words, the derivative of the natural logarithm of a variable is the reciprocal of that variable, times the derivative of that variable. From the discussion on the linear model above we know that

\[ \frac{ \partial a + b P_t + c D_t}{ \partial P_t } = b \]

Combining these two equations gives

\[ \frac {1}{S_t} \frac{\partial S_t}{\partial P_t} = b \; \text{or} \; \frac {\Delta S_t}{S_t} \approx b \]

So a $1 change in price leads to a \(100 \times b\%\) change in sales. Note that this approximation is only appropriate for small changes in the explanatory variable and may be affected by the scaling used (e.g., price in cents, dollars, or 1,000s of dollars). The approach outlined below for dummy variables is more general and can also be applied to continuous variables.

Because a dummy variable such as \(D\) is not continuous we cannot use differentiation and will again compare sales levels when \(D = 1\) to sales levels when \(D = 0\) to get \(\frac {\Delta S_t}{S_t}\). To get \(S_t\) rather than \(ln S_t\) on the left hand side we take the exponent of both sides. This gives \(S_t = e^{a + b P_t + c D_t}\). The percentage change in \(S_t\) when \(D_t\) changes from 0 to 1 is then given by:

\[ \begin{aligned} \frac {\Delta S_t}{S_t} &\approx \frac{ e^{a + b P_t + c\times 1} - e^{a + b P_t + c \times 0} } {e^{a + b P_t + c \times 0} }\\ &= \frac{ e^{a + b P_t} e^c - e^{a + b P_t} }{ e^{a + b P_t} }\\ &= e^c - 1 \end{aligned} \]

For the semi-log model \(100 \times \: (exp(c)-1)\) is the percentage change in sales when the product is on display. Similarly, for a $10 increase in price we would expect a \(100 \times \: (exp(b \times 10)-1)\) increase in sales, keeping other variables constant.

To illustrate the interpretation of coefficients in a log-log regression model we start with the following equation:

\[ ln S_t = a + b ln P_t + \epsilon_t \]

where \(ln P_t\) is the (natural) log of sales at time \(t\). Ignoring the error term for simplicity we can rewrite this model in its multiplicative form by taking the exponent on both sides:

\[ \begin{aligned} S_t &= e^a + e^{b ln P_t}\\ S_t &= a^* P^b_t \end{aligned} \]

where \(a^* = e^a\) For a continuous variable such as price we can again take the partial derivative of the sales equation with respect to \(P_t\) to get the marginal effect.

\[ \begin{aligned} \frac{\partial S_t}{\partial P_t} &= b a^* P^{b-1}_t\\ &= b S_t P^{-1}_t\\ &= b \frac{S_t}{P_t} \end{aligned} \]

The general formula for an elasticity is \(\frac{\partial S_t}{\partial P_t} \frac{P_t}{S_t}\). Adding this information to the equation above we see that the coefficient \(b\) estimated from a log-log regression can be directly interpreted as an elasticity:

\[ \frac{\partial S_t}{\partial P_t} \frac{P_t}{S_t} = b \frac{S_t}{P_t} \frac{P_t}{S_t} = b \]

So a 1% change in price leads to a \(b\)% change in sales.

For an overview of related R-functions used by Radiant to estimate a
linear regression model see
*Model
> Linear regression (OLS)*.

The key functions used in the `regress`

tool are
`lm`

from the `stats`

package and `vif`

and `linearHypothesis`

from the `car`

package.