Log-Log Regression - Dummy Variable and Index

Question

I have the following log-log regression equation (natural log was used):

ln(Sales Index) = B0 + B1 * ln(advertising spend) + B2 * (January) .... + e

where advertising spend is a continuous variable (is never zero) and January is a dummy variable. The Sales Index is never zero either.

I understand how to interpret the coefficient B1. Where I am getting stuck is in interpreting coefficient B2 (when January = 1). I've looked at the following questions for guidance:

Interpreting logarithmically transformed coefficients in linear regression
Interpretation of log transformed predictor
How to use index in a multiple regression

Is it correct to say that when January = 1, B2 = 0.4 and ln(Sales Index) = 0.35 then:

1) firstly, you need to take exp(0.4) = 1.4918.
2) this is then the multiplier of ln(Sales Index), so 1.4918 * 0.35 means that if it is January this leads to a 52.21% increase in the Sales Index (holding all other x variables constant)?

Answer 1

The interpretation of a dummy variable in a model with a logged dependent variable is in a sense asymmetric: it depends on whether you're turning January "on" (from 0 to 1) or turning January "off."

Let $Y$ be your sales index and $X$ your January dummy. I think it's fair to describe your model as the following (I'm going to follow convention and play it loose with subscripts). I'm omitting the advertising spend component because you've mentioned that you understand how to interpret that. We've got:

$$ \log{Y} = \alpha + \beta X + \epsilon $$

If we exponentiate both sides, this is

$$ Y = e^{\alpha + \epsilon} $$ when January is "off", and $$ Y = e^{\alpha + \beta + \epsilon} $$ when January is "on"

Now if we want to compare the difference between two numbers, A and B, relative to B, we calculate (A-B)/B, right? I say 15 is 50% larger than 10 because (15-10)/10 = 0.5; let's use this framework on the expressions above.

Difference going from January "off" to January "on" =

$$ \frac{e^{\alpha + \beta + \epsilon} - e^{\alpha + \epsilon}}{e^{\alpha + \epsilon}} = e^\beta - 1 $$

Difference going from January "on" to January "off" =

$$ \frac{e^{\alpha + \epsilon} - e^{\alpha + \beta + \epsilon}}{e^{\alpha + \beta + \epsilon}} = e^{-\beta} - 1 $$

Perhaps you see where this is going. Let's choose a value for $\beta$, say $0.3.$ Then going from off to on increases $Y$ by $e^{0.3} - 1= 0.35$, an increase of 35 percent. But going the other way decreases $Y$ by $-(e^{-0.3} - 1) = 0.26$, a decrease of 26 percent. Notably, neither of these matches your coefficient of 0.3!

And of course, this has all ignored the the error in estimating your regression parameters. That's fine if you're really interested in estimating changes not in $Y$, but in $\log{Y}$. It seems however that you care about $Y$ itself, and that invokes another problem: retransformation bias in log-linear models. That will bias your predictions of $Y$ downwards, and is independent of whether your independent variables are continuous or not, so I won't discuss that topic here.

Answer 2

When you include dummy variables for categories, you generally leave one dummy out so that the design matrix $X$ is not rank deficient. (This is an extremely important, somewhat abstract concept.)

• Let's say you have dummies for Jan, Feb, ..., Nov, that is, you leave out the dummy for December.
• The coefficient on Jan is then the effect of being in Jan relative to December.
• Eg. if $b_2$ were .02, it would imply that sales are about 2% higher in January than December.

Changes in logs is conceptually similar to percent changes in levels.

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Side note:

For $y_2$ near $y_1$, you have:

$$\log y_2 - \log y_1 \approx \frac{y_2 - y_1} {y_1}$$

That is, the log difference is close to the percent change (you can prove this using first order Taylor expand to linearize the log near 1). Eg. $\log(2.02) - \log(2) = .01$. As $y_2$ and $y_1$ get farther apart though, that approximation breaks down. Eg. $\log(1.4) - \log(1) = .3365$. 1.4 is 40% higher than 1, but the log difference is only .3365.

Answer 3

Also worth having a look at "Halvorsen, R., & Palmquist, R. (1980). The interpretation of dummy variables in semilogarithmic equations. American economic review, 70(3), 474-475."

Answer 4

On way to interpret the coefficient with the dummy variable is as a variable intercept. Something similar is explained here: Confusion of "slope" and "intercept" in linear regression

If you back transform to a linear scale then the intercept plays a role as a multiplicative constant.

$$\ln(\text{Sales Index}) = B_0 + B_1 * \ln(\text{advertising spend}) + B_2 (January) + \dots + \epsilon$$

which is equivalent to

$$\text{Sales Index} = \text{advertising spend}^{B_1} * \ln(B_0 + B_2 (\text {January}) + \dots) + \epsilon^\prime$$

This term $\ln(B_0 + B_2 (\text {January}) + \dots)$ will be having different values for different months

$$ln(B_0 + B_2 (\text {January}) + \dots) = \begin{cases} \ln(B_0 + B_2) & \quad \text{if it is January} \\ \ln(B_0 + B_3) & \quad \text{if it is Febuary} \\ \ln(B_0 + B_4) & \quad \text{if it is March} \\ \ln(B_0 + B_5) & \quad \text{if it is April} \\ \dots\\ \ln(B_0 + B_{13}) & \quad \text{if it is December} \\ \end{cases}$$