# Log of a Negative Market-Book Ratio in a Regression Model

As noted in this post, the book value of a company is its assets minus its liabilities.

I am reading this paper: "The price of sin: the effect of social norms on market" by Hong and Kacperczyk. One of the paper's hypothesis is that institutional investors are less likely to hold sin stocks. To demonstrate this, the following initial (Fama-Macbeth) regression was run as a benchmark:

$$IO \sim Logsize + Beta + LOGMB + PRINV + STD+ RET$$

IO is institutional ownership Logsize is the market cap Beta is the beta of the sector the stock belongs LOGMB is the log of the market to book ratio. PRINV is the inverse of price STD is standard deviation of monthly return RET is log of arithmetic of previous year's monthly return.

My question here is that, because the Market book ratio can be negative, how can we take log of this?

It is not specified in the paper how this is treated. I have the strong suspicion that it is likely any company with negative book value for whatever reason is ignored. Surely this would introduce some sort of bias. Does anyone know what is the standard practice?

Well the problem can be dodged by the use of some simple solution of just adding a constant to all values which circumvent issues of taking logarithms of negative numbers. Usually the mean is used as a centering point. If the mean is negative though some other number must be chosen.

Interpretation of such a transformed variable would be different from its regular logged counter part.

These are just some hunches on what could have been done. Its odd that methods aren't mentioned for this case.

• guess they probably chucked away the data with negative BM value... in this case, it certainly makes no sense to square it. I know you can add a constant but this is a hacky and not data agnostic. Aug 16 '17 at 18:22
• @Lost1 Ah, I see why you have an issue with squaring it. I messes up ordinality of numbers. Editing now
– EconJohn
Aug 16 '17 at 18:24
• @Lost1 how is adding a constant not data agnostic? it maintains ordinality of the data.
– EconJohn
Aug 16 '17 at 18:25
• if you use a different set of data, with a negative number of bigger magnitude, suddenly your current model breaks again. Aug 16 '17 at 18:26
• @Lost1 so the constant ends up changing based on the data? That's how all regression models work. estimates and their statistical significance can change as data does. Quantile regression would probably be the best tool in this case to check consistency of the estimates.
– EconJohn
Aug 16 '17 at 18:31

Market value cannot be negative. Book value can, but then again, these are outlier companies that can reasonably be taken out of the sample in order to estimate the "normal"/ "average" structure.

• this create a bias against company with negative book market value. I don't know how you can think this is okay and just dismiss them as outlier. Aug 16 '17 at 19:59
• @Lost1 Is a negative-book-value company part of a sample that is representative of the population, as regards the phenomenon we want to study? This is the question that needs to be answered. Aug 16 '17 at 20:04
• what does "representative of the population" really mean here? it is like studying IQ of the population, but we discard people who identify themselves as gay. You can claim people who are gay not representative of the population. Does it have an effect on the result? Perhaps it does not. However, does this mean we discard them from the population we study? Aug 16 '17 at 20:08
• in any sort of statistics analysis, it is generally safer to discard values when things when they are missing at random. However, here, we are selective discarding them depending on a parameter of the regression. That seems like a very bad practice. Aug 16 '17 at 20:10
• @Lost1 This is a very big methodological matter. Your concerns are legitimate. The only thing I am pointing out is that the statistical concept of "population" is obtained by certain criteria applied. In this case, why should the population be "all companies for which we have data"? The concept of "outliers" exists exactly in order to capture the possibility that certain observations reduce the degree that our sample is "representative of the population". Certainly, researchers should clearly lay out their argument for excluding observations as "outliers". (CONTD). Aug 16 '17 at 20:34

Negative book value is actually a 'bug' that remains unsolved by general accounting principles. It is a problem that we've better not think isolatedly. Any companies that have negative book values can actually 'transfer to' positive ones if alternative accounting principles are considered. But firms are not that flexible to change their ways of recording financial information so that this kinda anomaly exists. Correct me if I am wrong.