Natural Logarithm of a variable hving negative value

[Pranshu Tripathi]

As a variable, I have the Natural Logarithm of sales growth (Lnsalesgrowth). But the sales growth can be negative, and then Lnsalesgrowth is not defined. I have more than 1500 firms. So, I am dropping those observations which are undefined. Is this the right way to handle this, or is there any alternative?

[Carlo Lazzaro]

Pranshu:
the first question to pose is: why going ln?

[Nick Cox]

Strictly, logarithms of negative numbers are defined, just as complex numbers but that's not useful statistically, at least here.

See e;g; ,https://math.stackexchange.com/quest...egative-number

But we know what you mean. If the motive is to tame a skewed distribution, so-called neglog, namely sign(x) ln(1 + |x|)

sign(x) * log1p(abs(x))

can be useful as can

asinh(x)

(economists in particular are probe to call this IHS for inverse hyperbolic sine)

as can cube root, meaning

 sign(x) * abs(x)^(1/3)

All these transformations have in common

1. preserving sign

2. puttling in tails relatively speaking.

In short, omitting negative values is quite the wrong way to do it. Use the data as they come, or use a transformation fit for purpose.

[Pranshu Tripathi]

The sales growth range is -378,289 to 488,741 in my dataset. To reduce this variation, I am considering using the log natural of this variable. It is a proxy for investment opportunity in the existing literature. So, I am following the literature.

[Carlo Lazzaro]

Pranshu:
if you're going to do a regression on this regressand, why not considering a -glm-, with a log link and gamma family?

[Nick Cox]

I'd like to see the distribution not just the range. @Carlo Lazzaro's excellent idea does carry with it a presumption that the mean function is positive despite any negative values in the data.

[Carlo Lazzaro]

Nick is obviously correct.
My preference for -gml- with a log link and gamma family comes from several dreaedful experiences with healthcare cost data logged and then back-transformed via Duan's smear (https://www.jstor.org/stable/2288126) with disappointing results when contrasted against their raw scale.
This (painful) issue is well covered in https://www.stata.com/bookstore/heal...cs-using-stata , pages 96-99.

[Stephen Jenkins]

See also "The inverse hyperbolic sine transformation and retransformed marginal effects" by Ed Norton, The Stata Journal (2022) 22, Number 3, pp. 702–712, DOI: 10.1177/1536867X221124553. Examples with respect to health care costs, I recall.

[Pranshu Tripathi]

Mr. Lazzaro and Mr. Cox, thanks for your input. I am new to research, so I am unaware of GML . I will explore this and will come back if any query arises.

Thanks and Regards

[Nick Cox]

Carlo Lazzaro meant GLM -- generalized linear models.

[Pranshu Tripathi]

Thanks, Mr. Stephen Jenkins. I will go through it.

[Carlo Lazzaro]

Nick is obviously right again.
I'm progressively losing all the letters on my keyboard and sometimes typing the right letter is just probabilistic (time to buy a more decent keyboard!).

[Pranshu Tripathi]

Hey, I forgot to mention that the variable mentioned above is an independent variable.

[Nick Cox]

There might be other reasons for GLMs.

[Carlo Lazzaro]

Pranshu:
if the variable you mentioned is a predictor, I would leave it in its original metric, especially if logging produces a remakbale reduction of the original sample.