How to handle negative values in log transformations in a regression analysis?

Unfortunately, most economic data are non-invariant to the shift-transformation-type (when someone adds or subtracs some constant to the data). So your variant of transformation in many cases could directly lead to the wrong (skewed) results.If you have some negative values of the responce variable the log-transformation could be applied in two differents ways: 1) you can drop these values from the sample and estimate your model on such "truncated" sample; here of course you can lost some degrees of freedom and get results of less statistical sufficiency; 2) you can use so-called censored regression approach (for instance, tobit-models). It is much more correct case but it requires from you some special efforts in understanding the idea of tobit-type models. There are loads of books on the poblem, hope you will be successful.

Sometimes negative values can be removed by reformulating the problem or correcting errors. Do the negative values make sense in the context of the problem? What is the independent variable that is causing problems? Is it income, percent growth, coffee consumption?

@Vladislav: could you provide a citation to the observation that economic data are non-invariant to shift transformations. I am interested to see what it is about economic data that produces this outcome.

Dear  Morteza,

In this case, instead of the log transformation is better to use other transformations, for example, Johnson translation system or a two-parameter Box-Cox transformation.

Adding   or  subtracting a constant  affects   the mean  but   does  not affect  variance  . Therefore  it is  recommended  to add a constant .  The best constant to add in case of Cobb Douglas production function is to add the same constant to all values of the same variable which makes all values of the variable positive.  Suppose you have three negative values  such as -6  and -9  and   minus   2   then   Adding the constant   10  to all values  will make all values positive and greater than zero . The transformation such as log becomes possible without affecting R SQR   or  elacticity  etc...

you can change origin so that all observation to be positive. then you cat transfer into log form

Here is another option if you can assume that the reason you have zeros is because your sample size is insufficient to get a non-zero value. In biology this might be something like measuring insect mortality due to the application of an insecticide at 1, 10, 100, and 1000 grams/hectare. The 1 gram/hectare rate results in a mortality rate of 0.002%. Given that your sample size was 10, it is unlikely that any of the ten insects will die. So your estimate is zero mortality, but that is only because your sample size was inappropriate for measuring such a low mortality.

If a similar situation applies in your case, what happens if you just delete the zero values?

Hie.Surely negative values are common in regression.Adding a constant to make the minimum value positive has no harm to analysis. If the variable concerned is the depended variable, adding a constant will only alter the constant, but the parameter estimates will be the same.For independent variables, adding a constant does not change the parameters neither.Try it!

One could use the "Bi-Symmetric Log transformation", which performs a log-like transformation on numbers that are negative and doesn't exaggerate the 0-1 region.

Please is there any reference to back up this formula "log(Y+a)" for log transformation of negative numbers?

=log(sqrt((X^2)+1))

A common approach to handle negative values is to add a constant value to the data prior to applying the log transform. The transformation is therefore log(Y+a) where a is the constant. Some people like to choose a so that min(Y+a) is a very small positive number (like 0.001). Others choose a so that min(Y+a) = 1. For the latter choice, you can show that a = b – min(Y), where b is either a small number or is 1.

A criticism of this method is that some practicing econometricians don't like to add an arbitrary constant to the data. The argument is that a better way to handle negative values is to use missing values for the logarithm of a non-positive number.

i. Adding a constant fixed arbitrary value to a variable, in order to somehow render the argument of the logarithmic function positive, may be reasonable (or not) depending on the scientific support (possibly, the physical meaning) that may justify this option. Note that such constant does not necessarily add to the mentioned argument as a whole. It may possibly add just to a variable from several possibly considered. It may be, alternatively, found preferable to add a parameter, to be fitted after the data, rather than a constant of fixed arbitrary value.

ii. A quantity (q ≥ 0 u) can be logarithmized after dividing by its unit (u); except for the origin of the scale, where discontinuity is expected because ln(0) is undefined. This discontinuity does not occur if the said quantity is restricted in its domain to (q ≥ u), so that it is also ensured that the argument of the logarithmic function is q/u ≥ 1. This may be found convenient if negative logarithmic values are found undesirable. We sometimes encounter the case where the quantity q is a monotonic increasing function of time, q(t), so that some 'hidden' time (θ) can be added to displace the origin of the scale, possibly from q/u = 0 to q/u = 1, or as otherwise can be found justifiable. This figure (θ) is sometimes called induction time in some kinetic studies, and it can be possibly adopted as (an additional) correlation parameter.

As example for this kind approach; you may check my post here:

You may also find useful to check: https://www.researchgate.net/post/How_to_calculate_the_logarithm_of_a_negative_number2

Thats good contribution and way out without affecting the variance