I would like to use a linear form of Cobb-Douglas production function in my project. However, there are some negative values in one of my independent variables. As far as I know, there are some transformations as follows:
log(1+Y-min(Y))
or
- log( -y + 1) if Y is negative
log( y + 1) if Y is non-negative
Which is better? What are the consequences of these transformations in a regression analysis? Is there any better solution (e.g., a symmetric transformation that pulls in extreme values and preserves sign)?
Any comment would be appreciated.
Thanks in advance
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...
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.
Article A bi-symmetric log transformation for wide-range data
Please is there any reference to back up this formula "log(Y+a)" for log transformation of negative numbers?
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