I encountered a problem with linearity and transformation in linear regression

[Dejan Toscano]

Hello everyone, I'm having some troubles with the following regression analysis.
As my last post did not get much of a reaction at all, and I was advised to be more specific, I'm trying again for a similar question. If I got any technicalities wrong, please tell me, I'm eager to get this right.
I'm using stata 13

My variables are as follows:
Dependent:
par30 - portfolio at risk > 30 (percentage of loans overdue more than 30 days)
Independent:
perfem - percentage of female borrowers
TAK - total assets (in $1000)
PSK - average portfolio size per borrower (in $1000)
MFIage = indicates if an MFI is new, young or mature (1 = 1-4 years old, 2 = 4-8 years old, 3 = 8+ years old)

• I used a different variable beforehand (0 for new, +1 for each year the MFI is active) however I encountered huge problems due to linearity, so I decided to use this one

Group variable: numMFI
MFI = Microfinance Institution, also called financial service provider

Concerning the group variables, my data contains a variable called mfiname which has the individual name of each MFI so I did

. egen numMFI = group(mfiname)

. xtset numMFI fiscalyear
       panel variable:  numMFI (unbalanced)
        time variable:  fiscalyear, 2003 to 2012, but with gaps
                delta:  1 unit

As I want to do a linear regression analysis on my data I tried to follow this advice, so first I tested for linearity with -nlcheck- and as I wanted to have a look at it graphically aswell to include it in my thesis, I did another test following this advice.

. quietly xtreg par30 perfem TAK PSK MFIage

. nlcheck perfem

Nonlinearity test:

           chi2(  9) =   10.11
         Prob > chi2 =    0.3414

. nlcheck TAK

Nonlinearity test:

           chi2(  9) =    7.91
         Prob > chi2 =    0.5434

. nlcheck PSK

Nonlinearity test:

           chi2(  9) =   20.61
         Prob > chi2 =    0.0145

. nlcheck MFIage

Nonlinearity test:

           chi2(  1) =    0.13
         Prob > chi2 =    0.7229

This implies perfem, TAK and MFIage are linear, PSK however isn't.

. quietly regress par30 perfem TAK PSK MFIage

. predict r, resid

Then I checked linearity with the following command

. acprplot perfem , lowess

. acprplot TAK , lowess

. acprplot PSK , lowess

. acprplot MFIage , lowess

TAK and MFIage looked great, however

. acprplot perfem , lowess

. acprplot PSK , lowess

. kdensity PSK, normal

. summarize PSK, detail

                             PSK
-------------------------------------------------------------
      Percentiles      Smallest
 1%     .0266504       .0149792
 5%     .0484809       .0195255
10%      .061839       .0266504       Obs                 209
25%     .1161648       .0330702       Sum of Wgt.         209

50%     .2528287                      Mean            .431633
                        Largest       Std. Dev.      .4882636
75%     .5544006       2.138109
90%     1.126634       2.336616       Variance       .2384014
95%     1.406009        2.36487       Skewness       2.467618
99%     2.336616       3.398852       Kurtosis       11.24819

Now I have two questions

1. -nlcheck- indicated perfem was linear, however to me that doesn't look too great, any thoughts on that?
2. Any ideas on transforming PSK? I tried to log it on e which gave me the following results

. generate lnPSK = ln(PSK)

. quietly regress par30 perfem TAK lnPSK MFIage

. predict r, resid
(81 missing values generated)

. acprplot lnPSK , lowess

. kdensity lnPSK, normal

Which does look better I guess, however -nlcheck- still rejects the null

. quietly xtreg par30 perfem TAK lnPSK MFIage

. nlcheck lnPSK

Nonlinearity test:

           chi2(  9) =   20.58
         Prob > chi2 =    0.0146

Any ideas on how to properly transform PSK to get it linear, or is there no chance on doing this regression linearly with this model?
Thanks in advance!

[Clyde Schechter]

I think you're being far too finicky about PSK. My view of the acprplot is that things are rather close to linear, except for a rather tiny blip when PSK is close to zero. Predictably enough, when the problem lies near zero, log-transforming makes matters worse. If any such transformation were likely to be helpful in this setting, it would be something like 1/x or 1/sqrt(x) or 1/x², but in all honesty I think that any simple transformation is likely to just wreck the rather good linearity you are seeing everywhere but that tiny region. Bear in mind also that the lowess fit that you are using as a guide here is going to be less stable at the edges of the data than it is in the middle, and you have some data points at the left edge that are rather outlying and may be distorting the lowess picture, so I'm not sure how seriously I would even take this.

I suppose the p-value from -nlcheck- bothers you, but it wouldn't bother me: I very rarely let p-values influence my model specifications.

If it really irks you to have that degree of possible non-linearity near zero (and I can't fault you for wanting to dot your i's and cross your t's on the model specification), then I might use a linear spline with a knot at around 0.1 or something like that.

[Dejan Toscano]

Clyde thank you so much for your quick response!
I'll take your advice and use PSK as it is. As you understood correctly the p-value from -nlcheck- really bothered me, so its great to know I don't have to rely on it completely. I'll also use a spline at around 0.1 to run the model a second time and check how the results differ.
Thanks again, your reply really helped me a great amount!