discrepancies in sign/direction between absolute and inverse hyperbolic sine transformed values

[Karen Arulsamy]

Hi all,
I am analysing the effect of schooling on wealth. I am running regressions using mean financial wealth and the inverse hyperbolic sine of mean financial wealth (to get the % change in wealth with a 1 year increase in schooling).

I noticed that the signs change (positive to negative) between the two regressions which doesn't really make sense. I would like to get some advise on how to address this discrepancy.

Below is the output with mean financial wealth as outcome:

Below is the output with the inverse hyperbolic sine transformed outcome:

I use the formula in Bellemare and Wichman (2020) to convert the coefficient on Z_5 into a semi-elasticity coefficient. Specifically, I run:

predictnl Z_5b = _b[Z_5]*xbar_5a*((sqrt(mean_fw2+1))/mean_fw_main),se(Z_5b_se)

where xbar_5a is the mean of Z_5 and mean_fw2 is the squared mean wealth. The coefficient I get is -10.008 and standard error is 8.209.

The data extract below accounts for the sample restrictions i.e. mainsample & bw_5 above.

* Example generated by -dataex-. To install: ssc install dataex
clear
input float(mean_fw_main asinhfw_mean_main Z_5 distance sex_main xbar_5a mean_fw2)
     -2696 -8.592671 11.131124   7 1 10.797332   7268416
     -2696 -8.592671 11.131124   7 1 10.797332   7268416
     -2567  -8.54364  10.48713  -5 1 10.797332   6589489
     -2250 -8.411833 10.357333 -15 1 10.797332   5062500
     -2250 -8.411833  10.46117  -7 1 10.797332   5062500
     -2190 -8.384804 10.292435 -20 1 10.797332   4796100
     -2190 -8.384804 10.371334 -18 2 10.797332   4796100
     -2190 -8.384804 10.318394 -18 1 10.797332   4796100
     -2190 -8.384804  10.48713  -5 1 10.797332   4796100
     -2190 -8.384804 11.145125   4 2 10.797332   4796100
     -2190 -8.384804 10.305414 -19 1 10.797332   4796100
     -2182 -8.381145 11.144104   8 1 10.797332   4761124
     -2182 -8.381145 11.144104   8 1 10.797332   4761124
     -2182 -8.381145 11.144104   8 1 10.797332   4761124
     -2000 -8.294049 10.331373 -17 1 10.797332   4000000
-1623.6666 -8.085589 11.144104   8 1 10.797332   2636293
-1623.6666 -8.085589 11.144104   8 1 10.797332   2636293
-1623.6666 -8.085589 11.144104   8 1 10.797332   2636293
     -1570 -8.051978 11.118145   6 1 10.797332   2464900
     -1570 -8.051978 11.118145   6 1 10.797332   2464900
     -1500 -8.006368 11.119165   2 2 10.797332   2250000
     -1470 -7.986165 11.223004  10 2 10.797332   2160900
     -1470 -7.986165 11.223004  10 2 10.797332   2160900
-1326.6666 -7.883572  10.59199  -1 2 10.797332 1760044.4
-1326.6666 -7.883572  10.59199  -1 2 10.797332 1760044.4
-1326.6666 -7.883572  10.59199  -1 2 10.797332 1760044.4
   -1308.5 -7.869784 11.247943  16 1 10.797332 1712172.3
   -1308.5 -7.869784 11.247943  16 1 10.797332 1712172.3
     -1304 -7.866339 11.106186   1 2 10.797332   1700416
-1016.6667 -7.617432 10.501132  -8 2 10.797332 1033611.1
-1016.6667 -7.617432 10.501132  -8 2 10.797332 1033611.1
-1016.6667 -7.617432 10.501132  -8 2 10.797332 1033611.1
 -983.3333 -7.584095 11.184065   7 2 10.797332  966944.4
 -983.3333 -7.584095 11.184065   7 2 10.797332  966944.4
 -983.3333 -7.584095 11.184065   7 2 10.797332  966944.4
      -950  -7.54961  10.56603  -3 2 10.797332    902500
    -843.5 -7.430707 11.093206   0 2 10.797332  711492.3
    -843.5 -7.430707 11.093206   0 2 10.797332  711492.3
      -785 -7.358831 10.462193 -11 2 10.797332    616225
 -756.6667 -7.322071 10.383293 -13 1 10.797332  572544.5
 -756.6667 -7.322071 10.383293 -13 1 10.797332  572544.5
 -756.6667 -7.322071 10.383293 -13 1 10.797332  572544.5
 -666.6667 -7.195438 11.234962  15 1 10.797332  444444.5
 -666.6667 -7.195438 11.234962  15 1 10.797332  444444.5
 -666.6667 -7.195438 11.234962  15 1 10.797332  444444.5
 -464.1038 -6.833257 10.475172 -10 2 10.797332  215392.3
 -464.1038 -6.833257 10.475172 -10 2 10.797332  215392.3
 -464.1038 -6.833257 10.475172 -10 2 10.797332  215392.3
      -400 -6.684613  10.52607  -2 1 10.797332    160000
      -400 -6.684613 11.157084   9 1 10.797332    160000
end

Many thanks
Karen

[Nick Cox]

The inverse of asinh() is sinh()

. di sinh(asinh(42))
42

. di sinh(asinh(-42))
-42

I would need to read that paper again to discover what it recommends, but it's clear that no inverse recipe is good enough unless it is correct for arguments of any sign.

[Clyde Schechter]

Actually, you don't have a real sign change here. Look at the 95% confidence intervals for the Z_5 effect in both models. In each case, zero is included. So, in fact, neither model actually specifies a sign for the effect of Z_5. Both models are consistent with both positive and negative effects of Z_5.