# Ideal transformation for consumption variable in a probit model

I am trying to figure out the best transformation of my consumption variable. I am running a probit regression to look at whether or not a household enrolls in health insurance. Consumption per capita is an independent variable and in my current model I use both consumption and consumption squared (two separate variables) to show that consumption increases but with diminishing returns. This makes for fairly straightforward interpretation. However, using the log of consumption is a slightly better fit because it normalizes the distribution and contributes a bit more to the overall R2 for the model but it is more difficult to interpret. Which would you suggest I use - log of consumption or consumption plus the quadratic function? My research is focused on health economics so I'm not sure what the preference is in that discipline. Any insight would be much appreciated. Thank you!

Using variables in logs is actually quite common in economics, since the estimated coefficients can be interpreted as sensitivities to relative changes in RHS variables (or elasticities, if both LHS and RHS variables are in logs). For example, say that you have model y = b ln(x), and x changes to x(1+r). Then you can use the approximation $ln(1+t) \approx t$ to see how y changes: $$y = b \ln(x(1+r)) = b \ln(x) + b \ln(1+r) \approx b \ln(x) + b r.$$ So if r is 0.01 (x increases by 1%), y increases by b r = 0.01 b (of course, this works only for small r). In case of your probit model, if coefficient for log-consumption is b, it can be interpreted so that increase in consumption by 1% would increase probability of enrollment by b %.

I am not sure I understand your interpretation: a log-transformed predictor would imply that the effect is increasing with diminishing returns, while a quadratic function as a predictor would imply the existance of a peak in the effect (for ax^2+bx+c the peak is at -b/(2a)). I would assume the latter is less realistic in your context, and is also not supported by the better R-squared for the log-transformed predictor.

You can also use the HST (mentioned here)

How should I transform non-negative data including zeros?

If $x$ is p.c. consumption, create a variable $x'=x-\bar{x}$ (i.e. de-mean x). Then use $f(x',\theta=1)$ as your explanatory variable (where $f()$ is the inverse hyperbolic sin transform).

for positive values of x' (i.e. people who consume more than the average) $f(x',\theta=1)$ behaves as $\log(x')$.

for negative values of $x'$ (i.e. people who consume less than the average) $f(x',\theta=1)$ behaves as $-\log(-x')$.

($f(x',\theta=1)$ looks like a big 'S', passing by the origin).

It seems to me that you already have a 'partial' statistics answer (better $R^2$ as to the decision as to what to choose: log vs quadratic). You could use other data-driven metrics (e.g., out-of-sample hit rates, whether the parameters are reasonable etc) to judge which model structure is 'better'.

PS: By the way, are these two models consistent with economic theory? Just asking as I do not know this area. Another way to select a model is to check its consistency with theory.

• Thank you so much. It seems that the log transformation is widely practiced in economics. It is not part of economic theory but is widely accepted. I appreciate your very useful response. – Sarah Aug 10 '10 at 0:56