Interpreting regression coefficients when the outcome variable is an inverse hyperbolic sine function I just learned that when there are zero or negative values, a good alternative to using a logged function is the inverse hyperbolic sine function.
When using log transformation on the dependent variable, as in log(y)=βx+ε, the interpretation of β is 1 unit change in x = (100β)% change in y.
Likewise, what would the interpretation of β be if the regression is asinh(y)=βx+ε?
 A: The interpretation is not as easy as with the log-linear semi-elasticity case, but there is a nice working paper by Marc Bellemare and Casey Wichman on Elasticities and the Inverse Hyperbolic Sine Transformation that covers some of this in Section 2.2.
In a model like
$$
\DeclareMathOperator{\asinh}{asinh} 
\asinh ( y)= \alpha + \beta \cdot x+ \varepsilon,$$
you can recover $y$ as
$$\DeclareMathOperator{\sinh}{sinh} y= \sinh ( \alpha +\beta \cdot x+ \varepsilon).$$
Then taking the derivative with respect to $x$,  $$\frac{\partial y}{\partial x} = \beta\cdot \cosh(\alpha +\beta \cdot x+ \varepsilon) = \beta\cdot \cosh(\asinh(y))= \beta \cdot \sqrt{y^2+1}.$$
Multiplying that by $\frac{x}{y}$, we get the standard elasticity of $y$ with respect to $x$
$$\epsilon=\frac{\partial y}{\partial x}\cdot\frac{x}{y}=\beta \cdot x \cdot \frac{\sqrt{y^2+1}}{y}=\beta \cdot x \cdot \sqrt{\frac{{y^2+1}}{y^2}}=\beta \cdot x \cdot \sqrt{1+\frac{1}{y^2}}.$$
This looks messy, but when $y$ gets large in absolute value, $\sqrt{1+\frac{1}{y^2}} \rightarrow 1,$
 so you can ignore the last term and just focus on $$\epsilon \approx \beta \cdot x.$$ You can plug the average $x$ into that or evaluate it as function of $x$ for some interesting values of $x$. Since this is just a linear transformation of $\beta$, calculating the SEs is pretty easy.
You can also calculate the average elasticity:
$$\epsilon=\sum_i^N \hat \beta \cdot x_i \cdot \frac{\sqrt{\hat y_i^2+1}}{\hat y_i},$$
or evaluate it at interesting values of $x$ and other covariates. This has the interpretation of a percent change in expected $y$ for a 1% change in $x$. Since this is a non-linear function, SEs are more challenging.
If you want to retain the semi-elasticity interpretation, the
$$\epsilon^*=\frac{\partial y}{\partial x}\cdot\frac{1}{y}=\beta \cdot \sqrt{1+\frac{1}{y^2}}.$$
You may want to multiply that by 100 to get the expected % change in $y$ for a 1 unit increase in $x$.
