Meaning of 'independence' in regression model I am familiar with the following condition for statistical independence where $x$ and $y$ are two different (random) variables: 
\begin{equation}
E(xy)=E(x)\times E(y)
\end{equation}
Now, I came across a paper (Appendix B, p.1311) where it mentions that if $h$ and $^{\partial h}/_{\partial z}$ are independent then following holds: 
\begin{equation}
E\left(h \cdot \frac{\partial h}{\partial z}\right)=E(h)\times E\left(\frac{\partial h}{\partial z}\right)
\end{equation}
I would like to know: What does it mean if $h$ is a dependent, continuous variable--such as smoking expenditure (in dollars)--and $z$ is a continuous independent variable--say income in dollars--in regression model. Note that $^{\partial h}/_{\partial z}$ is the partial derivative of $h$ with respect to $z$ which is equal to $dh/h$ (percentage change in $h$)
 A: The marginal change of a variable w.r.t. to another variable cannot be equal to the corresponding percentage change of the former. The (point) percentage change equals the total differential of the logarithm of a variable, $dln(y) = dy/y$ 
The appendix you refer to writes (using your letters) $E[h\cdot (dh/h)]$, where $(dh/h)$ is the relative (marginal percentage) change in $h$ due to a change in some explanatory variable $z$. $d$ symbolizes total differential.
Then the paper says that if $h,\; (dh/h)$ are independent, then as usual the expected value can be broken.
For these two components to be independent, it must be the case that $h$ does not really appear in $(dh/h)$. This requires that $dh$ is a linear function of $h$, something like
$$dh = ahdz \Rightarrow dh/h = adz \qquad (\Rightarrow \frac{dh}{dz} = ah)$$
Related to the economic meaning of this, it is that the total marginal effect $z$ has on $h$ depends on the level of $h$.
Using your example, if $h$ is smoking expenditure and $z$ is income, then:  
-if $a$ is positive, it would mean that the higher smoking expenditure is, the more it increases with income.  
-if  $a$ is negative, it would mean that the higher smoking expenditure is, the less it increases with income. 
