Log-linear ordinary least squares specification coefficient loses its interpretation when we change an explanatory variable by a unit? Ok, so something that's common to the econometrics literature is that we interpret the coefficients in OLS log-linear models like such.
To spell it out in the main body:
$ln(y_i)=\beta_0+\beta_1X+u_i \Rightarrow \text{if } \Delta x = 1, \text{then } \text{%}\Delta y \approx 100\beta_1 $
I think that this is a very bad approximation, although my reasoning is probably incorrect (although I do understand the derivation of why this approx holds).
Ok, so an aside:
$\frac{\dot{y(t)}}{y(t)} = g \Rightarrow ln(y_t) = gt + c $
It also follow that:
$y_{t+1} \approx y_t (1+g \Delta t) $
So here, if I plug in a change of t = 1, and let g = 1, y would be doubling with each unit change in t, and so we should approx y as 2^x instead of something of the form e^x. Of course, large-ish changes in x mess up the calculus.
However, isn't plugging in a unit change in t (in the econometrics textbook, x is t) what the econometrics textbooks are doing? A unit change in x -&gt; 100% change in y (g and beta_1 are analogous, so g = 1 -&gt; beta_1 =1) -&gt; y approx doubles with each change in x -&gt; y should be modelled as something of the form 2^x, not e^x, and there's a sizeable difference between the two, and so this contradicts that fact that the specification implies that y is in the form e^x (rather than 2^x).
I hope this makes sense.
When
$$\ln y = \beta_0 + \beta_1 x + u \implies y = \exp\{ \beta_0 + \beta_1 x + u\}$$
$$\implies \partial y / \partial x = \beta_1 y \implies \frac{\partial y / \partial x}{y} = \beta_1.$$
So we see that $\beta_1$ is the marginal change in $y$ due to infinitesimal changes in $x$ as a proportion of its level. Therefore the accuracy of the approximation
$$\beta_1 = \frac{\partial y / \partial x}{y} \approx \frac{\Delta y / \Delta x }{y}$$
$$\implies \Delta x = 1: \beta_1 \approx \frac{\Delta y  }{y}$$
is nothing else than the general approximation inaccuracy issue that arises when we replace a derivative (infinitesimal change) with a disrete-interval proportional change.
In the specific case we have
$$y(x+1) - y (x) = \exp\{ \beta_0 + \beta_1 x + \beta_1 +u\} - \exp\{ \beta_0 + \beta_1 x + u\}$$
$$ = y(x)\cdot (e^{\beta_1 -1}) \implies \frac{\Delta y(x+1)}{y(x)} = (e^{\beta_1} -1).$$
We get

and we verify the folk wisdom that the approximation is accurate enough for purpose of economic analysis if $\beta_1 \in [-0.1,\; 0.1]$, and maybe for a larger interval, if one percentage point is not critical for the purposes of the specific research.