Log-linear ordinary least squares specification coefficient loses its interpretation when we change an explanatory variable by a unit?

Question

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 -> 100% change in y (g and beta_1 are analogous, so g = 1 -> beta_1 =1) -> y approx doubles with each change in x -> 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.

Answer 1

The source of the approximation:

Given, $\ln(Y_i)=\beta_0+\beta_1X_i+u_i$, for a unit change in $X$, i.e., $X_{i+1}-X_i=1$, we have:

\begin{align} \frac{Y_{i+1}-Y_i}{Y_i} &= e^{\beta_1+\Delta u_i}-1 \end{align}

For small $x$, we use taylor expansion to say: $e^x \approx1+x$. Using this above we get:

$$\frac{\% \Delta Y}{100} \approx \beta_1+\Delta u$$

This approximation is good when $\beta_1$ is quite small. In your example, you have taken $\beta_1=1$ which makes this a bad approximation.

Consider your example with $g=0.1$.

\begin{align} \frac{y_{t+1}-y_t}{y_t} &= e^g-1 \\ &=1.1052 - 1 \tag{for $g=0.1$} \\ & \approx g \end{align}

Answer 2

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.