Economics Asked by threelinewhip on June 16, 2021
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 100beta_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.
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}
Correct answer by Dayne on June 16, 2021
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.
Answered by Alecos Papadopoulos on June 16, 2021
Get help from others!
Recent Questions
Recent Answers
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP