Logs as percentage changes in a log-level-model

In the Econometrics class I taught last winter term, I explained why the coefficient of a regressor in levels can be interpreted as approximately the percentage increase of the dependent variable in logs. I have searched the web and textbooks for a concise, straightforward derivation but found none, so I made my own. I guess it’s worth sharing. If this helps you, let me know in the comments.

With Taylor’s theorem (see also Taylor series) we can approximate the natural logarithm around some $$a>0$$ by:
[ ln(t) approx ln(a) + ln'(a) cdot (t-a) ]
Recall that the first derivative of the natural logarithm is $$ln'(t)=frac{1}{t}$$ with $$t>0$$.
Then the above approximation becomes:
[ln(t) approx ln(a) + frac{1}{a} cdot (t-a)]
Note that this approximation becomes the worse, the larger the difference of $$t-a$$, i.e. the farther one is away from the expansion point $$a$$.

We are interested in an approximation for the log of a percent increase $$ln(1+p)$$, so let $$t=1+p$$:
[ln(1+p) approx ln(a) + frac{1}{a} cdot (1+p-a)]
Further, let $$a=1$$:
[ln(1+p) approx ln(1) + frac{1}{1} cdot (1+p-1)\ln(1+p) approx 0 + 1 cdot p\ln(1+p)approx p]

The percentage change $$ p $$ is given by the new  value minus the initial value all divided by the initial value.
[p=frac{(z+Delta z) – z}{z} = frac{z+Delta z}{z} – frac{z}{z} = frac{z+Delta z}{z} – 1 = frac{Delta z}{z}]

Now consider the log-level model:
[ln(hat{y}) = hat{beta}_0 + hat{beta}_1 x]

Increasing $$x$$ by one:
[ln(hat{y}_text{new}) = hat{beta}_0 + hat{beta}_1 (x+1)]

The difference is then:
[ ln(hat{y}_text{new}) – ln(hat{y}) = hat{beta}_0 + hat{beta}_1 (x+1) – hat{beta}_0 – hat{beta}_1 x \ ln(frac{hat{y}_text{new}}{hat{y}}) = hat{beta}_1 \ ln(frac{ hat{y} + Delta hat{y}}{hat{y}}) = hat{beta}_1 \ ln(1 + frac{Deltahat{y}}{hat{y}}) = hat{beta}_1 \ ln(1+p) = hat{beta}_1\ p approx hat{beta}_1 ]

From this, we can also see that the exact percent change is:
[ ln(1+p) = hat{beta}_1 \ p = e^{hat{beta}_1} – 1 ]

A quick comparison of the approximation and the exact value shows that the approximation is less than 5% off of the exact value if $$|p|<0.1$$, that is, if the change is less than $$pm 10%$$.

Leseempfehlung: “NBER Summer Lectures”

Über einen Tweet von @MarkThoma bin ich auf einen Blogartikel von Francis X. Diebold gestoßen, der die – wohl nicht ganz so bekannten – NBER Summer Lectures verlinkt hat.

Das NBER veranstaltet jeden Sommer ein Summer Institute über (zumeist) ökonometrische Methoden und hat die Vorträge samt Präsentationen online zur Verfügung gestellt. Dort, wo die Vorlesungen über Vimeo bereitgestellt werden, sind sie wohl auch als Download verfügbar, um sie offline – etwa im Zug – anzuschauen.

http://www.nber.org/econometrics_minicourse_2013
http://www.nber.org/econometrics_minicourse_2012
http://www.nber.org/econometrics_minicourse_2011
http://www.nber.org/econometrics_minicourse_2010
http://www.streamingmeeting.com/webmeeting/matrixvideo/nber/index.html
http://www.nber.org/minicourse_2008.html
http://www.nber.org/minicourse3.html

Die Vorlesungsreihe ist wohl insbesondere für fortgeschrittene Masterstudenten oder Doktoranden interessant, weil vieles einfach vorausgesetzt wird.

Ein bisschen weitergeklickt: Auch die Fed hat solche Lectures bei Vimeo: http://vimeo.com/album/2509117