Only half a plan

In a recent column at VoxEU, Francesco Giavazzi and Guido Tabellini argue for a concerted European stimulus policy to bring the Eurozone out of recession and low-inflation. Their plan consists of three parts, cutting taxes by about 5 per cent of GDP in each country, financing the resulting budget deficits by new long-term (30 year) bonds that are to be bought by the ECB in a sort of limited Quantitative Easing. while temporarily suspending the deficit criteria of the stability and growth pact. Giavazzi and Tabellini argue that tax-cuts are less likely to cause misallocations and corruption and that the ECB will start a QE programme anyway in roughly half a year's time, when they see that this is the Eurozone's last means to overcome the slump. While I in principle agree with their analysis, I consider their advice only half plan because they do not even consider supply side reforms.


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Rotation im EZB-Rat

In der FAZ wird berichtet, Bundesfinanzminister Schäuble akzeptiert das Rotationsprinzip im EZB-Rat, weil er "Heftigeres" befürchtet, wenn man jetzt über die Abstimmungsmodalitäten nachdenkt. Damit müssen die Chefs der fünf größten Zentralbanken (Deutschland, Frankreich, Italien, Spanien, Niederlande) der Reihe nach bei jeder fünften Abstimmung aussetzen, weil die Anzahl der Stimmen der NZBn im Rat 18 nicht überschreiten soll, um die Handlungsfähigkeit zu gewährleisten. Weiterlesen

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More trade, rather than carbon tariffs

Paul Krugman suggests an interesting solution to the problem of China's carbon emissions. Since their exporting sector is the main driving force behind their growth, it is crucial for China to have access to Western consumer markets. The suggestion is now to use this dependence and imposing a carbon tariff on imports from China. Paul reckons that this will encourage China to cut its emissions, thus lowering the tariff and staying competitive. Sounds good but I'm not so sure whether such a scheme will be effective, let alone efficient. Weiterlesen

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Sorry, mate, but it's the law

In one of its recent issues „The Economist“ obviously had a bad day. At least Charlemagne had, though usually his commentary on Europe and its politics are right, this time Charlemagne is plain-wrong in blaming German „legalism“ for the crisis and the Eurozone's slip into the dangerous low-inflation territory.


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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{\Delta\hat{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\%.

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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.

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:

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