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I am doing some research on a potential U.S. Nationwide Sales tax. I found per-capita sales tax revenues and sales tax rates for each of the 50 states and conducted a quadratic regression on them. I got a quadratic regression with a $R^2$ of approximately 0.4. Is this $R^2$ sufficient to state a good correlation and are there any other statistical tests I may need to do? Thank you in advance.

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  • $\begingroup$ Is your question a version of "Is this good enough to use regression results to characterize the US Laffer curve?"? $\endgroup$ – BKay May 14 '15 at 17:56
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    $\begingroup$ To answer your question. An R squared of 0.4 is perfectly fine. You will need to do other hypothesis testing as well. These can vary from simple tests such as the t-test to more complicated tests like Wald or Hausman. The problem you will find is in the experiment design. As @FooBar says, the problem isn't in the hypothesis testing it is in experiment design. There are way too many confounding variables and elements of complexity in the US tax system to estimate it in that way. $\endgroup$ – Jamzy May 15 '15 at 0:55
  • $\begingroup$ @Jamzy Thanks for the help! I will look into t-tests as well. Yes, I do understand that there are variables that are unaccounted for which I will research further. $\endgroup$ – Howsikan May 15 '15 at 2:24
  • $\begingroup$ It would be useful if you wrote in the question the equation that you used, i.e. the "quadratic regression". $\endgroup$ – Alecos Papadopoulos May 15 '15 at 3:03
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    $\begingroup$ @Howsikan You might just as well regress revenues on a quadratic equation of "average height of state population" and argue you are finding the revenue maximizing population height. As I tried to argue in my answer (and you insist on ignoring), your tax rates just coincide with the outcome. They are not causing it. There is no sense of optimality, or a maximum what-so-ever. $\endgroup$ – FooBar May 16 '15 at 15:02
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The Laffer Curve is a theoretical argument that feels strongly true the moment one hears about it. The complexities around it once one wants to have a closer look are many (just for a taste the wikipedia article can be consulted).

Even in the simplest econometric study, one has to make sure that what one does is consistent with the theory one wants to test (or the one on which one relies), otherwise it won't be an econometric study but an exercise in pure mathematical approximation devoid of economic content (after all any bounded continuous function can be approximated by polynomials as per Stone-Weierstrass theorem).

So what does the Laffer Curve (very) basic theory (and in its strongest form) asserts?
1) That output is a negative function of the tax rate.
2) That a tax rate of $100\%=1$ should result in zero output, and so zero tax revenue.
3) Moreover, logic says that if the tax rate is zero, tax revenues should be zero. it appears then that we can start by an assumption that somewhere in between there is a point of maximum tax revenues (although one of the complexities mentioned earlier is that the curve may not be unimodal, i.e. it may also have a local maximum).

Let's see which conditions should the simplest "quadratic regression" satisfy in order to be consistent with the theory. We have

$$T(\tau, Y(\tau)) = \beta_0 + \beta_1\tau + \beta_2\tau^2 + u$$

where $Y$ is output (or sales), $\tau$ is the tax rate, $T(\tau, Y(\tau))$ is Total tax revenues, and $u$ is an error term, to account from deviations from the mean-value.

We can deduce the following:
A) In order to satisfy point 3) above on average, $T(0, Y(0)) = 0$, it must be the case that $\beta_0 =0$.

B) This equation, in deterministic terms has derivatives

$$\frac {{\rm d}T(\tau)}{{\rm d} \tau} = \beta_1 + 2\beta_2\tau, \;\; \frac {{\rm d^2}T(\tau)}{{\rm d} \tau^2} = 2\beta_2$$

and so it will have a maximum at the point $\tau^* = -\frac {\beta_1}{2\beta_2}$ if and only if $\beta_2 < 0$.

C) Finally, in order for the equation to satisfy point 2) above, $T(1, Y(1)) = 0$, it must be the case that $\beta_1 = -\beta_2$.

But then we also obtain that the maximizer will be $\tau^* = 0.5$. This last consequence is not good: our tool (a specific mathematical functional form that appears convenient for our purposes), forces on us a very specific conclusion that is not predicted by the theory, something that should be left for the data to decide, even in the case were the data do indeed exhibit "Laffer-curve" properties. In other words, the quadratic regression postulates not "general Laffer curve properties", but a very specific Laffer curve (this essentially comes from the fact that the independent variable $\tau$ is restricted to lie in $[0,1]$).

But run the regression we must: so we found that for this specific "quadratic regression specification" to be valid, theory imposes three restrictions on the parameters,

$$\beta_0 =0,\;\;\; \beta_2 < 0,\;\; \beta_1 = -\beta_2$$

Since one of the restrictions is an inequality, one could use Inequality-Constrained Least-squares which requires an iterated algorithm, but I won't go there.

Otherwise, one should first run a regression with the coefficients unconstrained, and see whether at least, $\hat \beta_2 <0$. Then run the regression imposing the two equality constraints, hope that in this restricted regression $\hat \beta_2$ remains negative, and conduct an $F$-test on the two specifications to determine which fits the data better.

Now assume that the restricted model does have a $\hat \beta_2 <0$ and that it outperforms the unrestricted model: one can then argue that this data set behaves like the Laffer-curve theory would predict, but also, that the tax revenue maximizer is around the value $0.5$. One cannot escape this last conclusion, if the restricted model prevails.

If it does not prevail, other aspects of Laffer-curve theory are not supported by the data, so to mechanically calculate a tax revenue maximizer based on the estimates from the unrestricted model will be rather shaky.

And there are also important statistical misspecification issues:
1) Can we argue that the sales and the tax rates of the various states come from the same distribution, and are independent? (this would be needed to have the "i.i.d. sample" framework)

and especially (which I believe is what @FooBar 's answer focused upon)
2) Is the tax rate uncorrelated with the error term, or is it endogenous? Are there factors that affect directly output/sales and are also correlated with the tax rate (e.g.: if a productivity or other mid-term range shock affects output, is there pressure to change the tax rates also, and do they adjust?)

In light of the above, I would say that the $R^2$ "fit criterion" is not the first thing to check, nor the primary method, in order to judge the quality of such a regression estimation.

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  • $\begingroup$ Alecos, what if there was a simple polynomial expansion (4th degree, perhaps?) to fit revenue vs taxes? Can one restrict it still to cross the correct points with this technique mentioned? I imagine this would allow for maxima other than 0.5, making the project more interesting as a description of the existing relation, though admittedly still non-casual. $\endgroup$ – RegressForward May 21 '15 at 2:25
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    $\begingroup$ @RegressForward Consider the function $$T (\tau) = \tau\cdot (1-\tau)\cdot g(\tau)$$, with $0\geq \tau \geq 1$, and $g(\tau) >0$. It will cross the horizontal axis at $0$ and $1$. You can play around with the form of $g(\tau)$ to see what happens. $\endgroup$ – Alecos Papadopoulos May 21 '15 at 22:37
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What are you trying to do? I hope you're not after finding evidence for or against the Laffer Curve, because your approach pretty much will not be able to say anything about that.


The Laffer curve is a causal relationship between taxation and government revenue, stating that there are two effects from higher tax rates: (i) they increase the rate of revenue (ii) they might disturb production, and hence reduce the tax base.

The Laffer curve, while denoted as a correlation, is hence in fact a causal relationship. Individuals respond to a higher tax rate.

What you will find is merely a correlation without any causal implications. While you may observe that tax revenue correlates non-linearly with tax rates, you cannot state whether the former is caused by the latter. Hence you cannot speak towards the Laffer curve.

So what you're saying is that I can't use the Laffer Curve to develop a causal relationship between per capita revenue and sales tax rate? Do you have any other techniques that I can use to find a revenue maximizing tax rate? Also is it possible to use correlation to estimate a revenue maximizing tax rate?

It is the other way around: The Laffer curve is a causal relationship between per capita revenue and sales tax rate. If you want to estimate this causal relationship, you need to identify it. What you instead are identifying is a correlation, but not the causal relationship. Don't be too upset, your issue is the issue of all Macroeconomics. We know too little about many effects just because it is hard to filter out the causal part from the mere correlation.

To identify causality, you need either to

  • (i) be 100% sure you are controlling for anything else that could affect both tax rates and revenue, or
  • (ii) find random variation in the tax rates.

You most likely will not convince any academic that you managed to do (i), just because there's too many channels through which omitted variables could work here (google "omitted variable bias). The second channel gives you clean identification and is easy to do, the hard part is finding that random variation.

The easiest and most-likely-successfull way would be to collect dates at which sales tax were changed at the state level and then do a second-order diff-in-diff (after tax change-before tax change, compared to other states). Then, your identification assumption relies only on "changes in the tax rate are independent to the error term", which is still not great, but much - much - better than what you're doing now. Look at this paper, who look at the impact of introduction in female voting (women's suffrage) onto outcomes.

Instead of "Introduction of female voting", you would do "change in sales tax rate".

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    $\begingroup$ One additional issue is that because people can and do shift consumption between states, there's no reason to believe that a pattern seen in state sales tax revenue would generalize to a national program. $\endgroup$ – dismalscience May 14 '15 at 21:29
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    $\begingroup$ @FooBar I am not trying to develop a causal relationship but instead find the peak of the Laffer corrleation as the optimum tax rate. $\endgroup$ – Howsikan May 14 '15 at 23:30
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    $\begingroup$ @dismalscience Good observation. I will kepp that in mind. $\endgroup$ – Howsikan May 14 '15 at 23:32
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    $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. $\endgroup$ – Lumi May 15 '15 at 1:27
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    $\begingroup$ @FooBar That didn't come over unfortunately - and I'm not disagreeing about the (many) abuses of regression analysis - but I think it's important to take the time to do the formal discussion as Alecos did, which helps build understanding and intuition, rather than offhand 'I hope you're not...' type comments. $\endgroup$ – Lumi May 16 '15 at 16:14

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