Rothbard on statistics

Economics Economics Questions
1

This paper discusses, on pages 6-7, Rothbard’s comments on statistical inference. He says that the theory of statistics rests on the assumption that all samples will be distributed on a normal curve.

How is it possible to take this as anything other than a mistaken understanding by Rothbard of the Central Limit Theorem? That theorem does not say that all samples are distributed normally, but rather that for any non-normal distribution, the distribution of the averages of various groups can be made to approach as closely to normal as desired.

Furthermore, the CLT is not an assumption, as Rothbard said, but a theorem, logically derived from the axioms of probability theory.

I don’t want Rothbard’s statement to be absurd. Can any mathematicians think of an interpretation of Rothbard’s words that makes sense?

2

You should probably send this question to someone who really knows Rothbard’s writings as well as a lot of statistics…so email Robert Murphy or Walter Block.

bob.murphy.ancap@gmail.com

wblock@loyno.edu

3

I just looked at the PDF quickly, and I found it strange. I’m only a first year economic student and I wouldn’t make such an elemntary mistake, which seems odd since Rothbard was very proficient in maths.

4

the fellow that wrote the pdf about what Rothbard said, then goes on to comment about it

It may still
remain true, however, that in contexts where random
collectives do not exist (that is, contexts characterized
by lack of independent repetitions), as will often be the
case in economics, objective probabilities cannot be
used. Given that Rothbard embraced an objective,
frequency interpretation of numerical probability, his
rejection of statistics is a defensible and logically
consistent corollary. Moreover the rejection of the use
of objective probabilities in economics is in agreement
with the conclusions of some of the most recent research
about these matters, and with general arguments for
interpreting probabilities in economics as
epistemological rather than objective.8

5

How does epistemological contrast with objective? [:S]

6

im sure if i had a PhD i could explain,

but i dont… so… [:S]

7

Yet none of this is Rothbard’s argument. It’s one thing to give a different reason for Rothbard’s conclusion, but something else entirely to explain Rothbard’s reasoning.

8

is it not analgous to other differences between austrians and neo-classicals.?

yes, you have models and given the assumptions you make, the models are good models. but contrariwise your assumptions are nonsense so i dont find your models useful. etc. etc.

im really speculating as i know next to nothing about statistics aside from basic stuff about averages and whatnots.

9

I believe the difference is that objective probability results from uncertainty in the case, whereas epistemological uncertainty results from incomplete knowledge. We say that the outcome of a pool shot is not likely to put a ball in the hole, but it might, which seems to express epistemological probability, since (many believe anyway) if we knew all the forces, the outcome could be calculated in advance. An example of the distinction seems to be in the various interpretations of quantum mechanics - whether the Heisenberg uncertainty principle refers to uncertainty in the thing (the electron doesn’t have a certain position and velocity), which would be objective, or to uncertainty of knowledge (to measure the position, we use a wave of such a high energy that it changes the velocity) which would be epistemological. Thus, when we give the probability cloud, the probability of it being in some portion of the cloud is, in the first view, objective, and in the second, epistemological.

10

Ah I see, so it’s basically a term of art then.

11

Kling on probabitily.

12

Argh, yet another article to look into for my paper on methodology! Luckily it’s brief enough. [:P]

13

it was very informative article, i like how he categorised the three approaches to probability. (i wonder are there more?)

he said this:

Frank Knight was famous for describing a notion of uncertainty that could not be reduced to an objective probability distribution.

does that mean it could not be reduced to something that can be translated into the normal distribution that was mentioned earlier? is this then the rothbardian position?

14

Knight’s uncertainty is something like the probability that Hans Hermann Hoppe will oppose a proposed bailout bill. There’s no calculation or formula for figuring it out, and it’s just silly to say that Hoppe is 90% certain to oppose the bill. What does the 90% stand for? It isn’t his past record of positions on similar bills (frequentist.) It isn’t logically derivable from some set of axioms (axiomatic.) What is it? The best we can say is that he’s very likely to oppose it, but we cannot give a number. Some might want to give a number of 100%, but that’s not true either. People do change their positions, and an Austrian could decide tomorrow to support socialism. Giving a probability of 100% would imply that was impossible.

Hoppe relates Knightian uncertainty directly to human action.

Although it’s possible, I don’t think this is what Rothbard had in mind. My reason for thinking that is the he was talking in the context of a statistics class, and so he should have in mind the kinds of things that people claim statistical inference works with. An example would be GDP figures over long time frames, measured annually. Such numbers are not at all normally distributed. As far as I can tell, Rothbard is claiming that, for their calculations, statisticians simply assume that GDP is distributed normally, find the average, find sigma, and assign probabilities to the future from there. But everyone knows it’s not normal, and no one does this. Instead, statisticians make use of the CVT, and do this kind of procedure:

  1. Find the distribution annually. Notice that it isn’t normal, or close to normal.
  2. Break the interval into 2-year periods, and assign each 2-year period the average of the GDPs. Notice that it still isn’t close to normal.
  3. Break the interval into 3-year periods, averaging in each interval.
  4. Continue the process. The CVT says that the distribution will approach normality.
15

under what conditions would it be incorrect to apply the CLT when wanting to run some statistical analysis? do economic matters of the kind that concerned rothbard share these conditions or no?

its just that this on the wiki page for clt caught my eye>>

let X1, X2, X3, … Xn be a sequence of n independent and identically distributed (i.i.d) random variables each having finite values of expectation µ and variance σ2 > 0. etc.

i am a layman so forgive me if what im saying seems particularly stupid.

16

To see the preconditions for the CLT, let’s look at it first in the other direction. Suppose I have a population with mean mu and standard deviation sigma. Repeat: mu is the mean of the population - the “real” mean, and sigma is the “real” standard deviation. I can only draw a normal curve with these features if mu and sigma are both finite, so that’s a precondition for the CLT. Now, the trivial case is that the population is distributed normally. Then for any sample size, we expect the sample distribution (the distribution of the things we selected out of the population in some random sample) to approximate a normal curve with mean my and standard deviation sigma. So we won’t bother with this case.

Now, if I draw a simple random sample (another precondition for CLT) and measure some trait of the things selected, and the trait is itself a random variable (another precondition) I can then find the average of my sample. So, I can have a population of 10M people, select an SRS of 1000 people, measure their heights, and find the average height, call it h. What does this average height tell me about the population? I can make a guess that mu is about h, and there are statistical tests for determining my confidence in that and finding the range covered by “about” but I’m also interested in the distribution of heights. To get information about this, I begin by finding the distribution of average heights. That is, if I drew this sample many times, each time randomly selecting 1000 people, how would the various h values be distributed?

This is where CLT comes into play. Suppose that instead of 1000 people, I drew 10 people. Now I repeat this many times, each time finding the average, then determine how often each number comes up as the average (give ranges instead of numbers, though.) This distribution would be expected to have an average around mu, but not to be normal. Now, as I increase the sample size from 10 to 100 to 1000, the distribution is guaranteed to approach normality. Notice that at the limit, I have a sample size of 10M. Thus, each sample drawn is exactly the same, and we’d expect that for every sample, we’ll get the same average. This isn’t true, though - we can make mistakes measuring, or have measurements that call for judgement. Hence, at the limit we’ll have a normal distribution with a very small standard deviation. When the sample size is large enough (subject to definition) we can even describe what the normal distribution looks like - the mean is approximately mu, and the standard deviation is approximately sigma/sqrt(sample size)

So yes, there are situations where CLT doesn’t apply. In essence:

  1. Infinite population

  2. Infinite mean or standard deviation

  3. Zero standard deviation (hence zero variance - i.e. a constant population)

  4. Trait not describable as a random variable (such as political party preference, in Rothbard’s example - hence statistics makes no claim at all about normality of this trait)

  5. No way to draw simple random samples, or drawing simple random samples is misleading (such as determining the average number of testicles in a population of both males and females - similar to the man with his head in the oven and his feet in the freezer who is quite comfortable on average)

But this has nothing to do with Rothbard’s claim. Rothbard claimed that statistics claims that all traits whatsoever are distributed normally, and that this is an assumption of statistics. This seems wrong because:

  1. There are traits about which statistics makes no normality claims

  2. Regarding traits which statistics does make normality claims about, the CLT doesn’t say that the distribution is normal, only the sample distribution for a large enough sample size

  3. The CLT is not an assumption of statistics, it is a theorem.

17

if it helps or no, here is the longer article from which rothbards statements were pulled

perhaps your disagreement with rothbard is just that he was unaware of other statistical techniques aside from what you have outlined, and mistook one dominant technique for the whole of the field? but then again it leaves the question of how useful statistics are to economists unaddressed.

18

I don’t agree with Taleb’s politics, but I love his skeptical mathematical mind:

The Fourth Quadrant

Inverse Problems. It is the greatest epistemological difficulty I know. In real life we do not observe probability distributions (not even in Soviet Russia, not even the French government). We just observe events. So we do not know the statistical properties—until, of course, after the fact. Given a set of observations, plenty of statistical distributions can correspond to the exact same realizations—each would extrapolate differently outside the set of events on which it was derived. The inverse problem is more acute when more theories, more distributions can fit a set a data.

This inverse problem is compounded by the small sample properties of rare events as these will be naturally rare in a past sample. It is also acute in the presence of nonlinearities as the families of possible models/parametrization explode in numbers.

19

The assumption that Rothbard has a problem with is the following:

“In the science of statistics, the way we move from our known samples to the unknown population is to make one crucial assumption: that the samples will, in any and all cases, whether we are dealing with height or unemployment or who is going to vote for this or that candidate, be distributed around the population figure according to the so-called “normal curve.”

And according to the Wikipedia article, which cites John Rice in this definition:

the central limit theorem (CLT) states conditions under which the sum of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed

First of all, this definition of the theorem seems very different from yours, because it doesn’t say anything about making distributions “approach as closely to normal as desired.” Secondly, it seems to me, Rothbard doesn’t have a problem with the CLT itself as far as it goes, but rather a problem with the statistician’s habit of assuming it applies to the real world, given that the derived CLT itself only purports to state conditions under which the sum of certain numbers are normally distributed, and not that those conditions always exist in reality, or that the statistician can even know when and where those conditions are present in reality.

The critic of Rothbard quoted in the text himself admitted the limited nature of the CLT’s application:

Alternatively, normal distribution can be strictly derived by the Central Limit Theorem, which shows that where some variable is influenced by a large number of unrelated random variables, that variable will be normally distributed. This result holds subject to certain conditions, which are very widely, but not universally, encountered. Statisticians are open to the possibility of non-normal distributions where these conditions don ́t apply.

My question is, how do statisticians know these conditions occur “widely”?

20

I have never come across the definition in the wikipedia article. I’ve always learned, and taught, the way I detailed in a later post: That as the sample size increases, the sampling distribution(not the observed distribution in the sample, but rather the distribution of the average X value on the samples) approaches normality. Perhaps one is a correlary of the other; I’m not a statistician.

It’s an empirical question, but certainly by my statement, it doesn’t appear up to much debate. The conditions just aren’t that hard to meet. Finite mean and sd just means that all values are finite, and the population is finite. Having variables amenable to simple random samples, and independent, are harder to meet (height doesn’t quite work because it can be correlated to sex, and if we allow for sex variations, it’s not an SRS) but there still seem to be plenty of examples.