derivatives

[From Rick Marken (2004.09.18.0950)]

Marc Abrams (2004.09.18.1308)--

Yes, I made a transcription mistake, because I used h, rather that
then the
triangle and x to indicate a small change in x, but that still doesn't
change the fact that my equation was the right one and Powers was
wrong. Get
over it.

Your assertion that "Powers was wrong" makes it clear that your was
more than a "transcription mistake". If it was just a transcription
mistake you would know that the correctly written equation is exactly
equivalent to the one Bill posted. Bill was simply using a different
form of the equation. You were trying to write the f'(x) form of the
equation that defines the derivative, which is:

f'(x) = lim f(x+h) - f(x)
           h->0 --------------
                             h
Bill was using the dy/dx form, which is

dy/dx = lim f(x2) - f(x1)
              x2-->x1 --------------
                                      x2-x1

In this form of the equation, f(x) = y, x2 = x + h and x1 = x so x2 -
x1 = h and x2 -> x1 is equivalent to h - > 0.

What is the basis you use for accepting the sample of 15 quarters of
data as being representative of the entire _population_, that is, of
_all_
quarters of data?

I don't understand the question. What 15 quarters was I using as a
sample to estimate a population of quarters?

Was it a _random_ sample?

I still don't know what data you are talking about.

Could you frame your questions in terms of the data I presented,
please. What correlations? What pronouncement about growth in GNP
representing what? What did I say growth in GNP represents other than
growth in GNP?

No, you equated the deficit and debt as being 'bad' and the 'surplus'
as
being 'good'. I want to know why responsible debt is a 'bad' thing.
Deficits
are intolerable but debts are indeed needed.

It seems like you are talking about the quarterly data on growth and
budget balance over the last 15 years. If so, there was no sampling,
random or otherwise, involved in this data analysis. I didn't equate
the deficit with "bad" and the surplus with "good". And I didn't
present any data on the debt at all. All I did was show that the budget
balance (quarterly government deficit/surplus) was apparently under
control through the Clinton administration.

What is, and what was, the purpose of showing the amount of growth in
the
GNP?

I think you said -- but I have also heard it in the news media -- that
the Bush II deficits were the result of economic sluggishness or
recession that started at the end of the Clinton term and were
exacerbated by 9/11. The purpose of showing the GNP growth data was to
show that economic fluctuations as large as those that were occurring
at the beginning of Bush II (and after 9/11) had occurred through the
Clinton years and had no obvious effect on the budget balance.

Are you really attributing the 'growth' in the economy to a President?

No. I am considering the fluctuations in growth to be a disturbance to
the budget balance which, because it is barely affected by these
fluctuations, was apparently, under control during the Clinton
administration by not during Bush II. So I am assuming that presidents
have little or no control over fluctuations in economic growth.

RSM

···

---
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

[From Rick Marken (2004.09.18.1130)

Bruce Abbott (2004.09.18.1210 EST)--

Just a point about your statistical analysis. Your significance tests
have
meaning if, and only if, the 15 quarters of data you have represent a
sample from a larger population of quarters, and the sample is taken
randomly from that population. In the first paragraph above you appear
to
be implying that your data are the entire population of quarters of
interest. And even if they are a sample, they do not appear to be a
random
sample. If this is a population, then the observed correlation is what
it
is. You don't need a significance test to tell you that it isn't
likely to
be zero.

I assume you are talking about my analysis of the lagged correlations
between growth and investment (which involved considerably more than 15
quarters of data) since that's the only analysis I reported that
involved the computation of statistics (the correlation coefficient, in
that case).

What I found (to the apparent dismay of many) is a negative correlation
between investment and growth when investment preceded growth and a
positive correlation when investment followed growth.
Representativeness was not an issue in this analysis but precision was
(see my discussion of sampling in Ch. 5 of _Methods in Experimental
Psychology_). The correlations can be considered to be based on a
representative sample of measures of growth and investment taken during
the period of analysis (1953-present): the measures were taken at equal
intervals (quarters) throughout the period. Because the correlations
are based on a representative sample of data they can be considered
_unbiased_ estimates of the "true" lagged correlations between
investment and growth during this period. That is, the correlations
are not likely to systematically over or under estimate the "true"
correlations.

My statistical analysis, which involved calculating confidence
intervals around the lagged correlations between investment and growth,
was aimed at estimating the precision of these estimates. The
precision of the estimates (how close they are likely to be to the
"true" correlation value) depends on how accurately the sample values
of growth and investment are being measured. Since quarterly measures
of investment and GDP (on which the measures of growth are based) are
themselves based on sampled data entered by humans, they are likely to
by somewhat inexact. And, indeed, these figures are regularly revised
based on updated data. So the precision of correlations based on such
measures is definitely a concern.

I measured the precision of the correlations by computing 95%
confidence intervals around them. What I found is that the confidence
interval around the negative correlations, where investment preceded
growth by a couple quarters, did not overlap the confidence interval
around the positive correlations, where investment followed growth by a
couple quarters. Based on this result, I concluded that the negative
correlations, when investment preceded growth, were significantly
different from the positive correlations, when investment followed
growth. The difference was significant in the sense that it was seen
to exist within the precision (or lack thereof) of the growth and
investment data available.

Regards

Rick

···

---
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400