Re: independent variables - ln


J W wrote:
rastompk@xxxxxxxxx wrote:
Hi there,
If x1,...,xn are random independent variables,
it is true that ln(x1),...,ln(xn) are also independent?


I substituted "independent random variables" for "random independent
variables" in the above, in which case I believe my response is correct.

As we'll see, your language is still ambiguous. You were correct in
the sense you later stated, but incorrect in another sense which I'll
explain and illustrate.


I thought that QUESTION wss so infantile that was not worth
even a one line comment, such as:

What is the meaning of your "independent" in
"random indepdent variables"?

I would argue that *all* serious questions, no matter how trivial, are
worthy of a reply. The question posed seemed plain enough to me,

Therein lies my QUESTION and COMMENT in my reply

RF> What is the meaning of your "independent" in
RF> "random indepdent variables"?

which is both serious and non-facetious, in view of recent discusions
of the meaning of "independent variables" in a regression (the lr
in the subject).


and I
think we risk driving the statistically curious away if we demand that
posters to this group phrase (or re-phrase) queries in technically
correct, completely unambiguous language. The very reason most
part-timers stop here is because they're *uncomfortable* with statistics
and statistical language!

That being said, perhaps I should have clarified my assumptions about
the question being asked (i.e. independent = statistically independent)
in my response.

-J

Indeed, if you and/or the OP meant "statistically independent",
then the mention of lr (linear regression) is both unnecessary and
misleading and ambiguous.

The OP should have simply ASKED If x1, ... xn are random variables
that are independent,

it is true that ln(x1),...,ln(xn) are also independent?

then the only possible meaning of "independent" is "statistically
independent" or "stochastically independent" as you interpreted,
and the answer is trivially "YES", which is what I called in my
post the "dead duck".

But if the "random independent variables" in OP's question meant
"independent variables" in a lr (linear regression), then the ONLY
valid meaning of "independent" is "Linearly independent" in the
Linear Algebra sense, whether the X's are randomly observed
or fixed numbers chosen by design in the linear regression, then
the answer to the question

it is true that ln(x1),...,ln(xn) are also independent?

is NO, that is, ln(x1) ... ln(xn) MAY or MAY NOT be linearly
independent.

To put it another way, if the X's are valid variables in a
regression of Y on X1, ... Xn (whether the X's are fixed and
given or randomly observed), or X1, ... Xn are linearly
independent,

A regression of Y or ln(Y) on ln(X1), ln(X2), ..., ln(Xn) may
be impossible because ln(X1), ... ln(Xn) may be linearly
dependent so that no OLS solution is possible because the
matrix to be inverted for the solution may be singular or
rank deficient (to the extent of "worse than singular" <g>).

In other words, if the OP asks the question you took it to be,
it would be a no-brainer because the statistical independence
of random variables is invariant under all non-degenerate
transformations, and ln(X) is just any one of infinitely many
transformations that will leave the resultant transformed
variables statistically independent.

On the other hand, if the "independent variables" are only
linearly independent, as is to be understood in the independent
variables in a regression, then ln(X) transformation MAY leave
the variables inappropriate for a regression on the transformed
variables.

That would be the nontrivial answer that takes just a little bit
of thinking as to why the ln(Xi) may be linearly dependent
while the Xi are linearly independent.

If the general reason escaped you, here is a special small
set of X1, X2, and X3 that has the property than they ARE
linearly INDEPENDENT while ln(X1), ln(X2), and ln(X3) are
linearly DEPENDENT, to help you think through the general
situation.

For my ease of not having to line numbers up, these are three
independent variables of a regression shown as ROWS

X1: 2.71828182845905 7.38905609893065 20.0855369231877
54.5981500331443 148.413159102577

X2: 7.38905609893065 148.413159102577 8103.0839275754
59874.1417151979 162754.791419004

X3: 20.0855369231877 1096.63315842846 162754.791419004
3269017.37247212 24154952.7535753

It doesn't matter if the Xs are fixed and given or randomly observed.
They can easily be shown to be linearly INDEPENDENT.

Ln(X1), Ln(X2), and Ln(X3) are linearly DEPENDENT.

-- Reef Fish Bob.

.

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