A few things to keep in mind.

1.) Taylor's series is an approximation to exp(x).  Also it's a non-linear
approximation. x^2 is not linear and neither is x^3 and sum(x^i/i!) for
i=0->inf is certainly not.

2.) Taylor's series only equals the approximated function as the terms go
out to infinity (x^->inf/->inf! -> 0)

Thus it's quite possible that your approximated function (with a few
Taylor's series) gives you statistically significant results whereas your
linear model doesn't. You've, in essence, chosen a lower order approximation
to exp(x).

The best way to find out, and the quickest is probably just to try it out.

>Thank you for your comments.
>
>Maybe I could specify my question more clearly. For example, a linear model
is y = alpha + beta*x + epsilon, and this beta is not statistically
significant. Then a non-linear model, e.g., y = eta + gamma*exp(x) + omega,
since exp(x) = 1 + x + (x^2)/(2!) + (x^3)/(3!) + ..., then y = eta +
gamma*(1+x+x^2/2+...) + omega. But beta in the former formula is not
statistically significant, is there probability that gamma is statistically
significant?

>And I will post this to r-help also, thanks.

>HC

> -----Original Message-----
> From: markleeds@verizon.net [mailto:markleeds@verizon.net]
> Sent: 2008?10?7? 23:10
> To: Hsiao-nan Cheung
> Subject: RE: [R-SIG-Finance] Statistically significant in linear and
> non-linear model
>
> Hi:  it's an interesting question and probably something that you
> should
> send to R -help also. One reasonis  the following but
> I don't know if it's a big one ?
>
> if one has the linear model log(y) = X*beta + epsilon, then, this can
> clearly be transformed to y = exp(Bx) and minimized using
> nls. But, I think it's possible that beta can turn out significant in A)
> and not in B) because of the assumption about the error term.
>
> In A) the error term is assumed to be additive and this assumption is
> used HEAVILY in standard OLS theory.
>
> In B) The error, term if A is true, is multiplicative and, however, nls
> works out the standard errors ( I guess it's estimates the Hessian
> and uses that )  could cause Beta to be not significant.
>
> And, the argument can also probably go the other way so that, one could
> have significance in nls bt not OLS.
>
> As I mentioned,  I don't know what the constribution of above is to
> your
> question but it's a thought. If you get any offline replies,
> could you send them to me because i'd be interested. I bet you would
> get
> a lot of responses from R-help if you sent it there also.
>
>
>
>
>
> On Tue, Oct 7, 2008 at 10:38 AM, Hsiao-nan Cheung wrote:
>
> > Hi,
> >
> >
> > I have a question to ask. if in a linear regression model, the
> > independent
> > variables are not statistically significant, is it necessary to test
> > these
> > variables in a non-linear model? Since most of non-linear form of a
> > variable
> > can be represented to a linear combination using Taylor's theorem, so
> > I
> > wonder whether the non-linear form is also not statistically
> > significant in
> > such a situation.
> >
> >
> > Best Regards
> >
> > Hsiao-nan Cheung
> >
> > 2008/10/07
> >
> >
> >     [[alternative HTML version deleted]]
> >
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-- 
Theodore Van Rooy
http://greentheo.scroggles.com

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