From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Subject: Re: Estimation of spline join points? References, suggestions?
Keywords: Splines, join points, Maximum-Likelihood
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota
Date: Wed, 16 Dec 1992 20:33:29 GMT

gyan@unixg.ubc.ca (Gyan P. Sinha) writes:

>	I am estimating the parameters of a censored regression model
>(panel tobit), and one of the variables has a non-linear effect on
        ^^^^^
There are other estimators available that do not require the normality
assumption even when you have fixed effects in your panel data model.
See below.

>the dependent variable. I am trying to use a spline to model this, 
>and its defined as:

>	V1 = THETA + MIN(DIFF-THETA,0)
>	V2 = MAX(DIFF-THETA,0)

>so that if DIFF is less than THETA, then the effect on the dependent
>variable is different than when it is larger, and V1 and V2 are the
>two secondary independent variables created from the DIFF variable.
>Instead  of imposing a value on THETA, I'd like to be able to estimate
>it from the data itself.

>My questions are:

>(1) OPTIMIZATION STRATEGY: ML estimation will require some type of
>non-gradient  technique (?), so how do I get the asymtotic t-stats and
>such? 

Your model is continuous in THETA, and the only issue of
differentiability occurs when THETA = DIFF.  The right and left
derivatives are well defined even at this point, and the difference
between them is everywhere bounded by |alpha_2 - alpha_1|, where
alpha_1 and alpha_2 are the coefficients on V1 and V2 respectively.
You should be able to use the right and left derivatives when
implementing your maximization calculation.  The situation is similar
to the one that occurs when you calculate a median by minimizing the
sum of absolute deviations.  You might also have to deal with the
possibility of multiple local maxima to the likelihood.  (I haven't
checked.)

As for the asymptotics: All of the usual properties of ML estimation
should hold, except that you can't use the second derivative
formulation of the information matrix.  Instead, define the
information matrix as the variance of the score vector and all will be
well, provided that there is zero probability that THETA = DIFF at the
true model, so that you can ignore the points of non-
differentiability.  (The second derivative formulation of the
information matrix will not be valid since the right and left first
derivatives are not even continuous.)

It may happen that THETA = DIFF at the estimated model for at least a
few of the sample points even if it is known that this has zero
probability at the true model.  (The proportion of observations where
this happens should be asymptotically negligible.)  You can use any
number between the right and left derivatives of the likelihood when
evaluating the score vector for these observations.

>(2) MEANING OF T-STATS: If I am estimating the join point from the
>data itself, does that require some special tweaking on the t-ratios
>to get legitimate tests of significance?

Not that I can see, unless Pr{ THETA = DIFF } > 0 at the true model.
It will not be true that the information is equal to minus the
expectation of the second derivatives of the log-likelihood, however,
which is why you need the more general formulation in terms of the
variance of the first derivatives.

>(3) REFERENCES: Can anyone suggest references that I can explore on
>this problem?

You might get some insight into a general method for handling
non-differentiabilities of your objective function from the analysis
of the least absolute deviations estimator in:

"Asymptotics for Least Absolute Deviations Regression Estimators" by
David Pollard, Econometric Theory, Vol. 7 No. 2 (June 1991), pp.
186-199.

You might also want to look at

"Trimmed LAD and Least Squares Estimation of Truncated and Censored
Regression Models with Fixed Effects" by Bo Honore, Econometrica, Vol.
60, No. 3 (May, 1992), pp. 533-565.

This paper presents several estimators with continuous and piecewise
differentiable objective functions.

You should probably look at Honore's paper anyhow, since he shows how
to estimate a censored regression model from panel data without making
any normality assumptions.  The normality assumption can lead to
biased and inconsistent estimates in the censored regression setting,
unless the true errors really are normally distributed.

>(4) ALTERNATIVES: Can anyone suggest alternatives that would allow for
>similar effects, other than introducing a polynomial in DIFF, of which
>I'm already aware of.

There are lots of ways to smooth out the kink in your model.  Here is
one approach.  Your model is equivalent to one in which the two
regressors are:

	V1 = DIFF
	V2 = (DIFF - THETA)*1{ DIFF - THETA > 0 }

where 1{ } is the binary indicator function for whether or not the
event in braces is true.  V2 can be written:

	V2 = (DIFF-THETA)*g(DIFF-THETA)

where g(z) = 1{ z > 0 }.  If you want a smoother function simply
replace this g(z) with any other monotone increasing function from
R->[0,1].  You can make your likelihood as smooth as you want in this
way.  The function g might even have its own parameters.  Since it is
bounded, you won't get the increased sensitivity to extreme values of
DIFF that you are forced to live with when you use a polynomial.

>Seasons Greetings!

Same to you!
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209