Problem – 1. Asymmetric Loss
Suppose that is an IID sequence satisfying for each
,
where is independent of , a random vector. Suppose also that all moments of and are finite and that is nonsingular. Suppose that and are defined to be the values of and that minimize
over some set , where for some
Describe the asymptotic behavior of the estimators and as . If you need to make additional assumptions be sure to specify what these are and why they are needed.
In this problem it will be convenient to define a shorthand notation
The first issue to consider is whether the true parameters solve the optimization problem
,(1)
and whether this solution is unique.
Denote . Here variable is independent of random variable (which is the function of only).
Function is twice continuously differentiable and convex:
The first order conditions for the problem (1) are:
(Sorry for the mess with notations: the prime sign means transposition everywhere but next to the function, where it means the derivative).
The true parameters solve this system only if
(The second equation of the system holds true automatically because of the independence of and random variable ).
So, given the distribution of errors ’s, there exist the only value of asymmetry factor , which guarantees the consistency of estimators and .
The matrix of second derivatives, taken at the point , is:
The matrix should be non-singular, so the necessary condition is that
(obviously, if then some coefficients in the expression cannot be determined from the regression, the problem is unidentifiable).
To sum up, under the additional assumptions made the values solve problem (1), and this solution is unique both locally and globally (due to the convexity of function ). Therefore, the estimators and
are consistent () by the Law of Large Numbers, and are the standard extreme estimators, which means that their asymptotic behavior is:
where
Finally, the result is:
when and .
Problem – 2.Nonlinearity at LHS
An IID sample is available for the nonlinear model
,
where the parameters and are scalars. Show that the NLLS estimator of and
is in general inconsistent. What feature makes the model differ from a nonlinear regression where the NLLS estimator is consistent?
The NLLS estimators would be consistent only if true parameters solved the optimization problem
Let’s write down the corresponding first order conditions:
Noticing that , the first of these conditions simplifies to
Obviously, this equation cannot hold in general, since and is correlated with . Therefore, estimators and are not consistent.
What is the source of this inconsistency? The first straightforward difference is that in usual NLLS setting the non-linearity was confined in the variable (which error of regression does not depend on). In this problem, on the contrary, non-linearity is in the dependent variable , which correlates with the error . Because of this non-linearity, the marginal influence of parameter becomes dependent on the variable . Together with the fact that correlates with , this generates an inconsistency of .
Problem – 3
The purpose of this exercise is to find out what optimization routines are better to use in GAUSS. In a Monte Carlo framework, investigate how well three different ways of finding maximum likelihood estimates work in estimating the parameters of the Student’s distribution with the density
.
Try several parameter combinations and various starting parameter values. The three ways of obtaining the ML estimates are the following:
(a)Procedure maxlikfrom the library maxlik for obtaining maximum likelihood estimates. Described in \SRC\maxlik.src of the GAUSS directory.
(b)Procedure optmum from the library optmum for doing general non-linear optimization. Described in \SRC\optmum.src of the GAUSS directory.
(c)Procedure eqSolve from the library nlsys for for solving non-linear systems of equations. The system of equations will be FOC for the ML estimates (also known as zero expected score). Described in \SRC\eqsolve.src.
A brief description of maxlikand optmumcan also be found at the course Webpage. To attach a library in GAUSS, use the command lineжlike library maxlik;