I'm looking to obtain standard errors for constrained linear model estimation. I'd like to be able to estimate standard errors and confidence intervals for the coefficients of $\beta'$ estimated via:

$$ \beta' = \underset{\beta}{\operatorname{arg\,min}} \, (\mathbf{y} -\mathbf{X}\mathbf{\beta})^2 \ \ \ s.t.\ \ \ \mathbf{A^T} \mathbf{\beta} \leq \mathbf{c} $$

My understanding is that I can exploit GMM (Generalized method of moments) and the Delta Method to estimate my standard errors $\ SE(\ \beta' - \beta_0'\ )$ as $\sqrt{diag(\ \Omega \ )}$, where

$ f(\mathbf{X}, \mathbf{\beta}) := $ my quadratic loss function

$\mathbf{d} = \frac{\partial\ E_T(\ f(\mathbf{X}, \mathbf{\beta})\ )}{\partial \mathbf{\beta}}\bigg|_{\mathbf{\beta}'}, \ \mathbf{S} = E_t[\ f(\mathbf{X}, \mathbf{\beta})\ f(\mathbf{X}, \mathbf{\beta})^T\ ]$

$\Omega = \frac{1}{T}(\mathbf{d}^T\ \mathbf{S}^{-1}\ \mathbf{d})$

and $E_t[ \cdots ]$ is an expectation operator over my $T$ datapoints. My questions are:

  1. How should I handle the partial derivative of my loss function $\ f(\cdots)?$ It seems that my constraint condition should factor in somehow. Is it valid to use a logarithmic barrier function, i.e. $g(\mathbf{X}, \mathbf{\beta}) = f(\mathbf{X}, \mathbf{\beta}) - log(\mathbf{c} - \mathbf{A^T}\mathbf{\beta})$

  2. How do I handle the singularity when the constraint inequality is tight? Can I manually plug in a largely negative values for the Jacobian in the cases where my barrier function blows up?

  3. Is this approach generally valid? Does this approach work for general linear and quadratic programs? Also, I suspect that I'm going to have to take a pseudo-inverse in the case where the matrix is (nearly) singular.

I think I'm close... A related question (for unconstrained GLS) was asked here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.