Changes

<math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \boldsymbol{\Omega}^(-1)(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math>

<math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \boldsymbol{\Omega}^(-1)(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math>

And the variance-covariance matrix of these statistics is (see Appendix A for details):

We further derive the variance-covariance matrix of these statistics as

                        V=(G-G ̅ )^T (Ω ̂^(-1)-Ω ̂^(-1) X(X^T Ω ̂^(-1) X)^(-1) X^T Ω ̂^(-1) )(G-G ̅).

                      <math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\boldsymbol{\Omega}^(-1)-\boldsymbol{\Omega}^(-1) \mathbf{X}(\mathbf{X^T}\boldsymbol{\Omega}^(-1)\mathbf{X})^(-1) \mathbf{X^T} \boldsymbol{\Omega}^(-1))(\mathbf{G}-\bar{\mathbf{G}}) </math>.

Under the null, test statistics T_i=(U_i^2)/V_ii  is asymptotically distributed as chi-squared with one degree of freedom.

Under the null, test statistics T_i=(U_i^2)/V_ii  is asymptotically distributed as chi-squared with one degree of freedom.