I know how to derive the GLS estimator of beta (theoretical GLS), but there a slight change to the question and i am not quite sure how to go about it.

A researcher has reason to believe that the disturbances in the standard model are heteroscedastic, with a threefold block-diagonal structure, such that:

$$Y_i = X_i\beta + U_i, \;\;\ i=1,2,3$$

where $Y_i$ is a $T_i\times 1$ column vector $X_i$ is a $T_i\times K$ regressor matrix and $U_i$ isa $T_i\times 1$ column vector of disturbances. $\beta$ is a $K \times 1$ of population regression coefficients, assumed to be the same for each of the blocks.

Assume that the only departure from the standard assumptions is that:

$$E(U_iU_i') = \sigma^2_iI_{T_i}$$

where $I_{T_i}$ is the identity matrix of order $T_i$ .

My question is simply how to obtain the GLS estimator of $\beta$ assuming that the $\sigma^{2}_{i}$'s are known.

  • $\begingroup$ It appears you are estimating a system of three equations. In such a case you must specify the relation between the three error terms. $\endgroup$ Nov 27 '14 at 14:24
  • $\begingroup$ Well, in my notes nothing as been specified in regard to the relation between the error terms. I just know that the general expression for GLS estimator of beta which is found by beforehand finding a matrix P. That is Py = PXbeta + Pu Could you tell me more about how you would start with please? $\endgroup$
    – user304663
    Nov 27 '14 at 15:14
  • $\begingroup$ I will, but in a while. $\endgroup$ Nov 27 '14 at 16:04
  • $\begingroup$ Any chance someone can shed some light on this, or Alecos if you are still there? $\endgroup$
    – user304663
    Nov 27 '14 at 21:36
  • $\begingroup$ Be patient. I need to get near a keyboard first $\endgroup$ Nov 27 '14 at 22:05

This is essentially a "groupwise heteroskedastic" model, and it doesn't matter whether the three $Y_i$'s and the three $X_i$'s may contain different random variables (and not just different realizations of the same random variables). All that matters is that
a) The coefficient vector is assumed identical,
b) The regressor matrices have the same column dimension, and
c) The $\mathbf U_1, \mathbf U_2, \mathbf U_3$ are assumed independent.

Then it is business as usual.

To solve it, we stack our data set per equation, i.e. all observations pertaining to $Y_1$ equation first, etc, obtaining a $T\times K$ regressor matrix, $T= T_1+T_2+T_3$. We have the model

$$\mathbf y = \mathbf X\beta + \mathbf u \tag{1}$$

Mind you, we don't really care if each equation has different regressors The model is heteroskedastic, with $T \times T$ conditional covariance matrix

$$E(\mathbf u \mathbf u' \mid \mathbf X)=\mathbf V = \text{diag}(\sigma^2_1,...,\sigma^2_2,...,\sigma^2_3,...) \tag{2}$$ with appropriate sub-dimensions. This matrix is symmetric and positive definite, so, as a general matrix-algebra result, there exists a decomposition

$$\mathbf V^{-1} = \mathbf C'\mathbf C = \text{diag}(1/\sigma^2_1,...,1/\sigma^2_2,...,1/\sigma^2_3,...) \tag{3}$$

which also gives

$$\mathbf V = \mathbf C^{-1} (\mathbf C')^{-1} \tag{4}$$

So in our case $$\mathbf C = \text{diag}(1/\sigma_1,...,1/\sigma_2,...,1/\sigma_3,...) \tag{5}$$

where $\mathbf C$ is non-singular $T \times T$ square matrix. Consider the linear transformation of the model by premultiplication by $\mathbf C$:

$$\mathbf {\tilde y} = \mathbf C \mathbf y, \;\; \mathbf {\tilde X}= \mathbf C \mathbf X,\;\; \mathbf {\tilde u}= \mathbf C \mathbf u$$

This transformation makes the error term conditionally homoskedastic, since

$$E[\mathbf {\tilde u}\mathbf {\tilde u}'\mid \mathbf X] = \mathbf C E(\mathbf u\mathbf u'\mid \mathbf X) \mathbf C' = \mathbf C \mathbf V \mathbf C' = \mathbf C \mathbf C^{-1} (\mathbf C')^{-1} \mathbf C' = \mathbf I_T$$

and one can verify that all other properties of the linear regression model are satisfied. So running OLS on the transformed model is optimal (in the usual sense). We will obtain

$$\hat \beta^{(tr)}_{OLS} = \left(\mathbf {\tilde X}'\mathbf {\tilde X}\right)^{-1}\mathbf {\tilde X}'\mathbf {\tilde y} = \left(\mathbf X'\mathbf C' \mathbf C\mathbf X\right)^{-1}\mathbf X' \mathbf C' \mathbf C\mathbf y = \left(\mathbf X'\mathbf V^{-1}\mathbf X\right)^{-1}\mathbf X'\mathbf V^{-1}\mathbf y $$

But this is equivalent to run GLS on the un-transformed model,

$$\beta_{GLS} = \left(\mathbf X'\mathbf V^{-1}\mathbf X\right)^{-1}\mathbf X'\mathbf V^{-1}\mathbf y $$

and its conditional variance is easy to derive as

$$\text{Var}\left[\hat \beta_{GLS}\right] = \left(\mathbf X'\mathbf V^{-1}\mathbf X\right)^{-1}$$

  • $\begingroup$ Thank you Alecos for your working. Yes indeed i did not realize that having several regressors doesn't differ from the general case. However i think i have spotted a typo here if i may: very last line you said : $\text{Var}\left[\hat \beta_{GLS}\right] = \left(\mathbf X'\mathbf V^{-1}\mathbf X\right)^{-1}$ but i think it is $\text{Var}\left[\hat \beta_{GLS}\right] = \sigma^{2}\left(\mathbf X'\mathbf V^{-1}\mathbf X\right)^{-1}$ isn't it? And also, i assume that for the feasible GLS estimator of $\beta$ using a 2-step procedure, the threefold block-diagonal structure does not change anything do $\endgroup$
    – user304663
    Nov 28 '14 at 13:51
  • $\begingroup$ No, because there is no single $\sigma^2$ in the model. $\endgroup$ Nov 28 '14 at 20:11
  • $\begingroup$ Yes i know sorry about that but the formulaes wouldn't show up properly when i tried to post in comments for some reason.. $\endgroup$
    – user304663
    Nov 30 '14 at 15:17

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