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Your fallacy is thinking that "efficiency" merely boils down to number of observations. SUR does not "utilize more observations" than OLS. The relative efficiency of two estimators is the ratio of their squared standard errors. You must also consider the bias of the estimators and distinguish between attributes of regressors ($\beta$) and the resulting ...

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\begin{aligned} \min\limits_{\boldsymbol{b}} \boldsymbol{e}^T\boldsymbol{e} = (\boldsymbol{Y}-\boldsymbol{Xb})^T(\boldsymbol{Y}-\boldsymbol{Xb}) \\ \end{aligned} FONC: let $\boldsymbol{u} = \boldsymbol{Y}-\boldsymbol{Xb}$ \begin{aligned} \frac{\partial \boldsymbol{e}^T\boldsymbol{e}}{\partial{\boldsymbol{b}}} &= 0 \\ \frac{\partial [\boldsymbol{Y}^T-\...

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Because OLS belongs in a subfield of mathematical optimization called convex optimization, and that field has some nice properties, such as every local minimum is a global minimum

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Or how else is such a non-linear relationship without closed-form solution approached? For maximum likelihood we want the density of $y$ conditional on $x$ (because this is our sample), $f_{y|x}(y|x)$. We start with making an assumption on the density of $\varepsilon$ conditional on $x$, $f_{\varepsilon|x}(\varepsilon|x)$. When the relation is not ...

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Assume that the distributions here are such that $$E[y\mid (x,z)] = \beta_2 x+\beta_3 z$$ Then we can write $$y = E[y\mid (x,z)] + e_{xz}$$ where $e_{xz}$ is the conditional expectation function error (of the specific conditional expectation). By design, as can be easily verified, $$E[e_{xz} \mid (x,z)] = 0 \implies E[e_{xz} x] = E[e_{xz} z]=0$$ ...

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$U$ is an orthonormal matrix, with columns having norm $1$ and orthogonal to each other. The solution is not unique, so coming up with a solution should suffice for you I guess, if there are no other restrictions. You can create a random matrix, and then orthonormalize its columns using gram-schmidt process. The following R script does it for you: library('...

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For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Facebook Prophet, and others. The tricky part is time independent regressors: Most people don't realize that time independent regressors are of no use ...

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You could do that, or you could use a method that weights the composites (on both sides) based on the relationships between predictor and outcome variables.Back in the day, Jacob Cohen (1982) described what he called "set correlation," a generalization of regression that allowed the LHS to include a set of variables. However, there are a range of techniques ...

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