# Questions tagged [weighted-least-squares]

This tag is for questions relating to weighted least squares, a generalization of ordinary least squares and linear regression in which the errors covariance matrix is allowed to be different from an identity matrix.

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### Weighted Hotelling T2

Suppose I have a covariance matrix defined by: $\hat{\mathbf{\Sigma}}=\frac{1}{n-1} \sum_{i=1}^{n}\left(\mathbf{x}_{i}-\overline{\mathbf{x}}\right)^{T}\left(\mathbf{x}_{i}-\overline{\mathbf{x}}\right)$...
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### Closed form solution for Restricted Weighted Least Squares

From Greene, we know that the closed-form solution of a restricted least squares is: $\beta_{Constrained} = \beta_{Uncon} - (X'X)^{-1}R'[R(X'X)^{-1}R']^{-1}(R\beta_{Uncon}-r)$. Is there any similar ...
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### Convexity conditions for an Infimum operation

I came across these 2 seemingly contradictory statements in the book 'Convex Optimization' by Boyd. and So the top image is said to be concave. For the simplest case where $n=1$ the function is the ...
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### Please help me compare the variances of linear regression parameter estimates obtained by OLS and WLS

If the errors from the error vector $\varepsilon$ are independent, but have distinct variances, so that $Var(\varepsilon|X)=\Sigma=diag(\sigma _1^2,...,\sigma _N^2)$. Variance-covariance matrix of ...
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### How to calculate weight for least squares if both values error is known?

Usually minimizing $$\chi^2 = \sum_i w_i (y_i - f(a_0, a_1, \dots, a_n, x_i))^2$$ where $a_k$ are parameters is done by taking $w_i = 1/\sigma_i^2$ where $\sigma_i$ is a observation error of $y_i$ ...
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### Deriving the weighting matrix for the efficient GLS estimator

$u_{it} = \nu _{it} - \theta \nu _{i\left ( t-1 \right )}$ for $t>1$ $u_{i1} = \nu _{i1}$ and the $\nu _{it}$ are white noise with variance equal to $\sigma^{2}$. This is for a system of $T$ ...
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### Restricted Weighted Linear Regression in R

I have the following issue. I would like to run a linear regression imposing a constraint on the weighted coefficients. Let me construct an example: Consider the following cross-sectional regression ...