Questions tagged [gauss-markov-theorem]

The Gauss-Markov theorem gives the conditions for the best (minimum-variance) linear unbiased estimator of a linear model.

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Gauss-Markov theorem explanation (linear regression)

I have attatched an excerpt from my linear modelling lecture notes, this is the statement of the Gauss-Markov theorem, trouble is it goes into no more detail after this (not even explaining what the ...
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Gauss-Markov with $p>n$

Let $p$ be the number of parameters in a linear regression model, let $n$ be the number of observations, and let $p>n$. $$\mathbb E[Y\vert X] = \beta_0 +\beta_1X_1 +...+\beta_pX_p$$ Does the Gauss-...
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If the error terms in a regression setting are not observed, how can we ensure they're normally distributed?

According to the G-M assumptions, we should assume spherical errors. But my understanding is the errors -- as measured by the vertical distance from the true line of best fit to the response ...
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OLS is BLUE (or BUE) to minimize MSE. Is quantile regression is BLUE to minimize MAE?

The Gauss-Markov theorem considers "best" as "lowest mean square error (MSE)" and a recent version of the theorem shows OLS is not only BLUE but also BUE: https://www.ssc.wisc.edu/~...
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What is the benefit of regression with student-t residuals over OLS regression?

Sometimes I see advice to fit regressions with student-t residuals rather than using OLS (which is equivalent to assuming normally distributed residuals) if the distribution of the residuals is heavy-...
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Proof of Gauss-Markov Theorem: Unaccountable Line

I'm trying to follow Faraway's proof of the Gauss-Markov model in his book Linear Models with R, 2nd Ed., on pages 22-23, but I have been having numerous difficulties. The latest difficulty is thick ...
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Estimable Function Definition: Why $\forall \beta?$

I am reading Linear Models with R, 2nd Ed., by Julian J. Faraway, and on page 22, section 2.8 on the Gauss-Markov Theorem, he defines an estimable linear combination as follows: A linear combination ...
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Unbiased least squares estimate for GM Theorem

In order to prove the Gauss-Markov Theorem, we first have to show that the OLS estimate $\hat{\theta}$ is an unbiased estimator. From what Im reading on Internet and some textbooks, these are the main ...
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Gauss Markov theorem conditions as sufficient, but not necessary, for ‘BLUENESS’

In my understanding, the GM theorem posits a set of sufficient conditions (that a regression model must satisfy) in order for the estimator (of the coefficients in that model) to be the best linear ...
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Is any Gauss-Markov assumption violated by the simple OLS regression transformation?

I am creating a simple linear model with the following form: $$ y_i/x_i = \alpha + \beta x_i + u_i $$ The response variable has different name other than $y/x$, but it is essentially normalized by X ...
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Generalized Least Square When Disturbance Covariance Matrix Is Rank Deficient

I cannot find any general results on the following Generalized Least Square (GLS) problem. Let $Y = X\beta + E$, where $X$ is deterministic and of full column rank $k$, and $E$ is of zero mean, with a ...
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(Why) Are stepwise regression coefficients biased?

Issues with stepwise regression are known to statisticians. It yields R-squared values that are badly biased to be high. The F and chi-squared tests quoted next to each variable on the printout do ...
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linearity in parameter assumption - implication for the Gauss-Markov theorem [closed]

Even if a similar question has been asked many times, I have not been able to understand the consequences of non linearity in parameter in relation to the Gauss-Markov theorem. In this question ...
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Quadratic Form in Gauss-Markov Theorem

The following is the derivation of the Gauss-Markov Theorem in Greene's Econometric Analysis (8th ed.) I am trying to understand two points about the passage in yellow about the quadratic form. Why ...
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Gauss-Markov and Asymptotic Properties

Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $\sigma^2 I$) can imply "consistency" and "asymptotic normality" of the OLS estimator? ...
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Is Least Squares estimator for linear model the unique minimum variance unbiased estimator for a linear model?

I am following Linear Modfels in Statistics, Rencher & Schaalje, 2nd Edicition for the proof of Gauss-Markov Theorem (Theorem 7.3d, Page 146). I understand how least squares estimator $\mathbf{\...
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Using the unbiasedness assumption in the proof of the Gauss-Markov Theorem

In what follows $y = (y_1,\dots,y_n)$ is a $n\times 1$ vector of random variables and $X = (x_{ij})$ is a $n\times d$ random matrix ($n>d$ tipically) with $\text{rank}(X)=d$ with probability 1. ...
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Does the Gauss-Markov Theorem state that OLS is the only BLUE estimator?

I was reading through the proof on wiki. Which is the following. \begin{align} \operatorname{E} \left[ \tilde\beta \right] &= \operatorname{E}[Cy] \\ &= \operatorname{E} \left [\left ((X'X)^{-...
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Random Sampling: Weak and Strong Exogenity

$Y \ = \ X' \beta \ + \ e $ Where $Y = (y_1, ..., y_n)$ and $\beta = (\beta_0,..., \beta_k)$. Why would Weak Exogenity under random sampling produce Strong Exogenity? I know that weak exogenity is ...
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Exogenity: What does E(eX) really mean and why is it used?

What does it mean to talk about the expectation of the product of the error term and an independent variable? Like, why do we even need to mention $E(e_i X_{ik})$? What is it actually describing or ...
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Finding Best Linear Unbiased Estimator

I have the doubt that if Gauss Markov theorem is applicable here since the Variance is not constant in the model. Without Gauss Markov Theorem, how can we obtain BLUE?
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