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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If the errors are homogenous but non-normal, can the linear estimator be BLUE?
Under the Gauss-Markov assumptions, the requirements for OLS to be BLUE are:
Linearity: The estimator must be a linear function of the data
Unbiasedness: The expected value of the estimator should ...
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OLS vs MLE when errors are not normally distributed (Laplace distributed)
We say that under assumptions of the Gauss-Markov theorem, OLS is BLUE. The Gauss-Markov theorem doesn't mention the normality of errors.
If the errors are distributed as per the Laplace distribution,...
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Question about OLS estimator (BLUE proof)
We know that the OLS estimator of $\beta$ is the unique BLUE.
The proof goes as follows. Consider the general linear estimator
$$\hat{\boldsymbol{\beta}}_\mathbf{A}
= \hat{\boldsymbol{\beta}}_\text{...
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Is zero condition mean preserved after transforming the conditioned variable?
In my econometrics class regarding multiple linear regression, we learned that one of the Gauss-Markov assumptions is the zero conditional mean, expressed as $ E(y|\boldsymbol{x}) = 0$.
My question is:...
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Why don't standard estimation methods consider the distribution of the residual?
Consider a linear model in which a vector of data $\mathbf{d} \in \mathbb{R}^M$ is related to an unknown parameter of interest $\mathbf{x} \in \mathbb{R}^N$ via $$\mathbf{d} = \mathbf{A}\mathbf{x} + \...
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Why would bootstrap OLS standard errors differ from ML estimate?
Let's say I have a regression dataset (paired x and y) such that the response variable (y) has an unknown distribution (but definitely not Gaussian) and is large enough such that the central limit ...
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What are the consequences of the violation of Gauss-Markov assumptions?
If these are the assumptions:
A(1) E(ϵi)=0 for all i
A(2) ϵi and xi´ are independent for all i,i´
A(3) var(ϵi)=σ² < ∞ for all i
A(4) cov(ϵi,ϵi´)=0 for all i ≠ i´
What does the violation of these ...
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Gauss Markov Theorem for linear combinations [duplicate]
I know that the Gauss Markov theorem implies that under some conditions, OLS estimates have the smallest variance of all unbiased linear estimators.
In particular if I have a model like $ y = \alpha + ...
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Properties of OLS under non-stationary variables
I estimate the following equation using OLS:
$y_{t} = a + b*x_{t} + u_t$.
I performed ADF tests for both y and x series and found that H0 (the existence of unit root) can not be rejected. I also ...
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What does $R(G:X)$ mean in Rao's book?
I was reading Rao's chapter 4 of his Linear Statistical Inference, 2nd ed. He uses the notation $R(G:X)$ in section 4.i (p. 294, formula (e), p. 296 formula (4i.1.21)) and that notation appears again ...
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Conditional exogeneity when regressors are causally, linearly related
Let's say there is a multiple regression:
$$
Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 + \beta_5X_5+U.
$$
I know that conditional exogeneity implies that the error terms are mean ...
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Is this regression impossible because of full rank assumption violation? [duplicate]
If there are four variables each with five observations:
X1 X2 X3 X4
5 3 0 2
0 9 -9 0
3 1 0 2
7 3 0 4
5 2 0 3
Why can't you regress a dependent ...
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Minimize MSE for only some parameters
Suppose I have a structural model parameterized by some $\theta = (\beta_i)_{i=1}^n$, but I am only interested in obtaining an unbiased/consistent/low variance estimator for $\beta_1$. For example, ...
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Where does the linear regression assumption that the errors are uncorrelated enter into the proof of Gauss Markov and that Least Squares is BLUE?
I often see that the "Spherical Error" assumption is invoked for Gauss Markov. One of the parts of the assumption is that the variance is constant given $X$. The other is usually that the ...
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How is zero conditional mean assumption "stronger" than uncorrelated assumption?
I am trying to understand what the zero conditional mean assumption ($\mathbb{E}[u\vert X]=0 $) encompasses in addition to a zero-correlation assumption ($\text{Corr}(X,u)=0$). I assume it must be &...
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Normality test vs Gauss Markov assumption for panel data
I am doing fixed effect regression after conducting hausmann test on panel data.
I received significant results in line with what's expected for my model. My data set has around 6000 observations and ...
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How Does Random Sampling Imply No Serial Correlation with Cross-Sectional Data?
I'm trying to prove that, under the random sampling, linear in parameters, and exogeneity Gauss-Markov assumptions, the error terms in $y=\beta_0+\beta_1x_1+\dots+\beta_kx_k+u$ are uncorrelated when ...
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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? [duplicate]
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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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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Difference between $Cov(x, y)$ and $Cov(x_i, y_i)$
I have seen the Gauss Markov assumption of uncorrelated dependent variables and error term presented in three different ways. I want to make sure that I am correctly interpreting the underlying ...
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