I was trying to determine which of the Gauss-Markov assumptions allow us to see that $b_1$ is an unbiased estimator of $\beta_1$. I have a feeling it's that $X_{i}$ is not random, but is there anything else that I'm missing?
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3 Answers
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The LS-Estimator is:$$b=\beta + (X'X)^{-1}X'e$$ The estimator is unbiased if $(X'X)^{-1}X'e$ converges to zero, and this is the case, if the designmatrix $X$ is not correlated with the error $e$.
So, the necessary assumption is: $$E[X_{t,k}*e_t]=0$$
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$\begingroup$ Thanks! To clarify, at the moment I am doing the one-dimensional case. Is this to say that the errors and $X_{i}$ values are uncorrelated? $\endgroup$ Commented May 27, 2013 at 0:34
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$\begingroup$ Yes, this is also valid for one-dimensional case. $\endgroup$ Commented May 27, 2013 at 0:39
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$\begingroup$ But is it correct to say that what you're talking about is a correlation of zero between Xi and an error term $e_i$ (not a residual, the actual error)? And my apologies, I accepted, I made the same mistake on the last one but went back and accepted. $\endgroup$ Commented May 27, 2013 at 1:09
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$\begingroup$ Actually between the variable and errors. But you can only observe the residuals (differences between the observations and the estimated function), while the errors are deviations between the observations and the "true" function, which is unknown. So, you solve the problem using the estimators of errors - the residuals. $\endgroup$ Commented May 27, 2013 at 1:25
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$\begingroup$ Okay that makes sense. One final question though as I still am not sure what the decision matrix is. It is a multi-dimensional version of the Yi? $\endgroup$ Commented May 27, 2013 at 1:31
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The Gauss-Markov Theorem is actually telling us that in a regression model, where the expected value of our error terms is zero, $E(\epsilon_{i}) = 0$ and variance of the error terms is constant and finite $\sigma^{2}(\epsilon_{i}) = \sigma^{2} < \infty$ and $\epsilon_{i}$ and $\epsilon_{j}$ are uncorrelated for all i and j the least squares estimator $b_{0}$ and $b_{1}$ are unbiased and have minimum variance among all unbiased linear estimators. Note that there might be biased estimator which have a even lower variance.
Extensive information about the Gauss-Markov Theorem, such as the mathematical proof of the Gauss-Markov Theorem can be found here http://economictheoryblog.com/2015/02/26/markov_theorem/
However, if you want to know which assumption is necessary for $b1$ to be an unbiased estimator for $\beta1$, I guess that assumption 1 to 4 of the following post (http://economictheoryblog.com/2015/04/01/ols_assumptions/) must be fulfilled to have an unbiased estimator.
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$$\hat{\beta} = ([inv(X'X)]X')(X\beta + \epsilon)$$ $$\hat{\beta} = \beta + ([inv(X'X)]X')\epsilon$$
$\hat{\beta}$ is an unbiased estimator of $\beta$ under two conditions:
$X$ is non-stochastic $$E(\hat{\beta}) = \beta + E[([inv(X'X)]X')\epsilon]$$ if $X$ is deterministic, this would reduce to: $$E(\hat{\beta}) = \beta + ([inv(X'X)]X') E[\epsilon]$$ The second term on right hand side, $E[\epsilon]$ is zero under one of the Gauss markov assumption.
$X$ is stochastic but independent of error ($\epsilon$) Using this, we can reduce the equation to: $$E(\hat{\beta}) = \beta + inv(X'X)] E[(X')\epsilon]$$ where $E[(X')\epsilon] = 0$ from an assumption that comes from one of the OLS's properties, $E[X'e] = 0$.
Reference:
https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf
Thanks
Anurag
