β_hat = ([inv(X'X)]X')(Xβ + epsilon) β_hat = β + ([inv(X'X)]X')epsilon

β_hat is an unbiased estimator of β under two conditions:

1. X is non-stochastic

E(β_hat) = β + E[([inv(X'X)]X')epsilon]

if X is deterministic, this would reduce to;

E(β_hat) = β + ([inv(X'X)]X') E[epsilon]

The second term on right hand side, E[epsilon] is zero under one of the Gauss markov assumption.

2. X is stochastic but independent of error(epsilon)

Using this, we can reduce the equation to;

E(β_hat) = β + 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

$$\hat{\beta} = ([inv(X'X)]X')(X\beta + \epsilon)$$ $$\hat{\beta} = \beta + ([inv(X'X)]X')\epsilon$$

$\hat{\beta}$ is an unbiased estimator of $\beta$ under two conditions:

1. $X$ 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.

2. $X$ 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