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True Model:

$$Y = \alpha + \beta \bf{X} + \gamma W + V$$

Model to be regressed on:

$$Y = \alpha + \beta \bf{X} + U $$

Where:

$$U + \gamma \bf{W} + V$$

So, in this model, if Cov(X,W) $\neq 0$ then $\hat{\beta}$ will be biased, assuming $\gamma \neq 0$ too.

$$\hat{\beta} = \frac{\sum_{i=1}^{n}\left[(X_{i}-\bar{X})(Y_{i}-\bar{Y})\right]}{\sum_{i=1}^{n}\left[(X_{i}-\bar{X})^2\right]}$$

How can I compute what the bias is? I believe I should analyze:

$$\mathbb{E}[\hat{\beta}] - \beta = \mathbb{E}\left[ \frac{\sum_{i=1}^{n}\left[(X_{i}-\bar{X})(Y_{i}-\bar{Y})\right]}{\sum_{i=1}^{n}\left[(X_{i}-\bar{X})^2\right]} \right] -\beta $$

However, I also know plim($\hat{\beta}) -\beta$ = $\gamma \frac{Cov(X,W)}{Var(X)}$

So, is $\gamma \frac{Cov(X,W)}{Var(X)}$ the answer for asymptotic bias? If not, what approach should I be taking to evaluate $\mathbb{E}[\hat{\beta}] - \beta$?

Thank you; and nuanced corrections to my notation and reasoning are appreciated.

user43395
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  • when you write $\mathbb{E}[\hat{\beta} = f(X)] $ you are assuming a certain probability distribution for $X$. so do you assume that $X,W,V$ follow centered gaussian distributions with three different unknown variances ? – reuns Jan 16 '16 at 15:39
  • (X & Y are from a random sample)

    I am somewhat unable to exactly clarify this; but I believe yes.

     – user43395 Jan 16 '16 at 15:47
  • from the hypothesis that your data follow some gaussian distributions, you build an estimator for $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ with $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ and $\bar{Y} = \frac{1}{n} \sum_{i=1}^n Y_i$. from these you build an estimator for $\mathbb{E}[XY]$ and $\mathbb{E}[X^2]$ with $\hat{r}{XY} = \frac{1}{n} \sum{n=1}^n (X_i-\bar{X})(Y_i-\bar{Y})$ and $\hat{r}{XX} = \frac{1}{n} \sum{n=1}^n (X_i-\bar{X})(X_i-\bar{X})$. finally, you are asking if $\hat{B} = \frac{\hat{r_{XY}}}{\hat{r_{XX}}}$ is an unbiased estimator for $\frac{\mathbb{E}(XY)}{\mathbb{E}(XX)}$ ? – reuns Jan 16 '16 at 16:00
  • from http://www.uv.es/uriel/2%20Simple%20regression%20model%20estimation%20and%20properties.pdf (p.25) your $\hat{B}$ would be the maximum likelihood estimator so yes it should be unbiased – reuns Jan 16 '16 at 16:11
  • Perhaps I am not explaining this correctly- but there should be a bias. assuming Cov(X,W) $\neq$ = 0. It is an OLS regression where there is correlation between $X$ and the error term, $U$ (so $X$ is endogenous). An OLS performed on that should give a biased $\hat{\beta}$ – user43395 Jan 16 '16 at 16:47

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