Timeline for Looking for a OLS-Equation if one Regressor is correlated with the error
Current License: CC BY-SA 3.0
18 events
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| Dec 5, 2012 at 4:36 | comment | added | Druss2k | sry for late respone. thx again for great help! | |
| Nov 28, 2012 at 19:59 | comment | added | tchakravarty | @Druss2k I have updated the answer to include the DGP from the commentspace. I think this answers your question in full. | |
| Nov 28, 2012 at 19:58 | history | edited | tchakravarty | CC BY-SA 3.0 |
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| Nov 28, 2012 at 14:58 | comment | added | tchakravarty | @Druss2k No need. I just missed that part in the comments. I will update my answer when I have the time. | |
| Nov 28, 2012 at 14:36 | comment | added | Druss2k | Hey, thx. By the different setup i get different results because my $u_2$ is heteroscedastic ill include some graphical evidence in my first question. | |
| Nov 28, 2012 at 12:33 | comment | added | tchakravarty | @Druss2k I updated my answer from the original question. Now, in the comment here, I note that you have a different model for heteroskedasticity. I will update for that model later. | |
| Nov 28, 2012 at 12:32 | history | edited | tchakravarty | CC BY-SA 3.0 |
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| Nov 28, 2012 at 7:52 | comment | added | Druss2k | Did you get a chance to edit? | |
| Nov 23, 2012 at 11:43 | comment | added | tchakravarty | @Druss2k I will update my answer to include this information. | |
| Nov 23, 2012 at 11:40 | comment | added | Druss2k | To explain my model more detailed: I take the DGP as $y_1 = \beta_1 z_1 + \beta_2 z_2 + \beta_3 y_2 + u_1$ with $y_2 = \alpha_1 z_1 + \alpha_2 + u_2$ and $z_2=\begin{cases}1,&\text{for }i=1,..,\frac{1}{2}n\\ 2,&\text{for }i=\frac{1}{2}n+1,..,n\end{cases}$ The error $u_2$ is heteroscedastic as $Var(u_2|z_2 = 1)=1$ and $Var(u_2|z_2 = 2)=1/q$. I know take q from 1 to 5. Further I set $Cor(u_1,u_2)=Cor(u_1,u_2|z_2=1)=\rho$ and $Cor(u_1,u_2|z_2=2)=>\rho$. Since I cannot fix $Cor(u_1,u_2|z_2=2)=\rho$ and $Var(u_2|z_2 = 2)$ decreases with q I want to examine the impact on the ols-bias. | |
| Nov 23, 2012 at 2:46 | comment | added | Druss2k | Hey, I would like to use this estimator to check the impact of heteroskedasticity in one variable on the bias. In my experiment i increase the variance of one part of a variable to worsen the induced heteroskedasticity. If I do that the bias gets smaller and the Est $\boldsymbol{\widetilde{\beta}}$ converges to the true value. My question is to seek out why this happens. | |
| Nov 22, 2012 at 12:52 | comment | added | tchakravarty | Furthermore, the asymptotic bias of the OLS estimator is easily written as $\mathbb{E}(\widehat{\boldsymbol{\beta}} \mid \mathbf{X}) - \boldsymbol{\beta} = (\mathbf{X}'\mathbf{X})^{-1}\boldsymbol{\rho}$, where $\boldsymbol{\rho} = \mathbb{E}(\boldsymbol{X}U)$. | |
| Nov 22, 2012 at 12:48 | comment | added | tchakravarty | @Druss2k Do you want to show that if you use the estimator $\widetilde{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol{Y} - (\mathbf{X}'\mathbf{X})^{-1}\widehat{\boldsymbol{\rho}}$, where $\widehat{\boldsymbol{\rho}} = \mathbf{X}'\boldsymbol{U}$ -- you will get an unbiased estimator? This is not particularly hard to prove. | |
| Nov 22, 2012 at 11:23 | comment | added | Druss2k | I just noticed that my approach does work but im not quite sure why. If i write $\boldsymbol{\hat{\beta} = (X^{T}X)^{-1}X^Ty - (X^{T}X)^{-1}\rho}$ with $\boldsymbol{\rho = X^Tu}$ then ill get the unbiased results. | |
| Nov 22, 2012 at 11:01 | comment | added | Druss2k | Sry for the late response. Thx for your detailed answer but it was not quite what I was looking for. I awarded you anyway^^. I was more looking for a analytic way to determine how the impact of correlation biases the ols-estimator because in a simulation i want to calculate this particular bias as exact as possible. For my simple case i could easily caclulate that bias i was looking for. For a more advanced problem not so much. | |
| Nov 22, 2012 at 10:59 | history | bounty ended | Druss2k | ||
| Nov 22, 2012 at 10:59 | vote | accept | Druss2k | ||
| Nov 19, 2012 at 7:07 | history | answered | tchakravarty | CC BY-SA 3.0 |