I am looking for methods which can be used to estimate the "OLS" measurement error model.

$$y_{i}=Y_{i}+e_{y,i}$$ $$x_{i}=X_{i}+e_{x,i}$$ $$Y_{i}=\alpha + \beta X_{i}$$

Where the errors are independent normal with unknown variances $\sigma_{y}^{2}$ and $\sigma_{x}^{2}$. "Standard" OLS won't work in this case.

Wikipedia has some unappealing solutions - the two given force you to assume that either the "variance ratio" $\delta=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ or the "reliability ratio" $\lambda=\frac{\sigma_{X}^{2}}{\sigma_{y}^{2}+\sigma_{X}^{2}}$ is known, where $\sigma_{X}^2$ is the variance of the true regressor $X_i$. I am not satisfied by this, because how can someone who doesn't know the variances know their ratio?

Anyways, are there any other solutions besides these two which don't require me to "know" anything about the parameters?

Solutions for just the intercept and slope are fine.

I am looking for methods which can be used to estimate the "OLS" measurement error model.

$$y_{i}=Y_{i}+e_{y,i}$$ $$x_{i}=X_{i}+e_{x,i}$$ $$Y_{i}=\alpha + \beta X_{i}$$

Where the errors are independent normal with unknown variances $\sigma_{y}^{2}$ and $\sigma_{x}^{2}$. "Standard" OLS won't work in this case.

Wikipedia has some unappealing solutions - the two given force you to assume that either the "variance ratio" $\delta=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ or the "reliability ratio" $\lambda=\frac{\sigma_{X}^{2}}{\sigma_{x}^{2}+\sigma_{X}^{2}}$ is known, where $\sigma_{X}^2$ is the variance of the true regressor $X_i$. I am not satisfied by this, because how can someone who doesn't know the variances know their ratio?

Anyways, are there any other solutions besides these two which don't require me to "know" anything about the parameters?

Solutions for just the intercept and slope are fine.

I am looking for methods which can be used to estimate the "OLS" measurement error model.

$$y_{i}=Y_{i}+e_{y,i}$$ $$x_{i}=X_{i}+e_{x,i}$$ $$Y_{i}=\alpha + \beta X_{i}$$

Where the errors are independent normal with unknown variances $\sigma_{x}^{2}$ and $\sigma_{y}^{2}$. "Standard" OLS won't work in this case.

Wikipedia has some unappealing solutions - the two given force you to assume that either the "variance ratio" $\delta=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ or the "relaibility ratio" $\lambda=\frac{\sigma_{x}^{2}}{\sigma_{y}^{2}+\sigma_{x}^{2}}$ is known. I am not satisfied by this, because how can someone who doesn't know the variances know their ratio?

Anyways, is there any other solutions besides these two which don't require me to "know" anything about the parameters?

Solutions for just the intercept and slope are fine.

I am looking for methods which can be used to estimate the "OLS" measurement error model.

$$y_{i}=Y_{i}+e_{y,i}$$ $$x_{i}=X_{i}+e_{x,i}$$ $$Y_{i}=\alpha + \beta X_{i}$$

Where the errors are independent normal with unknown variances $\sigma_{y}^{2}$ and $\sigma_{x}^{2}$. "Standard" OLS won't work in this case.

Wikipedia has some unappealing solutions - the two given force you to assume that either the "variance ratio" $\delta=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ or the "reliability ratio" $\lambda=\frac{\sigma_{X}^{2}}{\sigma_{y}^{2}+\sigma_{X}^{2}}$ is known, where $\sigma_{X}^2$ is the variance of the true regressor $X_i$. I am not satisfied by this, because how can someone who doesn't know the variances know their ratio?

Anyways, are there any other solutions besides these two which don't require me to "know" anything about the parameters?

Solutions for just the intercept and slope are fine.