Instrumental variable method and measurement error model

General
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Hi，I want to use instrumental variable method(IV) to estimate the measurement error model. Now I have a two variables which one have measuring error .The equation is

\begin{eqnarray} y &=& b_0+b_1x_1+b_2 x_2 + u_1\\ w &=& x_2+u_2 \end{eqnarray}

The x2 is latent variables .Now I want to use IV methods estimate the parameters ,what I should do ? Moreover how to evaluated the error？Could you please recommend me some related literature to study? Thank you a lot!

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Welcome to the discourse @xiaodi321! You can use Latex on here for pretty formatting. I went ahead and edited your post so those equations are easier to read.

For a good resource on IV methods I recommend the instrumental variables chapter of Gelman and Hill. That helped me to understand causal analysis from a probabilistic perspective.

Instrumental variable models are easy to implement in Stan. Can you tell us more about your problem? Which variables are data and which are unknown and what are the instruments?

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Thank you very much. My question is as follows:
y=\beta_0+\beta_1x_1+\beta_2x_2+u_1 ,u_1~N(0,\sigma_1)
w=x_2+u_2 ,u_2~N(0,\sigma_2)
which y is dependent variable ,x_1 and x_2 are covariate variables. Moreover,x_1 is observable . x_2 is unobservable but have the observed value w which have the error u_2. On the other hand ,u_1 and u_2 obey nomal distribution,which have the variance \sigma_1 ,\sigma_2. It is usually assumed that \sigma_1 and \sigma_2 are known.To estimate the \beta_1 and \beta_2,we use the instrumental variable method.We use m as instrumental variable ,and x_2=\alpha_0+\alpha_1*m+u_3,u_3~N(0,\sigma_3).
To sum up, x_1 ,w,y are observered,use m as instrumental variable to estimate the \beta_0,\beta_1,\beta_2,and most importantly how to calculate the standard deviation and the standard error.

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Hi Xiao -

It sounds like you are looking for frequentist/MLE inference on instrumental variables. If so there are plenty of good R packages that implement what you describe, i.e. R package AER.

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