I am interested in how the variance for the conditional average treatment effect (CATE) is calculated for the doubly robust pseudo-outcome approach. Below are the exact details of the problem and my question

Let $A$ be the treatment of interest, $Y^a$ be the potential outcome under treatment $a$, and $Y$ be the observed treatment. There is some set of confounders $W$ and we are interested in estimating the CATE by $V$, where $V$ is a subset of $W$. Specifically, the CATE we are interested in is $$E[Y^1-Y^0|V]$$ In the context I am interested in, I am using a parametric model for the above quantity. So something like $$E[Y^1-Y^0|V] = \beta_0 + \beta_1V$$ is sufficient. The doubly-robust pseudo-outcome approach works like the following:

1. Estimate $E[Y|A,W,V]$
2. Estimate $\Pr(A|W, V)$
3. Calculate the pseudo-outcomes using the augmented inverse probability weighting formula
4. Calculate $\hat{Y}^1_i - \hat{Y}^0_i$ from the pseudo-outcomes in step 3, and estimate the $\beta$'s in the previous model by fitting some parametric model (OLS in this case)

My question is how is the variance for the CATE / $\beta$ calculated?. As far as I am aware, there isn't any reason to use the standard standard error estimates from the regression.

Most papers I have come across use machine learning to estimate step 4, but that is not what I am interested in at this point. Any sources on the variance (or code) would be deeply appreciated. Thanks!

I am interested in how the variance for the conditional average treatment effect (CATE) is calculated for the doubly robust pseudo-outcome approach. Below are the exact details of the problem and my question

Let $A$ be the treatment of interest, $Y^a$ be the potential outcome under treatment $a$, and $Y$ be the observed treatment. There is some set of confounders $W$ and we are interested in estimating the CATE by $V$, where $V$ is a subset of $W$. Specifically, the CATE we are interested in is $$E[Y^1-Y^0|V]$$ In the context I am interested in, I am using a parametric model for the above quantity. So something like $$E[Y^1-Y^0|V] = \beta_0 + \beta_1V$$ is sufficient. The doubly-robust pseudo-outcome approach works like the following:

1. Estimate $E[Y|A,W,V]$
2. Estimate $\Pr(A|W, V)$
3. Calculate the pseudo-outcomes using the augmented inverse probability weighting formula
4. Calculate $\hat{Y}^1_i - \hat{Y}^0_i$ from the pseudo-outcomes in step 3, and estimate the $\beta$'s in the previous model by fitting some parametric model (OLS in this case)

My question is how is the variance for the CATE / $\beta$ calculated?. As far as I am aware, there isn't any reason to use the standard standard error estimates from the regression.

Most papers I have come across use machine learning to estimate step 4, but that is not what I am interested in at this point. Any sources on the variance (or code) would be deeply appreciated.

I am interested in how the variance for the CATE is calculated for the doubly robust pseudo-outcome approach. Below are the exact details of the problem and my question

Let $A$ be the treatment of interest, $Y^a$ be the potential outcome under treatment $a$, and $Y$ be the observed treatment. There is some set of confounders $W$ and we are interested in estimating the CATE by $V$, where $V$ is a subset of $W$. Specifically, the CATE we are interested in is $$E[Y^1-Y^0|V]$$ In the context I am interested in, I am using a parametric model for the above quantity. So something like $$E[Y^1-Y^0|V] = \beta_0 + \beta_1V$$ is sufficient. The doubly-robust pseudo-outcome approach works like the following:

1. Estimate $E[Y|A,W,V]$
2. Estimate $\Pr(A|W, V)$
3. Calculate the pseudo-outcomes using the augmented inverse probability weighting formula
4. Calculate $\hat{Y}^1_i - \hat{Y}^0_i$ from the pseudo-outcomes in step 3, and estimate the $\beta$'s in the previous model by fitting some parametric model (OLS in this case)

My question is how is the variance for the CATE / $\beta$ calculated?. As far as I am aware, there isn't any reason to use the standard standard error estimates from the regression.

Most papers I have come across use machine learning to estimate step 4, but that is not what I am interested in at this point. Any sources on the variance (or code) would be deeply appreciated. Thanks!

I am interested in how the variance for the conditional average treatment effect (CATE) is calculated for the doubly robust pseudo-outcome approach. Below are the exact details of the problem and my question

Let $A$ be the treatment of interest, $Y^a$ be the potential outcome under treatment $a$, and $Y$ be the observed treatment. There is some set of confounders $W$ and we are interested in estimating the CATE by $V$, where $V$ is a subset of $W$. Specifically, the CATE we are interested in is $$E[Y^1-Y^0|V]$$ In the context I am interested in, I am using a parametric model for the above quantity. So something like $$E[Y^1-Y^0|V] = \beta_0 + \beta_1V$$ is sufficient. The doubly-robust pseudo-outcome approach works like the following:

1. Estimate $E[Y|A,W,V]$
2. Estimate $\Pr(A|W, V)$
3. Calculate the pseudo-outcomes using the augmented inverse probability weighting formula
4. Calculate $\hat{Y}^1_i - \hat{Y}^0_i$ from the pseudo-outcomes in step 3, and estimate the $\beta$'s in the previous model by fitting some parametric model (OLS in this case)

My question is how is the variance for the CATE / $\beta$ calculated?. As far as I am aware, there isn't any reason to use the standard standard error estimates from the regression.

Most papers I have come across use machine learning to estimate step 4, but that is not what I am interested in at this point. Any sources on the variance (or code) would be deeply appreciated. Thanks!