# causal impact estimation

say i have following causal model:

• outcome variable: Y (e.g. sales)
• treatment variable: T (e.g. price)
• covariate variable: x2 (e.g. traffic)
• unobserved variables: U (unobserved)

causal relation:

how can I estimate the casual effect of T on Y which includes both T cause Y directly and T cause Y through x2? the chanllenge is that x2 may also be impacted by some other unobserved factors. is there any methodology to do this?

--update.

the below answer seems not enough. Regression Y on T alone can't remove the effect from U which is unmeasured and unobserved.
is there any method to remove impact from U?

• Can you describe your data (time series, panel, cross-section, etc) and how the variation in price works in your setting (experiment, equilibrium, etc)? Oct 22, 2020 at 14:06

Thank you for including a causal diagram!

Answer: Simply regress $$Y$$ on $$T$$ like this: $$Y=aT+b.$$ There is no backdoor path from $$T$$ to $$Y,$$ so you don't need to condition on anything. In fact, if you want the full causal effect of $$T$$ on $$Y,$$ you need to NOT condition on $$x_2.$$

You have a mediation situation, so there are other numbers in which you might be interested. You can consult Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell, for more information on mediation.

• thanks for you comment. If I only regression on T, how to remove the impact of x2(essentially U on x2) on Y? e.g. if I keep T unchanged, then change U which will change x2 and again x2 will change Y. the regression will attribute change of Y to T which is wrong since T is unchanged and the root cause should be U, right? Oct 23, 2020 at 8:14
• You could condition on $T$ to find the causal impact of $x_2$ on $Y$, then find the causal impact of $U$ on $x_2$. Perhaps subtract or divide out what you don't want? I'm thinking along the lines of a two-stage linear regression, like an instrumental variable. Is $U$ measured? Oct 23, 2020 at 20:00
• U is unmeasured and un observed. Oct 24, 2020 at 11:37
• do you have any comments? I think the answer can't remove impact from U which is un measured and unobserved? Oct 30, 2020 at 3:03
• Well, I guess I have two comments. 1. Do you really need to remove $U?$ 2. The only ways I can see to remove the effects of $U$ are one of the following: a. Measure $U$ so you can condition on it. b. Insert an instrumental variable between $U$ and $x_2$ and do 2-stage linear regression. c. Insert a variable between $x_2$ and $Y,$ thus allowing you to use the front-door adjustment formula. Oct 30, 2020 at 12:36

To simplify, I am going to make the problem linear in parameters. You have a structural-form equation for the outcome $$y$$, the intermediate outcome equation for $$x$$, and an independence assumption:

\begin{align*} y_i &=\beta_1+\beta_t \cdot t_i + \beta_x \cdot x_i + \varepsilon_i \\ x_i &= \alpha_1+\alpha_t \cdot t_i + u_i \\ (t,x) & \perp \!\!\! \perp \varepsilon \\ \end{align*}

Plugging the second into the first gets you the reduced-form equation for the outcome:

$$y_i = (\beta_1 + \beta_x \cdot \alpha_1) + (\beta_t +\beta_x \cdot \alpha_t) \cdot t_i + (\beta_x \cdot u_i + \varepsilon_i)$$

You have two effects: \begin{align*} \text{Total Effect: }& E[y \vert t=1]-E[y \vert t=0] = \beta_t +\beta_x \cdot \alpha_t \\ \text{Direct Effect: }& E[y \vert t=1,w]-E[y \vert t=0, w] = \beta_t \\ \end{align*}

You can use the reduced-form outcome equation to estimate the first, and you can use the structural-form equation to estimate the second. A difference of the two recovers the indirect effect.

Here's a toy example using Stata where the indirect effect dominates:

. clear

. sysuse auto, clear
(1978 Automobile Data)

. quietly reg price i.foreign

. estimates store rf

. quietly reg price i.foreign c.mpg

. estimates store sf

. suest rf sf

Simultaneous results for rf, sf

Number of obs     =         74

------------------------------------------------------------------------------
|               Robust
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
rf_mean      |
foreign |
Foreign  |   312.2587   696.9581     0.45   0.654    -1053.754    1678.271
_cons |   6072.423   428.2447    14.18   0.000     5233.079    6911.767
-------------+----------------------------------------------------------------
rf_lnvar     |
_cons |    15.9902   .2260545    70.74   0.000     15.54714    16.43325
-------------+----------------------------------------------------------------
sf_mean      |
foreign |
Foreign  |   1767.292   599.3555     2.95   0.003     592.5771    2942.007
mpg |  -294.1955   59.50419    -4.94   0.000    -410.8216   -177.5695
_cons |   11905.42   1343.753     8.86   0.000     9271.709    14539.12
-------------+----------------------------------------------------------------
sf_lnvar     |
_cons |    15.6727   .2476991    63.27   0.000     15.18722    16.15818
------------------------------------------------------------------------------

. nlcom indirect_effect:[rf_mean]_b[1.foreign] - [sf_mean]_b[1.foreign]

indirect_e~t:  [rf_mean]_b[1.foreign] - [sf_mean]_b[1.foreign]

---------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
indirect_effect |  -1455.034   488.1763    -2.98   0.003    -2411.841   -498.2255
---------------------------------------------------------------------------------


If you don't care about the standard errors, this can be done with two separate regressions rather than Seemingly Unrelated Estimation.

• Don't you mean $(t,x)$ independent on $\epsilon$ and not $(d,x)$. And do you not also want to add $u$ such that $(t,x)$ is independent also of $u$. Otherwise there is a problem in the reduced form. Dec 8, 2020 at 20:53
• @JesperforPresident You are right on the first point. The second is implicit in the DAG, if I am not mistaken. Dec 8, 2020 at 22:07
• Yes I agree it is implicit in the DAG. It was only because in the comments above there is a discussion on whether it is necessary to "do something about the influence of U". Obviously it is not and I think that a virtue of your answer is that your reduced form shows this very clearly as soon as one sees that $u$ is independent of $t$ and that the standard conditions for OLS estimation is then satisfied. It was only to make it completely explicit given the apparent confusion expressed in comments above. Dec 8, 2020 at 22:15
• '+1' for adding the reduced form. Dec 8, 2020 at 22:15