# A quick note on causal effect

This post is a quick note as I have been reading papers and books on causal inference recently. I am sure that the materials are extremely intuitive to most social scientists, but I hope some of my notes could help the beginning grad students to quickly understand the notion of causal effect.

I recently came across a paper by Barnow, Cain, and Goldberger, that was published thirty-six years ago. In their paper, they talked about how to operationalize the following equation

$y=\alpha z+w+\varepsilon,$

where $\alpha$ is true treatment effect, $z$ is treatment status, $y$ is the outcome, and $w$ is an unobserved variable, with random term $\varepsilon.$ The basic idea is to introduce observable variables that determine the assignment in the equation:

“Assume that an observed variable, $t$, was used to determine assignment into the treatment group and the control group… [S]ince $t$ is the only systematic determinant of treatment status, $t$ will capture any correlation between $z$ and $w$. Thus, the observed $t$ could replace the unobserved $w$ as the explanatory variable.”

To understand their argument, let’s quickly review the conditional independence assumption (CIA). The CIA states that conditional on $t,$ the outcomes are independent of treatments, that is, $\{y_{0},y_{1}\}\perp\!\!\!\perp z|t.$

Suppose we have
$\underbrace{ E[y|z=1] - E[y|z=0]}_{\text{observed difference}} = \underbrace{ E[y_{1}-y_{0}|z=1]}_{\text{treatment effect}} + \underbrace{ (E[y_{0}|z=1] - E[y_{0}|z=0])}_{\text{selection bias}}.$

However, the selection bias is undesirable in our contexts. Conditional on $t,$ we obtain
${ E[y|t,z=1] - E[y|t,z=0]} = { E[y_{1}-y_{0}|t}].$

In this way, selection bias disappears! Now, let’s go back to Barnow, Cain, and Goldberger’s equation —

$y=\alpha z+w+\varepsilon.$

With the CIA, we can decompose $w$ into $w=\beta t + \varepsilon^*,$ where $\beta$ is a vector of population regression coefficients that is assumed to satisfy

$E[w|t]=\beta t .$

That is,

$y=\alpha z +\beta t + \varepsilon,$

where $\alpha$ is the causal effect.

# Living in La Jolla

DMV and Smog Test

I drove to San Diego from Indiana. It was an awesome road trip — I stopped by Springfield (IL), Des Moines (IO), Omaha (Nebraska), Denver and Glenwood Springs (Colorado), Escalante (Utah), and  Las Vegas. However, after the trip, I noticed two things. First, gas is expensive in California; I came from Crawfordsville, a small town that is mentioned in news only because of its cheap gas, and school lockdown (it’s our Dean in the picture with… SWAT). So I definitely do not appreciate the gas price in San Diego after the move. The second thing is that I probably need a new license/car registration. California requires you to have a smog certificate in order to register your car, and although I am not sure if it was due to the law or the awesome staff at the DMV, my Indiana smog certificate was not accepted. I recommend Clairemont Auto Care if you need a smog certificate — it is close to the DMV, and the people there are friendly and professional.

Here is another piece of advice: do make appointment with the Clairemont DMV before your visit, but still plan to wait at least 30 minutes even though you arrive on time. On a normal day, it will take you 2 hours for waiting if you just walk in. But here is the awesome point — we can game the system. Get a ticket, and instead of waiting for 2 hours in a crowded room, you can do your smog test, get your breakfast, and enjoy San Diego sun for a while before you go back to the DMV again. Also, this DMV does not accept credit card. Bring cash or debit card.

Car detailing

Back to the road trip — I killed at least 1,000 bugs on the way. (Do not open the previous link if you don’t like seeing a lot of dead bugs. And my car in fact looked 4-5 times worse than the picture.) I would give The Detail Shop La Jolla (6860 La Jolla Blvd) a 10 out of 5 because this place is just awesome. Reasonable price, accept credit card, friendly staff, and so on. I recommend this place to anyone who cares about his/her car!  I think the Google Map might show a different name because of change of ownership, but the address (6860 La Jolla Blvd) definitely works.

On Campus Parking

Hopkins Parking Structure is where I park because it is outrageously close to the Social Sciences Building. By outrageously close, I mean 50 steps from parking lot to my office (which is on the third floor of SSB). I have heard that the shuttle is great — but I am bad at finding which bus should I take, and I do not want to walk a ton every day, so I opt for paying for the parking permits. And since not many people are on campus during the summer, getting a Student pass (instead of B pass for graduate students) is much cheaper. The only problem is you can’t park at Gilman Parking Structure with a Student pass at UCSD, but I never feel the need to park there honestly. You can also buy a permit for an entire quarter — the price is actually not that bad if you carpool with someone. That being said, if you live at One Miramar St or Mesa, and want to carpool with me once the quarter starts, please let me know!

Haircut (men)

I just went to 18|8 Fine Men’s Salons today, and it has become one of my favorite places in San Diego now! My stylist is James, and he takes great care of my hair (whoever fixes my whorl is my hero)! They have a really easy-to-use online appointment system, and did I mention complimentary garage parking in La Jolla? I feel it’s a luxury because I am really bad at street parking and I hate the hot car during the summer. They also have a student price (only \$30 per cut), which is more than reasonable in the La Jolla area. Also, if you mention my name to them, you will get 50% off your first visit.