Causal Inference
===
Data
---

+ Happiness U(1,10)
+ Therapy or not (binary treatment var)
+ Did you take it on your free will (binary var)
+ Life expectation
+ Income Beta(2, 5)*4000
+ Age U(20, 80)
+ Length of therapy for DiD
+ (unobserved) Drug
+ (unobserved) Confession
+ (unobserved) Openness to
+ ~~Cultral acceptance~~
+ ~~Motiv U(0,1) ~~
+ ~~gender~~

$H = \mu(0) + \beta_{a1} \text{age} + \beta_{a2} \text{age}^2 + \beta_{i} \text{income} + \beta_{r} \text{resilience} + \Delta_i D_i + v_i$

### Confession
- if he goes - Happiness increase
- whether or not one goes to a priest is unobserved
-
-

### Depression Drug
- Light drug users may not reveal the drug usage
- But would have a rise in happiness (while taking therapy)
- The rise in happiness might be wrongly arrtibuted as an effect of therapy
- Most likely, an unhappy person that is not a drug addict would go to therapy BUT this person would be happier compared to an unhappy and drug addict person that does not go to therapy.
### Treatment Effectiveness on Individual
- The effetiveness of therapy on each individual is unobserved
-

### Resiliance
- Psychological resilience is the ability to mentally or emotionally cope with a crisis or to return to pre-crisis status quickly. Resilience is not easy to quantify or determine, yet, a resilient person is more likely to recover from a crisis regardless of a therapy has been pursued.

### Acceptance

- the willingness/acceptance to therapy is unobserved

- Most likely, person that is not open to therapy but go to therapy would be in a better mental health than a not open person that would go to therapy.

- An unobeservable factor is the individual's internal acceptance of or willingness to take psychological treatment. Most likely, persons that are accepting of psychological treatment and can properly judge their mental health state, are more likely to enter therapy early with an open mindset which might favor the therapies success. Yet, if this open mindset in general would anyhow improve even without therapy can not be known. On the other hand, peaople that are either not accepting of psychological treatment or can not judge their state of mind (e.g. due to a very guarded nature), are more likely to refuse therapy and refuse to take a survey on mental health. Therefore it can not be known how therapy would have affected them. In case their mental state is critical and they might be forced to take therapy (e.g. by cort order), this will also influence the probability of therapies effect.

## Set-up of the causal question

The curfew and various closings of meet-up places caused by the COVID-19 pandemic are having strong effects on the mental health of the youth. In France, the government started the "Psychic check" initiative. It consists of a voucher for three free psychologist consultations. University students only have the right to claim their checks. We examine the effect of the "Psychic check" initiative on the mental health of students in French Universities.

We observe that only a subgroup of our sample takes the treatment (use their voucher to see a psychologist). Since all university students are entitled to the treatment but decide by themselves whether or not they want to take it, selection into the treatment is not random. Students with the most fragile mental health are the most likely to use their check. But they are also likely to be the ones with the lowest mental health to begin with. The initial fragility of mental health is an individual's characteristic we cannot observe. This is a threat to our causal question because it consists of an omitted variable bias.

a) An ideal experiment would assign students randomly to the same psychologist for three sessions. It could capture the causal effect by comparing the mental health of the treated with the untreated students.

b) The optimal dataset would contain information on all personal traits that affect selection into treatment and mental health outcomes. For example, it would reveal the students' age, parents' income, universities, drug and alcohol consumption, medication, and religious beliefs. It would also inform us of the ability of each psychologist.

c) The most important threat to our identification strategy is that selection into treatment in the real world is not random. It might bias the results. The second most important threat is the validity of our instrument.

$H = \mu(0) + \beta_{a1} \text{age} + \beta_{a2} \text{age}^2 + \beta_{i} \text{allowance} + \beta_{f} \text{fragile} + \Delta_i D_i + U_i(0)$