Samenvatting
Summary Methods: Econometrics 1 (310153-M-3) 2022/2023
- Instelling
- Tilburg University (UVT)
Summary of 26 pages for the course Methods: Econometrics 1 at UVT
[Meer zien]Voorbeeld 3 van de 26 pagina's
Voorbeeld 3 van de 26 pagina's
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Voorbeeld van de inhoud
Lecture 1 IntroductionEconometrics is about building other universes, so-called counterfactuals; what would have
happened in the absence of an intervention or event, the road not taken.
The ideal experiment
Unit causal effect
Each subject has two potential outcomes, one for treatment condition, one for control
condition. The potential outcomes for unit i thus are:
• Outcome if unit were assigned to treatment
• Outcome if unit were assigned to control
The difference between a unit’s potential outcomes is called the unit causal effect.
However, the problem with causal inference is that only one potential outcome is observed,
which makes us unable to estimate the unit causal effect.
Average causal effect
We can however calculate the average causal effect, which is the difference between:
• The average potential outcomes if all units were assigned to treatment and
• The average of potential outcomes if all units were assigned to control
To estimate the average causal effect, all we need to do is to randomly assign some units to
treatment and the rest to control, so that the control group has on average similar
characteristics as the treatment group; the only difference is that one group is treated and
the other isn’t (figure 1). The mean of a random sample from the population of units is an
unbiased estimator for the mean of the population. So, thanks to randomization, we can
estimate the average causal effect
Figure 1
It is always important to look how units (people, counties, countries, …) are selected into
treatment. Did they select themselves or did someone else select them?
If we belief that assignment into treatment is as-good-as-random conditional on an
observed confounder, then we only need to adjust for this confounder to estimate the
effect of the treatment on the outcome.
,The ideal experiment
The control group shows what would have happened in the treatment group if the
treatment had not been administered. Given random assignment of units to treatment and
control, the so-called ‘exchangeability’-assumption holds. A comparable control group is
always based on random selection into treatment vs control. If this is the case we talk about
an ideal experiment. However, often there is non-random selection into treatment which
leads to selection bias. We will discuss this below.
For estimation of a treatment effect one can rely on an analysis of observational data rather
than controlled experimental data from a randomized trial if selection into treatment is
solely based on known observables for which you have data. This is called selection on
observables. There are however two reasons to go beyond selection on observables:
1. No data on known confounder u; if this is the case, de-confounding by adjusting for
u is not possible. u is an omitted variable in a regression of x on y.
2. Uncertainty about absence of confounders; it is impossible to be sure that there
aren’t any confounders anymore.
Non-random selection into treatment (selection bias)
There are two naïve ways of getting at counterfactuals:
1. Before/after comparison
You just look at the difference between the state before the intervention/event and
after the intervention/event. The state before the intervention is the counterfactual.
The problem with before/after comparison is that is suffers from selection bias, also
called the omitted variable bias; more variables should be included to get closer to
the causal affect. Selection bias leads to under- or overestimation of the treatment
effect. More on the selection bias and the direction of it later.
2. Cross-sectional comparison
At cross-sectional comparison, you just look at the difference between a subject who
is not treated and a subject who is treated. The not-treated subject is the
counterfactual. However, often there is non-random selection into treatment which
leads to the selection bias. We can visualize non-random selection into treatment
with a causal diagram which we will show below.
Causal diagram
Causal diagram = a graph that displays the causal relationships between the variables in a
causal model (figure 2). How to draw a causal diagram:
Figure 2
, Confounding bias = A type of non-random selection into treatment (selection bias): when x
and y have a common cause. Confounder = omitted variable = source of selection bias/ a
reason for selection into treatment.
In figuur 3 heeft de treatment private college invloed op de outcome later earnings. De
confounder (common cause for x and y) ability zorgt er echter voor dat het effect van
private college groter lijkt (over-estimation). Mensen met een hogere ability worden eerder
toegelaten tot een private college en verdienen daarna door hun ability later meer. Echter
lijkt dit te komen door de private college omdat alle mensen met een hoge ability daar
verzameld zijn.
Figure 3
Mediator = een indirect effect. Occupation in figuur 3 is een mediator. Private college zorgt
namelijk voor een betere baan wat weer zorgt voor hogere earnings. This is not a non-causal
path and there is no selection bias here. You should not adjust for a mediator.
The path private_college ability → later_earnings is a non-causal path between
private_college and later_earnings. We need to block this path by adjusting for the
confounder along the path. If your causal diagram is correct and a non-causal path between
x and y can be blocked by adjusting for a ‘confounder’, then you can identify the effect of x
on y because we assume the variation that remains after adjusting for caring attitude is now
as-good-as-random.
Solving confounding bias in regressions
The confounding bias can be solved by using control variables in your regression. The choice
of control variables is determined by a causal diagram. Control variables are only included to
help identify the effect of treatment on outcome; they should not be interpreted as causal.
For instance in the following regression, private_college has a causal interpretation, while
ability does not have a causal interpretation:
𝑙𝑎𝑡𝑒𝑟_𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 = 𝛼 + 𝛽 × 𝑝𝑟𝑖𝑣𝑎𝑡𝑒_𝑐𝑜𝑙𝑙𝑒𝑔𝑒 + 𝛾 × 𝑎𝑏𝑖𝑙𝑖𝑡𝑦
In figuur 4 heeft pokon effect op green leaves en green leaves op pokon. Om dit op te lossen
kan je gebruik maken van timing (t, t + 1).
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