Slides_Quan201_Topic1_Causation slides

QUAN201 - Introduction to Econometrics¶

Topic 1: Causation and Experiments¶

2020¶

Intro: What is econometrics?¶

Causation?¶

Differences in Means¶

Randomized Experiments¶

Reference: Angrist and Pischke, Chapter 1.

What is econometrics?¶

Econometrics is a set of statistical tools that allow us to learn about the world using data.
Difference from most Statistics and Machine Learning?
Economists/Econometricians put much greater emphasis on causation.

Causal effect¶

Causal effect for a specific individual: Health with insurance – health without insurance

$Y_{1i} - Y_{0i}$

1 or 0 indicate treatment:

1 = with health insurance
0 = without health insurance

i states that it is a specific individual:

$Y_{1,Khuzdar}-Y_{0,Khuzdar}$
$Y_{1,Maria}-Y_{0,Maria}$

Difficulty¶

Causal effect for a specific individual: Health with insurance – health without insurance

$Y_{1,Maria}-Y_{0,Maria}$

We only see one of these two!¶

People either have insurance or they don’t. We only observe one potential outcome.¶

What is the causal effect of health insurance for Khuzdar and Maria?

What is the difference between actual health between Khuzdar and Maria?

$Y_{Khuzdar} - Y_{Maria}=-1$

What is driving the actual differences in health?¶

Difference because of the treatment and difference because of other factors.

$\begin{align} Y_{Khuzdar}-Y_{Maria} &= Y_{Khuzdar,1}-Y_{Maria,0} \\ &= \underbrace{Y_{Khuzdar,1}-Y_{Khuzdar,0}}+\underbrace{Y_{Khuzdar,0}-Y_{Maria,0}} \\ &\quad \text{Treatment Effect} \quad + \quad \text{Selection Bias} \end{align}$

$\begin{align} \text{Difference in Group Means }&=\text{ Average Y of Treated }-\text{ Average Y of Control} \\ &= Avg_n[Y_i | D_i=1] - Avg_n[Y_i | D_i=0] \\ &= Avg_n[Y_{1,i} | D_i=1] - Avg_n[Y_{0,i} | D_i=0] \\ &= (Avg_n[Y_{0,i} | D_i=1] + \kappa) - Avg_n[Y_{0,i} | D_i=0] \\ &= \kappa + (Avg_n[Y_{0,i} | D_i=1] - Avg_n[Y_{0,i} | D_i=0]) \\ &= \text{(Average) Causal Effect} + \text{Selection Bias} \end{align}$

This is just the same as we said earlier, but with equations. And averages!

1st-to-2nd line: Definitions.
2nd-to-3rd line: Follows from treatment/control.
3rd-to-4th line: Constant effects assumption: $Y_{1,i}=Y_{0,i}+\kappa$ .
4th-to-5th line: Just changes grouping of terms.
5th-to-6th line: Definitions.

$D_i = \left\{ \begin{align} & 1 \text{ if } i \text{ is insured} \\ & 0 \text{ otherwise} \end{align} \right.$

Expected Value¶

The expected value of $Y$ , denoted $E[Y]$ , is the population average of the variable $Y$ .

Example: If $Y$ is the age of New Zealand citizens, then $E[Y]$ is the average age of New Zealand citizens.

Conditional expectation¶

The conditionally expected value of $Y$ given $x$ occurred, denoted $E[Y|X=x]$ , is the population average of the variable $Y$ after conditioning on $x$ occurring.

Example: If $Y$ is the age of New Zealand citizens, and $X$ is cities in New Zealand, and $x$ is Wellington, then $E[Y|X=x]$ is the average age of New Zealand citizens living in Wellington.

If $Y$ is generated by a random process, such as rolling a dice... ( $Y$ is the number rolled)

The Expected Value, $E[Y]$ , is the average value of infinitely many rolls.

If $X$ is family members and $x$ is mother, then the Conditional Expectation, $E[Y|X=x]$ is the average value of infinitely many rolls by mother.

Random assignment eliminates selection bias¶

$D$ is treatment status ( $D = 1$ if insurance, $D = 0$ if no insurance).

If D is randomly assigned, treatment and control group will be in expectations equally healthy in absence of insurance.
Formally: $E[Y_0 | D=1] = E[Y_0|D=0]$

The difference between treatment and control group can in expectations therefore be attributed to the average causal effect (ACE) of the treatment.
Formally, with randomization: $E[Y_1 | D=1] - E[Y_0|D=0]$ = Average Causal Effect

(Reminder: Selection Bias = $E[Y_0 | D=1] - E[Y_0|D=0]$ . So equals zero.)
(Notice: Expectations, so this is about the population, not the sample.)

Law of Large Numbers¶

The Law of Large Numbers (LLN) states that as the sample gets larger, the sample average becomes closer to the expected value.

$Avg_n[Y] \to E[Y]$ as $n$ goes to infinity.