Introduction¶
Econometric Analysis¶
Machine Learning \(\to\) Statistics \(\to\) Econometrics
The focus is on using the predictions of Machine Learning to analyse future causalities. This is more insight-oriented, as identifying patterns without understanding causes and implications is useless.
Statistical vs Causal¶
| Statistical Prediction | Causal Prediction | |
|---|---|---|
| \(\hat y\) | \(E \big [y \vert x = a \big]\) | \(E \big [y \vert \text{do}(x = a) \big]\) |
| What will be \(y\) if I ___ \(x=a\)? | observe | set |
| characteristic | natural outcome | manual action |
| statement | if \(x\) is correlated with \(y\) without any causal effect on \(y\), then we can only observe the correlation without the ability to change \(y\) by changing \(x\) | if \(x\) has a causal effect on \(y\), then we can change \(x\) and expect it to cause a change in \(y\). |
| True understanding enables predictions under a wide range of circumstances, including new hypothetical situations | ||
| Unstable | Stable | |
| Example 1 | What is the expected health status of someone who has received hospitalization? | What will the health status of a person if they receive hospitalization? |
| Example 2 | What is the expected sales of a company with a given amount of TV ad spending? | How much will my sales increase if I increase my TV ad spending by a certain amount? |
see vs do¶
- \(\text{see}(x= 1)\) means that you observe \(x\) as 1
- \(\text{do}(x= 1)\) means that you manually perform some action to set \(x\) as 1
If there is no way to manipulate a variable (for eg, \(\text{do}(x= 1)\) is not possible), then it is hard to define what its causal effect means
- A thought experiment that is often used to determine whether a variable x is manipulable in principle is to imagine a hypothetical experiment that assigns different values to x
- Research questions that cannot be answered by any experiment are FUQâd: Fundamentally Unidentified Questions
Correlation \(\centernot \implies\) Causation¶
Correlation doesnât always imply causation. correlation is useful for prediction, but not for understanding exactly why.
For eg, labor wage and their years of education has a strong correlation, but the reason for that could be
- education actually helps
- education just acts as the signal/proof for the employees that you possess knowledge
- or, it could just be that well-off people get better jobs, and coincidentally, they are getting more educated
Rain¶
We can now conclude that the barometer reading itself has no causal effect on rainfall. It is infact the atmospheric pressure that has causal effect on the rainfall.
Hospital¶
Letâs say there are 2 hospitals
| A | B | |
|---|---|---|
| Recovery Rate | 0.6 | 0.4 |
It seems like A is better than B. But A may not necessarily be the better hospital for me to go to. What if A is a regular community hospital and B is a speciality hospital for cancer patients. Obviously, A will have a better recovery rate. So, statistical prediction (is not wrong) is accurate saying that recovery rate for a patient going to hospital A is 0.6, because we are seeing the patient going there; the patient that goes there is a patient who chose to go there. If they were a serious patient, then they wouldâve gone to B.
Ad-Sales¶
Sales and Advertisement have a high correlation. But doesnât necessarily mean that Increasing advertisement will increase sales. This is because, the observed advertisment could be due to previous sales. So the observed advertisment here is a natural outcome, not an active decision; ie, last year i got high sales profits, so i increased my advertisment budget this year, or i got bad profit so i increased my ad budget to somehow increase the sales. itâs like a reaction, not an active decision.
However, when you actively decide to increase the ad budget, the change in sales depends on the causal effect of ad budget on sales. Why donât companies do this causal analysis???
- they donât know their math and stats
- causal analysis is harder
Scope¶
Scope is the set of populations in which a causal effect applies. A causal effect is only meaningful if we can define its scope.
Populations¶
| Study Population | Target Population | |
|---|---|---|
| Population where | experiment/observational study is conducted | weâre interested in learning the causal effect |
| Sample | specific | diverse |
| Population | specific | diverse |
| Social, cultural and economic environment | specific | diverse |
The causal effect of \(x\) on \(y\) can differ in 2 populations because:
- Causal mechanism is different in both populations
eg: Consider a country where oil prices are determined by market, and another country where prices are determined by govt. The effect of decreasing oil supply on gas price will be different
- Distribution of effect modifier \(P(s)\) is different
Eg: Consider 2 countries with the same economic structure, but different population age structures. The effect of raising retirement age will be completely different.
Properties of Results¶
Validity¶
| Internal | External | |
|---|---|---|
| Results valid for | study population | Other populations |
| Transportability of results | â | â |
Transportability¶
The ability of the result can be generalized/extrapolated correctly from one population to another.
A causal effect learnt from a study is transportable from study population to target population if both are within the scope.