Today¶
- Any Qs before we start?
- Regressions
- vocab
- statistically vs. economically significant
- a (brief) discussion of causality
Vocab via example¶
Last class, we estimated a model:
$$ intrate = a + b * CredScore + u$$- The null hypothesis we are testing is: "Credit score is NOT related to the interest rate" ($\beta=0$)
- Testable: Can we reject the null? (Establish $\beta \neq 0$?)
- Not testable: Accepting the null (Establish $\beta = 0$?)
Example: A covid test
- Null hypo: I don't have Covid
- Reject: I do have Covid :(
- Negative test means "I can't reject the null that I don't have covid"
- Negative test does not mean: "I don't have covid"
Our model gave us this output:
| var | coef | std err | t | P>t |
|---|---|---|---|---|
| Intercept | 11.5819 | 0.046 | 253.270 | 0.000 |
| CredScore | -0.0086 | 6.14e-05 | -139.198 | 0.000 |
- The coefficient column is the estimates of $\hat{a}$ and $\hat{b}$, (hats denote estimates)
- There is uncertainty in those estimates. Again:
Our model gave us this output:
| var | coef | std err | t | P>t |
|---|---|---|---|---|
| Intercept | 11.5819 | 0.046 | 253.270 | 0.000 |
| CredScore | -0.0086 | 6.14e-05 | -139.198 | 0.000 |
- The coefficient column is the estimates of $\hat{a}$ and $\hat{b}$, (hats denote estimates)
- Std err estimates std deviation of the coef
- t-stat: beta/se
- p-value (P>|t|): what is the probability that my non-zero beta is not zero, by random chance?
- the lower it is, the more "certain" we can be that the relationship isn't zero
- In tables, 1 star means p<10%, 2 means p<5%, 3 means p<1%
A "statistically significant relationship between X and Y" DOES NOT MEAN SIGNIFICANT¶
"Economic significance" matters: Stat sig but economically trivial = yawn
Loose definition: Is a "reasonable" change in X assoc with a "large" change in y?
- Former: $\beta$ captures a one unit change, which might be tiny or huge
- Good trick: Scale continuous variables by their STD so that a one unit change in X is a STD.
- Latter: Compare to avg and std of y
So you run a regression¶
... and a variable has a p-value below 5%.
... and the relationship is large enough to be meaningful.
Party?!
Not yet!
Everyone who confuses correlation with causation eventually ends up dead¶
More commonly: "Correlation is not causation"
Reasons your (significant) correlation ain't causation:
- You p-hacked
- Your focus should be on testing and evaluating a hypothesis, not "finding a result"
- Omitted variables ("CEO ability","firm quality")
- One version: Simpson's paradox can be found many places and sometimes can be fixed with fixed effects, which looks like this
- Mismeasurement of an X variable (IQ for "CEO ability", MTB for marginal Q)
- Look out for "proxies"
- Mismeasurement in y just raises SE
- (Classical) Mismeasurement in X causes attenuation
- Simultaneity (think: "equilibrium outcomes")
Everyone who confuses correlation with causation eventually ends up dead¶
More commonly: "Correlation is not causation"
Reasons your (significant) correlation ain't causation:
- It's spurious
- You p-hacked
- Omitted variables ("CEO ability","firm quality")
- Mismeasurement of an X variable (IQ for "CEO ability", MTB for marginal Q)
- Simultaneity (think: "equilibrium outcomes")
- Reverse causality
- Sample selection (effect of diversification of a firm on profitability)
- Selection on X is bad
- Selection on Y is superbad
The goal here is to understand what variables matter ($\hat{\beta}$ focused):
- For which X is the relationship $\hat{\beta}$ non-zero?
- Detect rejection of the null via P-value
- Is the relationship economically important?
- Size of coefficient
- Usually: Is the relationship causal?
- Requires (psuedo) random variation in X
- If causal, why? (What is the "mechanism" of causality?)
- If not causal: We say it is "descriptive"