Identification: endogeneity, instruments and panel methods

A coefficient is *identified* when the data can distinguish it from the alternatives. Regression alone identifies a causal effect only when the regressor is as good as randomly assigned, and it usually is not — so the field is a catalogue of designs for recovering identification from observational data.

Source	Mechanism	Example
Omitted variable	Something affects both $x$ and $y$	Ability affects both schooling and wages
Measurement error	$x$ is observed with noise	A proxy for a fundamental, attenuating the coefficient
Simultaneity	$y$ also causes $x$	Price and quantity determined together

Table 5.1 · Three ways to break exogeneity. All three produce the same symptom — the regressor is correlated with the error — and all three are invisible in the residuals. Which one you have determines the fix.

Definition 5.2

Instrumental variables

A valid instrument, $\mathrm{Cov}(z, x) \ne 0 \quad \text{and} \quad \mathrm{Cov}(z, \varepsilon) = 0$ — An instrument $z$ moves the regressor without affecting the outcome through any other channel. The first condition — relevance — is testable from the first stage. The second — exclusion — is not testable at all, and must be argued from how the world works. Almost every dispute about an instrumental-variables study is about the second.

Derivation 5.3

Two-stage least squares

Use only the part of $x$ that the instrument explains.

$\hat{x} = \text{fitted values from regressing } x \text{ on } z$
The first stage isolates the exogenous variation.
$y = \beta\hat{x} + u$
The second stage uses only that part, which is uncorrelated with the error by construction.

\hat\beta_{\text{IV}} = \frac{\mathrm{Cov}(z, y)}{\mathrm{Cov}(z, x)}

Figure 5.4 · What a weak instrument costs. The curve is steep exactly where instruments usually live. The rule of thumb is

F > 10

, and the reason is visible: below it the standard error is not merely large, it is rising fast enough that the estimate is dominated by whatever bias the instrument still carries.

Proposition 5.5

Weak instruments are worse than none

When $\mathrm{Cov}(z,x)$ is small, the instrumental-variables estimator divides by a number near zero: it becomes wildly variable and, worse, biased *toward* the OLS estimate it was supposed to correct. A weak instrument therefore reproduces the bias you were trying to remove while adding enormous variance.

Holds when

First-stage F below about 10 is the conventional warning sign.
The bias of IV is roughly the OLS bias divided by the first-stage F, so a weak instrument fixes almost nothing.
Adding more weak instruments makes it worse, not better.

Identification: endogeneity, instruments and panel methods

After this lesson you should be able to

Instrumental variables

Two-stage least squares

Weak instruments are worse than none

The rest of this lesson is in Premium