2. Instruments and Panel Data

Instrumental Variables

Use an external assignment to isolate treatment variation while defending exclusion

Instrumental Variables

The instrument changes the comparison

Scholarship receipt $D$ is selected: motivated applicants may be more likely to obtain it and complete a degree. Pathways randomly assigns an offer $Z$ , which changes receipt.

Rendering diagram…

IV uses only the component of $D$ moved by $Z$ . The crucial omitted arrow is $Z\rightarrow Y$ outside receipt; that omission is the exclusion restriction.

Three empirical objects

For a binary instrument:

Object	Calculation	Pathways meaning
first stage	$E[D\mid Z=1]-E[D\mid Z=0]$	offer changes receipt
reduced form	$E[Y\mid Z=1]-E[Y\mid Z=0]$	offer changes completion
Wald ratio	reduced form / first stage	receipt effect for compliers under assumptions

If receipt is 70% versus 20%, and completion is 58% versus 50%,

\hat\tau_{Wald}=\frac{0.58-0.50}{0.70-0.20}=0.16.

The 16-point estimate scales the ITT by the offer-induced 50-point change in receipt.

Four assumptions do different work

Relevance: $Z$ changes $D$ .
Assignment/independence: $Z$ is independent of relevant potential outcomes or as-if assigned conditional on stated variables.
Exclusion: $Z$ affects $Y$ only through the defined treatment.
Monotonicity: the instrument does not make some units do the opposite of everyone else’s direction.

Randomising the offer supports assignment. It does not guarantee exclusion: the letter may provide encouragement, information or status even when no money is received.

2SLS is a computation, not the argument

With controls $X$ , two-stage least squares can be written as:

D_i=\pi_0+\pi_1Z_i+X_i'\pi_2+v_i,

Y_i=\alpha+\tau\widehat D_i+X_i'\beta+u_i.

Standard software estimates both stages and the correct joint uncertainty. Manually replacing $D$ by $\widehat D$ and reading ordinary second-stage standard errors is incorrect.

Exclusion is substantive

Proposed instrument	Plausible direct path to outcome
scholarship offer letter	information, encouragement or stigma
distance to university	local labour market and family location
judge assignment	judge characteristics affect multiple case decisions
class-size rule	sorting, other resources or score manipulation

Negative-control outcomes, institutional details and alternative channels can challenge exclusion, but no single statistical test proves it.

The classic LATE framework is formalised by Angrist, Imbens and Rubin, Identification of Causal Effects Using Instrumental Variables.

Report the complete chain

instrument assignment and timing;
first-stage magnitude and uncertainty;
reduced form in outcome units;
2SLS/Wald estimate and weak-IV-robust interval;
exclusion channels and falsification evidence;
complier population and treatment version;
whether effects can be transported to the policy population.

Quick check

The offer has a strong first stage but also includes free advising unavailable to non-recipients. If treatment is defined only as cash receipt, what fails?

Answer

Exclusion is doubtful because the offer can affect completion through advising as well as cash receipt. Redefine the treatment package, estimate the ITT of the offer, or supply a design that separates the channels.

Next: Weak IV and LATE

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2. Instruments and Panel Data

Isolate treatment variation through external shifts or within-unit change

Weak Instruments and LATE

Diagnose unstable IV ratios, weak-identification inference and complier-specific interpretation