Distance to College as an Instrument

Why Distance to College Might Matter

Living near a college can make attending less costly: travel may be shorter, housing may be unnecessary, and information or application support may be easier to obtain. That makes distance a plausible source of variation in college attendance. But a plausible first stage is not enough. Distance is also tied to where people live, and place can affect earnings through many routes besides education. The central task is to assess each assumption needed for a causal interpretation, not merely to show that nearby people attend college more often.

For a focused analysis, suppose the treatment is binary college attendance and the outcome \(Y\) is annual earnings in dollars at a specified age. Define \(Z=1\) for living near a college and \(Z=0\) for living farther away. This binary coding is a simplification of actual distance, which is usually measured continuously. The conclusions concern the particular near/far definition, population, and outcome period under study.

Four Assumptions, Mapped to Place

Relevance requires that the instrument change treatment probability: \(E[D\mid Z=1]\ne E[D\mid Z=0]\) in the population being analyzed. For the proposed coding, the expected direction is more college attendance among those living near a college. This is a first-stage claim, not evidence that near and far residents are otherwise comparable.

Independence requires instrument assignment to be independent of the relevant potential outcomes and potential treatment statuses. In a randomized encouragement study, assignment may support this assumption by design. Residential distance is not ordinarily randomized. Family resources, parental education, academic preparation, urban or rural residence, and preferences for education can influence both where a person lives and whether they attend college. These factors can also predict earnings. Conditioning on measured covariates \(X\) may make conditional independence more plausible, but it does not establish that all relevant common causes have been addressed.

Exclusion requires that distance affect the outcome only through the defined treatment. Formally, \(Y(d,1)=Y(d,0)\) for each treatment status \(d\). For earnings, this is demanding. Residence near a college may also mean residence near more jobs, different wages, or different costs of living. It may affect access to internships, local networks, or other education and training. If the treatment is simply “any college attendance,” proximity could change the type, quality, or duration of college education while leaving this binary indicator unchanged. Those pathways violate exclusion for that treatment definition if they affect earnings.

Monotonicity requires \(D(1)\ge D(0)\) for every person: moving from the far category to the near category does not make anyone less likely to attend, when the two potential statuses are binary. People with \(D(1)=1\) and \(D(0)=0\) are compliers; those with \(D(1)=D(0)\) do not change attendance in response to this instrument. The assumption rules out defiers, who would attend when far but not when near. It is a claim about how the instrument changes treatment, not a claim that everyone responds in the same way.

Finally, consistency requires that observed treatment and outcome agree with the potential outcomes for the person’s actual instrument and treatment values. The treatment must be sufficiently well defined: “attended college” may combine very different durations and institutions. No interference requires one person’s distance category not to alter another person’s attendance or earnings in a way relevant to the analysis. For example, crowded local campuses or shared household decisions could make this assumption less credible.

Why the Wald Ratio Has This Interpretation

The connection between the assumptions and the interpretation can be shown directly for a binary instrument and binary treatment. Independence allows the observed near/far outcome difference to represent a difference in potential outcomes across instrument values. Under exclusion, the outcome depends on the instrument only through attendance, so that difference can be written as:

For an always-taker or never-taker, \(D(1)=D(0)\), so that person’s contribution to this expression is zero. Under monotonicity, the only people whose attendance changes are compliers, with \(D(1)=1\) and \(D(0)=0\). Their contribution is \(Y(1)-Y(0)\). Thus the outcome difference equals the proportion of compliers times their average treatment effect:

The treatment difference is the same proportion of compliers, because only they change attendance and each changes it by one:

Dividing the first equality by the second gives the complier average treatment effect. This derivation also shows why a direct effect of proximity, dependence between distance and potential outcomes, or defiers can undo the usual interpretation. A nonzero denominator alone is not sufficient.

Selection, Diagnostics, and Failure Modes

Distance to college is a residential characteristic, so independence deserves particular scrutiny. Covariate comparisons can reveal warning signs. Compare near and far residents on pre-college family income, parental education, academic preparation, and urbanicity. Large differences indicate that the instrument groups are not obviously comparable. Balance on recorded covariates is reassuring only in a limited sense: it cannot demonstrate balance on unmeasured characteristics or prove independence.

The following hypothetical counts show how an apparent first stage can arise from composition alone. Suppose attendance depends on socioeconomic stratum but not on proximity within a stratum. Among 100 high-resource people, 80 live near and 20 far, and 80% attend in either location. Among 100 low-resource people, 20 live near and 80 far, and 20% attend in either location. The pooled near attendance rate is \((80\times0.80+20\times0.20)/100=0.68\). The pooled far rate is \((20\times0.80+80\times0.20)/100=0.32\). The apparent difference is 0.36, although within each resource stratum the near/far difference is zero. Resource composition has created the pooled association. This example does not establish that adjusting for one resource measure solves the problem; it shows why an observed first stage must be interpreted alongside the assignment process.

For exclusion, list plausible routes from location to the outcome that do not pass through the chosen treatment. Examine whether the study population, outcome period, and treatment definition make those routes credible. Pre-treatment outcomes and outcomes that the proposed education pathway should not affect can serve as warning checks when suitable data exist. Passing such checks does not prove exclusion: an undetected direct pathway may remain, and a check can be uninformative if its outcome is measured poorly.

Monotonicity cannot generally be confirmed by comparing average attendance rates. A positive average first stage is compatible with some people being defiers, offset by a larger number of compliers. Consider whether proximity could deter attendance for some people—for example, because a nearby option changes their choices in a way that makes them forgo college, or because moving closer changes access to alternatives. These are substantive questions about individual responses; the group-level first stage does not answer them.

Alternatives and Limits

A randomized encouragement to apply or enroll can offer a more defensible basis for independence than residential distance, although exclusion and monotonicity still require justification. Another instrument may avoid some location-based pathways but introduce different threats. Direct adjustment or matching instead relies on measured covariates being adequate to address confounding; it does not supply the same instrument-based complier interpretation. These approaches answer causal questions under different assumptions, so choosing among them requires considering the design and the target effect rather than selecting the method with the largest estimate.

A continuous distance measure also raises choices hidden by a near/far coding: which contrast is being used, whether treatment changes monotonically over the relevant range, and whether the assumed relationship is appropriate. Changing the cutoff can change the population contrast and the people whose attendance responds. Likewise, an estimate for college attendance does not automatically describe the effect of years of schooling, institution quality, or degree completion. Those are different treatments with potentially different causal pathways and interpretations.

Common mistake: “People who live nearer attend more often, so distance is a valid instrument.” The reasoning establishes, at most, an observed difference in attendance. Correction: relevance is only one condition. Residential selection threatens independence, and location-based job access, networks, or education quality can threaten exclusion. A positive pooled first stage also does not prove monotonicity. State which condition the evidence addresses and which still rests on an argument.