Why Compare Instrument-Specific Estimates?
When several candidate instruments are available, each can produce its own just-identified IV estimate. Comparing those estimates is useful because it makes the ingredients of each ratio visible: how strongly the instrument predicts the endogenous regressor, how it predicts the outcome, and how those two relationships combine. The comparison is a diagnostic for reasoning about assumptions and populations—not a test that certifies an instrument as valid.
A fair comparison requires a common setup. Use the same outcome, endogenous regressor, control variables, and preferably the same observations. If those differ, a change in the estimate may reflect the change in analysis rather than the choice of instrument. The discussion below focuses on one endogenous regressor and one candidate instrument at a time, with exogenous controls permitted.
The Ratio and Its Assumptions
Write the structural equation as \(Y=\beta D+X'\gamma+u\), where \(D\) is endogenous, \(X\) contains exogenous controls, and \(\beta\) is the structural effect of interest. For each candidate \(Z_j\), the comparison relies on the following assumptions: the structural relationship is adequately represented by this equation with a common \(\beta\); the controls are exogenous for the structural equation; the instrument satisfies the exclusion and orthogonality condition after accounting for the controls; and the instrument is relevant, meaning its partial association with \(D\) is nonzero. Relevance is necessary to form the ratio, but does not establish validity.
Let \(\tilde Y,\tilde D,\tilde Z_j\) denote variables after removing their linear projections on the same controls \(X\). Define the population reduced-form and first-stage slopes as \(\rho_j=E[\tilde Z_j\tilde Y]/E[\tilde Z_j^2]\) and \(\pi_j=E[\tilde Z_j\tilde D]/E[\tilde Z_j^2]\). The corresponding population just-identified ratio is \(\beta_j=\rho_j/\pi_j\), provided \(\pi_j\ne 0\). In a sample, the same construction uses the estimated regression slopes, giving \(\hat\beta_{IV,j}=\hat\rho_j/\hat\pi_j\).
This identity gives the central comparison. If the instrument satisfies \(E[\tilde Z_j\tilde u]=0\), then \(\rho_j=\beta\pi_j\), so its population ratio is \(\beta\). If multiple instruments are valid and the same homogeneous structural effect applies, their population ratios agree, even if their first stages and reduced forms differ in size. In finite samples, estimates need not match exactly. Conversely, agreement cannot prove validity: instruments might share a violation, or their errors might happen to offset in the ratios.
How to Read the Three Components
Start by examining the first-stage and reduced-form slopes separately, with their units and signs. The first stage describes how the instrument shifts \(D\); the reduced form describes how it shifts \(Y\). The IV ratio translates the latter relationship into an effect per unit of \(D\). A larger reduced-form slope alone does not imply a larger IV effect: the first-stage slope may be larger in the same proportion.
The magnitudes depend on the scale of the instrument. Multiplying \(Z_j\) by a nonzero constant multiplies both slopes by that constant, leaving their ratio unchanged. Thus, compare ratios for effect estimates, and use the two slopes to understand how the estimate is constructed. Always check that the denominator is not near zero; a ratio with a small first-stage slope can be highly sensitive to modest changes in the reduced form. Assessing weak-instrument inference requires methods beyond this descriptive comparison.
Worked Example: Different Ratios from Two Candidate Instruments
Hypothetical data summary. In one study sample, the outcome is annual earnings measured in thousands of dollars, the endogenous regressor is years of schooling, and both regressions include the same exogenous controls. Candidate instrument \(Z_1\) has an estimated first-stage slope of \(0.40\) schooling years per instrument unit and a reduced-form slope of \(0.24\) thousand dollars per instrument unit. Candidate \(Z_2\) has corresponding slopes of \(0.20\) schooling years and \(0.14\) thousand dollars per instrument unit. Calculate and compare the two IV estimates.
For \(Z_1\), the ratio is
For \(Z_2\), it is
The estimates are \(600\) and \(700\) dollars of annual earnings per additional schooling year, respectively. The reduced-form slopes are not the same, nor are the first stages; the relevant disagreement is between the ratios, which differ by \(0.10\) thousand dollars per schooling year. Under a common homogeneous effect and valid instruments, the population ratios should coincide. These sample estimates therefore prompt scrutiny of the assumptions, sampling variation, and whether both instruments target the same population effect. They do not by themselves show which instrument, if either, is invalid.
The shared sample and controls make this comparison more interpretable, but do not establish that the instruments are relevant, excluded from the structural error, or orthogonal to it. Those claims require substantive arguments and appropriate diagnostics. The values here are hypothetical summaries; they do not provide evidence about any real instrument.
Similar Ratios Can Arise from Different Slopes
An instrument that moves the endogenous regressor more may also move the outcome more. If the reduced-form and first-stage slopes change proportionally, the resulting ratio is the same. This is why looking only at reduced forms can lead to mistaken conclusions about the size of the structural effect.
Worked Example: Stronger Slopes, the Same IV Estimate
Hypothetical data summary. Consider a common sample with the same controls, where \(D\) is a treatment dose in units and \(Y\) is an outcome measured in points. Candidate \(Z_1\) has a first-stage slope of \(0.10\) dose units per instrument unit and a reduced-form slope of \(0.20\) outcome points per instrument unit. Candidate \(Z_2\) has slopes of \(0.25\) dose units and \(0.50\) outcome points per instrument unit. Calculate both estimates.
For \(Z_1\), the reduced-form change divided by the first-stage change is
For \(Z_2\), the same calculation gives
The second instrument has both the larger first-stage slope and the larger reduced-form slope, but the two ratios agree. This pattern is consistent with a common effect, provided the model, relevance, exclusion, and orthogonality assumptions hold for both instruments. The arithmetic alone cannot verify those assumptions. The estimates are hypothetical, and their units show why dividing the outcome response by the dose response yields points per dose unit.
Disagreement Can Reflect Different Populations
Even when instruments are substantively plausible, they may induce variation in \(D\) for different groups or in different parts of the population. If the effect of \(D\) is not constant, their ratios may then summarize different responses rather than a single common \(\beta\). This possibility is especially relevant when instruments are binary or when the institutional mechanisms behind them reach different people. A difference should therefore lead to questions about both instrument validity and the population whose treatment changes because of each instrument.
Worked Example: Comparing Binary-Instrument Ratios
Hypothetical data summary. In a common sample with no additional controls, \(D\) is treatment receipt, recorded as zero or one, and \(Y\) is a symptom score in points. For binary instrument \(Z_1\), the treatment rate is \(0.12\) higher among those with \(Z_1=1\) than among those with \(Z_1=0\); the corresponding mean symptom score is \(0.036\) points higher. For binary instrument \(Z_2\), the treatment-rate difference is \(0.08\), and the mean symptom-score difference is \(0.016\) points. Calculate the Wald ratios and interpret them.
With a binary instrument and no controls, the reduced-form and first-stage slopes are the outcome and treatment differences between the two instrument groups. Thus, for \(Z_1\),
For \(Z_2\),
These hypothetical ratios differ by \(0.100\) points per treatment. Both instruments have nonzero first-stage differences in the summaries, but relevance alone is not enough. If a common homogeneous treatment effect and valid instruments were appropriate, the population ratios would agree. If treatment effects vary and the instruments induce treatment changes among different groups, the ratios may instead reflect different instrument-specific responses. Alternatively, one or both instruments may fail exclusion or orthogonality. The summaries cannot distinguish these explanations; that requires careful knowledge of the assignment mechanisms, the populations affected, and the substantive validity arguments.
A Disciplined Comparison
A useful comparison proceeds in a fixed order. First ensure that the estimates use the same outcome and endogenous regressor, the same controls, compatible samples, and consistent units. Next report each instrument’s first-stage and reduced-form slopes, then calculate its just-identified ratio. Finally, interpret any disagreement as a reason to examine assumptions and target populations, not as a verdict about instrument validity.
Use the same specification and observations where possible. If samples or controls differ, state that the comparison is not instrument-only.
Record each reduced-form slope and first-stage slope with units and signs. Confirm that the first-stage slope is nonzero before forming a ratio.
Divide the reduced-form slope by the first-stage slope, then express the result in outcome units per unit of the endogenous regressor.
Consider sampling variability, exclusion and orthogonality, specification choices, and whether the instruments induce changes for different populations.
Common Mistake: Treating Agreement as Proof
Wrong reasoning: “The two IV estimates are close, so both instruments must be valid.” Similar ratios are compatible with valid instruments under a common effect, but they are also compatible with shared violations, offsetting biases, or imprecise estimates. Correction: Treat agreement as one piece of context, then evaluate each instrument’s relevance and validity using its assignment mechanism and a defensible exclusion argument.
The converse overstatement is also a pitfall: different estimates do not automatically prove that one instrument is invalid. They may differ through finite-sample variation or because the instruments speak to different populations under heterogeneous effects. A comparison is informative precisely because it identifies what must be explained; it does not settle that explanation on its own.
Check Your Understanding
Answer each question by referring to the reduced form, first stage, ratio, and assumptions where relevant.
- An instrument has a reduced-form slope of \(0.30\) outcome units per instrument unit and a first-stage slope of \(0.15\) treatment units per instrument unit. What is its just-identified IV estimate, with units?
- Two instruments have different reduced-form slopes but identical IV ratios. Explain how that pattern can arise.
- Why should candidate-instrument comparisons use the same controls and, where possible, the same observations?
- Two IV estimates disagree. Give two explanations other than “one instrument must be invalid.”
- Why does close agreement between two candidate-instrument estimates not prove that either instrument satisfies exclusion?