Continuous Instruments and First-Stage Scaling

Why Instrument Units Matter

A continuous instrument might be recorded in points, tens of points, or standard-deviation units. The numerical value of a regression slope involving that instrument changes with the chosen units. That change is a matter of scale, not evidence that the instrument has become more or less causally informative. In a just-identified linear IV model, the first-stage and reduced-form slopes change together, so their ratio is unchanged.

The key is to keep track of what each slope measures. The first-stage slope describes how the treatment changes with the instrument. The reduced-form slope describes how the outcome changes with the instrument. Dividing the latter by the former expresses the instrument-related outcome change per unit of instrument-related treatment change.

Definition: Let \(Z\) be a continuous instrument, \(D\) a treatment, and \(Y\) an outcome. In linear regressions with intercepts, the first-stage slope \(\pi_1\) is the slope in \(D=\pi_0+\pi_1 Z+v\). The reduced-form slope \(\rho_1\) is the slope in \(Y=\rho_0+\rho_1 Z+u\). Their units are, respectively, treatment units per instrument unit and outcome units per instrument unit.

These equations define regression relationships; their slopes are not automatically causal. For a causal IV interpretation, assume a structural equation \(Y=\alpha+\beta D+\varepsilon\), where \(\beta\) is a constant treatment effect in outcome units per treatment unit. The instrument must be relevant, so its population first-stage slope is nonzero. It must also be independent of the structural causes of the outcome, and it must affect the outcome only through \(D\) (the exclusion restriction). In this linear model, these requirements imply \(\operatorname{Cov}(Z,\varepsilon)=0\). Under them, the population reduced-form slope is \(\beta\) times the population first-stage slope.

For a continuous instrument, the ratio of slopes is therefore \(\beta\) under this constant-effect structural model and the IV assumptions. Without a constant effect, the ratio does not automatically represent one universal treatment effect: its causal interpretation requires additional structure. A continuous instrument does not, just by being continuous, make the ratio an average effect for every person or every treatment change.

From the Two Slopes to the IV Ratio

With an intercept, the population linear-projection slopes can be written as covariances divided by the variance of the instrument. This representation makes the ratio and the role of the assumptions explicit.

\pi_1=\frac{\operatorname{Cov}(Z,D)}{\operatorname{Var}(Z)},\qquad \rho_1=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Var}(Z)}.

Substitute the structural equation \(Y=\alpha+\beta D+\varepsilon\) into the reduced-form slope. Since a constant has zero covariance with \(Z\), covariance is linear, and the IV independence condition gives \(\operatorname{Cov}(Z,\varepsilon)=0\), we obtain:

\rho_1 =\frac{\operatorname{Cov}(Z,\alpha+\beta D+\varepsilon)}{\operatorname{Var}(Z)} =\frac{\beta\operatorname{Cov}(Z,D)+\operatorname{Cov}(Z,\varepsilon)}{\operatorname{Var}(Z)} =\beta\pi_1.

If \(\pi_1\ne 0\), division gives \(\beta=\rho_1/\pi_1\). In data, the corresponding just-identified IV estimate is the estimated reduced-form slope divided by the estimated first-stage slope. The assumptions justify the target of this estimator; the arithmetic alone does not establish that they hold.

Worked Example: Calculating a Continuous-Instrument IV Ratio

Suppose a hypothetical study uses an encouragement score \(Z\), measured in score points, as an instrument for treatment hours \(D\). The outcome \(Y\) is a test score measured in points. The following five hypothetical observations are available:

Person	Encouragement \(Z\) (points)	Treatment \(D\) (hours)	Outcome \(Y\) (points)
1	-2	4	12
2	-1	5	14
3	0	6	15
4	1	7	17
5	2	8	17

The means are \(\bar{Z}=0\), \(\bar{D}=6\), and \(\bar{Y}=15\). With an intercept, each sample regression slope is the sum of products of centered values divided by the sum of squared centered instrument values. Here the denominator is:

\sum_{i=1}^{5}(Z_i-\bar{Z})^2 =(-2)^2+(-1)^2+0^2+1^2+2^2=10.

For the first stage, the numerator is \((-2)(4-6)+(-1)(5-6)+0(6-6)+1(7-6)+2(8-6)=10\). Thus the estimated first-stage slope is \(10/10=1.00\) treatment hour per encouragement point. For the reduced form, the numerator is \((-2)(12-15)+(-1)(14-15)+0(15-15)+1(17-15)+2(17-15)=13\). Its estimated slope is \(13/10=1.30\) outcome points per encouragement point.

\widehat{\beta}_{IV} =\frac{\widehat{\rho}_1}{\widehat{\pi}_1} =\frac{1.30\ \text{outcome points per encouragement point}} {1.00\ \text{treatment hour per encouragement point}} =1.30\ \text{outcome points per treatment hour}.

The ratio is an IV estimate, not a direct comparison of outcomes among people with different treatment hours. Its causal interpretation depends on the instrument assumptions and, for interpreting it as a single \(\beta\), the constant-effect structural model. This small hypothetical data set cannot by itself verify those assumptions.

Changing the Instrument’s Units

Consider an affine recoding \(Z^*=a+bZ\), where \(a\) is a constant and \(b\ne 0\). The new variable may represent a shifted origin, a different unit, or a reversed scale. Solving for the original instrument gives \(Z=(Z^*-a)/b\). Substitute this expression into the first-stage regression:

D=\pi_0+\pi_1\left(\frac{Z^*-a}{b}\right)+v =\left(\pi_0-\frac{\pi_1a}{b}\right)+\frac{\pi_1}{b}Z^*+v.

Thus the first-stage slope using \(Z^*\) is \(\pi_1^*=\pi_1/b\). Applying the same substitution to the reduced form gives \(\rho_1^*=\rho_1/b\). Both slopes change by the same factor, including when \(b\) is negative. Consequently:

\frac{\rho_1^*}{\pi_1^*} =\frac{\rho_1/b}{\pi_1/b} =\frac{\rho_1}{\pi_1}.

This is an exact algebraic result for the population slopes. In a sample, ordinary least-squares regressions with intercepts also have this property: replacing \(Z\) by \(a+bZ\) divides both estimated slopes by \(b\), leaving their ratio unchanged. The intercepts can change, but the ratio does not.

Worked Example: Reporting the Same Instrument in Tens of Points

Suppose a hypothetical analysis reports a first-stage slope of \(0.08\) treatment hours per encouragement point and a reduced-form slope of \(0.12\) outcome points per encouragement point. The IV ratio in the original units is:

\frac{0.12\ \text{outcome points per encouragement point}} {0.08\ \text{treatment hours per encouragement point}} =1.50\ \text{outcome points per treatment hour}.

Now record the instrument in tens of encouragement points, so that \(Z^*=Z/10\). In \(Z^*=a+bZ\), this has \(a=0\) and \(b=0.1\). Each slope is divided by \(b\), or equivalently multiplied by 10. The first-stage slope becomes \(0.08/0.1=0.80\) treatment hours per ten-point unit, and the reduced-form slope becomes \(0.12/0.1=1.20\) outcome points per ten-point unit. Their ratio is:

\frac{1.20\ \text{outcome points per ten-point unit}} {0.80\ \text{treatment hours per ten-point unit}} =1.50\ \text{outcome points per treatment hour}.

The slopes are ten times as large numerically because one unit of \(Z^*\) represents ten original points. The ratio and its outcome-per-treatment units are unchanged. Neither slope should be compared across reports without first checking the instrument units.

Worked Example: Recentring and Reversing the Instrument Scale

Return to the hypothetical slopes \(0.08\) treatment hours per original point and \(0.12\) outcome points per original point. Suppose the instrument is recoded as \(Z^*=100-2Z\). This shifts the origin and reverses the direction, with \(a=100\) and \(b=-2\). The transformed slopes are:

\pi_1^*=\frac{0.08}{-2}=-0.04 \quad\text{treatment hours per new unit},\qquad \rho_1^*=\frac{0.12}{-2}=-0.06 \quad\text{outcome points per new unit}.

Both slopes are negative because higher values on the reversed scale correspond to lower values on the original scale. But their ratio is still positive:

\frac{\rho_1^*}{\pi_1^*} =\frac{-0.06}{-0.04} =1.50\ \text{outcome points per treatment hour}.

The shift of 100 changes the regression intercepts but not either slope. Reversing direction changes the signs of both slopes, not the ratio. A negative first-stage slope is not, by itself, evidence against relevance: relevance concerns whether the slope is nonzero, not whether its sign is positive.

What Scaling Does—and Does Not—Change

A change of units preserves the information in the instrument: when \(b\ne0\), \(Z^*=a+bZ\) uniquely determines \(Z\), and vice versa. In the simple regressions considered here, fitted values are unchanged by that recoding. The first-stage slope’s numerical size alone is therefore not a scale-free measure of instrument strength. Its magnitude must be read with its units, and rescaling cannot make a substantively weak relationship strong.

The invariance result applies to a linear change of units, not to every transformation of \(Z\). For example, replacing a raw instrument by its square, logarithm, or categories is generally not an affine recoding. Such a change can alter the fitted relationships and the ratio; it may also change the substantive instrument being used. Treating a nonlinear transformation as a harmless unit conversion is incorrect.

Key takeaway: For a continuous instrument, the first-stage slope has treatment-per-instrument units and the reduced-form slope has outcome-per-instrument units. Their just-identified IV ratio has outcome-per-treatment units. An affine change of instrument units rescales both slopes equally, leaving the ratio unchanged, but it does not establish relevance, independence, exclusion, or a causal interpretation.

Common Mistakes

Comparing raw slope magnitudes without checking units. A slope of \(0.08\) per point and \(0.80\) per ten-point unit can describe the same relationship. Convert to common units before comparing.
Calling the first-stage slope the treatment effect. It describes treatment change per instrument unit. The IV ratio, not the first-stage slope alone, has outcome-per-treatment units.
Thinking a negative first-stage slope invalidates the ratio. If the instrument scale is reversed, both slopes reverse sign and their ratio remains the same. Relevance requires a nonzero first stage; validity requires separate assumptions.
Assuming scale invariance proves instrument validity. The algebra shows that a unit conversion preserves the ratio. It says nothing about whether the instrument is independent of outcome causes or satisfies exclusion.
Applying the result to any transformation. Shifting or multiplying by a nonzero constant preserves the ratio. Nonlinear transformations can change the regressions and do not generally preserve it.

Check Your Understanding

Answer each question using the units and assumptions in the setup.

A first-stage slope is \(0.05\) treatment doses per instrument point, and the reduced-form slope is \(0.20\) outcome units per instrument point. What is their ratio, including its units?
If \(Z^*=3+4Z\), by what factor do the first-stage and reduced-form slopes change? Does their ratio change?
Why does a first-stage slope’s numerical magnitude depend on the instrument’s units?
If an instrument is recoded in the reverse direction, what happens to the signs of the first-stage and reduced-form slopes, and what happens to their ratio?
Does the unchanged ratio after rescaling establish the exclusion restriction? Explain briefly.