Rescaling Treatment and Outcome in IV Regression

Why Units Matter in IV Regression

A 2SLS coefficient is expressed in the units of the outcome per unit of treatment. If treatment is recorded in tens of hours rather than hours, or an outcome in thousands of dollars rather than dollars, the numerical coefficient changes even though the underlying fitted relationship has not. Clear interpretation therefore requires tracking both the coefficient’s numerical scale and its units.

Standardization also changes the numerical coefficient, but it is not usually a simple physical unit conversion: it expresses changes in standard-deviation units and depends on the standard deviations used. This tutorial derives the relevant transformations, works through both kinds of changes, and shows how to translate estimates without changing their substantive meaning.

Model and Rescaling Rule

Consider a linear structural equation with one endogenous treatment $D$, outcome $Y$, and included exogenous covariates $W$:

Y = \alpha + \beta D + W'\gamma + u.

The 2SLS specification includes an intercept and $W$ as exogenous regressors, along with excluded instrument(s) $Z$ that provide variation in $D$. The instrument set must have enough relevant variation to identify the coefficient, so the 2SLS design matrix has full rank. For a causal interpretation, the usual IV requirements must also be credible: instruments are relevant, independent of the structural error given the included covariates, and affect the outcome only through treatment, under the specified model. The algebraic rescaling result below is a property of the estimator; it does not establish those causal requirements.

Rescaling rule: If the outcome and treatment are transformed as $Y^* = c + dY$ and $D^* = a + bD$, where $b$ and $d$ are nonzero, then the 2SLS coefficient on $D^*$ in a model with an intercept and the same covariates and instruments is $$ \hat{\beta}^*_{\text{2SLS}} = \frac{d}{b}\hat{\beta}_{\text{2SLS}}. $$ The shifts $a$ and $c$ change the intercept, not the slope. The coefficient’s units become units of $Y^*$ per unit of $D^*$.

The rule applies to changes such as dividing a measurement by a constant or changing its origin. A change in treatment scale divides the coefficient by that scale factor; a change in outcome scale multiplies it by that factor. If a scale factor is negative, the coefficient’s sign changes accordingly. In ordinary unit conversions, scale factors are positive.

Why the Rule Holds

A short derivation makes clear why the estimator transforms in this way. First remove the intercept and included exogenous covariates from the variables, as in partialling out covariates. Let $\widetilde D$ and $\widetilde Y$ denote the residualized treatment and outcome, and let $\widehat D$ be the fitted value of $\widetilde D$ from the first-stage regression on the residualized excluded instruments. For one endogenous treatment, the 2SLS coefficient can be written as

\hat{\beta}_{\text{2SLS}} = \frac{\widehat D'\widetilde Y}{\widehat D'\widetilde D}.

Because the model includes an intercept, adding constants to $D$ or $Y$ has no effect on their residualized values. Multiplying $D$ by $b$ multiplies both its residualized values and its first-stage fitted values by $b$; multiplying $Y$ by $d$ multiplies its residualized values by $d$. Thus the transformed estimate is

\hat{\beta}^*_{\text{2SLS}} = \frac{(b\widehat D)'(d\widetilde Y)} {(b\widehat D)'(b\widetilde D)} = \frac{bd\,\widehat D'\widetilde Y} {b^2\,\widehat D'\widetilde D} = \frac{d}{b}\hat{\beta}_{\text{2SLS}}.

This argument requires the transformed model to use the same observations, covariates, and instruments, with the intercept included, and requires nonzero scale factors and an identified 2SLS coefficient. The estimator’s units change, but the fitted structural relationship, expressed in the correspondingly transformed variables, does not.

Worked Example: Converting Dollars per Hour to Thousands of Dollars per Ten Hours

Suppose a hypothetical IV analysis estimates that one additional hour of treatment is associated, under the structural model, with a 2SLS change of $18$ dollars in the outcome. Thus $\hat{\beta}_{\text{2SLS}}=18$ dollars per hour. Now record treatment in units of ten hours, $D^*=D/10$, and outcome in thousands of dollars, $Y^*=Y/1000$.

Here $b=1/10$ and $d=1/1000$. Apply the rescaling rule:

\hat{\beta}^*_{\text{2SLS}} = \frac{1/1000}{1/10}(18) = \frac{10}{1000}(18) = 0.18.

The result is $0.18$ thousand dollars per ten treatment-hours. Converting back to the original scale gives $0.18 \times 1000 = 180$ dollars per ten hours, or $18$ dollars per hour. The numerical coefficient is different, but the implied change is consistent across both unit systems.

Shifts, Unit Conversions, and Interpretation

A shift changes where zero is placed, not the distance between values. For example, measuring temperature in a scale with a different zero involves both a shift and a scale change; measuring a treatment as years since a chosen baseline rather than years of age may involve only a shift. With an intercept in the regression, a shift in $D$ is absorbed by the intercept: the coefficient on treatment remains the same. A shift in $Y$ also changes the intercept but leaves the slope unchanged.

A unit conversion changes the size of a one-unit step. If treatment is converted from minutes to hours, one treatment unit now represents 60 minutes, so the coefficient per hour is 60 times the coefficient per minute. If outcome is converted from dollars to cents, the coefficient is multiplied by 100. These conversions affect the reported slope exactly as the rescaling rule specifies.

Worked Example: Changing the Treatment’s Origin

Suppose a hypothetical 2SLS estimate is $4.2$ outcome points per year of treatment. For convenience, redefine treatment as years since age 40: $D^*=D-40$. The treatment has been shifted, but not rescaled, so $a=-40$ and $b=1$; the outcome is unchanged, so $d=1$.

Therefore,

\hat{\beta}^*_{\text{2SLS}} = \frac{1}{1}(4.2) = 4.2 \quad\text{outcome points per year}.

A person or unit with $D^*=0$ now has $D=40$, so the intercept refers to a different treatment value. The estimated change per additional year remains $4.2$ outcome points. This interpretation assumes the same linear specification and that the intercept is included.

Standardization Is a Different Scale Change

A common standardization subtracts a variable’s mean and divides by its standard deviation. If the observed treatment and outcome are standardized using their respective means and standard deviations, define

D^* = \frac{D-\bar{D}}{s_D}, \qquad Y^* = \frac{Y-\bar{Y}}{s_Y}.

Here $s_D$ and $s_Y$ are the standard deviations used for the variables in that analysis. Applying the rescaling rule with $b=1/s_D$ and $d=1/s_Y$, the standardized 2SLS estimate is

\hat{\beta}^*_{\text{2SLS}} = \frac{s_D}{s_Y}\hat{\beta}_{\text{2SLS}}.

Its interpretation is in standard-deviation units: the fitted change in outcome, measured in outcome standard deviations, for a one-standard-deviation change in treatment, under the IV model. The numerical value depends on the standard deviations used, so it is not a universal, unit-free measure of causal importance. In particular, it should not be reported as though it were the original effect in dollars, hours, or other natural units.

Worked Example: Computing a Standardized 2SLS Coefficient

In a hypothetical analysis, suppose the 2SLS estimate is $240$ dollars per treatment-hour. In the analysis sample, the treatment standard deviation is $2.5$ hours and the outcome standard deviation is $1800$ dollars. Standardize both variables using these standard deviations.

The conversion is

\hat{\beta}^*_{\text{2SLS}} = \frac{2.5\ \text{hours}}{1800\ \text{dollars}} \left(240\ \frac{\text{dollars}}{\text{hour}}\right) = \frac{600}{1800} = 0.3333.

The interpretation is a fitted change of about $0.3333$ outcome standard deviations per one-standard-deviation change in treatment, under the model. The original estimate remains $240$ dollars per hour. It would be incorrect to call $0.3333$ dollars per hour, or to say that the 2SLS estimate in original units is $0.3333$.

Checking the Transformation With First-Stage and Reduced-Form Slopes

In a just-identified model with one instrument and no additional complications, the 2SLS coefficient can also be expressed as the reduced-form slope divided by the first-stage slope. This gives a useful arithmetic check on unit conversions: changing the outcome scale changes the reduced-form slope, while changing the treatment scale changes the first-stage slope. Their ratio must transform in the same way as the 2SLS coefficient.

Worked Example: Verifying a Rescaled Estimate From the Two Slopes

Suppose a hypothetical just-identified analysis has a reduced-form slope of $120$ dollars per instrument unit and a first-stage slope of $0.5$ treatment-hours per instrument unit. The 2SLS estimate is

\hat{\beta}_{\text{2SLS}} = \frac{120\ \text{dollars per instrument unit}} {0.5\ \text{hours per instrument unit}} = 240\ \text{dollars per hour}.

Now measure the outcome in hundreds of dollars and treatment in four-hour units: $Y^*=Y/100$, $D^*=D/4$. The transformed reduced-form slope is $120/100=1.2$ hundreds of dollars per instrument unit. The transformed first-stage slope is $0.5/4=0.125$ four-hour units per instrument unit. Their ratio is

\frac{1.2}{0.125} = 9.6 \quad\text{hundreds of dollars per four-hour unit}.

Since $9.6$ hundreds of dollars equals $960$ dollars, and one four-hour unit represents four hours, the result is $960/4=240$ dollars per hour. Equivalently, the rescaling rule gives $(1/100)/(1/4)\times240=9.6$. Both checks agree.

Common Mistakes and Practical Checks

Reversing the treatment conversion. If treatment is divided by 10, a one-unit increase in the new treatment variable represents 10 original units. The coefficient per new unit is 10 times the original coefficient when the outcome scale is unchanged.
Scaling only one part of an estimate. When both treatment and outcome change units, use $d/b$, not $d$ or $1/b$ alone.
Confusing a shift with a scale change. Subtracting a constant from treatment changes the intercept but not the slope when an intercept is included. Dividing treatment by a constant changes the coefficient.
Calling a standardized coefficient a natural-unit effect. A coefficient of $0.33$ after standardization is in outcome standard deviations per treatment standard deviation, not dollars per hour.
Treating a scale check as evidence of instrument validity. A coefficient that transforms correctly is an arithmetic consistency check. It does not establish relevance, independence, or exclusion.

Before reporting a rescaled estimate, state the original units, the new units, the scale factors, and the resulting units of the coefficient. If the purpose is comparison across variables, also specify which standard deviations were used. These details make it possible to verify the calculation and interpret the estimate correctly.

Check Your Understanding

Work through these questions by identifying the treatment and outcome scale factors before transforming the coefficient.

A 2SLS estimate is $5$ outcome units per treatment-unit. Treatment is re-expressed in units of five original treatment-units, with the outcome unchanged. What is the new coefficient and its units?
If an outcome measured in dollars is converted to thousands of dollars while treatment units remain fixed, what happens to the coefficient?
Why does redefining treatment as $D-12$ leave the slope unchanged in a model with an intercept?
A 2SLS estimate is $30$ dollars per hour, with treatment standard deviation $3$ hours and outcome standard deviation $900$ dollars. What is the standardized coefficient, and how should it be interpreted?
Does obtaining the expected rescaled coefficient establish that the instrument satisfies the exclusion restriction? Explain.