Comparing OLS, Reduced Form, and 2SLS

Three Coefficients, Three Questions

In an instrumental-variables analysis, it is useful to distinguish three quantities that can all appear in the same dataset. The OLS coefficient describes the sample relationship between treatment and outcome. The reduced-form coefficient describes the relationship between the instrument and outcome. The 2SLS coefficient uses instrument-related variation in treatment to estimate a treatment–outcome relationship. These coefficients need not be numerically equal, and they do not answer the same statistical question.

We will focus on a model with one treatment \(D\), one instrument \(Z\), and one outcome \(Y\), including an intercept and no covariates. The treatment and outcome are measured on specified numerical scales. The instrument is relevant when it predicts treatment. For a causal interpretation of the IV estimate, the instrument must also satisfy the appropriate independence and exclusion assumptions; a numerical calculation alone cannot establish that it does.

Definition: In the simple linear setting, the OLS slope is the slope from regressing \(Y\) on \(D\) with an intercept. The reduced-form slope is the slope from regressing \(Y\) on \(Z\) with an intercept. The first-stage slope is the slope from regressing \(D\) on \(Z\) with an intercept. With one instrument and a nonzero first-stage slope, the just-identified 2SLS slope is the reduced-form slope divided by the first-stage slope.

In a sample of \(n\) observations, the OLS slope is

\widehat{\beta}_{OLS} = \frac{\sum_{i=1}^n (D_i-\bar{D})(Y_i-\bar{Y})} {\sum_{i=1}^n (D_i-\bar{D})^2}.

The analogous regression slopes for the reduced form and first stage replace the predictor \(D\) with \(Z\), or the response \(Y\) with \(D\), respectively. These are descriptive sample regression coefficients whether or not the instrument assumptions hold. Their causal interpretations require more.

Worked Calculation on One Dataset

Consider the following hypothetical data on eight individuals. \(Z\) is a binary encouragement, \(D\) indicates treatment dose on a numerical scale, and \(Y\) is an outcome measured in outcome units. The data are illustrative, not observations from a real study.

Worked Example: Calculate All Three Coefficients

Data. Suppose the observations are:

Individual	Instrument \(Z\)	Treatment \(D\)	Outcome \(Y\)
1	0	0	0
2	0	0	4
3	0	1	3
4	0	1	7
5	1	1	10
6	1	1	2
7	1	2	6
8	1	2	8

Step 1: OLS. The sample means are \(\bar{D}=1\) and \(\bar{Y}=5\). The sum of cross-products is

\sum_{i=1}^8(D_i-\bar{D})(Y_i-\bar{Y}) =5+1+0+0+0+0+1+3=10.

The sum of squared treatment deviations is

\sum_{i=1}^8(D_i-\bar{D})^2 =1+1+0+0+0+0+1+1=4.

Therefore, the OLS slope is \(10/4=2.5\) outcome units per treatment unit. It summarizes the linear association of \(D\) with \(Y\) in these eight observations.

Step 2: Reduced form and first stage. Among observations with \(Z=0\), the mean treatment is \(0.5\) and the mean outcome is \(3.5\). Among those with \(Z=1\), the corresponding means are \(1.5\) and \(6.5\). Because \(Z\) is binary, the regression slope with an intercept equals the difference between the two group means:

\widehat{\pi}_{RF}=6.5-3.5=3 \quad\text{outcome units per unit of }Z, \qquad \widehat{\pi}_{FS}=1.5-0.5=1 \quad\text{treatment units per unit of }Z.

Step 3: 2SLS. The just-identified estimate is

\widehat{\beta}_{2SLS} = \frac{\widehat{\pi}_{RF}}{\widehat{\pi}_{FS}} = \frac{3}{1} = 3 \quad\text{outcome units per treatment unit}.

Thus OLS is \(2.5\), the reduced-form slope is \(3\), and 2SLS is \(3\). The reduced-form number is not directly a treatment effect: it measures the outcome difference associated with a one-unit instrument change. Dividing by the first-stage change expresses the outcome change per instrument-induced treatment change. These calculations do not, by themselves, establish that the instrument is valid or that the 2SLS value has a causal interpretation.

Why the Just-Identified Ratio Works

The ratio formula follows directly from the regression slopes when there is one instrument and each regression includes an intercept. Define the centered sample cross-product \(S_{AB}=\sum_{i=1}^n(A_i-\bar{A})(B_i-\bar{B})\). The reduced-form and first-stage slopes are \(S_{ZY}/S_{ZZ}\) and \(S_{ZD}/S_{ZZ}\). If \(S_{ZD}\ne 0\), their ratio is

\frac{\widehat{\pi}_{RF}}{\widehat{\pi}_{FS}} = \frac{S_{ZY}/S_{ZZ}}{S_{ZD}/S_{ZZ}} = \frac{S_{ZY}}{S_{ZD}}.

This algebra is the just-identified 2SLS result: the same coefficient is obtained by using the instrument to isolate the component of treatment associated with \(Z\), then relating that component to the outcome. In the first stage, the fitted treatment is \(\widehat{D}_i=\widehat{\pi}_0+\widehat{\pi}_{FS}Z_i\). Regressing \(Y\) on \(\widehat{D}\) with an intercept gives a slope of \(S_{\widehat{D}Y}/S_{\widehat{D}\widehat{D}}\). Since \(\widehat{D}_i-\bar{\widehat{D}}=\widehat{\pi}_{FS}(Z_i-\bar{Z})\), its numerator is \(\widehat{\pi}_{FS}S_{ZY}\) and its denominator is \(\widehat{\pi}_{FS}^2S_{ZZ}\). The slope is therefore \(S_{ZY}/(\widehat{\pi}_{FS}S_{ZZ})=S_{ZY}/S_{ZD}\), as claimed.

This derivation also identifies a requirement for the calculation: the first-stage slope cannot be zero. If the instrument does not predict treatment in the sample, the ratio is undefined. The formula is an algebraic identity for one instrument; it does not say that the instrument has been shown to be relevant in a broader population, nor does it establish the assumptions needed for causal inference.

Worked Examples: Units and Instrument Coding

Worked Example: Interpret the Reduced Form Before Dividing

In a second hypothetical dataset, \(Z=0\) and \(Z=1\) identify two instrument groups. Suppose the group means are: for \(Z=0\), mean treatment \(0.20\) and mean outcome \(10.0\); for \(Z=1\), mean treatment \(0.50\) and mean outcome \(13.5\). Treatment is coded in treatment units, and outcome is measured in outcome points.

The reduced-form slope is the outcome mean difference, \(13.5-10.0=3.5\) outcome points per unit of \(Z\). The first-stage slope is \(0.50-0.20=0.30\) treatment units per unit of \(Z\). The just-identified 2SLS coefficient is

\widehat{\beta}_{2SLS} = \frac{3.5\ \text{outcome points per unit of }Z} {0.30\ \text{treatment units per unit of }Z} = 11.67\ \text{outcome points per treatment unit}.

The value \(3.5\) answers, “How much does the outcome differ per instrument unit in this sample?” The value \(11.67\) rescales that outcome difference by the instrument-associated treatment difference. It is not correct to report \(3.5\) as the treatment effect, because it remains in outcome points per instrument unit.

Worked Example: Rescale the Instrument Without Changing 2SLS

Use the same group means as in the preceding example, but code the instrument as \(Z'=10Z\), so its values are 0 and 10 instead of 0 and 1. The difference in mean outcome per unit of \(Z'\) is \(3.5/10=0.35\) outcome points. The first-stage difference per unit of \(Z'\) is \(0.30/10=0.03\) treatment units. Hence

\widehat{\beta}_{2SLS} = \frac{0.35}{0.03} = 11.67 \quad\text{outcome points per treatment unit}.

Both instrument-based slopes changed because the instrument’s unit changed, but their ratio did not. This is a useful dimensional check: in a just-identified model, changing the scale of \(Z\) alone changes the reduced form and first stage by the same factor, leaving the 2SLS coefficient unchanged. The OLS slope is also unchanged by this instrument recoding because OLS regresses \(Y\) on \(D\), not on \(Z\).

Why OLS Can Differ From 2SLS

OLS uses all observed treatment variation, including variation that may be related to unmeasured causes of the outcome. 2SLS uses only the part of treatment variation associated with the instrument in the first stage. When treatment and unmeasured outcome determinants are associated, the OLS slope may not have a causal interpretation, even though it is calculated correctly as a regression coefficient. A valid instrument is intended to provide a different source of variation; it does not make OLS and 2SLS estimates interchangeable.

The distinction can be seen in a small hypothetical dataset. Suppose an instrument raises treatment by one unit on average and raises the outcome by five units on average. The reduced form is then five, and the just-identified 2SLS coefficient is five outcome units per treatment unit. Yet the OLS slope also depends on how treatment and outcome covary within and across instrument groups. It need not equal that ratio.

Worked Example: The Same Instrument Contrast, a Different OLS Slope

Consider four hypothetical observations: \((Z,D,Y)=(0,0,0),(0,1,4),(1,1,5),(1,2,9)\). For \(Z=0\), the means of \(D\) and \(Y\) are \(0.5\) and \(2\). For \(Z=1\), they are \(1.5\) and \(7\). Therefore the first-stage slope is \(1.5-0.5=1\) treatment unit per instrument unit, and the reduced-form slope is \(7-2=5\) outcome units per instrument unit. The 2SLS slope is \(5/1=5\) outcome units per treatment unit.

For OLS, \(\bar{D}=1\) and \(\bar{Y}=4.5\). The cross-product sum is

\sum_{i=1}^4(D_i-\bar{D})(Y_i-\bar{Y}) = 4.5+0+0+4.5 = 9.

The treatment-deviation sum of squares is \(1+0+0+1=2\). Thus \(\widehat{\beta}_{OLS}=9/2=4.5\) outcome units per treatment unit, while 2SLS is \(5\). Both calculations are correct: OLS uses the observed \(D\)-\(Y\) covariance, whereas 2SLS uses the instrument’s outcome contrast relative to its treatment contrast. The difference alone does not prove that either coefficient is causally correct; that judgment depends on the assumptions and estimand.

Assumptions and Interpretation

The regressions and ratios above are defined from the observed data. A causal interpretation of the 2SLS coefficient requires a defensible IV design. In this simple setting, the instrument must be relevant to treatment, independent of the relevant unobserved causes under the design, and excluded from the outcome equation except through treatment. The structural linear model writes this last relationship as \(Y_i=\alpha+\beta D_i+u_i\), with a valid instrument satisfying the required orthogonality condition \(E[Z_i u_i]=0\) (or its centered version when an intercept is included). These assumptions are substantive and cannot be verified merely by computing the three slopes.

The 2SLS coefficient’s causal meaning also depends on the model and treatment-effect structure. It should not automatically be described as the average effect for every individual or as a population-wide effect. With treatment-effect heterogeneity, the relevant interpretation requires additional assumptions and may target a particular IV estimand. In contrast, the reduced form remains an instrument–outcome association unless the instrument assignment and exclusion assumptions support a causal interpretation of the instrument’s effect on the outcome.

These quantities also have different units. OLS and 2SLS slopes are outcome units per treatment unit. The reduced-form slope is outcome units per instrument unit. If the instrument is binary, its slope with an intercept is a difference between instrument groups; the first stage is the corresponding treatment difference. The ratio converts the outcome contrast to a per-treatment-unit scale. Stating units makes it harder to confuse the reduced form with a treatment effect.

Common Mistakes

Calling the reduced form the treatment effect. The reduced form is the change in outcome per instrument unit. In the just-identified case, divide it by the first-stage slope to obtain the 2SLS coefficient.
Assuming OLS and 2SLS estimate the same quantity by definition. They use different sources of treatment variation, and their coefficients can differ. Causal interpretation depends on assumptions, not on whether the estimates are numerically close.
Ignoring units. A reduced-form slope and a treatment slope generally have different denominators. Report the predictor scale in the units of each coefficient.
Treating the ratio as proof of instrument validity. The ratio can be calculated whenever the sample first stage is nonzero. Relevance, independence, and exclusion require separate justification.

Key takeaway: OLS summarizes the observed treatment–outcome relationship; the reduced form summarizes the instrument–outcome relationship; and just-identified 2SLS divides that reduced form by the first stage to express the instrument-based outcome change per treatment unit. Keep the calculations, units, assumptions, and causal claims distinct.

Check Your Understanding

Answer each question by identifying the coefficient, calculation, or assumption involved.

A binary instrument changes mean outcome by 4 points and mean treatment by 0.5 units. What are the reduced-form and just-identified 2SLS slopes, including their units?
Why does the reduced-form coefficient not generally equal the 2SLS coefficient?
In the one-instrument model, what happens to the reduced-form and first-stage slopes if the instrument is recoded from \(Z\) to \(10Z\)? What happens to their ratio?
Does a numerical difference between OLS and 2SLS prove that the instrument is valid? Explain.
What variation does OLS use that 2SLS is designed not to use for identifying its coefficient?