Instrumental Variables for Endogenous Prices

Why Instrumental Variables Are Used

A quoted price is not always assigned independently of a customer's likelihood of buying insurance. For example, an insurer may quote higher prices to risks whose characteristics are also associated with demand. A regression of purchase on quoted price can then combine the effect of price with the effect of those underlying characteristics. The earlier material on confounding and adjustment in observational demand addresses measured confounders; an instrumental-variable approach can help when relevant sources of price variation remain unmeasured.

An instrument is a variable that shifts the quoted price but, under stated assumptions, does not otherwise carry information about the demand outcome. In a hypothetical insurance setting, an administrative pricing rule might create price differences among otherwise comparable quotes. Whether any particular rule is a credible instrument depends on how it affects both price and demand; calling a variable an instrument does not establish its validity.

Definition: Let $Y$ be a demand outcome, such as an indicator equal to 1 if a customer buys a policy and 0 otherwise; let $P$ be the quoted price; let $X$ be observed control variables; and let $Z$ be a proposed instrument. An instrument is relevant if it predicts price after controlling for $X$. It satisfies conditional independence if, given $X$, it is unrelated to unobserved causes of demand. It satisfies the exclusion restriction if it affects demand only through its effect on price, not through a separate pathway.

The Linear Model and Its Assumptions

For a basic instrumental-variable model, suppose the conditional mean of demand is linear in price and controls:

Y=\alpha+\beta P+\gamma^\mathsf{T}X+\varepsilon.

Here $\beta$ is the causal change in the conditional mean of $Y$ for a one-dollar increase in $P$, under the model's assumptions. When $Y$ is a purchase indicator, $\beta$ is measured in probability points per dollar. The error term $\varepsilon$ includes unobserved influences on demand. If price responds to those influences, ordinary least squares may not estimate $\beta$ consistently.

Instrumental variables require more than a correlation between $Z$ and price. The conditions below state what is needed for the usual linear interpretation.

Relevance: after controlling for $X$, $Z$ predicts $P$. In a linear model with one instrument, this means the coefficient on $Z$ in the price regression is not zero.
Conditional independence: given $X$, $Z$ is unrelated to $\varepsilon$. The instrument is not systematically assigned according to unobserved demand shocks.
Exclusion: given price and $X$, changing $Z$ has no direct effect on $Y$. The instrument changes demand only by changing price.
Model and measurement conditions: the variables must be defined consistently, and the linear model must be an appropriate description for the causal effect being estimated. For the standard constant-slope interpretation, the causal price effect is assumed to be the same over the relevant observations.

In practice, conditional independence and exclusion are substantive assumptions, not facts established just because an instrument is statistically significant. Relevance is assessable from observed data; exclusion generally requires a convincing account of the instrument's mechanism and its possible direct pathways to demand.

Two-Stage Least Squares

With one instrument and controls, two-stage least squares (2SLS) proceeds by first predicting price from the instrument and controls, then relating demand to that instrument-supported price variation. With multiple instruments, the first stage predicts price from all instruments and controls, and the second stage uses the fitted price in place of observed price. The coefficient and its uncertainty must be calculated as a 2SLS estimator; treating fitted price as an ordinary observed predictor and using ordinary second-stage standard errors does not generally give valid inference.

Estimate the first stage.
Regress quoted price $P$ on the instrument $Z$ and controls $X$. Check that the instrument shifts price in the relevant population.

Estimate the demand response.
Use the price variation predicted by the instrument, while retaining the controls. The 2SLS price coefficient estimates the response associated with instrument-induced price variation.

Interpret conditionally.
Give the effect in the outcome's units, per unit of price, and state that the causal interpretation depends on relevance, conditional independence, exclusion, and the model assumptions.

Why the Single-Instrument Ratio Works

The core calculation is especially clear with a binary instrument and no controls. Write the first-stage difference as the difference in average price between the $Z=1$ and $Z=0$ groups, and the reduced-form difference as the corresponding difference in average demand. Under the linear model and valid-instrument assumptions, the demand difference is caused by the price difference, not by a separate demand pathway.

\begin{aligned} E[Y\mid Z=1]-E[Y\mid Z=0] &=\beta\{E[P\mid Z=1]-E[P\mid Z=0]\},\\ \widehat{\beta}_{IV} &=\frac{\text{reduced-form effect of }Z\text{ on }Y} {\text{first-stage effect of }Z\text{ on }P}. \end{aligned}

The first equality follows by taking conditional expectations of $Y=\alpha+\beta P+\varepsilon$ in each instrument group and subtracting. The intercept cancels. Conditional independence makes the expected error equal across groups, and exclusion rules out an additional effect of $Z$ on $Y$; the remaining demand difference is therefore $\beta$ times the price difference. Dividing by a nonzero first-stage difference gives the ratio. With controls, the same logic applies to variation after accounting for those controls. In a linear model with one instrument, the controlled 2SLS coefficient is the reduced-form coefficient of $Z$ in the demand regression divided by the first-stage coefficient of $Z$ in the price regression.

The ratio is not justified if the instrument does not move price, is related to unobserved demand shocks, or directly affects demand. Nor does it automatically identify a universal response when causal price effects vary across customers: the linear constant-slope interpretation requires that additional modeling assumption.

Worked Example: A Binary Instrument and Purchase Probability

In a hypothetical portfolio study, an administrative rule changes the quoted price for one group of otherwise comparable quotes. Assume for this calculation that the rule satisfies relevance, conditional independence, and exclusion. The average quoted price is $120 when $Z=1$ and $100 when $Z=0$. Purchase rates are 0.28 and 0.36, respectively. Estimate the causal price slope under a constant linear effect and state the implied effect of a $10 increase.

Method. Use the single-instrument Wald ratio: divide the difference in purchase rates by the difference in average price, always taking the $Z=1$ group minus the $Z=0$ group.

First stage. The price difference is $120-100=\$20$. Thus the instrument is relevant in these hypothetical data.

Reduced form. The purchase-rate difference is $0.28-0.36=-0.08$, an 8-percentage-point decrease.

IV slope. Divide the reduced form by the first stage: $-0.08/20=-0.004$. The estimate is a decrease of 0.004 in purchase probability per dollar, or 0.4 percentage points per dollar.

$10 change and check. A $10 increase implies $10(-0.004)=-0.04$, or a 4-percentage-point decrease. Multiplying the estimated slope by the observed $20 price difference gives $20(-0.004)=-0.08$, exactly the observed reduced-form difference. This calculation avoids the common mistake of reporting the 8-percentage-point group difference as the effect of a $1 price increase.

Worked Example: Controlled 2SLS From Regression Coefficients

In a hypothetical quote-level analysis, $Y$ is a purchase indicator, $Z$ is an administrative price adjustment, and $X$ is a centered risk score included as a control. The estimated first-stage equation is $\widehat{P}=900+24Z+35X$. The reduced-form regression of $Y$ on $Z$ and $X$ gives a coefficient of $-0.072$ on $Z$. Assume the instrument assumptions hold and the constant-slope model is suitable. Find the 2SLS price coefficient and interpret a $10 increase.

Method. With one excluded instrument and the same controls in the first stage and reduced form, take the reduced-form coefficient on $Z$ divided by the first-stage coefficient on $Z$.

First-stage effect. Holding $X$ fixed, changing $Z$ by one unit changes fitted price by $24$, so the instrument raises price by $24.

Reduced-form effect. Holding $X$ fixed, changing $Z$ by one unit changes purchase probability by $-0.072$, or 7.2 percentage points.

2SLS coefficient. The ratio is $-0.072/24=-0.003$. The estimated causal slope is a decrease of 0.003 in purchase probability per dollar, or 0.3 percentage points per dollar.

$10 change and check. For a $10 increase, the estimated change is $10(-0.003)=-0.030$, a 3-percentage-point decrease. The corresponding $24 instrument-induced price change predicts $24(-0.003)=-0.072$, matching the reduced form. This interpretation is conditional on $X$; it does not say that every $10 price change for every customer necessarily has the same effect unless the constant-slope assumption is appropriate.

Validity Checks and Common Traps

The first stage should be examined before interpreting a 2SLS result. If the instrument barely changes price, the ratio divides by a small number and the estimate can be highly unstable. A large estimated 2SLS effect in that situation is not evidence of a large true price response. Report the first-stage relationship and use inference designed for instrumental-variable estimation; do not rely on an ordinary second-stage standard error.

Also ask how the instrument could affect demand other than through price. For example, if a hypothetical administrative rule also changes how a quote is presented, then it may affect purchase decisions directly. That would violate exclusion even if the rule strongly predicts price and is unrelated to demand shocks.

Common exam trap: Do not regress demand on observed price and call the coefficient 2SLS merely because an instrument is available. The ordinary price coefficient can remain confounded. The 2SLS calculation uses instrument-supported price variation. For the one-instrument ratio, divide the reduced-form effect on demand by the first-stage effect on price—not the other way around—and check that the denominator is nonzero.

Worked Example: A Direct Effect Invalidates the Ratio

Suppose a hypothetical binary instrument $Z$ occurs for half the observations, so $\operatorname{Var}(Z)=0.25$. It raises average quoted price by $12, giving $\operatorname{Cov}(Z,P)=0.25(12)=3$. Suppose the true causal price slope is $-0.004$ per dollar, but the instrument also directly reduces purchase probability by 0.02 through a change in quote presentation. Calculate the observed Wald ratio and explain whether it is a valid causal estimate.

Method. The direct effect violates exclusion. To show its impact, write the outcome model as $Y=\alpha+\beta P+\gamma Z+\varepsilon$, with $\beta=-0.004$, $\gamma=-0.02$, and $\operatorname{Cov}(Z,\varepsilon)=0$.

Demand covariance. The price pathway contributes $\beta\operatorname{Cov}(Z,P)=(-0.004)(3)=-0.012$. The direct pathway contributes $\gamma\operatorname{Var}(Z)=(-0.02)(0.25)=-0.005$. Therefore $\operatorname{Cov}(Z,Y)=-0.012-0.005=-0.017$.

Observed ratio. The Wald ratio is $-0.017/3=-0.005666\ldots$, or approximately $-0.0057$ probability per dollar.

Conclusion and check. This ratio is not a valid estimate of the stated price slope of $-0.004$, because it includes the direct effect of $Z$. Indeed, $(-0.004)(3)=-0.012$ is only the price-related covariance; the additional $-0.005$ comes from the prohibited direct pathway. Reversing the ratio would not fix the problem. A credible causal interpretation requires an instrument whose effect on demand operates through price alone.

From a Price Slope to an Elasticity

A 2SLS coefficient is a price slope in the units of the outcome and price; it is not automatically an elasticity. For a linear conditional mean $E[Y\mid P,X]$, the local price elasticity at a specified price and expected outcome is the slope multiplied by price and divided by the expected outcome. If the outcome is a purchase indicator, this expression uses the modeled purchase probability. State the evaluation point and the outcome scale when converting a slope to an elasticity. The causal interpretation still depends on the instrument assumptions and the demand model.

Key takeaway: 2SLS isolates price variation predicted by an instrument. The coefficient is interpretable as a causal price response only when the instrument is relevant, conditionally unrelated to demand shocks, excluded from demand except through price, and used with an appropriate model. A strong first stage alone does not establish validity.

Check Your Understanding

For each problem, identify the relevant instrumental-variable calculation and check the assumptions stated in the problem.

In a hypothetical study, a binary instrument raises average quoted price from $65 to $75 and lowers purchase rates from 0.47 to 0.42. Assuming a valid instrument and a constant slope, find the price slope and the implied change for a $10 increase. (Answer: $-0.005$ probability per dollar; $-0.05$, or a 5-percentage-point decrease, for $10.)
A controlled first stage has coefficient $15$ on $Z$, while the reduced-form regression of purchase probability on $Z$ and the same controls has coefficient $-0.045$. Under the valid-instrument and constant-slope assumptions, what is the 2SLS price coefficient, and what is the estimated change for a $20 increase? (Answer: $-0.003$ probability per dollar; $-0.06$, or a 6-percentage-point decrease, for $20.)
After controlling for $X$, the covariance between the residualized instrument and residualized price is $4$, while the covariance between the residualized instrument and residualized demand is $-0.012$. Compute the single-instrument slope ratio. (Answer: $-0.003$ probability per dollar.)
An instrument raises fitted price by only $1 and changes purchase probability by $-0.008$. Compute the ratio and state the main concern. (Answer: $-0.008$ probability per dollar; the very small first-stage price change makes the estimate potentially weak-instrument-sensitive.)
A binary instrument has variance $0.25$, $\operatorname{Cov}(Z,P)=2$, and the true price slope is $-0.003$. It also directly reduces demand by 0.016, so $\gamma=-0.016$ in the outcome model. Assuming no other violation, calculate the observed Wald ratio and explain its status. (Answer: $\operatorname{Cov}(Z,Y)=(-0.003)(2)+(-0.016)(0.25)=-0.010$, so the ratio is $-0.010/2=-0.005$ probability per dollar; it is not a valid causal price estimate because exclusion fails.)

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