Instrumental Variables and 2SLS: Why OLS Is Biased When Price Is Endogenous
October 2026 | Applied Economics | Econometrics / Causal Inference | ← Back to Blog
An OLS regression of demand on price can produce a clean, statistically significant coefficient that is still wrong. This post explains why price endogeneity causes that, what a valid instrument actually has to satisfy, and how two-stage least squares uses a valid instrument to identify the causal effect.
The Core Problem: Endogenous Price
A regression of demand on price assumes that price is uncorrelated with whatever is left in the error term, the part of demand the model cannot explain. In many pricing and demand settings, that assumption fails. Price is often correlated with something the researcher cannot observe but the firm can.
Consider product quality. A firm knows how good its product actually is. A researcher estimating demand from transaction data typically does not observe quality directly. It does not show up as a column in the dataset. If the firm sets a higher price for a higher-quality product, then in the data, high price and high demand move together for a reason that has nothing to do with how consumers respond to price itself. They move together because both are driven by the same unobserved quality.
Ordinary least squares cannot tell these two sources of correlation apart. It estimates one coefficient on price, and that coefficient absorbs both the true causal effect of price on demand and the correlation between price and the unobserved quality term. The result is an estimate that no longer isolates how consumers actually respond to price changes. This is what endogeneity means: the regressor of interest is correlated with the error term, and the exogeneity assumption OLS depends on no longer holds.
"The coefficient looked clean. It was also wrong, because price was endogenous."
What an Instrument Actually Needs to Satisfy
An instrument addresses the endogeneity problem by introducing a second source of variation in price, one that is unrelated to the unobserved demand shock. For a variable to work as a valid instrument, it has to satisfy two separate conditions.
The first is relevance. The instrument has to be correlated with price. This is a condition about predictive power, and it can be checked directly in the data. The first-stage regression of price on the candidate instrument and the other exogenous controls reveals whether the instrument provides enough independent variation in price to be useful.
The second is the exclusion restriction. The instrument must affect demand only through its effect on price, not through any other channel. Unlike relevance, exclusion cannot be verified directly in the data. There is no test that proves an instrument satisfies exclusion. It has to be argued, based on institutional knowledge of how the instrument operates in the setting being studied. A classic example is a cost shifter, something like an input cost that changes what a firm charges, without being tied to why consumers want more or less of the product in a given period. The argument for exclusion has to explain specifically why that cost shifter has no direct channel into demand other than through price.
A credible exclusion argument is harder to make, and it is the part that gets challenged.
How Two-Stage Least Squares Recovers the Effect
Once a valid instrument is in hand, two-stage least squares, 2SLS, is the standard way to use it.
The first stage regresses price on the instrument, along with any other exogenous controls already in the model. This produces a predicted value of price, the portion of price explained by the instrument and the model's other exogenous variables. Because the instrument satisfies exclusion, this predicted price is no longer correlated with the unobserved demand shock that was contaminating the original OLS estimate.
The second stage regresses demand on this predicted price, rather than on the actual observed price. The resulting coefficient reflects the variation in price explained by the instrument and the model's other exogenous variables. Under the exclusion restriction, that variation has nothing to do with the firm's private information about quality or any other unobserved demand shifter. Under the relevance and exclusion assumptions, that coefficient is a consistent estimate of the causal effect of price on demand.
The mechanics are straightforward. The identifying assumptions behind them are not. Two-stage least squares does not make an invalid instrument valid. It uses whatever variation in price the instrument provides, and the quality of the final estimate depends entirely on whether that variation really is exogenous.
When This Breaks: Weak Instruments
An instrument that is only weakly correlated with price creates a different problem. If the first-stage correlation is small, the predicted price used in the second stage carries very little real variation, and very little information to work with. The resulting IV estimate can be imprecise, and in finite samples, a weak instrument can produce an estimate biased in the same direction as OLS, sometimes even more severely.
This is not a hypothetical concern. It is common enough that checking instrument strength, usually through the first-stage F-statistic, is a standard step before reporting an IV result. An instrument that satisfies relevance and exclusion in principle but barely moves price in practice can still leave you with an estimate you cannot trust.
Why This Matters Beyond the Regression
In applied demand estimation, the choice of instrument is rarely a settled, mechanical step. It is usually the most contested part of the analysis. A strong first-stage F-statistic demonstrates relevance, but it says nothing about exclusion. The harder, more important task is explaining why the instrument has no direct channel into demand other than through price, and defending that argument against the specific institutional details of the market being studied.
A regression that reports a statistically significant price coefficient without that argument has not actually solved the endogeneity problem. It has only produced a number that looks resolved.
Further Reading
Angrist, J. and Pischke, J.S. (2009). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press. Standard reference for identification strategies including instrumental variables.
Berry, S. (1994). Estimating Discrete-Choice Models of Product Differentiation. RAND Journal of Economics, 25(2), 242–262. A foundational reference for the unobserved product-quality mechanism behind price endogeneity in differentiated-product demand estimation.
Stock, J.H. and Yogo, M. (2005). Testing for Weak Instruments in Linear IV Regression. In Identification and Inference for Econometric Models. Cambridge University Press. The standard diagnostic for instrument strength.
