Why Product Attributes Can Instrument for Price
In differentiated-product demand, a product’s price is often related to its unobserved demand quality. For example, firms may set higher prices for products they expect consumers to value more than the recorded attributes suggest. A demand regression that treats price as exogenous can then confound the effect of price with unobserved quality. Product attributes of other products in the market offer a potential source of price variation: they can affect firms’ competitive environment and pricing incentives without directly appearing in the focal product’s utility.
These instruments are not automatically valid because their values are observable or because they belong to other products. Their relevance depends on how characteristics and competition shape prices; their exclusion depends on whether those characteristics are unrelated to the focal product’s unobserved demand quality, conditional on the model’s controls. Both claims require economic justification.
Each sum is computed separately for every product and market. It can be calculated characteristic by characteristic. For instance, if \(x_{jt}\) contains horsepower and fuel economy, the resulting instruments contain sums of other products’ horsepower and sums of their fuel economy. When market sizes or the number of products vary substantially, researchers may also consider averages or include product-count information. Changing a sum to an average changes the instrument’s scale and economic interpretation; it does not by itself establish validity.
Constructing the Aggregates
Worked Example: Own-Firm and Rival-Firm Sums
Consider one hypothetical market with two firms and four products. Each product has horsepower and fuel economy, measured in miles per gallon (mpg).
| Product | Firm | Horsepower | Fuel economy (mpg) |
|---|---|---|---|
| A1 | A | 150 | 32 |
| A2 | A | 180 | 28 |
| B1 | B | 130 | 36 |
| B2 | B | 210 | 25 |
For A1, the other product from firm A is A2, so its own-firm instrument is \((180,28)\). The products from the rival firm are B1 and B2, so its rival-firm instrument is \((130+210,\;36+25)=(340,61)\).
For A2, the own-firm instrument is A1’s attributes, \((150,32)\), and the rival-firm instrument is again \((340,61)\). For B1, the own-firm instrument is B2’s attributes, \((210,25)\), while the rival-firm instrument is \((150+180,\;32+28)=(330,60)\). For B2, the corresponding values are own-firm \((130,36)\) and rival-firm \((330,60)\).
The calculations make the distinction clear: the own-firm sum excludes the focal product and includes the firm’s other products; the rival-firm sum includes products sold by all other firms. In this market, rival sums are identical for products from the same firm, while own-firm sums vary across the firm’s products.
The vectors can supply multiple candidate instruments, but not every characteristic aggregate will be useful. A characteristic that barely varies across products or markets may contribute little predictive information. Aggregates can also be redundant: if two columns of an instrument matrix move almost identically, adding both may provide little independent variation.
Relevance: A Pricing Channel, Not a Guarantee
An instrument is relevant for price when it predicts price after accounting for the included exogenous regressors and other controls. Differentiation-based instruments have a plausible pricing channel. Characteristics of other products affect the competitive setting: how closely products substitute for one another, the strength of rival offerings, and the incentives firms face when choosing prices. Attributes of a firm’s other products can also matter through the firm’s product portfolio and interactions among its offerings.
This logic does not imply a fixed direction. A more attractive rival product might constrain a focal firm’s price in one setting, while changes in substitution patterns, costs, or product positioning could lead to a different response in another. Nor does a plausible channel prove relevance in the available data. Relevance is a conditional relationship, not merely a story about competition.
Worked Example: A Small-Market Relevance Diagnostic
Return to the four hypothetical products above, and suppose their prices are A1: $9, A2: $11, B1: $10, and B2: $12. Use horsepower alone to illustrate how the aggregate instruments vary with price. This is a descriptive diagnostic, not a formal first-stage analysis.
The own-firm horsepower sums are 180, 150, 210, and 130. Their mean is \((180+150+210+130)/4=167.5\). The mean price is \((9+11+10+12)/4=10.5\). The sum of products of deviations is $$ (180-167.5)(9-10.5)+(150-167.5)(11-10.5)+(210-167.5)(10-10.5)+(130-167.5)(12-10.5)=-105. $$ Thus their sample covariance with price is \(-105/(4-1)=-35\) horsepower-dollar units.
The rival-firm horsepower sums are 340, 340, 330, and 330, with mean 335. The corresponding deviation-product sum is $$ (340-335)(9-10.5)+(340-335)(11-10.5)+(330-335)(10-10.5)+(330-335)(12-10.5)=-10. $$ Their sample covariance with price is \(-10/3=-3.333\) horsepower-dollar units. Both aggregates vary in this example, but the raw covariances do not establish their conditional relevance. That requires examining how they predict price after accounting for the included characteristics and controls, using data with adequate variation across products and markets.
The example also shows why instrument strength should not be inferred from the construction rule alone. For instance, rival sums are constant within each firm in this market, so their variation comes only from differences between firms. In a larger data set, useful variation may come from changes in the market’s product mix over time, but such changes must still be compatible with the exclusion restriction.
Exclusion: What Must Be Unrelated to Demand Quality
A common demand representation writes product utility’s systematic component as \(x_{jt}'\beta-\alpha p_{jt}\), with unobserved demand quality \(\xi_{jt}\). The demand residual \(\xi_{jt}\) captures determinants of product demand not represented by the observed regressors. If price responds to that quality, then price is endogenous. Candidate instruments \(z_{jt}\) must predict price and satisfy an exclusion condition: conditional on the included exogenous variables and controls, they must be uncorrelated with \(\xi_{jt}\).
This condition has a direct moment implication. If \(z_{jt}\) satisfies the condition, multiplying the demand residual by \(z_{jt}\) and taking its conditional expectation gives zero. If the instrument is correlated with unobserved demand quality, the moment does not vanish, even if the instrument is strongly associated with price.
Worked Example: A Threat to Exclusion
Suppose a hypothetical market has a demand shock that raises consumers’ preference for fuel-efficient vehicles. If this shock is not captured by the included controls, products’ fuel-economy attributes may be systematically related to the unobserved component of demand. A rival-firm instrument formed by summing rival fuel economy could then covary with the focal product’s unobserved demand quality, for example if firms choose product attributes in response to that market-wide preference.
In this case, the proposed instrument may still predict price, but its exclusion condition is questionable: the market-level demand shock can connect the rival sum to \(\xi_{jt}\). Adding appropriate market-level controls could address some common variation if those controls are part of a defensible model. It cannot be assumed, however, that controls remove every source of correlation. The researcher must justify the remaining conditional exclusion assumption.
Related concerns arise when products from the same firm share unobserved quality or when firms choose their product portfolios in response to anticipated demand. Those mechanisms can make own-firm aggregates correlated with a focal product’s demand residual. Rival-firm aggregates are not automatically safe either: competitors may respond to common market shocks or similar information. Firm and market structure, timing, controls, and the process determining product characteristics all matter for the argument.
Comparing Candidate Instrument Sets
A rival-only set uses aggregates of characteristics from other firms’ products as excluded instruments. A combined set adds aggregates from the same firm’s other products. The combined set can contribute additional predictive variation, but it also adds assumptions and can introduce redundant or weak columns. The focal product’s own characteristics ordinarily remain included regressors and are not counted as excluded instruments simply because they are also valid sources of exogenous variation.
| Candidate set | Potential advantage | Key concern |
|---|---|---|
| Rival-firm aggregates only | Captures variation in the competing product mix and competitive environment. | May vary little within firm or be related to common market demand shocks. |
| Own-firm and rival-firm aggregates | Adds portfolio variation that may predict firms’ pricing incentives. | May be redundant or weak; same-firm portfolio choices may share unobserved demand quality. |
| Included product characteristics only | Uses observed attributes that enter demand directly. | These are controls, not excluded sources of price variation for identifying the price coefficient. |
The appropriate comparison is not “more instruments is always better.” A defensible set must have a credible exclusion argument and sufficient conditional predictive variation. In applications, researchers examine first-stage evidence, consider whether instruments add distinct variation, and report how estimates respond to plausible alternative sets. A large collection of similar aggregates can make inference fragile rather than informative.
Record product characteristics and firm identity for each product in each market.
Exclude the focal product from its own-firm sum, and include products from all other firms in the rival-firm sum.
Retain the focal product’s observed characteristics as demand controls; assess the other-product aggregates as candidate excluded instruments.
Ask whether the aggregates predict price conditional on controls and whether they remain unrelated to unobserved demand quality.
Common Mistake and Key Takeaway
A common mistake is to argue, “These are other products’ characteristics, so they cannot affect the focal product’s unobserved demand.” That conclusion does not follow. The characteristics can be related to shared market shocks, firms’ responses to anticipated demand, or unobserved features common across a portfolio. The correction is to state the conditional exclusion restriction and defend it using the economic setting; the fact that an attribute belongs to another product is not enough.
Differentiation-based instruments turn observed product portfolios into candidate sources of price variation. Own-firm and rival-firm aggregates encode different aspects of the market and may have different relevance and exclusion risks. Their usefulness depends on the data and assumptions, not simply on whether the sums can be calculated.
Check Your Understanding
Answer each question by referring to the definitions and assumptions in this tutorial.
- For a focal product, which products enter its own-firm aggregate, and which enter its rival-firm aggregate?
- Why is the focal product’s own characteristic usually an included demand regressor rather than an excluded instrument?
- What economic mechanism could make rival-product characteristics predict the focal product’s price?
- Give one reason why a rival-firm characteristic aggregate could violate the exclusion restriction.
- In the four-product example, why do the rival horsepower sums vary only between firms rather than between products of the same firm?
- Why does a nonzero raw covariance between an aggregate and price not, by itself, establish instrument relevance for the demand equation?