Estimating Simple Logit Demand with IV

From Choice Shares to an Estimable Demand Equation

In a differentiated-products market, prices may respond to demand conditions that are not fully observed by the analyst. A product with unusually high unobserved appeal may command a higher price and also sell more. A regression of sales or market shares on prices can therefore confuse the effect of price with the effect of unobserved appeal. The simple logit model provides a way to express observed shares as a linear demand equation, while instrumental variables (IV) address the endogeneity of price.

Consider markets indexed by $t$ and products indexed by $j$. Let $s_{jt}$ be product $j$'s share of all potential market purchases, including the outside option, and let $s_{0t}$ be the outside-option share. Let $p_{jt}$ be price, $x_{jt}$ a vector of observed product characteristics, and $\xi_{jt}$ unobserved product appeal. The outside option represents not buying any of the inside products.

Definition: The linearized simple-logit demand equation is

y_{jt} = \log(s_{jt})-\log(s_{0t}) = x_{jt}'\beta-\alpha p_{jt}+\xi_{jt},

where $y_{jt}$ is the log share relative to the outside option, $\beta$ is the vector of characteristic effects, and $\alpha$ is the price-sensitivity parameter. A demand interpretation typically requires $\alpha>0$, so the price coefficient is $-\alpha<0$.

The share inversion requires positive observed shares for each inside product and a positive outside share. Shares must use the same market denominator: the number of potential purchases, not just the number of purchases of inside products. The log ratio $y_{jt}$ is the observed utility index relative to the outside option under the simple-logit choice-probability model.

Why Price Requires Instrumental Variables

The regression error $\xi_{jt}$ captures demand-relevant determinants not included in $x_{jt}$. If firms set prices in response to those determinants, price will generally be correlated with $\xi_{jt}$. For example, a firm may charge more for a product it knows consumers value unusually highly. In that case, the conditional relationship between price and the regression error is not zero, and ordinary least squares does not isolate the price effect.

An instrument $z_{jt}$ must provide variation in price that is unrelated to the demand shock, after accounting for included controls. In this setting, excluded instruments are variables that shift price but have no direct effect on demand once $x_{jt}$ and other specified controls are held fixed. Differentiation-based instruments for product prices are one possible source of such variation, but their validity depends on the economic setting and exclusion argument, not on their name or construction alone.

IV conditions: The demand equation must be correctly specified for the intended interpretation; the excluded instruments must be orthogonal to $\xi_{jt}$ conditional on included controls; and the instruments must predict the endogenous price variation left after those controls. In a linear IV model, identification additionally requires enough independent instrument variation to distinguish the price coefficient from the coefficients on the included regressors.

Write $w_{jt}=(x_{jt}',p_{jt})'$ for the full regressor vector, and let $Z$ contain the included exogenous regressors and the excluded instruments. Including the exogenous regressors among the instruments ensures that the moment conditions also account for their relationship with the outcome. In population notation, the identifying conditions are $E[Z_{jt}\xi_{jt}]=0$ and a full-rank condition on the relationship between $Z_{jt}$ and $w_{jt}$. With one endogenous price and one excluded instrument, the simplest IV estimate uses the part of price variation associated with that instrument, conditional on the included regressors.

Estimating the Price Coefficient

Two-stage least squares (2SLS) is a standard estimator. In the first stage, regress price on the included exogenous variables and the excluded instruments. In the second stage, use the price variation predicted by that regression, together with the included regressors, to estimate the demand equation. The first stage is not a separate causal model of demand; it isolates the instrument-predicted component of price for the IV calculation.

\widehat{w}_{jt}=P_Z w_{jt}, \qquad \widehat{\theta}_{2SLS} =(W'P_ZW)^{-1}W'P_Zy, \qquad P_Z=Z(Z'Z)^{-1}Z'.

Here $W$ is the matrix of demand regressors, $y$ is the vector of log-share outcomes, and $P_Z$ projects onto the columns of $Z$. In practice, the matrix formula requires the relevant inverses and rank conditions to exist. The estimated coefficient on price is the estimate of $-\alpha$; report price sensitivity as $\hat{\alpha}$, the negative of that coefficient. If the estimate of the price coefficient is positive, the fitted demand relationship has the wrong sign for the usual downward-sloping interpretation and should prompt careful investigation rather than an automatic sign change.

Worked Example: Inverting Hypothetical Market Shares

In a hypothetical market, three products have shares $s_A=0.20$, $s_B=0.30$, and $s_C=0.10$, measured among all potential purchases. The remaining share is the outside option. Calculate the linearized-logit outcome for each product.

The outside share is $s_0=1-0.20-0.30-0.10=0.40$, which is positive. For product A, $y_A=\log(0.20)-\log(0.40)=\log(0.20/0.40)=\log(0.50)\approx-0.6931$.

For product B, $y_B=\log(0.30/0.40)=\log(0.75)\approx-0.2877$. For product C, $y_C=\log(0.10/0.40)=\log(0.25)\approx-1.3863$. These log share ratios—not the shares themselves—are the dependent-variable observations in the linearized demand equation. The calculation uses a common market denominator and a positive outside share, as required.

Worked Example: A Two-Stage IV Estimate After Controls

Suppose four hypothetical market-product observations have been residualized with respect to all included exogenous characteristics. Thus the entries below are the residualized price $p$, excluded instrument $z$, and demand outcome $y$; the included controls have already been accounted for. Estimate the price coefficient using $z$ as the single excluded instrument.

Observation	Residualized $z$	Residualized $p$	Residualized $y$
1	-1.5	-1	2
2	-0.5	0	0
3	0.5	0	0
4	1.5	1	-2

First stage: regress residualized price on residualized $z$. Both variables have mean zero, so the slope is $\sum z_i p_i/\sum z_i^2=[(-1.5)(-1)+(-0.5)(0)+(0.5)(0)+(1.5)(1)]/[2.25+0.25+0.25+2.25]=3/5=0.6000$. The fitted prices are therefore $\hat p=(-0.9,-0.3,0.3,0.9)$.

Second stage: regress residualized $y$ on $\hat p$. The slope is $\sum \hat p_i y_i/\sum \hat p_i^2=[(-0.9)(2)+(-0.3)(0)+(0.3)(0)+(0.9)(-2)]/[0.81+0.09+0.09+0.81]=-3.6/1.8=-2.000$. Thus the estimated price coefficient is $-2.000$, and the implied price-sensitivity estimate is $\hat\alpha=2.000$ in the price units used. This calculation illustrates the mechanics of IV; whether $z$ is valid still depends on its exclusion and orthogonality assumptions.

From Price Sensitivity to Elasticities

An elasticity expresses a proportional change in share for a proportional change in price. In the simple-logit model, holding other products' characteristics and prices fixed, product $j$'s share has the form $s_j=\exp(\delta_j)/(1+\sum_k\exp(\delta_k))$, where $\delta_j=x_j'\beta-\alpha p_j+\xi_j$. Differentiating with respect to its own price gives

\frac{\partial s_j}{\partial p_j} =-\alpha s_j(1-s_j), \qquad \varepsilon_{jj} =\frac{\partial s_j}{\partial p_j}\frac{p_j}{s_j} =-\alpha p_j(1-s_j).

The own-price elasticity is negative when $\alpha>0$, price is positive, and the share is below one. Its magnitude depends on price and share as well as on price sensitivity. The formula uses price in the same units used to estimate $\alpha$; changing price units changes the numerical value of $\alpha$, but not the implied elasticity when the conversion is done consistently.

Worked Example: Calculating an Own-Price Elasticity

Suppose the estimated price sensitivity is $0.0800$ per dollar, a product's price is $\$10.00$, and its share is $0.200$. Under the simple-logit model, its own-price elasticity is $\hat\varepsilon_{jj}=-\hat\alpha p_j(1-s_j)=-(0.0800)(10.00)(1-0.200)=-(0.800)(0.800)=-0.6400$.

At these values, a 1% increase in this product's price is associated, locally in the model, with an approximately 0.64% decrease in its share, holding other products' prices and characteristics fixed. This is an elasticity interpretation, not a statement that the share falls by 0.64 percentage points. The price sensitivity is per dollar, so multiplying it by a price expressed in dollars is consistent.

Instrument Choice and What the Estimate Uses

Different valid instruments can identify price sensitivity from different components of price variation. In a finite sample, those components need not produce identical estimates. Instruments can also differ in relevance: an instrument with little conditional relationship to price supplies little identifying variation. An estimate based on a weak relationship can be unstable, even if the instrument's exclusion argument is plausible.

The following calculation compares two candidate instruments using hypothetical data that have already been residualized with respect to included controls. The single-instrument IV slope is the covariance of the instrument with the outcome divided by its covariance with price. For demand, the slope estimates the price coefficient $-\alpha$.

Worked Example: Comparing Two Candidate Instruments

Suppose the residualized price values are $(-1,0,0,1)$ and the residualized demand outcomes are $(1.8,0.3,-0.4,-1.7)$. Candidate instruments are $z_1=(-1.5,-0.5,0.5,1.5)$ and $z_2=(-1,1,-1,1)$. Both are centered, so their cross-products can be used directly for the scalar IV slopes.

For $z_1$, the cross-product with price is $\sum z_{1i}p_i=(-1.5)(-1)+(1.5)(1)=3$. The cross-product with the outcome is $\sum z_{1i}y_i=(-1.5)(1.8)+(-0.5)(0.3)+(0.5)(-0.4)+(1.5)(-1.7)=-5.6$. Thus the IV price coefficient is $-5.6/3=-1.867$, giving $\hat\alpha=1.867$.

For $z_2$, the cross-product with price is $(-1)(-1)+(1)(1)=2$, and with the outcome it is $(-1)(1.8)+(1)(0.3)+(-1)(-0.4)+(1)(-1.7)=-2.8$. The IV price coefficient is $-2.8/2=-1.400$, giving $\hat\alpha=1.400$. The estimates differ because the instruments weight the observed price-outcome variation differently in this sample. These calculations do not establish that either instrument is valid; that requires a substantive argument that it predicts price but has no direct relationship with the demand shock after controls.

Interpretation, Assumptions, and Common Pitfalls

A fitted price coefficient is not automatically a causal demand effect. Its IV interpretation depends on the demand equation and instrument conditions being credible. In particular, a variable that predicts price because it also predicts unobserved product appeal does not satisfy the exclusion condition. Adding more instruments does not repair invalid instruments. Nor does a large first-stage relationship by itself establish that the instruments are exogenous.

The simple-logit specification also gives the coefficient a specific structural meaning: price enters the systematic utility index linearly with a common coefficient. The estimated $\alpha$ therefore feeds directly into the model's implied elasticities. Elasticities are model-based implications of that coefficient and the shares, not separate facts established by the IV calculation alone. Their interpretation also depends on the accuracy of measured shares, prices, and the assumed outside option.

Common mistake: “The instrument is correlated with price, so the IV estimate is valid.” Relevance is only one requirement. The correction is to establish both conditional relevance and exclusion: the instrument must predict price variation after controls and be uncorrelated with the unobserved demand shock. A strong but invalid instrument can still produce a misleading demand estimate.

Construct the outcome.
Compute each inside product's log share relative to the positive outside share using consistent market shares.

Specify the demand equation.
Include observed characteristics and price, and state why price may be correlated with unobserved demand appeal.

Defend the instruments.
Explain the economic source of price variation and why the excluded instruments are relevant and orthogonal to demand shocks conditional on controls.

Estimate and translate.
Use IV for the price coefficient, take its negative to obtain $\hat\alpha$, and compute elasticities using prices and shares in consistent units.

Check Your Understanding

Work through these questions using the definitions and calculations above.

Why must the outside-option share be positive to construct the log share ratio?
In the linearized demand equation, what does the estimated price coefficient represent, and how is it related to $\alpha$?
State the two distinct requirements an excluded instrument must satisfy for IV estimation.
If price is recorded in cents instead of dollars, what must happen to the numerical price-sensitivity coefficient for the implied elasticity to remain unchanged?
Two candidate instruments give different IV estimates. Why does that difference alone not establish which instrument is valid?