Micro Moments for Demand Estimation

Why Add Consumer-Level Moments?

Market shares summarize how many consumers choose each product, but they do not reveal which consumers choose them. Two demand models can fit the same aggregate shares while implying different patterns across income, age, or other observed characteristics. Consumer-level data provide additional targets for these patterns. In differentiated-product demand estimation, those targets are called micro moments.

A micro moment can describe, for example, the fraction of high-income consumers choosing a product, the fraction of a demographic group choosing the outside option, or average expenditure within a group. Such moments can constrain taste parameters that are difficult to learn from market shares alone. They supplement, rather than replace, the market-level moments used in random-coefficient demand estimation.

Definition: A micro moment is a population expectation or conditional expectation involving consumer-level outcomes and characteristics, matched to its model-implied counterpart. Its moment residual is the observed target minus the model prediction. A collection of such residuals can be combined with market-level residuals in a single moment vector.

Model Predictions and Observed Targets

In market \(t\), let \(j=1,\ldots,J_t\) index inside products and let \(j=0\) denote the outside option. Consumer \(i\) has observed demographic characteristics \(z_{it}\). In a random-utility model, the consumer chooses the option with the highest utility. Write the utility of product \(j\) as

u_{ijt}=\delta_{jt}+\mu_{ijt}(\theta)+\varepsilon_{ijt},

where \(\delta_{jt}\) is mean utility, \(\mu_{ijt}(\theta)\) captures consumer-specific utility components and taste parameters, and \(\varepsilon_{ijt}\) is a choice shock. The parameter vector \(\theta\) includes the parameters being estimated. Given the model and the consumer’s observed characteristics, \(P_{ijt}(\theta)\) denotes the model-implied probability of choosing option \(j\). These probabilities sum to one over the inside products and the outside option.

Let \(Y_{it}\) be the observed choice and \(h(Y_{it},z_{it})\) a specified function of the choice and demographics. The observed micro target is an average of this function in the relevant consumer sample. Its model counterpart averages the same function using the model’s choice probabilities and the appropriate demographic distribution. For a binary indicator of choosing product \(j\) among consumers in group \(G\), the model prediction is the group average of \(P_{ijt}(\theta)\), not the probability for an arbitrarily selected consumer.

m^{\mathrm{pred}}_{j\mid G}(\theta) = \frac{\sum_{i,t} w_{it}\,\mathbf{1}\{z_{it}\in G\}\,P_{ijt}(\theta)} {\sum_{i,t} w_{it}\,\mathbf{1}\{z_{it}\in G\}}, \qquad g^{\mathrm{micro}}_{j\mid G}(\theta) = m^{\mathrm{obs}}_{j\mid G}-m^{\mathrm{pred}}_{j\mid G}(\theta).

Here \(w_{it}\) are the survey or population weights, \(\mathbf{1}\{\cdot\}\) is an indicator, and the denominator makes this a conditional group average. If consumers are sampled with unequal probabilities, weights should reflect the target population. The observed target and model prediction must refer to the same population, markets, demographic definition, and weighting scheme.

The same construction accommodates expenditure. If a consumer chooses at most one product and pays its price, expenditure can be written as \(E_{it}=p_{Y_{it},t}\) for an inside choice and zero for the outside option. The model prediction conditional on demographics is \(\sum_{j=1}^{J_t}p_{jt}P_{ijt}(\theta)\), averaged over consumers in the target group. More general expenditure definitions require using the same outcome definition in both the data and the model.

Worked Example: A High-Income Choice Moment

Suppose a hypothetical market has three options: product A, product B, and the outside option. For one high-income consumer, the utilities used to calculate the model choice probabilities are \(0.8\), \(0.2\), and \(0\), respectively. Assume the standard multinomial logit choice probabilities apply to these utilities. In a hypothetical consumer survey, the observed fraction of high-income consumers choosing A is \(0.5500\). Calculate the model prediction and the micro residual.

The probability of choosing A is its exponentiated utility divided by the sum for all three options:

P_A = \frac{\exp(0.8)} {\exp(0.8)+\exp(0.2)+\exp(0)} = \frac{2.2255409}{2.2255409+1.2214028+1} = \frac{2.2255409}{4.4469437} \approx 0.5005.

For this example, suppose the high-income group’s model prediction is represented by that probability. The observed-minus-predicted micro residual is

g^{\mathrm{micro}} = 0.5500-0.5004653 \approx 0.0495.

Thus, the observed high-income choice share for A exceeds the model prediction by about \(0.0495\), or 4.95 percentage points. The unrounded probability is used in the subtraction; subtracting the rounded display value \(0.5005\) would give a slightly different last digit.

How Demographics Can Identify Taste Parameters

Demographic variables contribute identifying variation when their relationship to choice differs across groups in a way the model attributes to taste parameters. A simple illustration uses product A and the outside option. Let \(z\) equal zero for the low-income group and one for the high-income group, and suppose the utility difference between A and the outside option is \(\alpha+\gamma z\). Under the binary logit model, the probability of choosing A is the logistic transformation of that utility difference. Here \(\gamma\) measures how the utility difference changes with the demographic indicator, conditional on the model specification.

This example uses a binary choice to make the identification logic transparent. In a differentiated-products model, the same logic applies to choice probabilities across several inside products and the outside option, with demographic characteristics interacting with product attributes or taste parameters.

Worked Example: Recovering a Demographic Taste Difference

In a hypothetical sample with equal numbers of low- and high-income consumers, 40% of the low-income group choose A and 70% of the high-income group choose A. Suppose the binary logit specification above is correct, both groups face the same alternatives, and their group-specific observed choice fractions estimate the corresponding population probabilities. Find the parameter values implied by these two targets.

The binary logit probability satisfies \(P(A\mid z)=\exp(\alpha+\gamma z)/(1+\exp(\alpha+\gamma z))\). Inverting it gives the log odds. For \(z=0\), the target probability is \(0.40\), so

\alpha = \log\left(\frac{0.40}{1-0.40}\right) = \log\left(\frac{2}{3}\right) \approx -0.4055.

For \(z=1\), the target probability is \(0.70\), so the high-income log odds are

\alpha+\gamma = \log\left(\frac{0.70}{1-0.70}\right) = \log\left(\frac{7}{3}\right) \approx 0.8473.

Subtracting the low-income equation from the high-income equation gives

\gamma = 0.8473-(-0.4055) \approx 1.253.

The two conditional choice targets imply an intercept of approximately \(-0.4055\) and a high-income utility difference approximately \(1.253\) units larger than the low-income utility difference. This is a model-based association, not by itself a causal effect of income. If only the pooled share were observed, it would provide one restriction for the two unknown parameters and generally would not determine both.

In richer demand models, micro moments can be selected to target particular dimensions of heterogeneity: for example, product choice by income group or spending by age group. Demographic variation helps identify a taste parameter only when the model maps that parameter into observable differences that are not indistinguishable from changes in other parameters. A good fit to a micro target is not enough to establish identification; the mapping from parameters to the full set of moments must contain sufficiently distinct information.

Combining Micro and Market-Level Moments

Let \(g^{\mathrm{market}}(\theta)\) collect the market-level moment residuals, such as instrument-weighted discrepancies involving observed market shares and the structural demand model. Let \(g^{\mathrm{micro}}(\theta)\) collect consumer-level residuals. The joint moment vector stacks both sets:

g(\theta) = \begin{bmatrix} g^{\mathrm{market}}(\theta)\\ g^{\mathrm{micro}}(\theta) \end{bmatrix}.

The restrictions require the relevant population moments to be zero at the true parameter under the model and the maintained sampling and instrument assumptions. In estimation, the stacked sample moments are evaluated at candidate parameter values and used together, with a weighting matrix, in the GMM criterion. The weighting matrix affects how moment discrepancies contribute to estimation; it does not make an invalid or mismatched target valid.

Worked Example: Constructing a Joint Moment Vector

Consider a hypothetical market with two equally sized demographic groups, each containing 500 consumers. The model predicts that low-income consumers choose A with probability \(0.30\) and B with probability \(0.20\); high-income consumers choose A with probability \(0.60\) and B with probability \(0.10\). The remaining probability in each group is for the outside option. Suppose the observed market shares are \(0.47\) for A and \(0.14\) for B. A hypothetical consumer survey records 310 high-income consumers choosing A and 90 low-income consumers choosing B. For illustration, define the market-level moments as instrument times share residual, with fixed instruments \(z_A=2\) and \(z_B=1\). Define the two micro moments as the high-income A share and low-income B share, respectively.

First calculate the model market shares by averaging across the equally sized groups:

s_A^{\mathrm{pred}}=\frac{0.30+0.60}{2}=0.45, \qquad s_B^{\mathrm{pred}}=\frac{0.20+0.10}{2}=0.15.

The observed-minus-predicted market-share residuals are \(0.47-0.45=0.02\) for A and \(0.14-0.15=-0.01\) for B. Applying the stated instruments gives the market-level moments \(2(0.02)=0.04\) and \(1(-0.01)=-0.01\).

The observed high-income A share is \(310/500=0.62\), compared with a model prediction of \(0.60\), so its micro residual is \(0.62-0.60=0.02\). The observed low-income B share is \(90/500=0.18\), compared with a model prediction of \(0.20\), so its micro residual is \(0.18-0.20=-0.02\). The joint vector, in the order market A, market B, high-income A, low-income B, is therefore

g(\theta) = \begin{bmatrix} 0.04\\ -0.01\\ 0.02\\ -0.02 \end{bmatrix}.

These targets are internally compatible with the hypothetical data: the observed market shares correspond to 470 choices of A and 140 of B out of 1,000 consumers, while the micro counts specify subsets of those choices. The vector summarizes discrepancies at this candidate parameter value; it is not by itself an estimate or a standard error. In an actual estimation, all moments would be computed from the same defined population or appropriately aligned samples.

Why the Moment Restrictions Are Justified

The justification for a correctly specified micro moment is the same basic population-expectation logic used for other moment restrictions. Let \(h(Y,Z)\) be the outcome function defining a target. Conditional on observed characteristics \(Z=z\), the choice model implies

E[h(Y,Z)\mid Z=z] = \sum_{j=0}^{J_t} h(j,z)P_{jt}(z;\theta_0),

where \(\theta_0\) is the true parameter and \(P_{jt}(z;\theta_0)\) is the model choice probability for option \(j\). This equality follows by summing the possible outcomes \(j\), each weighted by its conditional probability. Averaging both sides over the same demographic distribution, or conditioning both sides on the same group, gives equality between the population target and its model counterpart.

If the consumer sample represents that target population and the relevant averages obey a law of large numbers, the sample target and the corresponding model average converge to these population expectations. Consequently, the observed-minus-predicted sample micro moment approaches zero at \(\theta_0\). If the survey is selected in a way that changes the demographic or choice distribution and the selection is not corrected, this argument need not hold. Nor does it hold if the model’s choice probabilities or the outcome definition are inconsistent with the data being matched.

Common Pitfalls and Practical Checks

Using the wrong denominator. A high-income choice share is the number of high-income consumers choosing the product divided by the number of high-income consumers in the target sample, not the total number of consumers.
Comparing different populations. A survey target and model prediction should use compatible market coverage, demographic definitions, and weights. A population-weighted prediction should not be compared uncritically with an unweighted convenience sample.
Confusing a probability with a group average. When consumers within a group have different predicted probabilities, the group prediction is their properly weighted average, not one representative consumer’s probability.
Overstating identification. Adding a micro moment provides another restriction, but it identifies a parameter only if its response to that parameter is distinguishable from its response to other parameters.
Reading a residual as a causal effect. A positive observed-minus-predicted residual indicates that the model underpredicts the stated target at the candidate parameters. It does not establish that the demographic characteristic caused the choice difference.

Micro moments matter because aggregate fit can conceal important differences in who chooses which products. Consumer-level restrictions can discipline heterogeneity and improve the model’s ability to reproduce observed choice patterns. Their value depends on a clear target definition, a credible link between the sample and the modeled population, and a parameterization that can generate the observed variation.

Key takeaway: Define the consumer-level target precisely, compute its model counterpart using the same demographic composition and weights, and stack its observed-minus-predicted residual with the market-level moments. Demographic moments can add identifying information, but only when the model and data support that interpretation.

Check Your Understanding

Answer each question using the definitions and calculations in this tutorial.

A survey target is the fraction of low-income consumers choosing the outside option. What denominator belongs in the observed target?
Why should a model prediction for a demographic group average probabilities across that group rather than use one consumer’s probability?
In the binary logit example, what would the high-income choice probability be if \(\alpha+\gamma=0\)?
Suppose an observed micro target is \(0.41\) and its model prediction is \(0.46\). What is the observed-minus-predicted residual, and what does its sign indicate?
Why can adding one micro moment help identify a taste parameter without guaranteeing identification?