How do you estimate price elasticity and its uncertainty?

For the observed product, market and period, what relative quantity change is associated with a 1% price change, with what uncertainty and over what support?

The unit is a product-market week. The data contain 24 weeks, prices from €10 to €11 and no promotions. Simulation horizons must remain close to this support.

The default target is β in E[ln(Q)|ln(P),S]=α+βln(P)+γS. Counterfactual demand Q(p) separately requires a credible identification design.

Use strictly positive quantity and price, time identifier, promotions, seasonality, competitor prices, availability, distribution and common drivers of price and demand. Declare each variable as pre- or post-treatment.

Assume an adequate log-log form, errors compatible with the chosen inference, no unmodelled break and, for causal reading, no residual confounding or justified exogenous variation. R² does not prove these assumptions.

The simple model is ln(Q)=α+βln(P)+ε. The reference model adds season index S. Promotion is constantly zero and must be excluded to avoid a no-variation column.

Validate 24 rows, log-transform price and quantity, estimate both OLS specifications, calculate residual standard error, 95% CI and exact change (1.05)^β−1. The script fails when expected support is absent.

Simple model: β=−2.4107, SE=0.1198. With seasonality: β=−2.4586, SE=0.0251, 95% CI [−2.5078, −2.4093]. Under constant elasticity, +5% price corresponds to −11.3039% conditional volume.

The conventional CI is conditional on the model and 24 observations. It includes neither functional-form error, endogeneity nor regime-change uncertainty. A real time series requires variance compatible with serial correlation.

Managers may change price in response to unobserved expected demand. Price and error then correlate; OLS is associational and potentially biased. MSC-P-020 addresses instrumental-variable identification.

Plot residuals and fits, compare specifications, inspect leverage and influence, test temporal stability, serial correlation and collinearity, document scenario support and investigate the price-setting rule.

Common errors are calling β causal without a design, reading −2.46 as units, applying β to a large change without exponentiation, controlling a post-price variable, ignoring zeros or extrapolating beyond support.

The estimate describes highly price-sensitive demand in this small constructed dataset. Its narrow precision reflects artificial regularity; it is neither an industry benchmark nor a guarantee of real response.

Use the associational estimate to compare nearby scenarios, show conditional volume sensitivity and decide what additional data or experiments are needed.

Do not set an optimal price, promise a sales change, transport β to another market or causally attribute volume decline to price without identification and economic constraints.

Include question, unit, support, associational/causal status, specification, variables, coefficient, SE, CI, diagnostics, sensitivity, bounded scenario, supported/unsupported decision and reproducible files.