You are hired to analyse the price elasticity of a fashion retailer. You carry out different log-log models to analyse the impact of two predictors on monthly sales units (log-transformed) for jeans using a sample of 256 stores. The input variables/ regressors are (average) Price (log-transformed) and DisplayPromo, a dummy variable, which was not log-transformed and indicates whether a Store Display Promotion was active for the month the data was collected.

You conduct three different IV/ 2SLS estimators, each with different instruments. The results, including diagnostic tests, are shown below.

IV regression model 1 IV regression model 2 IV regression model 3
Variable Coefficient P-value Coefficient P-value Coefficient P-value
Intercept 1.843 0.011 1.894 0.001 1.444 0.009
DisplayPromo 0.923 0.001 0.901 0.002 0.898 0.001
Price -0.845 0.049 -0.656 0.001 -1.121 0.003

Weak instrument test 13.4 0.003 18.5 0.001 6.5 0.032
Wu-Hausman test 7.81 0.049 10.89 0.002 6.89 0.044
Sargan test 0.43 0.123 7.5 0.032 NA NA

What is the price elasticity result you would report based on the available information?

Question

You are hired to analyse the price elasticity of a fashion retailer. You carry out different log-log models to analyse the impact of two predictors on monthly sales units (log-transformed) for jeans using a sample of 256 stores. The input variables/ regressors are (average) Price (log-transformed) and DisplayPromo, a dummy variable, which was not log-transformed and indicates whether a Store Display Promotion was active for the month the data was collected.

You conduct three different IV/ 2SLS estimators, each  with different instruments. The results, including diagnostic tests, are shown below.

IV regression model 1		IV regression model 2		IV regression model 3
Variable	Coefficient	P-value		Coefficient	P-value		Coefficient	P-value
Intercept	1.843	0.011		1.894	0.001		1.444	0.009
DisplayPromo	0.923	0.001		0.901	0.002		0.898	0.001
Price	-0.845	0.049		-0.656	0.001		-1.121	0.003
								
Weak instrument test	13.4	0.003		18.5	0.001		6.5	0.032
Wu-Hausman test	7.81	0.049		10.89	0.002		6.89	0.044
Sargan test	0.43	0.123		7.5	0.032		NA	NA

What is the price elasticity result you would report based on the available information?

BlackTom AI · Accepted Answer

We start by restating the key setup: we are estimating a log-log model where the dependent variable is log monthly sales units for jeans, and the regressors are log-average price (Price) and DisplayPromo (a dummy, not log-transformed). In a log-log specification, the coefficient on a log-transformed regressor represents an elasticity: a 1% change in the regressor is associated with a corresponding percentage change in the dependent variable equal to the coefficient.

Now, examine the provided options and the IV/2SLS results across three models. The core question is about the price elasticity implied by the results, so we focus on the Price coefficient in each model:
- IV regression model 1: Price = -0.845, p = 0.049. This suggests that a 1% increase in price is associated with a 0.845% decrease in sales, all else equal. The p-value indicates statistical significance at conventional levels, though just barely.
- IV regression model 2: Price = -0.656, p = 0.001. Here, a 1% price rise is linked to a 0.656% fall in sales, with a more robust statistical significance (p-value much smaller).
- IV regression model 3: Price = -1.121, p = 0.003. This implies a larger elasticity: a 1% price increase corresponds to a 1.121% drop in sales, with a significant p-value as well.

Next, consider instrument strength and validity diagnostics, because these affect how much we trust the IV estimates:
- Weak instrument test: p-values are 0.003, 0.001, and 0.032 for models 1–3, respectively. All indicate weak instruments to some degree, with model 1 and 2 showing stronger concerns, and model 3 being the least weak among the three but still not ideal. Weak instruments can bias IV estimates toward OLS and inflate finite-sample distortions.
- Wu-Hausman test: p-values 0.049, 0.002, and 0.044. These tests compare IV/2SLS estimates to OLS to assess endogeneity. Small p-values suggest that endogeneity may be present, implying that IV is providing different (potentially more consistent) estimates than OLS.
- Sargan test: p-values 0.123 for model 1 and 0.032 for model 2; model 3 NA. A non-significant Sargan test (as in model 1) suggests instruments are not rejected (i.e., plausibly valid), whereas model 2 shows potential instrument validity concerns given p = 0.032. model 3 cannot be evaluated with Sargan in the provided output.

Given the above, what elasticity should you report? A careful approach would:
- Recognize that all three models yield negative elasticities, consistent with higher price reducing sales.
- Note that the magnitudes differ substantially across models (-0.656 to -1.121). The choice of which magnitude to report hinges on instrument strength/validity and endogeneity concerns.
- Acknowledge that weak instruments in all models reduce confidence in the IV estimates, with model 3 offering the largest absolute elasticity but still subject to instrument weakness, and model 2 showing the strongest statistical significance but potential validity concerns per the Sargan test.
- Therefore, you should avoid presenting a single definitive elasticity figure without caveats. Instead, report the range of IV-estimated elasticities observed (-0.656 to -1.121) and explicitly discuss the caveats: instrument weakness across all models, mixed instrument validity signals (Sargan), and endogeneity indications from Wu-Hausman tests. If forced to select a single point estimate, the model with the least compromised validity signals would be preferred, but here the evidence does not clearly endorse one model as decisively superior.

In practical terms, the explicit price-elasticity figures you can cite from the models are: Model 1 elasticity approximately -0.845, Model 2 elasticity approximately -0.656, and Model 3 elasticity approximately -1.121. Each of these is an elasticity interpretation given a 1% price increase, but due to instrument weaknesses and endogeneity concerns highlighted by the diagnostics, treat any single point estimate as tentative and discuss the range and caveats when presenting findings to stakeholders.

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