A. Descriptive statistics

Table 1 provides a description of the variables and the summary statistics. In the empirical analysis, we will rely on the real price in constant 2019 terms, ranging between 5 and 5,487 dollars, with an average of 49. Wine quality is measured by a score between 50 and 100, with a mean of 87. In addition to the vintage and year of issue, we have data on horizontal differentiation—whether the wine is ready to drink and the type of wine—and on the collective reputation of the wine appellation. This latter information is available only for France and Italy and comes from the Hugh Johnson wine guide, which has been published since 1977. Table A1 in the Appendix presents the distribution of wines by country. Since WS is an American guide, it is understandable that 31.8% of the bottles rated are from the United States. France and Italy are the second and third countries, with 24.2% and 17.3%. The remaining countries have minor shares, with Canada having a symbolic 0.1% presence.

Table 1. Description of the variables and summary statistics

Figure 1 shows the average real price by score value. The average real prices look flat and grow exponentially once the score exceeds 90 points. Regressing the average real prices over a cubic trend provides an R² equal to 0.86. Figures A1 and A2 show the actual and normal distribution of score values and the natural logarithm of the real price, which will be used in the econometric analysis. A few bottles get less than 70 and more than 98 points.

Figure 1. Average real price in $ and quality score.

B. OLS hedonic regressions

The empirical analysis relies on hedonic price regressions. In Table 2, which uses the full database with all countries, the natural logarithm of the real price is regressed over the variables listed in Table 1. Regressions include the vintage year and country dummy variables that are not shown for space reasons but are available upon request. Standard errors are clustered at the country level. We start with a regression that does not include the quality score (Column 1), then add the score in its linear (Column 2), quadratic (Column 3), and cubic (Column 4) specifications, while the last regression uses a set of dummy variables for each score value (Column 5).

Table 2. First stage, determinants of ln of real wine price, all countries

Notes: Regressions include vintage and country DVs. Robust standard errors clustered at the country level in parentheses. Regression in Column (5) uses a set of dummy variables for each score value instead of the linear, quadratic, or cubic functions (results are omitted for reasons of space but are available upon request and are visually represented in Figure 2).

*** p < 0.01, ** p < 0.05, * p < 0.1.

Quality is always strongly significant in all the specifications. The R² from Column (5), which includes dependent variables (DVs) for all the score values and is the most complete of the regressions proposed, is very close to that of Column (4), which uses a simple cubic function.Footnote ¹ The coefficients of the DVs of Column (5) are not shown for reasons of space but are available upon request; Figure 2 plots the values of the coefficients with their confidence intervals. The coefficients of the DVs for scores from 50 to 90 are not statistically different. Above 90 points, the effect of quality on price grows exponentially, and the coefficients of the DVs become gradually higher than those from 50 to 90 points at a 95% significance level.

Figure 2. Coefficients and 95% confidence interval of the score DVs.

Note: Results come from regression 5 in Table 2.

In line with previous literature, we find a positive effect of quality on price. However, descriptive statistics and econometric evidence show that the price premium attached to quality is concentrated at the upper end of the distribution. On average, the prices of wines with scores of between 50 and 90 points are statistically not different.

In Table A2 in the Appendix, we repeat regression 4 of Table 2 for each country, and the cubic effect of the score on the ln of price is confirmed for most countries.

C. Heteroskedastic hedonic regressions

In order to describe potential non-linear effects, we adopt a flexible approach: we repeat the hedonic regressions by describing the association with the score, vintage year, and age with restricted cubic splinesFootnote ² with two internal knots at score = (85, 90), year = (2000, 2011), and age = (5, 10). With the described definition, the design matrix X = (x₁, x₂, …, x_n)T is composed of 26 columns: the constant term, four indicators of dessert, red, rosé, and sparkling wine (reference = White), the “drinknow” indicator, 11 country dummy variables (reference = Argentina), and the three-dimensional splines associated with the score, vintage year, and age. The mean regression model, that is, the standard “linear” model with constant variance largely used in the literature (see Castriota (Reference Castriota2020, Ch. 2) for a review of empirical studies of wine quality and price), was not optimal since it was immediately evident the presence of data heteroskedasticity. Therefore, we considered two alternative solutions: a Normal heteroskedastic model and a quantile regression approach.

The Normal heteroskedastic model is the natural generalization of the standard normal model, that is, the “linear” (mean) regression model. The idea is to describe with a function of the predictors not only the mean but also the variance of a normal distribution, such that Y ~ N(μ(x|β), σ ²(x|θ)). We defined

(1)$$\mu ( {\bf x}\vert {\bf \beta }) = {\bf x}^T{\bf \beta }, \;{\rm log}\{ \sigma ^2( {\bf x}\vert {\bf \theta }) \} = {\bf x}^T{\bf \theta }.$$

This modeling approach allows for accounting for the presence of heteroskedasticity directly. The conditional mean is given by x^T β, which is also the value of the median. Other quantiles, for example the quartiles or the deciles, are defined as Q(p|x, β, θ) = μ(x|β) + σ(x|θ)z(p), where z(.) is the quantile function of a N(0,1) distribution. With our specification of the design matrix, the model had 26 + 26 = 52 parameters, estimated by maximum likelihood. Standard errors were computed by applying standard asymptotic theory, that is, using the square root of the diagonal elements of the negative inverse of the Hessian matrix of the log-likelihood, evaluated at its maximum.

Results in Figure 3a show the predicted mean and median, the quartiles, and the 5th and 95th percentiles of ln(price) as a function of the quality score. All functionals are approximately J-shaped or U-shaped, with all quantiles increasing rapidly for scores above 85. Interestingly, dispersion parameters also show a U-shaped association with the score. In Figure 4, we report the difference between the estimated 95th and 5th percentiles, which achieves a global minimum of approximately 87.5 points.

Figure 3a. Normal heteroskedastic model.

Note: Predicted mean and median, quartiles, and 5th and 95th percentiles of ln(price), as a function of the quality score, as estimated by applying a Normal heteroskedastic model.

In our analysis, we assumed that different types of wines (white, red, rosé, sparkling, and dessert) have the same price-score relationship, up to an additive constant. This assumption may not be plausible and is rejected by the data, where we found a significant interaction (p < 0.0001) between the type of wine and the spline function that describes the price-score association. However, the effect of the score on price is qualitatively the same in all types of wines. The plot in Figure 3b is obtained by applying the Normal heteroskedastic model to each type of wine separately. It can be seen that the functions differ by more than just an additive constant; however, they all show a similar behavior: approximately flat for scores below 80, with a sharper increase for scores between 85 and 90. While this “justifies” our simplifying assumption to pool all wines together, we also understand that a stratified analysis could provide additional useful information; for example, red and white wines show a sharper increase in price than the other three groups.

Figure 3b. Normal heteroskedastic model, by type of wine.

Note: Predicted mean of ln(price), as a function of the quality score, as estimated by applying a Normal heteroskedastic model to each type of wine separately.

Next, in Table A3, we applied the Normal heteroskedastic model using dummy variables for the score in place of spline functions. We report the estimated effects on the conditional mean (β) and log-standard deviation (θ). All the coefficients are highly significant. Results are consistent with those from the spline-based model: the minimum price is found in scores of 75–79, and a sharp increase is found for scores above 90. Moreover, the highest variance around the mean is found at very low and very high scores.

D. Quantile hedonic regressions

In the Normal model, specific quantiles of interest can only be extrapolated using the assumed parametric distribution. An alternative approach is to apply a quantile regression model (e.g., Koenker, Reference Koenker2005). This method avoids global parametric assumptions and allows to directly describe how the p-th quantile depends on the observed predictors, using a regression model of the form

(2)$$Q( p\vert {\bf x}, \;{\bf \beta }) = {\bf x}^T{\bf \beta }( p ) .$$

In this model, β(p) is a quantile-dependent vector of regression coefficients. For example, the coefficients that describe the conditional median (p = 0.5) differ from those that characterize the first quartile (p = 0.25) or the 95th percentile (p = 0.95).

Estimation is carried out by minimizing a loss function

(3)$$L( {\bf \beta }) = \Sigma _i( {\,p- u_i} ) ( y_i- {\bf x}_i^T {\bf \beta }( p ) ) , \;$$

where $u = I( y \le {\bf x}^T{\bf \beta }( p) )$ (e.g., Koenker, Reference Koenker2005), while standard errors can be computed by using non-parametric bootstrap. In our analysis, we estimated quantiles of order p = {0.05, 0.25, 0.5, 0.75, 0.95}. Results are illustrated in Figure 4 and are very similar to those obtained with the Normal heteroskedastic model. Some minor differences can be obtained in the estimated 5th and 95th percentiles, especially for low-quality score values. This may suggest that the tails are not well represented by a normal distribution. In Figure 5, we can also see that the difference between the 95th and the 5th percentile found with quantile regression deviates slightly from that obtained using the Normal heteroskedastic model.

Figure 4. Difference between Q(0.95) and Q(0.05).

Note: Estimated difference between the 95th and the 5th percentile of ln(price), expressed as a function of the quality score. Continuous line: Normal heteroskedastic model. Dashed line: quantile regression model.

Figure 5. Quantile regression model.

Note: Predicted median, quartiles, and 5th and 95th percentiles of ln(price), as a function of the quality score, as estimated by applying a Quantile regression model.