1 Introduction
Sales response functions describe the relationship between the sales of a product (or an entire product category) as dependent variable and predictors that are believed to influence sales as independent variables. In brand sales models based on store-level data, these predictors typically represent prices of (substitute) brands and other marketing variables related to promotional activities (like displays and feature advertising) as well as trend or seasonal indicators. In this context, retailers and academic researchers face several challenges, for example how to process the usually large amount of information (predictors) to arrive at a parsimonious model, how to specify the functional relationships between metric predictors (like prices) and sales, and/or how to accommodate dynamic effects in the model. More specifically, several streams of research for modeling (store) sales response to price variations have developed over the last 40 years. Among them, researchers have focused on choosing the right functional form to adequately capture the relationship between (own or competitive) prices and sales. Non- or semiparametric regression models have been proposed here in recent years in order to capture strong and/or complex nonlinearities in price response that could actually be proven for frequently purchased consumer goods in many empirical studies and are difficult to handle with parametric models. A second stream has addressed price dynamics in (store) sales response models by adding variables for lagged (or even lead) price effects, by including price terms for reference price or price-change effects, or by considering time-varying price parameters. Reference price effects are more commonly studied with disaggregate consumer data (i.e., household-level data), and have been less frequently incorporated into response models based on aggregate sales data. Obviously, reference price effects are much more difficult to model with aggregate data compared to disaggregate data. In the latter case, purchase incidence, brand choice, and purchase quantity decisions of consumers can be more easily separated at the individual consumer or household level, and reference price effects can in principle influence all three decisions (although they are most popular in models that have its focus on brand choice only). Beyond reference price effects, other forms of dynamics like stockpiling, state dependence (brand loyalty) or customer holdover, or consumer learning have been shown to be also very relevant at the individual consumer or household level, see for example Neslin and van Heerde (
2009) and van Heerde and Neslin (
2017) for an overview. Sales response models lack this micro-foundation: both the three different consumer decisions and the possibly individually different dynamics are confounded in aggregate data, making it challenging to disentangle them (see, e.g., Neslin and Shoemaker
1989; van Heerde et al.
2000,
2004). Using aggregate data, reference price effects can therefore only be interpreted for an aggregate of households in the sense that they refer to prices paid or observed in previous periods (e.g., weeks) rather than to prices paid or observed at previous individual purchase occasions. This may be less of a problem if goods like in the food sector are purchased on a regular (weekly) basis and are frequently advertised. On the other hand, if consumer goods have longer interpurchase times household data can also suffer from some problems like for example a different composition of consumers across periods when modeling dynamic effects over time (e.g., weeks). And, modeling reference price effects with aggregate data has also its pros due to the greater managerial relevance of aggregate data compared to household-level data. Although household-level data are richer for explaining customers’ purchasing behavior (as indicated above), they have been often criticized by managers for their potential lack of representativeness, which may cause share estimates to differ from those based on store-level data (cf. van Heerde
1999, p. 21). While household-level data cover only a subset of all customers purchasing at a retail store, store-level data cover all these customers, which is a weighty argument from the perspective of a store manager in favor of using aggregate data. For a comprehensive discussion on the pros and cons of the different data types, see van Heerde (
1999, pp. 20–21). As a result of the discussion above, it is however important to separate reference price effects from other dynamic effects like stockpiling or customer holdover when relying on aggregate data.
1
In this paper, we combine nonparametric sales response modeling with the estimation of price dynamics where the latter are captured by reference price effects. The fact that no other study has tackled this frontier up to now can probably be explained by the much higher popularity of studying reference price effects with household-level data and the greater difficulties to disentangle different sources of dynamics with aggregate data, as discussed above. We try to fill this research gap and propose a semiparametric model to flexibly estimating price-change or reference price effects based on store-level sales data. We compare different representations for accommodating symmetric versus asymmetric and proportional versus disproportionate price-change effects following adaptation-level and prospect theory, and further compare our flexible autoregressive model specifications to parametric benchmark models. Since management decisions should be based on the model with the highest predictive performance (van Heerde et al.
2002), our primary focus is on the predictive model performance rather than on solving the problem how to tease out different dynamic effects with aggregate data. Actually, focusing on prediction as our main goal relaxes the problem that it is more difficult to disentangle different dynamic effects with aggregate sales data compared to disaggregate consumer data. Nevertheless, we address this problem and separate reference price effects from other dynamic effects like stockpiling and customer holdover by including lagged sales as autoregressive model component.
In an empirical study, we demonstrate that our semiparametric dynamic models can provide more accurate sales forecasts compared to competing benchmark models that either ignore price dynamics or just include them in a parametric way. The main benefit of the proposed model is therefore to help managers to predict sales better. It has been shown before for other models that semiparametric modeling can provide (much) better predictions than parametric modeling (see the literature review in Sect.
2 below); as such, one main academic contribution of the paper is to show that this also holds for reference price models. In addition, we discuss likely implications of our model for related optimal pricing decisions in our outlook onto future research perspectives at the end of the article.
The rest of the paper is organized as follows. Section
2 provides a compact review of the relevant literature on price response modeling based on aggregate data, reflecting the road from parametric to more flexible model specifications as well as the different options of addressing price dynamics, including reference price effects in particular. In Sect.
3, we introduce our Bayesian model estimation framework. Using scanner data for refrigerated orange juice brands sold by a large supermarket chain we compare different model specifications (nonparametric vs. parametric, dynamic vs. static, alternative options of specifying reference price effects) for predictive performance and discuss implications regarding estimated price elasticities in Sect.
4. We conclude in Sect.
5 with a summary of the most important findings, managerial implications, and an outlook on future research opportunities.
2 Literature review
This section provides an overview of relevant literature for our proposed approach, referring to the functional form of price response models, the incorporation of price dynamics in such models, and the few approaches that have so far combined functional flexibility and the estimation of dynamic price effects in sales response models.
Early approaches used strictly parametric modeling to estimate sales/price response functions, as a rule using the sales variable in logarithmic form (e.g., Hruschka
1997; Montgomery
1997; Foekens et al.
1999; Kopalle et al.
1999; van Heerde et al.
2000,
2002; Hruschka
2006a,
b; Andrews et al.
2008). Using log sales instead of sales enables to capture nonlinearities in sales response, however the observed data are still projected “into a Procrustean bed of a fixed parameterization” (Härdle
1990; as cited in van Heerde
1999, p. 28).
In other words, parametric models only provide consistent estimates if the a priori assumed functional form is correct (e.g., Leeflang et al.
2000). The use of more flexible
semi- or
nonparametric models can help to overcome this problem, as these allow to ‘extract’ the shape of functional relationships directly from data without prior knowledge about the functional form (e.g., van Heerde
2017). van Heerde et al. (
2001) have shown for several food categories that the use of a kernel regression approach can improve the predictive performance of brand sales models based on store-level data compared to parametric modeling. Hruschka (
2006a) and Hruschka (
2007) used neural nets (multilayer perceptrons) to capture nonlinearities in sales response and reported much better log marginal densities as well as high posterior model probabilities or superior cross-validated predictive densities compared to strictly parametric modeling for all brands considered. Other researchers proposed spline approaches to model sales response and could reveal strong nonlinearities in price effects which in addition were shaped very differently at the individual brand level. Kalyanam and Shively (
1998) used stochastic cubic splines, Haupt and Kagerer (
2012) and Haupt et al. (
2014) applied B-splines, Hruschka (
2000) considered both B-splines and cubic smoothing splines, and Steiner et al. (
2007), Weber and Steiner (
2012), Lang et al. (
2015), and Weber et al. (
2017) employed Bayesian P-splines. Except for Kalyanam and Shively (
1998) and Hruschka (
2000), who focused on model fit (the first also used marginal posterior model probabilities, the latter applied AIC), the mentioned spline applications provided further evidence of (much) more accurate sales predictions when using nonparametric instead of parametric response models. These findings are of great importance since (store) managers should prefer the model specification with the best possible predictive performance (van Heerde et al.
2002). More flexible specifications have the potential to work better and to provide superior forecasts than parametric ones if the sales data at hand include complex nonlinear relationships in price response which are difficult to ‘read out’ with parametric models (e.g., Lang et al.
2015).
Beyond the choice of the right functional form, one can think about the incorporation of time-dependent (price) effects leading to
dynamic instead of static sales or price response models. In the context of price promotions there is empirical evidence that lags or leads of prices can have an impact on current brand or current category sales volumes. van Heerde et al. (
2000) and van Heerde et al. (
2004) used leads and lags of price indices reflecting promotional price cuts with different types of promotional support, and reported significant and in parts also very substantial dynamic effects. Nijs et al. (
2001) and Horváth and Fok (
2013) accounted for price dynamics by fitting VARX models. Nijs et al. (
2001) examined category-demand effects and found that the strong positive short-term effects of price promotions almost completely dissipate over time. Horváth and Fok (
2013) analyzed cross-price effects and found evidence of preemptive switching in a way that a brand’s price promotion in one period can decrease a substitute brand’s sales in subsequent periods. Foekens et al. (
1999) and Kopalle et al. (
1999) proposed varying parameter models to account for dynamic (pricing) effects in store-level sales response models, both using the widespread multiplicative functional form for modeling price response. Foekens et al. (
1999) reparameterized a brand’s own-price elasticity as to depend on cumulated previous price discounts (amount and time) for both the brand considered and competing brands, and reported that the magnitude and timing of preceding price cuts can have a significant impact on own-price elasticities at the current period. Kopalle et al. (
1999) reparameterized own- and cross-price parameters as functions of geometrically-weighted averages of past discounts and in addition developed a normative model for related pricing decisions.
Talking about lagged prices and dynamic price response modeling is further closely connected to the topic of
reference prices. Adaptation-level theory, as proposed by Helson (
1964), states that the perception of a new stimulus is performed relative to an ‘adaptation level’. Applied to a pricing context, the adaptation level for judging a newly observed price information is called reference price. In other words, the reference price constitutes an internal price standard of the consumer and works as the adaptation level the consumer compares the current price of a brand observed or paid to. Prospect theory also suggests that consumers evaluate alternatives based on a comparison to a standard or reference point, but further distinguishes how consumers value potential gains versus losses from making a decision (Kahnemann and Tversky
1979). The value of a new price information therefore not only depends on the (absolute) difference to a reference point but also on whether the price difference represents a
gain or a
loss for the consumer. Consequently, prospect theory offers an option to evaluate price-change effects much more differentiated compared to adaptation-level theory (see Sect.
3.1 for a more detailed description of the value function underlying prospect theory).
In the pricing literature, adaptation-level and prospect theory are most frequently applied in the context of (brand) choice modeling, i.e., in models that are based on disaggregate consumer data (for an introduction to this topic see Neslin and van Heerde
2009, for a detailed literature review see Mazumdar et al.
2005 and Neumann and Böckenholt
2014, and for a recent application see Boztuğ et al.
2014 and Baumgartner et al.
2018). Exceptions are for example Kucher (
1987) and Natter and Hruschka (
1997), who incorporated reference price effects into market share models (i.e., using aggregate data). At this point, it is important to note that the reference price as a construct to model dynamic effects of past prices can be either operationalized as the price of the previous purchase occasion (referred to as price-change effect) or determined via more complex reference price formation mechanisms based on several past prices (referred to as price-deviation effect), compare Kucher (
1987).
Gedenk (
2002, p. 249) has provided an overview of studies for either stream, and Briesch et al. (
1997) discussed different reference price formation mechanisms. Accordingly, because reference prices of consumers cannot be directly measured or be determined in aggregate sales data, some authors used either the price of the last period or the average of several previous prices as proxy for reference prices in aggregate response models (also see Gedenk
2002, pp. 247–249).
Referring to adaptation-level theory, Simon (
1982, pp. 208–213) still a little earlier argued that it seems realistic to assume not only an absolute price effect but also a price-change effect on brand sales, i.e., that the price of the last period should have an impact on the response to the current period’s price. In a first step, he proposed two versions of linear price-change response models, one where a brand’s sales depend on the difference between its current price and its previous price, and another where the relative price change instead of the absolute price change was used as independent variable. Both linear price-change response models assume a symmetric and proportional sales response to price increases and decreases. In addition, he also proposed a hyperbolic sine sales response function to accommodate non-proportional (but still symmetric) price-change effects, with the relative price change as its argument. He motivated the use of this functional form using the same behavioral rationale as is inherent to the well-known Gutenberg price response model: small price changes may have only marginal (below average) effects on sales, while large price changes should yield disproportionately large sales effects (e.g., Hruschka
2000). Note that the three models did not include a contemporaneous (static) price effect, however Simon (
1982, p. 211) mentioned that the model could be extended accordingly in case of a sufficiently large data base (as it is given today with store-level scanner data). Later, Simon (
1992, pp. 253–254) also considered the possibility of an asymmetric price-change response by expanding his linear price-change response model (with the absolute price difference as argument) to capture sales effects from price increases (losses) and price decreases (gains) separately. He motivated the model extension with an own empirical study where he observed a significant price-change effect for price decreases (but not for price increases) on the one hand, as well as by referring to prospect theory, which vice versa suggests a higher price elasticity for price increases compared to price decreases due to loss aversion of consumers on the other hand.
In the context of price assessment, Diller (
2008, pp. 140–143) linked prospect theory to reference prices and proposes (among others) a reverse s-shaped decreasing function to capture asymmetric price-change or price-deviation effects. The shape of the function looks similar to the logistic price response function but shows a steeper progression for losses than for gains, following prospect theory. As an alternative, he suggested an s-shaped decreasing response function in order to accommodate the existence of possible lower and upper price thresholds. Consequently, the latter function is not in line with prospect theory but resembles the shape of the Gutenberg function with a flatter middle part on the one hand and much more elastic parts for larger deviations from the reference point on the other hand. Both functions allow to capture a disproportionate price response pattern. Importantly, Diller (
2008, pp. 360–361) pointed out that the choice of the right functional form depends on the empirical data at hand which makes it necessary to compare different parametric approaches in empirical applications. As mentioned above, using nonparametric estimation techniques can remedy this dilemma by letting the data determine the shape of price-deviation or price-change effects without a priori assumptions about functional forms.
There are certainly pros as well as cons to decide upon whether to use only the price of the last period or a more complex reference price formation mechanism based on prices of several previous periods in an aggregate sales response model. Rinne (
1981, pp. 29–30) already used the previous price as reference price (in terms of the last seen price) in his sales response model, as well as Kucher (
1987, p. 179) did due to the “exceptional position” of the previous price among all past prices. In addition, there is empirical evidence that individual consumers would not access price information that lies much beyond the immediate past purchase occasion, simply due to difficulties in accurately remembering prices further back (see Krishnamurthi et al.
1992, and the literature cited therein). If consumers buy products on their shopping trips on a weekly basis, this restricted memory capacity argument with its focus on the immediately last price as reference price is also applicable to aggregate data. Also, established reference price formation models for disaggregate data use periodical (weekly) updates for a brand’s reference price at the individual consumer level even if a consumer did not buy that brand in the last period (cf. Erdem et al.
2010). Accordingly, “updating reference prices only when households make purchases would underestimate the reference price” (Erdem et al.
2010, p. 310). This assumes that consumers are monitoring brand prices over periods and therefore are aware of a brand’s price in the previous period, which is realistic for frequently purchased consumer goods (at least for segments of consumers). Based on these arguments, we focus on price-change response models using aggregate store-level sales data and the price of the last period as proxy for the reference price of an aggregate of consumers shopping at a retailer, hence we use the more parsimonious option to operationalize reference prices.
Beyond the approaches discussed above, only very few authors have explicitly considered prospect theory for modeling price effects in store-level sales response models. Based on the reference price model of Greenleaf (
1995), Kopalle et al. (
1996) assumed that demand for a brand is a linear function of price(s) and a price-deviation effect, the latter which is operationalized with two additively separable terms representing gains and losses. The authors developed optimal dynamic pricing strategies and showed that when (a sufficiently large number of) consumers weigh losses stronger than gains, as suggested by prospect theory, every day low pricing (i.e., setting constant prices) is optimal for a retailer. Conversely, if (enough) consumers weigh gains stronger than losses, a hi-lo strategy (cyclical pricing) would be the optimal retailer strategy (for a similar result see Fibich et al.
2007). Assuming asymmetric reference price effects with loss-averse consumers, Fibich et al. (
2003) demonstrated that for an infinite planning horizon the optimal pricing strategy ‘converges’ at a steady-state price, which turns out slightly lower than without considering reference price effects. Pauwels et al. (
2007) proposed smooth transition regression models to explore threshold-based price elasticities and found evidence for larger threshold sizes for gains than for losses.
Van Heerde et al. (
2004) addressed both functional flexibility and price dynamics in brand sales models (with store-level data). Based on van Heerde et al. (
2000), who used leads and lags of price discount variables to capture price dynamics, and based on van Heerde et al. (
2001), who applied kernel regression to flexibly estimate price discount effects, the authors combined both features (leads and lags, local polynomial regression) in order to decompose the sales effect of promotions into the three different sources cross-brand effects, cross-period effects, and category expansion effects. Finally, Natter and Hruschka (
1997) have been previously the only ones who estimated reference price effects within a flexible approach (via a neural network) and based on aggregate data, however their approach was directed on market share modeling rather than sales response modeling. To the best of our knowledge, no study so far has employed nonparametric regression to flexibly estimate asymmetric reference price (price-change) effects in store-level sales response models, and we attempt to fill this research gap in the literature with our study.
Table
1 summarizes the literature on (store-level) sales response models with focus on estimating price effects discussed above, distinguishing between approaches that have addressed either functional flexibility, or price dynamics (in the form of using lead or lagged prices, reference prices, or time-varying parameters), or both features. In the Appendix
A, we provide an overview of advantages of using nonparametric regression techniques in general and especially for capturing gain and loss effects and summarize further convenient properties of applying Bayesian P-splines as we do in this article.
Table 1
Overview of studies accommodating price dynamics (e.g., reference price effects) and/or functional flexibility in aggregate sales response models
| Reference prices (gains and losses) | – |
| Reference prices (price difference) | – |
| Reference prices (gains and losses) | – |
Kalyanam and Shively ( 1998) | – | Stochastic splines |
| Time-varying parameters | – |
| Time-varying parameters | – |
| – | B-splines, cubic smoothing splines |
| Leads and lags | – |
| Lags | – |
| – | Kernel regression |
| Reference prices (gains and losses) | – |
| Leads and lags | Local linear regression |
| – | Neural nets |
| Reference prices (gains and losses) | – |
| – | Neural nets |
| Reference prices (gains and losses) | – |
| – | Bayesian P-splines |
Brezger and Steiner ( 2008) | – | Bayesian P-splines |
| – | B-splines, quantile regression |
| – | Bayesian P-splines |
| Lags | – |
| – | B-splines, quantile regression |
| – | Bayesian P-splines |
| – | Bayesian P-splines |
This study | Reference prices (price difference, gains and losses) | Bayesian P-splines |
In the following, we present a semiparametric brand sales model which accounts for price dynamics via (asymmetric) price-change effects. Our model therefore combines reference price effects with functional flexibility. Our model specification is based on Weber et al. (
2017), who assumed a brand’s sales to depend on the brand’s own price and prices of substitute brands, further marketing covariates representing the use of displays and odd prices, as well as on store-specific and holiday effects. We additionally consider price dynamics via price-change effects in different ways, that way accommodating adaptation-level and prospect theory. Our focus is on the predictive performance of the proposed approach in comparison to several benchmark models, including linear, exponential, and multiplicative models with or without price dynamics. From a modeling perspective, we want to demonstrate the capability of nonparametric regression to estimate any kind of price-change response from data (symmetric vs. asymmetric response, proportionate vs. disproportionate response). From a managerial perspective, our primary goal is to provide an econometric model that can improve sales predictions from incorporating reference price effects into a flexible sales response model compared to simpler model specifications that either ignore price dynamics, functional flexibility, or both. Likely implications of our proposed model for related optimal pricing decisions will be discussed in Sect.
5.
5 Conclusions
In this article, we proposed a semiparametric approach to flexibly estimating reference price effects in brand sales models. In particular, we focused on the so-called price-change response of consumers (prominently introduced by Simon
1982), using aggregate store-level sales data and the price of the last period as proxy for the reference price of (an aggregate of) consumers. We compared different options to capture this dynamic price-change effect, following adaptation-level and prospect theory. While adaptation-level theory states that consumers evaluate a new price information for a brand relative to an adaptation level (which is the brand’s price of the last period in our context), prospect theory goes one step further and claims that consumers should value losses of a certain amount stronger than gains of the same amount (corresponding to price increases and price decreases of the same amount in our context), and that the value function is convex for losses and concave for gains. Accommodating functional flexibility for price effects via nonparametric regression helps to simultaneously analyze a potential asymmetry and/or disproportionality of the price-change response without the need to assume a certain functional form for it a priori. In other words, by letting the data determine the shape of the price-change effect we can easily verify if the implications of these behavioral theories hold for the data at hand. We further compared the semiparametric approach to parametric benchmark models in order to assess the added value of using nonparametric regression for estimating price (change) effects.
To compare the predictive performance of our models, we conducted an empirical study using store-level scanner data of the Dominick’s Finer Foods (DFF) data base for refrigerated orange juice. For model specification, we assumed a brand’s sales to depend on the brand’s own price, the brand’s sales of the previous period, prices of substitute brands, promotional activities, store-specific and holiday effects, as well as on the brand’s previous price to capture the price-change or reference price effect in the following ways: via a single price-difference term versus two separate price terms for perceived gains and losses, where the price change with respect to the previous price is measured in absolute monetary units or as a percentage change, respectively. We further estimated two nested variants of our semiparametric model in order to evaluate the impact of accounting for (price) dynamics on the predictive model performance: a static variant without the reference price term and without the lagged sales effect, and a simpler dynamic variant without the reference price term but including one-period lagged sales as autoregressive part. To assess the added value of employing nonparametric regression for estimating the price-change effect flexibly (as well as own- and cross-price effects), we also compared our semiparametric approach to the exponential (log-linear) and the multiplicative (log-log) sales response function (as well as to the simple linear model) as parametric benchmark models.
5.1 Results and managerial implications
The main results of our empirical study can be summarized as follows: first, accounting for price-change or reference price effects can substantially improve the predictive performance of brand sales models (as measured by the cross-validated average root median or mean squared errors, \(\mathrm {ARMedSE}\) or \(\mathrm {ARMSE}\)). For the parametric models (linear, exponential, and multiplicative), accommodating gain and loss effects with two separate price-terms (abs-gl, rel-gl) largely improves \(\mathrm {ARMedSE}\) values (i.e., reduces prediction errors) in holdout samples, whereas using a single price-difference term (abs-diff, rel-diff) provides only small (marginal) improvements or even decreases the predictive accuracy measured by \(\mathrm {ARMedSE}\) compared to the static model. A look at the estimated effects and the corresponding partial residuals of the competing dynamic model specifications reveals the reason for the clearly worse performance of the abs-diff and rel-diff models: obviously, the price-change effect is asymmetric for nearly all orange juice brands analyzed, however using a single price-difference term for both gains and losses does not allow the detection of asymmetrical effects of price changes. The use of separate variables for perceived gains versus perceived losses helps to overcome this limitation. In contrast, the semiparametric models do not have this shortcoming: due to their much greater flexibility they are able to capture such asymmetries and therefore possibly different shapes for gain and loss effects, even if gain and loss effects are not separated from each other with two different price terms. This explains why the four different dynamic specifications for the reference price effect perform similarly well here. In other words, once price (change) effects are accommodated flexibly the form of the specification of the reference price term gets secondary. The predictive validity results based on the \(\mathrm {ARMSE}\) measure closely resemble those for the \(\mathrm {ARMedSE}\) measure with one noticeable exception. For the parametric models, modeling gain and loss effects via two separate reference price terms did no consistently improve the predictive performance for all brands compared to using a single price-difference term only. In these cases, however, taking price dynamics into account did improve the predictive accuracy either only marginally or not at all, independent of the form of including the reference price term.
Second, as the spline functions are furthermore able to account for disproportionate effects of any shape, each of the semiparametric model variants provided more accurate sales predictions than its linear, exponential, or multiplicative counterparts for all brands considered but one (for each of the two predictive validity measures). For the one brand, the semiparametric model predicted similarly well or marginally better nevertheless. Interestingly, even the static semiparametric model leads to lower \(\mathrm {ARMedSE}\) values than each of the two dynamic exponential or multiplicative models when capturing the price-change effect with only a single price-difference term. This also held for all but one brand (two brands) for the exponential (multiplicative) model when the predictive accuracy was evaluated by the \(\mathrm {ARMSE}\) measure. This underlines the power of nonparametric estimation techniques in the present context. Overall, improvements in predictive accuracy from accommodating price effects flexibly over the best dynamic exponential or multiplicative models ranged up to \(-11\)% (\(-12\)%) in terms of \(\mathrm {ARMedSE}\) and up to \(-15\)% (\(-14\)%) in terms of \(\mathrm {ARMSE}\) at the individual brand level.
Third, referring to the shapes of the flexibly estimated price-change effects, we observed rather steep, disproportionate gain effects while rather flat loss effects for nearly all brands. Accordingly, consumers seem to weigh gains much stronger than losses in the refrigerated orange juice category, which contradicts prospect theory. Loss-gain ratios smaller than 1 as a rule underlined this finding. For the gain effect, decreasing returns to scale (as suggested by prospect theory) were found only for two of the national brands. For most of the other brands, the estimated gain effect curves turn out neither strictly concave nor strictly convex, showing more or less complex nonlinearities, which in addition differed across brands. This is exactly the strength of nonparametric modeling: there is no need to search for the right functional specification(s) in advance, the shape of effects is estimated directly from the data. Also note that we controlled in our models for other dynamic effects like stockpiling or customer holdover by including one-period lagged sales. Still, it could be that larger stockpiling effects may have gone undetected because post-promotion dips are generally hard to detect in aggregate response models. In this case, the negative effect of losses might have been undervalued. On the other hand, refrigerated orange is a less stockable product which suggests that the estimation bias in the estimated loss effects should not be that large, if one existed.
From a managerial point of view, using our more complex semiparametric approach seems worth the effort as it provides several advantages. First, as already discussed above, predictions turned out never worse, often better and sometimes considerably superior to those from any of the parametric models compared to. Second, semiparametric modeling especially pays off if price effects involve complex nonlinearities which are difficult or not at all to capture via parametric models. Even if complex nonlinearities (e.g., strong kinks or several thresholds) are not at work and improvements from using the more complex model would be not that big or only small, one need not care about the problem which parametric model to use for which brand to arrive at the best possible brand sales predictions. Our study has shown that nonlinearities for gain effects may be complex and may further turn out very differently at the individual brand level which favors the use of a flexible estimation techniques. Third, free software (BayesX) is available to easily estimate the semiparametric model (as well as the parametric models).
5.2 Limitations and outlook
Our study of course has some limitations. First, our empirical findings relate to only one product category (refrigerated orange juices) which does not yet allow a generalization of the results. More studies for different product categories and based on aggregate data are necessary to confirm or to complement our findings and to augment existing findings on loss aversion (or gain-seeking behavior), which so far have almost exclusively related to a brand choice modeling context, i.e., to disaggregate data.
Second, our article has its focus on functional flexibility
and reference price effects (i.e., exploring asymmetries and disproportionalities of the price-change effect), and contributes to the stream of nonparametric models in marketing. Except for the random store intercept, which captures heterogeneity in baseline sales across stores (e.g., due to differently sized store sales areas) in all of our models, we did not address potential heterogeneity of marketing effects across stores. Weber and Steiner (
2012) have shown that accounting for heterogeneity may be not helpful per se to improve the predictive performance of a store-level sales response model, whereas accommodating functional flexibility can substantially reduce prediction errors. Only few approaches exist that have accounted simultaneously for both functional flexibility and heterogeneity in store-level sales response models (e.g., Hruschka
2006a; Lang et al.
2015; Weber et al.
2017). The latter two studies have shown that extending an already flexible model to additionally accommodate store heterogeneity in marketing effects may further improve the predictive model performance. Transferred to our context, it could be interesting to analyze if and how strong the price-change effect differs across stores. On the other hand, existing (flexibly estimated) nonlinear effects might also just be a form of heterogeneity, as existing differences in price effects across stores can ‘add up’ to a
nonlinear (homogeneous) effect.
10 In the latter case, explicitly considering heterogeneity in addition to functional flexibility might not necessarily further improve predictive accuracy. Finally, accommodating time-varying parameters as an alternative or in addition to considering heterogeneity and/or functional flexibility could also accomplish improvements in predictions.
Third, we did not address the issue of price endogeneity, a point of increasing controversy in the relevant literature. A number of different approaches have been proposed to treat endogeneity in prices (for an overview of endogeneity in aggregate market response models, see Hruschka
2017), however Rossi (
2014, p. 671) has pointed out that using an invalid instrument to accommodate price endogeneity can “cause the estimates to differ even when there is no endogeneity bias”. Typical candidates for appropriate instrument variables in aggregate response models are lagged prices and costs (or wholesale prices). The former, however, are not exogenous in case of our dynamic models with reference price terms, and hence cannot be used as instruments here (also cf. Hruschka
2017). In addition, standard techniques (e.g., 2SLS) to account for endogeneity do not necessarily work in (flexible) nonlinear models, as we have with our flexibly estimated nonlinear price and reference price effects. Generalized Methods of Moments estimation (if a valid instrument is available) or the copula-based method as instrument-free alternative could be promising ways out to accommodate endogeneity within in our semiparametric approach (e.g., Hruschka
2017).
Fourth, we followed the stream of researchers who proposed using the price of the last period as reference price (cf. Sect.
2). Alternatively, the reference price could be built based on prices of several previous weeks, following as example the concepts of adaptive or extrapolative expectations (e.g., Natter and Hruschka
1997; Briesch et al.
1997; Baumgartner et al.
2018).
Finally, a further challenge would be to develop optimal dynamic pricing strategies for the different models. Here, one could at first lean on the research of Kopalle et al. (
1996) or Fibich et al. (
2003) (cf. Sect.
2) and compare pricing implications obtained from models that ignore asymmetries of the price-change effect versus models that do capture asymmetries in gain and loss effects. This could be especially interesting for the semiparametric approach where using either a single price difference term or separate terms for gain and loss effects performed similarly well in our empirical study regarding predictive validity. Based on the findings of Kopalle et al. (
1996) or Fibich et al. (
2003), we would expect a hi-lo or pulsing strategy (cyclical pricing) as optimal pricing strategy for most refrigerated orange juice brands, since gain effects turned out to be (much) larger than loss effects as a rule. For the two brands “Tree Fresh” and “Florida Gold”, we did find loss aversion for moderate and/or larger price differences, making predictions about the expected dynamic pricing strategies for these two brands more difficult. Furthermore, Lang et al. (
2015) reported higher expected total chain profits for their semiparametric (static) sales response models compared to the multiplicative sales response function. The expected loss in profit for the multiplicative model was accompanied by a larger number of lower optimal price levels across weeks than suggested by the semiparametric model. In other words, semiparametric modeling led to a larger number of higher price levels compared to parametric modeling in their study. Weber et al. (
2017) analyzed expected category profits and basically confirmed the findings of Lang et al. (
2015) that flexible (static) price response modeling offers a huge potential for increasing a retailer’s profits compared to parametric modeling. Moreover, we obtained (as a rule) lower price elasticities for our dynamic models (
abs-diff,
abs-gl, compare Sect.
4.3.2) compared to using a static model (except for very low price levels), independent from the type of model (parametric or semiparametric). This suggests that static sales response models might overestimate price sensitivities of consumers and therefore can lead to suboptimal pricing strategies. We leave the issue of optimal dynamic pricing for future research.