HIERARCHICAL MOVEMENTS IN DEODORANT PRICES
An application of the Dynamic Hierarchical Factor Model (DHFM) to the
regular prices of deodorant brands in the Netherlands
Author:
JASPER HIDDING
HIERARCHICAL MOVEMENTS IN DEODORANT PRICES
An application of the Dynamic Hierarchical Factor Model (DHFM) to the regular prices of deodorant brands in the Netherlands
Master thesis, MSc Marketing, specialisation Marketing Intelligence University of Groningen, Faculty of Economics and Business
Submitted: June 26, 2017
Author Supervisor/University
Jasper Hidding Dr. Keyvan Dehmamy
Studentnumber: 2737035 University of Groningen
Akeleistraat 15
9945 VD Wagenborgen Second supervisor
Tel.: +31 (0)6-10114392 Dr. Felix Eggers
e-mail: j.j.hidding@student.rug.nl University of Groningen
“The most incomprehensible thing about the world is that it is comprehensible”
HIERARCHICAL MOVEMENTS IN DEODORANT PRICES
An application of the Dynamic Hierarchical Factor Model (DHFM) to the regular prices of Deodorant Brands in the Netherlands
Jasper Hidding
ABSTRACT
Do prices move differently over time for different brands at different supermarket chains? In this thesis, a Dynamic Hierarchical Factor Model – as introduced by Moench, Ng, and Potter (2009) – is used to analyse the differences in variation across different deodorant brands in the Netherlands. Not only do the prices of deodorant brands differ in the way they move in relation to the common movement of all prices, the factors that account for the variance are also unique to some brands. This study assesses how deodorant prices respond to an industry-wide shock using an Impulse Response analysis, and find that high-end brands are more affected by such a shock than low-end brands. The effects for individual brands differ at the chain-level.
Finally, the usefulness and value of the Dynamic Hierarchical Factor Model in terms of predictive power is demonstrated by comparing a Factor Augmented VAR analysis that includes the common movement F with a normal Auto Regression. The FAVAR is found to vastly outperform the AR, demonstrating that the DHFM is useful tool to concentrate a large amount of information in a compact factor.
Keywords: Dynamic Hierarchical Factor Model, DHFM, variance decomposition, impulse
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FOREWORD
A four month journey comes to an end with the finishing of this thesis on Dynamic Hierarchical Factor Models. I started out blank and without any idea what a “DHFM” even was, and experienced a steep learning curve that was riddled with Eureka moments. There were times I felt like I was just doing something without knowing what, but these moments steadily grew sparse. Although the use of DHFM models is mostly restrained to macro-economic and monetary topics, I hope this thesis makes a case for a more broad application of the theory.
Of course this has not been a solitary journey and I duly acknowledge the help and support I have received along the way.
First and foremost I would like thank my supervisor, dr. Keyvan Dehmamy, for always being there when we needed help and got stuck in the theory and application. When we felt like we were lost, Keyvan would make time to discuss the matter and provide us with guidance and the apt knowledge to continue.
A large part of my gratitude also goes out to my parents – Elzoderk and Marina – and my brother and sister – Sebastiaan and Charlotte – who must have gotten really tired of all the times I tried to explain to them what I was doing, and who were faced with my irritable moods and frustration when things did not work out the way I wanted. It is their support, love, and encouragement that is responsible for where I am now, and for the endeavours I pursue.
In addition, I would like to thank Dana Ilie – who has been a great sparring partner and has really helped advance my work – and Rosanne Heijligers – who has provided me with valuable feedback on my work – for their help and support.
Finally, I would like to express my appreciation to dr. Felix Eggers for taking the time to read my work and act as second supervisor.
Wagenborgen, 26th of June 2017
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LIST OF ABBREVIATIONS AIC………...Akaike’s Information Criterion
AR………….Autoregression
DHFM…. ….Dynamic Hierarchical Factor Model FAVAR…….Factor Augmented Vector Autoregression
M…………...Mean (statistical)
MCMC……..Markov Chain Monte Carlo PCA………...Principal Component Analysis
SD…………..Standard Deviation (statistical)
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INTRODUCTION
Benjamin Franklin once wrote that nothing in this world can be said to be certain, except death and taxes. In a similar fashion, there are a number of products of which one can say with relative certainty that people will always keep buying and using them. One such product category that is deeply embedded in worldwide personal care routines, is deodorants. In 2016, the global antiperspirant and deodorant market was estimated to be worth approximately $18,9 billion, which is expected to grow by nearly $1 billion in 2017 (“Size of the global antiperspirant and deodorant market 2012-2021”, n.d.). Deodorants are so called “high penetration low frequency” products, which means that they are consumed slowly but regularly (Fader & Lodish, 1990). Given its significant size and important place in daily routines, it is interesting to take a closer look at the development of deodorant prices and how much of their variations can be explained by common dynamics.
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deodorant price development, this study seeks to identify differences in price development between the industry as a whole and individual brands and brands at different supermarket chains. The data stretches over a period that contains the start and end of a price war, which adds to the value of this analysis. By comparing the movements of prices before, during, and after the price war, this study provides extensive insights into the effects of an industrywide negative shock on regular price development of different brands at different supermarkets, as well as their potential deviation from the common price movement. Prior studies have shown that for another commodity product, beer, prices developed differently across brands during weeks in which there was a football game in the highest German football league; the Bundesliga. At the level of individual brands, some beer brands showed significant price increases during while other beer brands showed significant price decreases in response to a brand-level shock (Empen, & Hamilton, 2013). It is therefore reasonable to expect that a similar effect – once that differs per brand – could be observed with deodorants. An article by Van Heerde, Gijsbrechts, and Pauwels (2008) states that price wars by definition constitute market disruptions and major strategy changes, which go so far as to completely change the pricing strategy (e.g. from premium and high-service to a lowest-price formula at Albert Heijn). It is therefore interesting to analyse how prices of different brands at different supermarket chains respond to a market-wide shock like a price war, and how these return to their original price points after some time. It could be that the supermarket chains use the opportunity to reassess their pricing strategies and change these.
8 CHAPTER 1 – METHODS
1.1 Data
The data that will be used for this research is a dataset with time-series data of eight deodorant brands for five supermarket chains over a period of 124 weeks. The first data point is week 46 in 2003 and the final data point is week 12 in 2006. The regular prices of the deodorants are expressed in euros per 100 ml. The eight brands in the dataset are Axe, Dove, FA, Nivea, Rexona, Sanex, Vogue, and 8x4.
The time series are assumed to be stationary, mean-zero, standardized, and to have a unit variance (Moench, Ng, and Potter, 2009). Because some of the observations have a value of 0 for the regular price in several cases, the data was adjusted so that these values take the value of the surrounding regular prices. This way the 0 values do not cause strong positive or negative peaks in the analysis and their value is equal to the surrounding ones.
Based on the regular prices in the dataset, the brands are divided in high-end and low-end brands. The mean prices of the brands over the entire period in the dataset are calculated, and based on these means the brands are divided in end and low-end brands. The high-end brands are Dove (M = €2.45), Nivea (M = €2.76), Rexona (M = €2.45), and Axe (M = €3.02). These brands have the highest mean regular price. The low-end brands are then the remaining four brands; Fa (M = €1.70), Sanex (M = €2.22), Vogue (M = €2.28), and X8X4 (M = €1.88), which have the lowest mean regular price. Table 1 shows the descriptive statistics for each brand across all chains.
Brand N Mean price SD Median price Maximum price
Dove 620 €2.45 0.37 €2.37 €3.07 FA 620 €1.70 0.17 €1.70 €2.44 Nivea 620 €2.76 0.29 €2.69 €3.48 Rexona 620 €2.44 0.38 €2.41 €3.23 Sanex 620 €2.22 0.29 €2.09 €2.89 Vogue 620 €2.18 0.52 €2.11 €2.75 X8X4 620 €1.88 0.41 €1.70 €2.54 Axe 620 €3.02 0.33 €2.90 €3.70
Table 1: Descriptive statistics for the regular price of each brand
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it highlights the large difference between the maximum prices of the different brands (and supports the notion of a high-end and low-end segment). The aforementioned results in a four-level dynamic factor model consisting of 2 blocks b – which correspond to the high-end and low-end categories – and 8 subblocks s that correspond to the eight deodorant brands. Each subblock subsequently consists of 𝑁𝑏𝑠time series – corresponding to the individual brands at individual supermarket chains.
For additional information, a graph depicting the development of the average price per brand across all chains has been made:
Figure 1: Average regular price per brand across all chains
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products, including several deodorant brands. It is interesting to note that the other supermarket chains did not respond to Albert Heijn’s aggressive price reductions until around week 64 (C1000). The other three chains in this dataset – Edah (week 70), Super de Boer (week 67), and Jumbo (week 67) waited even longer before also joining in.
1.2 Factor Analysis
The Dynamic Hierarchical Factor Model (DHFM) is an adaptation of the basic factor analysis. Therefore, the basic factor analysis will first be discussed here.
When confronted with a large number of variables, one way of creating more parsimonious data is by identifying underlying factors. This is based on the assumption that a factor structure underlies the data and thus that they have a common-idiosyncratic decomposition. A factor analysis therefore provides a way of defining what type of variation is relevant for the data as a whole (Boivin, & Ng, 2006). Widaman (1993) adds to this that factor analysis is a procedure that is used to reduce the dimensions of a set of observed variables, creating factor solutions that share common variance. The covariation that exists among these variables can then be described by a small number of common factors and a unique factor for each of the variables (Malhotra, 2007).
Borrowing from Malhotra (2007), if the variables are standardized the factor model can expressed as:
𝑋𝑖 = 𝐴𝑖1𝐹1+ 𝐴𝑖2𝐹2+ 𝐴𝑖3𝐹3+ ⋯ + 𝐴𝑖𝑚𝐹𝑚+ 𝑉𝑖𝑈𝑖
where 𝑋𝑖 is the ith standardized variable, 𝐴𝑖𝑗 is the standardized multiple regression coefficient of variable i on common factor j, F is the common factor, 𝑉𝑖 is the standardized regression coefficient of variable i on unique factor j, 𝑈𝑖 is the unique factor for variable i, and
m is the number of common factors. These unique factors are not correlated with each other or
with the common factors, the latter which can be expressed as linear combinations of the variables observed:
𝐹𝑖 = 𝑊𝑖1𝑋1+ 𝑊𝑖2𝑋2+ 𝑊𝑖3𝑋3+ ⋯ + 𝑊𝑖𝑘𝑋𝑘
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variable belongs. The idea is that the total number of variables can then be redistributed over a lower number of factors or dimensions. Interpretation of the factors is done by identifying the variables that have large loadings on the same factor. So the factors are estimated based on the common variance they share. Because the first factors already explain a large percentage of the shared common variance, the dimensionality can be reduced by going with a the number of factors that explains the largest part of the variance, instead of keeping all individual variables. In doing so, the factors will still explain most of the total variance, while simultaneously becoming more parsimonious, which reduces the dimensionality in the data (Malhotra, 2007).
1.3 Dynamic Hierarchical Factor Models
However, the basic factor analysis does not incorporate hierarchical dynamical effects in the data. That is, in many analyses there is a block-like structure that separates variations between different blocks and the common movement in the data, as well as idiosyncratic noise in large dynamic panels. The Dynamic Hierarchical Factor Model (DHFM) allows for dimension reduction in the same way as the basic factor model does, while simultaneously taking into account the hierarchical structure of the factors. In this way, the models accounts for covariations that are not sufficiently prevalent to be treated as individual factors (Moench, Ng, and Potter, 2009). In this research, it is argued that at each time t, series n in a given block
b has four sources of variations. The first is idiosyncratic (or unsystematic) variation, which
pertains to a specific brand at a specific supermarket chain (e.g. Axe at Albert Heijn). The second is subblock-specific variation, which is the variance of a specific brand in our case (e.g. the variation of Axe). The third source of variation is the block-specific variation, which in this case is the variation for a block of high-end and low-end brands. The final source of variation is then the common-movement, which captures the market-wide variation.
This division is based on the idea that, although all brands are expected to share common variation, some of the dynamics may be specific to high-end or low-end brands. In similar fashion, although all high-end and low-end brands are expected to share common variation, it is also highly likely that some of the variation pertains to the individual brands, et cetera (Moench, Ng, and Potter (2009).
Based on the above, a four-level dynamic factor model is constructed. The time series
i in a given block b (e.g. brand Axe at supermarket chain Albert Heijn) is decomposed into a
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variation that is unique for the time series i at subblock s and block b. The common component is denoted by Λ𝐻.𝑏𝑠𝑖(𝐿)𝐻𝑏𝑠𝑡, with Λ𝐻.𝑏𝑠𝑖(𝐿) being the polynomial of lag order L, and describes the variation that time series i at block b shares with the other time series at block b. Taken together, this can be described by the level-one equation:
𝑍𝑏𝑠𝑖𝑡 = Λ𝐻.𝑏𝑠𝑖(𝐿)𝐻𝑏𝑠𝑡+ 𝑒𝑍𝑏𝑠𝑖𝑡
The variation of each subblock s in a given block b (𝐻𝑏𝑠𝑡) can be decomposed into a serially correlated specific component and a common component. The subblock-specific component describes the variation that is unique for the given subblock and can be denoted by 𝑒𝐻𝑏𝑠𝑡. The common component describes the variation that the subblock shares with other subblocks and is denoted by Λ𝐺.𝑏𝑠(𝐿)𝐺𝑏𝑡, with Λ𝐺.𝑏𝑠(𝐿) being the polynomial of lag order L. This together leads to the level-two equation:
𝐻𝑏𝑠𝑡 = Λ𝐺.𝑏𝑠(𝐿)𝐺𝑏𝑡+ 𝑒𝐻𝑏𝑠𝑡
Moving up to the block-specific level (𝐺𝑏𝑡) the variation can again be decomposed into a serially correlated specific component and a common component. The block-specific component describes the variation that is unique for any given block (which is either the high-end or the low-end block) and is denoted by 𝑒𝐺𝑏𝑡. The common component describes the variation of the blocks that all blocks share with each other, and is denoted by Λ𝐹.𝑏(𝐿)𝐹𝑡, with Λ𝐹.𝑏(𝐿) being the polynomial of lag order L. Adding these together to get an expression that captures the variation at the block-specific level then results in the level-three equation:
𝐺𝑏𝑡 = Λ𝐹.𝑏(𝐿)𝐹𝑡+ 𝑒𝐺𝑏𝑡
The final level at the top of the hierarchy – the level-four equation – is then 𝐹𝑡, an expression that describes the variation of economy-wide factors (prices in this case) and which are assumed to be serially correlated. The stochastic process of 𝐹𝑡 can be denoted by:
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In these equations, i stands for 𝑖 = 1, … 𝑁𝑏 time series, b is an indicator for the block (𝑏 = 1, 2), and s is an indicator for the subblock (𝑠 = 1, … , 8).
These expressions can be summarized in the following hierarchical model:
𝑍𝑏𝑠𝑖𝑡= Λ𝐻.𝑏𝑠𝑖(𝐿)𝐻𝑏𝑠𝑡 + 𝑒𝑍𝑏𝑠𝑖𝑡 𝐻𝑏𝑠𝑡 = Λ𝐺.𝑏𝑠(𝐿)𝐺𝑏𝑡+ 𝑒𝐻𝑏𝑠𝑡
𝐺𝑏𝑡= Λ𝐹.𝑏(𝐿)𝐹𝑡+ 𝑒𝐺𝑏𝑡 𝜓𝐹.𝑘(𝐿)𝐹𝑘𝑡= 𝜖𝐹𝑘𝑡.
For a more detailed treatment of the mechanisms behind this dynamic hierarchical model, this article refers to Moench, Ng, and Potter (2009).
1.4 Factor Augmented Vector Autoregressive Models (FAVAR)
The Factor Augmented Vector Autoregression (hereafter FAVAR) is a combination of the factor model for the variables and the Vector Autoregression (VAR) for the factors. Normal VARs are an often used technique to identify structural shocks and to analyse how they propagate (Marcellino, & Sivec, 2016). After first generating a number of factors, these can subsequently be modelled with a VAR in order to identify structural shocks and how they affect the variables under analysis. Let 𝑌𝑡 be an 𝑀 × 1 vector of an observable economic variable that is assumed to drive the dynamics of the economy. Usually, one would estimate a Vector Autoregressive model using data for 𝑌𝑡 only. However, it often occurs that additional information that cannot fully be captured by 𝑌𝑡 is needed to appropriately model the dynamics. Assume that 𝐹𝑡 is a 𝐾 × 1 vector of unobserved factors that summarizes the additional information that is required in modelling the dynamics, with K being small. The joint dynamics of (𝐹𝑡, 𝑌𝑡) can then be given by the following expression:
[𝐹𝑡
𝑌𝑡] = 𝜇 + Φ(𝐿) [ 𝐹𝑡−1 𝑌𝑡−1] + 𝜈𝑡
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observed (Stock, & Watson, 2016). In this research, 𝐹𝑡 consists of the output of the DHFM and 𝑌𝑡 consists of the weekly market shares of the different deodorant brands.
1.5 Procedure and estimation
The model presented in paragraph 2.3 will be estimated using Gibbs sampling, which is a Markov Chain Monte Carlo (MCMC) iterative method. First, initial values for the factors and model parameters will be obtained using a simple principal component analysis (PCA). However, because these are static, their effect vanishes throughout the sampling process. The draws from the Gibbs sampling method will therefore be used for the Impulse Response analysis. The Markov Chain Monte Carlo is often used to deal with the issue of not being able to sample from the desired posterior distribution immediately. The MCMC samples each unknown parameter in turn, cycling sequentially through each unknown many times. In these draws, the subsequent draw is always conditional on the latest draws for all the other parameters. This sampling technique will reach a stationary distribution for the parameters under general conditions (Leeflang, Wieringa, Bijmolt, & Pauwels, 2013). In attempting to achieve convergence to the target posterior, a large number of iterations is run for the MCMC. Because initial iterations are not from the target posterior, these are commonly ‘thrown away’ in the so-called ‘burn-in’ period, so that their effect on the posterior inference is minimized.
After running the Markov chain, the posterior distribution will be used to identify the common movement of the regular price across all brands at all supermarket chains. Once this has been established, the variance decomposition and impulse response will be analysed. The decomposition of variances provides insights into the variations on block level, sub-block level, and on the level of idiosyncratic components. It allows for an evaluation of the importance of the variations at each level in the DHFM. All these decomposed variances add up to the total variance in regular prices. The impulse responses show how an economy-wide negative shock affects the regular prices of the different brands, both at the generic brand and at the supermarket chain level. In addition to this, the model shall be used to forecast variation in the regular price after a negative economy-wide shock.
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each week and these are then divided by the total sales per week. This results in the percentage of sales that belongs to each brand, and which will be treated as the market share in this study. Basically this FAVAR will test whether adding the extracted common factor F to a VAR analysis of market share leads to better predictions of this market share.
CHAPTER 2 – RESULTS
In this section the empirical results of the Dynamic Hierarchical Factor Model will be presented. The model was estimated using a four-level factor model consisting of one common movement and two blocks, corresponding to the high end and low end brands. Each of these blocks consists of four sub-blocks that correspond to the four brands in the high end and low end category respectively. Finally, within each of the sub-blocks there are five variables that represent the regular price of the brand at the five supermarket chains. This leads to a total of N = 40 time series with 124 observations of the regular deodorant price measured in euros. For the Markov Chain Monte Carlo 40,000 draws were made, of which 20,000 were left out as a burn-in to make sure that the sampler has converged. Every 20th draw was stored, which leads to 1,000 draws which can be used to compute the posterior distribution.
2.1 Common Movement
After extracting the values for F and sampling a distribution using the Gibbs sampler, the following graph is obtained:
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The graph shows the common movement of the regular prices of deodorant across all brands and all chains. Hence, this figure captures the common factor F with 5% confidence bands. As was to be expected, the common movement of regular prices shows a steep decline around the time the price war between the supermarket chains started (week 57). Throughout the year following the start of the price war, the common movement of the prices fails to return to pre-price war levels.
2.2 Variance decomposition
One of the aims of this study is to observe how the fluctuations in regular price are influenced on the different hierarchical levels (i.e. on the economy-wide level, the high-end/low-end level, the brand level, and the supermarket level). In order to facilitate this, a variance decomposition analysis has been constructed. The variance decomposition analysis is a widely used analysis in strategic management and economics and can be used to explain relative variance explained by period, industry, firm, or firm characteristics for example. In this study, variance decomposition shall be used to explain relative variance explained by industrywide price development, the high-end/low-end segment, the brand level, and the chain-specific variations per brand. In order to obtain these relative variances, the total variance is split to ascribe a part of the total variance to the segment the brand belongs to (i.e. high-end or low-end), the brand-specific characteristics, and the chain-specific characteristics per brand (e.g. the price variation of Axe at Albert Heijn) (Kim, & Patel, 2017). The following tables show the variance decomposition for each block at the different levels of the hierarchy, as described above. The numbers in the tables are the posterior means over the mean shares for the series of each of the brands (Dehmamy, & Halberstadt, 2015). The standard errors are included between parenthesis.
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It is interesting to find that apparently the different brands in the high-end segment differ in the way the variance is decomposed over the different hierarchical levels. All brands do have in common that only a very small part of the total variance is explained by the segment level (0% to 2.4%). When looking at the common movement, this only explains a large part of the total variance for Dove (24%) while it is very low for the other brands in the high-end segment (0.3 – 1.55%). The brand-specific component explains most of the total variance for Rexona (64.3%) and Axe (67.2%). For Nivea it explains a large part of the variance (39.1%), while for Dove it only explains a tiny part of the total variance (2.2%). The chain level accounts for an important part of the total variation for all high-end brands, however also here there are clear differences. Dove is again the odd one out, with 71.4% of the variance in its regular price explained by the chain level. For Nivea the chain level accounts for 59.3% of the total variance and for Rexona and Axe this is slightly lower (35.1% and 32.5% respectively). This means that the brands differ in the way the different hierarchical levels contribute to the total variance of the regular price. An shock to the common factor will therefore have a larger effect on Dove than on the other brands. Similarly, a shock to the brand-specific component will have a larger effect on the other three brands. whereas the other brands are mainly affected by the brand-specific and chain-specific components. Finally, none of the brands is really affected by a shock to the block level, while all brands are susceptible to a shock to the chain-specific level. Low-end segment Brand 𝑆ℎ𝑎𝑟𝑒𝐹 𝑆ℎ𝑎𝑟𝑒𝐺 𝑆ℎ𝑎𝑟𝑒𝐻 𝑆ℎ𝑎𝑟𝑒𝑋 FA 0.007 (0.009) 0.097 (0.017) 0.032 (0.005) 0.864 (0.016) Sanex 0.000 (0.000) 0.000 (0.000) 0.236 (0.029) 0.763 (0.029) Vogue 0.000 (0.000) 0.001 (0.002) 0.193 (0.011) 0.806 (0.011) X8X4 0.000 (0.000) 0.001 (0.002) 0.652 (0.060) 0.347 (0.060) Table 3: Variance decomposition of the low-end segment
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of the total variance). At the brand-specific level something interesting happens, as X8X4 shows a much larger explained variance than the other three brands in the segment (65.2%) and FA shows a much smaller explained variance (3.2%). Sanex and Vogue show comparable explained variances at 23.6% and 19.3% respectively. Finally, three brands show a large explained variance on the chain-specific level (76.3% to 86.4%) whereas X8X4 shows a much smaller proportion of the variance explained here (34.7%). Hence, none of the brands would be really affected by a shock to the common factor, and only FA would be slightly affected by a shock to the block-level component. All brands are heavily affected by shocks to the brand-specific and chain-brand-specific components. More precisely, FA, Sanex, and Vogue are mostly affected by shocks to the chain-specific component and X8X4 is mostly affected by shocks to the brand-specific component.
2.3 Impulse Response
Now that it is known how the total variance is decomposed over the different hierarchical levels, it is interesting to look at how the regular price of the deodorant brands responds to changes in the deodorant market. As a means to do so, an impulse response analysis is performed to examine how segment-, brand-, and chain-specific factors respond to a shock to the first common factor. Figure 3 shows the effect of a one standard deviation shock to the first common factor.
Figure 3: Impulse responses of the common factor and block specific factors with a shock on the first common factor
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levels for the common factor. For the block-specific factor of the high-end segment, this would take approximately 85-90 weeks (the line shows a very asymptotic movement in the tail). No significant impulse response to a shock on the common factor was found for the block-specific factor of the low-end segment, as can be seen from the distribution of the factors.
When looking at the impulse response of the individual brands to a shock to the common factor, Dove is the only brand that shows a significant effect. In the case of a positive shock to the common factor, the price of Dove is positively affected and takes approximately 75 weeks to return to the normal price level, with the peak of the effect lying around the fifth week. The impulse response graphs are presented in figure 4 and show the effects across all chains.
Figure 4: Impulse responses of the subblock-specific factors (brand level) with a shock on the first common factor
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Figure 5: Impulse responses of the subblock-specific factors (brand level) with a shock to the first common factor at Albert Heijn
A positive shock to the first common factor leads to a positive impulse response for Dove at Albert Heijn. The peak of the impulse response lies around week 5 and it takes approximately 100 weeks before the effect dies out. Again, the other brands show no significant effect as the distribution of the factors spans both positive and negative values.
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Similarly to Albert Heijn, a positive shock to the first common factor also leads to a positive impulse response for Dove at Super de Boer. At Super de Boer, the peak of the impulse response lies around week 10, which is markedly later than at Albert Heijn. This indicates that it takes longer for the regular price of Dove to respond to a shock at Super de Boer than it does at Albert Heijn. Similarly to Albert Heijn though, the impulse response tends to 0 after approximately 100 weeks. Unsurprisingly, the other brands again show no significant effect, as was to be expected based on the variance decomposition.
At Edah, Jumbo, and C1000 no significant impulse responses were found for any of the brands with a positive shock to the first common factor. This can be explained by the fact that this analysis only considers 1 common factor. If more common factors, or on a different hierarchical level, more segment or subblock-specific factors would have been included, this would likely have led to a better representation of the real impulse responses. The graphs of these brands have been included in appendix A.
In addition to shocking the common factor F, another analysis tested the effect of applying a shock to the low end segment. This specific shock was chosen because the variance decomposition showed that most of the brands were not affected by the segment factor, except for FA. With FA being in the low-end segment, it is interesting to see whether the same effect as in the variance decomposition can be found in the impulse response analysis. A shock to the low-end segment resulted in the following impulse response:
Figure 7 : Impulse responses of the subblock-specific factors (brand level) with a shock on the block-specific low-end segment.
22 2.4 Forecasting
Next, the extracted factors for the common movement F shall be used to forecast the regular price per brand. The extracted factors of the first 100 weeks will be used to make the forecast for the next period. On the brand- and chain-specific levels these forecasts are also compared to the actual price in the last 24 weeks after 𝑡 = 100. Figure 7 shows the forecasts for the common movement and the high-end and low-end segments based on F. These forecasts show an expectation for the development of the regular price based on the data that is available, it does not include the effect of a shock like in the impulse response.
Figure 8: Forecast for the common movement, high-end segment, and low-end segment with 95% confidence bands
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200 weeks the movement of the price is forecast to tend to 0 again. The bottom right graph shows an entirely different picture, as the movement in the low-end segment fluctuates heavily over time. The response to the price war is less pronounced in this segment and prices seems to recover rather quickly. According to the forecast, the variance in the price is expected to tend to 0 as early as in week 130, which is markedly faster than in the high-end segment. We can therefore say that the movement of the regular price at the block-specific level returns to 0 much slower in the high-end segment than in the low-end segment. Furthermore, the high-end segment is closely linked to the forecast of the common movement
F. Figure 8 shows the forecasts for the different brands at Albert Heijn for the last 40 periods.
The purple line illustrates the actual development of the regular price, the red line shows the forecast price development, and the yellow and blue lines denote the 95% confidence bands.
Figure 9: Forecast of price development for all brands at Albert Heijn with 90% confidence bands
The forecast appears to be quite accurate for all brands at Albert Heijn, as judged by the purple line being within the confidence bands most of the time. The only one that is really off is X8X4, of which the actual development appears to be very close to 0 while the forecast expects a massive growth (especially compared to the other forecasts at Albert Heijn). In addition there are some peaks in the data that fall outside the confidence bands, such as for Vogue. Since this model is pretty basic and only extracts one common factor, it is to be expected that not all real variance is accounted for.
24
confidence bands. The only odd one out is X8X4, which consistently predicts large increases whereas the actual variance mostly remains stationary. The probable explanation for this is that a lot of the variance for X8X4 at Albert Heijn can be explained by the idiosyncratic level. This means that it does not take a lot of variance from the common movement. Because the prediction is mostly off, the actual data rarely lie within the 90% confidence bands for X8X4 and it can be argued that the forecasts based on the extracted common factor do not perform well for X8X4. Another version of the forecast was also made, where F was not included. This so-called in-sample prediction can be found in appendix C.
2.5 Factor Augmented VAR
In order to test the usefulness of the common factor that was extracted and the entire DHFM analysis, a Factor Augmented Vector Autoregression (FAVAR) is performed. The FAVAR is actually a basic Vector Autoregression (VAR) that is enhanced with a factor. In this case, the factor is the common factor that was extracted in the DHFM. Hence, the VAR is enhanced by a factor that can be treated as observed. The FAVAR will be used to make predictions for the market share of all eight brands, on the basis of the market share and common factor F in period 𝑡 − 1. In addition to simply using one lag, the FAVAR analysis also computes more lags to see whether this improves the model. The selection of the most appropriate number of lags is done by R studio based on Akaike’s Information Criterion (AIC). A standard Autoregression (AR) will also be performed in order to have a benchmark to compare the FAVAR to. This then allows for a comparison to see to what extent the enhancement with the common factor F improves predictions. The FAVAR analysis is computed per brand across all chains, which means that the market share is simply the market share of the brand across all chains.
For each brand, the FAVAR returns four coefficients from the following two equations:
Equation 1: 𝑌𝑡= 𝑎0+ 𝑎 ∙ 𝑌𝑡−1 + 𝑏 ∙ 𝐹𝑡−1+ 𝜀1 Equation 2: 𝐹𝑡 = 𝑐0+ 𝑐 ∙ 𝑌𝑡−1+ 𝑑 ∙ 𝐹𝑡−1+ 𝜀2
25 Dove
𝑌𝑡 𝐹𝑡
𝑌𝑡−1 0.2553 (0.036) 1.0459 (1.371)
𝐹𝑡−1 -0.0088 (0.006) 0.9710 (0.088)
Table 4: FAVAR coefficients for Dove, with the standard deviation in parentheses
The FAVAR analysis showed that for Dove, a 1 period lag leads to the optimal predictive model. Therefore the coefficients shown above relate to a 1 period lag.
These coefficients result in the following four histograms:
Figure 10: The histograms of the four coefficients reported above, showing the distribution of the posterior for 1 period lag for Dove
26
Figure 11: Graph showing the predictions for market share of Dove by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 1 period lag)
The bold black line depicts the prediction of market share and the bold dashed lines show the 95% confidence interval. The solid black line shows the prediction of market share by the FAVAR and the dashed line that quickly returns to 0 shows the prediction of market share by the VAR. It quickly become obvious that the FAVAR delivers a much better performance in terms of prediction. Although large parts still lie outside the confidence bands, the prediction vastly outperforms the VAR. The VAR seems to predict a negative trend that ultimately regresses to 0, whereas in reality this would not be very realistic. This shows that simply adding the extracted common factor F from the DHFM hugely increases the predictive value of a very simple autoregressive analysis.
For Nivea, another interesting result was found. The FAVAR model showed that the prediction for Nivea was optimal with 4 periods lag. The below tables shows the coefficients for a model with 1 period lag, as well as the coefficients of the first and fourth lag in a model with 4 periods lag:
Nivea
1 period lag 𝑌𝑡 𝐹𝑡
𝑌𝑡−1 0.1563 (0.006) 0.6249 (0.841)
𝐹𝑡−1 0.0044 (0.008) 0.9560 (0.010)
27 Nivea 4 periods lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.1852 0.9489 𝐹𝑡−1 -0.0035 0.7786 𝑌𝑡−4 -0.0680 (0.014) -2.4860 (0.861) 𝐹𝑡−4 0.0000 (0.007) 0.3102 (0.090) Constant 0.0761 0.2379
Table 5: FAVAR coefficients of 1 period lag and 4 periods lag for Nivea, with the standard deviation in parentheses
As these coefficients already show, the distribution strongly changes when using 4 lags instead of 1. The histograms corresponding to 4 periods lag are shown below, the histograms for the original 1 period lag can be found in appendix E.
Figure 12: The histograms of the four coefficients reported above, showing the distribution of the posterior for 4 periods lag for Nivea
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lines show the 95% confidence bands for the FAVAR prediction, and the lowest dashed line shows the prediction of the normal VAR.
Figure 13: Graph showing the predictions for market share of Nivea by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 4 periods lag)
Although it is hard to judge from this graph, compared to the 1 period lag model (which can be found in appendix F) it seems to perform slightly better in that the real values fall within the confidence bands slightly more often.
29 CHAPTER 3 – CONCLUSION
The aim of this study was to explore the use of Dynamic Hierarchical Factor Models (DHFM) in analyzing the movement of the regular price of deodorant brands. By applying the DHFM, this study sought to analyze the movement of the regular price on four different levels: the common level, the block-specific segment level, the subblock-specific brand level, and the idiosyncratic/chain level. In analyzing the movement of price on these different hierarchical levels, some interesting findings were reported. The common movement behaved as one would expect, showing a steep decrease during the price war between supermarkets. However, the extent of the effect varied across brands. Some brands were unique in their variance, like Dove being strongly affected by the common and segment level, where other brands showed next to no effect by these factors. Or X8X4 being the only low-end brand that is affected by the block-specific segment factor. Overall however, the most relevant factors were the subblock-specific brand level and the idiosyncratic chain level. These levels generally accounted for the largest share of the variance in the regular price of deodorant brands.
The impulse response analysis showed that only for Dove a significant impulse response could be reported after a one standard deviation shock to the common factor, and that this was only the case at Albert Heijn and Super de Boer. This was not surprising, as Dove was the only brand with a large share of the variance explained by the common movement. When moving down one level, a one standard deviation shock to the block-specific low-end segment was found to only affect X8X4. This, like for Dove, could be explained by the variance decomposition, which already pointed to X8X4 being affected by the block-specific segment factor. Subsequently, for all brands except X8X4, forecasts were found to be very good and for nearly all brands at all chains the observed values fell within the 90% confidence bands.
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31
CHAPTER 4 – LIMITATIONS AND FURTHER RESEARCH
The present study is of a highly explorative nature and there are plenty of other possibilities and research angles to pursue. First of all, this paper uses only a small number of deodorant brands at a limited number of supermarket chains. Of course there are many more retail points selling deodorants and plenty of additional deodorant brands. So while this study provides interesting insights, they are bound by the limited number of brands and supermarket chains, and certainly do not represent the entire deodorant market. Another limitation of this analysis lies in the fact that it only looks at the movement of regular prices. These only make up for one part of the equation of course, as there are also promotional prices – which have a strong symbiotic relation with regular prices – and other forces such as promotional activities like feature and display advertising. Another issue that becomes clear when looking at the data and the way the variance is spread across brands, is that even within the two segments there appear to be different movements and developments. For example, prices of X8X4 and FA actually increase during the price war, while the other two brands in the low-end segment decrease in price. Similarly, for some brands most of the variance (>75%) is accounted for by the subblock-specific brand level, while for others the largest share is explained by the idiosyncratic chain level. Therefore, other distinctions than the high-end/low-end segment could be made and also an extra level could be added to account for these different movements. When looking at the overall model, this study only uses a simplified version of the DHFM and therefore, its performance could be improved by orders of magnitude if more factors would be considered. In addition, the FAVAR analysis treated the extracted common factor F as an endogenous variable. Future research could attempt to repeat this analysis and treat the factor F as an exogenous variable.
32 REFERENCES
Bernanke, B.S., Boivin, J., & Eliasz, P. (2005). Measuring the effects of monetary policy: a factor-augmented vector autoregressive (FAVAR) approach. The Quarterly Journal of
Economics, pp. 387-422
Bhasin, K. (2011, October 10). How Axe became the top-selling deodorant by targeting nerdy losers. Business Insider. Retrieved from http://www.businessinsider.com/axe-advertising-unilever-brandwashed-2011-10?international=true&r=US&IR=T
Boivin, J., & Ng., S. (2006). Are more data always better for factor analysis? Journal of
Econometrics, 132, pp.169-194
Dehmamy, K. & Halberstadt, A. (2015). Modeling the term structure of interest rates in a dynamic hierarchical factor model. Essays on Bayesian modeling in marketing and
economics, pp. 1-43
Empen, J., & Hamilton, S.F. (2013). How do supermarkets respond to brand-level demand shocks? Evidence from the German beer market. American Journal of Agricultural
Economics, 95(5), pp. 1223-1229
Fader, P.S., & Lodish, L.M. (1990). A cross-category analysis of category structure and promotional activity for grocery products. Journal of Marketing, 54(4), pp. 52-65
Gupta, R., Jurgilas, M., & Kabundi, A. (2010). The effect of monetary policy on real house price growth in South Africa: a factor-augmented vector autoregression (FAVAR) approach.
Economic Modeling, 27(1), pp. 315-323
Kim, K.Y., & Patel, P.C. (2017). Employee ownership and firm performance: a variance decomposition analysis of European firms. Journal of Business Research, 70, pp. 248-254
Leeflang, P.S.H., Wieringa, J.E., Bijmolt, T.H.A., & Pauwels, K.H. (2013). Modeling
Markets: analyzing marketing phenomena and improving marketing decision making. New
33
Malhotra, N.K. (2007). Marketing research: an applied orientation. Upper Saddle River, NJ: Pearson/Prentice Hall
Marcellino, M., & Sivec, V. (2016). Monetary, fiscal and oil shocks: evidence based on mixed frequency structural FAVARs. Journal of Econometrics, 193, pp. 335-348
Moench, E. (2008). Forecasting the yield curve in a data-rich environment: a no-arbitrage factor-augmented VAR approach. Journal of Econometrics, 146, pp. 26-43
Moench, E., Ng, S., & Potter, S. (2009). Dynamic Hierarchical Factor Models. Federal
Reserve Bank of New York Staff Reports, 412, pp. 1-44
Raftery, A.E., & Lewis, S. (1991). How many iterations in the Gibbs sampler? Bayesian
Statistics, 4(2), pp. 763-773
Size of the global antiperspirant and deodorant market from 2012 to 2021. (n.d.). Retrieved from https://www.statista.com/statistics/254668/size-of-the-global-antiperspirant-and-deodorant-market/
Stock, J.H., & Watson, M.W. (2016). Factor Models and Structural Vector Autoregressions in Macroeconomics. Forthcoming in the Handbook of Macroeconomics.
Van Heerde, H.J., Gijsbrechts, E., & Pauwels, K. (2008). Winners and losers in a major price war. Journal of Marketing, 45(5), pp. 499-518
34 APPENDICES
Appendix A: Impulse Response at Edah, Jumbo, and C1000
Figure 14: Impulse responses of the subblock-specific factors (brand level) with a shock to the first common factor at Edah
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36
Appendix B: Forecasts for all brands at Edah, Super de Boer, Jumbo, and C1000
Figure 17: Forecast of price development for all brands at Edah with 90% confidence bands
Figure 18: Forecast of price development for all brands at Super de Boer with 90% confidence bands
37
38
Appendix C: Forecasts without F for all brands at all chains
Figure 21: Forecast of price development for all brands at Albert Heijn without the common factor F (with 90% confidence bands)
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Figure 23: Forecast of price development for all brands at Super de Boer without the common factor F (with 90% confidence bands)
Figure 24: Forecast of price development for all brands at Jumbo without the common factor F (with 90% confidence bands)
40
Appendix D: FAVAR coefficients Rexona 1 period lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.0269 (0.003) -0.1498 (0.351) 𝐹𝑡−1 -0.0038 (0.002) 0.9588 (0.009) Constant 0.2078 0.0205 5 periods lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.0269 -0.1498 𝐹𝑡−1 -0.0038 0.9588 𝑌𝑡−5 -0.0909 (0.016) 0.3641 (0.354) 𝐹𝑡−5 0.0584 (0.023) -0.0875 (0.099) Constant 0.2211 -0.2904
Table 6: FAVAR coefficients for Rexona at 1 period lag and at 5 periods lag, with the standard deviation in parentheses
Axe
1 period lag 𝑌𝑡 𝐹𝑡
𝑌𝑡−1 0.2088 (0.014) -0.2998 (0.504)
𝐹𝑡−1 -0.0057 (0.009) 0.9567 (0.076)
Constant 0.1721 0.0536
Table 7: FAVAR coefficients for Axe at 1 period lag, with the standard deviation in parentheses
FA
1 period lag 𝑌𝑡 𝐹𝑡
𝑌𝑡−1 0.3552 (0.017) -0.5945 (0.837)
𝐹𝑡−1 -0.0052 (0.006) 0.9543 (0.085)
Constant 0.0696 0.0525
Table 8: FAVAR coefficients for FA at 1 period lag, with the standard deviation in parentheses
Sanex
1 period lag 𝑌𝑡 𝐹𝑡
𝑌𝑡−1 0.4889 (0.040) 0.2304 (0.684)
𝐹𝑡−1 0.0028 (0.009) 0.9578 (0.094)
Constant 0.0570 -0.0371
41 Vogue 1 period lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.1356 (0.010) 1.0326 (1.025) 𝐹𝑡−1 0.0022 (0.004) 0.9572 (0.076) Constant 0.0577 -0.0811
Table 10: FAVAR coefficients for Vogue at 1 period lag, with the standard deviation in parentheses
X8X4 1 period lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.5344 (0.030) 0.6327 (0.874) 𝐹𝑡−1 0.0087 (0.001) 0.9484 (0.018) Constant 0.0527 -0.0843 4 periods lag 𝑌𝑡 𝐹𝑡 𝑌𝑡−1 0.2047 1.1057 𝐹𝑡−1 -0.0082 0.7925 𝑌𝑡−4 0.0132 (0.098) -0.4123 (1.722) 𝐹𝑡−4 0.0100 (0.024) 0.3640 (0.098) Constant 0.0377 0.3693
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Appendix E: Histograms posterior distribution FAVAR Nivea
Figure 26: The histograms of the four coefficients reported above, showing the distribution of the posterior for Nivea at 1 period lag Rexona
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Figure 28: The histograms of the four coefficients, showing the distribution of the posterior for Rexona at 5 periods lag
Axe
44 FA
Figure 30: The histograms of the four coefficients reported above, showing the distribution of the posterior for FA at 1 period lag
Sanex
45 Vogue
Figure 32: The histograms of the four coefficients reported above, showing the distribution of the posterior for Vogue at 1 period lag
X8X4
46
47
Appendix F: Graphs comparing FAVAR and VAR per brand
Figure 35: Graph showing the predictions for market share of Nivea by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 1 period lag)
Figure 36: Graph showing the predictions for market share of Rexona by the FAVAR and VAR, as well as the observed values with 95%
48
Figure 37: Graph showing the predictions for market share of Rexona by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 5 periods lag)
Figure 38: Graph showing the predictions for market share of Axe by the FAVAR and VAR, as well as the observed values with 95%
49
Figure 39: Graph showing the predictions for market share of FA by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 1 period lag)
Figure 40: Graph showing the predictions for market share of Sanex by the FAVAR and VAR, as well as the observed values with 95%
50
Figure 41: Graph showing the predictions for market share of Vogue by the FAVAR and VAR, as well as the observed values with 95%
confidence bands (modeled with 1 period lag)
Figure 42: Graph showing the predictions for market share of X8X4 by the FAVAR and VAR, as well as the observed values with 95%
51
Figure 43: Graph showing the predictions for market share of X8X4 by the FAVAR and VAR, as well as the observed values with 95%
