The Timing and Speed of New Product Price Landings

A New Model for the Timing and Speed of New
Product Price Landings
Carlos Hernandez-Mireles, Dennis Fok, Philip Hans Franses
March 27, 2009
Abstract
Many high-tech products, information goods and durable goods exhibit exactly one
significant price cut some time after their launch. We call this transition of high to
low prices price landing. In this paper we present a new model that describes two
important features of price landings: their timing and their speed.
Prior literature suggests that price landings might be driven by sales, product line
pricing, competitors sales or simply by time. We propose a mixture specification to
find out which of these explanations best describe the pricing patterns we observe in
our data. In addition, we observed that price landings differ across products and we
explicitly allow for heterogeneity in the timing and speed of their landings. We model
this heterogeneity with a hierarchical structure for each mixture component.
To our knowledge, we are the first to present an empirical study of price landings.
We estimate our model using a rich dataset containing the sales and prices of 1195
newly released video games. In contrast with previous literature, our findings suggest
that it is not product line pricing or sales but that it is mainly competition and time
itself that best describes price landings. Finally, we find substantial heterogeneity in
the timing and speed of landing across firms and product types.
KEYWORDS: PRICING, PRICING MODELS, NEW PRODUCTS
1
1
INTRODUCTION
It is well known that the sales life cycle of many durable products seems to exhibit a moment
of take-off, that is, from that moment onwards sales seem to grow almost explosively. See, for
example, the studies of Tellis et al. (2003), Stremersch and Tellis (2004), Golder and Tellis
(1997) and Sood et al. (2009). On the other hand, the prices of many durable products follow
a pattern that mimics that of sales itself but to the inverse of the sales’ S-shape. That is,
many products’ prices are cut permanently at a certain moment. We will call this transition
from an initially high price to a lower price level the price landing. Our study focuses on
modeling these transitions and we particularly focus on the moment and speed at which
they occur. Hence we offer a complement to studies like those of Tellis et al. (2003) and
Golder and Tellis (1997) about sales take-off but we study different phenomenon. In contrast
with the heuristics applied in the sales take-offs studies, we propose a novel approach that
identifies price landings relying only on statistical modeling.
We are aware of many studies on the time to take-off for sales but we are not aware
of any empirical study of price landings. This is quite unexpected because the timing of a
permanent price cut for a new product is without a doubt an important managerial decision.
Last year during April, Steve Jobs received several emails written by American customers
who purchased Apple’s iPhone and who witnessed a $200 price drop, 33% of the launch
price, just 66 days after the release of the iPhone. The complaints from customers were
taken very seriously, even such that Apple issued $100 store credit to everyone who had
already purchased an iPhone. While in the UK, Apple cut the iPhone price by £100 and
many journalists indicate this is a final rebate before the introduction of a new iPhone
version. Some reporters hypothesized that the price cut by Apple was timed too early given
that Christmas was only two months away from the price cut while others argued that
the iPhone’s price cut was due to lower than expected sales, stiff competition or due to a
possible new launch. See BusinessWeek Online (2007) and BusinessWeek Online (2008) for
more details on Apple’s story. Manufacturers of products like video games, apparel, PCs,
movies, and so on, face similar price cut decisions. For example, some academic studies point
that managers at apparel retailers in NYC report the timing of price cuts as an important
decision variable and the size of the price cut in their industry is typically between 25 and
50%, see Feng and Gallego (1995) and Gupta et al. (2006).
In this paper we develop a model that is useful to identify the timing and speed of price
landings and simultaneously it is useful to characterize the driving forces behind the price
landing patterns. We apply our model to the market of video games and to a rich data set
that concerns 1195 newly released video games.
The plan of the paper is as follows. In Section 2 we present our literature review. In
Section 3 we present our data and market context. Next in Section 4 we present our modeling
approach and in Section 5 we present our results and discussion. Details about our modeling
approach are reported in Section A and all figures and tables can be found at the end in
Section 7.
2
2
LITERATURE REVIEW
Below we review the literature related to the pricing of new products. In Section 2.1 we
review the literature on the timing of price cuts. Next in Section 2.2 we review the literature
on the speed of landings. Finally in Section 2.3 we review the literature regarding the drivers
of pricing policies.
2.1
The Timing of Price Landings
To our knowledge, there is no empirical study of the timing of price landings available and
academic literature offers still a very limited view on when firms cut prices. Among the
extensive list of articles we surveyed there are 9 empirical pricing studies among which only
Simon (1979) offers some insights on when to cut prices. Clements and Ohashi (2005), Nair
(2007) and Chintagunta et al. (2006) are three relevant empirical studies of pricing and
diffusion in the video game industry. However, their studies do not give insights about when
prices are significantly cut by video game or game console manufacturers.
In contrast, there are 25 analytical pricing studies among which 11 offer some insights
on when prices should be cut. These 11 studies are Feng and Gallego (1995), Ferguson and Koenigsberg (2007), Franza and Gaimon (1998), Gupta et al. (2006), Gupta and
Di Benedetto (2007), Krishnan et al. (1999), Chandy et al. (2006), Padmanabhan and Bass
(1993), Rajan et al. (1992), Teng and Thompson (1996) and Zhao and Zheng (2000). In
Table 1 we present a summary of the literature we reviewed. The objective of these latter
studies is in most cases to derive optimal price policies. However, we find that authors give
some but not full attention to the topic of when exactly prices should be cut. The common
suggestion of these authors is that firms should drop prices dramatically after competitive
entry, after a certain high-value consumer is not buying anymore or after the market gets
saturated. That is, the accepted academic suggestion is that prices should change (in most
cases drop) after an event modifies the market. For example, in 8 of these 11 studies prices
are allowed to jump from one high level to a lower level after a certain threshold or after a
market event. In contrast, empirical studies on pricing, in most cases, completely ignore the
timing of price cuts and the events that might trigger them. See for example Bayus (1994),
Horsky (1990), Kalish (1985) and Nair (2007). This, we believe, is a limitation of current
empirical studies and we attempt to fill this literature gap by characterizing price landings,
their timing and the events that mark them.
2.2
The Speed of Price Landings
In Table 1 we report 34 academic studies on pricing and we find that in 22 studies prices
are allowed to change gradually. In most cases the price changes are linear and monotonic
while in only 6 studies prices are allowed to move either non-linearly or to alternate between
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high and low states. In all empirical studies authors assume that prices move gradually; in
some cases prices will reflect demand changes, like in Bayus (1994) and Rao and Bass (1985)
while in other cases prices will be a function of past prices or the demand of complementary
products, like in Nair (2007), Chintagunta et al. (2006) and Clements and Ohashi (2005). In
contrast, all analytical studies derive an optimal price path but little attention is given to the
optimal price decrease rate. In some cases the optimal decrease rate is derived and reported,
like in Dockner and Gaunersdorfer (1996), Raman and Chatterjee (1995) and Bayus (1994),
but most authors do not provide insights on how this rate changes in different settings.
Additionally, it is very hard to make cross model comparisons on the optimal decrease rate
of prices at the current state of the literature. For example, we do not know if previous
literature suggests many different price decrease rates. In contrast, our data suggests a huge
diversity of the decrease rates as we will show in more detail in Section 3. We contribute to
the empirical literature by applying a model that is useful to capture the speed of landing
and its heterogeneity across many products.
2.3
The Drivers of Price Landings
Table 1 provides an overview of the main price drivers covered in the literature: saturation
(14 studies), market entry and sequential innovations (8 studies), learning curves (5 studies),
consumer heterogeneity (6 studies) and indirect network effects (2 studies). Nair (2007) and
Clements and Ohashi (2005) documented the saturation effects on video game prices and the
price elasticity during the life cycle of game consoles. On the other hand, Ravindran (1972)
and DeVries (1964) are concerned with the season effects and the efffects of the planning
horizon on the optimal pricing policies. In this study we put to an empirical test three of
these main drivers and our objective is to find out how well saturation, market entry and
simply time describe the price patterns for each of the 1195 products in our dataset. The
approach we present below is new given that we simultaneously test multiple explanations
that have been offered by the existing literature and for those that we have information
about.
3
VIDEO GAME PRICES
In this section we first describe our data and next we present a brief description of the Video
Game market.
3.1
Data
The database we analyze consists of monthly time series of unit sales and prices for 1195
PlayStation2 (PS2) video games released between September 1995 and February 2002 in the
US. This data was collected by NPD Group from retailers that account for 65% of the US
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market. We used the first two years of data for each video game and left out video games with
less than 12 monthly observations. This time frame is justified by the fact that most video
games stay on store shelves for less than two years and their sales drop very rapidly to zero
afterwards. Binken and Stremersch (2009) use the same data as ours and they assume that
a video game is in a so-called dead regime after its sales drop below 5000 units. Therefore,
Binken and Stremersch (2009) do not use any observation after this cut-off point and hence
they leave out 32 % of their observations. In our case the 24 month cut-off point leaves out
38 % of the observations. We compared our results against a 30 and a 36 month cut-off point
that leave out 28 and 20 % of the observations respectively and our results are the same.
Our final sample consists of 1075 video games.
In Figure 1 we show the price landing of 50 randomly selected video games. This figure
clearly shows great diversity of price patterns. The introductory prices range from 40 to
approximately 60 USD while their landing level is between 15 and 30 USD. Similarly, there
is great diversity in the timing of price landings. It is easy to notice some video games prices
dropping right after the second month while others land around the 10th, 12th or 15th month
or even later. Finally we notice that some prices drop very fast, see the lines almost parallel
to the vertical axis, while in many other cases they land at slower rates and with more noise
around them.
In Figure 2 we show the price landing of one of the most popular video games in our
data, the Spider-Man game. In each of the panels we plot the price of the Spider-Man game
on the vertical axis but we use a different scale on the horizontal axis. In the upper-left
panel we use time on the horizontal axis, in the upper right panel we use the cumulative
sales of Spider-Man and in the lower panel we use the cumulative number of video games
launched to the market after the introduction of Spider-Man. We choose these axis because
later we identify each of these variables as a potential driver of price landings. More details
on this are given in Section 4.3. These graphs of course show very similar price patterns.
That is, we could say that the price cut of the Spider-Man occurred approximately at the
10th month after its introduction (upper-left panel); or just after reaching 600 thousand
unit sales (upper-right panel); or after 250 video games were launched (lower panel). The
price landings in these figures are extremely similar but the interpretation of the different
thresholds are very different. In all cases, these thresholds represent an event after which
prices drop, that is the timing of price landings. Finally, if we look closely at the different
price landing patterns we discover that the speed of landing varies across these panels. Prices
drop much faster when we use cumulative sales than when we use time on the horizontal
axis.
In the analysis that follows we show how we select one of these price drivers for each of
the products in our sample. That is, we find the driver that best matches each video game
price landing. It is easy to conclude, based on Figure 2, that each of these drivers results
in the same price pattern hence they should be highly correlated or it should be possible
to express them all simply as a function of time. This is, however, a visual effect given all
horizontal axis are linear and their scale increases at a constant rate in all panels. However,
for most video games these drivers grow non-linearly as we can see in Figure 3. The sales
5
of the Spider-Man are not a linear function of time neither are the cumulative competitive
introductions. The same applies for the rest of the video games in our sample and there is a
similar diversity in sales paths as there is in price landings. Hence, in this study we focus on
price landings and how well these drivers might be useful to characterize them. Developing a
joint model for prices, sales and competitive entry is beyond the scope of this paper and we
consider it as an area for future research. We explain more details on our modeling approach
in Section 4 and next we present the market context of our application.
3.2
The Video Game Market
The video game market is highly competitive and there are 78 video game publishers who
design games for PS2. On average, they released 29 new VGs per month between 1992
and 2005. The main publisher of these VGs is Sony and it has a market share of 16%.
Acclaim and Electronic Arts follow Sony with market shares of 11% and 6%, respectively.
In the upper left panel of Figure 4 we present the distribution of the market shares across
all publishers. We notice that 20 publishers have about 80% of the market while the 58
remaining publishers cover the next 20% of the market. In the upper right panel of Figure 4
we depict the monthly time series of the number of newly released video games. There is an
upward trend in the number of VGs being released. In 1996 less than 11 VGs were released
per month while in 2002 this volume has increased to 40 monthly releases.
The lower left panel of Figure 4 shows the industry’s sales pattern. Total VG sales are
extremely seasonal and they peak every December when they may reach numbers like 14
million copies. This last number is especially high if we compare it against the 24.1 million
units of PS2 consoles sold between 1995 and 2002. Finally, in the lower right panel we show
the average number of video games released from 1995 to 2002 and the average sales per
month. An interesting fact is that most new VGs are released during November and January
but sales peak in between these two months. From 1995 to 2002, December VG unit sales
are on average 14 million and in January sales decrease to less than 3 million copies while on
average 18 new VG are released on December, 27 in November and 34 in January. In Figure
5 we can see the distribution of the type of video games sold. For example, sports games
account for 21.5 %, Action 14 % while Strategy games account for 4 % of all video games in
our data.
The consumers in this market concern 40 million US-based consumers who buy video
games each year. Figure 6 shows a histogram of the total sales across all video games.
Preferences clearly differ across VGs as we observe substantial heterogeneity in the market
potential across the video games. We follow the tradition of diffusion research by labeling
the cumulative sales reached by a video game the market potential. From Figure 6 we can
learn that sales above one million units for a single game seem to occur only rarely. The
average market potential for the video games in our sample is around 254.75 thousand units.
However, approximately for half of the video games in our sample (to be precise: for 504
video games) the market potential is less than 66 thousand units.
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4
PRICE LANDING: MODELING
Our model consists of two parts. First we present an equation to describe the price landing,
that is the underlying price of product i at time t, which we call Pi∗ (t). Next we specify
an equation that relates the pricing landing to the actually observed prices, what we call
Pi (t). As we observe in Figure 1 prices follow a general inverse S-shape but they do not
follow it very smoothly and in most cases the prices we observe are noisy. Hence, in the first
equation we capture the price landing and its main two features (timing and speed) and in
the second we capture deviations from it. In the Section 4.1 we present these two equations.
Next, as each video-game is allowed to have its own price landing speed, timing, initial price
and landing price parameters in Section 4.2 we specify how we model their heterogeneity. In
Section 4.3 we briefly discuss the mixture specification that allows us to identify the driver
that best describes each video game. Finally, in Section 4.5 we present details regarding the
co-variates in the hierarchical structure of the model.
4.1
Price Landing Model
The price landing of game i is Pi∗ (t) and we assume it depends on a driver denoted by Di (t).
That is, prices change according to
(P ∗ (t) − κi )(ρi − Pi∗(t))
dPi∗(t)
= i
dDi (t)
(κi − ρi )νi
(1)
where ρi is the starting price level, κi is the final pricing level, and νi a constant that
modifies the rate of change dPi∗ (t)/dDi (t). For ease of interpretation, dDi (t) might be for
example time and then dPi∗(t)/dDi (t) = dPi∗ (t)/dt. However, dDi (t) might be set to be
a driver that we are interested in, like sales or competition. In the last equation we note
that a smaller νi implies a faster rate of change. Here, the time index t will in each case be
relative to the launch date of the particular product. In other words for each product t = 0
corresponds to the time of launch. In the numerator of (1) we have that the closer Pi∗ (t)
is to its initial or final levels, the slower prices would change and that if Pi∗ (t) < ρi ,νi > 0,
Pi∗ (t) > κi , ρi > κi for all t then dPi∗ (t)/dDi (t) < 0. These last conditions are exactly the
phenomenon we are observing in high-tech product prices.
dP ∗ (t)
dDi (t)
Equation (1) is unusual in the sense that it models dDii (t) instead of dP
. The former is
∗
i (t)
the typical solution proposed by analytical studies while the latter is the typical form assumed
in empirical studies. One of the possible reasons why empirical studies have assumed this
latter form is that many of them focus on a single-firm, usually a monopolist, that controls
dP ∗ (t)
how dPi∗ (t) evolves. In contrast, in this study we observe the dDii (t) for hundredths of products
dDi (t)
across
launched by 78 firms. Hence, our objective is to characterize the differences of dP
∗
i (t)
many products and to capture two of its features, the timing (λi ) and speed (νi ) of significant
7
price cuts. In addition, the advantage of equation (1) is that we can solve it analytically and
test it empirically, just like its inverse has been tested elsewhere. It can be shown that (1)
is a separable differential equation and that its solution is
Pi∗ (t) = κi + (ρi − κi )hi (t)
(2)
with
“
e
hi (t) = 1 −
Di (t)−λi
νi
“
1+e
”
Di (t)−λi
νi
”
(3)
That is, we propose that the price of product i is composed of two parts, that is, a fixed
landing price (κi ) plus a mark-up ρi − κi that evolves in time proportionally to hi (t). We
do not observe marginal costs in our data but in most previous literature it is assumed that
perishable products are sold at the marginal cost at the end of their life cycle. Hence, in this
paper we use the terms marginal costs or price landing level interchangeably. The function
hi (t) gives the percentage of the markup at time t, 0 ≤ hi (t) ≤ 1. From (2) it is clear
that hi (t) follows a logistic shape and that λi can be interpreted as the location of the price
landing for product i in terms of the driver Di (t), and νi is the speed at which the landing
occurs. That is, we observe a price drop after Di (t) reaches the threshold λi .
In principle, Di (t) can be any monotonously increasing or decreasing function. The
simplest choice for Di (t) is simply time (Di (t) = t). However, the choice of our driver
changes the interpretation of λi and νi . If we set Di (t) to be the cumulative sales of product
i then λi can be interpreted as the limit of the cumulative sales of product i after which a
price cut occurs, that is the size of the segment that buys at high prices. Furthermore, if we
define Di (t) as the number of products introduced after launch of product i then λi becomes
a competitive threshold after which prices are cut. In all cases νi is a scaling constant that
marks the transition speed of prices as we set in equation (1) and it of course might depend
on the scale of Di (t).
The advantage of a logistic function for the pricing equation is that we can interpret its
parameters in a natural way in our application. We plot equation (1) for Di (t) = t and
different values of λi and νi in Figure 7. As can be noticed from the graph, the effect of
an increase (decrease) of λi is to shift the complete function to the right (left) and νi has
the role of smoothing the function or of making it steeper. That is, νi is a parameter that
determines how fast prices are falling and λi captures the moment (event) when prices are
dropping.
As discussed above, Pi∗ (t) aims to capture the underlying price pattern of product i,
that we call price landing. In practice we observe this pattern plus noise. The observed
prices may therefore differ from Pi∗(t). Furthermore, we only observe the prices at regularly
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spaced intervals. We adopt the convention that we observe the prices for product i at
t = 0, 1, 2, . . . , Ti . We denote the observed price at time t by Pit . We model the relation
between the observed prices and price landing pattern using an auto-regressive specification
of order one. In terms of the observed price this gives
Pi (t) = Pi∗ (t) + αi [Pi (t − 1) − Pi∗ (t − 1)] + εi (t)
t = 1, 2, 3, . . . , Ti
(4)
where εi (t) denotes the source of the random deviation at time t from the underlying
price landing pattern, and αi determines the memory in the deviations from the underlying
pattern. We assume that εi (t) ∼ N(0, σi2 ). If αi = 0 there is no memory, and (4) then states
that the deviations are independent over time. If αi > 0, a positive deviation at time t is
likely to induce a positive deviation at time t + 1. For the first observation we set
Pi (0) = Pi∗ (0) +
r
1
× εi(0)
1 − α2
(5)
The variance factor is set such that the variance of the random term equals the unconditional variance of Pi (t) in (4).
We note that the specification in equation (4) addreses two important modeling issues.
The first is what we mentioned earlier and it is that observed prices deviate from the underlying price pattern of equation (2). The second is that some of the Di (t) drivers might
be endogenous to Pi (t) and hence we follow the usual approach of including past prices to
control for such endogeneity. However, we note that there is no theory that relates the form
of Pi∗ (t) rather than Pi (t) to be endogeneusly determined by Di (t). In addition, we believe
endogeneity is not a major modeling concern given the descriptive nature of our study.
4.2
Heterogeneity in Main Parameters
In the above discussion of the model we have explicitly allowed for heterogeneity, that is, all
parameters and the pricing driver Di (t) are product-specific. In this section we discuss how
we model the heterogeneity in all parameters.
In the model we will allow for K different drivers, which are denoted by D1i (t), D2i (t),
. . ., DKi(t). We denote Si = k if the driver k is selected for product i. We can now write
the driver that best fits the price landing as DSi (t). The relationship between the observed
price and the price landing in (2) remains unchanged. In addition, for each price driver we
define a specific price landing equation Pki∗ (t) that depends on the driver k, that is,
9
Pki∗ (t) = κi + (ρi − κi )hki(t)
“
hki(t) = 1 −
e
Dki (t)−λki
νki
“
1+e
(6)
”
Dki (t)−λki
νki
”
Note that this definition is very similar to those in (2) and (3). The parameters λki and
νki have the same interpretation as before and they give the threshold value and the speed
of the price landing for the driver k. Note that only the price-landing process depends on
the driver k and that the initial price level ρi and the price landing level κi are independent
of the driver variable that is used.
The marginal cost (κi ), the initial price level (ρi ), and the threshold value (λki ) and the
speed of adjustment (νki) are defined to vary across products. For each of these parameters
we specify a second-level model. For the marginal costs (price landing level) and the launch
prices we specify
κi = Zi′ γ κ + ωiκ
ρi = Zi′ γ ρ + ωiρ
with (ω κ , ω ρ ) ∼ N(0, Σ)
(7)
κ
where Zi denotes a vector of product specific characteristics, ω κ = (ω1κ , ω2κ...ωN
) and ω ρ =
ρ
ρ
ρ
(ω1 , ω2 ...ωN ) . Among these product-specific variables could be product type, manufacturer
variables and a seasonal factor. So in (7) we assume that the landing price (and launch
prices) are additive separable in a number of parts and we give more details about them in
Section 4.5. For each driver variable k we specify
λ
ln λki = Zi′ γkλ + ηki
ln νki =
Zi′ γkν
+
ν
ηki
with (ηkλ , ηkν )′ ∼ N(0, Ωk ).
(8)
λ
λ
λ
ν
ν
ν
where ηkλ = (ηk1
, ηk2
...ηkN
) and ηkν = (ηk1
, ηk2
...ηkN
). The log transformation in (8) is
used to ensure that λki and νki are positive. We allow the random terms of λki and νki to
be correlated. For example, it might be that products that stay at high prices for longer
periods might have a certain speed. Hence, these correlations might help us capture such
relationships between the timing λki and the speed parameters νki .
10
4.3
Choice of Driver and Mixture Specification
The driver of the price landing is of course unobserved to the researcher. We denote this
(unobserved) variable as Si , that is, we denote Si = k if the driver k is selected for product
i. We complete this part of the model by specifying probabilities for each driver, that is,
the driver k is selected with probability πk for k = 1, 2, . . . , K. In our application k = 1
would mean that time is the driver, k = 2 means that cumulative sales are the driver and
k = 3 means that cumulative competitive introductions are the main driver of equation
(2). We provide more details on how we measured each driver in the next subsection. The
probabilities πk will reflect the overall likelihood of each of the different drivers. Note that
conditionally on the observed prices, the probability of Si = k is different across games.
In Figure 8 we describe the intuition about how drivers are selected. For this purpose
we need two main elements. The first element consists of the distributions of the threshold
parameters for each of the different drivers. That is, the distribution of λik and νik across
all i and for each k. For example, if we collect the parameter λi,k=1 for all i we obtain the
distribution of λ for the first driver. As we defined in equation (8) the distribution of λik
and νik depend on co-variates Zi and hyper-parameters γk and the variance term associated
to them. The second element we need is the relative distance (probability) of a video game
i parameters λik and νik to be drawn from the overall distribution in (8).
In Figure 8 we plot the prices of two video-games, the Spider-Man and the NFL Madden
2001. In addition, we plot the hypothetical distribution of the threshold parameters λki for
each of the mixture components k. The distribution of λi,k=1 in the upper left panel, λi,k=2
in the upper right panel and λi,k=3 in the lower left panel. Note that λi,k=1 is the time (in
months) after which the price drops; λi,k=2 is the cumulative number of sales after which the
price drops; and λi,k=3 is the cumulative number of competitive introductions after which
the price drops. We selected the Spider-Mann and the NFL Madden because their price cut
thresholds are easy to detect visually. It is easy to see (in the upper left panel) that the
NFL Madden price drop occurs at the 10th month and that the mean of the distribution of
λi,k=1 is very close to this month. The same is not true for the NFL Madden in the other two
frames. Hence, the most likely driver for the NFL Madden is time according to this graph.
In contrast, the most likely driver for the Spider-Mann is cumulative sales; as we can see in
the upper right panel is the graph where the price drop of the Spider-Mann is closer to the
mean of the distributions and hence it is the most likely.
4.4
Heterogeneity in Mixture Probabilities
We suspect that there might be heterogeneity in the mixture probabilities across games.
Hence, as an extension to the model we allow the probabilities of Si = k to depend on
a set of product specific variables. To model this dependence we specify a Multinomial
Probit Model for Si . Hence, we introduce additional latent variables yi∗ for i = 1, . . . , N and
k = 1, . . . , K These latent variables are related to Si by
11
Si = k
∗
if and only if yik
= max(yil∗ )
(9)
l=1...K
∗
We specify yik
as
∗
yik
= Zi′ δk + ϑik
with ϑk ∼ N(0, 1)
(10)
where ϑk = (ϑ1k, ϑ2k, , ..., ϑN ) and we set δ1 = 0 for identification. In principle the set of
variables used in this specification may differ from that in (6) and (7). The probability that
the driver k is used for product i now becomes
∗
∗
πik = Pr[yik
= max(yik
)]
(11)
This concludes our model specification. For inference we will rely on MCMC and Bayesian
analysis and treat all product- specific parameters as latent variables and we sample these
together with the parameters in (6), (7) and (8). A complete description of the sampling
steps in this Markov Chain can be found in the Appendix.
4.5
Model Specifics for Video Games Pricing Model
We estimated two versions of our model. The first version consists of the model with three
driver and the second consists of four drivers. We define Dik (t), for k = 1, 2, 3, 4 where
Di1 (t) = Ait , Di2 (t) = Cit and Di3 (t) = Iit and Di4 (t) = Rit . Ait is defined as the age
of a video game in months, that is, the same as the time between launch and t. Cit is the
cumulative sales of video game i between release date and t. Iit is defined the cumulative
number of video games introduced between the launch date of video game i and t. Rit is
defined as the release schedule of the firm that released product i. We know the number
of games a firm released at every point in time. We use a time window that sums the
introductions before time t up to the introductions coming in the next three months after t
and that is our fourth driver Rit .
The interpretation of the λik and νik varies depending on the driver k. Hence, λ1i can
be interpreted as the price landing time, λ2i as a competitive threshold, λ3i as the hard-core
gamer segment size and λ4i as a release limit after which we observe a price drop. For each
of these drivers, the parameter νki for k = 1, 2, 3, 4 can be interpreted as a scaling constant
that changes the speed at which the price landing occurs.
In all what follows we discuss the model with three drivers, that is k = 1, 2, 3 and we
leave out Rit . The reason for this is that Rit is selected with a probability very close to zero
12
when we include it as the fourth driver. We present the discussion regading the 4th driver
in our results in section 5.3.
The hierarchical structure of the corresponding threshold λki, speed νki and ρi and κi
parameters for each mixture component will depend on a set of Zi variables that contain
game type, publisher and seasonal effects plus the launch price and the time to introduce a
new game consoles as co-variates. Seasonal dummies are defined by the month of launch of
each video game i. The launch price is the observed price of video game i at launch time,
that is at t = 0. We include this variable in order to test if our co-variates remain significant
after including past prices in the equation for the timing and speed of launch. It might be
that the price at launch of a VG might contain information regarding the timing of the price
landing and its speed. We believe it is reasonable to include the launch price because of
the very likely uni-directional relationship between launch price and timing of price landing.
That is, it is very hard to argue that a firm decides how to price a VG based on its decision
on when to permanently cut its price; on the other hand, it might be that firms decide to
cut prices based on the launch price. For example, firms might cut the price of expensive
games after more time than the time they wait to cut the price of cheaper VGs. And finally,
the launch price is an instrumental variable for quality and hence we test if our covariates
remain significant after we control for it.
The time to console launch measures the time between a game release and the launch
of VG console to be released next. The PlayStation2 with DS controllers was introduced in
June 1998 and other versions of the PS2 console were released in February 1999 and January
2002. For example, a VG released in January 1998 will face a console introduction after 6
months; a video game released in January 1997 will face a release in 18 months, and so on.
We decided to include this variable to test whether the price landing pattern is different if a
console launch is near the release date of a video game.
From the seasonal fixed effects we excluded January, from the game types we excluded
Adventure games. The remaining game type categories are: Action, Arcade, Children,
Driving, Family, Fighting, Role playing, Shooter, Sports, Strategy and Compilations. The
remaining publisher dummies are Electronic Arts (EA), Acclaim, Infogames, Konami, Activision, Midway,Eidos Interactive, THQ, Capcom, Namco, Agetec, Interplay, Hasbro, 2nd
group, 3rd group and 4th group. The 2nd group is composed by six publishers that have at
least 1% market share, the 3nd group is composed by 14 publishers that account for the next
10% market share and the 4th group is composed by 43 publishers that account for less than
1% of the market share. In all our Tables we sorted publishers by their market share and
in descending order. The main publishers (EA, Acclaim, etc.) account for 80% of the video
games in our sample while the dummies for 2nd , 3rd and 4th publishers group the next 20%
of the market share. We left Sony as the reference.
13
5
RESULTS
In this section we present our results in three subsections. In the first we present results
regarding the heterogeneity of the parameters, next we present the results regarding driver
selection and finally we discuss the model performance.
5.1
Heterogeneity of Landing Time and Speed
Our results indicate that there is heterogeneity in the model parameters. This is the first
contribution we have to offer is that we find significant firm effects on both the timing
and speed parameters across all mixtures. That is, firms might be deciding not only on
when to cut the price but on how fast to cut it. To our knowledge, this result is new
and to our knowledge we are the first to show it empirically. In Table 2 and in Table 3
we can see the different firm effects across mixtures and model parameters. For example,
Acclaim’s landing time (λi ) coefficient in the time mixture is −0.196 and this means that
video games of Acclaim face a price drop 1.17 months before video games of Sony; because
exp(1.887)−exp(1.887−0.196) = 1.17 assuming all other Zi co-variates are zero. In addition,
we find several of the firm effects on the landing speed (νi ) to be significant. For example,
electronic Arts has a ν constant of 3.40 (that is exp(1.226)) while Agetec speed is 8.55 (that
is exp(2.147)). In the last four columns of Table 3 we report the results for the hierarchical
specification of (7). In both cases we observe very important firm effects. For example,
Konami sets the landing prices 2.535 USD above the landing prices of Sony, 17.34. While
the launch prices of Konami are not significantly different than those of Sony that start
at 40.49; see the 0.586 coefficient. In summary we find that firm effects are important to
describe the price landing timing and its speeds and the launch and landing prices of the
video games in our sample.
There are several other numbers reported in Table 2 and 3 and we will not attempt to
discuss them all. The main message of these tables is that there is very clear firm and game
type effects present in many of the main parameters of the model while seasons are important
but not across all main parameters. Seasons are more important for the starting and landing
levels of prices rather than the price landing timing and speed. We also find that for some
mixtures the effect of the launch price and the time to launch a new console are significant
for some of the main parameters.
We graph the posterior distribution of the parameters of the three mixture model in
Figure 9, Figure 10 and the distribution of the auto-regressive term of equation (4) in Figure
11. The distribution of the timing and speed parameters is reported in Figure 10. We can see
that each mixture has quite different threshold and speeds. For example, the time mixture
mean is around 7 months. That is, firms cut VG prices mainly at the 7th month after their
release. The timing parameters for all mixtures are graphed in the left frames while in the
right frames we present the speed parameter distribution. In addition, in Figure 9 we see
14
the distribution of the starting price level ρi for all i in the left frame and the distribution
of the κi in the right frame. These parameters show that the starting level might be as low
as 20 USD and as high as 70 USD while the landing level is as low as 5 USD and as high as
35 USD.
5.2
Drivers of Price Landings
Our second contribution is that we find that the drivers that best describe price landings
are competitive introductions and time and not cumulative sales. In Figure 12 we report the
posterior probability of each of the drivers in the three-mixture version of our model. Our
results are in contrast with the findings of Clements and Ohashi (2005) and Chintagunta et al.
(2006) who posit that competition does not explain the prices in the VG industry. However,
our specification is radically different from them and we posit that competition matters for
at least a 25.7 % of the VGs in our sample. Furthermore, the academic convention is that
past sales should should be a price driver. However, we find that the mixture that includes
cumulative sales as main driver has the lowest posterior probability and its mean is 12.05%.
The most likely driver is time itself or in other words, the most probable driver is simply
the age of a video game. The time mixture has a posterior mean probability of 62.21%.
A limitation of our study is that we tested only four drivers, taking into account the four
mixture version of our model. We believe that there might be other important drivers that
could potentially outperform time (a video game age) and our model provides a framework
useful to test their relative importance and overall likelihood.
In the fourth mixture version of our model we tested a fourth driver without much success.
The additional mixture included the release schedule of firms as driver Di . The idea was to
test whether firms release schedule determines the timing and speed of price landings. We
know the number of games a firm released at every point in time and the amount of games
it will release after each point in time. Therefore, we added the introductions before time t
up to the introductions coming in the next three months after t. For example, if the video
game i is launched by firm x we use the release schedule of this firm as a driver of price
landings. We decided to use a three months time window because most online sources of
VG releases cover, as a maximum, the upcoming three months. That is, today we know the
VGs to be released during the next three months. Of course, in our database we just know
the release schedule perfectly and that is why we tested it as it seems a reasonable driver.
However, we find that the probability of this latter driver mixture is on average zero. Our
conclusion is that prices are better described by the whole market introductions rather than
the release schedule of any single firm. This makes some sense given that the 78 VG firms
in our sample face on average 29 releases per month. Hence, firms might be more likely to
monitor all market introductions rather than to their own product introductions.
The estimates of the hierarchical structure for the mixture probabilities is reported in
Table 4. In contrast with the heterogeneity in the main parameters we do not find heterogeneity in the mixture probabilities. That is, we know that there is heterogeneity in the
15
timing and speed of price landings but we do not know why a driver is more likely than the
others. We consider this an area for further research.
5.3
Model Performance
We compared our model against two models: A naive model for prices, that is an AR(1); and
against our same model but replacing all drivers with time (what we call the single driver
model). Hence, the single driver model uses the same specification and the same number of
mixtures as our model but we replace all drivers with time. That is, Di (t) = time for all k
mixture components. These comparisons are reported in Table 5 and in Table 6.
Our model preforms extremely well when compared against the AR(1) model and reasonably well when compared against the single driver model. In Table 5 we see that our
model forecasts prices better than a naive AR(1) model for 40 out of 50 randomly chosen
games. We report the root mean square forecast error and the log of the predictive density
for all 50 VGs. More details on how we compute the predictive are given in Section A.3.
Moreover, the model performed better than the model with three time mixtures for 27 out
of the same 50 games and in 9 other cases it performed equally well as the alternative specification. That is in total 38 out of 50 games where our specification performs at least as well
as the alternative or better.
The reason why our model outperforms the AR(1) is that it captures the timing of
significant price cuts and the speed at which the price cut occurs while we can not demand
the same from an AR(1). At the moment and to our knowledge, we are the first to propose
an empirical model that captures these price dynamics. Therefore, we believe our model is a
first attempt to describe these price dynamics usually seen in high-tech products and it sets
the first benchmark to test against and to break with further research.
6
CONCLUSIONS
Our aim with this paper is to model and explain the dynamics of new products price patterns,
what we call now price landings. Price landings are strikingly simple and they usually follow
the inverse of the well known S-shape of sales. Nonetheless, there are no empirical studies
dealing with these regularities of new product prices.
In this paper we are concerned with products that face one significant price cut during its
life cycle. Several online price trackers report similar dynamics to a wider range or products
like mobile phones, cameras, storage media, books, etc. Our data was collected by NPD
Group but several websites like www.pricescan.com or www.streetprices.com let their users
plot price trends and indeed it is relatively easy to find many other products facing a single
and significant price drop during their lifetime. That is, knowing when a price is cut or when
16
to significantly cut the price of a product permanently is an exciting area of further research
and one with wide managerial implications across different industries.
In this paper we provide evidence that there is heterogeneity in the timing and speed
of prices landings. We find that most of these heterogeneity is driven by firm effects. That
is, firms seem to be deciding on the time to permanently cut the price of a product and
as well on how fast to cut it. Our model captures this heterogeneity and it is flexible and
useful to forecast and describe the price landing patterns in our data. In addition, our model
performance is superior to other model specifications. Finally, we find that it is the age of a
video game what is best describing the price landings patterns, the next most likely driver
is competition and the least likely is cumulative sales. These latter finding goes against the
academic convention that sales are the main driver of prices; at least for our application we
find evidence that this is not the case.
17
0
10
20
30
40
50
60
FIGURES AND TABLES
Price
7
5
10
15
20
Time in Months
Figure 1: Price Landing Pattern for 50 Randomly Selected Games
18
Figure 2: Price Landing Pattern
19
10
5
0
Time, Sales or Introductions
15
Time (Months)
Sales (Hundredth Thousands)
Cumulative Introductions (Hundredths)
0
10
5
Time (Months)
Figure 3: Drivers Path for the Spider-Man
20
15
Figure 4: The Video Games Market
21
Figure 5: What do publishers sell?
22
Figure 6: Total Sales Distribution
23
Figure 7: Main Pricing Function at Different Parameter Values
24
Sales Mixture
40
15
30
0.030
35
10
5
0
Months after Introduction
200
400
40
35
0.08
0.00
Price
30
20
25
Spider-Man Price
NFL Madden Price
Dist. of Entry Threshold
100
150
200
250
600
800 1000
1400
Cumulative Sales after Introduction (In Thousands)
Market Entry Mixture
50
0.015
0.000
25
0.6
0.0
20
0
Spider-Man Price
NFL Madden Price
Dist. of Sales Threshold
20
1.2
Price
30
Spider-Man Price
NFL Madden
Dist. of Landing Time
25
Price
35
40
Time Mixture
300
Number of Video Games Launched after Introduction
Figure 8: Identification of Drivers
25
Figure 9: Starting and Landing Price Distributions
26
Figure 10: Threshold and Speed Parameters
27
Figure 11: Pricing Dynamics Across Video Games
28
Figure 12: Probability of Alternative Price Drivers
29
NEW PRODUCTS PRICING STUDIES
30
Author (Journal, Year)
Bass and Bultez (1982)
Bayus (1994)
Bayus (1992)
Bayus (1992)
Clements and Ohashi (2005)
Chandy et al. (2006)
Chintagunta et al. (2006)
Dockner and Gaunersdorfer (1996)
Dockner and Jorgensen (1988)
Dolan and Jeuland (1981)
Eliashberg and Jeuland (1986)
Feng and Gallego (1995)
Ferguson and Koenigsberg (2007)
Franza and Gaimon (1998)
Gupta and Di Benedetto (2007)
Gupta et al. (2006)
Horsky (1990)
Kalish (1985)
Kalish (1983)
Kalish and Lilien (1983)
Kornish (2001)
Krishnan et al. (1999)
Nair (2007)
Nascimento and Vanhonacker (1993)
Padmanabhan and Bass (1993)
Parker (1992)
Rajan et al. (1992)
Rao and Bass (1985)
Raman and Chatterjee (1995)
Robinson and Lakhani (1975)
Schmalen (1982)
Simon (1979)
Teng and Thompson (1996)
Zhao and Zheng (2000)
Notes
Approach
Analytical
Empirical
Analytical
Analytical
Empirical
Analytical
Empirical
Analytical
Analytical
Analytical
Analytical
Analytical
Analytical
Analytical
Analytical
Analytical
Empirical
Empirical
Analytical
Analytical
Analytical
Analytical
Empirical
Analytical
Analytical
Empirical
Analytical
Empirical
Analytical
Analytical
Analytical
Empirical
Analytical
Analytical
Price Changes
Speed
–
Gradual
Gradual
Gradual
Gradual
Jumps
Gradual
Gradual
Gradual
Gradual
Jumps
Jumps
Jumps
Gradual
Gradual
Jumps
Gradual
Gradual
Gradual
Gradual
Gradual
Gradual
Gradual
Gradual
Jumps
–
Jumps
Gradual
Gradual
Gradual
Jumps
Gradual
Jumps
Jumps
Price Mimics
Diffusion
Yes
No
No
No
No
No
No
No
Yes
No
No
No
No
No
No
No
Yes
No
Yes
Yes
No
No
No
No
No
No
Yes
Yes
Yes
No
No
No
No
No
When to
cut prices
No
No
No
No
No
Yes
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
Yes
Yes
No
Yes
No
Yes
–
No
No
No
Yes
Yes
Yes
Table 1: Literature Review
Main Price
Driver
Saturation
Saturation
Learning Curve + Consumer Heterogeneity
Consumer Heterogeneity
Saturation + Indirect Network Effects
Consumer Heterogeneity
Saturation + Marketing Mix
Saturation
Learning Curve + Saturation
Learning Curve
Entry
Saturation
Entry
Entry
Entry
Saturation
Saturation
Advertising
Learning Curve + Saturation
Saturation
Entry
Saturation
Consumer Heterogeneity
Consumer Heterogeneity
Entry
Saturation
Saturation
Entry + Learning Curves
Saturation
Saturation
Entry
Saturation
Saturation
Consumer Heterogeneity
31
Intercept
Game Type
Action
Arcade
Children
Driving
Family
Fighting
Role playing
Shooter
Sports
Strategy
Compilations
Publisher
Electronic Arts
Acclaim
Infogames
Konami
Activision
Midway
Eidos Interactive
THQ
Capcom
Namco
Agetec
Interplay
Hasbro
2nd Publishers
3rd Publishers
4th Publishers
Season
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Launch Info
Launch Price
Time to Console Launch
Mixture (1) D(t)= VG Age
Landing Time
Launch Speed
log(λi )
log(νi )
1.887***
(0.266)
-0.575
(0.627)
Mixture (2) D(t) = Cumulative VG Introductions
Competitive
Landing Speed
Threshold log(λi )
log(νi )
1.273***
(0.389)
2.419**
(0.914)
-0.234*
-0.248
-0.266
-0.350**
-0.378**
-0.295*
-0.056
-0.646***
-0.276**
-0.269
-0.342
(0.122)
(0.186)
(0.217)
(0.133)
(0.178)
(0.145)
(0.143)
(0.134)
(0.123)
(0.176)
(0.220)
-0.172
-0.715
-0.250
-0.400
0.459
-0.120
-0.493
-0.006
-0.249
-0.383
1.116**
(0.307)
(0.500)
(0.526)
(0.334)
(0.417)
(0.347)
(0.381)
(0.334)
(0.313)
(0.393)
(0.503)
-0.050
-1.909
0.380
-0.308
0.504*
-0.218
-0.060
-0.182
-0.096
-0.142
-2.450***
(0.205)
(1.517)
(0.286)
(0.195)
(0.281)
(0.224)
(0.207)
(0.188)
(0.179)
(0.212)
(0.473)
-0.197
-2.508
-1.575**
-0.200
-0.602
-0.607
-0.060
-0.085
-0.056
-0.329
-5.903***
(0.378)
(2.354)
(0.695)
(0.393)
(0.529)
(0.543)
(0.502)
(0.357)
(0.382)
(0.470)
(1.389)
-0.014
-0.196*
-0.234
-0.387***
-0.280**
0.039
-0.791***
-0.320**
-0.109
0.218
-0.132
-0.758***
-0.097
-0.461***
-0.299***
-0.399***
(0.079)
(0.105)
(0.147)
(0.122)
(0.111)
(0.124)
(0.123)
(0.157)
(0.128)
(0.159)
(0.183)
(0.130)
(0.150)
(0.099)
(0.096)
(0.109)
1.226***
1.106***
1.016***
0.446
0.693**
1.665***
1.434***
1.621***
0.657
2.109***
2.147***
1.470***
0.941**
1.335***
0.942***
1.334***
(0.211)
(0.288)
(0.327)
(0.344)
(0.291)
(0.345)
(0.308)
(0.414)
(0.410)
(0.457)
(0.429)
(0.310)
(0.392)
(0.254)
(0.269)
(0.267)
-0.220
-0.507***
-0.690***
-0.042
-0.020
-0.281
-0.697***
-0.693*
-0.108
0.416**
-4.106***
-1.798***
-1.016
-0.388**
-0.636***
-0.386**
(0.133)
(0.162)
(0.196)
(0.179)
(0.201)
(0.221)
(0.187)
(0.336)
(0.199)
(0.206)
(1.190)
(0.423)
(0.970)
(0.182)
(0.142)
(0.176)
-0.022
0.253
-0.020
0.144
0.146
0.115
-0.248
0.338
0.356
-0.648
-5.610***
-1.436
0.910
0.154
0.066
0.410
(0.298)
(0.328)
(0.345)
(0.412)
(0.445)
(0.423)
(0.522)
(0.529)
(0.416)
(0.965)
(1.620)
(1.074)
(1.626)
(0.488)
(0.336)
(0.382)
-0.183
-0.111
-0.056
-0.142
0.050
-0.224
0.045
-0.014
-0.129
-0.109
-0.229*
(0.150)
(0.140)
(0.162)
(0.154)
(0.154)
(0.173)
(0.153)
(0.130)
(0.137)
(0.128)
(0.140)
-0.561
-0.564
-0.218
-0.549
-0.947**
-0.354
-0.474
-0.469
-0.402
-0.393
-0.420
(0.429)
(0.389)
(0.452)
(0.456)
(0.443)
(0.530)
(0.397)
(0.387)
(0.393)
(0.373)
(0.421)
0.014
-0.125
-0.421
-0.199
-0.259
-0.249
-0.154
0.023
-0.042
0.054
-0.287
(0.235)
(0.223)
(0.267)
(0.247)
(0.253)
(0.265)
(0.276)
(0.213)
(0.211)
(0.194)
(0.231)
-0.364
-0.403
-0.514
0.110
-1.087
-1.023
-0.550
-0.215
-0.358
-0.309
-0.198
(0.767)
(0.627)
(0.810)
(0.701)
(0.718)
(0.907)
(0.741)
(0.673)
(0.677)
(0.678)
(0.789)
0.018***
-0.005
(0.005)
(0.004)
0.003
-0.002
(0.012)
(0.009)
0.048***
-0.021***
(0.007)
(0.006)
-0.025
0.027**
(0.019)
(0.011)
Notes: Standard deviation between parentheses. *,**,*** indicate zero is not contained in the 90, 95 and 99% highest posterior density region.
Table 2: Estimation Results Part I
32
Intercept
Game Type
Action
Arcade
Children
Driving
Family
Fighting
Role playing
Shooter
Sports
Strategy
Compilations
Publisher
Electronic Arts
Acclaim
Infogames
Konami
Activision
Midway
Eidos Interactive
THQ
Capcom
Namco
Agetec
Interplay
Hasbro
2nd Publishers
3rd Publishers
4th Publishers
Season
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Launch Info
Launch Price
Time to Console Launch
Mixture (3) D(t)=Cumulative Sales
Hard-core
Segment Size
Landing Speed
log(λi )
log(νi )
2.579
(1.810)
4.133***
(1.363)
Landing Price
in USD
κi
17.34***
All Mixtures
Launch Price
in USD
ρi
(1.510)
40.79***
(2.046)
-0.431
-1.171
-1.696
0.342
0.705
-1.416
2.021
-0.823
-1.147
-1.545
-0.603
(1.179)
(1.567)
(1.887)
(1.203)
(1.641)
(1.129)
(1.730)
(1.230)
(1.301)
(1.640)
(1.732)
0.161
-1.866
0.551
1.040
-0.012
-0.974
0.067
0.170
-1.369*
-0.462
2.599**
(0.829)
(1.145)
(1.245)
(0.718)
(1.243)
(0.814)
(1.356)
(0.901)
(0.845)
(1.271)
(1.185)
0.751
0.543
-1.128
-0.243
-1.111
2.061*
3.162***
-0.310
-0.987
2.364*
-7.490***
(0.987)
(1.492)
(1.456)
(1.007)
(1.225)
(1.109)
(1.180)
(1.056)
(0.983)
(1.240)
(1.534)
-1.793
-5.244***
-10.46***
-1.363
-5.322***
0.378
1.033
2.050
-1.450
1.290
8.880***
(1.297)
(1.969)
(2.092)
(1.299)
(1.627)
(1.443)
(1.460)
(1.372)
(1.301)
(1.592)
(3.272)
3.395***
0.904
0.049
2.240**
2.886**
1.346
1.674
5.719***
2.154
-2.462
0.806
1.134
0.816
0.398
-0.212
1.356*
(0.647)
(0.782)
(0.664)
(1.090)
(1.078)
(1.018)
(1.142)
(1.126)
(1.326)
(4.770)
(1.182)
(1.241)
(1.009)
(0.933)
(0.597)
(0.779)
2.077***
0.039
0.119
-0.037
1.683***
-0.030
1.537***
2.341***
2.476***
-2.518
-0.472
2.043***
-1.433
-0.035
0.007
0.408
(0.442)
(0.619)
(0.577)
(0.656)
(0.651)
(0.875)
(0.747)
(0.888)
(0.869)
(2.743)
(0.714)
(0.814)
(1.460)
(0.909)
(0.454)
(0.567)
1.135*
-1.572*
-1.766**
2.535***
-1.105
-1.617
-2.009**
-0.919
1.540
1.433
-3.267**
-0.778
-3.545***
-3.134***
-1.124
-3.216***
(0.671)
(0.865)
(0.888)
(0.979)
(0.962)
(1.054)
(1.013)
(1.172)
(1.070)
(1.554)
(1.427)
(1.127)
(1.289)
(0.767)
(0.702)
(0.815)
1.712**
1.617
1.262
0.586
-0.439
2.344*
5.029***
2.005
-0.874
2.235
3.487
9.382***
-4.731***
1.892*
1.947**
0.771
(0.807)
(1.124)
(1.289)
(1.199)
(1.196)
(1.339)
(1.421)
(1.656)
(1.324)
(1.597)
(2.106)
(1.741)
(1.720)
(1.081)
(0.943)
(1.212)
-1.274
0.364
-10.56***
-12.00***
-11.45***
-3.430**
1.129
0.739
0.849
0.588
0.109
(2.237)
(1.099)
(3.330)
(2.095)
(2.627)
(1.819)
(1.519)
(0.877)
(0.892)
(0.990)
(0.982)
-0.508
-0.344
-8.255***
-8.820***
-8.422***
-3.950***
0.173
0.840
0.652
0.597
0.358
(1.474)
(0.890)
(2.689)
(1.870)
(1.864)
(1.306)
(1.202)
(0.709)
(0.761)
(0.720)
(0.766)
-0.121
2.038*
2.306*
1.499
3.648***
1.227
0.236
1.397
1.112
0.617
2.150*
(1.254)
(1.123)
(1.375)
(1.281)
(1.230)
(1.367)
(1.235)
(1.088)
(1.124)
(1.073)
(1.173)
-2.178
-2.663*
0.148
-1.118
-3.058*
-1.847
-3.719**
-2.537
-0.832
-0.965
0.534
(1.747)
(1.559)
(1.878)
(1.811)
(1.717)
(1.926)
(1.655)
(1.536)
(1.573)
(1.498)
(1.682)
0.155***
-0.122***
(0.018)
(0.025)
0.068***
-0.059***
(0.015)
(0.018)
–
0.127***
–
(0.025)
–
0.401***
–
(0.033)
Notes: Standard deviation between parentheses. *,**,*** indicate zero is not contained in the 90, 95 and 99% highest posterior density region.
Table 3: Estimation Results Part II
Intercept
Game Type
Action
Arcade
Children
Driving
Family
Fighting
Role playing
Shooter
Sports
Strategy
Compilations
Publisher
Electronic Arts
Acclaim
Infogames
Konami
Activision
Midway
Eidos Interactive
THQ
Capcom
Namco
Agetec
Interplay
Hasbro
2nd Publishers
3rd Publishers
4th Publishers
Season
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Launch Info
Launch Price
Time to Launch
Latent
Utility
of Sales
Mixture
3.727***
(0.579)
Latent
Utility
of Time
Mixture
0.549
(0.445)
0.699*
-0.032
0.481
0.708*
-0.100
1.153***
-0.304
0.809
-0.010
0.342
0.930*
(0.360)
(0.543)
(0.479)
(0.402)
(0.451)
(0.427)
(0.552)
(0.567)
(0.372)
(0.613)
(0.530)
0.260
0.404
0.005
0.288
-0.577*
0.357
-0.146
0.716**
0.057
-0.055
-0.258
(0.269)
(0.451)
(0.456)
(0.278)
(0.333)
(0.372)
(0.355)
(0.297)
(0.252)
(0.362)
(0.438)
-0.689
-0.723
0.723
1.110**
1.060**
0.884*
0.002
-0.287
-0.355
-0.043
0.592
0.648
-0.364
0.203
0.324
0.510
(0.514)
(0.583)
(0.456)
(0.469)
(0.450)
(0.454)
(0.668)
(0.531)
(0.451)
(0.649)
(0.558)
(0.626)
(0.512)
(0.427)
(0.402)
(0.374)
-0.137
-0.365
-0.332
0.068
0.356
-0.119
0.265
-0.017
-0.821**
-0.503
0.592
0.800*
0.303
0.387
-0.025
0.164
(0.239)
(0.284)
(0.360)
(0.343)
(0.325)
(0.384)
(0.365)
(0.459)
(0.357)
(0.426)
(0.512)
(0.430)
(0.380)
(0.305)
(0.272)
(0.301)
0.159
0.002
0.448
-0.309
0.067
0.149
-0.197
0.008
0.156
0.070
0.666
(0.467)
(0.454)
(0.575)
(0.499)
(0.533)
(0.547)
(0.479)
(0.373)
(0.410)
(0.357)
(0.418)
-0.026
0.441
0.296
-0.361
0.709**
0.007
0.355
-0.214
0.285
0.069
0.329
(0.361)
(0.297)
(0.431)
(0.344)
(0.381)
(0.417)
(0.367)
(0.259)
(0.293)
(0.243)
(0.307)
-0.162***
0.010
(0.013)
(0.015)
0.004
-0.012
(0.007)
(0.008)
Table 4: Results of Hierarchical Structure for Mixture Probabilities
33
Log
of
Forecast
PreRMSE
Forecasted St. Dev.
Forecast
dicted
AR(1)
Game Title
Months
Price
RMSE
Density
NHL 2001
10
0.18
0.17
0.15
-0.89*
JJ’S VR FOOTBALL 98
8
2.39
1.76*
5.19
-5.06*
HIGH HEAT BSBALL 2002
18
7.14
2.18*
11.92
-10.25*
MADDEN NFL 98
12
4.56
2.55*
5.03
-11.05*
MR DOMINO
18
8.33
3.10*
12.34
-17.22*
THE CROW CITY ANGELS
18
14.53
3.24*
26.27
-12.15*
PITBALL
18
13.49
3.72*
27.92
-12.50*
FROGGER 2
18
10.48
3.86*
22.53
-12.30*
BIG OL’ BASS 2
18
14.45
3.92*
22.63
-15.22*
MK & ASHLEY WINNER’S
18
11.88
4.11*
17.46
-11.90*
CIVILIZATION 2
18
9.73
4.25*
12.52
-8.94*
PONG
18
11.43
4.37*
20.81
-14.54*
ROGUE TRIP
14
1.80
4.38
1.90
-16.24
RESIDENT EVIL 3:NEMES
18
10.16
4.70*
10.34
-15.91*
ETERNAL EYES
18
8.31
4.92*
9.45
-24.18*
TEKKEN 2
18
7.39
5.13*
8.54
-15.13*
TEST DRIVE 4
18
11.50
5.39*
28.29
-12.53*
F1 WRLD GRAND PRIX 00
18
7.11
5.63*
7.52
-29.76*
FADE TO BLACK
18
9.09
5.88*
8.58
-21.99*
SHEEP RAIDER
18
9.35
6.03*
11.20
-81.57*
G POLICE2:WPN JUSTICE
9
10.79
6.04*
24.87
-8.60*
RISK
10
9.50
6.55*
12.39
-13.02*
SYNDICATE WARS
18
8.93
6.66*
12.83
-16.71*
JUGGERNAUT
18
9.33
6.71*
16.33
-51.62*
KISS PINBALL
10
8.47
6.73*
13.79
-11.21*
BACKYARD SOCCER
18
16.59
6.74*
23.35
-24.94*
OLYMPIC SUMMER GAMES
18
8.57
7.02*
12.52
-16.79*
NECTARIS:MILITARY MAD
18
13.34
7.06*
19.63
-16.75*
T.CLANCYS ROGUE SPEAR
18
5.38
7.88
4.54
-21.11
TOCA 2 CAR CHALLENGE
18
13.75
7.97*
13.93
-23.36*
NFL XTREME 2
18
14.35
8.27*
24.53
-20.69*
ARENA FOOTBALL
17
3.40
8.35
4.08
-14.76*
FINAL FANTASY IX
13
6.23
8.43*
12.81
-10.78*
SHEEP
18
3.47
8.83
3.54
-22.08
SIMPSON’S WRESTLING
12
8.66
8.87*
12.12
-16.69*
POCKET FIGHTER
18
10.51
9.02*
17.25
-17.09*
POWERBOAT RACING
18
10.50
9.19*
24.08
-14.13*
GRAND SLAM 97
18
11.09
9.53*
11.69
-24.66*
RAMPAGE WORLD TOUR
6
2.88
9.67
4.58
-1245.7
EAGLE ONE: HARRIER
13
11.43
10.5*
13.02
-50.33*
STRIKER PRO 2000
9
10.66
10.6*
20.87
-10.52*
NEWMAN/HAAS RACING
16
3.88
11.31
4.63
-38.44
DISCWRLD 2:MRTLY BYTE
18
6.31
11.42
5.95
-87.15*
CROSSROAD CRISIS
18
9.77
12.4*
15.33
-93.93*
SLAM N JAM 96
18
10.23
13.3*
19.53
-18.82*
NBA LIVE 2002
18
9.23
14.4*
21.76
-80.22*
ARMD COR 2 PRJ PNTSMA
15
8.71
15.0*
16.21
-11.77*
CRASH TEAM RACING
18
15.92
15.8*
17.91
-342.07
DISNEY’S DINOSAUR
18
5.00
16.35
5.68
-27.82
NFL BLITZ 2000
18
5.63
17.57
6.41
-30.46*
Notes: * Means the RMSE or the predictive likelihood is smaller in our model than in the AR(1)
Table 5: Forecasting Performance
34
Log
Likelihood
of
predicted
AR (1)
-3.47
-6.79
-142.23
-27.81
-188.45
-369.19
-264.59
-2421.81
-485.79
-493.25
-1009.23
-646.32
-0.58
-71.64
-82.53
-85.26
-511.42
-50.37
-38.53
-112.25
-386.84
-267.31
-55.53
-60.97
-128.41
-652.10
-38.77
-292.34
-16.62
-177.86
-467.21
-23.53
-32.38
-4.11
-31.70
-169.35
-466.81
-121.27
-6.05
-847.83
-139.91
-4.55
-41.79
-458.48
-280.41
-466.39
-263.19
-117.05
-15.57
-130.64
Game Title
NHL 2001
JJ’S VR FOOTBALL 98
HIGH HEAT BSBALL 2002
MADDEN NFL 98
MR DOMINO
THE CROW CITY ANGELS
PITBALL
FROGGER 2
BIG OL’ BASS 2
MK & ASHLEY WINNER’S
CIVILIZATION 2
PONG
ROGUE TRIP
RESIDENT EVIL 3:NEMES
ETERNAL EYES
TEKKEN 2
TEST DRIVE 4
F1 WRLD GRAND PRIX 00
FADE TO BLACK
SHEEP RAIDER
G POLICE2:WPN JUSTICE
RISK
SYNDICATE WARS
JUGGERNAUT
KISS PINBALL
BACKYARD SOCCER
OLYMPIC SUMMER GAMES
NECTARIS:MILITARY MAD
T.CLANCYS ROGUE SPEAR
TOCA 2 CAR CHALLENGE
NFL XTREME 2
ARENA FOOTBALL
FINAL FANTASY IX
SHEEP
SIMPSON’S WRESTLING
POCKET FIGHTER
POWERBOAT RACING
GRAND SLAM 97
RAMPAGE WORLD TOUR
EAGLE ONE: HARRIER
STRIKER PRO 2000
NEWMAN/HAAS RACING
DISCWRLD 2:MRTLY BYTE
CROSSROAD CRISIS
SLAM N JAM 96
NBA LIVE 2002
ARMD COR 2 PRJ PNTSMA
CRASH TEAM RACING
DISNEY’S DINOSAUR
NFL BLITZ 2000
Notes:
Forecast
Horizon
10
8
18
12
18
18
18
18
18
18
18
18
14
18
18
18
18
18
18
18
9
10
18
18
10
18
18
18
18
18
18
17
13
18
12
18
18
18
6
13
9
16
18
18
18
18
15
18
18
18
St. Dev.
Price
0.18
2.39
7.14
4.56
8.33
14.53
13.49
10.48
14.45
11.88
9.73
11.43
1.80
10.16
8.31
7.39
11.50
7.11
9.09
9.35
10.79
9.50
8.93
9.33
8.47
16.59
8.57
13.34
5.38
13.75
14.35
3.40
6.23
3.47
8.66
10.51
10.50
11.09
2.88
11.43
10.66
3.88
6.31
9.77
10.23
9.23
8.71
15.92
5.00
5.63
Log of Predictive
Density
Original Model
-0.89
-5.06
-10.25
-11.05
-17.22
-12.15
-12.50
-12.30
-15.22
-11.90
-8.94
-14.54
-16.24
-15.91
-24.18
-15.13
-12.53
-29.76
-21.99
-81.57
-8.60
-13.02
-16.71
-51.62
-11.21
-24.94
-16.79
-16.75
-21.11
-23.36
-20.69
-14.76
-10.78
-22.08
-16.69
-17.09
-14.13
-24.66
-1245.7
-50.33
-10.52
-38.44
-87.15
-93.93
-18.82
-80.22
-11.77
-342.07
-27.82
-30.46
Log of Predicted
Density
Time Model
-0.90
-4.89
-9.92
-11.12
-15.86
-11.29
-12.06
-12.51
-16.29
-12.11
-11.36
-13.10
-30.14
-17.48
-25.35
-16.17
-12.40
-26.22
-23.64
-121.60
-8.56
-33.94
-13.63
-98.83
-11.05
-21.56
-14.71
-16.17
-22.90
-26.08
-17.40
-14.56
-9.54
-23.30
-16.84
-16.58
-14.76
-23.81
-2072.64
-77.20
-11.26
-54.93
-38.19
-34.08
-15.18
-82.06
-14.41
-397.18
-18.46
-34.48
Table 6: Comparison with Alternative Model
35
LPD Original
>
LPD Time
**
*
*
**
*
*
**
**
**
**
**
**
**
**
*
**
**
**
**
**
*
*
**
**
*
**
**
*
**
*
**
**
**
**
**
**
**
**
A
ESTIMATION METHODOLOGY
To draw inference on the parameters we will rely on a Bayesian analysis and the Gibbs sampler. Whenever it is possible we use Gibbs sampling with block updating and the MetropolisHastings algorithm when there are no close form sampling distributions. We run a Markov
Chain for 200 thousand iterations of which the first 100 thousand are discarded for burn-in
and we keep a tenth of total draws. This Markov Chain has the posterior distribution of the
parameters τi , θ and the latent decision variable indicators Si i = 1, . . . , N as the stationary
distribution.
In all what follows we collect the first level model parameters in the blocks: τi = (ρi ,κi ,αi ,
2
σi ,λki,νki ), ρ = (ρ1 , ..., ρN ), κ = (κ1 , ..., κN ), α = (α1 , ..., αN ), σ 2 = (σ12 , ..., σN
), λk =
(ln(λik ), ...,ln(λN k )) and finally νk = (ln(νik ), ..., ln(νN k )).
We further collect all hyper-parameters in the following blocks: θ = (γ P , γkL , Π, Ω).
Where Ω = (Ω1 , ..., ΩK ), Π = (π1 , ..., πK ). We have that γ P = (γ κ , γ ρ ) where γ κ =
ρ
κ
λ
λ
(γ1κ , ..., γM
) and γ ρ = (γ1ρ , ..., γM
). Finally, γkL = (γkλ , γkν ) where γkλ = (γk1
, ..., γkM
) and
ν
ν
ν
γk = (γk1 , ..., γkM ). M refers to the number of variables in Z, k refers to the number of mixtures (same as number of drivers), and N refers to the total number of products. Therefore
Z=(Z1 , ..., ZM ). In addition, x|y refers to the conditional mean of x given y, σ x|y refers to the
conditional variance of x given y, and φ(x; µ, σ 2 ) means that x has normal distribution with
b N)
mean µ and variance σ 2 . Finally, p() denotes a general density function and Ω ∼ IW (Ω,
b and N degrees
denotes that Ω follows and inverted Whishart distribution with scale matrix Ω
of freedom.
Note that in this context we treat the product specific parameters τi as latent variables.
We consider the log of λik and νik k = 1, ...K, i = 1, . . . , N as focal parameters strictly for
convenience and to impose that they are positive. This has no impact on the results. In this
Markov Chain we will sample the latent variables alongside with the parameters.
The complete data likelihood for product i is
p(Pi , Si , τi |θ) = πSi × p(Pi |Si , τi , θ) × p(τi |θ)
(A-1)
where Pi = (Pi (0), ..., Pi(T )) and p(Pi |Si , τi , θ) is equal to
p(Pi(0)|Si , τi , θ) ×
t=T
Y
p(Pi (t)|Pi (t − 1)Si , τi , θ)
t=1
Furthermore, we have that the first observation likelihood is
36
(A-2)
p(Pi (0)|Si, τi , θ) = φ Pi (0); Pi∗(0),
1
σ2
1 − α2 i
(A-3)
and all other observations have as likelihood
p(Pi (t)|Pi(t − 1)Si , τi , θ) = φ (Pi (t); Pi∗ (t) + αi [Pi (t − 1) − Pi∗ (t − 1)], σi )
(A-4)
Next, we have
p(τi |θ) = p(ρ, κ|τ, θ)
K
Y
p(λki , νki |τ, θ)
(A-5)
k=1
where
P′
p (ρ, κ)|τ −(ρ,κ) , θ = φ (ρ, κ)′ ; γ Z, Σ ⊗ (Z′ Z)−1
(A-6)
and
p (λki , νki)|τ−(λki ,νki) , θ = φ (λk , νk )′ ; γkL′ Z, Ωk ⊗ (Z′ Z)−1
(A-7)
Plugging these two last two equations in (A-5) results in
K
Y
P′
′
′
−1
×
φ (λk , νk )′ ; γkL′ Z, Ωk ⊗ (Z′ Z)−1
p(τi |θ) = φ (ρ, κ) ; γ Z, Σ ⊗ (Z Z)
(A-8)
k=1
We impose flat priors on all almost all parameters, for αi we set a uniform prior on the
interval (-1,1) to impose stationarity. This completes the main model specification and next
we discuss how we sample from the posterior distribution for all parameters.
A.1
Sampling distributions
If πk is fixed across products, the density of Si on Pi (0), P , τi , and θ equals a multinomial
distribution with probabilities proportional to
37
πSi × p(Pi |Si , τi , θ) × p(τi |θ)
(A-9)
Equation (A-9) can be used to sample αi and σi2 . The full conditional distribution for
αi is a truncated normal on the interval [-1,1], where the mean and variance are given
by applying the Ordinary Least Squares formulas to a regression of Pi (t)-Pi∗(t) on its lag
with known variance of the disturbance term σi2 . A draw for σi2 can be obtained using the
Metropolis-Hastings sampler and taking as candidate
σi2cand
=
T
P
(ε̂i(t))ˆ2
t=1
where w ∼ χ2(T −1)
w
(A-10)
where ε̂i(t) is the estimated residual of equation (4). We evaluate this candidate and the
current sampler value for σi2 in the conditional distribution of the first observation given in
equation (A-3). Hence we take the candidate as the next drawn value of σi2 with probability
min 1,
1
2
φ Pi (0); Pi∗(0), 1−α
2 σicand
!
1
2
φ Pi (0); Pi∗(0), 1−α
2 σicurrent
(A-11)
To derive the full conditional distribution of κi and ρi we first rewrite equations (4) and
(5) as
q
q
q
2
2
1 − αi Pi (0) = [ 1 − αi hSi (0)] × κi + [ 1 − αi2 hSi (0)] × ρi + εi (0)
(A-12)
and
Pi (t) − αi Pi (t − 1) = [1 − hSi (t) − αi (1 − hSi (t))] × κi + [hSi (t) − αi hSi (t)] × ρi + εi (t) (A-13)
These equations should be combined with the specification for the hierarchical layer in
(7) as follows:



  A
εi
Xi XiB
Yi
ρi
 ρi  =  1
0 
+  ωρ 
κi
ωκ
κi
0
1

(A-14)
where ρi and κi is the current draws for ρi and κi from their hierarchical specification
defined in equation (A-6). Next, we define XiA and XiB as
38
XA
i



=

p
1 − αi2 (1 − hSi (0))
1 − hSi (1) − αi (1 − hSi (1))
..
.
1 − hSi (Ti ) − αi (1 − hSi (Ti ))





and
XB
i



=

p
1 − αi2 hSi (0)
hSi (1) − αi hSi (1)
..
.
hSi (Ti ) − αi hSi (Ti )



 (A-15)

and Yi as



Yi = 

p
1 − αi2 Pi (0)
Pi (1) − Pi (0)
..
.
Pi (T ) − Pi (T − 1)





(A-16)
Finally, we can draw κi and ρi from
′∗
′∗
′∗
∗ −1
∗
∗ −1
β ∼ N (Wi Wi ) Wi Y , (Wi Wi )
(A-17)
where β = (ρi , κi )′ and


XiA XiB
Wi =  1
0 
0
1

and E  εi ω ρ

2
ε
i
σi 0
ρ 
κ

ω
=Γ
=
ω
0 Σ
κ
ω

(A-18)
with Γ−1/2 Γ−1/2 = Γ−1 and Wi∗ = Γ−1/2 Wi and Yi∗ = Γ−1/2 Yi .
Due to the non-linearity in the price patterns, the conditional distributions of λk and νk
are not of a known form. We will sample each parameter one at a time using a random walk
Metropolis Hastings sampler. Given the current draw of one of these parameters we draw
a candidate by adding a draw from a normal with mean zero and a fixed variance. This
candidate draw for λk and νk is accepted with probability
p(λcand
ki |νki , ...)
min 1,
p(λcurrent
|νki ...)
ki
and
cand
p(λki |νki
, ...)
min 1,
current
p(λki|νki
, ...)
respectively. The posterior of the i′ th element of λk is
39
(A-19)
p(λki|νki , ...) = p(Pi (0)|Si , τi , θ)
t=T
Y
t=1
λ |ν
p(Pi (t)|Pi (t − 1)Si , τi , θ)φ λki ; λki|νki , Ωkki ki (A-20)
and the posterior of the i′ th element of νk is
p(νki |λki, ...) = p(Pi (0)|Si, τi , θ)
t=T
Y
t=1
νki |λki
(A-21)
p(Pi (t)|Pi(t − 1)Si , τi , θ)φ νki; νki |λki, Ωk
These are conditional posterior distributions because we allow λk and νk to be correlated
to each other. In other words, the timing of the price cut and the speed of the price cut
might be correlated and these correlation is different across mixtures. The variance of the
proposal density is chosen such that we obtain an acceptance rate close to approximately
25%, that is the optimal rate for high-dimensional models (see Robert and Casella (2004,
page 316), Carlin and Louis (2000, page 154) or Gamerman and Lopes (2006, page 196)).
The
of πi ,. . . , πK is a Dirichlet distribution with parameters
P conditional distribution
P
1 + i 1[Si = 1],. . . , 1 + i 1[Si = K]; thatP
is, we draw each πk proportional toP
the number
of products assigned to mixture m, that is 1 1[Si = k], and naturally restrict k πk =1.
Given the latent variables in τi sampling the hyper-parameters of the hierarchical part
for the marginal costs, launch price, and price landing characteristics is relatively straightforward. We draw γ P from a normal
P
γ ∼N
(A-22)
k = 1, . . . , K
(A-23)
(Z′ Z)−1 Z′ κ
, Σ ⊗ (Z′ Z)−1
(Z′ Z)−1 Z′ ρ
and γkL |Ωk from
γkL |Ωk ∼ N 0, g[Ωk × (Z′ Z)]−1
Next we draw (λki, νki ) from
λki
νki
∼N
1
(Z′ Z)−1 Z′ λki
1+g
1
(Z′ Z)−1 Z′ νki
1+g
40
1
Ωk ⊗(Z′ Z)−1
,
1+g
!
(A-24)
The factor g comes from the g-prior which states that the variance of (λki, νki ) is proportional to the variance of the data. This prior is used when the researcher does not know
the parametric form of the variance covariance matrix but assumes that it might be proportional to other know variance covariance matrix. See Fernandez et al. (2001) for a detailed
discussion.
b N where
Finally, we draw Σ ∼ IW Σ,
b=
Σ
ω
bκ
ω
bρ
(b
ω κ ,b
ω ρ)
(A-25)
b
and we draw Ωk ∼ IW Ωk + I2 , N + M + 5 where
bk =
Ω
A.2
ηbkλ
ηbkν
(b
ηkλ ,b
ηkν )
(A-26)
Hierarchical Structure in the Mixture Probabilities
The previous steps give the methodology to analyze our model without a hierarchical specification on the mixture probabilities πk . As discussed in this paper, the model can be easily
expanded to include a hierarchical specification on the mixture probabilities. As before, we
will assume that πki differs across products but here we test if a multinomial probit specification that depends on Z is useful to explain their heterogeneity. For that we need to define
first K latent variables for each product i
∗
yki
∼ N(Zi δk , 1)
(A-27)
∗
is the largest of all
where δ1 = 0 for identification. Product i belongs to mixture m if yim
∗
k = 1, . . . , K. Given (A-27) , we can write the conditional distribution of yim
given the
∗
other latent utilities (-m), denoted as yi,−m , as follows:
∗
yim
∗
∗
∗
∗
p(yi,m
|yi,−m
...) = p(yi,m
> max(yi,−m
)) × p(Pi , Si = m, τi |θ)
∗
∗
+p(yi,m < max(yi,−m ))p(Pi , Si = m∗ , τi |θ)
(A-28)
where m∗ = argmax (yik∗ ) . Based on (A-28) we can apply the inverse cdf technique to
m6=k
∗
draw yim
from its full conditional distribution. Note that in this specification the indicator
41
∗
variable Sik is determined based on yim
and the δm parameters can be obtained from a normal
with mean (Zi′ Zi )−1 Zi′ δk and variance (Zi′ Zi )−1 for m = 2, . . . , K. We programmed all our
routines in Ox (see Doornik (2007)) and our graphs in R (see R Development Core Team
(2005)).
A.3
Posterior Predictive Density
We used two measures to compare predictive performance in Table 5: the root mean squared
error and the log of the posterior predictive density for observations after t > 7. The
predictive density log(p(Pi (7),...,Pi (T )|Pi(1),...,Pi (6))) is defined as:
log
Z Z Z
p(Pi (7),...,Pi (T )|Pi(1),...,Pi (6), Si, τi , θ)×p(Si , τi , θ|Pi (1),...,Pi (6))dSidτi dθ (A-29)
That is, we compute the log of the density for the forecast sample given the six observations included in the model and the posterior of all model parameters given these latter
observations. The posterior predictive density can easily be obtained from the MCMC output
by taking the log of the average out-of-sample likelihood over all draws.
42
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