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\title{\large Web appendix for \\ {\LARGE Smooth quantile based modeling of brand sales, price and
promotional effects from retail scanner panels}, {\em Journal of Applied Econometrics}}

\author{
{\large Harry Haupt\footnote{Prof.\ Dr.\ Harry Haupt, Department of
Statistics, University of Passau, Innstr.~27, 94032 Passau, Germany.
Tel.\ +49 851 509-2560 \newline Corresponding author:
harry.haupt@uni-passau.de \vspace*{1.5ex}}, \, Kathrin
Kagerer\footnote{Dr.\ Kathrin Kagerer, \mbox{Faculty of Business,
Economics and Management Information Systems}, University of
Regensburg, Universit\"atsst.~31, 93053 Regensburg, Germany.
\newline kathrin.kagerer@wiwi.uni-regensburg.de
\vspace*{1.5ex}} \,\, and \, Winfried J.\ Steiner\footnote{Prof.\
Dr.\ Winfried J.\ Steiner, Department of Marketing, Clausthal
University of Technology, Julius-Albert-Str.~2, 38678
Clausthal-Zellerfeld, Germany, Tel.\ +49 5323 72-7650. \newline
winfried.steiner@tu-clausthal.de} } }

\date{}

\begin{document}


\maketitle


\section{Computational issues}
\label{sec:Computations}

All of the computations in the paper are implemented using the open-source
software {\sf R} (version 2.15.0, 32-bit). B-spline basis functions
are generated using the function \texttt{splineDesign} from the
\texttt{spline} package. For the estimations of the conditional mean
with monotonicity constraint
\begin{equation*}
\alpha_{j-1} (\vartheta) - \alpha_{j} (\vartheta) \geq 0,
\quad \textrm{for } j = -(k-1), \ldots, m,
\label{eq:inequality}
\end{equation*}
[equation (5) in the paper] (i.e.\ for the
constrained B-spline model and the smoothing spline model) we employ
the function \texttt{pcls} from the \texttt{mgcv} package
(\citealt{Woo2012}). This function is based on an optimization
process using quadratic programming with respect to the monotonicity
constraint which is imposed by means of an inequality constraint.
For quantile regression, routines from the \texttt{quantreg} package
from \citet{Koe2012} are used, based on the respective works of
\citet*{KoeNgPo1994}, \citet*{HeShi1994, HeShi1998},
\citet*{HeNgPo1998}, \citet*{HeNg1999}, \citet*{KoeMiz2004}, and
\citet{Koe2005, Koe2010}. Monotonicity constraints can be
easily implemented in the optimization process via inequality
constraints (see e.g.\ \citealt{KoeNg2005}), where we apply the
option \texttt{method="fnc"} for the \texttt{rq}-function of the
\texttt{quantreg} package. A detailed description of the {\sf
R}-functions we use is given in Table~\ref{tab:R_fct}.

\vspace{2ex}
\begin{table}[htb!]
\begin{center}
{\renewcommand{\baselinestretch}{1.05}
\large \small
\begin{tabular}{l|l|l|l}
model   & LS estimations & QR estimations & chosen parameters    \\
\hline
1: parametric& \texttt{ls} (base)               & \texttt{rq} (quantreg)         & - \\
            & \texttt{summary} (base)          & \texttt{summary.rq} (quantreg) & \\
\hline
2: B-spline & \texttt{ls} (base)               & \texttt{rq} (quantreg)         & knot sequence: equidistant, \\
            & \texttt{summary} (base)          & \texttt{summary.rq} (quantreg) & \hspace*{1ex} $m=4$ \\
            & \texttt{splineDesign} (splines)  & \texttt{splineDesign} (splines)&  \\
\hline
3: monotone & \texttt{pcls} (mgcv)             & \texttt{rq} (quantreg, method="fnc") & knot sequence: equidistant, \\
B-spline    & \texttt{splineDesign} (splines)  & \texttt{summary.rq} (quantreg) & \hspace*{1ex} $m=4$ \\
            & bootstrapped std.\ err.          & \texttt{splineDesign} (splines)& inequ.\ restr.: constr.\ matrix: \\
& & & \hspace*{1ex} $\left(\begin{smallmatrix} \mathbf C & \mathbf 0 \\ \mathbf 0 & \mathbf 0 \end{smallmatrix}\right)$, $\mathbf C = \left(\begin{smallmatrix} -1 & \phantom{-}1 & 0 & \\ \phantom{-}0 & -1 & 1 & \\[-1ex] & & \ddots & \ddots \end{smallmatrix}\right)$ \\
\hline
4: smoothing& \texttt{pcls} (mgcv)             & \texttt{rqss} (quantreg)       & knot sequence: distinct obs. \\
spline      &  \texttt{splineDesign} (splines) & \texttt{qss} (quantreg, constr.="D") & inequ.\ restr.: see 3 \\
            & bootstrapped std.\ err.          & \texttt{summary.rqss} (quantreg) &  smoothing parameter: $\lambda = 1$ \\
\end{tabular}
\caption{Applied {\sf R}-functions (and package they are contained in, partly with function option) for model estimations.}
\label{tab:R_fct}
}
\end{center}
\end{table}


\section{Model Specification and Data}
\label{sec:DataModel}

We illustrate the quantile regression methods outlined in Section 2 of the paper
in an empirical application using weekly
store-level scanner data from Dominick's Finer Foods, a large retail
chain in the Chicago metropolitan area. The data is publicly
available at \texttt{http://research.chicagobooth.edu/marketing/databases/dominicks},
provided by the James M.\ Kilts Center, University of Chicago Booth School of
Business. In particular, the data set includes weekly unit sales,
retail prices and display activities on the store level for nine
brands in the refrigerated orange juice category (package size 64
oz) and contains a balanced (i.e. no product introductions occurred
during the considered time span) panel data set for a cross-section
of 46 stores of the retail chain over a time span of 88 weeks
resulting in a total of 4048 observations. Among the nine brands are
three premium brands, five national brands and the supermarket's own
store brand. For those nine brands, Table~\ref{tab:sumstat} provides
summary statistics for average weekly prices and market shares
(pooled across stores). Differences in quality across the three
tiers are well represented by higher (lower) average prices for
higher (lower) quality tier brands. For brands that comprise several
sub-brands (e.g., with different levels of pulp, with/without
Calcium) those sub-brands were merged in advance to prevent
multicollinearity problems. In this case, prices represent
sales-weighted averages of the sub-brands' prices. In the following,
we exemplarily study the national brand Minute Maid.

\vspace{2ex}
\begin{table}[htb!]
\begin{center}
{\renewcommand{\baselinestretch}{1.05}
\large \small
\begin{tabular}{ll|rrr|rrr}
 & & \multicolumn{3}{c}{average prices (\$)} & \multicolumn{3}{c}{average market shares (\%)} \\
\cline{3-8}
brand & abbreviation    &   [min, max]      &   mean    &   sd      &   [min, max]      &   mean    &   sd \\
\hline
Florida Natural & flona      &   [1.56, 3.16]    &   2.85    &   0.33    &   [0.57, 47.79]       &    4.55       &    6.12 \\
Tropicana Pure & tropu       &   [1.58, 3.50]    &   2.94    &   0.50    &   [3.24, 55.26]       &   14.18       &   11.24 \\
Minute Maid Premium & mmprm    &   [1.58, 3.50]    &   2.94    &   0.51    &   [1.24, 20.28]       &    5.40       &    4.13 \\
\hline
Minute Maid & mmaid          &   [1.29, 2.88]    &   2.22    &   0.39    &   [3.26, 85.54]       &   20.18       &   20.41 \\
Citrus Hill & cithi          &   [1.08, 2.78]    &   2.29    &   0.31    &   [1.28, 75.19]       &    7.51       &   10.90 \\
Florida Gold & flogo         &   [1.31, 2.78]    &   2.17    &   0.35    &   [0.20, 34.26]       &    2.68       &    4.52 \\
Tree Fresh & trefr           &   [1.06, 2.44]    &   2.13    &   0.27    &   [0.80, 39.78]       &    4.01       &    5.04 \\
Tropicana & tropi            &   [1.49, 2.70]    &   2.19    &   0.33    &   [2.06, 71.24]       &   20.08       &   21.82 \\
\hline
Dominick's & domin           &   [0.99, 2.44]    &   1.75    &   0.39    &   [1.38, 78.24]       &   21.41       &   20.28 \\
\hline
\end{tabular}
\vspace{2mm}
\caption{Summary statistics for average weekly prices and market shares (pooled across stores, i.e.\ summary of 88 weekly observations which are each pooled from 46 store observations).}
\label{tab:sumstat}
}
\end{center}
\end{table}

To account for the price-quality tier structure of the data, one
could capture cross-price effects at the tier-level (e.g.,
\citealt{SteBreBe2007}).
This would yield a more parsimonious model on the one hand, but may affect the
estimates of own-price elasticities and/or the predictive
performance of a model due to that aggregation on the other hand.
One solution for this trade-off is to model at least separate
cross-price effects for the direct competitors of Minute Maid in its
national brand tier (Florida Gold, Tree Fresh, Tropicana, Citrus
Hill). In addition, it is reasonable to model the cross-price effect
of Minute Maid Premium separately, as a Minute Maid manager who is
responsible for both Minute Maid brands would be especially
concerned with this effect. The cross-price effect for the other two
premium brands (Florida Natural, Tropicana Pure) is instead captured
jointly by defining \texttt{cross\_premium}$_{s,t}$ to represent the
price of the respective cheaper brand in week $t$ and store $s$.
Using this minimum rule to aggregate the cross-prices of Florida
Natural and Tropicana Pure seems to be a natural choice since a
national brand buyer is usually attracted by price cuts in the
premium brand tier which is well-reflected by the lowest price of
premium brands rather than a mean price, for instance.
We therefore include the
following cross-price covariates in our model:
\texttt{cross\_premium}, \texttt{cross\_mmprm},
\texttt{cross\_cithi}, \texttt{cross\_flogo}, \texttt{cross\_trefr},
\texttt{cross\_tropi}, \texttt{cross\_domin} (please compare Table~\ref{tab:sumstat}).
Given this cross-price specification, a standard
parametric model for analyzing store-level data can be stated as
follows:
\begin{align}
\notag
\log(\texttt{sales})_{s,t}^b &=
\beta_0^b + \beta_1^b \log(\texttt{price})_{s,t}^b + \beta_2^b \log(\texttt{cross\_premium})_{s,t}^b + \beta_3^b \log(\texttt{cross\_mmprm})_{s,t}^b \\
\notag
&\phantom{=} + \beta_4^b \log(\texttt{cross\_cithi})_{s,t}^b + \beta_5^b \log(\texttt{cross\_flogo})_{s,t}^b + \beta_6^b \log(\texttt{cross\_trefr})_{s,t}^b \\
\notag
&\phantom{=} + \beta_7^b \log(\texttt{cross\_tropi})_{s,t}^b + \beta_8^b \log(\texttt{cross\_domin})_{s,t}^b \\
\notag
&\phantom{=} + \beta_9^b \texttt{display}_{s,t}^b + \beta_{10}^b \texttt{end\_99}_{s,t}^b + \beta_{11}^b (\texttt{display} \cdot \texttt{end\_99})_{s,t}^b + \beta_{12}^b \texttt{holiday}_{t} \\[-1ex]
&\phantom{=} + \beta_{13}^b \texttt{summer}_t + \beta_{14}^b \texttt{fall}_t + \beta_{15}^b \texttt{winter}_t
+ \sum_{o=2}^{46} \delta_o^b \texttt{store} \, o_{s} + u_{s,t}^b,
\label{eq:model A1}
\tag{$A,1$}
\end{align}
where
for brand $b$ (here: Minute Maid) in store $s$ and week $t$,
$\texttt{sales}_{s,t}^b$ are the unit sales,
$\texttt{price}_{s,t}^b$ is the observed own-price,
$\texttt{cross\_premium}_{s,t}^b, \ldots, \texttt{cross\_domin}_{s,t}^b$ are cross-prices (as outlined above),
$\texttt{display}_{s,t}^b$ is a dummy indicating whether a display is used,
$\texttt{end\_99}_{s,t}^b$ is a dummy indicating whether the own-price ends in .99 cents,
$\texttt{holiday}_t$ is a dummy indicating whether a holiday is in week $t$,
$\texttt{summer}_t$, $\texttt{fall}_t$, and $\texttt{winter}_t$ are seasonal dummies indicating whether week $t$
belongs to the summer season, the fall season, or the winter season, respectively,
$\texttt{store} \, o_{s}$ are store dummies (for stores 2 to 46) indicating whether the observations refer to store $s$,
and $u_{s,t}^b$ is an error term.
The store dummies are used to account for cross-sectional heterogeneity in the baseline sales
of Minute Maid across the different stores (e.g., due to different store sizes), while the seasonal
and holiday effects accommodate time-specific shifts in baseline sales for Minute Maid.
This log-log or multiplicative model is the most widely used parametric sales
response model to relate brand sales to marketing instruments (\citealt*{HanParSc2001}).

We study three semiparametric models (denoted as models ($A,2$),
($A,3$), and ($A,4$)), where the parametric part ($\beta_0^b +
\beta_1^b \log(\texttt{price})_{s,t}^b$) is replaced by the B-spline
functional
\begin{equation*}
\sum_{j=-(k-1)}^{m} \alpha_j^b B_j^{\boldsymbol\kappa, k}( \log(\texttt{price})_{s,t}^b)
\end{equation*}
in order to estimate own-price response flexibly, respectively.
Models ($A,2$) and ($A,3$) have in common that they use a cubic
($k=4$) B-spline basis with an equidistant knot sequence
$\boldsymbol\kappa$ with $m=4$ interior knots (cf.\ Section 2.4 in the paper).
However, models ($A,2$) and ($A,3$) differ
in that the latter has an additional monotonicity constraint imposed
on the nonparametric price term to obtain a monotonically decreasing
own-price effect and hence a negative own-price elasticity. Finally,
model ($A,4$) is estimated using a linear smoothing spline with an
additional monotonicity constraint imposed on the nonparametric
price term (cf.\ Section 2.4 in the paper)\footnote{We
estimated models ($A,1$) to ($A,4$) with several different
cross-price specifications, including flexibly estimated cross-price
effects using B-splines and smoothing splines. Details on estimation
results for these models can be obtained from the authors upon
request.}.

In Section 5 of the paper where we also study the predictive
performance of models with proxy cross-prices approximated from
historical data (instead of ex post observed cross-prices considered
in models ($A,1$) to ($A,4$)), we mimic the situation of a manager
who wants to predict the unit sales for her/his brand, but does not
have complete information on future competitive pricing ex ante. In
analogy, the proxy-based alternatives are referred to as models
($B,1$) to ($B,4$). Table~\ref{tab:models} provides an overview of
all models considered for estimation and prediction in the
following.

\vspace{2ex}
\begin{table}[htb!]
\begin{center}
{\renewcommand{\baselinestretch}{1.05}
\large \small
\begin{tabular}{m{23ex}m{15ex}m{15ex}m{15ex}m{18ex}}
covariate versions                               & \multicolumn{4}{c}{functional forms}                             \\
\hline
        & parametric      &  unconstrained \newline B-spline   & constrained \newline B-spline & constrained \newline smoothing spline \\
observed cross-prices                   & ($A,1$)     & ($A,2$)         & ($A,3$)    & ($A,4$)          \\
proxy cross-prices                & ($B,1$)  & ($B,2$)         & ($B,3$)    & ($B,4$)          \\
\end{tabular}
\caption{Models studied.}
\label{tab:models}
}
\end{center}
\end{table}

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