Modeling heterogeneous effective advertising stock using single-source data

Abstract

This paper presents a nonlinear modeling of market response between advertising stock and direct utility with heterogeneous parameters using single-source data. We examine advertising threshold effects and measure the effective advertising stock at the individual consumer level. Two kinds of information, i.e., TV advertising exposure data and consumer’s purchase history data, are combined for the modeling. The former is used for constructing advertising stock over calendar time via heterogeneous carryover parameters and the latter is applied to the choice model. The Markov chain Monte Carlo (MCMC) method is applied to estimate these heterogeneous parameters. Compared to other possible nonlinear specifications, it is shown that the proposed threshold utility function model with discontinuity at the threshold performs better than other smooth market response models. The empirical results support the existence of an advertising threshold and suggest the pulsing or “on/off” policy for our datasets. In terms of the effective reach, implying the reach after suspending the ad exposure to investigate how it is damping out for a possible “on/off” advertising policy, the optimal “off” interval was measured to be quite short to support a high-frequency pulsing policy, because the carryover parameter as well as the difference of ad stock and threshold are not large enough for our datasets.

Appendices

Appendix A

1.1 Markov chain Monte Carlo (MCMC) procedure

The specifications of the model proposed above and the setting appropriate to the prior distribution for each parameter engender Bayesian inference. We evaluate posterior densities using the MCMC method. Starting from the initial values of all parameters, the MCMC procedure has two stages: (1) generating advertising stocks using ad exposure data, and (2) sampling from full-conditional posterior densities using purchase data.

• 1. Advertising exposure data

  Consumer h’s advertising stock [A _jht ] data for brand j are generated by the following steps. Conditional on the draw of ρ _h :

  • (a) Generate the advertising stock of brand j by $A_{{jhw}} = a_{{jhw}} \prime + \rho _{h} A_{{jh,w - 1}}$ over the entire observation period w = 1, …, T: {A _jhw , w = 1, …, T|ρ _h }, and

  • (b) Select the subset at consumer h’s purchase time t = 1, …, T _h as {A _jht , t = 1, …, T _h |ρ _h }.

• 2. Purchase data

  Conditional on the draw of other parameters: $\left( {\theta _h^{\left( i \right)} ,\Sigma ^{\left( i \right)} ,i = 1,2} \right)$, r _h , (λ ⁽ⁱ⁾, Λ⁽ⁱ⁾, η, ξ, ζ, ν),

  • (a) Generate latent utility $\left[ {u_{jht}^{\left( 1 \right)} ,u_{jht}^{\left( 2 \right)} } \right]$, and

  • (b) Draw from conditional posterior density

    • (a) Within subject parameters $\theta _h^{\left( i \right)} | -$ and Σ⁽ⁱ⁾|–, i = 1, 2.,

    • (b) Threshold parameter r _h |–,

    • (c) Carryover parameters ρ _h |–, and

    • (d) Between subject parameters λ ⁽ⁱ⁾|–, Λ⁽ⁱ⁾|–, η|–, ξ|–, ζ|–, and ν|–,

  where the notation “|–” means conditional posterior density given data and other parameters. The iteration of steps 1. and 2. approximates the required joint posterior density of parameters of our model. Details of each step are described in Appendix B.

Appendix B

2.1 Markov chain Monte Carlo (MCMC) algorithm

2.1.1 Likelihood function

Given the values of threshold {r _h }, the data of covariates are allocated to each regime according to the level of individual advertising stock at each purchase occasion, and the corresponding latent utility vector is generated based on the choice data using the algorithm of the Bayesian probit model for each regime. This operation constitutes two different likelihoods. For that reason, no identification problem arises across regimes.

$$\begin{array}{*{20}l} {\prod\limits_{t \in R^{\left( 1 \right)} \left( {r_h } \right)} {\left| {\Sigma ^{(1)} } \right|^{ - \frac{1}{2}} } \exp \left\{ { - \frac{1}{2}\left( {u_{ht}^{\left( 1 \right)} - x_{ht}^{\left( 1 \right)} \theta _h^{\left( 1 \right)} } \right)\prime \Sigma ^{\left( 1 \right) - 1} \left( {u_{ht}^{\left( 1 \right)} - x_{ht}^{\left( 1 \right)} \theta _h^{\left( 1 \right)} } \right)} \right\}} \hfill \\ { \times \prod\limits_{t \in R^{\left( 2 \right)} \left( {r_h } \right)} {\left| {\Sigma ^{(2)} } \right|^{ - \frac{1}{2}} } \exp \left\{ { - \frac{1}{2}\left( {u_{ht}^{\left( 2 \right)} - x_{ht}^{\left( 2 \right)} \theta _h^{\left( 2 \right)} } \right)\prime \Sigma ^{\left( 2 \right) - 1} \left( {u_{ht}^{\left( 2 \right)} - x_{ht}^{\left( 2 \right)} \theta _h^{\left( 2 \right)} } \right)} \right\}} \hfill \\ \end{array} $$

In those equations, R ⁽¹⁾(r _h ) ∪ R ⁽²⁾(r _h ) = T _h . Furthermore, assuming independent choice behavior across consumers, we obtain an overall likelihood function by taking products over respective consumers as

$$\begin{array}{*{20}c} {L\left( {\left\{ {r_h } \right\};\left\{ {I_{ht} } \right\},\left\{ {x_{ht} } \right\}\left| {\theta ^{\left( \cdot \right)} } \right.,\Sigma ^{\left( \cdot \right)} } \right) \propto \prod\limits_{h = 1}^H {\left\{ {\prod\limits_{i = 1}^2 {\left\{ {\prod\limits_{t \in R^{\left( i \right)} \left( {r_h } \right)} {\left| {\Sigma ^{\left( i \right)} } \right|^{ - \frac{1}{2}} } \exp \left\{ { - \frac{1}{2}\left( {u_{ht}^{\left( i \right)} - x_{ht}^{\left( i \right)} \theta _h^{\left( i \right)} } \right)\prime \Sigma ^{\left( i \right) - 1} \left( {u_{ht}^{\left( i \right)} - x_{ht}^{\left( i \right)} \theta _h^{\left( i \right)} } \right)} \right\}} \right\}} } \right\}} } \\ { \equiv \prod\limits_{i = 1}^2 {l\left( {\left\{ {u_{ht}^{\left( i \right)} } \right\}\left| {\left\{ {\theta _h^{\left( i \right)} } \right\},\Sigma ^{\left( i \right)} ,\left\{ {r_h } \right\}} \right.} \right),} } \\ \end{array} $$

where {I _ht } means the personal observed choice data and $\theta ^{\left( \cdot \right)} = \left\{ {\theta _h^{\left( 1 \right)} ,\theta _h^{\left( 2 \right)} } \right\}$, $\Sigma ^{\left( \cdot \right)} = \left\{ {\Sigma ^{\left( 1 \right)} ,\Sigma ^{\left( 2 \right)} } \right\}$.

2.1.2 Prior and full conditional posterior distributions

We summarize the prior and conditional posterior distribution used for model estimation below.

• 1. Prior distributions

  $$- \theta _h^{\left( i \right)} :\Sigma ^{\left( i \right)} ,\Lambda ^{\left( i \right)} ,\lambda ^{\left( i \right)} - $$

  • (a) $\Sigma ^{\left( i \right) - 1} \sim {\text{Wishart}}\left( {\nu ^{\left( i \right)} ,V^{\left( i \right)} } \right)$; $\nu ^{\left( i \right)} = N + 1$, V ⁽ⁱ⁾ = ν ⁽ⁱ⁾ I _N-1 .

  • (b) $\Lambda ^{\left( i \right)^{ - 1} } \sim {\text{Wishart}}\left( {\nu _{b0}^{\left( i \right)} ,V_{b0}^{\left( i \right)} } \right)$; $\nu _{b0}^{\left( i \right)} = f_0 ,\quad V_{b0}^{\left( i \right)} = \nu _{b0}^{\left( i \right)} I_k$

  • (c) $\lambda ^{ * \left( i \right)} = {\text{vec}}\left( {\lambda ^{\left( i \right)} } \right) \sim {\rm N}\left( {\overline d ,\left( {\Lambda ^{\left( i \right)} \otimes A_d^{ - 1} } \right)} \right)$; $\left( {\overline d = 0,\quad A_d = 0.01I_d } \right)$

  $$- r_h :\zeta ,\nu - $$

  • (a) $\zeta \sim N\left( {\overline \zeta ,\overline \Sigma _\zeta } \right)$; $\overline \zeta = 0,\overline \Sigma _\zeta = 0.01I_d$

  • (b) $\nu ^{ - 1} \sim {\text{Wishart}}\left( {\overline \nu ,\overline \Upsilon _\nu } \right)$; $\overline \nu = g0,\overline \Upsilon _\nu = \overline \nu I_d$

  $- \rho _h :\eta ,\xi -$

  • (a) $\eta \sim N\left( {\overline \eta ,\overline \Sigma _\eta } \right)$; $\overline \eta = 0,\overline \Sigma _\eta = 0.01I_d$

  • (b) $\xi ^{ - 1} \sim {\text{Wishart}}\left( {\overline \xi ,\overline \Upsilon _\xi } \right)$; $\overline \xi = s0,\overline \Upsilon _\xi = \overline \xi I_d$

  The likelihood function and the prior distributions given above lead to the following conditional posterior distributions.

• 2. Full conditional posterior distributions

  • I. Latent utility

    $$\left\{ {u_{ht}^{\left( i \right)} } \right\}\left| {\left\{ {I_{ht} } \right\}} \right.,\theta _h^{\left( i \right)} ,\Sigma ^{\left( i \right)} ,\left\{ {x_{ht}^{\left( i \right)} } \right\},r_h ;{\text{ Truncated normal distribution}}$$

  • II. Market response and variance

    $$\begin{array}{*{20}l} {\theta _h^{\left( i \right)} \left| {\left\{ {u_{ht}^{\left( i \right)} } \right\},\Sigma ^{\left( i \right)} ,\lambda ^{\left( i \right)} ,} \right.\Lambda ^{\left( i \right)} ,\left\{ {x_{ht}^{\left( i \right)} } \right\},\left\{ {Z_h } \right\},r_h ;\quad {\text{Normal}}\,{\text{distribution}}} \hfill \\ {\Sigma ^{\left( i \right) - 1} \left| {\left\{ {u_{ht}^{\left( i \right)} } \right\},\theta _h^{\left( i \right)} ,\left\{ {x_{ht}^{\left( i \right)} } \right\},\left\{ {Z_h } \right\}r_h ;\quad {\text{Wishart distribution}}} \right.} \hfill \\ \end{array} $$

  • III. Hierarchical coefficient and variance

    $$\begin{aligned}& \lambda ^{\left( i \right)} \left| {\left\{ {\theta _h } \right\}} \right.,\Lambda ^{\left( i \right)} ,\left\{ {Z_h } \right\},r_h ;\;{\text{Normal distribution}} \\& \Lambda ^{\left( i \right) - 1} \left| {\left\{ {\theta _h } \right\}} \right.,\lambda ^{\left( i \right)} ,\left\{ {Z_h } \right\},r_h ;\;{\text{Wishart distribution}} \\ \end{aligned} $$

    Specifications of these normal and Wishart distributions are available in the appendix of Rossi et al. (1996, pp. 338–339).

  • IV. Effective advertising stock level: [r _h ]

    • 1. $r_h \left| {u_{ht}^{\left( i \right)} } \right.$, Σ⁽ⁱ⁾, $\theta _h^{\left( i \right)}$, ρ _h , _h , ζ, ν

      The posterior density of r can be written as

      $$\begin{array}{*{20}c}{\pi \left( {\left\{ {r_h } \right\}\left| {\theta ^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\zeta ,\nu } \right) \propto \left[ {\prod\limits_{i = 1}^2 {l\left( {\left\{ {u_{ht}^{\left( i \right)} } \right\}\left| {\left\{ {\theta _h^{\left( i \right)} } \right\}} \right.,\Sigma ^{\left( i \right)} ,\left\{ {r_h } \right\}} \right)} } \right]P\left( {\left\{ {r_h } \right\}\left| \zeta \right.,\nu ,\left\{ {Z_h } \right\}} \right)} \\\end{array} .$$

      It does not constitute a conjugate family. We use Metropolis–Hastings with a random walk algorithm,

      $$r_h^{\left( k \right)} = r_h^{\left( {k - 1} \right)} + \omega _{3,h} \quad ,\omega _{3,h} \sim N\left( {0,0.1} \right),\quad \left( {0 <r_h^{\left( k \right)} } \right),\;r_h^{\left( 0 \right)} \sim N\left( {\frac{1}{{T_h }}\sum\limits_{t = 1}^{T_h } \;S_{ht}^ * ,0.5} \right)$$

      Acceptance probability α is defined as

      $$\alpha \left( {r_h^{\left( k \right)} ,r_h^{\left( {k - 1} \right)} \left| {\theta ^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\zeta ,\nu } \right) = \min \left[ {\frac{{\pi \left( {r_h^{\left( k \right)} \left| {\theta ^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\zeta ,\nu } \right)}}{{\pi \left( {r_h^{\left( {k - 1} \right)} |\theta ^{\left( \bullet \right)} ,\Sigma ^{\left( \bullet \right)} ,\zeta ,\nu } \right)}},1} \right].$$

      Then, using uniform random number u _3,h  ∼ U(0,1), we determine the rule of acceptance of random draws as

      $$r_h = \left\{ \begin{aligned}& r_h^{\left( k \right)} \quad {\text{accepted}}\quad {\text{if}}\quad u_{3,h} \leqslant \alpha \left( {r_h^{\left( {k - 1} \right)} ,r_h^{\left( k \right)} \left| {\theta ^{\left( k \right)} } \right.,\Sigma ,\zeta ,\nu } \right) \\& r_h^{\left( {k - 1} \right)} \quad {\text{accepted}}\;{\text{if}}\quad {\text{otherwise}} \\ \end{aligned} \right.$$

    • 2. ζ| r, ν $\left[ {r_h = Z_h \zeta _h + \psi _h ,\;\psi _h \sim N\left( {0,\nu } \right)} \right]$

      $$\begin{aligned} & \zeta \sim N{\left( {\zeta ^{ * } ,\Sigma _{{\zeta ^{ * } }} } \right)}; \\ & \zeta ^{ * } = {\left( {Z\prime Z + \overline{\Sigma } _{\zeta } ^{{ - 1}} } \right)}^{{ - 1}} {\left( {Z\prime Z\widehat{\zeta } + \overline{\Sigma } _{\zeta } ^{{ - 1}} \overline{\zeta } } \right)},\Sigma _{{\zeta ^{ * } }} = {\left( {Z\prime Z + \overline{\Sigma } _{\zeta } ^{{ - 1}} } \right)}^{{ - 1}} ,\widehat{\zeta } = {\left( {Z\prime Z} \right)}^{{ - 1}} Zr \\ \end{aligned} $$

    • 3. v| r, ζ

      $$\nu ^{ - 1} \sim W\left( {\overline \nu + H,\mathop {\left( {\overline \Upsilon _\nu ^{ - 1} + Q} \right)}\nolimits^{ - 1} } \right),\;Q = \sum\limits_{h = 1}^H \;\mathop {\left( {r_h - Z_h \zeta } \right)}\nolimits^\prime \left( {r_h - Z_h \zeta } \right)$$

  • V. Carryover parameters: [ρ _h ]

    • 1. $\rho _h |u^{ * \left( i \right)}$, Σ⁽ⁱ⁾, η, ξ, θ ⁽ⁱ⁾, τ, $r\left[ {u_{jht}^{ * \left( i \right)} = A_{jh,t - 1}^{ * \left( i \right)} \rho _h + \varepsilon _{jht}^{\left( i \right)} } \right]$

      We first note that $u^{{ * {\left( i \right)}}}_{{jht}} = A^{{ * {\left( i \right)}}}_{{jh,t - 1}} \rho _{h} + \varepsilon ^{{{\left( i \right)}}}_{{jht}}$, where $u_{jht}^{ * \left( i \right)} = u_{jht} - M_{jht} \beta _h^{\left( i \right)} - \left( {a_{hnt} - a_{hjt} } \right)\alpha _h^{\left( i \right)}$ and $A_{jh,w - 1}^{ * \left( i \right)} = A_{hn,w - 1} \alpha _h^{\left( i \right)}$ gives us a conditional likelihood for this parameter. Then, the posterior density of reparameterized $\rho _h^ *$ can be formulated as

      $$\begin{aligned}& \begin{array}{*{20}c}{\pi \left( {\left\{ {\rho _h^ * } \right\}\left| {\theta ^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\xi ,\eta } \right) \propto \left[ {\prod\limits_{i = 1}^2 {l\left( {\left\{ {u_{ht}^{\left( i \right)} } \right\}\left| {\left\{ {\theta _h^{\left( i \right)} } \right\}} \right.,\Sigma ^{\left( 1 \right)} ,\left\{ {r_h } \right\}} \right)} } \right]P\left( {\left\{ {\rho _h^ * } \right\}\left| \xi \right.,\eta ,\left\{ {Z_h } \right\}} \right)} \\\end{array} \\& \quad \quad \quad \quad \quad \quad \quad \propto \left[ {\prod\limits_{i = 1}^2 {l\left( {\left\{ {u_{ht}^{ * \left( i \right)} } \right\}\left| {\left\{ {\rho _h^ * } \right\}} \right.,\Sigma ^{\left( 1 \right)} ,\left\{ {r_h } \right\}} \right)} } \right]P\left( {\left\{ {\rho _h^ * } \right\}\left| \xi \right.,\eta ,\left\{ {Z_h } \right\}} \right)\left| {J_{\rho \to \rho ^ * } } \right| \\ \end{aligned} $$
      $$ \propto \exp \left\{ { - 0.5 * \left[ {\begin{array}{*{20}c} {\sum\limits_{t = 1}^{T_h^{\left( 1 \right)} } \,\mathop {\left( {{{u_{jht}^{ * \left( 1 \right)} - A_{h,t - 1}^{ * \left( 1 \right)} } \mathord{\left/ {\vphantom {{u_{jht}^{ * \left( 1 \right)} - A_{h,t - 1}^{ * \left( 1 \right)} } {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right)}\nolimits^\prime \Sigma ^{\left( 1 \right) - 1} \left( {{{u_{jht}^{ * \left( 1 \right)} - A_{h,t - 1}^{ * \left( 1 \right)} } \mathord{\left/ {\vphantom {{u_{jht}^{ * \left( 1 \right)} - A_{h,t - 1}^{ * \left( 1 \right)} } {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right)} \\ { + \sum\limits_{t = 1}^{T_h^{\left( 2 \right)} } \,\mathop {\left( {u_{jht}^{ * \left( 2 \right)} - {{A_{h,t - 1}^{ * \left( 2 \right)} } \mathord{\left/ {\vphantom {{A_{h,t - 1}^{ * \left( 2 \right)} } {\left( {1 + \exp \left( {\rho _h^* } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \exp \left( {\rho _h^* } \right)} \right)}}} \right)}\nolimits^\prime \Sigma ^{\left( 2 \right) - 1} \left( {u_{jht}^{ * \left( 2 \right)} - {{A_{h,t - 1}^{ * \left( 2 \right)} } \mathord{\left/ {\vphantom {{A_{h,t - 1}^{ * \left( 2 \right)} } {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \exp \left( {\rho _h^ * } \right)} \right)}}} \right)} \\ { + \mathop {\left( {\rho _h^ * - Z_h \eta } \right)}\nolimits^\prime \xi ^{ - 1} \left( {\rho _h^ * - Z_h \eta } \right)} \\ \end{array} } \right]} \right\}\left| {J_{\rho \to \rho ^ * } } \right|$$

      where $J_{\rho \to \rho ^ * }$ is the Jacobian.

      It does not constitute a conjugate family, and we use Metropolis–Hastings with a random walk algorithm.

      $$\rho _h^{ * \left( k \right)} = \rho _h^{ * \left( {k - 1} \right)} + \omega _{1,h} ,\;\omega _{1,h} \sim N\left( {0,0.01} \right),\;{\text{and}}\;\rho _h^{ * \left( 0 \right)} \sim U\left( {0,1} \right)$$

      Acceptance probability α is defined as

      $$\alpha \left( {\rho _h^{ * \left( k \right)} ,\rho _h^{ * \left( {k - 1} \right)} \left| {\theta _h^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\eta ,\xi } \right) = min\left[ {\frac{{\pi \left( {\rho _h^{ * \left( k \right)} |\theta _h^{\left( \bullet \right)} ,\Sigma ^{\left( \bullet \right)} ,\eta ,\xi } \right)}}{{\pi \left( {\rho _h^{ * \left( {k - 1} \right)} |\theta _h^{\left( \bullet \right)} ,\Sigma ^{\left( \bullet \right)} ,\eta ,\xi } \right)}},1} \right]$$

      Then, the sample uniform random number is u _1,h  ∼ U(0,1). We determine the rule of acceptance of random draws as

      $$\rho _h^ * = \left\{ \begin{aligned}& \rho _h^{ * \left( k \right)} \quad {\text{accepted}}\quad {\text{if}}\quad u_{1,h} \leqslant \alpha \left( {\rho _h^{ * \left( {k - 1} \right)} ,\rho _h^{ * \left( k \right)} \left| {\theta _h^{\left( \bullet \right)} } \right.,\Sigma ^{\left( \bullet \right)} ,\eta ,\xi } \right) \\& \rho _h^{ * \left( {k - 1} \right)} \quad {\text{accepted}}\;\;{\text{if}}\;\;{\text{otherwise}} \\ \end{aligned} \right..$$

      The inverse transformation $\rho _h^{\left( k \right)} = \frac{1}{{1 + \exp \left( {\rho _h^{ * \left( k \right)} } \right)}}$ leads to a required sample.

    • 2. $\eta \left| {\rho _h^ * } \right.,\xi \;\left[ {\rho _h^ * = Z_h \eta + \omega _h ,\quad \omega _h \sim N\left( {0,\xi } \right)} \right]$

      $$\begin{aligned}& \eta \sim N\left( {\eta ^ * ,\Sigma _{\eta ^ * } } \right); \\& \eta ^ * = \left( {Z\prime Z + \overline \Sigma _\eta ^{ - 1} } \right)^{ - 1} \left( {Z\prime Z\widehat\eta + \overline \Sigma _\eta ^{ - 1} \overline \eta } \right),\Sigma _{\eta ^ * } = \left( {Z\prime Z + \overline \Sigma _\eta ^{ - 1} } \right)^{ - 1} ,\widehat\eta = \left( {Z\prime Z} \right)^{ - 1} Z\rho ^ * \\ \end{aligned} $$

    • 3. $\xi \left| {\rho _h^ * } \right.$, η

      $$\xi ^{ - 1} \sim W\left( {\overline \xi + H,\mathop {\left( {\overline \Upsilon _\xi ^{ - 1} + R} \right)}\nolimits^{ - 1} } \right),\;R = \sum\limits_{h = 1}^H \,\mathop {\left( {\rho _h^ * - Z_h \eta } \right)}\nolimits^\prime \left( {\rho _h^ * - Z_h \eta } \right)$$

2.1.3 Continuous and restricted threshold utility function models

The continuous threshold utility function models and restricted utility function models have a common structure in the sense that they are partially linear. We follow the preceding steps to implement the threshold utility function model restricted by $\beta _h^{\left( 1 \right)} = \beta _h^{\left( 2 \right)} = \beta _h$. Denoting the linear part of the explanatory variable as x _1ht  = M _ht and corresponding parameter vector by θ _1h , we have k dimensional x _ht  = (A _ht , x _1ht ′)′ and $\theta _h^{\left( i \right)} = \left( {\alpha _h^{\left( i \right)} ,\theta _{1h} \prime } \right)\prime$ for i = 1,2.