Online Supplement. and. Pradeep K. Chintagunta Graduate School of Business University of Chicago

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1 Online Suppleent easuring Cross-Category Price Effects with Aggregate Store Data Inseong Song Departent of arketing, HKUST Hong Kong University of Science and Technology and Pradeep K Chintagunta pradeepchintagunta@gsbuchicagoedu Graduate School of Business University of Chicago Technical Appendix A Econoetric odel arginal Category Purchase Probabilities (Equation 4 in the paper As discussed in the paper, a consuer chooses a bundle that axiizes her utility in our odel The probability that a consuer purchases in category is nothing but the probability that the best bundle (utility axiizing bundle has d atheatically, it can be represented as follows: (A Pr(y Pr ax U(l, l,,l st I(l > > ax U(k, k,, k st I(k > (k,k,,k (l,l,,l In other words, it is the probability that the best bundle that contains d has a higher utility than the best bundles that contains d Then, what is ax U(l l,,l st I(l >? The (l,l,,l, axiu of Gubel rando variable is also a Gubel with the sae scale paraeter And the location paraeter of the axiu is given by (A log exp ( Γ (,d,,d + V,l + d V l l l l where d I(l > for,3,, Note that the first suation goes fro to while the rest suations go fro to By the sae way, ax U(k k,,k st I(k > is a (k,k,,k,

2 Gubel with the location paraeter given by (A3 log exp ( Γ (,f,,f + f V k k k where f I(k > for,3,, Define G ( and G ( as follows: (A4 G ( exp ( Γ (,d,,d + V,l + d V l, st d I(l >,,3,, l l l (A5 G ( exp ( Γ (,f,,f + f V k, st f I(k >,,3,, k k Then the category purchase probability is given by G( (A6 Pr(y G( + G( If not infeasible, the above expression is not coputationally convenient Define W exp(v We copute G ( first The last suation over k is divided into two j j parts, the part associated the category purchase and the other for k (A7 k k ( Γ + exp ( (,f k,,f, f Vk V k exp (,f,,f, fv k G ( k k + Γ + + Γ Γ { } e e + W e Fro the definition in (4, it is given that k f V (,f,,f, (,f,,f, Γ (d,,d,,d,,d Γ (d,,d,,d,,d +γ + γ (, kd + γ (k, d So, k k + k k + k < k > k And Γ (,f,,f, Γ (,f,,f, +γ + γ(,f Therefore we have > Γ (,f,,f, γ f + γ(, f f

3 fv k (,f (A8 { } Γ(,f,,f, γ + γ G ( e e + We Note that k k e e + W e k k (,f { } fv γ k f + γ(, ff > γ + γ k > k e + W e f V + γ f + γ(, f f γ + γ(,f e e + W e k f V + γ f + γ (, f f f V +γ + γ(, f γ +γ(,f + γ(,f k,k > fvk + γ f + γ(, f f > e * + W e + e + W e γ + γ (,f V,k +γ + (, f (, (,f γ γ +γ + γ k fvk + γ f + γ(, f f > e * + W e + W e + W e γ + γ(,f γ + γ(, f γ +γ(, + γ(,f fvk + γ f + γ(, f f > e * + W e + W e + W W e γ + γ(,f γ + γ(, f γ +γ +γ(, + γ(, f + γ(,f fvk + γ f + γ(, f f γ f +γ(, + γ(, f f > f f f f e * W W e Further recursion yields the following expression for G (: Γ d3 (A9 G ( e W W W By the sae token, we have (,d,,d d d 3 d {,} d 3 {,} d {,} Γ d3 (A G ( W e W W W (,d,,d d d 3 d {,} d 3 {,} d {,} 3

4 So the category purchase probability expression in (A6 can be written as (A W e W W W Γ(,d,,d d d3 d 3 d d Γ(,d,,d d d3 d Γ(,d,,d d d3 d W e W W 3 W + e W W 3 W d d d d Pr(y Brand Choice Probability (Equation 5 in the paper The conditional brand choice probability for j in category is the probability that the best bundle that contains j for category has a higher utility than the best bundle that does not contains j for category conditional that the best bundle subject to d has a higher utility than the best bundle subject to d For the conditional brand choice in category, it is given by Pr(y y j ax U(j,l,,l ax U(l l,,l st I(l > Pr > ax U(k, k,,k st k j > ax U(k, k,,k st I(k > (k,k,,k (k,k,,k (A, (l,,l (l,l,,l The ter (A3 ax U(j,l,,l is Gubel distributed with the location paraeter given by (l,,l Vj log e e l l Γ (,g,,g + g Vl where g I(l > for,3,, eanwhile, the ter also Gubel with the location paraeter given by (A4 ax U(k k,,k st k j is (k,k,,k log e + e k j, k k k k k >, Γ (,f,,f + Vk + f V l Γ (,f,,f + f V l where f I(k > for,,3,, Fro the earlier discussion we know that And k j, k k k > Γ (,f,,f + f V l (,d,,d d d e Γ e W W k k d {,} d {,} e k l Γ (,f,,f + V + f V 4

5 Γ (,f,,f + Vk + f V l Γ (,f,,f + Vj+ f V l e e k k k k k Γ(,d,,d d d V j Γ(,d,,d d d d {,} d {,} d {,} d {,} W e W W e e W W ( V j Γ (,d,,d d d d {,} d {,} W e e W W Again, the location of ax U(j,l,,l is given by (l,,l Γ(,d,,d d d (A5 log e W W d {,} d {,} And the location of (A6 ( (k,k,,k ax U(k k,,k st k j is given by, V j Γ(,d,,d d d Γ(,d,,d d d + d {,} d {,} d {,} d {,} log W e e W W e W W Therefore the unconditional brand choice probability is given by (A7 V j Γ(,d,,d d d e e W W d {,} d {,} j Γ(,d,,d d d Γ(,d,,d d d W e W W + e W W d {,} d {,} d {,} d {,} Pr(y The conditional brand choice probability is obtained by dividing the unconditional brand choice probability by the arginal category purchase probability Dividing (A7 by (A produces (A8 exp(v j Pr(yj y exp(v k k oint Purchase Incidence Consider the joint purchase event where the first categories are purchased while the other - categories are not purchased Since the consuer axiizes her utility by choosing the best brand cobination, the utility of such an event is given by: (A9 ax U(l l,,l st l >,,l >,l,,l (l,l,,l, + It is also a Gubel rando variable whose location paraeter is given by (A log exp Γ (,,,, + Vklk l l k 5

6 So the joint purchase incidence probability is given by (A ( Pr y,, y, y,, y + exp Γ (,,,, + V clc l l c + exp Γ (f,,f,f,f + V + ckc k k k+ k c where fc I(lc >,c,, Note that the probability is the su of the joint brand choice probabilities (A l l + ckc k k k+ k c ( l l + exp Γ (,,,, + V cl c c + l l exp Γ (f,,f,f,f + V Pr y,, y, y,, y Two Category Case (Equation 7 and 8 in the paper Since we use the arginal choice probabilities for estiating the odel with aggregate data, the expression for the joint purchase incidence is not utilized so we do not pursue any further siplification of the expression However, it is worth to explore the odel property based on joint purchase and conditional purchase incidence For this purpose, we again turn to the siple case where there are two categories only Using our notation W exp( Vj purchase incidence in the two-category case is given by (A Pr ( y, y (A Pr ( y, y (A3 Pr ( y, y (A4 Pr ( y, y, the joint j, γ γ γ +γ +γ(, e W e W e WW γ e W γ γ γ +γ +γ e W e W e WW (, γ e W γ γ γ +γ +γ e W e W e WW + + +, (, e WW γ+γ +γ(, γ γ γ +γ +γ(, e W e W e WW 6

7 The expression for the arginal purchase incidence in the paper can be siplified as follows: (A3 Pr ( y Pr ( y, y Pr ( y, y γ γ +γ +γ(, e W+ e WW γ γ γ +γ +γ e W e W e WW (, Γ(, Γ(, W( e W + e Γ(, Γ(, Γ(, Γ(, ( + + ( + W e W e e W e which is equation (7 in the paper Differentiating (A3 with respect to W yields Γ(, Pr(y We (A4 Γ(, Γ(, Γ(, Γ(, W W( e W + e + ( e W + e Γ(, Γ(, Γ(, Γ(, W( e W + e ( We + e Γ(, Γ(, Γ(, Γ(, { W ( e W + e + ( e W + e } We W( e W + e + ( e W + e W e W + e We + e Γ(, Γ(, Γ(, Γ(, W e W + e + e W + e { } ( ( { ( ( } Γ(, Γ(, Γ(, Γ(, Γ(, Γ(, Γ(, Γ(, Γ(, W { ( ( } ( Γ(, Γ(, Γ(, e e e Γ(, Γ(, Γ(, Γ(, W e W + e + e W + e which is the equation (8 in the paper Now let us consider the conditional purchase incidence γ+γ +γ(, γ γ +γ +γ(, e W+ e WW (A5 Pr( y y Pr( y,y /Pr( y e WW What is the difference between the arginal and the conditional? It can be easily shown that Pr y y Pr y A e γ (A6 ( ( where (, γ +γ(, e W γ +γ(, γ γ γ +γ +γ(, A * > e W e W e W e WW So if γ (, >, the conditional probability is larger than the arginal 7

8 B Estiation Steps The general idea behind our estiation procedure is identical to those by BLP(995 and Nevo The ain difference is that due to the ulticategory nature of our odel we use a category-by-category contraction apping procedure to invert the share equations in order to copute the unobserved ter, ξ st,k Our estiation proceeds in the following steps Step Divide the paraeters into two sets We will refer to one set as the set of linear paraeters and the other as the set of nonlinear paraeters The linear paraeters are { α k, β,, β, λ s,, λ t, } for all c and all j in each category and the nonlinear paraeters are { θ, θ k, ρ, µ, η, γ (, } for all and for all k within each category The rationale for these labels linear and nonlinear is as follows In the absence of the nonlinear paraeters (ie, when there is no heterogeneity, the linear paraeters can be estiated by linear regression ethods after coputing the log-odds ratios of the shares In particular, when there is neither heterogeneity nor copleentarity in the odel ( γ (,, the predicted share for brand j in store s and week t in category is given fro equation (A7 as: (B S exp( α +β p +β d +λ D +λ D +ξ j, st,j st,j s, s t, t st,j st,j + k α k +β, st,k +β st,k +λ s, s +λ t, t +ξ st,k exp( p d D D And the no-purchase share for category is given by (B S + exp( α +β p +β d +λ D +λ D +ξ st, k k, st,k st,k s, s t, t st,k Taking the log of the ratio of the above share expressions we obtain S st,j (B3 log α j +β,pst,j +β dst,j +λ s,ds +λ t,dt +ξ st,j S st, For a single category, estiation is then accoplished by estiating the paraeters of the above equation via OLS or IVR depending on whether or not one expects p st,j to be correlated with ξ st,j In the ulti-category cases, we need to stack up log(s st,j / S st, for all the categories as the dependent variable in the regression The regressor atrix then coprises all the right hand side variables in the above equation stacked up in blocks so a unique set of paraeters can be estiated for each category Once again OLS or IVR can be used to estiate the odel paraeters 8

9 However, the nonlinear paraeters cannot be estiated via such a transforation We decopose the utility into the linear part and nonlinear part according to its associated paraeters Specifically, (B4 α hk +β,hpst,k +β dst,k +λ s,ds +λ t,dt +ξst,k ( α k +β,pst,k +β dst,k +λ s,ds +λ t,d t +ξ st,k + [ θν k +θkν +ρν+ ( µ ω +ηωp st,k] δ + [ θ ν +θ ν +ρ ν+ ( µ ω +η ωp ] st,k k k st,k where δ st,k α k +β,pst,k +β dst,k +λ s,ds +λ t,dt +ξ st,k In equation (B4, δ st,k does not contain household specific ters As we can see, all the unknown paraeters enter these expressions linearly Step As the expression for the unconditional choice probability in equation (5 in the paper has no closed for for the integral, we evaluate it via onte Carlo siulation In particular, we ake R draws fro the distribution of v { ν k, ν, ν, ω, ω} in order to copute the integral in (5 In this step, we also ake initial guesses for the nonlinear paraeters { θ, θ k, ρ, µ, η, γ (, } Step 3 Nuerically copute δ ( δ st,k for all s,, k and t in (B4 that equates observed brand shares to predicted brand shares (S st,k for the given values of { θ, θ k, ρ, µ, η, γ (, for all and k} Due to the existence of copletarity paraeters, γ (,, we cannot use the contraction apping procedure developed by BLP (995 in our case Instead, we copute category specific δ ( δ st,k for all s, k and t conditional on δ - Specifically, it consists of the following sequentially iterative substeps Substep 3 ake an initial guess on δ{δ,, δ,, δ } and set δ OLD δ Substep 3 Copute δ δ,, δ using BLP (995 procedure Then update δ Substep 3 Copute δ δ,, δ -, δ +,, δ and update δ 9

10 Substep 3 Copute δ δ,, δ - and update δ Substep 3+ Check if δ OLD updated δ If yes, go to step 4 Otherwise, set δ OLD δ and go to substep 3 Step 4 Recall fro (B4 that δ st,k α k +β,pst,k +β dst,k +λ s,ds +λ t,dt +ξ st,k Regress δ on brand duy variables, price, and prootional variables, and store and tie duies and to obtain estiates for the linear paraeters{ αk, β,, β, λs,, λ t,} However, given possible correlation between prices and ξ st,k we use the instruental variables ethod instead of ordinary least squares This regression is very siilar to that in Step for the no heterogeneity case with the difference being how the dependent variable is constructed The estiates of linear paraeters obtained here are conditional on the values of the chosen nonlinear paraeters Further, the residuals fro the regressions are also conditional on the values of the nonlinear paraeters Step 5 Interact the residuals coputed above with the instruents and copute the G objective function value Search the space of nonlinear paraeters to iniize the G objective function That is, Θ ˆ argin(z' ξθ ( 'A(Z' ξθ ( G where A is the weight atrix given by A (Z'Z Θ We perfor sensitivity analysis on the nuber of draws R