Web Appendix B Estimation. We base our sampling procedure on the method of data augmentation (e.g., Tanner and Wong,

Transcription

1 Web Appendx B Estmaton Lkelhood and Data Augmentaton We base our samplng procedure on the method of data augmentaton (eg anner and Wong 987) here e treat the unobserved ndvdual choces as parameters Specfcally e use the approach proposed by Musalem et al (008) for the case of ndependent samples (e each sumer only appears once n the data) and extend t to to-stage model In our data each sumer faces a ck-through decson after hs ntal search at tme t usng keyord Condtonal on ck-through the sumer also faces a verson decson We next defne N as the observed number of cks for keyord at tme t and as the observed number mp of versons for keyord at tme t hle represents the number of mpressons (searches) Note that the assgnment of ndces to sumers s arbtrary n ths case and thout mp loss of generalty e assgn the frst N ndces out of all N ndces to the sumes ho ck on keyord at tme t Out of these ndces e assgn the frst ndces to the N sumers ho vert after ckng on keyord at tme t (for more detals see Musalem et al 008) he data augmentaton method allos us to treat the unobserved ndvdual choces N N N z and z as parameters to be smulated from ther posteror dstrbutons he augmented lkelhood of our ntegrated ck-through and verson model s gven by the follong equaton: Note that these ndces reman fxed at all teratons of the Gbbs sampler hs allevates cerns th regards to label stchng (Musalem et al 008)

2 () W N zt z t Pu Pu t t L P u P u I mp W N z t P u t N W N z z t Z Z t S N here mp N N () S Z Z : z N z N Sampler Basc Model Recall that e observe data on W keyords for tme perods e days he dmenson of the vector x s l and of x s l In each tme perod t e observe ns searches for keyord ) Generate z z x x * We propose a ne vector tanng ns ndvdual-level coeffcents for keyord and tme t based on the current mean and varance of the sumer-level heterogenety dstrbuton (under the assumpton that the covarance structure holds true for the hole populaton) hs allos us to augment the parameter vectors of the nonckers accordng to the covarance structure and addresses selecton as ell as eometrcal problems due to dfferent sample sze We accept the proposal accordng

3 to the lkelhood as defned by () (3 rd Method of Chb and Greenberg 995) Note that non-ckers do not enter the verson lkelhood ) Generate b b 0 b ~ MVNb NS b NS NS b NS NS here 0 I b 0 and l l l l NS ns 3) Generate b NS b ~ IW NS S here l l b b' and S I l l W t 4) Generate z z x x We use a random-alk Metropols-Hastngs (MH) step to propose a ne vector * for each keyord and tme t and accept t accordng to the lkelhood as defned by () combned th the standard normal pror of as defned n equaton (3) 5) Generate and W ~ IG c t here c and W ' d d 3

4 Model Steps )-3) as defned above are unchanged n the model Recall that e defned a vector of nstruments of sze l for poston We need to adapt the dra of as ell as the x dra of based on the model defned n equaton (7) 4a) Generate z x x We use a random-alk Metropols-Hastngs (MH) step to propose a ne vector for each keyord and tme t and accept t accordng to the lkelhood as defned by () combned th the bvarate normal pror of and the nstrument resdual gven by L ~ MVN0 here pos x and 4b) Generate pos x pos x ~ MVN here 0 x x ' t x 0 0 pos 0 t 0 and I( l ) 5) Generate pos x 4

5 pos x W ~ IW W S t ' 0 0 pos x pos x here and S I L Model Steps )-3) as defned above are unchanged n the L model Recall that e defned C latent categorcal varables to correct for poston endogenety At tme t e represented the ass membershp of keyord by the vector tanng C zeros and one For example n the case of () C 3 the vector 0 ould ndcate that keyord at tme s a member of ass We need to adapt the dra of as ell as the dra of based on the L model defned n equaton (9) 4a) Generate z x c We use a random-alk Metropols-Hastngs (MH) step to propose a ne vector for each keyord and tme t and accept t accordng to the lkelhood as defned by () L combned th the bvarate normal pror of and the nstrument resdual as gven by L ~ MVN0 L here L pos and L L 4b) Generate pos 5

6 pos ~ MVN here 0 L ' t 0 0 pos 0 t 0 and I( C) 5) Generate pos c W ' pos ~ W 0 0 c IW S t pos pos x here and S I 6) Generate pos c We dra a as a categorcal varable th a posteror probablty gven by L C L c ( c) pc Pr c here denotes that keyord at tme t s ( ) p assgned to ass c L s the lkelhood functon of the L model evaluated at and pc s the pror probablty of ass c membershp 7) Generate p Dra ne ass probablty p based on p ~ Drchlet( K K ) here the C vector denotes the sum of all e K ( c k) K c c W t 6