From Argentina to Zimbabwe: Department of Statistics Texas A&M University 15 Feb 2010
Acknowledgments This is joint work with my coauthors Bani Mallick (Texas A&M University) Debu Talukdar (SUNY - Buffalo) K. Sudhir (Yale University). A special thanks to Craig Meisner (World Bank) and Azucena Pernia (United Nations World Tourism Organization). My research was funded by a grant from the National Science Foundation.
Outline Problem Description Model
Motivation Global marketing managers are deeply concerned with how well their products are adopted. Understanding this diffusion process allows them to better allocate their finite resources. My work improves the current understanding of that process.
Research Questions I answer the following five questions: 1. How does diffusion speed change throughout the life of a product? 2. Which covariates best describe diffusion speed? 3. How can cross-country dynamics be included beyond the standard word-of-mouth effects? 4. How should the distance between two countries be measured? 5. Can the diffusion predictions be improved?
To answer those questions, I use augmented versions of two models commonly used in the diffusion literature. The first is the logistic diffusion model: y(t) Y (t 1) = λ [ 1 ] Y (t 1) + ɛ t αm(t) y(t) = Number of adoptions at time t Y (t 1) = Cumulative number of adoptions at time t 1 M(t) = Population at time t α = Adoption ceiling λ = Speed parameter ɛ t N(0, σ 2 )
The second is the Bass diffusion model: [ ] Y (t 1) y(t) = [αm(t) Y (t 1)] p + q e ɛt. αm(t) y(t) = Number of adoptions at time t Y (t 1) = Cumulative number of adoptions at time t 1 M(t) = Population at time t α = Adoption ceiling p = Coefficient of innovation q = Coefficient of imitation ɛ t N(0, σ 2 )
Data Description Putsis et al. (1997) point out the need for research on a broader set of new products (than their four), countries (than their 10), and parameter covariates (than their two). A final contribution is my novel data set. Drawn from the IMF, UN, World Bank, and World Tourism Organization, my data set covers seven new product diffusions across 31 countries.
Data Description The seven products in my data set are: VCRs (introduced in 1976) CD players (1984) microwaves (1975) camcorders (1984) fax machines (1979) home computers (1980) cellular phones (1981)
Country %Pop. %GDP Country %Pop. %GDP Argentina 0.6 0.96 Italy 0.91 3.06 Australia 0.31 1.19 Malaysia 0.39 0.47 Austria 0.13 0.49 Mexico 1.6 1.96 Belgium 0.16 0.62 Netherlands 0.25 1.01 Brazil 2.88 2.75 Norway 0.07 0.34 Canada 0.5 1.9 Philippines 1.29 0.83 Chile 0.25 0.32 Portugal 0.16 0.39 China 20.19 15.87 Singapore 0.07 0.22 Denmark 0.08 0.33 South Korea 0.75 1.91 Finland 0.08 0.3 Spain 0.67 2.05 France 0.94 3.5 Sweden 0.14 0.53 Germany 1.28 4.44 Switzerland 0.12 0.52 Greece 0.17 0.46 Thailand 0.99 0.98 Hong Kong 0.11 0.43 United Kingdom 0.93 3.65 India 16.94 6.73 United States 4.59 22.3 Ireland 0.06 0.25 TOTAL 57.62 80.76
Data Description Time-Invariant Covariates Daily newspapers Ease of doing business index GINI index International migration stock Population growth rate Price for residential fixed line Individualism Index Uncertainty Avoidance Index Pump price for gasoline Households with television International tourism International voice traffic Time-Varying Covariates Age dependency ratio Electric power consumption Foreign direct investment GDP per capita Household final consumption Internet users Labor force participation rate, female Consumer price index Telephone mainlines Trade Unemployment Urban population
Outline Problem Description Model Problem Description Model
Research Questions Answered Model The logistic diffusion model (LDM) [ 1 y(t) Y (t 1) = λ ] Y (t 1) + ɛ t αm(t) has a single parameter, λ, to describe the diffusion speed. That single parameter allows me to answer the first two questions: 1. How does diffusion speed change throughout the life of a product? 2. Which covariates best describe diffusion speed?
Model Research Questions Answered 1. How does diffusion speed change throughout the life of a product? The current literature assumes that the λ parameter is constant over time. I allow my λ to vary over time, enabling me to determine how the speed changes.
Model Research Questions Answered 2. Which covariates best describe diffusion speed? I include the covariates through a hierarchical structure. Using Gibbs variable selection (Dellaportas and Forster 1999) I am able to determine which covariates are significant.
Model Augmentation Problem Description Model The y(t) Y (t 1) = λ [ 1 ] Y (t 1) + ɛ t αm(t) was extended by Van den Bulte (2000) to [ y i (t) Y i (t 1) = λ i 1 Y ] i(t 1) + ɛ it. α i M(t) I further extend the model to [ y in (t) Y in (t 1) = λ in(t) 1 Y ] in(t 1) + ɛ int. α in M i (t)
Hierarchical Structure Problem Description Model To borrow strength, I use a hierarchical structure. log (λ in (t)) = f (t) + A nt + B it + ɛ h A nt = k P n γ k X k β k + ɛ nt B it = k P i γ k X k β k + ɛ it ɛ h N(0, σ 2 h ) ɛ nt N(0, σ 2 A ) ɛ it N(0, σ 2 B ).
Model Estimation Steps Most parameters are estimated through N-IG Gibbs steps. Each γ k is a binary variable drawn using Gibbs variable selection. f (t) is estimated using Bayesian Adaptive Regression Splines (Kass et al. 2001).
Estimation Steps Problem Description Model The α in and λ in (t) terms require a M-H step. p(α in ) exp p(λ in (t) ) exp ( [ yin (t) Y in (t 1) λ in(t) 2σ 2 l ( [ yin (t) Y in (t 1) λ in(t) 2σ 2 l 1 Y in(t 1) α in M i (t) 1 Y in(t 1) α in M i (t) [log(λ in(t)) A nt B it f (t)] 2 2σ 2 h ]) 2 ]) 2 log(λ in (t)) }
Time-Varying Component Model f(t) 1.2 0.8 0.4 0.0 5 10 15 20 Lag Year
Variable Selection Model Time-Invariant Covariates Time-Varying Covariates International voice traffic 0.88 Internet users 1.00 Price for fixed line 0.78 Consumer price index 1.00 Daily newspapers 0.67 Trade 0.88 GINI index 0.25 Foreign direct investment 0.86 Ease of doing business 0.14 GDP per capita 0.78 Households with television 0.12 Household consumption 0.78 International tourism 0.12 Telephone mainlines 0.76 Uncertainty Avoidance 0.08 Electric consumption 0.32 Pump price for gasoline 0.05 Urban population 0.12 International migration 0.04 Female Labor 0.10 Individualism 0.01 Unemployment 0.09 Population growth rate 0.00 Age dependency ratio 0.06
Model Selected Covariate Groupings Ability to Purchase Consumer price index GDP per capita Household total consumption Price for fixed line Cross-country Interaction Foreign direct investment Trade Technology Daily newspapers International voice traffic Internet users Telephone mainlines
Outline Problem Description Problem Description Model
Research Questions Answered The Bass diffusion model (BDM) allows me to answer the final three questions: 3. How can cross-country dynamics be included beyond the standard word-of-mouth effects? 4. How should the distance between two countries be measured? 5. Can the diffusion predictions be improved?
Research Questions Answered 3. How can cross-country dynamics be included beyond the standard word-of-mouth effects? The only source of cross-country influence in the existing literature is the product-specific word-of-mouth effect from existing adopters. However, cross-country information flow is unlikely to be so restricted. For instance, observational learning among reference leaders and followers (Ger and Belk 1996). I apply an explicit reference leader-follower hierarchical structure among countries as another source of cross-country influence.
Research Questions Answered 4. How should the distance between two countries be measured? While it seems logical to use population centroid distance, there may be some other metrics better suited to new product diffusion. I develop and test several alternative measures of distance Bilateral flow of people (tourism) Bilateral flow of goods/service (trade) Cultural similarity (Hofstede)
Research Questions Answered 5. Can the diffusion predictions be improved? By incorporating a reference hierarchy and using a better distance measure, I will improve the predictions.
(BDM) To answer those three questions, I use the as my base model. It can be expressed as (Talukdar et al. 2002): y in (t) = [α in M i (t) Y in (t 1)] [ ] Y in (t 1) x p in + q in e ɛ int α in M i (t) I propose several modified versions of the BDM to better capture the influence dynamics.
Proposed : Model 2 Model 2 extends the BDM by adding an explicit reference leader-follower hierarchy among countries as another source of cross-country influence: y in (t) = [α in M i (t) Y in (t 1)] Y in (t 1) x p in + q in α in M i (t) + r in a ij L j e ɛ int j i where a ij = i w ij j i w ij and L j = L j j L j
Proposed : Model 2 w ij is a measure of the strength of the relationship between countries i and j. I investigate four surrogates for that relationship: Inverse Centroid Distance Bilateral Trade Flow Bilateral Tourism Flow Cultural Similarity
Proposed : Model 3 My next proposed model includes product-specific word-of-mouth as the source of external influence, similar to Albuquerque et al. (2007). y in (t) = [α in M i (t) Y in (t 1)] Y in (t 1) x p in + q in α in M i (t) + s Y jn (t 1) in b ij e ɛ int M j (t) where b ij = i j i v ij j i v ij v ij has the same four candidates (trade, tourism, culture, and distance).
Proposed : Model 4 Model 4 combines models 2 and 3, incorporating both the word-of-mouth and reference hierarchy terms. y in (t) = [α in M i (t) Y in (t 1)] Y in (t 1) x p in + q in α in M i (t) + r Y jn (t 1) in a ij L j + s in b ij e ɛ int M j (t) j i j i
Sub My three proposed models ( 2-4) each have variants. Specifically, 2 and 3 each have 4 sub models while Model 4 has 16 sub models based on how the basis of bilateral cross-country interactions (w ij or v ij ) are modeled. Taken together, my three proposed models represent 24 modified versions of the BDM.
Hierarchical Setup Problem Description After being transformed to the real line, the parameters are decomposed. αin = logit (α in ) pin = log (p in ) qin = log (q in ) α in α pin i + α ( n X T β ) = p qin i + pn ( αin + X T β ) π αin qi + qn ( pin X T β ) + π pin π qin qin π αin π pin MVN(0, Σ 1 ) π qin
Hierarchical Setup Problem Description With the country and product effects further decomposed. α ( i X T β ) pi αi ( = X T β ) π αi π αi qi ( pi X T β ) + π pi π pi MVN(0, Σ 2 ) π qi qi π qi α n π αn p n = π pn qn π qn π αn π pn MVN(0, Σ 3 ) π qn
Estimation Steps Again, the majority of the parameters are sampled using N-IG (or MVN-IW) Gibbs steps. α in, p in, q in, r in, and s in all require similar M-H steps.
Model Comparison Problem Description To determine which model is the best, I predicted the sales for one, two, and three years in the future. I calculated the mean squared prediction error for each model and parameter draw.
Model Comparison Problem Description Model 2 Model 3 Distance Measure Improvement over BDM Centroid Distance 40.3% Bilateral Tourism Flow 38.5% Hofstede Cultural Similarity 23.3% Bilateral Trade Flow 5.9% Hofstede Cultural Similarity 57.5% Centroid Distance 17.7% Bilateral Tourism Flow 5.9% Bilateral Trade Flow 4.2%
Model Comparison Problem Description W-o-M Hierarchy Improvement Improvement over BDM over Model 3 Model 2 N/A Centroid 40.3% Model 3 Cultural N/A 57.5% Trade Tourism 69.4% 27.9% Tourism Centroid 66.5% 21.1% Model 4 Cultural Centroid 65.2% 18.1% Cultural Cultural 63.1% 13.2% Cultural Tourism 62.1% 10.8%
Outline Problem Description Model
I have explored the new product diffusion process and answered the following five questions: 1. How does diffusion speed change throughout the life of a product? With the initial promotion, the speed parameter is higher at the beginning of the product s life. With increased globalization, the speed parameter increases as the product ages. 2. Which covariates best describe diffusion speed? The descriptive covariates fall into three categories: ability to purchase, technology, and cross-country interaction.
3. How can cross-country dynamics be included beyond the standard word-of-mouth effects? A reference hierarchy measures how close a country is to the influential countries. 4. How should the distance between two countries be measured? The true distance between countries is better described by the cultural similarity and the amount of trade and tourism flow. 5. Can the diffusion predictions be improved? My model which includes both a reference hierarchy and word-of-mouth effects improves predictive ability by 69% over the BDM and by 28% over model 3.
Future Work All my models presented today have the implicit assumption that the covariates affect each of the countries in the same way (i.e. all β i are the same). I am currently working (with David Dahl) to use a Dirichlet process as the prior distribution of the β i. Additionally, I am working to incorporate the distance between the countries into the prior.
From Argentina to Zimbabwe: Department of Statistics Texas A&M University 15 Feb 2010