36.2 Product Adoption and Diffusion
36.2.1 Background
Every new thing either diffuses through population or fails
Researchers are interested in the shape and processes of diffusion
Bass is the first to model in marketing
Diffusion in different fields:
Demography
Archaeology
Geography
Epidemiology
Sociology
Linguistics
Physics
Cosmology
Models of Diffusion
Negative Exponential
Bass
FDA
Network
Levels of analysis:
Class:
Category
Technology
Brand
Classic model
does not account fro marketing mix
requires peak sales for stable estimates (if you have the peaks, you don’t need the model)
no repurscrhsases
no multiple generation
does not fit viral patterns
36.2.1.1 (Chandrasekaran and Tellis 2007) A review of new products diffusion
Products = idea, person, good, or service
New product \(\neq\) innovation
| In econ | In marketing | |
|---|---|---|
| Diffusion | “the spread of an innovation across social groups over time (p. 39) | “the communication of an innovation through the population” |
| Phenomenon (spread of a product) \(\neq\) drivers (communication) | Phenomenon (spread of a product) = driver (communication) |
This paper focuses on the econ definition
Product’s life cycle stages:
- Commercialization: when the product was first sold
- Takeoff: dramatic and sustained increase in sales
- Introduction: between commercialization and takeoff
- Slowdown: decreasing in sales
- Growth: between takeoff and slowdown
- Maturity: Slowdown until decline.
Generalizations:
Shape of the Diffusion Curve: cumulative sales over time is S-shaped curve.
Parameters of the Bass model:
Coefficient of innovation or external influence (\(p\))
mean between 0.0007 and 0.03
mean for developed countries is 0.001 and developing countries is 0.0003
Coefficient of imitation or internal influence (\(q\))
mean between 0.38 and 0.53
industrial/medical innovation > consumer durables
0.51 for developed countries and 0.56 for developing countries
the market potential (\(\alpha\) or \(m\))
- 0.52 for developed countries and 0.17 for developing countries.
Cautions regarding the parameters:
Time to peak sales: 19 years for developing and 16 for developed countries.
Biases in parameter estimation: static models (e.g., Bass) lead to downward biases in market potential and innovation while upward bias in imitation.
Drivers: WOM, communication, economics, marketing mix variables (e.g., prices, consumer heterogeneity, consumer learning), purchasing power parity adjusted per capita income, international trade.
Turning points of the diffusion curve
Takeoff
Time to takeoff: 6-10 years (varies by countries,products, time).
Drivers: price decrease
Slowdown
Sales decline by 15-32%
Drivers: price decline, market penetration, wealth (GNP), and info cascades (fast takeoff = fast decline)
Findings across stages
Duration:
introduction: 6-10 years
growth: 8-10 years
early maturity: 5 years
duration of growth:
time saving products > non-time saving products
leisure enhancing products < non-leisure enhancing products
introduction and early maturity duration get shorter over time (but not growth)
Price: price reduction is getting larger as time progresses (for both introduction nd growth).
Growth rates:
Introduction: 31%
Takeoff: 428%
Growth: 45%
Slowdown: -15%
Early maturity: -25%
Late maturity: 3.7%
Future Research:
Measurement: When to start or stop, or takeoff, differentiation between first purchases and repurchases, demand is better than supply measure,
Theories: no reconciliation yet
Models: comprehensive (from commercialization to takeoff, growth, and slowdown)
Findings: More fine-tune subgroups, include failed diffusion, and consider other countries.
Specification
The probability that an individual will purchase at time \(T\) is a function of the number of previous buyers.
\[ P(t) = \frac{f(t)}{1 - F(t)} = p + \frac{q}{m} Y(t) \]
where
\(P(t)\) = hazard rate
\(Y(t)\) = cumulative number of adopters at \(t\)
\(p\) = probability of an initial purchase at time 0 (when \(Y(0) = 0\)) (also known as innovators importance).
\(\frac{q}{m} Y(t)\) = pressure of prior adopters on imitators
\(m\) = number of initial purchases before any replacement purchases (i.e., market size)
\(F(t)\) = cumulative fraction of adopters at time \(t\)
\(f(t)\) = likelihood of purchase at time \(t\)
Rearrange the formula to get the likelihood of purchase at time \(t\)
\[ f(t) = (p + q F(t) ) [1 - F(t)] \]
The number of adoptions at time \(t\) is
\[ S(t) = mf(t) = pm + (q - p) Y(t) - \frac{q}{m} Y^2(t) \]
then Bass solves the differential equation:
\[ dt = \frac{dF}{p + (q - p) F - qF^2} \]
to obtain cumulative adoption at time \(t\)
\[ F(t) = \frac{1 - e^{-( p + q)t}}{q + (q/p) e^{-( p + q)t}} \]
Hence, the cumulative number of adopters is
\[ Y(t) = m \frac{1 - e^{-( p + q)t}}{q + (q/p) e^{-( p + q)t}} \]
Rewriting the number of adoptions at time \(t\)
\[ S_t = a + bY_{t-1} + c Y^2_{t-1}, t = 2, 3, \dots \]
where
\(S_t\) = sales at time \(t\)
\(Y_{t-1}\) = cumulative sales through period \(t-1\)
\(a = p \times m\)
\(b = q - p\)
\(c = - q /m\)
Equivalently,
\[ p = a/m \\ q = -cm \\ m = (-b \pm (b^2 - 4 ac)^{1/2})/2c \]
Strengths
Good fit to the S-shaped curve (thank to the quadratic term)
Appealing interpretations:
\(p\) = coefficient of innovation (i.e., spontaneous rate of adoption in the population) or external influence (e.g., mass -media communications)
\(q\) = coefficient of imitation (i.e., effect of prior cumulative adopters on adoption) or internal influence (e.g., interpersonal communication influence from prior adopters).
Good application: time (\(t\)) or magnitude (\(S(t)\)) of peak sales.
\[ t^* = \frac{1}{p + q} \times \ln (\frac{q}{p}) \\ S(t)^* = m \times \frac{(p + q)^2}{4q} \]
Incorporated prior literature
If \(p =0\), the Bass model is a logistic diffusion function (driven only be imitation adoption)
If \(q = 0\), the Bass model is an exponential function (driven only innovation adoption)
Limitations
Bass requires 2 most important events that we want to predict in the first place: takeoff and slowdown to have stable estimates.
Unstable estimates after incorporating new observations.
Do not directly account for marketing mix variables (price, promotion), but indirectly capture by \(m, p\)
Assumes product definition is static (no growth or changes in product as time progresses)
Using OLS which can cause
Multicollinearity between \(Y_{t-1}, Y^2_{t-1}\) (making the estimates unstable)
Do not estimate the SE for \(p, q, m\)
Time interval bias (model uses discrete time series data to estimate a continuous model)
Hard to determine starting and ending points of the the sales time.
Supposedly, we need to use first adoptions of new product as sales (\(S_t\)), but data could not capture this, only both first purchases and repurchases
Sales should start from the first year of commercialization, but usually we only have reports when products are selling well already
No clear stopping rule for the time interval.
Improvements
Incorporating marketing mix
Price: affect market potential (\(m\)) and probability of adoption (\(P(t)\)) and heterogeneous across products
Advertising
Distribution: 2 adoption processes: retailer and consumer, where number of retailers who affect determine the market potential \(m\) for consumers
(Bass, Krishnan, and Jain 1994) incorporate both price and promotion to the Generalized Bass model
\[ \frac{f(t)}{1 - F(t)} = (p + q F(t) )x(t) \]
where \(x(t)\) is the current marketing effort (sum of advertising and price) on the conditional probability of product adoption at time \(t\) such that
\[ x(t) = 1 + \beta_1 \frac{\Delta P(t)}{P(t-1)} + \beta_2 \frac{\Delta A(t)}{A(t-1)} \]
where
\(\Delta P(t) = P(t) - P(t-1)\) rate of changes in price
\(\Delta A(t) = A(t) - A(t-1)\) rate of changes in advertising
When prices and advertising remain constant, GB model reduces to Bass model. But it seems like they only stop at 2 variables (not all marketing mix variables or macro and micro econ variables - income changes).
Incorporate supply restrictions
- Include another stage between potential adopter to adopters which is waiting applicants.
\[ \frac{d A(t)}{dt} = [p + \frac{q_1}{m}A(t) + \frac{q_2}{m} N(t)][ m - A(t) - N(t) ] - c(t) A(t) \\ = \text{[Waiting population + Adopters] - conversion rate of applicants to adopters}\\ \]
and
\(\frac{d N(t)}{dt} = c(t) A(t)\)
where
\(d(A)/dt\) is the rate of changes of waiting applicants
\(c(t)\) is the supply coefficient
the second equation is the impact of supply restrictions on adoption rate
The growth of new applicants is
\[ \frac{d Z(t)}{dt} = \frac{d A(t)}{dt} + \frac{dN(t)}{dt} \\ = (p + \frac{q_1}{m} A(t) + \frac{q_2}{m} N(t) ) (m - A(t) - N(t)) \]
To incorporate waiting applicants abandoning their adoption decision after some time see (Ho, Savin, and Terwiesch 2002)
Incorporate competitive effects
Instead of using product category as the unit of analysis, we can model at the brand level (different brand might have different rate of diffusion).
A new brand can
increase the entire market potential (\(m\)) (by increased promotion and product variety)
compete in the existing market potential (interfere the diffusion process of other brands)
Diffusion depends on the order of entry and competition.
Incorporate complementary effects
- In market that has indirect network externalities, co-diffusion exists and asymmetric
Incorporate technological generations for successive generations of the same product (i.e., substitution effects).
\[ S_1(t) = m_1F_1(t) - m_1 F_1(t) F_2(t - r_2) \]
where \(r_2\) is the introduction time of the next-generation product.
\[ S_2(t) = F_2(t- r_2) [m_2 + F_1(t) m_1] \]
where
\(S_i(t)\) = sales of generation \(i\)
\(F_i(t)\) = fraction of adoption for each generation
\(m_i\) = market potential for each generation
Leapfrogging behavior is possible (i.e., skip a generation to buy the next one) (Mahajan and Muller 1996)
Incorporate time-varying parameters
Model market potential (\(m\)) as a function of time-varying exogenous and endogenous variables (Mahajan and Peterson 1978)
Model coefficient of imitation to be time-varying (Easingwood, Mahajan, and Muller 1983)
\[ \frac{d F(t)}{dt} = [ p + q F(t)^\delta][ 1 - F(t)] \]
where \(\delta\) is the nonuniform influence
when \(\delta = 1\), the model becomes the Bass model
When \(\delta \in [0,1]\), means high initial coefficient of imitation,
When \(\delta >1\), means delay in influence -> lower and later peak.
Different adopters could influence later adopters differently (people who adopted more recently are more vocal) (Sharma and Bhargava 1994)
- Incorporate replacement and mufti-unit purchases
(Balasubramanian and Kamakura 1989)
\[ y(t) = [a + bX(t)][\alpha \text{Population}(t) P^\beta (t) - X(t)] + r(t) + e(t) \]
where
\(y(t)\) = sales
\(P(t)\) = price index
\(X(t)\) = total units in use at the beginning of year \(t\) with dead units are replaced already
\(r(t)\) = number o units that have died or need replacement at year \(t\)
\(a\) = coefficient of innovation
\(b\) = coefficient imitation
\(\beta\) = price change effect on ultimate penetration
\(\alpha\) = ultimate penetration (price is at its original level)
(Steffens 2003) models multiple units purchase by a single household.
Incorporate trail-repeat purchases
Incorporate variations across countries
Evaluation:
- All of the improvements still rest on the assumption of one driving mechanism: knowledge dispersion through WOM.
Improvements in estimation
MLE: avoid time-interval bias, but underestimates the SE (Schmittlein and Mahajan 1982)
Non linear least squares: (V. Srinivasan and Mason 1986) need lots of obs
Estimates are more flexible
No time-interval bias
valid SE
Hierarchical Bayesian method
Incorporate parameter updating
Problem with definition of similar products (fixed by (Bayus 1993) with product segmentation scheme)
Adaptive techniques: stochastic techniques (parameter vary over time) ((J. Xie et al. 1997)augmented Kalman filter)
Genetic algorithms:
can find global optimum
better estimate (less bias).
Alternative models of diffusion
Alternative drivers:
Affordability: (Peter N. Golder and Tellis 1998) model as Cobb-Douglas model:
\(S = P^{\beta_1} \times I^{\beta-2} \times CS^{\beta_3} \times MP^{\beta_4} \times e^\epsilon\)
Sales = product (price, income, consumer sentiment, market presence)
(Horsky 1990) incoproates both price and income and WOM on sales growth.
Heterogeneity: aggregate level diffusion models: (J. H. Roberts and Urban 1988), (Oren and Schwartz 1988), (Chatterjee and Eliashberg 1990), (Bemmaor 1984) (Song and Chintagunta 2003b), (Sinha and Chandrashekaran 1992) (Karshenas and Stoneman 1993)
Strategy: model supply side: (market entry, marketing mix, location) (Dekimpe, Parker, and Sarvary 2000), (Bulte and Lilien 2001),(Bart J. Bronnenberg and Mela 2004)
Alternative phenomena:
Spatial diffusion (Mahajan and Peterson 1979), (Redmond 2003), (Garber et al. 2004)
Contagious diffusion (infectious diseases)
Expansion diffusion (one source like wildfire)
Hierarchical diffusion (ordered series of classes)
Relocation diffusion:
Diffusion of entertainment products: follow exponential decay (Eliashberg and Sawhney 1994), (Eliashberg et al. 2000), (Elberse and Eliashberg 2003), (Moe and Fader 2002), (J. Lee, Boatwright, and Kamakura 2003)
Modeling the turning points in diffusion
Takeoff: follow (Peter N. Golder and Tellis 1997) definition: “point of transition from the introduction stage to the growth stage”
Measurement
(Peter N. Golder and Tellis 1997): threshold takeoff (compare to other in the categories)
Logistic curve rule: first turning point of the logistic curve (max of the 2nd derivative) (hindsight only)
Maximum growth rule: largest sales increases within 3 years (not size invariant)
(Agarwal and Bayus 2002) measure based on annual percentage change in sales
(Stremersch and Tellis 2004) adapted the threshold method for international markets
(Garber et al. 2004) rule of thumb: 10-20 market penetration
Drivers
(Peter N. Golder and Tellis 1997) price declines lead to takeoff
(Agarwal and Bayus 2002) increase in firm entry lead to better product quality, marketing infrastructures
(Tellis, Stremersch, and Yin 2003) venturesome culture lead to takeoff
Model: either proportional hazards (Peter N. Golder and Tellis 1997) or log-logistic hazard (Tellis, Stremersch, and Yin 2003)
Evaluation: Only model successful innovation so far.
Slowdown: point of transition from the growth stage to the maturity stage (Peter N. Golder and Tellis 1997)
Measurement: (Peter N. Golder and Tellis 2004) “operationalize as the first year of two consecutive years after takeoff in which sales are lower than the highest previous sales.” (p.72)
Explanation:
Dual-market phenomenon: early adopters vs. early majority (Goldenberg, Libai, and Muller 2001)
Informational cascades: negative cascades (Peter N. Golder and Tellis 2004)
Affordability(Peter N. Golder and Tellis 2004)
Modeling:
Cellular automata models: (Goldenberg, Libai, and Muller 2001)
Hazard models: (Peter N. Golder and Tellis 2004)
Evaluation: still new can have more research
36.2.1.2 (Bass 1969)
Assumption:
The timing of a consumer’s initial purchase is correlated with the number of previous
This paper looks at new class of products (not new brands or new models of older products)
Focus on infrequently purchased products
Theory of Adoption and Diffusion
Innovators: adopt independently (regardless of others’ opinions): pressure to adopt does not increase with the growth of the adoption.
Imitators (include early adopters, early majority, late majority): adoption depends on the timing of adoption (i.e., influenced by the decisions of others to adopt.
Laggards
“The probability that an initial purchase will be made at \(T\) given that no purchase has yet been made is a linear function of the number of previous buyers” (p. 216)
\[ P(T) = p + \frac{q}{m} Y(T) \]
where \(p\) and \(q/m\) are constants
\(Y(T)\) is the number of previous buyers.
When \(Y(T) = 0\), \(p\) represents the probability of an initial purchase at \(T = 0\)
\((q/m) Y(T)\) is the pressures on imitators to adopt.
Model Assumptions:
36.2.2 Discussion
36.2.2.1 (Sood, James, and Tellis 2009)
Functional regression
Contributions:
Theoretically sound (integrate info across categorizes)
Augmented Functional regression outperforms existing models
Product-specific effects are more helpful in predicting penetration than country-specific effects.
They use yearly cumulative penetration of each category as the unit of analysis (i.e., curve/ function).
3 functional data analysis techniques:
Functional principal components
functional regression
functional cluster analysis
To treat discrete intervals: use smoothing spline to generate continuous smooth curves
Even though the spline approach requires a lot of data to smooth, other appearances to create smoothness are still available. Hence, you can still use function regression and or cluster with 2 or 3 time points.
Advantage s of functional regression:
incorporate info from other products
nonparametric fitting procedure
uses the functional nature of the penetration curves.
Predictions on: number of years to take off, peak marginal penetration and the level of peak marginal penetration
Good: tell a story from simple to more sophisticated model to justify their improvements in the paper.
2 dimensions that are not captured by simple extrapolation models:
info from prior history of the new product
intrinsic info across products and countries.
Classic Bass model ignores:
other categories (fixed by meta-bass and augmented meta-bass)
uses parametric methods.
Questions:
Technically could redo the analysis with new dataset (including 2009 till now) to see the out of sample performance.
No hypothesis, just model and probable explanation
Use only curves under the same category to predict the new product (not all categories).
36.2.2.2 (Appel, Libai, and Muller 2019)
Growth, Popularity and the Long Tail: Evidence from Digital Markets
part of MSI’s working paper series and MSI insights
Context: digitized markets (long-tail markets)
Most popular products do have S-shaped curve, but lower-popularity products exponential-like decline (“slide”) or a combination of slide and bell (S&B) are more common.
Shortcomings of previous research:
- Pro-innovation bias: success correlates with importance in the new product development research
Data: SourceForge (exclude inactive and less than 200 downloads): 5 years with high Gini coefficient - 0.96 (i.e., high concentration).
Dominant patterns:
A bell-shaped pattern: bell (popular products)
- Caveat in the movie market: popular products decline over time.
An exponential-like decline beginning at launch: slide
Combination of the first 2: S&B
Proposed model: inception model (inception effect = heightened external growth).
Long-tail market:
Supply side: low cost of inventory, stocking, efficient delivery, and low cost of new products development.
Demand-side: easy to search, recommendation system, social networks and online communities.
Popularity = extent of demand = number of downloads.
The shape of new product growth: previous literature says S-shaped
Non-S-shaped markets:
r-shaped cumulative curve: because of
Large budget for promotion: movies
Pre-launch buzz: on social media
The role of popularity on the shape of growth: was ignored in the literature
Free and Open-Source Software (FOSS)
Data Analysis:
Stage 1: To facilitate comparison, scale pattern to a (0,1) by dividing each observation by the total sum of downloads, and smooth the graph using Hodrick-Prescott filter
Stage 2: Use peaks-and-troughs algorithm for the classification
Descriptive: the S-shaped curve is representative for more popular products, while for those that are not as popular, we have a blend of S&B and slide as well.
Try to observe the same pattern with smartphone app download (data provided by Mobility - an anonymous app providers for businesses)
Drivers of Multi-pattern Growth
Analogy to movies (characterized by an exponential decline): not similar because
different product types (utilitarian vs. entertainment)
Different pattern exhibited by popular and unpopular: while in movies the exponential decline is from blockbuster, and sleepers has a bell shape, under this dataset, less popular product has the exponential decline, while the popular products are bell-shaped.
Analogy to supermarkets: not good because FDP is affected by social influence, supermarkets are usually under large investments and not much social influence.
The inception alternative: 2 influences of new product growth
Internal: from previous adopters
External (not from previous adopters): marketing mix, social media posts, recommendation, expert opinions, influencers. Expected to stronger early on and decay. (i.e., inception effect - external influence as a function of time with an initial external influence parameter \(p(t) = pe^{\delta t}\))
The relationship between inception and popularity: The higher the product’s popularity, the lower the share of adoptions due to the inception effects (i.e., products with high initial investment that failed to reach critical is less popular).
Inception is typically a necessary but not sufficient condition to reach popularity.
36.2.2.3 (Tellis et al. 2020)
No awards (nominated only)
Emotion is more effective than information
brand hurts, but branding is used a lot
surprise and humor are good, but videos don’t use
Limitation: Because these emotions are rare, maybe that why they are effective. But if everyone starts using these tactics, maybe that they wont’ work anymore.
36.2.2.4 (Chandrasekaran, Tellis, and James 2020)
Was rejected 5 times.
Leapfrogging, Cannibalization, and survival during disruptive technological change
2 types of dilemma when it comes to new technology:
Incumbent: invest in new technology or old or both
Entrant: target niche or mass.
Solution: relation between new technology and old one (i.e., high rate of disengagement - cannibalization or low rate of disengagement- coexistence)
Data:
Successive technology penetration across multiple countries and years
Sales of contemporaneous pair across multiples countries
Case analyses
“Disruption occurs if the incumbent focuses on the old technology to the exclusion of the new one” (p. 4)
Definitions:
Successive/New technology: not new version/generations of the same product
Cannibalization: “the extent to which the successive technology”eats” into real or potential sales (or penetration) of the old technology due to substitution.”(p. 5)
Rate of disengagement \(F_{12}\): (account for partial substitution)
Adopter segments for a new successive technology:
Leapfroggers: adopt new, but would never have adopted the old
Switchers: Adopted old, but switch to new once it’s introduced
Opportunists: wait for the old, but end up with the new one.
Dual users: both technologies
Models: based on (J. A. Norton and Bass 1987)
\[ S_1 (t) = m_1 F_1(t) (1- F_{12}(t- \tau_2 + 1)) \\ S_2 (t) = F_2(T- \tau_2 + 1) (m_2 + m_1 F_1(t)) \]
where
- \(S_i(t)\) = penetration of technology \(i\) in period \(t\)
- \(m_1\) = long-run penetration for technology 1
- \(m_1 + m_2\) = long-run penetration for technology 2
The fraction of all potential technology_g consumers for each technology (g = technology 1 or 2)
\[ F_g(t) = \frac{p_g(1 - e^{-(p_g + q_g)^t})}{p_g + q_g e^{-(p_g + q_g)t}} \]
where
\(t \ge 0\)
\(g = 1, 2\)
\(p\) = innovation coefficient
\(q\) = imitation coefficient
\(p_{12}, q_{12}\) = disengagement coefficients
\(F_1, F_2, F_{12}\) = adoption rate of technology 1, technology 2, and disengagement rate at which technology 1 customers abandon to get technology 2
Model contributions:
Model the adoption rate of technology 2 different from disengagement rate of technology 1 (\(F_2 \neq F_{12}\))
Varying \(p, q\) (for different technologies)
\(F_1\) has the same function form as \(F_1, F_2\) (because it fits the data well, and reduces to previous model which matches previous literature)
Model can be applied to both generational and technology diffusion
Model Estimation
Using nonlinear least squares to estimate the parameters that that minimize
\[ \sum_{i = 1}^n (s_{i1} - m_1 F_1(t_i)) (1 - F_{12} (t_i - \tau_2 + 1))^2 \\ + \sum_{i=1}^n (s_{i2} - F_2 (t_i - \tau_2 + 1)(m_2 + m_1F_1(t_i)))^2 \]
Segments of adopters
\[ S_2(t) = L_2(t) + DU_2(t) + SW_2(t) + O_2(t) \]
while
\[ S_1(t) = L_1(t) - CAN_2(t) = L_1(t) - (SW_2(t) + O_2(t)) \]
where
\(SW\) = switchers
\(O\) = Opportunists
\(CAN\) = Canalization
\(L\) = Leapfroggers
\(DU\) = dual -users
Market growth segment = sum(leapfroggers, dual users)
Cannibalization = sum(switchers, opportunists).
36.2.2.5 (Prins and Verhoef 2007) Marketing effects on adoption timing
Studies the effects of direct marketing and mass marketing on adoption timing (in the context of a new e-service among existing customers)
Data: 6k customers of a Dutch telecom operator over 25 months
Findings:
advertising shortens the time to adoption (including those by competitors)
Mass marketing has a greater effect on loyal customers (compared direct marketing)
Related literature:
Adoption
customer management
Adoption timing is defined as “the time between the introduction and the adoption of the new service” (p. 170) following (Jan-Benedict E. M. Steenkamp and Gielens 2003)
Switchers to competitive services are considered as non-adopters (even if they adopt comeptitor’s new service). It’s valid when the focus is on the adoption of the folca company’s new service among existing customers.
(Donkers, Franses, and Verhoef 2003) demonstrates that if oversampling is not accompanied by stratfied sampling on the independent variables, it should not affect the parameter estimates or SE for are event in binnary choice models.
Meausres of Time to adoption: For each tiem period \(t\), a customer can either adopt the new serive or not. The time to adoption for each customer is the time elasped in \(t\) since the intro of the service. Dependent vairable = indivudal time to adoption.