Dynamic Models with Serial Correlation: Particle Filter Based Estimation

Transcription

1 Dynamic Models with Serial Correlation: Particle Filter Based Estimation April 6, 04 Guest Instructor: Nathan Yang

2 Class Overview ( of ) Questions we will answer in today s class: Why should we care about serial correlation in marketing research about dynamic behavior? Why should we use particle filtering in light of serial correlation? How do we actually use particle filtering?

3 Class Overview ( of ). Customer Goodwill Model. Serial Correlation Issues. Econometric Specification 4. Sample of Applications 5. Guide to Particle Filtering 6. Monte Carlo Analysis 7. Additional Remarks

4 MGT 756: Particle Filtering in Marketing CUSTOMER GOODWILL MODEL

5 Goodwill Formation ( of 5) Suppose that we are trying to model a customer i s purchase or no-purchase decision in transaction t Denote decision as Y it {0,} Customers form goodwill (Z it ) about the product, so they are more likely to purchase if they ve purchased in the past U it = α + Z it + ε it from purchasing (Y it = ) U it = 0 from no purchase (Y it = 0) What would be the ideal specification to estimate? t Z it = β s Y s Z it = δz it + βy it Z it = δz it + βy it + η it where η it ~N(0,)

6 Goodwill Formation ( of 5) Z it = δz it + βy it + η it Retention of goodwill Accumulation of new goodwill

7 Goodwill Formation ( of 5) Z it = δz it + βy it + η it Retention of goodwill Serial correlation Accumulation of new goodwill Y it+ Z it+ Y it

8 Goodwill Formation (4 of 5) Z it = δz it + βy it + η it Managerial implications regarding β (0,)? δ (0,)?

9 Goodwill Formation (5 of 5) Z it = δz it + βy it + η it Note that: Z it is a function of Z it Therefore a function of a sequence of η i,:t Y it is a non-linear function of Z it Therefore a function of complicated interactions between sequence of Y i,:t and η i,:t

10 MGT 756: Particle Filtering in Marketing SERIAL CORRELATION ISSUES

11 Serial Correlation ( of ) In demand and supply side models, often interested in identifying the link between past and current behavior Demand side Goodwill, learning, loyalty, state dependence, word-ofmouth, etc Supply side Advertising, entry costs, learning-by-doing, organizational forgetting, etc

12 Serial Correlation ( of ) When studying such linkages, we are essentially trying to estimate a model of behavior with lagged dependent variables Problem? If customers or firms exhibit persistent and unobserved heterogeneity, lagged dependent variables will accumulate these estimation errors Certain types of data variation help (Sudhir and Yang, 04) Therefore, our inferences about the impact of these lagged dependent variables will be confounded by serial correlation Lead to misguided managerial implications

13 Serial Correlation ( of ) Particle filtering techniques to deal with serial correlation Doucet, de Freitas, and Gordon (00) Linear and non-linear particle filtering used in economics, finance, and marketing Linear models: Bass, Bruce, Majumdar, and Murthi (007) Bruce, Foutz, and Kolsarici (0) Bruce, Peters, and Naik (0) Fernandez-Villaverde and Rubio-Ramirez (007) Nonlinear models: Blevins (04) Blevins, Khwaja, and Yang (04) Bruce (008) Fang and Kung (00) Gallant, Hong, and Khwaja (0a, b) Nishida and Yang (04) Note: Particle filtering assumes that the parameters in model are identified For non-parametric identification of dynamic models with serial correlation, refer to Hu, Shum, and Tan (00)

14 MGT 756: Particle Filtering in Marketing ECONOMETRIC SPECIFICATION

15 Setting up the Likelihood ( of ) Can use maximum likelihood (ML) estimation Parameters to estimate are θ = (α, β, δ) Likelihood for the sequence of decisions is denoted as P Y :T ; θ Goodwill is not observed, so integrate: P Y :T ; θ = P Y :T :T ; θ P(:T ; θ) d:t

16 Setting up the Likelihood ( of ) P Y :T :T ; θ P(:T ; θ) d:t? Monte Carlo integration Simulate many paths of :T (via many draws of serially correlated η it ), and compute integral of dimension T Computational burdensome Is there an easier way?

17 Setting up the Likelihood ( of ) P Y :T :T ; θ P(:T ; θ) d:t Note that integral above can be re-written as: T P Y t Z t, Y :t ; θ P(Z t Y :t ; θ) dz t t From a T-dimensional integral to a -dimensional integral Need to draw P(Z t Y :t ; θ), which is posterior distribution of unobserved random variable conditional on Y :t Can use sequential Monte Carlo (SMC) methods (a.k.a., particle filtering)

18 MGT 756: Particle Filtering in Marketing SAMPLE OF APPLICATIONS

19 Advertising Themes ( of 6) Bass, Bruce, Majumdar, and Murthi (007) Overview Portfolio of ads with different themes Stimulation, product offer, price offer, reconnect, and reassurance How to allocate advertising expenditures? Look at demand for residential telephone services Minutes of call time Number of calls Serial correlation Wear-in effects from ad exposure Goodwill depreciates because of forgetting But goodwill is often unobservable to researchers

20 Advertising Themes ( of 6) Nerlove-Arrow model dg(t) = qa t δg(t) dt q = effectiveness of ad spending δ = rate of decay of goodwill due to forgetting A t = advertising spending at time t G t = goodwill at time t

21 Advertising Themes ( of 6)

22 Advertising Themes (4 of 6) Dynamics of telephone usage Linear model Why?

23 Advertising Themes (5 of 6) Goodwill evolution Allows for interactions between different ad themes

24 Advertising Themes (6 of 6) Serial correlation Specification allows for noise to enter goodwill accumulation Need to integrate out contribution of w t Interpretation of w t?

25 Machine Replacement ( of 4) Blevins (04) extension of Rust (987) Overview Replace engine or not, based on observed and unobserved states What is observed and what is unobserved? Trade-off Larger sunk cost today, or larger maintenance costs in the future Serial correlation Quality of engine will be affected by Persistent features of the engine (i.e., unobserved quality) Bus itself (i.e., inexperienced or abusive driver) Typical routes taken (i.e., difficult terrain or traffic congestion) May influence distribution of observed mileage Bus serving heavy traffic area (which is unobserved) may accrue mileage at a lower rate than buses serving more rural and long-distance routes Route may be extended, shortened, or changed, thus changing distribution of mileage increments

26 Machine Replacement ( of 4) Utility of replacing versus not replacing Structural parameters are replacement cost (c 0 ), cost of mileage (c x ), and cost associated with unobserved latent state (c ξ )

27 Machine Replacement ( of 4) Distribution for observed change in mileage (Δx t ) Follows exponential distribution

28 Machine Replacement (4 of 4) Latent quality state follows AR() process for when machine not replaced When instead machine is replaced, latent state drawn anew from stationary distribution

29 Fast Food Expansion ( of ) Blevins, Khwaja, and Yang (04) Overview Fast food chains decide how much to expand or contract across geographic markets Their decision depends on competitive landscape, market characteristics, and unobserved profitability Serial correlation Market-time specific managerial shocks that persist over time and affect unobserved profitability Unobserved profitability may also be affected by existing stock of outlets Cost-based explanation» Learning-by-doing and/or scale economies Demand-based explanation» Accumulation of customer goodwill towards fast food

30 Fast Food Expansion ( of ) One-shot payoff of adding (or subtracting) n imt fast food outlets

31 Fast Food Expansion ( of ) Unobserved profitability Depends on past unobserved profitability and past stock of existing outlets

32 MGT 756: Particle Filtering in Marketing GUIDE TO PARTICLE FILTERING

33 Nonlinear Particle Filtering ( of 4) Objective Given collection of observations {Y it }, want to estimate distribution of unobserved state {Z it } Transition process depends on (δ, β) Estimating these parameters implies estimating the posterior distribution Bayesian point of view: Inference on {Z it } and (δ, β) essentially are the same problem, as (δ, β) can be thought of as time-invariant component of {Z it }

34 Nonlinear Particle Filtering ( of 4) What information do we need to begin? Initial distribution of Z t Can draw from N(0,) distribution Observation likelihood P Y t Y t, Z t Relates unobserved Z t to binary decision Y t Law of motion for Z t Given by goodwill accumulation equation

35 Nonlinear Particle Filtering ( of 4) Recovering P(Z t Y t ; θ) Filtering Use Baye s rule to obtain filtering distribution for transaction t P Y t Z t ; θ P(Z t Y t ; θ) P Z t Y t ; θ = P Y t Z t ; θ P Z t Y t ; θ dz t Greater weight on values with better estimated uncertainty Prediction Integrate w.r.t. to transition density for Z t given Z t and Y t P Z t Y t ; θ = Q Z t Z t, Y t ; θ P Z t Y t ; θ dz t Q Z t Z t, Y t ; θ = δz t + βy t + η t is transition density, given by goodwill accumulation equation

36 Nonlinear Particle Filtering (4 of 4) Initialization: Draw r from some distribution for each simulation draw r =,, R Recursion: Repeat the following steps for each t =,, T Importance sampling: Draw Z t r based on the transition equation, and set w t r = P Y t Y t, Z t r for each simulation draw r =,, R Resampling: For each r =,, R, draw Z t r from collection of {Z t r } in proportion to the weights computed in the previous step {w t r }

37 Comparison with EM Methods Popular method for integrating out persistent heterogeneity is Arcidiacono and Miller (0) Iterative EM algorithm for finite mixtures Comparative advantages of particle filtering: Unobservable can have continuous support Transition of unobservable can evolve endogenously (i.e., controlled stochastic process) Do not need to identify # of unobserved types Igami and Yang (04) apply Kasahara and Shimotsu (009) result to identify number of unobserved types for AM (0) Comparative disadvantages of particle filtering: Models with continuous unobserved (and observed) states are generally hard to solve, which makes counterfactuals tricky Transitions in unobserved heterogeneity are reduced form and may not be insightful

38 How to draw P(Z t Y t ; θ)? Initialization step - Let us draw initial particles to form a swarm

39 How to draw P(Z t Y t ; θ)? Updating step (initial) - Calculate the using and the goodwill equation

40 How to draw P(Z t Y t ; θ)? w w w Weighting step - Calculate weights based on w = P(Y )

41 How to draw P(Z t Y t ; θ)? w w w Resampling step - Draw a set of posterior by sampling with replacement the particles in proportion to their assigned weights

42 How to draw P(Z t Y t ; θ)? w w w Updating step - Calculate the Z using and the goodwill equation Z Z Z

43 How to draw P(Z t Y t ; θ)? w w w Iteration - Repeat weighting, resampling, and updating step until we reach T Z Z Z

44 How to Compute Weights? w w w Want to draw posterior distribution for goodwill (, Z, Z ) from (, Z, Z ) Suppose that w = P Y = 0.5, w = P Y Z = 0., and w = P Y Z = 0.8 Draw from (, Z, Z ) with the following resampling weights: Weight for = Weight for = Weight for = 0.5 = = = 0.6 = = 0.8 = Z Z Z

45 How to estimate θ? By approximating the posterior distribution of the particles, can use log-likelihood below Integrating out the Z it using draws from their approximated posterior distributions We can then use standard ML optimization N T ln L θ = ln i t R P(Y R r it Z r it, Y it ; θ) θ = arg max ln L θ

46 MGT 756: Particle Filtering in Marketing MONTE CARLO ANALYSIS

47 Particle Filtering in Action ( of ) Let us now consider the sample exercise Refer to application_notes_spring04.pdf Sample estimation code in MATLAB: main Runs simulations, estimation, and fit analysis simulate Generate the fake data loglik Likelihood for particle filtering

48 Particle Filtering in Action ( of ) Generate the data 00 customers who make 0 transactions each For the model parameters, set α = 0.9, β = 0., and δ = 0.5 Estimate the data Using the simulated data, estimate the main parameters using the particle filtering method Use,000 particles

49 How to draw P(Z t Y t ; θ)?

50 How to draw P(Z t Y t ; θ)?

51 How to draw P(Z t Y t ; θ)? w w w

52 How to draw P(Z t Y t ; θ)? w w w

53 How to draw P(Z t Y t ; θ)? w w w Z Z Z

54 How to draw P(Z t Y t ; θ)? w w w Z Z Z

55 How to estimate θ? N T ln L θ = ln R i t R r P(Y it Z it r, Y it ; θ)

56 Results from Monte Carlo Analysis Parameter SMC vs true value True SMC Intercept (α) New goodwill (β) Retention (δ) Parameters from SMC close to true value

57 Results from Monte Carlo Analysis Parameter SMC vs true value True SMC Intercept (α) New goodwill (β) Retention (δ) Parameters from SMC close to true value

58 Optional Exercise Generate the data 00 customers who make 0 transactions each Modify the goodwill accumulation equation to allow for individual heterogeneity ψ i Z it = ψ i + δz it + βy it + η it As before, set α = 0.9, β = 0., and δ = 0.5 For individual heterogeneity, set ψ :50 = 0. and ψ 5:00 = 0. Estimate the data Using the simulated data, estimate the main parameters using the particle filtering method Use,000 particles Assume that we know that customers to 50 are of type ψ :50, and customers 5 to 00 are of type ψ 5:00 In addition to customer-specific heterogeneity, what else do these new parameters ψ :50 and ψ 5:00 capture?

59 MGT 756: Particle Filtering in Marketing ADDITIONAL REMARKS

60 Linear Particle Filtering ( of ) Consider the similar model as before, except Y it = α + Z it + ε it Can be estimated using a linear particle filter Bass, Bruce, Majumdar, and Murthi (007) Two main components Forward filtering Simulation of Z it

61 Linear Particle Filtering ( of ) Forward filtering Posterior distribution for Z t Z t Y :t ~N(m t, C t ) Prior distribution for Z t Z t Y :t ~N(a t, R t ) Some terms a t = δm t + βy t R t = δ C t + m t = a t + R t (R t + )(Y t a t + α) C t = R t [ R t (R t + )]

62 Linear Particle Filtering ( of ) Simulation for Z t Step Compute posterior mean and variance (m t, C t ) for all t using forward filtering sequentially Step At end of series, sample Z T from the posterior distribution N(Z T m T, C T ) Step Moving backwards, sample Z t conditional on draw Z t Will get draws :T from full conditional posterior

63 Extensions ( of ) Nonlinear particle filtering methods be paired with a two-step estimation method like Bajari, Benkard, and Levin (007) Blevins, Khwaja, and Yang (04) First step: Reduced form policy function estimation and transition equation estimation The reduced form policy functions will have to be parametric Second step: With policy function and transition equation on hand, approximate the value functions Forward simulate the following» Transition of serially correlated unobservables» Actions as determined using policy function

64 Extensions ( of ) Chain store expansion model Chain decides how much to expand/contract in market m at time t, denoted as Y mt { K,,,0,,, K} K is the maximum # of stores chain can add or subtract The stock of active outlets is given by the following: N mt = N mt + Y mt Objective is to maximize it s one-shot payoff, as defined by: Π mt = α N mt + α N mt + Z mt + ε mt Z mt = δz mt + βn mt + η mt represents the evolution of unobserved profitability

65 Extensions ( of ) If K is large, how to estimate decision model? What is an interesting trade-off one could study if the chain is forward looking? How can particle filtering be applied? How could we highlight the persistence of unobserved profitability?

66 Key Takeaways Particle filtering can be used to estimate models with serially correlated unobservables That evolve both stochastically and endogenously That have continuous support Reduces the burden of high-dimensional integration Uses information more efficiently than typical numerical integration techniques Easily implemented, and can be applied to discrete choice Customer goodwill, retail expansion, May be paired with other dynamic structural methods Nested fixed point estimation, such as Rust model Two step estimation, such as BBL

67 Main References Bass, Bruce, Majumdar, and Murthi Wearout Effects of Different Advertising Themes: A Dynamic Bayesian Model of the Advertising-Sales Relationship. Marketing Science 6, Blevins. 04. Sequential Monte Carlo Methods for Estimating Dynamic Microeconomic Models. Working paper, OSU Department of Economics. Blevins, Khwaja, and Yang. 04. Market Share Evolution, Size Spillovers, and Organizational Forgetting in Retail Chain Dynamics. Working paper, Yale SOM.

68 Please direct all questions about this lecture to THANK YOU!