Econ 8208 Homework 2 Due Date: May 7

Transcription

1 Econ 8208 Homework 2 Due Date: May 7 1 Preliminaries This homework is all about Hierarchical Linear Bayesian Models (HLBM in what follows) The formal setup of these can be introduced as follows Suppose there are j = 1,, m regressions you would like to estimate Assume there are i = 1,, n j data points for each of the m regressions We specify equations as follows: y i,1 = X i,1 β 1 + ε i,1, ε 1 iid N ( 0, σ1i 2 ) n1 y i,j = X i,j β j + ε i,j, ε j iid N ( 0, σj 2 ) I nj y i,m = X i,m β m + ε i,m, ε m iid N ( 0, σmi 2 ) nm Denote by y j is n j 1 column vector that contains values for dependent variable in equation j, and X j is the n j k matrix of regressors for equation j We assume errors ε j are not correlated across j: ε = ε 1 0 σ1i 2 n1 0 0 ε j N 0, 0 σ j 2I n j σmi 2 nj ε m 1

2 Instead, we capture the correlation across equations via the prior structure: β 1,1 = δ 1,1 z 1,1 + δ 2,1 z 1,2 + + δ nz,1z 1,nz + v 1,1 β 2,1 = δ 1,1 z 2,1 + δ 2,1 z 2,2 + + δ nz,1z 2,nz + v 2,1 β m,1 = δ 1,1 z m,1 + δ 2,1 z m,2 + + δ nz,1z m,nz + v m,1 β 1,2 = δ 1,2 z 1,1 + δ 2,2 z 1,2 + + δ nz,2z 1,nz + v 1,2 β 2,1 = δ 1,2 z 2,1 + δ 2,2 z 2,2 + + δ nz,2z 2,nz + v 2,1 β m,2 = δ 1,2 z m,1 + δ 2,2 z m,2 + + δ nz,2z m,nz + v m,2 β 1,k = δ 1,k z 1,1 + δ 2,k z 1,2 + + δ nz,kz 1,nz + v 1,k β 2,k = δ 1,k z 2,1 + δ 2,k z 2,2 + + δ nz,kz 2,nz + v 2,k β m,k = δ 1,k z m,1 + δ 2,k z m,2 + + δ nz,kz m,nz + v m,k (1) where ν j iid N (0, V β ), j = 1,, m Equations (1) specify a normal prior on β This prior can be rewritten more succintly as a multivariate regression: B = Zδ + V, B = β 1, Z = z 1 [, δ = δ 1 δ k ], V = ν 1 (2) β m z m ν m In the above, B is m k, Z is m n z, and δ is n z k Each column of δ has coefficients which describe how the mean of k coefficients varies as a function of z It will be convenient to assume independent priors on each σ 2 j : To complete the hierarchical model, we need priors on δ and V β : σ 2 j ν js 2 0j χ 2 ν j (3) V β IW (v, V ), (4) vec [δ] V β N ( vec [ δ], Vβ A 1) Here IW stands for Inverse Wishart distribution, and vec [A] is the vectorize function that turns a matrix into a column vector: ξ = ξ 11 ξ 12 ξ 21 ξ 22 vec [ξ] = ξ 11 ξ 21 ξ 12 ξ 22 2

3 In principle, these priors depend on other parameters (which are v, V, vec [ δ], and A) and one could in turn impose priors on those parameters as well For practical purposes, we stop here and refer to these as hyperparameters in what follows We pick those priors since those are the natural conjugate priors for multivariate regression model (2) 2 Application to Simulated Data The HLBM from part 1 can be estimated if one has the following data on y, X and z For this exercise, the data comes in a csv file named hw2 data simulatedcsv This file was generated by the following model: m = 2, so there are two equations to estimate; k = 3, so there are 3 regressors in each main equation of interest (including the constant), and n z = 2, so there are 2 auxilliary regressors in the mean equations (again, including the constant) There are n j = 5000 data points for each j The following example will help to think about the data: a single firm, call it Southwest Airlines, is interested in estimating demands for two of its products: tickets in couch class (equation 1) and tickets in business class (equation 2) 1 There regressors in X j can be thought of as measures of ticket prices and other observable attributes like trip mileage or service quality measures Finally, the variables in Z represent some relevant demographics, that presumably has an impact on distribution of tastes for attributes There are 8 columns in the csv file, each representing a specific variable The first two colums contain y 1 and y 2, the dependent variables for each of the two equations (quantities demanded) The next two columns have x 1,2 and x 1,3, which are the regressors for the first equation You can think of these as the 2nd and the 3rd column of X 1 (the first column being the constant) In columns 5 and 6 you can find x 2,2 and x 2,3, the regressors for the second equation of interest The final two columns contain z 1,2 and z 2,2 These are the second columns of matrix Z j for each of the two equations (the first column will again be a constant) You will notice that the only values in these two columns that are not equal to zero appear in the first row The zeros are completely irrelevant and they are there just to facilitate the data reading and writing from/to Matlab Only the two numbers from the first rows of z are of interest to you Your main equations will thus look like: y 1,i = β 1,1 + β 1,2 x 1,2,i + β 1,3 x 1,3,i + ε 1,i y 2,i = β 2,1 + β 2,2 x 2,2,i + β 2,3 x 2,3,i + ε 2,i 1 I fully admit this example does not account for any possible kind of endogeneity between p and q in demand curves 3

4 and your prior on β will look (by equation) as: β 1,1 = δ 1,1 + z 1 δ 2,1 + v 1,1 β 1,2 = δ 1,2 + z 1 δ 2,2 + v 1,2 β 1,3 = δ 1,3 + z 1 δ 2,3 + v 1,3 β 2,1 = δ 1,1 + z 2 δ 2,1 + v 2,1 β 2,2 = δ 1,2 + z 2 δ 2,2 + v 2,2 β 2,3 = δ 1,3 + z 2 δ 2,3 + v 2,3 with all appropriate distributions and parametrizations from the previous section Your primary goal is to simulate the posterior distribution for each β and σ 2 You will find parts and 37 of the Rossi, Allenby, and McCullogh s book extremely helpful in doing this See part 4 at the end of the assignment for details on the form of answer that is expected from you 3 Application to Actual Data Your next task involves applying the HLBM from part 1 to some real data We are interested in promotional response modeling Borden, a company producing sliced cheese, has data on some of its key accounts, which are defined as a combination of retailer and market area The data comes in the csv file hw2 data cheesecsv, which has four columns: 1 First colimn is the key account identifier, call it retailer There is information on 88 retailers for an average of 65 weeks (that is, for a given value of retailer, each observation is data on weekly sales) 2 Second column is the sales volume, call it volume 3 Third column is the measure of display activity, call it disp (more about this below) 4 Last, fourth column is sales price in dollars, call it price Thus weekly observations on sales volume and dollar price, as well as for the measure of display activity, are available Displays are a form of in-store advertising that consists of displaying given merchandise in a particular hot location in the store The available measure of display activity is the percentage of inventory that was on display, which varies from 0 to 1 The data is described in somewhat greater detail in the Rossi s book on pp Your primary goal is to estimate a model similar to the one discussed in the book and come up with your own version of Figure 37 from the book See part 4 below for details on the form of answer that is expected from you 4

5 4 Submission Requirements After you are done with programming, I want you to write a 5-7 page document that succintly summarizes your work Describe briefly how your programs work, and demonstrate some program output that would convince the reader that your work had been successful In particular, for every β i,j and for every σj 2, do the following: 1 Report the simulated means and standard deviations of the corresponding posterior distributions 2 Plot the figure similar to the one in the bottom panel of Rossi s Figure 34 to demonstrate convergence of your simulated Markov chain 3 Construct the autocorrelation function to formally test for convergence For the posteriors obtained in part 3 of this homework, also do the following Assume that you happen to control the cheese manufacturer Borden Your job is to determine the optimal decision on promotional measures for your products Specifically, you want to maximize the following profit function: π (q, θ disp ) = p (q, θ disp ) q c (q) where q is the number of products you consider promoting (assume it is a continuous variable for simplicity) and θ disp is the parameter that measures demand response to display activity The functions p ( ) and c ( ) are the inverse demand function and the total cost function, respectively, and they are assumed to take the following form: p (q, θ disp ) = 100 θ disp q and c (q) = 20q In case θ disp is deterministic, this profit maximization problem would be fairly straightforward However, you only know the posterior distribution for θ disp (call it f (θ disp ), so you will need to maximize the expected profit: max q π (q, θ disp ) f (θ disp ) dθ disp Θ disp (5) 4 Solve the problem in (5) and report your results Also, plot the expected profits as a function of q 5