1. Estimating Logit Models using Household Level and Aggregate Data
K. Sudhir
PhD Seminar
Yale University

2. Three Estimation Models
Individual Level Models
Guadagni and Little (1983)
Heterogeneity Modeled exponential smoothing measure
Kamakura and Russell (1994)
Unobserved Heterogeneity using discrete segment approach
Gonul and Srinivasan (1993)
Unobserved Heterogeneity using continuous distribution approach
We will discuss the first two models
Aggregate Level Models
Sudhir (2001)
Chintagunta (2001) paper for its discussion on how without attributes, it uses a covariance matrix of the intercepts to model attribute similarities.

3. Guadagni and Little (1983) Utility Model Loyalty Normalization
Likelihood function
Optimization and computing Standard Errors

4. Loyalty Variable %Constructing Loyalty Variable
BrLoy=ones(NObs,NProd);
for i = 1:1:NCons;
BrLoy(PIDSt(i),:)=1/NProd;
for j=PIDSt(i)+1:1:PIDSt(i+1)-1;
BrLoy(j,:)= alph*BrLoy(j-1,:)+(1-alph)*Ch(j,:);
end;

5. Normalization %Subtracting out the normalized product's variables
for i=1:1:NProd;
Price1(:,i)=Price(:,i)-Price(:,NProd);
Feat1(:,i)=Feat(:,i)-Feat(:,NProd);
Disp1(:,i)=Disp(:,i)-Disp(:,NProd);
BrLoy1(:,i)=BrLoy(:,i)-BrLoy(:,NProd);
end;

6. Likelihood function function f= GLNegLogLik(x)
%Global Variables made available from the main function
global Price1 Feat1 Disp1 BrLoy1 Ch NObs NProd;
expU=ones(NObs,NProd);
for i=1:1:NProd-1;
expU(:,i)=exp(x(i)+x(NProd)*Price1(:,i)+x(NProd+1)*Feat1(:,i)+x(NProd+2)*Disp1(:,i)
+x(NProd+3)*BrLoy1(:,i));
end;
expU(:,NProd)=1;
SU=(sum(expU'))';
Prob=expU./[SU SU SU SU];
f = -sum(sum(Ch.*log(Prob)));

7. Maximizing Likelihood, Computing SEs
options=optimset('Display','iter','TolFun',1e-4,'TolX', 1e-4,'MaxIter',25, 'MaxFunEvals',560, 'LargeScale','off', 'HessUpdate', ‘bfgs’);
[x, fval,exitflag,output,grad,hessian] = fminunc('GLNegLogLik',x0,options);
se=sqrt(diag(inv(hessian)));
tstat=x./se;

8. Exercise Estimate the model without the loyalty variable
Report the estimates
Compute U2
Compute the U2 for the model with the loyalty variable as here.

9. Kamakura and Russell (1989)
Utility Model
Discrete Heterogeneity
Normalization
Likelihood function
Optimization and computing Standard Errors

10. expU=ones(NObs,NProd*NSeg);
SU=ones(NObs,1);
Prob=ones(NObs,NSeg);
Lik=ones(NCons,NSeg);

11. Computing Event Likelihood for Each Segment
for j=1:1:NSeg;
for i=1:1:NProd-1;
expU(:,(j-1)*NProd+i)=exp(x((j-1)*(NProd+NVar-1)+i)
+x((j-1)*(NProd+NVar-1)+NProd)*Price1(:,i)
+x((j-1)*(NProd+NVar-1)+NProd+1)*Feat1(:,i)
+x((j-1)*(NProd+NVar-1)+NProd+2)*Disp1(:,i));
end;
expU(:,j*NProd)=1;
SU=(sum(expU(:,(j-1)*NProd+1:j*NProd)'))';
Temp=(expU(:,(j-1)*NProd+1:j*NProd)./[SU SU SU SU]).^Ch;
Prob(:,j)=prod(Temp')';

12. Likelihood of each person’s purchase string
for i=1:1:NCons;
Lik(i,:)=prod(Prob(freq(i):freq(i+1)-1,:));
end;

13. Computing the Overall Log-Likelihood
siz=exp(x((NProd-1+NVar)*NSeg+1:(NProd-1+NVar)*NSeg+NSeg-1));
siz=[siz; 1];
siz=siz/sum(siz);
LLik=Lik*siz;
f=-sum(log(LLik));

14. Computing Standard Errors
%To get standard errors when we use transformed variables
Nx=size(x,1)
A=eye(Nx);
A(Nx-1,Nx-1)=(exp(x(NSeg*(NProd+NVar-1)+1))* (1+exp(x(NSeg*(NProd+NVar1)+2))))/ ((1+exp(x(NSeg*(NProd+NVar-1)+1))+exp(x(NSeg*(NProd+NVar-1)+2))))^2;
A(Nx,Nx)=(exp(x(NSeg*(NProd+NVar-1)+2))* (1+exp(x(NSeg*(NProd+NVar-1)+1))))/ ((1+exp(x(NSeg*(NProd+NVar-1)+1))+exp(x(NSeg*(NProd+NVar-1)+2))))^2;
se=sqrt(diag(A*inv(hessian)*A'));

15. Exercise Estimate the model with only preference heterogeneity
Estimate the optimal number of segments using the AIC or BIC criterion.
Report the estimates and standard errors
Estimate the model with both preference and response heterogeneity as in the code

16. Using Aggregate Data to Estimate Random Coefficients Logit Model
Several Issues
Simulation to integrate over random coefficients distribution
Drawing from a multivariate distribution
Endogeneity
Use of IV Estimation
Linearization of the mean utility to facilitate IV by using contraction mapping
Generalized Method of Moments
Two part approach to estimate linear and nonlinear parameters

17. Drawing multivariate normals
w1=randn(NObs1,NCons);
w2=randn(NObs1,NCons);
aw1=b(1)*w1+b(2)*w2;
aw2=b(3)*w1;

18. Contraction Mapping; Simulation based Integration
while (k >= km)
de=de1;
sh=zeros(NObs1,1);
psh=zeros(NObs1,1);
for i=1:1:NCons;
psh=exp(aw(:,i)+awp(:,i)+de); psh=reshape(psh',2,NObs)';
spsh=sum(psh')';
psh(:,1)=psh(:,1)./(1+spsh); psh(:,2)=psh(:,2)./(1+spsh);
sh=sh+reshape(psh',NObs1,1);
end;
sh=sh/NCons;
de1=de+log(s)-log(sh);
k=max(abs(de1-de));

19. Analytically estimating the linear parameters; Optimizing over nonlinear parameters
blin=inv(xlin'*z*W*z'*xlin)*(xlin'*z*W*z'*d);
er=d-xlin*blin;
f=er'*z*W*z'*er;

20. Exercise
Compute the standard errors for the estimates in the code using approach suggested in Greene
Optional Exercise:
Add a supply side equation assuming Bertrand pricing by the two firms