MATLAB® Script for Replicating Abbring and Daljord (2020)
This script computes the examples in Abbring and Daljord (2020). Its results are stored in .csv and .tex files that are read by LaTeX upon the paper's compilation to produce the graphs and documentation of the paper's examples.
Jaap H. Abbring and Øystein Daljord, May 2020
Contents
- Example in Which an Exclusion Restriction on Current Values Suffices for Identification but One on Primitive Utility Does Not
- Example in Which Magnac and Thesmar (2002)'s Rank Condition Fails, but an Exclusion Restriction on Primitive Utility Suffices for Identification
- Example of Data that are Consistent with an Exclusion Restriction on Current Values but Not with One on Primitive Utility
- Example with Two Moment Conditions of Which One Identifies the Discount Factor
- Example with Two Moment Conditions that Jointly Identify the Discount Factor but Individually Do Not
- Same Example with Noise in Choice Probabilities
- Monotonicity Example
- Dependencies
- References
Example in Which an Exclusion Restriction on Current Values Suffices for Identification but One on Primitive Utility Does Not
Qi = [0.25 0.25; 0 0.25]; QK = [0.9 0; 0 0.9; 0 1]; p = [0.5; 0.49; 0.1]; example(p,Qi,QK,'identificationFails',[1 2],'yes',[])
Name: identificationFails
Qi =
0.2500 0.2500 0.5000
0 0.2500 0.7500
QK =
0.9000 0 0.1000
0 0.9000 0.1000
0 1.0000 0
deltaQ =
-0.6500 0.9000 -0.2500
p =
0.5000 0.5000
0.4900 0.5100
0.1000 0.9000
m =
0.6931
0.6733
0.1054
lhs =
0.0400
Slope r.h.s. Magnac and Thesmar (2002):
0.1291
betaMT =
0.3098
Example in Which Magnac and Thesmar (2002)'s Rank Condition Fails, but an Exclusion Restriction on Primitive Utility Suffices for Identification
Use the same Qi and QK as in the previous example
p = [0.5; 0.48; 0.1]; example(p,Qi,QK,'moreEmpiricalContent',[1 2],'yes',[])
Name: moreEmpiricalContent
Qi =
0.2500 0.2500 0.5000
0 0.2500 0.7500
QK =
0.9000 0 0.1000
0 0.9000 0.1000
0 1.0000 0
deltaQ =
-0.6500 0.9000 -0.2500
p =
0.5000 0.5000
0.4800 0.5200
0.1000 0.9000
m =
0.6931
0.6539
0.1054
lhs =
0.0800
Slope r.h.s. Magnac and Thesmar (2002):
0.1116
betaMT =
0.7169
Example of Data that are Consistent with an Exclusion Restriction on Current Values but Not with One on Primitive Utility
Qi = [0.0 0.25; 0.25 0.25]; QK = [0 1;0 1;0 0]; p = [0.5; 0.48; 0.5]; example(p,Qi,QK,'needNoMT',[1 2],'yes',[])
Name: needNoMT
Qi =
0 0.2500 0.7500
0.2500 0.2500 0.5000
QK =
0 1 0
0 1 0
0 0 1
deltaQ =
-0.2500 0 0.2500
p =
0.5000 0.5000
0.4800 0.5200
0.5000 0.5000
m =
0.6931
0.6539
0.6931
lhs =
0.0800
Slope r.h.s. Magnac and Thesmar (2002):
0
betaMT =
Inf
Example with Two Moment Conditions of Which One Identifies the Discount Factor
Qi = [0.4000 0.3333 0.1905;
0.2647 0.2941 0.2647;
0.1786 0.3571 0.1786;
0.1818 0.2727 0.4545];
QK = [0.1739 0.2609 0.1304;
0.1333 0.0667 0.2000;
0.2000 0.3000 0.1000;
0.2500 0.1500 0.5000];
% Impose $u_1(x_1) = u_1(x_2)$ and $u_1(x_3) = u_1(x_4)$
u1 = 0.5;
u3 = 2;
beta = 0.3;
[~,~,~,p] = avf(QK,Qi,[u1 u1 u3 u3]',beta);
example(p,Qi,QK,'oneRedundant',[1 2;3 4],'yes',[])
Converged in 28 iterations
Name: oneRedundant
Qi =
0.4000 0.3333 0.1905 0.0762
0.2647 0.2941 0.2647 0.1765
0.1786 0.3571 0.1786 0.2857
0.1818 0.2727 0.4545 0.0910
QK =
0.1739 0.2609 0.1304 0.4348
0.1333 0.0667 0.2000 0.6000
0.2000 0.3000 0.1000 0.4000
0.2500 0.1500 0.5000 0.1000
deltaQ =
0.0947 -0.1550 -0.0046 0.0649
0.0468 -0.0656 0.1241 -0.1053
p =
0.5969 0.4031
0.5924 0.4076
0.8795 0.1205
0.8790 0.1210
m =
0.9086
0.8975
2.1160
2.1121
lhs =
0.0187
0.0045
Slope r.h.s. Magnac and Thesmar (2002):
0.0743
0.0238
betaMT =
0.2518
0.1882
Example with Two Moment Conditions that Jointly Identify the Discount Factor but Individually Do Not
Qi = [0.4286 0.3333 0.1905;
0.2647 0.2941 0.2647;
0.1786 0.3571 0.1786;
0.1818 0.2727 0.4545];
QK = [0.1739 0.2609 0.1304;
0.1333 0.0667 0.2000;
0.2000 0.3000 0.1000;
0.2500 0.1500 0.5000];
% Impose $u_1(x_1) = u_1(x_2)$ and $u_1(x_3) = u_1(x_4)$
u1 = 2;
u3 = 0.5;
beta = 0.9;
[~,~,~,p] = avf(QK,Qi,[u1 u1 u3 u3]',beta)
example(p,Qi,QK,'twiceTwo',[2 1;4 3],'yes',[])
Converged in 314 iterations
p =
0.9192
0.9197
0.6319
0.6323
Name: twiceTwo
Qi =
0.4286 0.3333 0.1905 0.0476
0.2647 0.2941 0.2647 0.1765
0.1786 0.3571 0.1786 0.2857
0.1818 0.2727 0.4545 0.0910
QK =
0.1739 0.2609 0.1304 0.4348
0.1333 0.0667 0.2000 0.6000
0.2000 0.3000 0.1000 0.4000
0.2500 0.1500 0.5000 0.1000
deltaQ =
-0.1233 0.1550 0.0046 -0.0363
-0.0468 0.0656 -0.1241 0.1053
p =
0.9192 0.0808
0.9197 0.0803
0.6319 0.3681
0.6323 0.3677
m =
2.5163
2.5226
0.9993
1.0005
lhs =
0.0068
0.0019
Slope r.h.s. Magnac and Thesmar (2002):
0.0490
0.0291
betaMT =
0.1385
0.0656
Same Example with Noise in Choice Probabilities
rng(23670,'twister'); pHat = p + (1/1000)*rand(4,1) example(pHat,Qi,QK,'twiceTwoNoise',[2 1;4 3],'yes',0.10)
pHat =
0.9202
0.9207
0.6321
0.6333
Name: twiceTwoNoise
Qi =
0.4286 0.3333 0.1905 0.0476
0.2647 0.2941 0.2647 0.1765
0.1786 0.3571 0.1786 0.2857
0.1818 0.2727 0.4545 0.0910
QK =
0.1739 0.2609 0.1304 0.4348
0.1333 0.0667 0.2000 0.6000
0.2000 0.3000 0.1000 0.4000
0.2500 0.1500 0.5000 0.1000
deltaQ =
-0.1233 0.1550 0.0046 -0.0363
-0.0468 0.0656 -0.1241 0.1053
p =
0.9202 0.0798
0.9207 0.0793
0.6321 0.3679
0.6333 0.3667
m =
2.5282
2.5342
1.0000
1.0031
lhs =
0.0066
0.0050
Slope r.h.s. Magnac and Thesmar (2002):
0.0493
0.0295
betaMT =
0.1330
0.1694
betaSet =
0.1023
0.2764
0.7898
0.9137
Monotonicity Example
J = 3; learnRate = 0.75; Qi = diag((1-learnRate)*ones(J,1))+diag(learnRate*ones(J-1,1),1); %Qi(end)=1; Qi = Qi(:,1:end-1); depRate = 0.5; QK = diag((1-depRate)*ones(J,1))+diag(depRate*ones(J-1,1),-1); QK(1)=1; QK = QK(:,1:end-1); beta=0.8; u=@(x)-0.5+(x>2).*(x-2); [~,~,~,p] = avf(QK,Qi,u(1:J)',beta) example(p,Qi,QK,'experience',[2 1],'yes',[]) name = 'experienceStructure'; % Save structural parameters to .tex file f1=fopen(['data//' name '.tex'],'w'); fprintf(f1,['\\def\\' name 'learnRate{%6.2f}\n'],learnRate); fprintf(f1,['\\def\\' name 'depRate{%6.2f}\n'],depRate); fprintf(f1,['\\def\\' name 'beta{%6.2f}\n'],beta); fprintf(f1,['\\def\\' name 'uBoldPrime{\\left[\\begin{array}{cccc}%4.2f&%4.2f&%4.2f\\end{array}\\right]}\n'],u(1:J)); fprintf(f1,['\\def\\' name 'uOne{%4.2f}\n'],u(1)); fprintf(f1,['\\def\\' name 'uTwo{%4.2f}\n'],u(2)); fprintf(f1,['\\def\\' name 'uThree{%4.2f}\n'],u(3)); fclose(f1);
Converged in 144 iterations
p =
0.4365
0.5588
0.7073
Name: experience
Qi =
0.2500 0.7500 0
0 0.2500 0.7500
0 0 1.0000
QK =
1.0000 0 0
0.5000 0.5000 0
0 0.5000 0.5000
deltaQ =
0.2500 -1.0000 0.7500
p =
0.4365 0.5635
0.5588 0.4412
0.7073 0.2927
m =
0.5736
0.8184
1.2285
lhs =
0.4918
Slope r.h.s. Magnac and Thesmar (2002):
0.2465
betaMT =
1.9953
Dependencies
References
- Abbring, Jaap H. and Øystein Daljord (2020). Identifying the discount factor in dynamic discrete choice models. Quantitative Economics 11(2), 471–501.
- Thierry Magnac and David Thesmar (2002). Identifying dynamic discrete choice processes. Econometrica 70, 801-816.