Dynare forums

by **Niklas** » Tue Oct 29, 2013 2:46 pm

Dear all,

It seems that I have experienced a strange interaction between the command identification and the ability of the optimizer csminwel (mode_compute = 4) to find the posterior mode.

If I estimate 3 distinct AR(1) processes (see the attached mod-file and data file) with the identification command, mode_compute = 4 leads to a positive definite hessian. If I run it without the identification command, mode_compute = 4 does not deliver the maximum, and the hessian is not positive definite. Also, the values of the posterior are slightly different. The log data density is higher (!) for the case with the not positive definite hessian. And this is the case although the plots following mode_check all look fine (see the attached plots 1 and 2)!

Does anybody has an explanation for this observation?

I am thankful for any help,
all the best,
Niklas

by **jpfeifer** » Tue Oct 29, 2013 6:16 pm

You are running identification for the prior. Thus, after running identification, the parameter values are set to the prior mean. In contrast, when running estimation directly, the parameters are initialized to the specified starting values. This difference in starting values explains the different outcome when running a numerical optimizer.

by **Niklas** » Wed Oct 30, 2013 10:14 pm

Dear jpfeifer,

as always, thank you very much for your help!

I am not quite sure whether I understood your answer correctly. Does you answer imply that running the model without specifying any starting values (hence starting from the prior mean by default) should lead to the same outcome as having the identification command before running mode_compute = 4? This is however not the case (as you can see below where I have added some extractions from the Matlab output): The initial values of the log posterior are quite different and the number of iterations in both cases are also quite different. More importantly, in the case without identification, I get the error message of "not positive definite hessian".

What I also really do not understand is, why csminwel (and also mode_compute = 5, I checked it) is having such a hard time in finding the posterior mode. The problem is very simple (3 AR(1) processes without any connection), so it should never lead to the problem of having a "not positive hessian"!? Do you have an explanation for this?

All the best!
Niklas

Run without the identification command, but with starting values at the prior mean:

Code: Select all: Initial value of the log posterior (or likelihood): 567.0646 ----------------- ----------------- f at the beginning of new iteration, -567.0645821956 Predicted improvement: 9394.239308087 lambda = 1; f = -565.2445182 lambda = 0.33333; f = -566.8751041 lambda = 0.11111; f = -567.0475026 lambda = 0.037037; f = -567.0637321 lambda = 0.012346; f = -919.0702508 lambda = 0.023866; f = -567.0644589 lambda = 0.01607; f = -567.0645819 lambda = 0.012676; f = -932.4183817 lambda = 0.014615; f = -988.3939319 lambda = 0.016851; f = -567.0645799 lambda = 0.015472; f = -948.2675865 lambda = 0.016285; f = -567.0645816 lambda = 0.015792; f = -567.0645822 lambda = 0.015314; f = -964.3574522 lambda = 0.015599; f = -567.0645822 lambda = 0.015427; f = -953.3961886 Norm of dx 1.3707 Cliff. Perturbing search direction. Predicted improvement: 14893.717837427 lambda = 1; f = -562.2121069 lambda = 0.33333; f = -566.5458539 lambda = 0.11111; f = -567.0134807 lambda = 0.037037; f = -567.0608057 lambda = 0.012346; f = -567.0645224 lambda = 0.0041152; f = -718.5544360 lambda = 0.0079555; f = -921.7019034 lambda = 0.015379; f = -567.0643882 lambda = 0.010356; f = -567.0645680 lambda = 0.008168; f = -567.0645822 lambda = 0.0080187; f = -921.1032345 lambda = 0.008108; f = -918.2936891 lambda = 0.0081982; f = -567.0645822 Norm of dx 2.2239 Cliff again. Try traversing Predicted improvement: 102642132.493013650 lambda = 1; f = 203262438.5730386 lambda = 0.33333; f = 22584121.2386550 lambda = 0.11111; f = 2508812.8774944 lambda = 0.037037; f = 278242.9741065 lambda = 0.012346; f = 30408.5104982 lambda = 0.0041152; f = 2873.5601311 lambda = 0.0013717; f = -185.1413799 lambda = 0.00045725; f = -524.7512971 lambda = 0.00015242; f = -562.4037969 lambda = 5.0805e-05; f = -566.5600962 lambda = 1.6935e-05; f = -567.0128166 lambda = 5.645e-06; f = -567.0601198 lambda = 1.8817e-06; f = -567.0643905 lambda = 6.2723e-07; f = -670.9653593 lambda = 1.2125e-06; f = -567.0645618 lambda = 8.1646e-07; f = -710.3305928 lambda = 1.0351e-06; f = -567.0645779 lambda = 8.9775e-07; f = -567.0645820 lambda = 8.2424e-07; f = -710.4171907 lambda = 8.6759e-07; f = -567.0645822 lambda = 8.4132e-07; f = -708.8439203 lambda = 8.5698e-07; f = -704.1474898 lambda = 8.7295e-07; f = -567.0645822 lambda = 8.6333e-07; f = -567.0645822 lambda = 8.5383e-07; f = -705.4318732 Norm of dx 14328 ---- Improvement on iteration 1 = 421.329349745 back and forth on step length never finished ----------------- ----------------- f at the beginning of new iteration, -988.3939319404 Predicted improvement: 19510.472534793 lambda = 1; f = -986.6782540 lambda = 0.33333; f = -988.2038025 lambda = 0.11111; f = -988.3729731 lambda = 0.037037; f = -988.3916582 lambda = 0.012346; f = -988.3936971 lambda = 0.0041152; f = -988.3939112 lambda = 0.0013717; f = -988.3939309 lambda = 0.00045725; f = -991.6241260 Norm of dx 1.9754 ---- Improvement on iteration 2 = 3.230194084 ... (I deleted the iterations 3-30) Improvement on iteration 31 = 0.000000247 back and forth on step length never finished ----------------- ----------------- f at the beginning of new iteration, -1023.4940295049 Predicted improvement: 0.000010945 lambda = 1; f = -1023.4940195 lambda = 0.33333; f = -1023.4940286 lambda = 0.11111; f = -1023.4940295 lambda = 0.037037; f = -1023.4940295 Norm of dx 8.0884e-05 ---- Improvement on iteration 32 = 0.000000022 improvement < crit termination Objective function at mode: -1023.494030 POSTERIOR KERNEL OPTIMIZATION PROBLEM! (minus) the hessian matrix at the "mode" is not positive definite! => posterior variance of the estimated parameters are not positive. You should try to change the initial values of the parameters using the estimated_params_init block, or use another optimization routine. Warning: The results below are most likely wrong! > In dynare_estimation_1 at 480 In dynare_estimation at 70 In mode_identification_interaction at 228 In dynare at 120 MODE CHECK

Run with speficied starting values (actually the mode itself) and with the identification command in the beginning (i.e. at the prior mean):

Code: Select all: Initial value of the log posterior (or likelihood): 1023.4939 ----------------- ----------------- f at the beginning of new iteration, -1023.4939473386 Predicted improvement: 0.391220249 lambda = 1; f = -1023.4939470 lambda = 0.33333; f = -1023.4939473 lambda = 0.11111; f = -1013.0549904 lambda = 0.037037; f = -1021.8991452 lambda = 0.012346; f = -1023.2984405 lambda = 0.0041152; f = -1023.4733102 lambda = 0.0013717; f = -1023.4923285 lambda = 0.00045725; f = -1023.4940044 lambda = 0.00015242; f = -1023.4940331 lambda = 0.00029465; f = -1023.4940529 Norm of dx 0.0088456 ---- Improvement on iteration 1 = 0.000105571 ----------------- ----------------- f at the beginning of new iteration, -1023.4940529095 Predicted improvement: 0.011833352 lambda = 1; f = -1023.4940528 lambda = 0.33333; f = -1018.6489259 lambda = 0.11111; f = -1022.9332580 lambda = 0.037037; f = -1023.4306419 lambda = 0.012346; f = -1023.4871290 lambda = 0.0041152; f = -1023.4933457 lambda = 0.0013717; f = -1023.4939959 lambda = 0.00045725; f = -1023.4940538 lambda = 0.00015242; f = -1023.4940554 Norm of dx 0.0015614 ---- Improvement on iteration 2 = 0.000002502 ----------------- ----------------- f at the beginning of new iteration, -1023.4940554110 Predicted improvement: 0.000897340 lambda = 1; f = -1023.4940554 lambda = 0.33333; f = -1022.8271337 lambda = 0.11111; f = -1023.4258643 lambda = 0.037037; f = -1023.4867148 lambda = 0.012346; f = -1023.4932614 lambda = 0.0041152; f = -1023.4939724 lambda = 0.0013717; f = -1023.4940478 lambda = 0.00045725; f = -1023.4940551 lambda = 0.00015242; f = -1023.4940556 Norm of dx 0.00043959 ---- Improvement on iteration 3 = 0.000000150 ----------------- ----------------- f at the beginning of new iteration, -1023.4940555607 Predicted improvement: 0.000002508 lambda = 1; f = -1023.4902628 lambda = 0.33333; f = -1023.4936388 lambda = 0.11111; f = -1023.4940098 lambda = 0.037037; f = -1023.4940506 lambda = 0.012346; f = -1023.4940551 lambda = 0.0041152; f = -1023.4940555 lambda = 0.0013717; f = -1023.4940556 lambda = 0.00045725; f = -1023.4940556 Norm of dx 1.6316e-05 ---- Improvement on iteration 4 = 0.000000002 improvement < crit termination Objective function at mode: -1023.494056 MODE CHECK Fval obtained by the minimization routine: -1023.494056

by **jpfeifer** » Thu Oct 31, 2013 9:32 am

From the initial value at posterior, you can see that the algorithm started from different values. I don't really know why derivative-based solvers have such problems, apart from the likelihood obviously having local maxima. My conjecture is that the problem has to do with a finite sample bias in AR-estimation if the process is persistent (described in some Econometrica paper in the 1980s I think).

by **Niklas** » Tue Nov 05, 2013 9:32 am

Dear jpfeifer,

again, thank you very much!

I still don't know whether I got you right. My feeling is that specifying starting values or not does not matter here. In the example below I have exactly the same starting value for both cases (at the prior mean). In both cases, I arrive at a posterior mode which is approximately the same (up to the fourth digit). However, in the case where I run the identification command before, the hessian is positive definite. In the case where I do not run the identification command, the hessian is not positive definite.
It is as if running the identification command improves the outcome of csminwel. Do you have an idea why this might be the case?

One side question, how do you know/see that there are local maxima...?

All the best,
Niklas

Run with identification command:

Code: Select all: ==== Identification analysis ==== Testing prior mean All parameters are identified in the model (rank of H). All parameters are identified by J moments (rank of J) ==== Identification analysis completed ==== Loading 55 observations from data_mode_identification_interaction.xls Initial value of the log posterior (or likelihood): 567.0646 ----------------- ----------------- f at the beginning of new iteration, -567.0645821956 Predicted improvement: 9395.110504760 lambda = 1; f = -565.2443467 lambda = 0.33333; f = -566.8750856 lambda = 0.11111; f = -567.0475008 lambda = 0.037037; f = -567.0637320 lambda = 0.012346; f = -919.0936673 lambda = 0.023866; f = -567.0644588 lambda = 0.01607; f = -567.0645819 lambda = 0.012676; f = -932.4421725 lambda = 0.014615; f = -988.3942773 lambda = 0.016851; f = -567.0645799 lambda = 0.015472; f = -948.1812928 lambda = 0.016285; f = -567.0645816 lambda = 0.015792; f = -567.0645822 lambda = 0.015314; f = -964.2985128 lambda = 0.015599; f = -567.0645822 lambda = 0.015427; f = -953.3184351 Norm of dx 1.3708 Cliff. Perturbing search direction. Predicted improvement: 13747.664098287 lambda = 1; f = -563.0339956 lambda = 0.33333; f = -566.6355882 lambda = 0.11111; f = -567.0229223 lambda = 0.037037; f = -567.0616785 lambda = 0.012346; f = -567.0645580 lambda = 0.0041152; f = -703.1734998 lambda = 0.0079555; f = -886.2257451 lambda = 0.015379; f = -567.0644745 lambda = 0.010356; f = -567.0645784 lambda = 0.008168; f = -892.9637700 lambda = 0.0094179; f = -567.0645813 lambda = 0.0086467; f = -567.0645822 lambda = 0.0082147; f = -893.9354671 lambda = 0.0084712; f = -893.5366052 lambda = 0.0087358; f = -567.0645821 lambda = 0.0085761; f = -567.0645822 lambda = 0.0084193; f = -894.6499136 lambda = 0.008513; f = -892.1084386 lambda = 0.0086078; f = -567.0645822 Norm of dx 2.029 Cliff again. Try traversing Predicted improvement: 265890718.828363960 lambda = 1; f = 521488220.8781180 lambda = 0.33333; f = 57942447.4042859 lambda = 0.11111; f = 6437484.2678855 lambda = 0.037037; f = 714751.5104909 lambda = 0.012346; f = 78905.9567007 lambda = 0.0041152; f = 8260.9978827 lambda = 0.0013717; f = 413.0737335 lambda = 0.00045725; f = -458.4125987 lambda = 0.00015242; f = -555.0760002 lambda = 5.0805e-05; f = -565.7602403 lambda = 1.6935e-05; f = -566.9286655 lambda = 5.645e-06; f = -567.0522530 lambda = 1.8817e-06; f = -567.0639071 lambda = 6.2723e-07; f = -775.7610827 Norm of dx 23060 ---- Improvement on iteration 1 = 421.329695144 back and forth on step length never finished ----------------- ----------------- f at the beginning of new iteration, -988.3942773400 Predicted improvement: 19548.357215532 lambda = 1; f = -986.6880608 lambda = 0.33333; f = -988.2051971 lambda = 0.11111; f = -988.3734343 lambda = 0.037037; f = -988.3920162 lambda = 0.012346; f = -988.3940438 lambda = 0.0041152; f = -988.3942568 lambda = 0.0013717; f = -988.3942763 lambda = 0.00045725; f = -991.6319344 Norm of dx 1.9773 ---- Improvement on iteration 2 = 3.237657047 … Improvement on iteration 30 = 0.000000335 ----------------- ----------------- f at the beginning of new iteration, -1023.4940568420 Predicted improvement: 0.000000002 lambda = 1; f = -1023.4940568 Norm of dx 1.7586e-06 ---- Improvement on iteration 31 = 0.000000002 improvement < crit termination Objective function at mode: -1023.494057 MODE CHECK Fval obtained by the minimization routine: -1023.494057 RESULTS FROM POSTERIOR MAXIMIZATION parameters prior mean mode s.d. t-stat prior pstdev rho_einG_a 0.750 0.7958 0.0719 11.0761 beta 0.1000 rho_einG_b 0.750 0.8473 0.0636 13.3122 beta 0.1000 rho_etauw_a 0.750 0.9478 0.0208 45.5896 beta 0.1000 rho_etauw_b 0.750 0.9016 0.0378 23.8494 beta 0.1000 rho_eg_a 0.750 0.7940 0.0723 10.9888 beta 0.1000 rho_eg_b 0.750 0.9140 0.0400 22.8438 beta 0.1000 standard deviation of shocks prior mean mode s.d. t-stat prior pstdev nua_einG 0.010 0.0019 0.0002 10.7694 invg 2.0000 nub_einG 0.010 0.0017 0.0002 10.7511 invg 2.0000 nua_etauw 0.010 0.0017 0.0002 10.4729 invg 2.0000 nub_etauw 0.010 0.0013 0.0001 10.5941 invg 2.0000 nua_ecG 0.010 0.0020 0.0002 10.7690 invg 2.0000 nub_ecG 0.010 0.0013 0.0001 10.6270 invg 2.0000 Log data density [Laplace approximation] is 963.396285. Total computing time : 0h00m18s

Run without identification command:

Code: Select all: Loading 55 observations from data_mode_identification_interaction.xls Initial value of the log posterior (or likelihood): 567.0646 ----------------- ----------------- f at the beginning of new iteration, -567.0645821956 Predicted improvement: 9394.239308087 lambda = 1; f = -565.2445182 lambda = 0.33333; f = -566.8751041 lambda = 0.11111; f = -567.0475026 lambda = 0.037037; f = -567.0637321 lambda = 0.012346; f = -919.0702508 lambda = 0.023866; f = -567.0644589 lambda = 0.01607; f = -567.0645819 lambda = 0.012676; f = -932.4183817 lambda = 0.014615; f = -988.3939319 lambda = 0.016851; f = -567.0645799 lambda = 0.015472; f = -948.2675865 lambda = 0.016285; f = -567.0645816 lambda = 0.015792; f = -567.0645822 lambda = 0.015314; f = -964.3574522 lambda = 0.015599; f = -567.0645822 lambda = 0.015427; f = -953.3961886 Norm of dx 1.3707 Cliff. Perturbing search direction. Predicted improvement: 14893.717837427 lambda = 1; f = -562.2121069 lambda = 0.33333; f = -566.5458539 lambda = 0.11111; f = -567.0134807 lambda = 0.037037; f = -567.0608057 lambda = 0.012346; f = -567.0645224 lambda = 0.0041152; f = -718.5544360 lambda = 0.0079555; f = -921.7019034 lambda = 0.015379; f = -567.0643882 lambda = 0.010356; f = -567.0645680 lambda = 0.008168; f = -567.0645822 lambda = 0.0080187; f = -921.1032345 lambda = 0.008108; f = -918.2936891 lambda = 0.0081982; f = -567.0645822 Norm of dx 2.2239 Cliff again. Try traversing Predicted improvement: 102642132.493013650 lambda = 1; f = 203262438.5730386 lambda = 0.33333; f = 22584121.2386550 lambda = 0.11111; f = 2508812.8774944 lambda = 0.037037; f = 278242.9741065 lambda = 0.012346; f = 30408.5104982 lambda = 0.0041152; f = 2873.5601311 lambda = 0.0013717; f = -185.1413799 lambda = 0.00045725; f = -524.7512971 lambda = 0.00015242; f = -562.4037969 lambda = 5.0805e-05; f = -566.5600962 lambda = 1.6935e-05; f = -567.0128166 lambda = 5.645e-06; f = -567.0601198 lambda = 1.8817e-06; f = -567.0643905 lambda = 6.2723e-07; f = -670.9653593 lambda = 1.2125e-06; f = -567.0645618 lambda = 8.1646e-07; f = -710.3305928 lambda = 1.0351e-06; f = -567.0645779 lambda = 8.9775e-07; f = -567.0645820 lambda = 8.2424e-07; f = -710.4171907 lambda = 8.6759e-07; f = -567.0645822 lambda = 8.4132e-07; f = -708.8439203 lambda = 8.5698e-07; f = -704.1474898 lambda = 8.7295e-07; f = -567.0645822 lambda = 8.6333e-07; f = -567.0645822 lambda = 8.5383e-07; f = -705.4318732 Norm of dx 14328 ---- Improvement on iteration 1 = 421.329349745 back and forth on step length never finished ----------------- ----------------- f at the beginning of new iteration, -988.3939319404 Predicted improvement: 19510.472534793 lambda = 1; f = -986.6782540 lambda = 0.33333; f = -988.2038025 lambda = 0.11111; f = -988.3729731 lambda = 0.037037; f = -988.3916582 lambda = 0.012346; f = -988.3936971 lambda = 0.0041152; f = -988.3939112 lambda = 0.0013717; f = -988.3939309 lambda = 0.00045725; f = -991.6241260 Norm of dx 1.9754 ---- Improvement on iteration 2 = 3.230194084 … Improvement on iteration 31 = 0.000000247 back and forth on step length never finished ----------------- ----------------- f at the beginning of new iteration, -1023.4940295049 Predicted improvement: 0.000010945 lambda = 1; f = -1023.4940195 lambda = 0.33333; f = -1023.4940286 lambda = 0.11111; f = -1023.4940295 lambda = 0.037037; f = -1023.4940295 Norm of dx 8.0884e-05 ---- Improvement on iteration 32 = 0.000000022 improvement < crit termination Objective function at mode: -1023.494030 POSTERIOR KERNEL OPTIMIZATION PROBLEM! (minus) the hessian matrix at the "mode" is not positive definite! => posterior variance of the estimated parameters are not positive. You should try to change the initial values of the parameters using the estimated_params_init block, or use another optimization routine. Warning: The results below are most likely wrong! > In dynare_estimation_1 at 480 In dynare_estimation at 70 In mode_identification_interaction at 227 In dynare at 120 MODE CHECK Fval obtained by the minimization routine: -1023.494030 RESULTS FROM POSTERIOR MAXIMIZATION parameters prior mean mode s.d. t-stat prior pstdev rho_einG_a 0.750 0.7958 0.0718 11.0853 beta 0.1000 rho_einG_b 0.750 0.8473 0.0635 13.3491 beta 0.1000 rho_etauw_a 0.750 0.9478 0.0199 47.6245 beta 0.1000 rho_etauw_b 0.750 0.9017 0.0367 24.5749 beta 0.1000 rho_eg_a 0.750 0.7939 0.0722 10.9989 beta 0.1000 rho_eg_b 0.750 0.9140 0.0391 23.3698 beta 0.1000 standard deviation of shocks prior mean mode s.d. t-stat prior pstdev nua_einG 0.010 0.0019 0.0000 0.0000 invg 2.0000 nub_einG 0.010 0.0017 0.0000 0.0000 invg 2.0000 nua_etauw 0.010 0.0017 0.0000 0.0000 invg 2.0000 nub_etauw 0.010 0.0013 0.0000 0.0000 invg 2.0000 nua_ecG 0.010 0.0020 0.0000 0.0000 invg 2.0000 nub_ecG 0.010 0.0013 0.0000 0.0000 invg 2.0000 Log data density [Laplace approximation] is 965.284555. Total computing time : 0h00m12s

by **jpfeifer** » Tue Nov 05, 2013 9:39 am

Could you provide me with the code for this example. I have to look close into this. You might be up to something.

by **Niklas** » Tue Nov 05, 2013 8:20 pm

Yes, of course! Here you are:

by **jpfeifer** » Thu Nov 07, 2013 10:48 am

The reason is that identification turns on

Code: Select all: analytic_derivation

by **Niklas** » Thu Nov 07, 2013 4:00 pm

I understand. Thank you very much, jpfeifer! As always, you are a great help to me!
Best
Niklas

Dynare forums

Strange interaction b/w mode_compute = 4 and identification?

Strange interaction b/w mode_compute = 4 and identification?

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

Re: Strange interaction b/w mode_compute = 4 and identificat

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