# Problem with second order approximation to the value functio

Hi everyone,

I am evaluating the effects of the different monetary policies on the households’ and entrepreneurs’ value functions in my two sector model. To do this I ran the code by taking second approximations to the value functions. However, I have some problems with this.

1. It takes too long (1 hour on average) to run the code when I do the 2nd order approximations to the value functions compared to 14 seconds when I do the first order approximation.

2. For some values of m, the parameter in front of the borrowing constraint, dynare is not able to calculate autocorrelation, correlation, variance, or mean for the both value functions. However, I do not have any problem when I run the code with first order approximation of the variables. In any scenario the model converges to true steady state but there is only problem with var-covariance matrix and correlations and means.

I did some research and the only information I get as to why this problem might occur is that this is caused by non-stationarity of the
resulting decision rule. I do not need to know autocorrelation, var, covar, etc. of those variables that are approximated by second order, however the problem I am facing is weird anyway.

I am wondering if anyone had any experience with a situation similar to what I described above.

Any help is greatly appreciated.

This is how dynare output looks like ( ve: the value function of the entrepreneurs). At the very end you will see that the mean, std dev, etc. of the value function is not defined.

mu 0.84069
ce -0.84069
lambda -3.76448
gint 1.0101
ginf 1
q -0.0731779
y 1.30886
x 0.154151
ke 4.48104
l -0.253566
w 1.00779
be 4.11013
a 0
c 1.51492
k 2.45729
fk 0.245729
z 1.60553
pnew 0
pb 3.64439
pa 4.25179
xss 0.154151
yss 1.30882
kess 4.481
lss -0.2536
kss 2.45763
fkss 0.245763
zss 1.60551
ue -0.84069
ve -42.0345

EIGENVALUES:
Modulus Real Imaginary

``````  4.476e-016       4.476e-016                0
9.073e-016      -9.073e-016                0
8.628e-009      -2.192e-015       8.628e-009
8.628e-009      -2.192e-015      -8.628e-009
0.8063           0.8063                0
0.95             0.95                0
1.014            1.014                0
1.02             1.02       9.835e-009
1.02             1.02      -9.835e-009
1.347            1.347                0
1.417            1.253           0.6628
1.417            1.253          -0.6628
Inf              Inf                0
Inf              Inf                0
Inf              Inf                0
``````

There are 9 eigenvalue(s) larger than 1 in modulus
for 9 forward-looking variable(s)

The rank condition is verified.

MODEL SUMMARY

Number of variables: 29
Number of stochastic shocks: 2
Number of state variables: 6
Number of jumpers: 9
Number of static variables: 14

MATRIX OF COVARIANCE OF EXOGENOUS SHOCKS

Variables eps epsm
eps 0.000074 0.000000
epsm 0.000000 0.000074

POLICY AND TRANSITION FUNCTIONS
ke k l q fk y z be ginf
Constant 4.481044 2.457291 -0.253566 -0.073178 0.245729 1.308856 1.605528 4.110134 1.000000
gint (-1) -10.407361 78.749069 -1.780409 -3.015897 0 -1.192874 -0.886648 -12.744440 -0.294715
ke (-1) 14.565943 -110.215687 2.257207 4.349435 0 1.842329 1.369380 17.965318 0.348255
be (-1) -10.512486 79.544514 -1.798393 -3.046360 0 -1.204923 -0.895604 -12.873172 -0.297692
a (-1) 7.768978 -58.785298 1.403521 3.015120 0 1.890359 1.405081 10.284924 0.184154
k (-1) 0.038475 -0.291126 -0.024002 0.021590 0.100000 -0.016081 0.013718 0.057555 -0.004131
eps 8.177872 -61.879261 1.477391 3.173811 0 1.989852 1.479032 10.826236 0.193847
epsm -8.463208 64.038302 -2.464620 -2.958880 0 -1.651295 -1.227387 -10.870077 -0.486475

MOMENTS OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE MEAN STD. DEV. VARIANCE SKEWNESS KURTOSIS
ke 4.482616 0.171177 0.029302 0.000471 -0.003151
k 2.445400 1.295243 1.677654 -0.000471 -0.003151
l -0.253263 0.034630 0.001199 0.001145 0.005314
q -0.072554 0.072996 0.005328 -0.002562 0.009921
fk 0.244541 0.129524 0.016776 -0.000501 -0.003136
y 1.309670 0.092961 0.008642 -0.003887 0.008082
z 1.605829 0.039560 0.001565 -0.003938 0.027749
be 4.112228 0.231530 0.053606 -0.000583 -0.000144
ginf 1.000010 0.004659 0.000022 0.004559 0.019301

CORRELATION OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE ke k l q fk y z be ginf
ke 1.0000 -1.0000 0.9747 0.9679 -0.8064 0.8989 0.8923 0.9968 0.3763
k -1.0000 1.0000 -0.9747 -0.9679 0.8064 -0.8989 -0.8923 -0.9968 -0.3763
l 0.9747 -0.9747 1.0000 0.9394 -0.6984 0.8299 0.8626 0.9703 0.5403
q 0.9679 -0.9679 0.9394 1.0000 -0.8246 0.9562 0.9771 0.9849 0.2498
fk -0.8064 0.8064 -0.6984 -0.8246 1.0000 -0.9391 -0.7997 -0.8178 0.1889
y 0.8989 -0.8989 0.8299 0.9562 -0.9391 1.0000 0.9574 0.9234 -0.0166
z 0.8923 -0.8923 0.8626 0.9771 -0.7997 0.9574 1.0000 0.9255 0.1297
be 0.9968 -0.9968 0.9703 0.9849 -0.8178 0.9234 0.9255 1.0000 0.3387
ginf 0.3763 -0.3763 0.5403 0.2498 0.1889 -0.0166 0.1297 0.3387 1.0000

AUTOCORRELATION OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE 1 2 3 4 5
ke 0.8064 0.6519 0.5254 0.4248 0.3435
k 0.8064 0.6519 0.5254 0.4248 0.3435
l 0.6790 0.5544 0.4489 0.3682 0.3006
q 0.8577 0.7241 0.6134 0.5234 0.4493
fk 0.8064 0.6519 0.5254 0.4248 0.3435
y 0.9181 0.7734 0.6548 0.5576 0.4782
z 0.9036 0.7943 0.7019 0.6246 0.5592
be 0.8214 0.6723 0.5498 0.4519 0.3724
ginf -0.0047 0.0086 0.0001 0.0109 0.0095

MODEL SUMMARY

Number of variables: 29
Number of stochastic shocks: 2
Number of state variables: 6
Number of jumpers: 9
Number of static variables: 14

MATRIX OF COVARIANCE OF EXOGENOUS SHOCKS

Variables eps epsm
eps 0.000074 0.000000
epsm 0.000000 0.000074

POLICY AND TRANSITION FUNCTIONS
c ce ve
Constant 1.434289 2.094342 -51.854272
(correction) -0.080635 2.935031 -9.819794
gint (-1) 0.487892 -15.380624 -75.691695
ke (-1) -0.554401 21.654871 106.065089
be (-1) 0.492820 -15.535984 -76.456257
a (-1) 0.379116 12.223465 69.888743
k (-1) 0.008637 0.067301 0.290264
eps 0.399069 12.866805 73.567098
epsm -0.109618 -13.013807 -62.058432
gint (-1),gint (-1) -20.559636 -727.990614 -5629.976881
ke (-1),gint (-1) 59.814170 1973.443426 15441.829721
ke (-1),ke (-1) -43.791968 -1323.873504 -10524.461352
be (-1),gint (-1) -41.046726 -1486.068733 -11449.382364
be (-1),ke (-1) 60.418353 1993.377198 15597.807799
be (-1),be (-1) -20.730670 -750.539764 -5782.516345
a (-1),gint (-1) 30.126793 1084.831414 8400.879073
a (-1),ke (-1) -44.274777 -1453.598765 -11439.036274
a (-1),be (-1) 30.431104 1095.789307 8485.736437
a (-1),a (-1) -11.120454 -398.818317 -3107.991244
k (-1),gint (-1) 0.168063 5.855970 43.838244
k (-1),ke (-1) -0.245318 -7.786665 -59.352158
k (-1),be (-1) 0.169761 5.915121 44.281054
k (-1),a (-1) -0.126944 -4.308191 -32.427026
k (-1),k (-1) 0.000237 -0.004677 -0.053599
eps ,eps -12.321832 -441.903952 -3443.757611
epsm,eps 25.600008 920.495212 7152.865745
epsm,epsm -12.699319 -468.310016 -3663.541431
gint (-1),eps 31.712414 1141.927804 8843.030603
gint (-1),epsm -32.519740 -1181.739080 -9149.187516
ke (-1),eps -46.605029 -1530.103963 -12041.090815
ke (-1),epsm 48.019066 1588.430627 12481.020385
be (-1),eps 32.032741 1153.462428 8932.354145
be (-1),epsm -32.848222 -1193.675838 -9241.603552
a (-1),eps -23.411482 -839.617509 -6543.139461
a (-1),epsm 24.320008 874.470451 6795.222458
k (-1),eps -0.133625 -4.534938 -34.133711
k (-1),epsm 0.151772 5.029169 36.874001

MOMENTS OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE MEAN STD. DEV. VARIANCE SKEWNESS KURTOSIS

ve NaN NaN NaN NaN NaN

CORRELATION OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE ve

ve NaN

AUTOCORRELATION OF SIMULATED VARIABLES (HP filter, lambda = 1600)

VARIABLE 1 2 3 4 5
ve NaN NaN NaN NaN NaN

Total computing time : 1h11m40s

You don’t need to use
periods = …
to get the results that you want.

It is a well known problem that simulation of second order approximation can explode.

Best

Michel