The paper studies the implication of initial beliefs and associated confidence under adaptive learning. We first illustrate how prior beliefs determine learning dynamics and the evolution of endogenous variables in a small DSGE model with credit-constrained agents, in which rational expectations are replaced by constant-gain adaptive learning. We then examine how discretionary experimenting with new macroeconomic policies is affected by expectations that agents have in relation to these policies. More specifically, we show that a newly introduced macro-prudential policy that aims at making leverage counter-cyclical can lead to substantial increase in fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect information about the policy experiment.

1. Introduction Medium-scale model Estimation Conclusion
Estimating Financial Frictions under Learning
Patrick A. Pintus1
Jacek Suda2
Burak Turgut3
1
CNRS-InSHS and Aix-Marseille University
2
SGH and FAME|GRAPE
3
CASE - Center for Social and Economic Research
SCE 27th International Conference
Computing in Economics and Finance
June 16-18, 2021

2. Introduction Medium-scale model Estimation Conclusion
Financial Crisis and Expectations
Change in collateral values or constraints are important amplification
mechanisms of the shocks into the economy:
Kiyotaki and Moore (JPE, 1997) and Iacoviello (AER, 2005)
Iacoviello and Neri (AEJ, 2010), Jermann and Quadrini (AER, 2012), Liu, Wang,
Zha (Ecta, 2013), Berger, Guerrieri, Lorenzoni, Vavra (ReStud, 2018)
Housing demand shock is the key force driving the variation in output,
consumption and investment
The common assumption in all these models is that agents are fully rational.

3. Introduction Medium-scale model Estimation Conclusion
Rational Expectations in Reality?
Can agents have rational expectations ?
Rational expectations
require detailed knowledge concerning nature of equilibrium in the economy or
economic situation,
assume agents know
as much as the modeler,
more than the econometrician.
Recent literature
Coibion, Gorodnichenko, Kamdar, (JEL, 2018): Full-Information Rational
Expectations assumption is rejected in data
Alternative approaches:
noisy information (Woodford, 2002) and sticky information (Mankiw and Reis, QJE
2002)
bounded rationality (Sargent, 1999; Gabaix, QJE 2014), diagnostic expectations
(Gennaioli and Shleifer, QJE 2010), and adaptive learning (Evans and Honkapohja,
1999 and 2012)
DSGE models with adaptive learning fits better to data (Milani, JME 2007;
Slobodyan and Wouters, JEDC 2012)

4. Introduction Medium-scale model Estimation Conclusion
What we do
NK model with collateral constraint and housing:
extension of Iacoviello and Neri (2010):
alternative sources of shocks
Replace rational expectations (RE) with adaptive learning.
Calibrate/estimate the model using US data from 1975Q1-2008Q4 period.
Compare the responses under both RE and learning.

5. Introduction Medium-scale model Estimation Conclusion
What we find
In a medium scale model learning still matters:
estimated parameters different under learning than under RE,
better fit of data under learning.
Learning behaviour generates time-varying dynamics in beliefs
it can result in deviations from RE for the system under AL
The impact of collateral shock on the economy is larger under learning:
more persistent collateral process,
time-varying dynamics in beliefs.

6. Introduction Medium-scale model Estimation Conclusion
NK Model - overview
We use medium scale NK model with housing and collateral constraint
The model is the extension of Iacoviello and Neri (2010) with
stochastic collateral process
adaptive learning
The model features both supply of and demand for housing
We can consider both monetary and macroprudential polices

7. Introduction Medium-scale model Estimation Conclusion
Two types of households: Patient and Impatient. Housing and consumption
enter utility
Housing, wholesale, and retailers. Housing can be used as collateral for loans
(by impatient households)
Entire capital are owned and accumulated by patient households
Fluctuations in house prices affect borrowing capacity (LTV, mt)

8. Introduction Medium-scale model Estimation Conclusion
Impatient Households
Impatient household (borrower) maximize
max E∗
0
X
t=0
βb
t
ztu(cb
t , jt, hb
t , τt, nb
t )
where
E∗
0 denotes possibly non-rational expectations
ct denotes consumption of patient function (subject to internal habit formation)
ht is housing
nt is labor (to both housing and consumption sector)
zt, τt, jt are intertemporal, preferences, labor supply, housing shocks
subject to the budget constraint
cb
t + qthb
t − bb
t = wb
t nb
t + qt(1 − δh)hb
t−1 −
Rt−1bb
t−1
πt
+ Divb
t
and borrowing constraint
bb
t ≤ mtE∗
t
qt+1hb
t
Rt/πt
Exogenous collateral process:
mt = (1 − ρm)m + ρmmt + εm
t

9. Introduction Medium-scale model Estimation Conclusion
Patient Households
Patient household (lender) maximize
max E∗
0
X
t=0
βl
t
ztu(cl
t, jt, hl
t, τt, nl
t)
with βl
βb
, subject to the budget constraint
cl
t + qthl
t + kt + pl,tll
− bl
t = wl
tnl
t + qt(1 − δh)hl
t−1+
Rk,tkt + (pl,t + Rl,t)lt−1 −
Rt−1bl
t−1
πt
+ Divl
t − φt − a(zk,t)kt
where
lt is land and pl,t is its price
kt is capital (in each sector) and Rk,t is rate of return (net of depreciation)
φt, a(zt) are capital adjustment and utilization costs (depends on utilization, zt)
φt capital adjustment cost
Own productive capital
lend it impatient households
supply it to firms

10. Introduction Medium-scale model Estimation Conclusion
Production: housing sector and consumption wholesale good
New houses are produced with labor, business capital, land, and intermediate
inputs in the housing sector
IHt =
Ah,t(nl
h,t)α
(nb
h,t)1−α
1−µh−µb−µl
(zh,tkh,t−1)µh
kµb
b,t lµl
t
Consumption wholesale goods are produced with labor and business capital
Yt =
Ac,t(nl
c,t)α
(nb
c,t)1−α
1−µc
(zc,tkc,t−1)µc
Wholesale firms maximize profits
max
Yt
Xt
+ qtIHt −
X
i=c,h
X
j=b,l
wj
i,thj
i,t −
X
i=c,h
Ri,tzi,tki,t−1 − Rl,tlt−1 − pb,tkb,t

11. Introduction Medium-scale model Estimation Conclusion
Production: consumption good
Retailers purchase wholesale good, differentiate it costlessly and sell it at
markup
Only fraction 1 − θπ of retailers can set prices optimally, others index prices to
inflation with elasticity ιπ
CES aggregate of these goods converted to homogeneous consumption and
investment goods (by households)
Consumption Phillips curve summarize the above
ln πt − ιπ ln πt−1 = β(E∗
t ln πt+1 − ιπ ln πt) − π ln(Xt/X) + up,t
Analogous formulas for wage Philips curves

12. Introduction Medium-scale model Estimation Conclusion
Monetary policy and market clearing
Nominal interest rate is set according to Taylor rule
Rt = RrR
t−1π
(1−rR)rπ
t
GDPt
GCGDPt−1
(1−rR)rY
rr1−rR
uR,t
Market clears
Ct + IKc,t/Ak,t + IKh,t + kb,t + φt = Yt
and
Ht + (1 − δh)Ht−1 = IHt
and
bl
t + bb
t = 0, lt = 1

13. Introduction Medium-scale model Estimation Conclusion
Finding the linear solution
Compute the first-order conditions
Linearize FOCs and market clearing conditions around the balanced growth
Write down a linearized expectational system
Xt = AXt−1 + BE∗
t [Xt+1] + Cξt, (1)
all variables Xt are expressed in percentage deviations from the steady state:
A, B, C are matrices expressed as functions of parameters.
The solution of the above system
Xt = TXt−1 + Sξt, (2)

14. Introduction Medium-scale model Estimation Conclusion
Adaptive learning
Instead of holding rational expectations (E∗
t = Et) agents forms expectations in
adaptive way
Marcet and Sargent (1989), Evans and Honkapohja (2001)
Perceived law of motion corresponds to MSV REE:
Xt = M1,t−1Xt−1 + M2,t−1ξt = Mt−1
Xt−1
ξt
, (3)
Agents behave as econometricians
Mt = Mt−1 + νR−1
t Xt−1(Xt − M0
t−1Xt−1) (4)
Rt = Rt−1 + ν(Xt−1X0
t−1 − Rt−1), (5)
Implied ALM
[I − BM1,t−1]Xt = AXt−1 + [BM2,t−1 + C] ξt (6)
ALM combines learning dynamics with structural parameters:
Xt = TtXt−1 + Stξt (7)

15. Introduction Medium-scale model Estimation Conclusion
Estimation procedure
Cast the model in the state-space form
RE :Xt = TXt−1 + Sξt,
AL :Xt = TtXt−1 + Stξt
Estimate the model using Bayesian MCMC method
The behavior of matrix Mt s fully determined by the data and past values of
Mt−1
Mt = Mt−1 + νR−1
t Xt−1(Xt − M0
t−1Xt−1) (4)
Can use standard Kalman Filter
Adaptive learning introduces gain parameter, ν, and prior beliefs M0 and R0.
Prior beliefs M0 and R0: RE solution at every parameter draw
Gain parameter ν: Estimated

16. Introduction Medium-scale model Estimation Conclusion
Estimation procedure
Extended Iacoviello and Neri (2010) data for 1975Q1-2008Q4 and leverage
from Pintus and Suda (2019)
Data on
real consumption, real residential investment, real business investment, real house
prices, nominal interest rate, inflation, hours and wage inflation in the consumption
sector, hours and wage inflation in the housing sector
Data on debt from Boz and Mendoza (2014) and Pintus and Suda (2019)
bb
t ≤ mtE∗
t
qt+1hb
t
Rt/πt
Two MCMC chain. Use 100 000 draws in each chain and drop first 20 000
Report posterior for RE and learning models

17. Introduction Medium-scale model Estimation Conclusion
Data
Figure 1: Selected Data for the Estimation
0
0.2
0.4
0.6
0.8
1975Q1
1985Q1
1995Q1
2005Q1
Real Consumption
-0.2
0
0.2
0.4
0.6
0.8
1975Q1
1985Q1
1995Q1
2005Q1
Real Nonresidential Investment
-0.3
0
0.3
0.6
0.9
1.2
1975Q1
1985Q1
1995Q1
2005Q1
Real Residential Invetment
-0.02
-0.01
0
0.01
0.02
0.03
1975Q1
1985Q1
1995Q1
2005Q1
Inflation
-0.1
0
0.1
0.2
0.3
0.4
0.5
1975Q1
1985Q1
1995Q1
2005Q1
Real House Prices
-0.4
0
0.4
0.8
1.2
1.6
1975Q1
1985Q1
1995Q1
2005Q1
Real Debt

18. Introduction Medium-scale model Estimation Conclusion
Estimation: Learning vs rational expectation
Table 1: Posterior means of parameters.
Adaptive learning RE
Description Parameter Mean 5% 95% Mean 5% 95%
Sd. of productivity shock in consumption σAC 0.0079 0.0071 0.0087 0.0105 0.0094 0.0116
Sd. of monetary shock σe 0.0028 0.0024 0.0031 0.0031 0.0027 0.0036
Sd. of productivity shock in housing σAH 0.0192 0.0175 0.0210 0.0200 0.0179 0.0223
Sd. of housing preference shock σAK 0.0511 0.0400 0.0630 0.0712 0.0500 0.0964
Sd. of productivity shock in non-residential σj 0.0131 0.0118 0.0146 0.0135 0.0116 0.0156
Sd. of cost push-upshock σp 0.0057 0.0051 0.0065 0.0053 0.0044 0.0063
Sd. of inflationary shock σs 0.0338 0.0262 0.0416 0.0379 0.0295 0.0477
Sd. of labor shock στ 0.0505 0.0445 0.0559 0.0212 0.0171 0.0269
Sd. of intertemporal preference shock σz 0.0307 0.0275 0.0342 0.0376 0.0274 0.0514
Sd. of collateral shock σm 0.0148 0.0135 0.0162 0.0150 0.0133 0.0168
Noise in hours in housing σNH 0.1823 0.1745 0.1905 0.1798 0.1628 0.1998
Noise in wage in housing σWH 0.0050 0.0045 0.0055 0.0052 0.0047 0.0058
AR productivity shock in consumption ρAC 0.9900 0.9814 0.9963 0.9440 0.9172 0.9693
AR productivity shock in housing ρAH 0.9777 0.9685 0.9861 0.9948 0.9897 0.9982
AR housing preference shock ρj 0.9478 0.9305 0.9607 0.9406 0.9006 0.9687
AR productivity shock in nonresidental ρAK 0.9766 0.9657 0.9860 0.9489 0.9230 0.9710
AR labor preference shock ρr 0.9345 0.9171 0.9515 0.9690 0.9455 0.9875
AR intertemporal preference shock ρz 0.9935 0.9913 0.9957 0.9990 0.9982 0.9996
AR collateral shock ρm 0.9965 0.9924 0.9991 0.9898 0.9773 0.9973

19. Introduction Medium-scale model Estimation Conclusion
Estimation: Learning vs rational expectation
Table 2: Posterior means of parameters
Adaptive learning RE
Description Parameter Mean 5% 95% Mean 5% 95%
share of patient labor α 0.8328 0.8012 0.8653 0.6957 0.6277 0.7613
habit formation for patient c 0.4361 0.3789 0.4867 0.4085 0.2886 0.5567
habit formation for impatient c1 0.2748 0.2247 0.3312 0.2467 0.1564 0.3594
disutility of labor patient ηc 0.4442 0.3896 0.5019 0.3855 0.2685 0.5161
disutility of labor impatient ηc1 0.3003 0.2349 0.3659 0.5062 0.3583 0.6932
capital adjustment costs ψk 16.9630 15.1484 18.6752 18.9890 15.9926 22.1731
capital adjustment costs ψh 10.7668 8.8997 12.4786 11.3855 7.7469 15.7668
inflation indexation ιp 0.8165 0.6929 0.9452 0.7442 0.5649 0.9070
wage indexation in consumption sector ιw,c 0.1276 0.0555 0.1937 0.0854 0.0299 0.1549
wage indexation in housing sector ιw,h 0.3832 0.2925 0.4732 0.3310 0.1669 0.5078
disutility of labor patient ξ -1.0248 -1.1275 -0.9199 -1.1257 -1.2716 -0.9885
disutility of labor impatient ξ0
-1.0437 -1.1280 -0.9770 -0.9835 -1.1437 -0.8187
Taylor rule inflation feedback Rp 1.4339 1.3303 1.5459 1.5228 1.4053 1.6518
Taylor rule AR parameter Rr 0.6764 0.6357 0.7175 0.6252 0.5631 0.6800
Taylor rule output gap feedback RY 0.3774 0.3301 0.4269 0.3568 0.2634 0.4460
Fraction of price non-optimizers θ 0.6850 0.6573 0.7135 0.7993 0.7598 0.8388
Fraction of wage non-optimizers in consumption θw,c 0.9025 0.8879 0.9143 0.8435 0.8136 0.8725
Fraction of wage non-optimizers in housing θw,h 0.9619 0.9579 0.9659 0.9765 0.9704 0.9822
Trend in consumption γAC 0.0059 0.0054 0.0064 0.0051 0.0048 0.0055
Trend in housing γAK 0.0063 0.0057 0.0071 0.0079 0.0059 0.0100
Trend in nonresidential investment γAH -0.0022 -0.0028 -0.0014 -0.0011 -0.0016 -0.0006
capacity utilization curvature ζ 0.8650 0.7940 0.9326 0.8817 0.7775 0.9631
constant-gain parameter υ 0.0280 0.0253 0.0310
Log Likelihood 4.,271.8 4,236.6
Note: Parameters for both models estimated with 200k draws of MH algorithm.

20. Introduction Medium-scale model Estimation Conclusion
Estimation: Convergence
Figure 2: Distribution of parameters under AL

21. Introduction Medium-scale model Estimation Conclusion
Estimation: Distribution
Figure 3: Distribution of parameters under AL

22. Introduction Medium-scale model Estimation Conclusion
Estimation: Impulse Response Functions
Figure 4: IRFs to Housing Preference Shock with beliefs at 2008Q4
0 5 10 15 20
0
0.5
1
1.5
Consumption
0 5 10 15 20
-2
-1
0
1
2
3
Real Business Investment
0 5 10 15 20
-1
0
1
2
3
4
5
Real Residential Investment
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
Real House Prices
0 5 10 15 20
-0.2
-0.1
0
0.1
0.2
Nominal Interest Rate
0 5 10 15 20
0
0.5
1
1.5
GDP
RE
AL-Beliefs 2008Q4

23. Introduction Medium-scale model Estimation Conclusion
Estimation: Impulse Response Functions
Figure 5: IRFs to Negative Collateral Shock with beliefs at 2008Q4
0 5 10 15 20
-1.5
-1
-0.5
0
Consumption
0 5 10 15 20
-3
-2
-1
0
1
2
Real Business Investment
0 5 10 15 20
-5
-4
-3
-2
-1
0
1
Real Residential Investment
0 5 10 15 20
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Real House Prices
0 5 10 15 20
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Nominal Interest Rate
0 5 10 15 20
-1.5
-1
-0.5
0
GDP
RE
AL-Beliefs 2008Q4

24. Introduction Medium-scale model Estimation Conclusion
Conclusion
Use a DSGE model with collateral-constrained borrowing.
Replace rational expectations with adaptive learning.
Consider initial beliefs to differ from RE
Find that learning changes the properties of stochastic processes even in
medium scale model.
Next:
contribution of beliefs and parameters in IRFs and FEVDs
how learning matters for policy