An Adaptive Expectations Approach to the Mechanisms of
Transmission Model of the Central Bank of Colombia
Fernando Arenas
Pontificia Universidad Javeriana
farenas @ puj.edu.co
Franz Hamann
Banco de la Republica
fhamansa @banrep.gov.co
ABSTRACT
Looking for the potential applications of system dynamics in macroeconomic
modeling at the Central Bank of Colombia, the Mechanisms of Transmission
Model (MTM) was recast in a system dynamics model. The forward-looking
function of the model that, in the case of the MTM is a rational expectations
based function, was approached by means of the TREND function. This
document describes the system dynamics model and shows comparative
impulse-response results between the models, when PULSE and STEP shocks
are applied to inflation target, monetary policy, food supply, nominal
depreciation rate, and risk premium.
Introduction
This document is the preliminary result of collaborative work made by Pontificia
Universidad Javeriana and the Central Bank of Colombia, as part of a research project on
potential applications of system dynamics to macroeconomic modeling at the Bank. The
Mechanisms of Transmission Model (MTM) of the Bank was chosen to be “recast” as a
system dynamics model due to the presence of feedback relationships between variables,
and the presence of autoregressive components that can be described by means of stock and
flow variables. The MTM has been recently developed, and is intended to be used as a
policy design model, specifically for evaluating the effectiveness of changes in the
inflation target and the monetary policy by the Bank, and what the Bank should do in the
event of external shocks to food supply, nominal depreciation rate and risk premium. The
Bank has become interested in system dynamics models because of the transparency of
their structure which facilitates public discussion, the possibility to incorporate stocks, and
their powerful graphic interface. The main goal of this first part of the research was to find
out if, by means of a system dynamics model, it was possible to obtain results similar to
those of the MTM. Although the MTM uses a forward-looking function based on rational
expectations, an adaptive expectations approach was used for the forward-looking function
of the system dynamics model, in order to faster reach preliminary results that could
provide ideas on how to proceed further with the research. Several commonly used
forward-looking functions, such as moving average, exponential smoothing and Holt’s
method, were tested. The best results were obtained with the TREND function (Sterman,
1987) and, therefore, this was the one used in the system dynamics model.The first part of
the document describes the Bank’s model and, in a parallel way, how the system dynamics
model was developed, and the second part presents comparative results between the models
when disturbed by a shock applied to inflation target, monetary policy, food supply,
nominal depreciation rate and risk premium.
The Mechanisms of Transmission Model
Currently, the Central bank of Colombia (Banco de la Reptblica) is managing its monetary
policy following an operative strategy known as inflation target. This strategy consists,
basically, of controlling the interest rate in such a way, that the inflation forecast be aligned
with the inflation target. If the inflation forecast is higher than the target, the Central Bank
will tend to increase interest rates. This increase in interest rates will activate the diverse
channels of transmission of monetary policy, driving inflation to the target. These channels
are the aggregate demand channel, the direct and indirect exchange rate channels, and the
expectations channel.
Aggregate demand channel
This channel transmits monetary policy to the inflationary pressures arising from aggregate
demand. Figure 1 shows an influence diagram of this channel. If the real interest rate
increases, GDP decreases, increasing GDP gap, and reducing demand and inflation.
potential GDP
+
GDP gap inflation
real interest -
rate core aggregate
ig - demand
Figure 1. Aggregate demand channel.
Exchange rate direct channel
This channel transmits the pass-through effect of variations in exchange rate on domestic
prices. Figure 2 shows an influence diagram of this channel. An increase in domestic
interest rates improves the profitability of investment inside the country compared with that
of investment outside the country. In a floating exchange rate system, the exchange rate
tends to appreciate. For a given level of long term exchange rate, this appreciation produces
an increase in expected devaluation that equals the profitability of investment inside and
outside the country.
The exchange rate appreciation reduces inflation of imports. This is the first stage of the
direct or pass-through channel. In its turn, the lower inflation of imports reduces inflation of
CPI. This is the second stage of pass-through.
profitability of
investment out of the
i +
country 3 . —. ,
imports CPI inflation
le La inflation
interest rate - gap
profitability of E *
——s investment in the
+ country i
expected
devaluation exchange tale
Es long term
exchange rate
exchange rate ee
gap +
Figure 2. Pass-through channel.
Exchange rate indirect channel
In the description of the pass-through channel, it was shown how an increase in interest rate
produces an appreciation in the nominal exchange rate. This nominal appreciation results in
a real appreciation that tends to reduce the competitiveness of exports and increase that of
imports and, as a consequence, reduces net exports. This causes a reduction in domestic
demand that tends to decrease GDP. Such decrease has an impact on inflation that depends
on how much is the gap between current GDP level and its potential level. Figure 3 shows
an influence diagram for this channel.
Expectations channel
By this channel, a decrease in inflation expectations tends to reduce the inflation itself. In
its turn, inflation expectations may depend on recent results of inflation, the performance of
exchange rate, and the monetary policy position.
interest rate————_» nominal —___» real exchange
- exchange rate Fra rate
( _
exports imports
competitiveness competitiveness
potential
: 4s GDP. * /
inflation, <«——— GDP- gap ‘ i A
Be domestic
b % 4 het exports
GDP.<—— cemans
Figure 3. The exchange rate indirect channel.
Model equations and system dynamics model
The inflation without food is modeled by means of a Phillips curve augmented by
expectations:
TE, 2 oT 1 + OM + (1 Oo OW" 1-8) + OY te (Eq. I)
where 1, denotes the inflation without food, 1°, the inflation expectations, 1, the
imports inflation, 8”, the equilibrium real depreciation rate, ¥; the product gap, and e", the
residual.
Since residuals were not used in the system dynamics model, they will be omitted in the
rest of the document. Figure 4 shows the system dynamics structure and equation for
inflation without food.
<inflation ao al a3
expectations> 1 te equlibrium real
“~*~ inflation depreciation rate
without food a
a
<inflationwo <GDP gap
food t-1> t-1>
; <TIME
imports Ss STEP>
inflation t-1 chg imports Pe
\ inflation t-1
inflation without food= a0*"inflation wo food t-1"+a1*("imports inflation t-1"-equlibrium
real depreciation rate)+(1-a0-al1)*inflation expectations+a3*"GDP gap t-1"
Figure 4. Structure and equation for inflation without food
The measurement of expectations used in the model is obtained by the equation:
Te, = OgMyr1+ OsE M41 (Eq. 2)
where 74; denotes the annualized quarterly inflation.
As can be seen the equation includes a backward-looking process (0474-1) and a forward-
looking process (Qs5E,%4141). In the Bank’s model, the forward-looking process is
approached by using the generalized Schur form to solve a multivariate linear rational
expectations model (Klein, 2000). In the system dynamics model, the TREND function
(Sterman, 1987) was used as an adaptive expectations approach. The input to the TREND
function in the model is the current inflation, and the output (Perceived Trend) is used to
calculate the change in expected inflation and, consequently, the expected inflation. The
TREND function parameters were set using the Vensim optimization function defining the
payoff function as a calibration against the results of inflation obtained with the Bank’s
model when a shock to the inflation target is applied, and setting one quarter (0.25 year) as
the minimum value for the parameters. The result obtained was of 0.25 year for each of the
three parameters (Time to Perceive Present Condition, Time Horizon for Reference
Condition and Time to Perceive Trend). The structure for the TREND function in the
system dynamics model is shown in Figure 5.
Reference
Change eR ference Condition
Condi fot Be
Time to
Perceive Trend
Time Horizon for
Reference Condition
SS coe ys Trend Trend
Change in
f ed
Indicated Perceived
Time to Perceive
Present Condition
Trend gap
Perceived
Chafige in Present expected
PPC Condition inflation
a ge chg in expected
inflation
Perceived Present
Condition gap
<total quarterly
inflation>
Figure 5. The TREND function.
The equations for the TREND function are as follow:
Perceived Present Condition= INTEG (Change in PPC, 0.03)
Perceived Present Condition gap= total quarterly inflation-Perceived Present Condition
Change in PPC= Perceived Present Condition gap/Time to Perceive Present Condition
Time to Perceive Present Condition= 0.25
Reference Condition= INTEG (Change in Reference Condition, 0.03)
Change in Reference Condition= (Perceived Present Condition-Reference Condition)/Time
Horizon for Reference Condition
Time Horizon for Reference Condition= 0.25
Indicated Trend= ((Perceived Present Condition-Reference Condition)/Reference
Condition)/Time Horizon for Reference Condition
Perceived Trend= INTEG (Change in Trend, 0)
Trend gap= Indicated Trend-Perceived Trend
Change in Trend= Trend gap/Time to Perceive Trend
Time to Perceive Trend= 0.25
chg in expected inflation= expected inflation*Perceived Trend
expected inflation= INTEG (chg in expected inflation, 0.03)
The imports inflation is modeled with a partial adjustment equation:
TW", = bon. .1 + (1-bo)( 7. + 8) (Eq. 3)
*
where TC ; denotes the foreign inflation and 6, the nominal depreciation. Figure 6 shows the
structure and equation for foreign inflation.
imports [age <TIME
inflation t-1 chg imports | STEP>
is ta
imports
po inflation foreign
inflation
nominal
> epreciation
~
imports inflation= b0*"imports inflation t-1"+(1-b0)*(nominal depreciation + foreign
inflation)
Figure 6. Structure and equation of imports inflation
Food inflation is modeled with a Phillips curve:
T, = CoM, + CW. + Co(U", - 8.) +03 ¥1 (Eq. 4)
The Vensim equation for food inflation is:
food inflation= c1*"food inflation t-1"+c2*(imports inflation - equlibrium real
depreciation rate)+cO*inflation without food+c3*GDP gap + food supply shock*food
supply shock switch
The total inflation is calculated as the weighted sum of inflation without food and food
inflation. The weighting factors correspond to the participation of each component in the
Consumer Price Index:
Ty = Osa, + (1 ~ Osa) Wr (Eq. 5)
The Vensim equation for total inflation is:
total quarterly inflation= inflation weighting factor*inflation without food+(1-inflation
weighting factor)*food inflation
The product gap is modeled with an IS curve:
Ye = oY 1 — dif t doz.1 (Eq. 6)
where r; denotes the real interest rate, 1, the long term real interest rate, f, the real interest
rate gap, Z , the real exchange rate, Z ,, the long term real exchange rate, and Z , the real
exchange rate gap. Figure 7 shows the structure and equation of product gap.
we do di
Lf® GDP gap
GDP ga chg GDP gap tl
long term real tl
interest rate F Teal _ TIME
lepreciation| ae sreps
real interest gap 1 4 oti) real a
rc , lepreciation gap t-
eee, real interest f0
rate gap t-1 J
real depreciation
real interest <TIME gap
fate STEP> P4 Ss
<real exchange rate <equlibrium real
depreciation> depreciation rate>
GDP gap= d0*"GDP gap t-1"-d1*"real interest rate gap t-1"+d2*"real depreciation gap
tl"
Figure 7. Structure and equation of GDP gap.
The transmission of interest rates is modeled with the equations:
P.= eg Pt (1- eo)( Pic + er un- Wa) +291 (Eg. 9)
P.i7,-i, (Eq. 10)
h= gohitgi M+ gPu (Eq)
Pr=iy-c (Eq. 12)
iti, (Eq.13)
in=mtay (Eq. 14)
where i?, denotes the policy interest rate, i°), the long term policy interest rate, 1°14, the
are the inflation target in the period t+k (k=1 for
expected inflation in the period t+k, 7
the system dynamics model), i, the policy interest rate gap, i, the nominal interest rate gap,
1; the long term nominal interest rate, C a constant, i, the nominal interest rate, and Tae
the inflation without food annualized. Figure 8 shows the structure of interest rates
transmission and Vensim equations for policy interest rate and nominal interest rate gap.
~ /
<inflation without
<long term real i
ees ‘omina e_ food annualized> <TIME
: - STEP>
constant ¢ ; interest rate 2
long term policy «——— jong term nominal gap tL chg nominal
interest rate
interest rate { interest rate gap t1
inflation target ©0
- tk Bh: of policy interest nominal interest _____—g2
Af rate gap fale Bip
ep
policy interest le 1
rae 8
Ts 20
<expected <GDP gap>
inflation t+k>
policy interest]
rate gap t-1
~ policy chg policy interest
interest rate rate gap t-1
chg policy cl 3 Maa
interest rate t-1 <TIME
STEP>
policy interest rate= e0*"policy interest rate t-1"+(1-e0)*(long term policy interest
ratet+e1*("expected inflation t+k" -"inflation target t+k")+e2*GDP gap)
Figure 8. Structure of interest rates transmission.
Depreciation and risk premium are determined in the following equations:
6 =i-i*,- iP", (Eq. 15)
men yo PPM + (1 - yo) PP (Eq. 16)
6.=84i"%-0", (Eq. 17)
Z=Ziit f(6- Ov (Eq. 18)
where 5°, denotes the expected nominal depreciation, i*, the foreign interest rate, i”,
the risk premium, i?" the long term risk premium, 6”, the real exchange rate
depreciation, and 5’, the long term real exchange rate depreciation. Figure 9 shows the
structure of depreciation and risk premium.
forci exthange SME
foreign real exchange rate change J. — STEP>
_— inflation’ depreciation rate gap t-1] chg real exchange STEP?
nominal ed! rate gap t-1 6
depreciation, <inflation foreign interest
without food> rate
expeuea real exchange
depreciation 4 rate gap
exedkexiiervratn <equlibrium real
long term nominal "ae cial é rate
premium risk interest rate Precia depreciation>
Figure 9. Structure of depreciation and risk premium.
Impulse-response results
Diverse shocks were applied to the system dynamics model and the results compared with
those of the MTM. The comparison is made between Vensim custom graphs for the system
dynamics model, and graphs of the MTM results, scanned from a document of the Central
Bank (Banco de la Reptiblica, 2003).
Inflation target
A reduction from 3% to 2% in the inflation target was set by means of a STEP function in
year 1. Figure 10 shows the comparative results for inflation (total, without food, and food),
Figure 11 the results for imports inflation and depreciation (nominal and real), Figure 12
the results for GDP gap and real interest rate gap, and Figure 13 the results for policy and
real interest rates.
Inflation
0.08 003
—— inflacién Total
0.027 0.028 —— beflacién cin Alimentos
— Infecién de Alimentos.
02a 02
021 on
0.018 022
o1r?s 4s 678 9 21 as
‘Time (Year) pe
total inflation : base v4e ;
inlaion wiltou food : base we gigi i i i
food inflation base vse ———— Zar aaa aT tT
Figure 10. Results for inflation after a shock to inflation target, for the SD model (left) and
the MTM (right). Notation of the right-hand graph must be read as year-quarter (for
example, 2Q1 must be read as second year-first quarter).
Depreciation
0.08
0.0625
0.045
0.0275
01
012345678 90 2B 4 15
Time (Year)
imports inflation : bass
nominal depreciation : base ve
real exchange rate depreciation : base v4e
Figure 11. Results for imports inflation and depreciation (nominal and real) after a shock to
inflation target, for the SD model (left) and the MTM (right). Notation of the right-hand
graph must be read as year-quarter (for example, 2Q1 must be read as second year-first
quarter).
GDP and real interest rate gap
0.028 001
fe Pi
aon }—-f Ne ‘ay
/
/
ob +b
0012s NPR
0.025
= Brecha dei Producio
== Brecha Tasa de interes Real
or23a45 678 9 WN 121 M15
‘Time (Year)
i H i
201 4Q1 601 eat 1001
GDP gap + base y4¢§ —$$$_$_
real imerest rate gap ‘base ve. 404
Figure 12. Results for GDP and real interest rate gap after a shock to inflation target, for
the SD model (left) and the MTM (right). Notation of the right-hand graph must be read as
year-quarter (for example, 2Q1 must be read as second year-first quarter).
Interest rates
0.095 O08 —+ 7
C —
08s Al ons
fi poss.
oors | ff ow
|
oss |_|
/ os
0055
or? 3 4s 678 9 RD wis O06
Time (Year)
policy ineres rate: base wie, ga i : H
teal inerest ae: base e—— 57 BS 1. aa 7o0T
Figure 13. Results for policy and real interest rates after a shock to inflation target, for the
SD model (left) and the MTM (right). Notation of the right-hand graph must be read as
year-quarter (for example, 2Q1 must be read as second year-first quarter).
Monetary Policy
The policy interest rate was increased 100 basic points during 4 quarters, beginning in year
1, by means of the Vensim RC STEP function. Figure 14 shows the comparative results for
inflation (total, without food, and food), Figure 15 the results for imports inflation and
depreciation (nominal and real), Figure 16 the results for GDP gap and real interest rate
gap, and Figure 17 the results for policy and real interest rates.
Inflation
0.0305,
0.0297
0.029
0.0282
0.0275,
012345678 9 ON 2 4 15
Time (Year)
{otal quarterly inflation : monetary policy shock ——— Wlacion de Aimentos |
inflation without food : monetary policy shock elt
food inflation : monetary policy shock, 201 401 601 801 1001
Figure 14. Results for inflation after a shock to monetary policy, for the SD model (left)
and the MTM (right). Notation of the right-hand graph must be read as year-quarter (for
example, 2Q1 must be read as second year-first quarter).
Depreciation
015
or2s 4s 678 9 WD WIS
‘Time (Year) oo ledlacion de Imporiados:
import inflation corrected : monetary policy shock Hectesscarel ga
nominal depreciation : monetary policy shock, ————————————— 20s i =
teal exchnge rate depreciation monly policy shock oa ee te
Figure 15. Results for imports inflation and depreciation after a shock to monetary policy,
for the SD model (left) and the MTM (right). Notation of the right-hand graph must be read
as year-quarter (for example, 2Q1 must be read as second year-first quarter).
GDP & Real Interest Rate Gap
0.015
0.0087
0.0025
0.0037
-0.01
ote 3s +s 6 7 8 9 eas
Tine (Yea
GDP sup: monetary poicy shock —————————————— gy Lit i i i i
Sa ——— eer eer aa)
Figure 16. Results for GDP and real interest rate gaps after a shock to monetary policy, for
the SD model (left) and the MTM (right). Notation of the right-hand graph must be read as
year-quarter (for example, 2Q1 must be read as second year-first quarter).
Interest Rates
0.075 0015,
one | Ooty
\ OMS
00562 XC 427 Sanne oF
2 3 4
0.005}.
0.05
o 1 36 7 8 9 Wi 12 13 14 15
‘Time (Year) 0.01
policy interest rate: monetary policy shock gy]
real interest rate : monetary policy shock
Figure 17. Results for policy and real interest rates after a shock to monetary policy, for the
SD model (left) and the MTM (right). Notation of the right-hand graph must be read as
year-quarter (for example, 2Q1 must be read as second year-first quarter).
Food Supply
A pulse increment of 0.25 was made to food inflation, beginning in year 1. Figure 18 shows
the comparative results for inflation.
Inflation - food shock
02s
025
] — infacién Total \|
os Hh a2, — Inftacién sin Amentos
ms TY \ 01s — Infiacitn de Akmantos
{\
00s [fA ——o| ott.
0.08 0.05
ols 8
or23a45 678 9 WD 4 1S aos.
‘Time (Year) .
ttl qual ination : foo nation sek, ————————_04f--4- dy
{ood inflation: food inflation shock, 18 307 701 tor sai 100
Figure 18. Results for inflation after a shock to food supply, for the SD model (left) and the
MTM (right). Notation of the right-hand graph must be read as year-quarter (for example,
2Q1 must be read as second year-first quarter).
Nominal Depreciation
The risk premium was increased 100 basic points during 4 quarters, beginning in year 1, by
means of the Vensim RC STEP function. Figure 19 shows the comparative results for
inflation.
Inflation - dep shock
0.0302 %
— Iniacién Total
— lnfiacién tin Alimentos
[= Inllacion de Alimentos
0.0301
0.0300
0.0299
0.0299
o123 45678 9 Wl 2 13 14 Is
‘Time (Year)
total quarterly inflation : nom dep shock i
inflation without food: wom dep sbock fj
i i i
food inflation : nom dep shoek $5 $$@§$€@—~——§!—_—— 201 4al aI bai
Figure 19. Results for inflation after a shock to nominal depreciation, for the SD model
(left) and the MTM (right). Notation of the right-hand graph must be read as year-quarter
(for example, 2Q1 must be read as second year-first quarter).
Risk Premium
A pulse increase of 0.01 was made to the risk premium, beginning in year 1. Figure 20
shows the comparative results for inflation.
Inflation - premium shock Pt
0.0305 8
= Wnltacign Total
— Inflaciéa ain Aiementos
0.03 —— Inftacién de Alimertos
0.0295
0.029
0.0285
or? 3 45678 WN RBIS gl
‘Time (Year)
total quarterly inflation : premium shock ; i i
inflation without food : premium shock qu —— i
food inflation : premium shoe, 201 4a) 601 Bat 001
Figure 20. Results for inflation after a shock to risk premium, for the SD model (left) and
the MTM (right). Notation of the right-hand graph must be read as year-quarter (for
example, 2Q1 must be read as second year-first quarter).
Conclusions
The pattern of performance the variables in the SD model, for shocks to the inflation target
and monetary policy, is quite similar to that of the MTM, but, looking carefully at the
MTM results in Figures 11 and 15, a downward jump of depreciation in year 1 can be
noticed. This jump is a consequence of the solution of the rational expectations component
of the MTM. This performance makes a noticeable difference between the SD and MTM
results when shocks are applied to food supply, nominal depreciation, and risk premium. In
spite of the differences, the TREND function seems to be an appropriate adaptive
expectations approach to this model.
Based on the results obtained, the Bank considers that system dynamics models represent a
potentially useful tool for policy design and will continue supporting the research that has
originated this document.
In the specific case of the MTM, the Bank considers the adaptive expectations approach as
interesting but too deterministic, and will support a rational expectations approach to the
model, from the system dynamics perspective.
References
Banco de la Reptiblica - Departamento de Modelos Macroeconémicos. 2003. El Modelo de
Mecanismos de Transmisién. Documento interno.
Klein P. 2000. Using the generalized Schur form to solve a multivariate linear rational
expectations model. Journal of Economic Dynamics and Control 24: 1405 — 1423.
Sterman J. 1987. Expectation formation in behavioral simulation models. Behavioral
Science 32: 190 - 211
