Samba: Stochastic Analytical Model with a BayesianApproach

DSGE Model Project for Brazil�s economy

Working in Progress - Preliminary results

DSGE team: Solange Gouvea, André Minella, Rafael Santos,Nelson Souza-Sobrinho, and Tomiê Sugahara

Banco Central do Brasil �Research DepartmentX Seminar on In�ation Targeting

August 4th, 2008

Outline

� Introduction

� Model

� Estimation results

� Challenges and next steps

Purposes of the project

� Provide the Banco Central do Brasil with a Dynamic Stochastic General Equi-librium (DSGE) model to be used as a tool for:

� policy analysis

� framework for policy discussions; qualitative and quantitative assessmentof shock e¤ects, monetary policy decisions and di¤erent scenarios, etc.

� medium-term forecast

� �All models are wrong! Some are useful�George Box

� Models and judgement are complements, not substitutes

Model features

� Microfounded model developed for the in�ation targeting period (started inmid-1999)

� Small open economy model

� Aggregate demand (C + I + G + X - M):

� Households -> private consumption and investment

� Firms -> import demand

� Government -> government consumption

� Rest of the world -> export demand

Model features

� Supply side (Y)

� Competitive �rms -> assemble di¤erentiated goods supplied by monopo-listic competitive �rms and sell them in

� Local markets (domestic consumption and investment goods)

� Abroad (export goods)

� Monopolistic competitive �rms -> production of di¤erentiated goods

� Inputs: labor, capital services, and imports

� Price rigidity (à la Calvo) with forward- and backward-looking behavior(Galí and Gertler, 1999)

Model features

� Government:

� Monetary policy: Taylor rule

� Fiscal policy rule

� Rest-of-the-world variables: interest rate, in�ation, world imports, and foreigninvestors�"risk aversion".

Main loglinear equations

� Aggregate Demand: Consumption

� Optimizing households

cot =�

1

1 + h

�Et�cot+1

�+�

h

1 + h

�cot�1 �

1

�

�1� h

1 + h

�Et (rt � �t+1) +

:::+1

�

�1� h

1 + h

�(1� �c) z

ct

� Rule-of-thumb households

crott = wrt + nrott

� Aggregate consumption

ct = (1�$c) cot +$cc

rott

rt - interest rate; �t - in�ation; zct - shock to consumption;wrt - real wages; n

rott - employment

� Aggregate Demand: Investment:

it =1

�s (1 + �)qIt +

�

1 + �Etit+1 +

1

1 + �it�1 +

1� �I�

1 + �

!zIt

Shadow price of capital

qIt = Etn� (1� �) qIt+1 + (1� �(1� �)) brkt+1 � (rt � �t+1)

o

� Aggregate Demand: Net Exports

� Exports

xt = m�t + {qt

� Imports

mt = yt � % (qt �mct)

brkt - rental rate of capital; zIt - shock to investment; m�t - world imports;qt - real exchange rate; yt - (gross) output; mct - real marginal cost

� Aggregate Supply

� Production function

yt = f (kt; ut; nt;mt; at)

� Labor market

� Labor supply

nt = (1�$n)not +$nn

rott

� Labor demand

nt = yt � [(1� %) + %sd] at � [�+ % (1 + sd) (1� �)]wrt +

+� [1� % (1� sd)] rkt + %(1� sd)qt

kt - physical capital; ut - rate of capital utilization; at - productivity shock

� Capital services

� Demand

kt + ut = yt � [(1� % (1� sd)] at � [(1� �) + �% (1� sd)] brkt +:::+ (1� �) [(1� % (1� sd)]w

rt + %(1� sd)qt

� Supply

ut =1

�abrkt

� Law of motion for capital

kt+1 = (1� �)kt +�I

K

�it

� Phillips curve

�t = �mct + �b�t�1 + �fEt�t+1

where:

mct = sdh�brkt + (1� �)wrt � at

i+ (1� sd)qt

��; �b; �f

�= f (�;$b; �)

� Financial variables

� Real exchange rate (UIP)

qt = Etqt+1 �h�rt � Et�t+1)� (r�t + �t � Et�

�t+1

�i� Country-risk premium

�t = � by�t+1 + �z

��t + z

�t

r�t - world interest rate; ��t - world in�ation;

z��t - international investors�risk averstion; z�t - shock to country-risk premium

� Government

� Monetary policy (Taylor rule)

rt = rrt�1 + (1� r)h �Et (�t+1 � �t+1) + �t + yy

V At

i+ zrt

� Fiscal policy rule

gyt = gg

yt�1 +

�1� g

� � sbsyt�1 � bb

yt

�+ z

gt

�t - in�ation target; zrt - shock to monetary policy;gyt - government consumption-to-GDP ratio; s

yt - primary �scal surplus target;

zgt - shock to �scal policy; bsyt�1 - primary �scal surplus deviation from the target

� Shocks and rest-of-the-world variables:

zt = �zt�1 + "t

� Value added (GDP) - Equilibrium:

yV At = scct + siit + sggt + sxxt � smmt

Estimation technique

� Bayesian estimation:

Estimated parameter distribution = prior distribution + likelihood information fromthe data

It is a bridge between calibration and maximum likelihood

Results: Model + Data + Priors

Estimation

� Sample period: 1999Q2 to 2008Q1 (36 obs)

� Data: 25 series:

� Data treatment: HP �lter

� Number of model parameters: 58

� 41 estimated: 17 structural parameters and 24 shock parameters

� 17 calibrated: 3 structural parameters and 14 steady-state relationships

Posteriors distributions for selected parameters

0 0.2 0.4 0.6 0.80

2

h

0 0.2 0.4 0.6 0.80

2

h

0.4 0.6 0.8 10

10

θ

0.4 0.6 0.8 10

10

θ

1 1.5 20

1

2

γπ

1 1.5 20

1

2

γπ

Impulse responses to a consumption shock

Impulse responses to a monetary policy shock

Impulse responses to a world GDP shock

Challenges

� Common to DSGE models and their estimation:

� Generation of slower and more persistent dynamics (enough propagationmechanisms, lags in the transmission mechanisms, etc.)

� Identi�cation of the main model channels in place

� Large number of parameters to be estimated �calibration versus estimation

Challenges

� Brazilian economic features:

� Small sample size

� Speci�c features: administered prices

� Large changes in some ratios over the sample (ex.: net external debt-to-GDP ratio)

Next steps

� Re�ning model setup:

� Add nominal and real rigidities: wage rigidity, price rigidity in the importand export sectors, �rm-speci�c capital

� Disaggregate CPI in�ation into administered and non-administered prices

� New estimation and model implementation

Thank you for your attention!