Escola de P os-Gradua˘c~ao em Economia - EPGE Funda˘c~ao ...

Escola de Pos-Graduacao em Economia - EPGE

Fundacao Getulio Vargas

Ensaios em Econometria Aplicada

Tese de submetida a Escola de Pos-Graduacao em Economia da Fundacao

Getulio Vargas como quesito para a obtencao Tıtulo de Doutor em Economia.

Aluno: Rafael Martins de Souza

Orientador: Joao Victor Issler

Rio de Janeiro

2009

Escola de Pos-Graduacao em Economia - EPGE

Fundacao Getulio Vargas

Ensaios em Econometria Aplicada

Tese de submetida a Escola de Pos-Graduacao em Economia da Fundacao

Getulio Vargas como quesito para a obtencao Tıtulo de Doutor em Economia.

Aluno: Rafael Martins de Souza

Banca Examinadora:

Joao Victor Issler (Orientador, EPGE/FGV)

Prof. Marco Antonio Bonomo (EPGE/FGV)

Caio Ibsen Rodrigues de Almeida (EPGE/FGV)

Marcelo C. Medeiros (DE/PUC-Rio)

Paulo Pichetti (EESP/FGV)

Rio de Janeiro

2009

iii

Abstract

This thesis has three chapters. Chapter 1 explores literature about exchange rate pass-through,

approaching both empirical and theoretical issues. In Chapter 2, we formulate an estate space

model for the estimation of the exchange rate pass-through of the Brazilian Real against the US

Dollar, using monthly data from August 1999 to August 2008. The state space approach allows us

to verify some empirical aspects presented by economic literature, such as coefficients inconstancy.

The estimates offer evidence that the pass-through had variation over the observed sample. The

state space approach is also used to test whether some of the “determinants” of pass-through are

related to the exchange rate pass-through variations observed. According to our estimates, the

variance of the exchange rate pass-through, monetary policy and trade flow have influence on the

exchange rate pass-through. The third and last chapter proposes the construction of a coincident

and leading indicator of economic activity in the United States of America. These indicators

are built using a probit state space model to incorporate the deliberations of the NBER Dating

Cycles Committee regarding the state of the economy in the construction of the indexes. The

estimates offer evidence that the NBER Committee weighs the coincident series (employees in non-

agricultural payrolls, industrial production, personal income less transferences and sales) differently

way over time and between recessions. We also had evidence that the number of employees in non-

agricultural payrolls is the most important coincident series used by the NBER to define the periods

of recession in the United States.

iv

Resumo

A tese esta dividida em tres capıtulos. O capıtulo 1 trata de uma revisao de literatura sobre

pass-through, abordando aspectos empıricos e teoricos. O segundo capıtulo trata da estimacao de

um modelo de espaco de estados para estimacao dos pass-through da taxa de cambio no Brasil de

agosto 1999 a agosto 2008. A abordagem espaco de estados permite contemplar alguns aspectos

empıricos apresentados pela literatura economica, tais como a inconstancia dos parametros. As

estimativas ofereceram evidencia de que o pass-through no Brasil variou no perıodo estudado.

Ainda, a abordagem por espaco de estados permite que se estude os“determinantes” (ou variaveis

associadas) do pass-through. Com isto tivemos evidencia de que a variancia da taxa de cambio, a

polıtica monetaria e o fluxo de comercio afetam o pass-through. O terceiro e ultimo artigo da tese

trata da construcao de um indicador coincidente e antecedente da atividade economica nos Estados

Unidos da America. Nele utiliza-se um modelo probit de espaco de estados para incorporar as

decisoes do NBER Dating Cycles Committee na construcao dos ındices. A estimativas ofereceram

evidencia de que o comite do NBER pondera as series coincidentes (total de empregados em

atividades nao agrıcolas, producao industrial, renda pessoal menos transferencias governamentais

e vendas) de maneira diferente ao longo do tempo e entre as recessoes. Tambem evidenciou-se que

a serie coincidente total de empregados em setores nao-agrıcolas e a principal serie considerada

para a definicao dos perıodos de recessao nos Estados Unidos.

v

Agradecimentos

Agradeco,

A Deus, por me permitir caminhar ate aqui, apesar de todas as dificuldades.

Ao meu orientador, Prof. Joao Victor Issler pelo apoio, incentivo e exemplo profissional. Ao

Prof. Pedro Cavalcanti Gomes Ferreira que abriu as portas da EPGE quando eu era, ainda, um

aluno de graduacao para fazer bolsa de iniciacao cientıfica.

A Fundacao Getulio Vargas, a CAPES e ao CNPq pelo suporte financeiro.

Aos amigos de San Diego, Daniel, Daniel Aiex e Eillen. O convıvio com voces foi inesquecıvel.

A gratidao e eterna.

A todos os amigos da EPGE. Em especial ao Jose Diogo, ao Orlando, ao James, ao Flavio,

ao Luiz Felipe, a Amanda, ao Pedro, ao Gustavo, ao Gabriel e ao Hilton. Obrigado por todos os

momentos.

Aos meus amigos de longa data, Aline e Ralph, que, mesmo quando estavam em paıses distantes,

estivem sempre proximos o suficente para me encorajar e me incentivar nos momentos difıcies.

Aos meus novos colegas de trabalho, Luisa e Gustavo, que pela paciencia, incentivo e forca na

reta final.

Aos meus pais, Marlene e Ronaldo, e ao meu irmao Samuel, por todo amor, incentivo, encora-

jamento, participacao... Descrever toda a importancia da nossa famılia e impossıvel. A gratidao e

infinita.

A Mozuca, pela sua paciencia, benevolencia, altruısmo, seu trato carinhoso, sua calma, sua

tolerancia, sua crenca, seu suporte,... Enfim, por todo o seu amor. O Amor nunca falha.

vi

Key words and phrases: Exchange rate pass-through, business cycles, indicators of economic

activity, state space models, Kalman filter.

Palavras-Chave: Pass-through da taxa de cambio, ciclo de negocios, indicadores de atividade

economica, modelos de espaco de estados, filtro de Kalman.

Contents

I Exchange Rate Pass-Through 1

1 A Discussion on Exchange Rate Pass-Through 2

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Empirical Evidences on Exchange Rate Pass-Through . . . . . . . . . . . . . . . . 4

1.3 Pass-Through Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Macroeconomic Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Output Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.3 Microeconomic Determinants of Pass-Through . . . . . . . . . . . . . . . . 11

1.4 The economic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Pass-Through Estimation in Brazil 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Econometric Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Linear state space models under restrictions . . . . . . . . . . . . . . . . . . 16

2.2.2 Model Selection and Inference . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Econometric Setting and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vii

CONTENTS viii

2.3.1 Time Varying Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

II Coincident and Leading Indexes of Economic Activity 37

3 A State Space Model for Indices of Economic Activity 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Determining a basis for the cyclical components of coincident variables . . . 41

3.2.2 Estimating a structural equation for the unobserved business cycle state . . 43

3.2.3 The iterated extended Kalman filter and smoother . . . . . . . . . . . . . . 49

3.2.4 The Kalman Filter and Smoother Instrumental Variables Index . . . . . . . 51

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 The Basis Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.3 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.4 Predicting Recessions in Real Time . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

List of Figures

2.1 IPA-OG smoothed coefficient of Δ log et, Δ log et−1 and Δ log yt−1. . . . . . . . . . 25

2.2 IPA-OGPA smoothed coefficient of Δ log et, Δ log et−1 and Δ log yt−1. . . . . . . . 26

2.3 IPA-OGPI smoothed coefficient of Δ log et, Δ log et−1 (top), Δ log yt−1 and Δ log yt−6

(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 IPA-OG long run pass-through. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 IPA-OGPA long run pass-through. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 IPA-OGPI long run pass-through. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Tested determinants of pass-through: Monetary policy, (solid line), variance of ex-

change rate (dashed line) and international trade (dotted line). . . . . . . . . . . . 31

3.1 Coincident Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Coincident Cycles (growth rate) plot. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Filtered and Smoothed weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Predicted and Smoothed probabilities using data from 1960:06 to 2007:03. . . . . . 61

3.5 Predicted and Smoothed probabilities using data from 1960:06 to 2007:03. . . . . . 63

3.6 Filtered and Smoothed weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

ix

LIST OF FIGURES x

3.7 1990 Recession. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.8 2001 Recession. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 2007 Recession. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

List of Tables

2.1 Some quality of fit statistics of the adjusted models. . . . . . . . . . . . . . . . . . 23

2.2 Information criteria observed values for incomplete, null and complete pass-through

exchange rate models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 P-values of the tests for null and complete pass-through exchange rate. . . . . . . . 24

2.4 Estimates of the IPA-OG, IPA-OGPA and IPA-OGPI series (p-values between paren-

thesis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Estimated parameters and corresponding p-values (in parenthesis). . . . . . . . . . 34

2.6 Estimated parameters and corresponding p-values (in parenthesis). . . . . . . . . . 34

3.1 Coincident and leading variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Squared canonical correlations and canonical-correlation test. . . . . . . . . . . . . 57

3.3 Descriptive statistics of the filtered (left) and smoothed (right) coefficients. . . . . 60

3.4 Accuracy of estimation based on a cut-off point of 0.5. . . . . . . . . . . . . . . . . 62

3.5 Descriptive statistics of the smoothed weights. . . . . . . . . . . . . . . . . . . . . . 65

3.6 Predicted probabilities associated for period from 2007:08 to 2008:07 . . . . . . . . 69

xi

Part I

Exchange Rate Pass-Through

1

Chapter 1

A Discussion on Exchange Rate

Pass-Through

1.1 Introduction

The exchange rate pass-through degree is the elasticity between exchange rate and domestic prices.

In other words, it is the percentage impact on 1% change in exchange rate into domestic prices.

In an open economy, domestic prices can be affected by external shocks, whether by currency

relative price adjustment or by movements in international supply and demand. The exchange

rate pass-through highlights how sensitive each market is to fluctuations in exchange rate.

The study of exchange rate pass-through has intensified since 1980. The literature focuses on

the behavior of the impact of exchange rate on prices and their determinants. However, the real

motivation for these studies was the study of the Purchase Power Parity Puzzle (PPP). The PPP

2

CHAPTER 1. A DISCUSSION ON EXCHANGE RATE PASS-THROUGH 3

assumption states that all variation in exchange rate is passed-through into prices. The major

conclusion from empirical studies in the last years is that the PPP is not valid in the short run;

therefore, exchange rate pass-through into prices is less than one. However, there is some evidence

in favor of the validity of PPP on long run.

The pass-through estimation could provide a test for the existence of the purchase power parity.

If the PPP were valid in the long term, the pass-through would be complete and the sum of the

pass-through coefficients would have to sum to one. Otherwise, the long run pass-through would

be incomplete and the effect of variations in exchange rate into prices would be restricted.

The importance of exchange rate pass-through has increased since the adoption of the inflation

targeting regime. Fraga, Goldfajn and Minella (2003) have shown that the problem of having a

high exchange rate pass-through degree is that it implies a greater difficulty for attaining inflation

targets. A greater exchange rate pass-through means that the domestic economy is more sensitive

to external shocks, consequently the impact of exogenous shocks into domestic prices is amplified.

Exchange rate pass-through also seens to affect the inflation forecast. According to Goldfajn

and Werlang (2000), the exchange rate pass-through into prices is directly associated with inflation

forecast error. With a smaller pass-through, the domestic economy is more stable and less affected

by external factors. Therefore, a smaller pass-through means that the difference between inflation

expectations and inflation targets is smaller. In other words, a small pass-through generates a

minor inflation forecast error. Consequently, a small pass-through is associated with a major

transparency of inflation path and a minor volatility in price variations in the economy, rising

social welfare and monetary policy efficiency.

The exchange rate pass-through into prices is one of the main drivers to optimal monetary

policy. According to Betts and Devereux (2000), the trade-off between output volatility and


inflation volatility is dependent on how sensitive prices are to exchange rate variations. They

begin their argument by explaining that the nature of the trade-off between different exchange

rate regimes is quite different in industrial countries from the trade-off in emerging ones. Using a

DGE Model (Dynamic General Equilibrium Model), they argue that the critical distinction is the

exchange rate pass-through into prices. With very high exchange rate pass-through, policies that

stabilize output require high exchange rate volatility, which implies high inflation volatility. But

with limited or delayed pass-through, this trade-off is less pronounced and a flexible exchange rate

policy that stabilizes output can do so without high inflation volatility.

Another study that emphasizes the importance of exchange rate pass-through in an inflation

targeting regime is that of Fraga, Goldfajn and Minella(2003). They have shown that the problem

of having a high exchange rate pass-through degree is that it implies a greater difficulty for attaining

inflation targets. The larger the exchange rate pass-through, the more sensitive the domestic

economy to external shocks, that is, the impact of exogenous shocks on domestic prices is amplified

by a larger exchange rate pass-through.

1.2 Empirical Evidences on Exchange Rate Pass-Through

A main factor in pass-through estimation is the difficulty of using aggregated data and the known

problem of aggregation bias. This factor favors a disaggregating process for prices, and tries to

capture the exchange rate pass-through for each good or each market. Campa and Goldberg (2005)

present results where estimates are better across industries than across countries with aggregate

data. These authors also say that the major source of pass-through variations are competition

issues in each sector. Yang (1996) and Olivei (2002) show that pass-through varies significantly


across industries. Menon (1996) supports these findings pointing to the aggregation bias, indicating

that disaggregated data provides more accurate estimates and captures the impact of exchange

rates on commodities prices more precisely.

Campa and Goldberg (2005) and Pollard and Coughlin (2005) follow this trend of disaggregated

estimation of pass-through. Their approach permits a more individual analysis for each market,

relating pass-through with market power and degree of competition. This type of analysis allows

more plausible explanations for aggregated pass-through behavior. Campa and Goldberg (2005)

observe that the US has seen a change in composition of certain industries in its import basket.

Industries with a bigger pass-through, such as energy and raw materials, have shown a decrease

in their share in the US imports basket, reducing the aggregate pass-through. The proportion of

tradable and non-tradable goods is important to analyze the aggregate exchange rate pass-through

because tradables are more sensitive to changes in exchange rate than the non-tradables. Therefore,

the greater the share of tradable goods, the higher the exchange rate pass-through.

Some results about pass-through estimation can be seen in Goldberg and Knetter (1997), where

the exchange rate pass-through to US inflation was approximately 50% after 6 months. Campa

and Goldberg (2005) estimate pass-through for 25 OECD countries. They found a pass-through of

26%, in the short term, and 41% in the long term for the US. The average pass-through estimated

for OECD countries in the short and long run was 61% and 77%, respectively.

Sekine (2006) estimated exchange rate pass-through for six developed countries (United States,

Japan, Germany, United Kingdom, France and Italy) by taking into account their time-varying

natures. The author incorporates that characteristic by allowing permanent shifts in pass-through

parameters. He found that pass-through has declined over time in all major industrial countries

and, in most cases, pass-through did not show the parameter shift envisaged by split sample


estimations.

Calvo and Reinhart (2000) have shown that the pass-through degree of emerging countries is

four times greater than that of developed countries. Additionally, these authors calculate that the

variance of inflation compared with the variation of exchange rate is 43% for emerging countries

and 13% for developed ones.

In the case of the Brazilian economy, there are few studies estimating the exchange rate pass-

through. Belaisch (2003) used a VAR specification controlling for petroleum shocks and estimated

the exchange rate pass-through into IPCA approximately 6% after 3 months. He also estimated

other price indeces and found that the pass-through to IPA (34%) was larger than to IGP (27%),

which is larger than IPCA. Carneiro, Monteiro and Wu (2002) used a non-linear estimation for

pass-through into IPCA, in an attempt to capture possible asymmetries in exchange rate variations

into prices. Their estimate was 6, 4%, on average.

Albuquerque and Portugal (2003) used a time varying estimation for the IGP, IPCA and IPA.

They found evidence of time varying pass-through in Brazil, although they used a complicated

period (1980-2002). Their data set was prejudiced by multiple exchange rate regimes, multiple

changes in economic policies and some financial crisis. Their state equation estimates for IPA

were not significant, with the exception of the persistence term. For IPCA, their estimates were

approximately 6% on average. However, since 1995 their exchange rate pass-through to IPCA was

not significantly different from null.

After the estimation of exchange rate pass-through, studies started to explain its behavior and

the reasons for so much variation across countries, across time and across industries. The reasons

could be in the exchange rate pass-through determinants. According to Goldfajn and Werlang

(2000), the exchange rate pass-through varies across countries, in a way that more stable countries


exhibit smaller pass-through. Another result is that the exchange rate pass-through changes with

time horizon, reaching its peak at 12 months in the case of Brazil. These authors analyzed four

variables as pass-through determinants: real exchange rate misalignment, initial inflation, output

gap and openness degree. Their results indicate that all variables have important correlations with

exchange rate pass-through, depending on the countries’ characteristics, although real exchange

rate misalignment and inflation environment were the most important.

Inflation is positively correlated to exchange rate pass-through. Empirical evidence suggests

that the larger the inflation persistence, the larger inflation rate. Hence, there is more volatily in

the macroeconomic variables than the exchange rate pass-through.

According to Taylor (2000), a low inflation environment implies a decrease in exchange rate

pass-through. He argues that low and more stable inflation should be associated with less persistent

inflation. Hence, the low inflation and the monetary policy that has delivered it have led to lower

pass-through by a reduction in expected persistence of cost and price movements.

Gagnon and Ihrig (2001) argue that recent adoptions of anti-inflationary policies and the rise in

central bank credibility are important factors to explain the diminishing effects of inflation on the

exchange rate pass-through. When inflation is low and the commitment of the central bank to keep

inflation stable has credibility, the economic agents become less inclined to quickly pass-through

costs variations to prices.

According to Choudri and Hakura (2003) there exists strong evidence of a positive and sig-

nificant relation between average inflation and pass-through. The authors argue that a limited

pass-through gives more freedom for a independent monetary policy, benefiting the implementa-

tion of a inflation targeting regime.


1.3 Pass-Through Determinants

Since there are many studies trying to identify the causes of the exchange rate pass-through, we

summarize some the possible exchange “determinants”, proposed by these studies. According to

Menon (1996), Goldfajn and Werlang (2000), Taylor (2000) and Campa and Goldberg (2002), the

main drivers of price sensibility to exchange rate changes can be inferred. From the Macroeconomic

point of view, the pass-through depends on the openness degree of the economy, the output gap,

inflation persistence and real exchange rate misalignments. From the standpoint of disaggregated

analysis, the exchange rate pass-through is associated with the competition degree of each industry

and with a firm’s market power (with the elasticity price-demand).

1.3.1 Macroeconomic Determinants

1.3.2 Output Gap

The output gap is defined by the deviation of a product in relation to its long term value; in other

words, the difference between observed product and the value it was supposed to be according to

its long term trend. The evidence of past studies shows a positive correlation between pass-through

and output gap. The larger the difference between GNP and its potential, the greater the demand

pressure over prices. This fact generates an inflation environment, raising the probability that

firms pass-through changes in costs into prices. Therefore, in an environment where the output

gap is increasing, the exchange rate pass-through’s effect on inflation is intensified.


Inflation Environment

According to Goldfajn and Werlang (2000), the variable inflation environment is defined as the

frequency which agents remark their prices based on past inflation. In countries with an inflationary

environment, it is easier for the agents to pass-through cost changes and increase prices. As a result,

the larger the inflationary environment – and the more persistent the inflation – the easier it is

for agents to pass-through exchange rate increases into prices. This reasoning is corroborated by

Taylor(2000), who suggests a correlation between inflation and exchange rate pass-through using

the inflation persistence as a channel of transmission. The model indicates that observed changes

in pass-through, or firms’ market power, are partly originated from changes in the persistence of

expected movements in cost and competitor prices. In this sticky price model, the pass-through

to prices depends on how permanent the increase of cost is. The greater the half-life of a rise in

marginal cost, the more firms will revise prices. For this reason, if exchange rate depreciation is

transitory, firms will pass-through to prices some of this increase in costs. However, the greater the

persistence of exchange rate depreciation, the greater the pass-through will be. Taylor(2000) argues

that persistence in cost changes is related to price stability. Therefore, in a stable environment, the

inflation persistence will be smaller. As a result, the half-life of cost changes will decrease causing

a smaller pass-through.

Openness Degree

The openness degree of an economy depends on the presence of tradable goods, which determine

how sensitive prices are to changes in exchange rates. This degree can be defined as the sum

of imports and exports as a proportion of GNP. In a more open economy, we expect that the

presence of goods more sensitive to exchange rate will be lager, which implies a larger exchange


rate pass-through to inflation.

Real Exchange ate Disalignment

According to Goldfajn and Valdes (1999), a real exchange rate over valuated results a mean factor

on future inflation composition. If the real exchange rate is below its long term value, agents make

up the expectation of future devaluations, adjusting relative prices. However, if exchange rate

variation are not adjusted by relative prices, it will imply an increase in internal inflation in relation

to external inflation. As a result, an over valuated real exchange rate implies future depreciations

because the exchange rate is supposed to meet its steady state in the future. The agents will

take on this expectation of future depreciation, amplifying the effect on prices. Consequently, the

exchange rate pass-through will be negative associated with the difference of real exchange rate

and its long run value. The more over valuated the real exchange rate, the greater the expectations

of future devaluations, which will lead to an increase in the prices.

Variance of Exchange Rate

Large movements in exchange rate are associated with a higher exchange rate pass-through to

prices. If the variance of exchange rate is large, then the cost of changing prices decreases and

price-makers have more incentive to pass-through cost changes to prices. The idea is that if the

cost variation is large, then it is easier for the price maker to pass-through this cost changes to

prices.

Changing listed prices entails menu costs, such as the cost of printing new price lists and

the cost of notifying consumers of new prices. In order to justify the cost of raising prices, the

anticipated profit from the price change must exceed the menu cost. If the currency depreciation is


substantial and is not likely to reverse itself, the cost of changing prices will be small in proportion

to the profit generated by the higher price. Furthermore, a significant cost increase affecting all

competitors simultaneously reduces the impact of a price hike on a company’s reputation. Price

changes therefore occur more frequently when exchange rate movements are large.

Devereux and Yetman (2002) developed a simple theoretical model of endogenous exchange

rate pass-through. The model ignores many factors that might limit pass-through, and focuses

exclusively on the role of price rigidities. Their main argument was that exchange rate pass-

through is determined by the types of shocks in the economy and the persistence of the shocks.

For a given size of the menu cost of price changes, firms will choose a higher frequency of price

adjustment if the average rate of inflation is higher and the nominal exchange rate is more volatile.

Thus, large movements in exchange rate and an inflationary environment are associated with a

higher exchange rate pass-through.

1.3.3 Microeconomic Determinants of Pass-Through

A main factor to analyze the exchange rate pass-through into disaggregated prices is the degree

of competition on the price setting sector. When the competition increases in an industry, the

market power of firms diminishes and the producers can pass-through less cost change to consumers

without losing market-share. Therefore, in a highly competitive environment, the exchange rate

pass-through will be limited and the producers will absorb cost increases – accepting less mark-

ups – and will not fully pass-through exchange rate variations to prices, with the intention to

protect market-share. Therefore, there is a negative relation between competition and exchange

rate pass-through.

Besides market power, the elasticity price-demand also impacts exchange rate degree. The


more elastic the demand, the more consumers will respond to price changes, which implies that

producers have a limited ability to pass-through costs changes. Therefore, the more inelastic the

demand, the more producers will pass-through exchange rate variations into prices. This implies

the existence of a negative correlation between pass-through and elasticity price-demand.

Campa and Goldberg (2005) also argue that the aggregated pass-through have declined because

of the change in composition of certain industries in the import basket. Industries with a larger

pass-through have shown a decrease in their share in the US imports basket. At the same time,

industries with prices that are less sensitive to exchange rate devaluations experience a growth in

market share. The authors give the example of the reduction in the US energy sectors share, which

has an exchange rate pass-through of 70%, and raw materials (pass-through of 64%).

1.4 The economic model

The theoretical framework used to formulate the econometric models in the next chapter is directly

inspired by articles as Olivei (2002), Pollard and Coughlin (2005) and Campa and Goldberg (2002),

among others. The law of one price says that the price of any good, say good x, denoted by a

common currency should be the same in any two markets:

PH = EPF , (1.1)

where H is the home country, F is the foreign country and E is the home currency price of the

foreign currency. Given some costs, such as transportation and barriers to trade, the absolute

version of the law of one price usually does not hold. Instead, another version may hold, for

example:


PH = �EPF , (1.2)

where � indicates the deviation from the law of one price.

The model is built assuming that the foreign price of a good x, PF , is determined by the

markup over marginal cost,

PF = Markup ⋅ mc, (1.3)

where mc is the marginal cost.

Markup is a function of industry-specific factors, �, and the general macroeconomic conditions,

proxied by the exchange rate, E, as follows:

Markup = �E�, (1.4)

where � is the elasticity of the exchange rate. Marginal cost mc is determined by the prices of

substitutes goods and services, PS , the cost of inputs of good x in the producer country, W , and

income, Y , as follows:

mc = P�SW 1Y 2 (1.5)

Rewriting the above equations, we have:

PH = ��E(1+�) ⋅ W 1P�S Y 1 , (1.6)

or, applying the logarithm,


logPH = log(��) ⋅ logE(1+�) ⋅ log(P�SW 1Y 2), (1.7)

we get a additive model, as follows:

pH = log(��) + (1 + �)e+ �pS + + 1w + 2y, (1.8)

where the small caps represents variables logs.

Goldberg and Knetter (1997) show that the econometric model specification generated in equa-

tion (1.8) is exactly the same as any other widely accepted theoretical approach to study prices and

exchange rate pass-through, such as the pricing-to-market model presented by Krugman (1997).

The econometric specification and estimation will be the subject of the next chapter.

Chapter 2

Pass-Through Estimation in Brazil

2.1 Introduction

The are few studies estimating the exchange rate pass-through in Brazil. In this chapter we present

a state space model to estimate the exchange rate pass-through in Brazil from August 1999 to

August 2008. The state space framework is suitable to build a econometric model based on the

economic model discussed in section 1.8 that is suitable to address some stylized facts presented

on the literature on exchange rate pass-through.

One of the motivations of the proposed econometric model is that, as argued by Parsley (1995),

stability of exchange rate pass-through is not well tested in common econometric specifications of

pass-through equations. Therefore, the state space formulation is suitable to build linear models

with time varying coefficients, allowing us to fill this gap in the literature. There are recent

contributions using state space models, such as Sekine (2002) and Albuquergue e Portugal (2005),

but they do not cover some key aspects of interest. For example, a importante contribuition of

15

CHAPTER 2. PASS-THROUGH ESTIMATION IN BRAZIL 16

our study is that we used novel the techniques presented by Pizzinga, Fernandes and Contreras

(2008) and Pizzinga (2009) to estimate the model with constraints in the time varying coefficients

to address the PPP puzzle in this framework.

We go further exploring the proprieties of state space models. We estimated a modified version

of the econometric model to test whether some “determinants” of exchange rate pass-through are

related to variations of the exchange rate pass-through over time. This is possible because we can

specify a movement equation to the time varying coefficients with explanatory variables. Although

our methodology does not allow to claim what is the direction of the casual effects, it offers new

evidence on the so called “determinants” by the literature.

With the purpose of controlling for aggregation bias, we estimate both specifications for different

levels of aggregation of the wholesales Brazilian price index used. The model is estimated for the

IPA-OG series, including its versions for industrial products, the IPA-OGPI, and agricultural

products, IPA-OGPA.

2.2 Econometric Framework

2.2.1 Linear state space models under restrictions

We define a linear Gaussian state space model by the following measurement equation, state equa-

tion and initial state vector:

Yt = Zt�t + dt + "t , "t ∼ NID(0, Ht)

�t+1 = Tt�t + ct + �t , �t ∼ NID(0, Qt)

�1 ∼ N(b1, P1).

(2.1)

The former equation linearly relates the observed time series Yt to the unobserved state �t and


the latter gives the state evolution through a Markovian structure. The random errors "t and

�t are independent (of each other and of 1), and the system matrices Zt, dt, Ht, Tt, ct and

Qt are deterministic. Notice that dt and ct are generally reserved to the inclusion of exogenous

explanatory variables.

For a given time series of size n and any t,j, ℱj ≡ � (Y1, . . . , Yj), �t∣j ≡ E (�t∣ℱj) and Pt∣j ≡

V ar (�t∣ℱj). The Kalman filtering consists of recursive equations for these first and second order

conditional moments. The formulae and their respective deductions corresponding to predicting

(j = t − 1), filtering (j = t) and smoothing (j = n), as detailed in the estimation of unknown

parameters in the system matrices by (quasi) maximum likelihood, can be found in Harvey (1989)

and Durbin and Koopman (2001).

Now, suppose the following: for each t, At�t = qt, where At is a known k × m fixed matrix

and qt = (qt1, . . . , qtk)′

is a k × 1 observable vector, may be random. Also suppose that qt is

ℱt-measurable. A restricted estimation of this type can be achieved under the restricted Kalman

filtering, presented in Pizzinga and Fernandes (2008) and summarized in the following algorithm:

Let t be an arbitrary time period.

1. Re-write the linear restrictions as

At,1�t,1 +At,2�t,2 = [At,1 At,2](�′t,1, �

′t,2

)′= qt, (2.2)

where At,1 is a k × k full rank matrix.

2. Solve (2.2) for �t,1:

�t,1 = A−1t,1 qt −A−1t,1At,2�t,2. (2.3)


3. Take (2.3) and replace it in the measurement equation of model (2.1):

Yt = Zt,1�t,1 + Zt,2�t,2 + "t

= Zt,1(A−1t,1 qt −A

−1t,1At,2�t,2

)+ Zt,2�t,2 + "t

= Zt,1A−1t,1 qt − Zt,1A

−1t,1At,2�t,2 + Zt,2�t,2 + "t

⇒ Y ∗t ≡ Yt − Zt,1A−1t,1 qt =(Zt,2 − Zt,1A−1t,1At,2

)�t,2 + "t

≡ Z∗t,1�t,2 + "t.

4. Postulate a transition equation for the unrestricted state vector �t,2 and finally get the

following reduced linear state space model:

Y ∗t = Z∗t,2�t,2 + "t , "t ∼ (0, Ht)

�t+1,2 = Tt,2�t,2 + ct,2 +Rt,2�t,2 , �t,2 ∼ (0, Qt,2)

�1,2 ∼ (a1,2, P1,2).

(2.4)

5. Apply the usual Kalman filter to the model in (2.4) and obtain t,2∣j , for all j ≥ t.

6. Reconstitute the estimates �t,2∣j :

�t,1∣j = A−1t,1 qt −A−1t,1At,2�t,2∣j . (2.5)

As Pizzinga and Fernandes (2008) claim, an interesting feature of this approach is that there is

no need to worry about specifying the state vector equation until the reduced form is achieved in the

4th step of the described algorithm. This avoids any risk of obtaining an augmented measurement

equation that is theoretically inconsistent with the original state equation. Another good property

that should be noted, and that will be used later in this paper, is that the reduced restricted

Kalman filtering enables us to investigate the plausibility of the assumed linear restrictions by

using information criteria (e.g. AIC and BIC).


2.2.2 Model Selection and Inference

One of the purposes of this study is to identify the most adequate number of lags in the exchange

rate. The hypotheses of completeness (or absence) of exchange rate pass-through will also be

verified. To accomplish this we use the following steps:

1. Diagnostic tests with the (standardized) residuals.

2. Information criteria, such as AIC and BIC.

3. Predictive power by comparing PseudoR2 and MSE measures.

Finally, the statistical significance of the parameters of measurement and state equation will

be tested under a likelihood ratio (LR) testing approach. Since both the reduced and the com-

plete model maintain the standards of good properties of maximum likelihood estimation (cf.

Pagan, 1980), it follows that, asymptotically, LR ≡ 2 [logLMax,Comp − logLMax,Red] ∼ �21, where

logLMax,Red represents the maximum of the log-likelihood for a model with a particular explana-

tory variable dropped from the specification.

2.3 Econometric Setting and Estimation

2.3.1 Time Varying Coefficients

The dependent series are the Wholesale Price Index, Global Supply (IPA-OG), Wholesale Price

Index, Global Supply - Industrial Products (IPA-OGPI) and Wholesale Price Index Global Supply

- Agricultural Products (IPA-OGPA) all created by the Getulio Vargas Foundation in Brazil. The

IPA are the best proxy for a producer price index in Brazil and for this reason they are used in

this work. We also used the controlled consumer price index, IPCA-MP to illustrate how useful


the Pizzinga and Fernandes (2008) technique can be. The monthly average commercial exchange

rate (bid), the monthly GDP, both published by Banco Central do Brasil (the institution that has

a similar role to the American FED in Brazil) and the American PPI, Industrial Commodities

are the explanatory variables. These variables are chosen because they are good proxies to the

variables presented in equation 1.8. The GDP is proxy for income, the PPI is proxy for the cost of

production in the foreign country. As the indexes analyzed work are aggregates of many different

goods, no substitute index prices was adopted in this study.

The sample has data from August 1999 to August 2008. A longer period would be desirable,

however Brazilian economic history lacks longer periods of economic stability. For example, from

March 1994 to January 1999 Brazil experienced the adoption of the Real Plan to fight high inflation.

Much of the strength of this new plan was set on the fixed exchange rate system. However,

a sequence of international crises in the nineties made Brazil change this regime for a floating

exchange rate with inflation targeting regime in February 1999. As it always takes time for economic

agents to adapt themselves to new environments and since we also need to use some lags in the

exchange rate to correctly specify our model, we decided to estimate the proposed model using

observations since August 1999.

Since we decided to investigate whether exchange rates had a contemporaneous effect on Brazil-

ian wholesale prices, it is necessary to correctly deal with a possible endogeneity between the log

difference of exchange rate and the price indexes. This is done using the results in Kim (2006).

The instrumental variables used for the growth rate of the exchange rate are lags in the growth

rate of the exchange rate itself, lags in growth rate of the American PPI, industrial commodities,

lags in growth rate of Brazilian consumer price index, IPC, and the Brazilian IPA-OG. Among

all these instrumental variables, only lags in the exchange rate were statically significant in the


auxiliary regressions.

We now present our state space model for the exchange rate pass-through for a given index

price. The model is composed by a measurement equation, namely,

Δ log pt =∑mk=0 �ktΔ log et−k + �(m+1)tΔ log pt−1 + 0 + 1Δ log ppit−1 + 2Δ log yt−1 + "t,

"t ∼ NID(0, �2)

(2.6)

and state equation, as follows:

�t+1 = �t + �t, �t ∼ NID(0, Q). (2.7)

The former equation linearly relates the observed monthly log-variation of the domestic price index

to the log-variation of exchange rate from time t to time t − m and to the American Producer

Price Index, ppi and to a demand variable, yt−1. The coefficients of Δlog et−k in equation (2.8)

are the state coefficients and their dynamics are given in equation (2.9).1 The lagged term pt−1 is

introduced to deal with persistence observed in the inflation indexes. As proposed by Kim (2006),

we added residual terms from the auxiliary regression in the measurement equation to control

for endogeneity. The matrix Qm×m is set diagonal for simplicity. As in Sekine(2006), the estate

equation implies that all shocks have permanent effect on the time varying coefficients. Although,

it seems a oversimplifying assumption, it has many advantages. For example, small variance terms

in matrix Q provides evidence that the constant coefficients is the most adequate formulation.

The exchange rate pass-through literature has many studies arguing that the exchange rate pass-

through is declining over time. Therefore, a stationary moving average formulation in the estate

1Many attempts were made with different lags structures in the explanatory variables. The lag structure adopted

in the presented formulation was the one with best results.


equation would imply a undesirable mean reversion behavior on the coefficients movement that is

not supported by the literature. Finally, specifications with up to 12 lag exchange rate terms were

tested. An AR(1) formulation for the coefficients would have (at least) 24 more parameters than

the adopted in this study, which would imply in a worthless computational enforce.

As proposed by Kim (2006), we added residual terms from the auxiliary regression in the mea-

surement equation to control for endogeneity. The reducing method from the previous subsection

has been used in order to impose the restrictions of the Purchasing Parity (PPP) or Producer

Currency Pricing (PCP) hypotheses, that is,∑m+1i=0 �it = 1, and of the Local Currency Pricing

(LCP) hypothesis, i.e. null pass-through∑mi=1 �it = 0. The completeness of the exchange rate

passing-through means that all the variation of the exchange rate is passed to the domestic prices.

This is a key question for Economic Theory, since accepting it is implies accepting the PPP hy-

pothesis. On the other hand, the accepting that null exchange rate pass-through model is the

most adequate scenario implies that the exchange rate movements do not have an effect in the

domestic prices, and it follows that the monetary authority need not be concerned with exchange

rate movements to make monetary policy with price indexes.

The proposed model shows a good fit for all IPA-OG cited series, as can be seen in table 2.1,

below. For IPCA-MP the goodness of fit was not good, as expected. Since ICPA-MP is a index

of controlled prices, its movements are determined by political decisions, contracts and other ways

not considered by our economic model. Besides that, the estimation results for this series are

present to illustrate the methodology propose by Pizzinga and Fernandes (2008). For IPA-OGPI

series we included a 6 lags term for the dependent variable to control for autocorrelation pattern

in the residuals.

As a first exercise, we test whether the the null (LCP) and full (PCP or PPP) exchange rate


Table 2.1: Some quality of fit statistics of the adjusted models.

Model Pseudo R2 MSE

IPA-OG 0.635 0.616

IPA-OGPA 0.378 3.796

IPA-OGPI 0.711 0.461

IPCA-MP 0.014 2.513

pass-through hypotheses are acceptable for the data, as seen in table 2.3. According to the values

of the AIC and BIC criteria, we have no evidence that both hypothesizes of none and full exchange

rate pass-though are the most adequate for the IPA-OG, IPA-OGPA and IPA-OGPI. Therefore, we

have evidence that there exist a partial exchange rate pass-through in Brazil in the sample period

for these series. The exception is the series IPCA-MP, the monitored consumer prices index. Since

the prices are controlled by the government, we do not expect to have a pass-through greater than

zero for this series, as we are using monthly data. 2. This is confirmed by the results. The model

for IPCA-MP shows some evidence that its exchange rate pass-through is zero in the long run, as

indicated by the AIC and BIC information criteria. This is a excepted result as the government

decisions regarding prices rely more on political aspects than on economic ones.

According to previous findings, the relation between exchange rate changes and inflation seems

to be statistically significant for different prices indexes series analyzed in this study, as can be seen

in Figures 2.1, 2.2 and 2.3. From these figures, we have evidence that coefficients have vared in

Brazil since 1999. Moreover, these figures indicate that this relation is declining over time, which

2In Brazil, the controlled prices, such as rent, public transportation, educational, among others, have a annual

schedule of readjustment. Additionally, the political agenda decides whether some cost raise will passed through

another group of prices (fuels are a good example of this group)


Table 2.2: Information criteria observed values for incomplete, null and complete pass-through

exchange rate models.

Series Criterium unrestricted no pass-through full pass-through

IPA-OGAIC 2.065 2.461 3.082

BIC 2.338 2.684 3.306

IPA-OGPAAIC 3.987 4.119 5.077

BIC 4.260 4.343 5.300

IPA-OGPIAIC 2.129 3.191 2.940

BIC 2.452 3.464 3.213

IPCA-MPAIC 2.685 2.683 4.049

BIC 2.958 2.907 4.272

Table 2.3: P-values of the tests for null and complete pass-through exchange rate.

suggests that the estimates with constant pass-through coefficients are not valid.

Our results suggest that the contemporaneous effect of dollar variation in the analyzed index

prices variations is greater than zero. Its estimated values are mostly constant over time, but

the lagged effects are varying for the IPA-OG, IPA-OGPA and IPA-OGPI series. Therefore, it

is important to point out that almost all of the exchange rate pass-through verified is due to the

amount of the lagged exchange rate variation passed through to prices. This is reinforced by the fact

that the confidence interval for the smoothed coefficient of lagged exchange rate variation contains

zeros in some periods and does not in others. This may favor the macroeconomic environment

effect over pass-through. Some possible explanations for this are that depending on the credibility

of the central bank, the inflationary environment or the economic growth, the price-maker could


2000 2001 2002 2003 2004 2005 2006 2007 2008

0.1

0.2

2000 2001 2002 2003 2004 2005 2006 2007 2008

−0.25

0.00

0.25

2000 2001 2002 2003 2004 2005 2006 2007 2008

0.55

0.60

0.65

0.70

Figure 2.1: IPA-OG smoothed coefficient of Δ log et, Δ log et−1 and Δ log yt−1.

change the speed of his price adjustment. These hypotheses are going to be investigated later.

The inclusion of the lagged dependent variable with a time varying parameter helps us inves-

tigate whether there is variability of inflation persistence. For example, there are some authors

that argue that persistence in the inflation rate is greater during high inflation periods. If were

the case, the higher long run pass-through during high inflation periods could be a consequence

of higher persistence. Our estimates show that this is not the case for the series analyzed. Al-

though constant, the persistence is very high and statistical significant, around 0.60, for every

series. The lagged 6 IPA-OGPI term in Figure 2.3 is not significant, however it controls for serial

autocorrelation in the residuals. Therefore, we decide to keep it in the model to avoid inconsistent

estimators.

The estimates shown in figures 2.4, 2.5 and 2.6 are the long run estimates for the pass-through.

They present the same picture seen above: the exchange rate pass-through seems to be declining


2000 2001 2002 2003 2004 2005 2006 2007 2008

0.00

0.25

0.50

2000 2001 2002 2003 2004 2005 2006 2007 2008

0.0

0.5

2000 2001 2002 2003 2004 2005 2006 2007 2008

0.5

0.6

Figure 2.2: IPA-OGPA smoothed coefficient of Δ log et, Δ log et−1 and Δ log yt−1.

over time. Although there are some periods in which it is not verified (i.e. from 2002 to 2003 and

from 2005 to 2006), its value keeps declining until where we see slightly increase 2008.

The high persistence produces high variations in the long run pass-through. For example, it

reaches values as high as 0.90 during the crisis period of 1999 and 2002 for all studied series. It

highlights the importance of the autoregressive term in the measurement equation for the long run.

The agricultural prices have had a step decline since 1999. Since the two peaks of 1999 and

2002, the long run pass-through has declined and converged to almost 0.20. One of the possible

explanations to this is a increase in the competition.

Industrial prices have a similar behavior, with more intense decline. After reaching values

around 1 in 2002, the exchange rate pass-through estimated values were around 0.1, 10% of the

value in the crisis period. It is important to notice that there was a peak in the decreasing trend

from 2004 to 2005. In 2004, Brazil faced a strong GDP growth and an increase in trade volumes. At


2000 2005

0.00

0.05

0.10

0.15

0.20

2000 2005

−0.2

0.0

0.2

2000 2005

0.55

0.60

0.65

2000 2005

0.00

0.05

0.10

0.15

Figure 2.3: IPA-OGPI smoothed coefficient of Δ log et, Δ log et−1 (top), Δ log yt−1 and Δ log yt−6

(bottom).

the same time, world demand was growing sharply. This led to an increase in demand of industrial

goods because of a lack of competition. Therefore, cost movements and exchange rate changes

were more easily passed through to prices, including exchange rate changes. That is a possible

reason why the pass-through increased in this period. The central bank was forced to implement

a strong restrictive monetary policy that caused a reversion in inflation expectations and reduced

the exchange rate pass-through.

An important fact is the increase in pass-through in 2003 before the Brazilian elections. The

fear of macroeconomic policy changes caused a decrease in foreign investments. The expectation

was that the exchange rate would be devalued for a long time, consequently, agents anticipated

this expected devaluation and changed their prices. However, they realized that economic policies

would be continued, then the exchange rate decreased to its long term value and agents passed


2000 2001 2002 2003 2004 2005 2006 2007 2008

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.4: IPA-OG long run pass-through.

2000 2001 2002 2003 2004 2005 2006 2007 2008

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.5: IPA-OGPA long run pass-through.


1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 2.6: IPA-OGPI long run pass-through.

through this valuation of domestic currency.

These results strongly reinforce the belief that the pass-through is declining in Brazil for whole-

sale prices. It is important to note that 1999 and 2002 were periods of much domestic uncertainty.

In January 1999, Brazil shifted from fixed to a flexible exchange rate regime and suffered a strong

crisis of credibility. According to the Calvo and Reinhart (2000) both lack of credibility and

volatility of exchange rate are linked to a high exchange rate pass-through.

We tested whether the explanatory variables are significant in the model and we found that none

of the coefficients in the measurement equation were significant at the usual levels of significance.

Likelihood tests were conducted for the constant coefficients in the model. The results in Table

2.4 show that for all coefficients, the null hypothesis that ( i = 0, i = 0, 1, 2), cannot be rejected


Table 2.4: Estimates of the IPA-OG, IPA-OGPA and IPA-OGPI series (p-values between paren-

thesis).

Model constant vt pibt−1 ppit−1

IPA-OG0.3233 -0.024 0.017 0.082

(0.000) (0.505) (0.249) (0.071)

IPA-OGPA0.475 -0.062 0.027 0.002

(0.021) (0.537) (0.508) (0.986)

IPA-OGPI0.276 -0.023 0.019 0.078

(0.003) (0.512) (0.197) (0.071)

IPCA-MP0.428 -0.040 -0.001 0.085

(0.000) (1.000) (0.962) (0.165)

at the usual significance levels3.

2.3.2 Determinants

Some of the changes in the Brazilian economy appear to have exacerbated fluctuations in exchange

rates. The liberalization of capital flows in the last two decades and the increase in the scale of

cross-border financial transactions have increased exchange rate movements. Currency crises in

emerging market economies are unique examples of high exchange rate volatility. In Brazil, these

large movements in exchange rate may be associated with greater pass-through to prices as seen

in Figure 2.7.

Given the results presented in the previous section, we reformulate the model. Since the

3Here we remark that none correction for the log likelihood ratio significance test were implemented for theresiduals of the auxiliary residuals. So, for its coefficients are used for illustration only.


2000 2002 2004 2006 2008

0.00

0.05

0.10

0.15

0.20

Time

Figure 2.7: Tested determinants of pass-through: Monetary policy, (solid line), variance of ex-

change rate (dashed line) and international trade (dotted line).

coefficients of the contemporaneous effect of the exchange rate on the wholesale price indexes are

mostly constant, we decide to introduce them with constant coefficients. The same was done to

the persistence coefficients. In addition, all variables statistically null in the former model were

excluded from the actual one. The only exception was the error term to control for endogeneity

of the contemporaneous log difference of exchange rate pass-through. Futhermore, we tried to test

the importance of some determinants of exchange rate pass-through. For this purpose, we made

changes in state equation and we introduced some explanatory variables in the state equation. As

will be clear in the next lines:


Δ log pt = �1tΔ log et−1 + 0 + 1Δ log e0 + 2Δ log pt−1 + "t, "t ∼ NID(0, �2) (2.8)

�t+1 = �t + dt + �t, �t ∼ NID(0, Q). (2.9)

The former equation linearly relates the observed monthly log-variation of price to the log-variation

of exchange rate from time t to time time t− 1 and to its own value at time t− 1. The coefficient

of Δlog et−1 in equation (2.8) is the state coordinate and its dynamics are given in equation (2.9).

This equation now has the explanatory variable (or “determinant”), dt, with coefficient and an

error term with variance Q.

Guided by the literature presented in the previous sections, we tested four explanatory variables

for the latent exchange rate pass-through coefficients: the difference between the exchange rate

variance of daily log returns from time t to time t − 1, dvdnert; the variation of the ratio of the

inflation expectation and the inflation target set by the central bank from time t to time t − 1,

dpmt; the log difference of the trade flow (given by the sum of exports and imports) divided by

the real GDP from time t to time t − 1, dlflowt; and the log difference of the Brazilian IPCA (a

consumer price index computed by the Brazilian Census Bureau, IBGE) from time t to time t− 1.

We also included one lag for each variable because all these variables are likely to be endogenous.

The coefficient gamma represents the effect of each “determinant” over the dynamic of exchange

rate pass-through.

The monetary policy measure is the change in inflation expectation over the inflation target.

In an inflation targeting regime, the central bank uses one monetary policy rule to accommodate

inflation expectations close to the target set before. In an attempt to identify the success of the

Brazilian Central Bank in stabilizing inflation expectations, we constructed PM as the moving


average of the difference of inflation expectations of 12 months ahead over the inflation target. To

control for the monetary policy’s forward looking behavior, we set a moving weight, as follows

pmt = (1/12)(Et(�t+12)− �t+12) + (1− j/12)(Et+1(t+ 13)− �t+13), (2.10)

for j = 1, . . . , 12.

Mishkin (2008) stated that the correlation between inflation and the rate of nominal exchange

rate depreciation can indeed be high in an unstable monetary environment in which nominal shocks

fuel both high inflation and exchange rate depreciation. Furthermore, the evidence suggests that

even countries where inflation and exchange rate depreciation appear to be fairly closely linked

over time, have experienced a sizable decline in pass-through following the adoption of improved

monetary policies. To test if the credibility of the Brazilian Central Bank has been decreasing the

exchange rate pass-through, we created a monetary policy variable as the inflation expectation (by

Boletim FOCUS) over the inflation target. The credibility and the efficiency of the monetary policy

tend to hinder the ability of price makers to adjust their prices. In an stable inflation environment,

the agents are less likely to adjust their prices.

Taylor (2000) argues that the exchange rate pass-through has a positive relation to the per-

sistence of costs changes. If the volatility of changes in exchange rates is associated with its

persistence, smaller volatility periods will be followed by a smaller degree of pass-through. The

volatility in exchange rate can represent uncertainty in the economy, where large exchange rate

movements could more be easily passed through to prices.

Our results show a statistical significant association between exchange rate volatility and pass-

through. In periods with high uncertainty, large variations in exchange rate are positively correlated

with pass-through. We obtained evidence that the 12 month variance for exchange rate explains the


Table 2.5: Estimated parameters and corresponding p-values (in parenthesis).

Series dvdnert dvdnert−1 dpmt dpmt−1 dflowt dflowt−1

IPA-OG2.230 2.273 2.977 −0.912 −0.304 −0.328

(0.001) (0.000) (0.051) (0.493) (0.265) (0.223)

IPA-OGPA3.493 3.687 4.097 −1.027 −0.416 −0.427

(0.002) (0.002) (0.231) (0.757) (0.028) (0.028)

IPA-OGPI1.717 1.646 −5.246 −5.247 −0.143 −0.198

(0.033) (0.045) (0.036) (0.036) (0.665) (0.551)

Table 2.6: Estimated parameters and corresponding p-values (in parenthesis).

Series dlntradeflow/GDPt dlntradeflow/GDPt−1 dlnIPCAt dlnIPCAt−1

IPA-OG−0.178 −0.179 −0.005 −0.005

(0.551) (0.569) (0.484) (0.423)

IPA-OGPA−0.535 −0.565 −0.007 −0.006

(0.033) (0.026) (0.322) (0.291)

IPA-OGPI−0.224 −0.136 −0.001 −0.002

(0.447) (0.666) (0.858) (0.817)

pass-through dynamics for all price indexes, more strongly agricultural prices. The trade openness

variable is only significant for exchange rate pass-through to agricultural prices. If an economy

is more open to foreign goods, it will face more competition and market power of producers will

decrease. For the whole IPA and for industrial products, the increase in imports share is less

pronounced and the effect over pass-through decline is not statistically significant. In the case of

agricultural prices, the sharp increase in imports and exports increased the competition level in


the sector and reduced the propensity to pass-through cost changes to prices.

For the monetary policy variable, we found mixed results. For agricultural prices, the monetary

policy did not explain the pass-through. However, the exchange rate pass-through to industrial

prices was negatively associated with the monetary policy, where a large misalignment between

inflation expectations and the target is related to a smaller pass-through. We found that credibility

and a smaller deviation of expectation over the inflation target decrease the incentives to readjust

prices for the IPA only.

The variable dlnIPCA was introduced as an attempt to capture the inflationary environment.

With this variable we did not obtain evidence that the inflation environment affects the exchange

rate pass-through, as shown by table 2.6.

2.4 Conclusions

In this paper we estimated the evolution of the exchange rate pass-through for some wholesale

indexes prices in Brazil with a Gaussian state space model.

Using our formulation we were able to investigate some important aspects as endogeneity be-

tween exchange rate pass-through and the indexes prices, aggregation effects and persistence vari-

ation over time. We were also able to investigate the significance of some possible “determinants”

of exchange rate pass-through.

The estimates shown suggest that the the short run and long run exchange rate pass-through

are declining over time. Around 2002, the presidential election year President Lula ran for office,

the short run pass-through had risen to approximates one. Since then, the short run pass-through

has followed its decreasing trend and stabilized in 2008.

Other results about inflation persistence show that estimated coefficient of the lagged dependent

variable are constant over time, indicating that the persistence of inflation is not varying. This

implies that if the long run pass-through changes over time, it is be due to the variation of the

short run pass-through.

We did not find strong evidence that there are important endogeneity from Brazilian wholesale

price indexes on exchange rate.

The data for the wholesales indexes does not support the null and the full exchange rate pass-

through hypotheses. This reinforces the belief that there exists a positive, although incomplete,

exchange rate pass-through in Brazil. For illustration propose, we estimated our model to a

Brazilian price index for monitored prices. In this case the estimates confirmed our previous belief

that there is no exchange rate pass-through for monitored prices.

Finally, we motivated and tested the importance of set exchange rate pass-through determinants

suggested by the literature on exchange rate pass-through. We obtained strong evidence that the

variance of exchange rate causes a greater pass-through to prices. We also obtained evidence that

some variables are able to explain the pass-through of some index prices but not of others. For

example, we found evidence that adjusting monetary policy led to a reduction in the pass-through

to industrial prices but not to agricultural prices. On the other hand, an increase in trade flow

results in a decrease the pass-through to agricultural prices but not to industrial prices. We had

no evidence that the inflationary environment was able to cause pass-through.

Part II

Coincident and Leading Indexes of

Economic Activity

37

Chapter 3

A State Space Model for Indices

of Economic Activity

3.1 Introduction

Traditionally, business-cycle research has focused on sophisticated econometric models aiming to

capture the main features of either GDP or of the four coincident variables that the NBER is

said to follow (employment, industrial production, income and sales) to estimate coincident and

leading indices of economic activity, establish business-cycle turning points, as well as to estimate

their respective probability of occurrence; see Stock and Watson (1988a, 1988b, 1989, 1991, 1993a),

Hamilton (1989), Kim and Nelson (1998), Harding and Pagan (2003), Hamilton (2003), and Chau-

vet and Piger (2008), inter-alia. Arguably, these models mis a key variable that should be included

in them – the NBER decisions on U.S. turning points as determined by its business-cycle dating

38

CHAPTER 3. A STATE SPACE MODEL FOR INDICES OF ECONOMIC ACTIVITY 39

committee. Although this information is usually available with a considerable lag, there is no

reason not to include it ex-post on econometric models. This point was forcefully made in Issler

and Vahid (2006).

There has been a recent trend of incorporating NBER dating-committee decisions into different

business cycle econometric models. Although some of these contributions are independent, they all

recognize that one should not discard the informational content of these decisions when constructing

econometric models; see Birchenhall et al. (1999), Dueker (2005), Issler and Vahid, and Chauvet

and Hamilton (2006). A key aspect of the NBER dating committee is that there are some changes in

its members through time. Additionally, shocks hitting the economy affect GDP and key economic

variables that the NBER is said to follow in a different manner, either happens because these

shocks vary across time (i.e., supply shocks in one recession and demand shocks in another) or

because some of these relationships are indeed not stable. Thus, in building econometric models

using the NBER-committee decisions we should consider the possibility of time-varying weights in

econometric relationships.

Our first original contribution is to propose a state-space model with time-variable weights

using the decisions to construct coincident and leading indices of economic activity for the U.S.

economy. Our model is a probit regression of NBER decisions on the coincident series, where

instrumental-variable techniques are needed to consistently estimate time-varying weights of this

index. In estimation, we apply the extended iterated Kalman filter and use the Rivers and Voung

(1988) procedure to correct for simultaneity. Also, we account for the fact that NBER decisions

on whether there is or not a recession at time t is made well into the future, i.e., in time t + ℎ,

ℎ > 0.

We use canonical-correlation analysis to extract the cycles from the coincident variables. With


these cycles, we run the instrumental-variable state-space probit regression mentioned above. Using

the probit regression we are able to estimate the time-varying coefficients related to each coincident

series: the so called kalman Filter Coincident Index (KFCI) and Kalman Smoother Coincident

Index (KSCI). Here, we integrate state-space research with the probit regression and instrumental-

variable techniques which proposes a unique algorithm to estimate a simple coincident index with

a good track record vis-a-vis NBER decisions. We employ the fact that the NBER Business Cycle

Dating Committee uses information available at time t + ℎ, where ℎ > 0, to decide whether the

economy is in recession or not at time t. This generates an MA(ℎ) structure in the error term of

the probit regression.

Despite the fact that the NBER decisions use future information, the econometrician cannot do

the same if he/she is interested in building models that are useful in real time. Because we want

our model to be useful in real time, our techniques use only lagged information to forecast current

and future variables in our model. This leads us to our second original contribution which is shows

that our real-time model performs well in predicting the probability of a recession in a real-time

setting. We illustrate the model’s predictive ability with data from the past three recessions: 1990,

2001, and 2007.

In an out-of-sample exercise, the parameters of the model are estimated using information up

to 1 year prior to each of the last 3 recessions. Variable weights are predicted into the future. Using

the real time value of the coincident series we then estimate the probability of the NBER declaring

a recession in real time. For all three recessions we show that our model is useful if the objective

is to have a reliable real-time estimate of the NBER decisions. For example, when using a cutoff

of 50% for probabilities, our filtered estimates predict 84.93% of the recession periods correctly.

We then present evidence that the weights of the coincident series of employment is very high,


greater than 0.5, in our sample period. This agrees with a memo from the Business Cycle Dating

Committee (Hall et al., 2002) which states that “employment is probably the single most reliable

indicator [of recessions]”.

Finally, based on our coincident index estimate, we construct an optimal variable-weight leading

index of economic activity using the results from the canonical-correlation analysis. The optimal

variable-weight leading index has an important role in our method. It extracts the cycles from the

coincident series that have common features with the “business cycle”, where by cycle we mean the

information that can be linearly predicted from the past. This step is important for two reasons: it

allows us to separate signal and noise in state-space estimation and it allows combining the present

and past by linking coincident and leading indexes, respectively.

3.2 The model

3.2.1 Determining a basis for the cyclical components of coincident vari-

ables

One of the innovations proposed by Issler and Vahid (2006) was the use of the statistical technique

in canonical correlation analysis (Hotelling, 1935 and 1936) to create a coincident and a leading

indices of economic activity. As stated before, the canonical correlation analysis is important for

the extraction of the non-cyclical features from the coincident series that could introduce noise

in state-space estimation (Chauvet, 1998); thus, allowing the combination of present and past

when linking coincident and leading indices, respectively. Also the canonical correlation analysis

is suitable in dealing with possible asymmetric cycles in coincident series. As the use of canonical

correlation analyses were well motivated by Issler and Vahid’s work, we will only mention some


essential points for understanding the remaining of paper.

Canonical analysis maximizes correlations among all possible linear combination in both sets

of the coincident and leading series. Denote the set of coincident variables with the vector xt =

(x1t, x2t, x3t, x4t)′ and the set of m (m ≥ 4) “predictors” by the vector zt (this includes lags of

xt as well as lags of the leading variables). Canonical-correlation analysis transforms xt into four

independent linear combinations A(xt) = (�′1xt, �′2xt, �

′3xt, �

′4xt) with the property that �1xt

is the linear combination of xt which is the most linearly predictable of zt, �2xt is the second

most predictable linear combination of xt from zt after controlling for �1, and so on. These linear

combinations are uncorrelated with each other and they are restricted to have unit variances, so

that they can be uniquely idenfified up to sign change. By-products of this analysis are four linear

combinations of zt = ( ′1zt, ′2zt,

′3zt,

′4zt) with the property that ′1zt is the linear combination

of zt, which has the highest squared correlation with �′1xt, for i = 1, 2, 3, 4. Again, the elements

of G(zt) are uncorrelated with each other, and they are uniquely identified up to a sign change

with the additional restriction that all four have unit variances. The regression R2s between

�ixt and ′izt zt for i = 1, 2, 3, 4 which we denote by the squared canonical correlations between

(r2l , r22, r

23, r

24) xt and zt.

We can use a simple statistical test procedure to examine whether the smallest canonical cor-

relation (or a group of canonical correlations) is statistically equal to zero. The likelihood ratio

test statistic for the null hypothesis that there are k significant cycles (i.e., there are 4 − k zero

canonical correlations) is

LR = −T4∑

i=k+1

ln(1− �2i )

which has an asymptotic �2 distribution with (4 − k)(m − k) degrees of freedom (see Anderson,


1984). It is customary to use (T − m) instead of T in the above statistic to improve its finite

sample performance. If the null is not rejected, then the linear combinations corresponding to the

statistically insignificant canonical correlations cannot be predicted from the past and therefore,

can be dropped from the set of basis cycles. In this case, we conclude that all cyclical behaviors in

the four coincident series can be written in terms of less than four basis cycles. Hence, the use of

linear combinations of xts that are not associated with a zero canonical correlation is equivalent

to using only the cyclical components of the coincident series. Any linear combination of the

significant basis cycles is a linear combination of coincident variables, which is convenient for our

purposes as it implies that our coincident index will be a linear combination of the coincident

variables themselves.

3.2.2 Estimating a structural equation for the unobserved business cycle

state

After we obtained the cycles estimates, we combine them to construct our coincident index of

economic activity. In building a model with this aim we consider a specification that accounts for

some technical difficulties. The data we would have liked to work with, the state of economy, is

not observable. Instead, we have a dummy variable, the NBER indicator, that signalizes whether

the economy is in a recession or not at time t to the NBER Business Cycle Dating Committee’s

best knowledge. A important characteristic of the NBER indicator is that the NBER Business

Cycle Dating Committee uses information available up to time t+ℎ, where ℎ is a positive integer,

to make its decision about the state of the economy at time t. These specific characteristics of the

available data brings technical difficulties that we explore in the next paragraphs.

There are key assumptions that enables us to estimate the coincident index. We start by the


following:

Assumption 1 There exists a time-varying linear index of (the cyclical parts of) the coincident

series that has the exact same correlation pattern with past information as the unobserved state of

the economy.

Although the index which has the same correlation pattern with the past only involves the

significant basis cycles (i.e., will not involve white noise combinations of the coincident series),

these basis cycles are themselves (time-varying) linear combinations of coincident series. Hence,

the index is ultimately a linear combination of coincident series.

Let yt denote the unobserved state of the economy and {c1t, c2t, c3t} denotes the significant

basis cycles of the coincident series at time t. Assumption 1 clearly implies that there must be a

time varying linear combination of yt and {c1t, c2t, c3t} that is unpredictable from the information

before time t. That is,

E(yt − �0 − �1tc1t − �2tc2t − �3tc3t∣It−1) = 0, (3.1)

where It−1 = (xt−1,xt−2, . . . ; zt−1, zt−2, . . .) is the set of all observed values of the coincident and

leading variables until time t− 1.

Rewriting Eq. (3.1), we obtain

E(yt∣It−1) = E(�0 + �1tc1t + �2tc2t + �3tc3t∣It−1). (3.2)

Assumption 2 �∗it and cit are independent given It−1.

Assumption 2 suggests that the way the NBER makes its decisions is not cyclical nor does it

depend on business cycles.


Using Assumption 2, we have

E(yt∣It−1) = �0 +E(�1t∣It−1)E(c1t∣It−1)+E(�2t∣It−1)E(c2t∣It−1)+E(�3t∣It−1)E(c3t∣It−1). (3.3)

It is important to remember that we do not observe E(cit∣It−1), but cit at time t, i = 1, . . . , 3.

Therefore, Equation 3.3 is rewritten as follows:

E(yt∣It−1) = �0 + E(�1t∣It−1)c1t + E(�2t∣It−1)c2t + E(�3t∣It−1)c3t + !t, (3.4)

where E(!t∣It−1) = 0 and !t is obviously correlated with cit, i = 1, . . . , 3.

As previously mentioned, y∗i is not observable. Instead we only observe the NBER indicator.

The NBER indicator is set to one when, to the best knowledge of the NBER Dating Cycle Com-

mittee at time t+ℎ, the economy was in recession in time t. That is, the indicator that makes use

of information available at time t+ ℎ is bellow a critical value:

NBERt =

⎧⎨⎩1, if E(yt∣It+ℎ) < 0,

0, otherwise.

(3.5)

As we can always write

E(yt∣It+ℎ) = E(yt∣It−1) + �t + �t+1 + . . .+ �t+ℎ, (3.6)

using equation 3.4, we obtain

E(yt∣It+ℎ) = �0+E(�1t∣It−1)c1t+E(�2t∣It−1)c2t+E(�3t∣It−1)c3t+!t+�t+�t+1+. . .+�t+ℎ, (3.7)


where �t+i is the “surprise” associated with the new information arriving in period t + i. ut =

�t + �t+1 + . . . + �t+ℎ is unforeseeable given information at time t − 1 and imposes a “forward”

MA(ℎ) structure in the model.

As mentioned in the introduction, we believe that the NBER Dating Cycles Committee either

changes the weights of the coincident cycles according to the its composition, its members cu-

mulated knowledge over time or the political environment. These components are not stored in

the set of past values of the coincident and leading variables. Therefore, consider the following

assumptions:

Assumption 3 The time-varying coefficients �it, i = 1, 2, 3, do not depend on the past values of

the coincident and leading series stored in It−1.

Assumption 4 The changing weights mechanism acts over the weights of the basis cycles accord-

ing to the following law of movement:

�it = �it−1 + "it,

where "i ∼ N(0, �2i ) are independent white noise error terms, i = 1, 2, 3.

Assumption 4 deserves special attention. It postulates that the weights movement is given by a

random walk process, which has strong persistence. As these movements are caused by the events

previously cited, we believe it is a reasonable way to describe the way the Committee weights each

cycle over time. Actually, none of those motivations are likely to be transient (even the political

motivations are likely to act for more than a couple of months). By the econometric standpoint, it

allows a direct comparison with a model with constant coefficients. If the constant coefficients are

the correct way to deal with the weights of the coincident variables, the variance of innovations "it


will be close to zero and, therefore, the coefficients will be mostly constant.1

This formulation, leads us to a state space econometric model for business cycle. A adequate

way to deal with this framework is the Kalman filter, as it will be described later. However, before

describing the Kalman filter, there are some issues we have to deal with.

As pointed out in early paragraphs, c1t, . . . , c3t are correlated to !t. Moreover, the cycles

c1t, . . . , ckt are not observed, but estimated by canonical correlation analysis and modeled as cit =

�i( ′zt) + �it. These factors generate endogeneity in our model. Using the ideas of Rivers and

Vuong (1988) the correlation is modeled as follows:

⎛⎜⎜⎝ !t

�t

⎞⎟⎟⎠ ∼ N⎛⎜⎜⎝0,

⎡⎢⎢⎣ �2! �′!�

��! Σ��

⎤⎥⎥⎦⎞⎟⎟⎠ (3.8)

where the �it, i = 1, . . . , k, are collected in to a vector �t, �t and ′izt for i = 1, . . . , k come from

the canonical-correlation analysis, �� is a k × k diagonal variance-covariance matrix of �t, and

��! is a k × 1 vector of covariances between ut and �t. Joint normality of !t and �t implies that

!t = �′t� + �t, (3.9)

where � = Σ−1��u, �t ∼ N(0, �2u − �′�uΣ−1��u) and �t is independent of �t. Substituting, for !t

in equation (3.7), we obtain

E(yt∣It+ℎ) = �0 +E(�1t∣It−1)c1t + . . .+E(�3t∣It−1)ckt + �t + �t+1 + . . .+ �t+ℎ + �′t� + �t, (3.10)

1Besides all these considerations, we obtained some estimates adopting a auto-regressive model for the movement

equation of the coefficients. The results were far worse than those obtained with a random walk process which

motivate us to consider only the simpler model.


Notice that, by construction, all the regressors in (3.10) are uncorrelated with the error term �t.

The normalization �2� = 1 make all the parameters identifiable. Since this transformation eliminates

the correlation between the error term and the regressors, we no longer have the endogeneity

problem.

We still have some estimation issues to deal with. First, there are many parameters and states

to be estimated what possibly would be too demanding computationally. Kim (2006) pointed

out this problem and adapted a two stage procedure to a Gaussian state space model with good

results. Second, as explained before, yt is not observed. Instead, we have the NBER indicator. As

the observable data is a dichotomic variable, we have to use limited information model. This is

done replacing the the Gaussian model by a probit one and, therefore, we have to estimate a state

space probit model under presence of endogeneity. Rivers and Vuong (1988) proposed a suitable

way of estimating probit models with endogenous regressors. They prove their procedure produces

strongly consistent estimates and their estimators are asymptotically normally distributed.

Adopting a probit model, the new equation to be estimated is given as follows:

Pr(NBERt = 1) = Φ(−(�0+E(�1t∣It−1)c1t+. . .+E(�3t∣It−1)c3t+�t+�t+1+. . .+�t+ℎ+�1v1t+�2v2t+�3v3t)).

(3.11)

Considering all drawbacks from the previous paragraphs, we use the ideas developed by Kim

(2006) and Rivers and Vuong (1988) to estimate the time varying coefficients, as shortly presented

in the following steps:

1. Regress cit, i = 1, . . . , k, on zt to get �it and Σ��, a consistent estimate of Σ��.

2. From �it i = 1, . . . , k , form �t and then obtain estimates of �t = (�0, �′) denoted by � and

of the states (�1t, �2t, �3t, �t, . . . , �t+ℎ).


3.2.3 The iterated extended Kalman filter and smoother

As we are interested in investigating the time varying structure of the coefficients, it is natural

to use a state space model to estimate �1t, �2t and �3t. State space models are well presented

in many textbooks,such as Durbin and Koopman (2002), Harvey (1989), among others. Among

many interesting aspects, one special feature of state space models is their ability to deal with

correlation patterns in the error term. This aspect will be used in our work to address the MA(ℎ)

structure in the residuals.

The Kalman filter, from to Kalman (1960), is a direct way to estimate state space models. It is

widely used to estimate Gaussian state space models. However, as we have a limited information

dependent variable, we must to use a probit state space model. Some methods of estimation of

non-Gaussian state space methods rely on Monte Carlo procedures. The method we decided to

use is the iterated extended Kalman filter and smoother, which is very well presented by Klein

(2003). Iterated extended Kalman filter and smoother is based on a Taylor expansion of the

probit equation, that, as an approximation, has less accuracy than other methods not based on

aproximations. Nonetheless, our results are evidence that we are able to predict the states of the

economy well. For instance, Klein (2003, chapter 4) compared the Markov Chain Monte Carlo

(MCMC) method with the iterated extended Kalman filter to estimate a state space model for

simulated data. He concluded that there were no important differences in favor of the MCMC

method.

The results of the Taylor expansion of equation (3.11), give us the following approximating

equation:

ˇNBERt = �( t)−1{NBERt − Φ( t)}+ t (3.12)

where � is the Gaussian density function, Φ is the Gaussian cumulative distribution function, and


t is the predicted value for NBERt, based on a initial guess (or some previous filtered value, if

iteration process was ran at least once) on the vector of states, as we clarify later. The measurement

equation of the approximating Gaussian state space process can now be written:

ˇNBERt = Z∗t �∗t + �t, �t ∼ N(0, Vt) (3.13)

were Z∗t = (1 v1t v2t v3t c1t c2t c3t 1 0 . . . 0) is a 1×7+ℎ vector, �∗t′ = (�0 �1 �2 �3 �1t �2t �3t �

′t)′

is a (7 + ℎ)× 1 vector,

�t =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ˇNBERt

�t + . . .+ �t−ℎ+2

�t + . . .+ �t−ℎ+3

...

�t

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(3.14)

is a ℎ× 1 vector and variance Vt = exp t(1 + exp t)−2�( t)

−2.

We write equation 3.13 using the notation and formulation proposed by Durbin and Koopman

(2002, p. 46 and p. 54) to address the MA(ℎ) structure in the model. The state equation is defined

as follows:

�∗t+1 = T ∗�∗t + �t, �t ∼ N(0, Q). (3.15)


where

T ∗ =

⎡⎢⎢⎣ I7 0

0 T

⎤⎥⎥⎦ , T =

⎛⎜⎜⎝ 0ℎ−1 Iℎ−1

0 0′ℎ−1

⎞⎟⎟⎠ , "t =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

07

"1

"2

"3

�t+1

...

�t+1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (3.16)

Q =

⎛⎜⎜⎜⎜⎜⎜⎝04×4 0 0

0 Σ" 0

0 0 Σ�

⎞⎟⎟⎟⎟⎟⎟⎠ , Σ" =

⎛⎜⎜⎜⎜⎜⎜⎝�2"1 �"1"2 �"1"3

�"2"1 �2"2 �"2"3

�"3"1 �"3"2 �2"3

⎞⎟⎟⎟⎟⎟⎟⎠ and Σ� = �2�Iℎ, (3.17)

noting that Ik is an identity k×k, 04×4 and 0k is a k×1 vector. The state space model is completed

with a initial distribution to the state vector: �∗t0 ∼ N(a0, Q0).

Finally, to obtain the desired estimates, we run the Kalman filter and smoother estimates using

the equations present above. The set of information that will feed the Kalman filter and smoother

will be {It, Dt}, where Dt is the set of all deliberations of the Committee stored until time t.

Again, the details of filtering ans smoothing procedures are well described by the textbooks cited

above.

3.2.4 The Kalman Filter and Smoother Instrumental Variables Index

In this section, we present the products of this paper. Using the state space approach, we gen-

erate two indexes. The first index is the Kalman Filter Instrumental Variable Coincident Index

(KFIVCI), namely,


KFIV CIt =

[E(�1t∣It, Dt) E(�2t∣It, Dt) E(�3t∣It, Dt)

] [�′1xt �′2xt �′3xt

], (3.18)

if n, the sample size is greater than t. For periods in which there is not a NBER Business

Cycles Committee deliberation (i.e., time n+q) there is available information about the coincident

variables the KFIVCI is written as

KFIV CIn+q =

[E(�1,n+q∣In, Dn) E(�2,n+q∣In, Dn) E(�3,n+q∣In, Dn)

] [�′1xn+q �′2xn+q �′3xn+q

].

(3.19)

However, we may be presented with a case where we want to use all information available at

time n to revise our previous estimates of the index. In this case we apply the Kalman Smoother

to obtain the K alman Smoother Instrumental Variable Coincident Index (KSIV CI), namely,

KSIV CIt =

[E(�1t∣In, Dn) E(�2t∣In, Dn) E(�3t∣In, Dn)

] [�′1xt �′2xt �′3xt

]. (3.20)

The probabilities associated to each of these indexes are straightforwardly computed using

Equation (3.11). The only modification we applied here is the entering of the cyclical components

of Equation (3.11) and their respective coefficients into the model. The outputs of this procedure

are the K alman Filter Instrumental Variables Probabilities (KFIVP) and the K alman Smoother

Instrumental Variables Probabilities (KSIVP). Additionally we use the same ideas and revisions

for creating the probabilities as those used in the index construction.

Another important output as a result of this methodology is a tool to analyze how the NBER

Business Cycles Dating Committee members make their decision through time. We extract at


each time t the weight of each coincident variable. For example, the K alman Filter Instrumental

Variables Weights, KFIVW , which are as follows:

KFIVWit =


]⎡⎢⎢⎢⎢⎢⎢⎣�1i

�2i

�3i

⎤⎥⎥⎥⎥⎥⎥⎦


]⎡⎢⎢⎢⎢⎢⎢⎣�11 �12 �13 �14

�21 �22 �23 �24

�31 �32 �33 �34

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

1

1

1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(3.21)

The Kalman Smoother Instrumental Variables Weights, KSIVW , is built using the same method,

but using the smoothed coefficients

KSIVWit =

[E(�1t∣In) E(�2t∣In) E(�3t∣In)

]⎡⎢⎢⎢⎢⎢⎢⎣�1i

�2i

�3i

⎤⎥⎥⎥⎥⎥⎥⎦

[E(�1t∣It) E(�2t∣It) E(�3t∣It)

]⎡⎢⎢⎢⎢⎢⎢⎣�11 �12 �13 �14

�21 �22 �23 �24

�31 �32 �33 �34

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

1

1

1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3.22)

Note that at every time t the weights are set to sum one. Therefore, we can check whether there

is difference in the importance of each coincident variable in the NBER decision at a given time t.


As it will be explored in the following section, this exercise is in which interesting in under-

standing the way the NBER Committee made their decision regarding the beginning and the end

of some recessions.

3.3 Results

3.3.1 The Data

The data used in this work are monthly observations from 1960:01 to 2008:07. The four coincident

variables, “Employment” (Et), “Industrial Production” (Yt), “Income” (It) and “Sales” (St)), and

the leading series are defined in Table 3.3.1. The growth rates of the coincident series are plotted

in Figure 3.1 (the shaded areas show the recessions as claimed by NBER). Figure 3.1 shows that

the employment and industrial production show the strongest cyclical behavior.

3.3.2 The Basis Cycles

The basis cycles were found using the same procedure used by Issler and Vahid (2006). In this

section we will be describing the results found. Conditional on a VAR(4), we calculated the

canonical correlations between the coincident series (Δ lnEt, Δ lnYt, Δ ln It, Δ lnSt) and the

respective conditioning set, which comprised of four lags (Δ lnEt, Δ lnYt, Δ ln It, Δ lnSt) and

four lags of the leading series. All the leading series were transformed by taking the log difference,

however the two interest rate series were transformed by taking the first difference.

The canonical-correlation tests results in Table 3.3.2 indicate that only the last of the four

basis cycles obtained is not correlated to the last linear combination obtained from the leading

variables. Only the null hypothesis which stated that the 4th correlation coefficient associated


Time

Em

ploy

ees

1960 1970 1980 1990 2000 2010

−0.

03−

0.01

0.01

0.03

Time

Indu

stria

l Pro

duct

ion

1960 1970 1980 1990 2000 2010

−0.

03−

0.01

0.01

0.03

Time

Inco

me

1960 1970 1980 1990 2000 2010

−0.

010

0.00

00.

010

0.02

0

Time

Sal

es

1960 1970 1980 1990 2000 2010

−0.

03−

0.01

0.01

0.03

Figure 3.1: Coincident Series.

with the related basis cycle is rejected. All other hypotheses are not rejected.

The obtained business cycles are given as follows:


Table 3.1: Coincident and leading variables.

Series definition

Coincident series

Employess in non-agricultural payrolls

Industrial production

Personal income less transferences

Manufacturing trade and Sales

Leading series

MFG Unfilled Orders: Durable Goods, Ind. Total

Manufacturing & Trade Inventory: Total

NPOHUA by Building Permits in PIP

Industrial Production, Durable Consumer Goods

10-Year Treasury Constant Maturity Rate

Interest rate spread: 10-Year - 3 months Treasury Constant Maturity Rate

Nominal Weighted Exchange Rate

All Employees: Service-Providing Industries

Number Unemployed for Less than 5 Weeks

[c1t c2t c3t

]=

[Δ lnEi Δ lnYi Δ ln Ii Δ lnSi

]×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

21.68 10.38 7.75

−0.56 −6.44 −4.32

0.66 −1.27 1.77

−1.14 3.64 −2.82

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(3.23)


Table 3.2: Squared canonical correlations and canonical-correlation test.

Sq. Canonical correlations Degrees of freedom �2j and all smaller �2j = 0

�2j P-values (df corrected test)

0.5534 208 0.0000

0.2454 153 0.0000

0.2026 100 0.0000

0.0973 49 0.1511

From the coefficients of equation 3.23 we notice that the employment series has the greatest

weight in all three relevant cycles (more so in the first cycle). This reinforces our comprehension

that employment is the most cyclical variable among the coincident variables. We could use

the basis cycles (or a combination of the cycles) as a coincident index of economic activity. In

Figure 3.2 we can observe that the ups and downs of the first three cycles (mainly the first cycle)

have a similar pattern as the NBER official dating of recessions. So, as the first three cycles have

cyclical behavior, it is natural to seek a linear combination of the three that provides the “optimal”

prediction. As mentioned before, Issler and Vahid (2006) used a probit regression to extract all

information contained in the NBER Business Cycle Committee on the construction of the their

index. In their work they noticed that weights of each coincident variables were likely to vary over

time. So in the next section we present the results of our time varying probit model.

3.3.3 Estimates

The time varying coefficients estimated are presented in Figure 3.3. Table 3.3.3 shows that there are

variations in the coefficients of the basis cycles. For example, the coefficient of the first dependent


Time

c1

1960 1970 1980 1990 2000 2010

−0.

10.

2

Time

c2

1960 1970 1980 1990 2000 2010

−0.

150.

10

Time

c3

1960 1970 1980 1990 2000 2010

−0.

150.

10

Time

c4

1960 1970 1980 1990 2000 2010

−0.

10.

2

Figure 3.2: Coincident Cycles (growth rate) plot.

variable varies from 64.98 to 71.31, which yields a variation of 9.47%. Figure 3.3 maps the variations

that occur. It is easy to notice that most of the change happened durring a period of recession.


Some of these more prominent changes can be seen in the recessions during the 1970’s, 1980 and

2001.

Time

C1

slop

e

1960 1970 1980 1990 2000 2010

5565

7585

Time

C2

slop

e

1960 1970 1980 1990 2000 2010

−20

020

Time

C3

slop

e

1960 1970 1980 1990 2000 2010

3040

5060

Figure 3.3: Filtered and Smoothed weights.


Table 3.3: Descriptive statistics of the filtered (left) and smoothed (right) coefficients.

Statistic c1 c2 c3

Min 64.98 67.54 -13.37 -9.76 38.62 41.36

First Quant. 69.01 69.23 -7.05 -7.02 40.92 41.76

Median 70.08 70.34 -5.45 -5.23 42.36 42.11

Mean 69.88 69.90 -5.76 -5.96 42.60 42.65

Third Quant. -0.05 70.64 -3.53 -4.75 43.60 43.45

Maximum 71.31 70.97 -0.52 -4.22 48.42 45.49

We can compare the filtered and smoothed probabilities obtained with our model to the official

recessions periods proclaimed by NBER Business Cycle Committee. The results are shown in

Figure 3.4. We decided to use a probability cutoff point of 0.5 claim whether the economy is in

recession or not. As it is a probability, this is a intuitive classification rule. If we decided to choose a

optimal cutoff point, we would have to choose a loss function, what is itself arbitrary. For example,

Kamisnky and Reinhart (1998) and Ito and Yabu (2006) use the noise-to-signal ratio. Adopting a

procedure like that would generate different optimal cutoff points for each probabilities estimation

method, which is not desirable. In this Figure we observe that all three estimated probabilities are

likely to show values greater than 0.5 in periods officially dated as recessions. Initially, there was a

draw back in our method: we are more likely to claim false recessions, as our model presents more

indications of recession in periods when there is not a recession.

Using a cutoff point of 0.5 in the estimated probabilities to claim recession and expansion,

we can compare the adjustments of our model to that of Issler and Vahid (2006). Comparing

our smoothed probabilities to theirs’ (which is fair as they use all the data to estimate their


Filt

ered

IVC

I

1960 1970 1980 1990 2000 2010

0.0

0.4

0.8

Sm

ooth

ed IV

CI

1960 1970 1980 1990 2000 2010

0.0

0.4

0.8

Time

Issl

er

1960 1970 1980 1990 2000 2010

0.0

0.4

0.8

Figure 3.4: Predicted and Smoothed probabilities using data from 1960:06 to 2007:03.

probabilities) we see in Table 3.3.3 that our model has a better performance in dating recessions.

Actually, our model is able to correctly date 92.59% of the recessions, compared to 69.13% in Issler


and Vahid (2006) model. Our filtered probability, which uses only the information available at time

t to predict the state of economy at time t, has the same performance 92.59%. Better performance

during recessions has a drawback of a worse fit during expansions. Issler and Vahid (2006) model

correctly dates 97.98% of the recession periods compared to 84.31% of our smoothed probabilities

and 83.90% of our filtered probabilities.

Table 3.4: Accuracy of estimation based on a cut-off point of 0.5.

Estimators State of Economy

Overall Recessions Expansions

KFIVCI 85.12 92.59 83.90

KSIVCI 85.47 92.59 84.31

IVCI 93.94 69.13 97.98

There are a few important things to note about these difference. Figure 3.5 confirms the good

performance of our KFIVIC and KSIVIC. Both indexes are likely to show negative growth rates

at the dated recessions. And, more important, although the index cut the origin line in periods in

which there is not recession, the magnitude of the growth rate of the indexes in these situations

are not comparable to the magnitudes when there is really a recession. This may be a suggestion

that the indexes are more reliable than the probabilities.

Figure 3.6 shows how the coincident variables weights vary over time. As it was reported by

other studies, the employment series seems to be the most important one to determine the status

of the economy. Most of time, its smoothed weight varies around 0.55 but a careful look at its

behavior reveals that its coefficients are likely to increase at the recession periods archiving its

larger value just after the recession period in each recession period “neighborhood”. This behavior


Time

KF

IVC

I

1960 1970 1980 1990 2000 2010

−0.

015

0.00

0

Time

KS

IVC

I

1960 1970 1980 1990 2000 2010

−0.

010

0.00

5

Figure 3.5: Predicted and Smoothed probabilities using data from 1960:06 to 2007:03.

occurs at all recessions but the last in 2001. Actually, the 2001 recession is the one that the model

presents worse prediction.


Time

Em

ploy

men

t

1960 1980 2000

0.51

0.53

0.55

0.57

Time

Indu

stria

l Pro

duct

ion

wei

ghts

1960 1980 2000

0.24

80.

252

0.25

6

Time

Per

sona

l Inc

ome

wei

ghts

1960 1980 2000

0.22

890.

2291

0.22

93

Time

Sal

es w

eigh

ts

1960 1980 2000

−0.

06−

0.02

Figure 3.6: Filtered and Smoothed weights.

The opposite behavior occurs with the Industrial Production weights, which varies around a

value close to 0.25. It seems to decrease during recessions periods, except the last recession. The


same behavior is presented by sales, which has coefficients that vary around a value close to 0.1.

The sales weights vary mostly around 0.23 and personal income varies around −0.05, therefore,

suggesting that they are not significantly considered by the Business Cycle Committee. Table 3.3.3

shows some descriptive statistics from the time-varying coefficients.

Table 3.5: Descriptive statistics of the smoothed weights.

Statistic Employees Ind. Production Personal Income Sales

Min 0.5121 0.5317 0.2472 0.2504 0.2289 0.2291 -0.06529 -0.04366

First Quant. 0.5450 0.5457 0.2518 0.2522 0.2291 0.2292 -0.04710 -0.04049

Median 0.5534 0.5550 0.2530 0.2535 0.2292 0.2292 -0.03559 -0.03766

Mean 0.5518 0.5513 0.2528 0.2530 0.2291 0.2292 -0.03380 -0.03346

Third Quant. 0.5635 0.5575 0.2544 0.2538 0.2292 0.2292 -0.02582 -0.02713

Maximum 0.5794 0.5603 0.2566 0.2542 0.2293 0.2292 0.01177 -0.01117

The changes in the weights could explain why our model has a better performance during

recessions. It is exactly during these periods that the weights change. For example, there is a

group of economists that claim that the 2001 recession lasted longer than the period that was

claimed by the NBER Committee. Actually, at that time, the model suggested that the committee

decreased the weights in the three main coincident variables.

3.3.4 Predicting Recessions in Real Time

As a final exercise, we show how this model can be useful in predicting recessions. We consider

the last three recessions. So, we estimate the parameters of the model using information up to 1

year prior to each recession. As the cycles weights are random walks, their best prediction will be


given by their values at the end of the estimation period. Using this estimates and the “real time”

value of the coincident series we can estimate the probability of the NBER declaring a recession

in real time – at least with the minimal lag of 2 months for data disclosure.

Figure 3.7 presents the results for the recession during the 1990’s. The model is able to strongly

indicate the occurrence of recessions. However, it indicated the start of the recession before its

occurrence and estimated the recessions to last longer than it was declared by the NBER.

Time

pred

icted

prob

aliliti

es

1989.0 1989.5 1990.0 1990.5 1991.0 1991.5

0.00.2

0.40.6

0.81.0

Figure 3.7: 1990 Recession.

Figure 3.8 generally shows the same picture before. Some remarks should be made here.

The model is able to indicate the recession in a timely manner. The model does indicate that

the economy was in a recession during the entire recession period declared by the NBER. However,

it also show that the recession actually started one month prior to that declared by the NBER.

The model shows some early signs the recession was going to start before the NBER officially


claimed the recession started. And, more important, is clearly suggests that the 2001 recession

should last longer than it lasted according to the NBER deliberations.

Time

pred

icted

prob

aliliti

es

2000.5 2001.0 2001.5 2002.0

0.20.4

0.60.8

1.0


Finally, Figure 3.9 presents the results for the last recession. The model is able to indicate

the recession in a timely manner. The model does indicate that the economy was in a recession

during the entire recession period declared by the NBER. However, it also show that the recession

actually started one month prior to that declared by the NBER. For comparison proposes, the

Issler and Vahid (2006) probabilities computed using the entire sample period did not indicate any

recessions for all last recession periods. Table 3.3.4 presents the estimated probabilities.

Time

pred

icted

prob

aliliti

es

2007.0 2007.5 2008.0 2008.5

0.20.4

0.60.8


3.4 Conclusion

The goal of this paper is to introduce a time varying structure on the IVCI and IVLI proposed

by Issler and Vahid since they suggest that found evidence that the weights of the coincident

variables are likely to vary over time. To do this we propose a state space probit model. We

also take advantage of the flexibility of the state space formulation to incorporate the time series

structure that exists on the residuals. With this model we targeted: the endogeneity of the model

and treated the dynamical structure of the residuals in an adequate manner.

With this structure we observed variations in the weights. For example, for the employment

series the weights vary from 0.96 to 0.52. We also had evidence that our index is able to predict the

recessions better. Our filtered probabilities predict 84.93% of the recessions, compared to 68.49%

obtained by Issler and Vahid.

Table 3.6: Predicted probabilities associated for period from 2007:08 to 2008:07

Year and month Predicted Probability of Recession

2007:08 0.2603

2007:09 0.3120

2007:10 0.1233

2007:11 0.5202

2007:12 0.8150

2008:01 0.7654

2008:02 0.9554

2008:03 0.8726

2008:04 0.5316

2008:05 0.6169

2008:06 0.5706

2008:07 0.6247

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