Structural Inference in Cointegrated Vector Autoregressive Models WW.pdf

Contents

0 Introduction

1 The reduced form 7

1.1 The stationary VAR model . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Deterministic terms . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Alternative representations of cointegrated VARs . . . . . . . . . 16

1.4 Weak exogeneity in stationary VARs . . . . . . . . . . . . . . . . 20

1.5 Identifying restrictions . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6 Estimation under long run restrictions . . . . . . . . . . . . . . . 29

1.7 Restrictions on short run parameters . . . . . . . . . . . . . . . . 39

1.8 Deterministic terms . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.9 An empirical example . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 Structural VARs 49

2.1 Rational expectations . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2 The identication of shocks . . . . . . . . . . . . . . . . . . . . . 53

2.3 A class of structural VARs . . . . . . . . . . . . . . . . . . . . . . 56

2.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.5 A latent variables framework . . . . . . . . . . . . . . . . . . . . . 61

2.6 Imposing long run restrictions . . . . . . . . . . . . . . . . . . . . 62

2.7 Inference on impulse responses . . . . . . . . . . . . . . . . . . . . 66

2.8 Empirical applications . . . . . . . . . . . . . . . . . . . . . . . . 76

2.8.1 A simple IS-LM model . . . . . . . . . . . . . . . . . . . . 76

2.8.2 The Blanchard-Quah model . . . . . . . . . . . . . . . . . 81

CONTENTS

2.8.3 The KPSW model . . . . . . . . . . . . . . . . . . . . . . 84

2.8.4 The causal graph model of Swanson-Granger (1997) . . . . 90

2.9 Problems with the SVAR approach . . . . . . . . . . . . . . . . . 93

3 Problems of temporal aggregation 101

3.1 Granger causality . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.2 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.3 Contemporaneous causality . . . . . . . . . . . . . . . . . . . . . 114

3.4 Monte Carlo experiments . . . . . . . . . . . . . . . . . . . . . . . 120

3.5 Aggregation of SVAR models . . . . . . . . . . . . . . . . . . . . 123

4 Inference in nonlinear models 129

4.1 Inconsistency of linear cointegration tests . . . . . . . . . . . . . . 132

4.2 Rank tests for unit roots . . . . . . . . . . . . . . . . . . . . . . . 136

4.3 A rank test for neglected nonlinearity . . . . . . . . . . . . . . . . 144

4.4 Nonlinear short run dynamics . . . . . . . . . . . . . . . . . . . . 147

4.5 Small sample properties . . . . . . . . . . . . . . . . . . . . . . . 154

4.6 Empirical applications . . . . . . . . . . . . . . . . . . . . . . . . 163

4.7 Appendix: Critical values . . . . . . . . . . . . . . . . . . . . . . 169

5 Conclusions and outlook

173

Chapter 0

Introduction

In one of the rst attempts to apply regression techniques to economic data,

Moore (1914) estimated the \law of demand" for various commodities. In his

application the percentage change in the price per unit is explained by a linear

or cubic function of the percentage change of the produced quantities. His results

are summarized as follows:

\The statistical laws of demand for the commodities corn, hay, oats,

and potatoes present the fundamental characteristic which, in the clas-

sical treatment of demand, has been assumed to belong to all demand

curves, namely, they are all negatively inclined".

(Moore 1914, p. 76). Along with his encouraging results, Moore (1914) estimated

the demand curve for raw steel (pig-iron). To his surprise he found a positively

sloped demand curve and he claimed he have found a brand-new type of demand

curve. Lehfeldt (1915), Wright (1915) and Working (1927) argued, however, that

Moore has actually estimated a supply curve because the data indicated a moving

demand curve that is shifted during the business cycle, whereas the supply curve

appears relatively stable.

This was probably the rst thorough discussion of the famous identication

problem in econometrics. Although the arguments of Wright (1915) come close to

a modern treatment of the problem, it took another 30 years until Haavelmo (1944)

suggested a formal framework to resolve the identication problem. His elegant

CHAPTER 0. INTRODUCTION

probabilistic framework has become the dominating approach in subsequent years

and was rened technically by Fisher (1966), Rothenberg (1971), Theil (1971) and

Zellner (1971), among others.

Moore's (1914) estimates of \demand curves" demonstrate the importance

of prior information for appropriate inference from estimated economic systems.

This is a typical problem when collected data are used instead of experimental

data that are produced under controlled conditions. Observed data for prices and

quantities result from an interaction of demand and supply so that any regression

between such variables require further assumptions to disentangle the eects of

shifts in the demand and supply schedules.

This ambiguity is removed by using prior assumptions on the underlying eco-

nomic structure. A structure is dened as a complete specication of the prob-

ability distribution function of the data. The set of all possible structures S is

called a model. If the structures are distinguished by the values of the parameter

vector that is involved by the probability distribution function, then the identi-

cation problem is equivalent to the problem of distinguishing between parameter

points (see Hsiao 1983, p. 226). To select a unique structure as a probabilistic

representation of the data, we have to verify that there is no other structure in

S that leads to the same probability distribution function. In other words, an

identied structure implies that there is no observationally equivalent structure

in S. In this case we say that the structure is identied (e.g. Judge et al. 1988,

Chapter 14).

In this thesis I consider techniques that enables structural inference (that is

estimation and tests in identied structural models) by focusing on a particular

class of dynamic linear models that has become important in recent years. Since

the books of Box and Jenkins (1970) and Granger and Newbold (1977), time series

techniques have become popular for analysing the dynamic relationship between

time series. Among the general class of the multivariate ARIMA (AutoRegressive

Integrated Moving Average) model, the Vector Autoregressive (VAR) model turns

out to be particularly convenient for empirical work. Although there are important

reasons to allow also for moving average errors (e.g. Lutkepohl 1991, 1999), the

VAR model has become the dominant work horse in the analysis of multivariate

time series. Furthermore, Engle and Granger (1987) show that the VAR model is

an attractive starting point to study the long run relationship between time series

that are stationary in rst dierences. Since Johansen's (1988) seminal paper, the

cointegrated VAR model has become very popular in empirical macroeconomics.

An important drawback of the cointegrated VAR approach is that it takes the

form of a \reduced form representation", that is, its parameters do not admit

a structural interpretation. In this thesis, I review and supplement recent work

that intends to bridge the gap between such reduced form VAR representations

and structural models in the tradition of Haavelmo (1944). To do this, I rst

discuss in Chapter 1 aspects of the reduced form model that are fundamental

for the subsequent structural analysis as well. In Chapter 2 I consider structural

models that take the form of a linear set of simultaneous equations advocated by

the inuential Cowles Commission. An alternative kind of structural models are

known as \Structural VAR models" or \Identied VAR models". These models

are considered in Chapter 3. Problems due to the temporal aggregation of time

series are studied in Chapter 4 and Chapter 5 deals with some new approaches to

analyze nonlinear models. Chapter 6 concludes and makes suggestions for future

work.

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