The Cointegrated VAR Model: Methodology and Applications

by Katarina Juselius

Overview

This valuable text provides a comprehensive introduction to VAR modelling and how it can be applied. In particular, the author focuses on the properties of the Cointegrated VAR model and its implications for macroeconomic inference when data are non-stationary. The text provides a number of insights into the links between statistical econometric modelling and economic theory and gives a thorough treatment of identification of the long-run and short-run structure as well as of the common stochastic trends and the impulse response functions, providing in each case illustrations of applicability. This book presents the main ingredients of the Copenhagen School of Time-Series Econometrics in a transparent and coherent framework. The distinguishing feature of this school is that econometric theory and applications have been developed in close cooperation. The guiding principle is that good econometric work should take econometrics, institutions, and economics seriously. The author uses a single data set throughout most of the book to guide the reader through the econometric theory while also revealing the full implications for the underlying economic model. To test ensure full understanding the book concludes with the introduction of two new data sets to combine readers understanding of econometric theory and economic models, with economic reality.

Product Details

ISBN-13:	9780191622960
Publisher:	OUP Oxford
Publication date:	12/07/2006
Series:	Advanced Texts in Econometrics
Sold by:	Barnes & Noble
Format:	eBook
File size:	21 MB
Note:	This product may take a few minutes to download.

About the Author

Katarina Juselius obtained her Ph.D from the Swedish School of Economics, Helsinki in 1983. In 1985 she became Associate Professor at the University of Copenhagen and in 1996 she was appointed the Chair of Macroeconometrics. She has published extensively on the methodology of Cointegrated VAR Models with applications to Monetary Transmission Mechanisms, Policy Control Rules, Price Linkages, Wage-, Price, and Unemployment Dynamics. She has been the leader of numerous research projects, and has been on the editorial boards of the International Journal of Forecasting, the Journal of Business and Economic Statistics, and is presently serving the Journal of Economic Methodology. In 1995-98 she was a member of the Danish Social Sciences Research Council and is presently a member of the EUROCORES committee at the European Science Foundation.

Table of Contents

Preface     vii
Bridging economics and econometrics     1
Introduction     3
On the choice of economic models     4
Theoretical, true and observable variables     6
Testing a theory as opposed to a hypothesis     7
Experimental design in macroeconomics     8
On the choice of empirical example     9
Models and relations in economics and econometrics     13
The VAR approach and theory-based models     14
Inflation and money growth     15
The time dependence of macroeconomic data     18
A stochastic formulation     21
Scenario analyses: treating prices as I(2)     27
Scenario analyses: treating prices as I(1)     32
Concluding remarks     32
The probability approach in econometrics, and the VAR     35
A single time-series process     35
A vector process     38
An illustration     40
Reviewing some useful results     42
Deriving the VAR     43
Interpreting the VAR model     46
The dynamic properties of the VAR process     48
The roots of the characteristic function     48
Calculating the eigenvalue rootsusing the companion matrix     50
Illustration     51
Concluding remarks     52
Specifying the VAR model     53
The unrestricted VAR     55
Likelihood-based estimation in the unrestricted VAR     55
The estimates of the unrestricted VAR(2) for the Danish data     59
Three different ECM representations     60
The ECM formulation with m = 1     61
The ECM formulation with m = 2     63
ECM representation in acceleration rates, changes and levels     64
The relationship between the different VAR formulations     65
Misspecification tests     66
Specification checking     66
Residual correlations and information criteria     66
Tests of residual autocorrelation     73
Tests of residual heteroscedasticity     74
Normality tests     75
Concluding remarks     77
The cointegrated VAR model     79
Defining integration and cointegration     79
An intuitive interpretation of [Pi] = [alpha][beta][prime]     80
Common trends and the moving average representation     84
From the AR to the MA representation     85
Pulling and pushing forces      88
Concluding discussion     90
Deterministic components in the I(1) model     93
A trend and a constant in a simple dynamic regression model     93
A trend and a constant in the VAR     95
Five cases     99
The MA representation with deterministic components     100
Dummy variables in a simple regression model     102
Dummy variables and the VAR     104
An illustrative example     109
Conclusions     112
Estimation in the I(1) model     115
Concentrating the general VAR model     115
Derivation of the ML estimator     117
Normalization     120
The uniqueness of the unrestricted estimates     120
An illustration     121
Interpreting the results     124
Concluding remarks     128
Determination of cointegration rank     131
The LR test for cointegration rank     131
The asymptotic tables with a trend and a constant in the model     134
The role of dummy variables for the asymptotic tables     139
Similarity and rank determination     139
The cointegration rank: a difficult choice     140
An illustration based on the Danish data     143
Concluding remarks     145
Testing hypotheses on cointegration     147
Recursive tests of constancy     149
Diagnosing parameter non-constancy     149
Forward recursive tests     151
The recursively calculated log likelihood     151
Recursively calculated trace test statistics     153
Recursively calculated eigenvalues [lambda][subscript i]     154
The fluctuations test     157
The max test of constant [beta]     159
Tests of '[beta][subscript t] equals a known [beta]'     160
Recursively calculated prediction tests     163
Backward recursive tests     164
Log likelihood function     165
The trace test statistics     166
The log transformed eigenvalues     166
Fluctuations tests     166
Max test of constant [beta]     167
Test of [beta][subscript t] equal to a known [beta]     167
Backward predictions tests     167
Concluding remarks     170
Testing restrictions on [beta]     173
Formulating hypotheses as restrictions on [beta]     173
Same restriction on all [beta]     175
Illustrations      178
Some [beta] vectors assumed known     183
Illustrations     185
Only some coefficients are restricted     186
Illustrations     187
Revisiting the scenario analysis     190
Testing restrictions on [alpha]     193
Long-run weak exogeneity     193
Empirical illustrations     196
Weak exogeneity and partial models     197
Illustration     198
Testing a known vector in [alpha]     200
Illustration     202
Concluding remarks     203
Identification     205
Identification of the long-run structure     207
Identification when data are non-stationary     207
Identifying restrictions     209
Formulation of identifying hypotheses and degrees of freedom     212
Just-identifying restrictions     216
Over-identifying restrictions     219
Lack of identification     221
Recursive tests of [alpha] and [beta]     224
Concluding discussion     228
Identification of the short-run structure     229
Formulating identifying restrictions     230
Interpreting shocks      231
Which economic questions?     232
Restrictions on the short-run reduced-form model     236
The VAR in triangular form     240
Imposing general restrictions on A[subscript 0]     243
Is a current effect empirically identifiable?     243
Illustration 1: Lack of empirical identification     245
Illustration 2: The problem of weak instruments     245
Illustration 3: The preferred structure     249
A partial system     252
Concluding remarks     252
Identification of common trends     255
The common trends representation     255
The unrestricted MA representation     258
The MA representation subject to restrictions on [alpha] and [beta]     262
Imposing exclusion restrictions on [beta][Up Tack]     266
Assessing the economic model scenario     268
Concluding remarks     272
Identification of a structural MA model     275
Reparametrization of the VAR model     275
Separation between transitory and permanent shocks     277
How to formulate and interpret structural shocks     279
An illustration     282
Are the labels credible?     286
The I(2) model      289
Analysing I(2) data with the I(1) model     291
Linking the I(1) and the I(2) model     292
Stochastic and deterministic trends in the nominal variables     293
I(2) symptoms in I(1) models     297
The characteristic roots of the model     298
The graphs of the cointegration relations     299
Is the nominal-to-real transformation acceptable?     302
Transforming I(2) data to I(1)     302
Testing long-run price homogeneity     303
Concluding remarks     308
The I(2) model: Specification and estimation     311
Structuring the I(2) model     312
Deterministic components in the I(2) model     314
Restricting the constant term and the trend     315
Restricting a broken trend and the dummy variables     316
ML estimation and some useful parametrizations     318
The two-step procedure     318
The ML procedure     318
Decomposing the [Gamma] and the [Pi] matrix     320
Estimating the I(2) model     322
Determining the two reduced rank indices     322
The unrestricted I(2) estimates     326
Concluding discussion     329
Testing hypotheses in the I(2) model     331
Testing price homogeneity     332
Long-run price homogeneity     332
Medium-run price homogeneity     333
Assessing the I(1) results within the I(2) model     335
Testing the restrictions of the I(1) model     335
A data consistent long-run structure     339
An empirical scenario for nominal money and prices     340
Concluding discussion     343
A methodological approach     345
Specific-to-general and general-to-specific     347
The general-to-specific and the VAR     347
The specific-to-general in the choice of variables     348
Gradually increasing the information set     349
Combining partial systems     352
Introducing the new data     354
Wage, price, and unemployment dynamics     359
Economic background     359
Centralized wage bargaining and an aggregate wage relation     361
The price wedge, productivity and unemployment     363
Phillips-curve type relations     365
The data and the models     368
Empirical analysis: the EMS regime     371
Specification testing     371
The overall tests      373
Exploiting the information in the [Pi] matrix     375
Identifying the long-run structure     377
Empirical analysis: The post-Bretton-Woods regime     380
Specification tests     380
Investigating the [Pi] matrix     382
An identified long-run structure     383
Concluding discussion     385
Foreign transmission effects: Denmark versus Germany     387
International parity conditions     388
The data and the models     395
Rank determination     396
Tests of a unit vector in [beta] and zero row and a unit vector in [alpha]     397
Analysing the long-run structure     399
Identifying the long-run relations     399
The common driving trends     401
Concluding remarks     401
Collecting the threads     403
The full model estimates     404
Some general results     406
A more detailed analysis     407
Comparing the two periods     408
What have we learnt about inflationary mechanisms?     410
Main findings     410
Do we now understand previous puzzles better?     412
Which theories seem empirically relevant?      414
About the VAR analysis and the theory model     415
Concluding discussion     416
The asymptotic tables for cointegration rank     419
A roadmap for writing an empirical paper     423
Bibliography     425
Index     439