Nonstationary Time Series Analysis and Cointegration

By Colin P. Hargreaves | Go to book overview

nonstationary. Again, following Durlauf and Hall ( 1989c), if current and lagged

and are used in the ht+1 projection, then the explosive part of the filtered noise will be recovered. However, this means that the ht+1 projection itself is nonzero. Therefore, the ht+1 projection will detect a deviation from the null model for any type of nonstationary noise except a rational bubble. Theorem 3 is therefore verified.


Proof of Theorem 2 and Theorem 4

Theorem 2 is an implication of Theorem 1 and can be verified by a straightforward generalisation of Theorem 1.3 in Durlauf and Hall ( 1989c). As before, all projections are well defined so long as the histories of Pt and Mt are included in Lt(x).

To verify Theorem 4, observe that when prices obey

, Δht+1 is an MA(1) process. The MA(1) coefficient is unrestricted as the model contains no implications for the variance/covariance of ζt, ξt, and ut. By a straightforward generalisation of Theorem 1, Δht+1 must be orthogonal to Lt-1(x).

Under the misspecification alternative, Δht+1 equals

. (5)

Following the argument given in the proof of Theorem 1, the only case where the projection of Δht+1 onto Lt-1 (x) is zero while the projection of

onto Lt-1(x) is nonzero occurs when Nt is a rational bubble. This verifies Theorem 4.


REFERENCES

BURMEISTER, E. and K. D. WALL ( 1982), "'Kalman Filtering Estimation of Unobserved Rational Expectations with an Application to the German Hyperinflation'", Journal of Econometrics, 20, pp. 255-84.

-- ( 1987), "'Unobserved Rational Expectations and the German Hyperinflation with Endogenous Money Supply'", International Economic Review, 28, pp. 15-32.

CAGAN, P. ( 1956), "'The Monetary Dynamics of Hyperinflation'" in Friedman, M. (ed.), Studies in the Quantity Theory of Money, University of Chicago Press, Chicago.

CASELLA, A. ( 1989), "'Testing for Price Level Bubbles with Exogenous or Endogenous Fundamentals: the German Hyperinflation Once More'", Journal of Monetary Economics, 24, pp. 109-22.

DIBA, B. and H. GROSSMAN ( 1988), "'Explosive Bubbles in Stock Prices?'", American Economic Review, 78, pp. 520-30.

DURLAUF, S. N. and R. E. HALL ( 1989a), "'Bounds on the Variances of Specification Errors in Models with Expectations'", Working Paper no. 2936, NBER.

-- ( 1989b), "'Measuring Noise in Stock Prices'", Working Paper, Stanford University.

-- ( 1989c), "'A Signal Extraction Approach to Recovering Noise in Expectations Based Models'", Working Paper, Stanford University.

-281-

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Nonstationary Time Series Analysis and Cointegration
Table of contents

Table of contents

  • Title Page iii
  • Contents v
  • List of Figures vii
  • List of Tables xiii
  • List of Contributors xv
  • Foreword xvii
  • 1 - Introduction 1
  • References 8
  • 2 - Towards a Theory of Economic Forecasting 9
  • Appendix 48
  • References 50
  • 3 - Bayes Models and Forecasts of Australian Macroeconomic Time Series 53
  • References 86
  • 4 - A Review of Methods of Estimating Cointegrating Relationships 87
  • References 129
  • 5 - A Test of the Null Hypothesis of Cointegration 133
  • References 151
  • 6 - Modelling Seasonal Variation 153
  • References 176
  • 7 - Cointegration, Seasonality, Encompassing, and the Demand for Money in the Uk 179
  • Appendix A. the Data 214
  • Appendix B. Sequential Reduction Analysis 216
  • References 220
  • 8 - Evaluating a Real Business Cycle Model 225
  • Appendix 252
  • References 254
  • 9 - Misspecification Versus Bubbles in the Cagan Hyperinflation Model 257
  • References 281
  • 10 Regime Switching with Time-Varying Transition Probabilities 283
  • Appendix 299
  • References 302
  • Name Index 303
  • Subject Index 307
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