Real Business Cycles: A Reader

By James E. Hartley; Kevin D. Hoover et al. | Go to book overview

CHAPTER 16

© 1991 American Statistical Association

Journal of Business & Economic Statistics, July 1991, Vol. 9, No. 3

Calibration as Testing: Inference in Simulated Macroeconomic Models

Allan W. Gregory and Gregor W. Smith

Department of Economics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

A stochastic macroeconomic model with no free parameters can be tested by comparing its features, such as moments, with those of data. Repeated simulation allows exact tests and gives the distribution of the sample moment under the null hypothesis that the model is true. We calculate the size of tests of the model studied by Mehra and Prescott. The approximate size of their test (which seeks to match model-generated, mean, risk-free interest rates and equity premia with historical values) is 0 although alternate, empirical representations of this model economy or alternate moment-matching tests yield large probabilities of Type I error.

KEY WORDS: Equity premium; Monte Carlo; Simulation; Type I error.

Calibration in macroeconomics is concerned primarily with testing a model by comparing population moments (or perhaps some other population measure) to historical sample moments of actual data. If the correspondence between some aspect of the model and the historical record is deemed to be reasonably close, then the model is viewed as satisfactory. If the distance between population and historical moments is viewed as too great, then the model is rejected, as in the widely cited equity-premium puzzle of Mehra and Prescott (1985). A drawback to the procedure as implemented in the literature is that no metric is supplied by which closeness can be judged. This leads to tests with unknown acceptance and rejection regions.

This article provides a simple way to judge the degree of correspondence between the population moments of a simulated macroeconomic model and observed sample moments and develops a framework for readily calculating the size (probability of Type I error) of calibration tests. We apply this method to the well-known equity-premium case. This article is not concerned with a “solution” to the equity-premium puzzle. Rather it evaluates the probability of falsely rejecting a true macroeconomic model with calibration methods. One finding is that the size of the test considered by Mehra and Prescott (which seeks to match mean risk-free interest rates and equity premia) is 0, so the model with their parameter settings is unlikely to have generated the observed historical moments. Some alternate versions of the consumption-based asset-pricing model or alternate moment-matching tests yield large probabilities of Type I error.

Section 1 characterizes calibration as testing. A simple formalization of calibration as Monte Carlo testing allows exact inference. Section 2 contains an application to the test conducted by Mehra and Prescott (1985). Section 3 concludes.


1. CALIBRATION AS TESTING

Calibration in macroeconomics has focused on comparing observed historical moments with population moments from a fully parameterized simulation model—that is, one with no free parameters. One might elect to simulate a model because of an analytical intractability or because a forcing variable is unobservable. In macroeconomics, examples of unobservable forcing variables include productivity shocks in business-cycle models or consumption measurement errors in asset-pricing models.

Consider a population moment θ, which is restricted by theory, with corresponding historical sample moment

for a sample of size T. Call the moment estimator Assume that is consistent for θ. The population moment is a number, the sample moment is the realization of a random variable (an estimate), and the estimator is a random variable. The calibration tests applied in the literature compare θ and and reject the model if θ is not sufficiently close to In some calibration studies, attempts are made to exactly match the population moment to the sample moment (there must be some leeway in parameter choice to make this attempt nontrivial). Such matching imposes unusual test requirements because θ and can differ even when the model is true due to sampling variability in Moreover, judging closeness involves the sampling distribution of the estimator Standard hypothesis testing procedures may be unavailable because the exact or even asymptotic distribution of the estimator is unknown.

One prominent advantage in the calibration of macroeconomic models that has not been exploited fully is that the complete data-generating process is specified. Thus the sampling variability of the simulated moment can be used to evaluate the distance between θ and

297

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Real Business Cycles: A Reader
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Acknowledgements xi
  • Part I - Introduction 1
  • Chapter 1 - The Limits of Business Cycle Research 3
  • Notes 34
  • Chapter 2 - A User's Guide to Solving Real Business Cycle Models 43
  • Part II - The Foundations of Real Business Cycle Modeling 55
  • Chapter 3 57
  • Chapter 4 83
  • References 96
  • Chapter 5 97
  • Chapter 6 102
  • Chapter 7 108
  • Part III - Some Extensions 147
  • Chapter 8 149
  • Chapter 9 168
  • References 178
  • Chapter 10 - Current Real-Business-Cycle Theories and Aggregate Labor-Market Fluctuations 179
  • Chapter 11 - The Inflation Tax in a Real Business Cycle Model 200
  • Part IV - The Methodology of Equilibrium Business Cycle Models 217
  • Chapter 12 219
  • Chapter 13 237
  • Chapter 14 254
  • Chapter 15 272
  • Part V - The Critique of Calibration Methods 293
  • Chapter 16 295
  • Chapter 17 - Measures of Fit for Calibrated Models 302
  • Chapter 18 333
  • Chapter 19 355
  • Part VI - Testing the Real Business Cycle Model 381
  • Chapter 20 - Business Cycles: Real Facts and a Monetary Myth 383
  • References 398
  • Chapter 21 399
  • Chapter 22 - Evaluating a Real Business Cycle Model 431
  • Chapter 23 462
  • Chapter 24 496
  • Chapter 25 513
  • Chapter 26 - Did Technology Shocks Cause the 1990-1991 Recession? 533
  • Part VII - The Solow Residual 541
  • Chapter 27 - Technical Change and the Aggregate Production Function 543
  • Chapter 28 552
  • Chapter 29 564
  • Chapter 30 - Output Dynamics in Real-Business-Cycle Models 571
  • Part VIII - Filtering and Detrending 591
  • Chapter 31 - Postwar U. S. Business Cycles: an Empirical Investigation 593
  • Chapter 32 609
  • Chapter 33 626
  • Index 652
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