scispace - formally typeset

Journal ArticleDOI

A Consistent Test of Stationary Ergodicity

01 Aug 1993-Econometric Theory (Cambridge University Press)-Vol. 9, Iss: 04, pp 589-601

AbstractA formal statistical test of stationary-ergodicity is developed for known Markovian processes on R^d. This makes it applicable to testing models and algorithms, as well as estimated time series processes ignoring the estimation error. The analysis is conducted by examining the asymptotic properties of the Markov operator on density space generated by the transition in the state space. The test is developed under the null of stationary-ergodicity, and it is shown to be consistent against the alternative of nonstationary-ergodicity. The test can be easily performed using any of a number of standard statistical and mathematical computer packages.

Topics: Z-test (58%), Statistical hypothesis testing (55%), State space (54%), Ergodicity (54%), Markov process (53%)

Summary (2 min read)

1 Introduction

  • Ergodicity conditions play an integral part in many estimation and modeling decisions.
  • Under that null hypothesis, the authors develop a consistent test.
  • Stationary ergodicity of the data generating process is equivalent to the convergence of Cesaro averages of the transition probabilities to a unique invariant measure.
  • This is a completely different version of an earlier paper ((9], based on Chapter 4 of (11]) that was presented in the 1988 North American Summer Meetings of the Econometric Society.
  • Financial support from the NSF is gratefully acknowledged.

2 T he Null Hypothesis of Stationary-Ergodicity

  • The authors assume that there exists at least one stationary density f* such that Pf*= f*.
  • Since ergodicity is mainly used to ensure that time series sample moments converge to the moments under the unique stationary measure, this limitation does not seem very severe.
  • The authors test will be shown to have asymptotic power 1 against the non stationary-ergodic alternative.
  • As seen from [10, example 4, p. 218], their weaker criterion of (not necessarily uniform) convergence can be satisfied in cases where condition (D) is violated.

3 An Operational Test of Stationary-Ergodicity

  • Notice that non-stationarity in this sense is different from the common usage of the term in time series contexts.
  • The same procedure can now be followed where the xJ's are drawn from g.
  • Notice that the mapping iii is itself one-to-one and onto, and is quite easy to implement in practice, and hence, the authors can limit attention to transitions on [O, 1].
  • This establishes the existence of a unique probability measure on ([O, 1], B([O, 1])) (namely 7r .,(.)) to which the Cesaro averages of transitions from any initial condition converge, which is their definition of the stationary-ergodicity of p.,(., .).

4 A consistent testing procedure

  • A test of the type described above can be performed for any pair of initial densities to obtain the required size.
  • See [18] for a number of those tests based on the empirical distribution function.
  • Their algorithm generates a discrete random variable Z from the multinomial distribution with the probability vector p0, ,Pk, and then generates x as 1.
  • Now, the authors know that under the alternative of non-stationary-ergodicity, Harris recurrence [13, p.115] must be violated.

5 Monte Carlo Investigation of small sample be

  • The authors report on Monte Carlo results investigating the small sample properties of their suggested testing procedure.
  • For random number generation, the authors used Press et al's [17], subroutine ran1{), and they initialized it with the clock time each time they ran a new Monte Carlo at different values of k, n, and s.
  • For the Kolmogorov-Smirnov test, the authors used Press et al's [17) subroutine kstwo, with its accompanying subroutines probks and sort(}.
  • The rest of the code was written in C, and compiled, vectorized, and run on a Cray YMP2E/116.

6 Concluding Remarks

  • By known, all the authors mean is that one can generate random draws from some stochastic transition x1+1 � p(x,, .).
  • This accommodates among other things simulations from models where closed form solutions cannot be explicitly written.
  • There is no reason in principle why one cannot use this test on estimated laws of motion PT(e, .) which are believed to be consistent estimators of some true p( e,.) under the maintained hypothesis of stationary-ergodicity.
  • Clearly, as T i oo, the stationary ergodic or otherwise behavior of the transition PT(e, .) will mimic that of the original p(e, .).
  • On the other hand, their test parameters k, n, and s are within their control, bounded only by computational limitations.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report




   
  
    
 
 
 
  
   
 
  

    



           
 
   
          
     




    

 

        
 

  





   



   
    

    

          
   
  
 
  
        
 
     
    
       
 


 
 
 







   
    

      

    




  
 


j








 

 




J




J

 
      






   

  



 







 









  

 
       

 
 
   
 

 

   

  


     
   


 
 
   

 
 


 
   

          

  
   



           


Citations
More filters

Journal ArticleDOI
Abstract: Two difficulties arise in the estimation of AB models: (i) the criterion function has no simple analytical expression, (ii) the aggregate properties of the model cannot be analytically understood. In this paper we show how to circumvent these difficulties and under which conditions ergodic models can be consistently estimated by simulated minimum distance techniques, both in a long-run equilibrium and during an adjustment phase.

104 citations


Journal ArticleDOI
Abstract: In this paper we introduce a class of nonlinear data generating processes (DGPs) that are "rst order Markov and can be represented as the sum of a linear plus a bounded nonlinear component. We use the concepts of geometric ergodicity and of linear stochastic comovement, which correspond to the linear concepts of integratedness and cointegratedness, to characterize the DGPs. We show that the stationarity test due to Kwiatowski et al. (1992, Journal of Econometrics, 54, 159}178) and the cointegration test of Shin (1994, Econometric Theory, 10, 91}115) are applicable in the current context, although the Shin test has a di!erent limiting distribution. We also propose a consistent test which has a null of linear cointegration (comovement), and an alternative of n C22

57 citations


Posted Content
TL;DR: This paper illustrates the use of the nonparametric Wald-Wolfowitz test to detect stationarity and ergodicity in agent-based models and shows that with appropriate settings the tests can detect non-stationarity and non-ergodicity.
Abstract: This paper illustrates the use of the nonparametric Wald-Wolfowitz test to detect stationarity and ergodicity in agent-based models. A nonparametric test is needed due to the practical impossibility to understand how the random component influences the emergent properties of the model in many agent-based models. Nonparametric tests on real data often lack power and this problem is addressed by applying the Wald-Wolfowitz test to the simulated data. The performance of the tests is evaluated using Monte Carlo simulations of a stochastic process with known properties. It is shown that with appropriate settings the tests can detect non-stationarity and non-ergodicity. Knowing whether a model is ergodic and stationary is essential in order to understand its behavior and the real system it is intended to represent; quantitative analysis of the artificial data helps to acquire such knowledge.

52 citations


Journal ArticleDOI
Abstract: We propose a set of algorithms for testing the ergodicity of empirical time series, without reliance on a specific parametric framework. It is shown that the resulting test asymptotically obtains the correct size for stationary and nonstationary processes, and maximal power against non-ergodic but stationary alternatives. The test will not reject in the presence of nonstationarity that does not lead to ergodic failure. The work is linked to recent research on reformulations of the concept of integrated processes of order zero, and we demonstrate the means to operationalize new concepts of "short memory" for economic time series. Limited Monte Carlo evidence is provided with respect to power against the non-stationary and non-ergodic alternative of unit root processes. The method is used to investigate debates over stability of monetary aggregates relative to GDP, and the mean reversion hypothesis with respect to high frequency data on exchange rates. The test also is applied to other macroeconomic time series, as well as to very high frequency data on asset prices. Both the Monte Carlo and data analysis results suggest that the test has very promising size and power.

26 citations


Journal ArticleDOI
Abstract: We propose a set of algorithms for testing the ergodicity of empirical time series, without reliance on a specific parametric framework. It is shown that the resulting test asymptotically obtains the correct size for stationary and nonstationary processes, and maximal power against non-ergodic but stationary alternatives. The test will not reject in the presence of nonstationarity that does not lead to ergodic failure. The method is used to investigate debates over stability of monetary aggregates relative to GDP, and the mean reversion hypothesis with respect to high frequency data on exchange rates. Both the Monte Carlo and data analysis results suggest that the test has good size and power performance.

20 citations


References
More filters

01 Jan 1994
TL;DR: The Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08.
Abstract: Note: Includes bibliographical references, 3 appendixes and 2 indexes.- Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08

19,744 citations


Book
01 Jan 1953

10,504 citations


Book
01 Jan 1982

3,152 citations


Book
16 Apr 1986
Abstract: This is a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods. Authors’ address: School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6. The authors’ research was sponsored by NSERC Grant A3456 and FCAR Grant 90-ER-0291. 1. The main paradigms The purpose of this chapter is to review the main methods for generating random variables, vectors and processes. Classical workhorses such as the inversion method, the rejection method and table methods are reviewed in section 1. In section 2, we discuss the expected time complexity of various algorithms, and give a few examples of the design of generators that are uniformly fast over entire families of distributions. In section 3, we develop a few universal generators, such as generators for all log concave distributions on the real line. Section 4 deals with random variate generation when distributions are indirectly specified, e.g, via Fourier coefficients, characteristic functions, the moments, the moment generating function, distributional identities, infinite series or Kolmogorov measures. Random processes are briefly touched upon in section 5. Finally, the latest developments in Markov chain methods are discussed in section 6. Some of this work grew from Devroye (1986a), and we are carefully documenting work that was done since 1986. More recent references can be found in the book by Hörmann, Leydold and Derflinger (2004). Non-uniform random variate generation is concerned with the generation of random variables with certain distributions. Such random variables are often discrete, taking values in a countable set, or absolutely continuous, and thus described by a density. The methods used for generating them depend upon the computational model one is working with, and upon the demands on the part of the output. For example, in a ram (random access memory) model, one accepts that real numbers can be stored and operated upon (compared, added, multiplied, and so forth) in one time unit. Furthermore, this model assumes that a source capable of producing an i.i.d. (independent identically distributed) sequence of uniform [0, 1] random variables is available. This model is of course unrealistic, but designing random variate generators based on it has several advantages: first of all, it allows one to disconnect the theory of non-uniform random variate generation from that of uniform random variate generation, and secondly, it permits one to plan for the future, as more powerful computers will be developed that permit ever better approximations of the model. Algorithms designed under finite approximation limitations will have to be redesigned when the next generation of computers arrives. For the generation of discrete or integer-valued random variables, which includes the vast area of the generation of random combinatorial structures, one can adhere to a clean model, the pure bit model, in which each bit operation takes one time unit, and storage can be reported in terms of bits. Typically, one now assumes that an i.i.d. sequence of independent perfect bits is available. In this model, an elegant information-theoretic theory can be derived. For example, Knuth and Yao (1976) showed that to generate a random integer X described by the probability distribution {X = n} = pn, n ≥ 1, any method must use an expected number of bits greater than the binary entropy of the distribution, ∑

3,112 citations


Book
01 Apr 1986
Abstract: Here is the first book to summarize a broad cross-section of the large volume of literature available on one-dimensional empirical processes. Presented is a thorough treatment of the theory of empirical processes, with emphasis on real random variable processes as well as a wide-ranging selection of applications in statistics. Featuring many tables and illustrations accompanying the proofs of major results, coverage includes foundations - special spaces and special processes, convergence and distribution of empirical processes, alternatives and processes of residuals, integral tests of fit and estimated empirical processes and martingale methods.

2,708 citations


Frequently Asked Questions (1)
Q1. What are the contributions in "Division of the humanities and social sciences" ?

Domowitz and El-Gama this paper developed a statistical test of stationary ergodicity for known Markovian processes on a density space.