# A Consistent Test of Stationary Ergodicity

Abstract: A 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.

## 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

##### Citations

104 citations

57 citations

52 citations

26 citations

20 citations

##### References

19,744 citations

3,112 citations

2,708 citations