An Introduction to Bootstrap Theory in Time Series Econometrics

While often simple to implement in practice, application of the bootstrap in econometric modeling of economic and financial time series requires establishing validity of the bootstrap. Establishing bootstrap asymptotic validity relies on verifying often nonstandard regularity conditions. In particular, bootstrap versions of classic convergence in probability and distribution, and hence of laws of large numbers and central limit theorems, are critical ingredients. Crucially, these depend on the type of bootstrap applied (e.g., wild or independently and identically distributed (i.i.d.) bootstrap) and on the underlying econometric model and data. Regularity conditions and their implications for possible improvements in terms of (empirical) size and power for bootstrap-based testing differ from standard asymptotic testing, which can be illustrated by simulations.

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Testing financial time series for autocorrelation: Robust Tests

CIENCIA ergo sum ◽

10.30878/ces.v27n3a6 ◽

2020 ◽

Vol 27 (3) ◽

pp. e96

Author(s):

Nelson Omar Muriel Torrero

Keyword(s):

Time Series ◽

Stock Returns ◽

Financial Time Series ◽

Real Data ◽

Small Sample ◽

Small Sample Properties ◽

Robust Tests ◽

Daily Stock Returns ◽

Portmanteau Statistics ◽

Empirical Size

Two modified Portmanteau statistics are studied under dependence assumptions common in financial applications which can be used for testing that heteroskedastic time series are serially uncorrelated without assuming independence or Normality. Their asymptotic distribution is found to be null and their small sample properties are examined via Monte Carlo. The power of the tests is studied under the MA and GARCH-in-mean alternatives. The tests exhibit an appropriate empirical size and are seen to be more powerful than a robust Box-Pierce to the selected alternatives. Real data on daily stock returns and exchange rates is used to illustrate the tests.

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Asymptotic behaviour for random walks in random environments

Journal of Applied Probability ◽

10.1239/jap/1032374457 ◽

1999 ◽

Vol 36 (2) ◽

pp. 334-349 ◽

Cited By ~ 33

Author(s):

S. Alili

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Transition Probabilities ◽

Invariant Probability Measure ◽

Random Environments ◽

Regularity Conditions ◽

Large Numbers ◽

Random Transition ◽

Independent Case ◽

Uniquely Ergodic

In this paper we consider limit theorems for a random walk in a random environment, (Xn). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (Xn)’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (Xn) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

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Asymptotic behaviour for random walks in random environments

Journal of Applied Probability ◽

10.1017/s0021900200017174 ◽

1999 ◽

Vol 36 (02) ◽

pp. 334-349 ◽

Cited By ~ 4

Author(s):

S. Alili

Keyword(s):

Random Walk ◽

Limit Theorems ◽

Transition Probabilities ◽

Invariant Probability Measure ◽

Random Environments ◽

Regularity Conditions ◽

Large Numbers ◽

Random Transition ◽

Independent Case ◽

Uniquely Ergodic

In this paper we consider limit theorems for a random walk in a random environment, (X n ). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (X n )’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (X n ) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.

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