A bootstrapped test of covariance stationarity based on orthonormal transformations

2025-04-30 1 0 522.27KB 25 页 10玖币

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arXiv:2210.14086v3 [math.ST] 21 May 2024

Submitted to Bernoulli

A bootstrapped test of covariance stationarity

based on orthonormal transformations

Jonathan B. Hill1and Tianqi Li2

1Dept. of Economics, University of North Carolina

2Dept. of Economics, University of North Carolina

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time

series. We work in a generalized version of the new setting in Jin, Wang and Wang (2015), who exploit Walsh

(1923) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict

stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric

estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonor-

mal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet

and Walsh functions, and we allow for linear or nonlinear processes with possibly non-iid innovations. This is

important in macroeconomics and ﬁnance where nonlinear feedback and random volatility occur in many settings.

We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation

difference statistic, where the maximum is taken over the correlation lag ℎand basis generated sub-sample counter

𝑘(the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and

counter H𝑇and K𝑇. Of particular note, our test is capable of detecting breaks in variance, and distant, or very

mild, deviations from stationarity.

Keywords: Covariance stationarity; max-correlation test; multiplier bootstrap; orthonormal basis; Walsh functions

1. Introduction

Assume {𝑋𝑡:𝑡∈Z}is a possibly non-stationary time series process in L2. We want to test whether 𝑋𝑡

is covariance stationary, without explicitly assuming stationarity under the null hypothesis, allowing

for linear or nonlinear processes with a possibly non-iid innovation, and a general memory property.

Such generality is important in macroeconomics and ﬁnance where nonlinear feedback and non-iid

innovations occur in many settings due to asymmetries and random volatility, including exchange rates,

bonds, interest rates, commodities, and asset return levels and volatility. Popular models for such time

series include symmetric and asymmetric GARCH, Stochastic Volatility, nonlinear ARMA-GARCH,

and switching models like smooth transition autoregression. See, e.g., Teräsvirtra (1994), Gray (1996)

and Francq and Zakoïan (2019).

Evidence for nonstationarity, whether generally or in the variance or autocovariances, has been

suggested for many economic time series, where breaks in variance and model parameters are well

known (e.g. Busett and Taylor,2003,Gianetto and Raissi,2015,Hendry and Massmann,2007,Perron,

2006). Knowing whether a time series is globally nonstationary has large implications for how ana-

lysts approach estimation and inference. Indeed, it effects whether conventional parametric and semi-

(non)parametric model speciﬁcations are correct. Pretesting for deviations from global stationarity

therefore has important practical value.

There are many tests in the literature on covariance stationarity, and concerning locally station-

ary processes. Tests for stationarity based on spectral or second order dependence properties have a

long history, where pioneering work is due to Priestley and Subba Rao (1969). Spectrum-based tests

with L2-distance components have many versions. Paparoditis (2010) uses a rolling window method

to compare subsample local periodograms against a full sample version. The maximum is taken over

the L2-distance between periodograms over all time points. An asymptotic theory for the max-statistic,

however, is not provided, although an approximation theory is (see their Lemmas 1 and 3). Furthermore,

conforming with many offerings in the literature, under the null 𝑋𝑡is a linear process with iid Gaus-

sian innovations. Dette, Preuß and Vetter (2011) study locally stationary processes, and impose linearity

with iid Gaussian innovations. Their statistic is based on the minimum L2-distance between a spec-

tral density and its version under stationarity, and local power is non-trivial against 𝑇1/4-alternatives.

Aue et al. (2009) propose a nonparametric test for a break in covariance for multivariate time series

based on a version of a cumulative sum statistic.

Wavelet methods have arisen in various forms recently. von Sachs and Neumann (2000), using

technical wavelet decomposition components from Neumann and von Sachs (1997), propose a Haar

wavelet based localized periodogram test of covariance stationarity for locally stationary processes (cf.

Dahlhaus,1997,2009), but neglect to characterize power. Haar wavelet functions form an orthonormal

basis on L2[0,1), but the proposed frequency domain tests are complicated, a local power analysis

is not feasible, and empirical power may be weak (see simulation evidence from Jin, Wang and Wang

(2015)).

Dwivedi and Subba Rao (2011) and Jentsch and Rao (2015) use the discrete Fourier transform

[DFT] 𝐽𝑇(𝜔𝑘)=(2𝜋𝑇)−1/2Í𝑇

𝑡=1𝑋𝑡exp {𝑖𝑡𝜔𝑘}at canonical frequencies 𝜔𝑘=2𝜋𝑘/𝑇and 1 ≤

𝑘≤𝑇.Dwivedi and Subba Rao (2011) generate a portmanteau statistic from a normalized sam-

ple DFT covariance, exploiting the fact that an uncorrelated DFT implies second order stationar-

ity. Nason (2013) presents a covariance stationarity test based on Haar wavelet coefﬁcients of the

wavelet periodogram, they assume linear local stationarity, and do not treat local power. See also

Nason, von Sachs and Kroisandt (2000).

In a promising offering in the wavelet literature, Jin, Wang and Wang (2015) [JWW] exploit so-

called Walsh functions (akin to “global square waves” although not truly wavelets; cf. Walsh (1923)

and their implied systematic samples for comparing sub-sample covariances with the full sample one.

They utilize a sample-size dependent maximum lag H𝑇and maximum systematic sample counter K𝑇,

and show their Wald test exhibits non-negligible local power against √𝑇-alternatives. They do not

consider any other orthonormal transformation because Walsh functions, they argue, have “desirable

properties” based primarily on simulation evidence, asymptotic independence of a sub-sample and

sample covariance difference (√𝑇(ˆ𝛾(𝑘1)

ℎ−ˆ𝛾ℎ),√𝑇(ˆ𝛾(𝑘2)

ℎ−ˆ𝛾ℎ)) across systematic samples 𝑘1≠𝑘2,

and joint asymptotic normality (JWW, p. 897). It seems, however, that such theoretical properties are

available irrespective of the orthonormal basis used, although we do not provide a proof. See Section

2.1, below, for deﬁnitions and notation. We do, however, ﬁnd in the sequel that the Walsh basis has

superlative properties vis-à-vis a Haar wavelet basis.

JWW’s asymptotic analysis is driven by local stationarity and linearity 𝑋𝑡=Í∞

𝑖=0𝜓𝑖𝑍𝑡−𝑖, with zero

mean iid 𝑍𝑡, and 𝐸|𝑍𝑡|4+𝛿<∞,𝛿 > 0, which expedites characterizing a parametric asymptotic covari-

ance matrix estimator. The iid and linearity assumptions, however, rule out many important processes,

including nonlinear models like regime switching, and random coefﬁcient processes, and any pro-

cess with a non-iid error (e.g. nonlinear ARMA-GARCH). JWW’s Wald-type test statistic requires an

inverted parametric variance estimator that itself requires ﬁve tuning parameters and choice of two ker-

nels.1Indeed, most of the tuning parameters only make sense under linearity given how they approach

asymptotic covariance matrix estimation.

1One tuning parameter 𝜆∈ (0, .5)governs the number 𝑄𝑇=[𝑇𝜆]of sample covariances that enter the asymptote covariance

matrix estimator (see their p. 899). The remaining four (𝑐1, 𝑐2;𝜉1, 𝜉2)are used for kernel bandwidths 𝑏𝑗=𝑐𝑗𝑇−𝜉𝑗,𝑗=1,2,

for computing the kurtosis of the iid process 𝑍𝑡under linearity (see p. 902-903). The authors set 𝑐𝑗equal to 1.2 times a so-called

"crude scale estimate" which is nowhere deﬁned.

A bootstrapped test of covariance stationarity 3

Now deﬁne the lag ℎautocovariance coefﬁcient at time 𝑡:

𝛾ℎ(𝑡) ≡ 𝐸[(𝑋𝑡−𝐸[𝑋𝑡]) (𝑋𝑡−ℎ−𝐸[𝑋𝑡−ℎ])],ℎ=0,1, ...

The hypotheses are:

𝐻0:𝛾ℎ(𝑠)=𝛾ℎ(𝑡)=𝛾ℎ∀𝑠, 𝑡,∀ℎ=0,1, ... (cov. stationary) (1.1)

𝐻1:𝛾ℎ(𝑠)≠𝛾ℎ(𝑡)for some 𝑠≠𝑡and ℎ=0,1, ... (cov. nonstationary).

Under 𝐻0𝑋𝑡is second order stationary, and the alternative is any deviation from the null: the autoco-

variance differs across time at some lag, allowing for a (lag zero) break in variance. The null hypothesis

otherwise accepts the possibility of global nonstationarity.

In this paper we do away with parametric assumptions on 𝑋𝑡, and impose either a mixing or physical

dependence property that allows us to bound the number of usable covariance lags H𝑇and systematic

samples K𝑇. The conditions allow for global nonstationarity under either hypothesis, allowing us to

focus the null hypothesis only on second order stationarity. We show that use of the physical depen-

dence construct in Wu (2005) is a boon for bounding H𝑇since it allows for slower than geometric

memory decay, and ultimately need only hold uniformly over (ℎ, 𝑘). The mixing condition imposed,

however, requires geometric decay, and must ultimately hold jointly over all lags ℎ=1, ..., H𝑇(ℎis the

covariance lag, and 𝑘is a systematic subsample counter). The latter leads to a much diminished upper

bound on the maximum lag growth H𝑇→ ∞. This may be of independent interest given the recent rise

of high dimensional central limit theorems under weak dependence (eg. Chang, Chen and Wu,2024,

Chang, Jiang and Shao,2023,Zhang and Wu,2017).

Rather than operate on a Wald statistic constructed from a speciﬁc orthonormal transformation of

covariances, our statistic is the maximum generic orthonormal transformed sample correlation coefﬁ-

cient, where the maximum is taken over (ℎ, 𝑘) with increasing upper bounds (H𝑇,K𝑇). By working

in a generic setting we are able to make direct comparisons, and combine bases for possible power

improvements.

We provide examples of Haar wavelet and Walsh functions in Sections 2.1 and 2.2, and show how

they yield different systematic samples. This suggests a power improvement may be available by using

multiple orthonormal transforms. As JWW (p. 897) note, however, clearly other orthonormal transfor-

mations are feasible, although simulation evidence agrees with their suggestion that the Walsh basis

works quite well.

We use a dependent wild bootstrap for the resulting test statistic, allowing us to sidestep asymptotic

covariance matrix estimation, a challenge considering we do not assume a parametric form, and the null

hypothesis requires us to look over a large set of (ℎ, 𝑘). We sidestep all of JWW’s tuning parameters,

and require just one governing the block size for the bootstrap. We ultimately achieve a signiﬁcantly

better upper bound on the rate of increase for (H𝑇,K𝑇)than JWW. Penalized and weighted versions

of our test statistic are also possible, as in JWW and Hill and Motegi (2020) respectively. There is,

though, no compelling theory to justify penalties on (ℎ, 𝑘)in our setting, and overall a non-penalized

and unweighted test statistic works best in practice.

Note that Hill and Motegi (2020) study the max-correlation statistic for a white noise test, and only

show their limit theory applies for some increasing maximum lag H𝑇, but do not derive an upper

bound. In the present paper we use a different asymptotic theory, derive upper bounds for H𝑇and K𝑇,

and of course do not require a white noise property under 𝐻0.

Jin, Wang and Wang (2015, Section 2.6) rule out the use of autocorrelations because, they claim, if

the sample variance were included, i.e. ℎ≥0, then consistency may not hold because the limit theory

neglects the joint distribution of ˆ𝛾0and the correlation differences. We show for our proposed test that

the difference between full sample and systematic sample autocorrelations at lag zero asymptotically

reveals whether 𝐸[𝑋2

𝑡]is time dependent. Further, our test is consistent whether non-stationarity is

caused by variances, or covariances, or both. See Section 3.3 and Example 3.5. Our proposed test is

consistent against a general (nonparametric) alternative, and exhibits nontrivial power against a se-

quence of √𝑇-local alternatives.

The max-correlation difference is particularly adept at revealing subtle deviations from covariance

stationarity, similar to results revealed in Hill and Motegi (2020). Consider a distant form of a model

treated in Paparoditis (2010, Model I) and Jin, Wang and Wang (2015, Section 3.2: models NVI, NVII),

𝑋𝑡=.08 cos{1.5−cos(4𝜋𝑡/𝑇)}𝜖𝑡−𝑑+𝜖𝑡with large 𝑑(JWW use 𝑑=1 or 6). JWW’s test exhibits

trivial power when 𝑑≥20, while the max-correlation difference is able to detect this deviation from

the null even when 𝑑≥50. The reason is the same as that provided in Hill and Motegi (2020): the

max-correlation difference operates on the single most useful statistic, while Wald and portmanteau

statistics congregate many standardized covariances that generally provide little relevant information

under a weak signal.

In Section 2we develop the test statistic. Sections 3and 4present asymptotic theory and the bootstrap

method and theory. We then perform a Monte Carlo study in Section 5, and conclude with Section

6. The supplemental material Hill and Li (2024) contains all proofs, an empirical study concerning

international interest rates, and complete simulation results.

We use the following notation. [𝑧]rounds 𝑧to the nearest integer. L2is the space of square integrable

random variables; L2[𝑎, 𝑏)is the class of square integrable functions on [𝑎, 𝑏).|| ·||𝑝and || · || are the

𝐿𝑝and 𝑙2norms respectively, 𝑝≥1. Let Z≡ {... −2,−1,0,1,2, ...}, and N≡ {0,1,2, ...}.𝐾 > 0 is

a ﬁnite constant whose value may be different in different places. 𝑎𝑤 𝑝1 denotes “asymptotically with

probability approaching one”. Write maxH𝑇=max0≤ℎ≤H𝑇. maxK𝑇=max1≤𝑘≤K𝑇and maxH𝑇,K𝑇=

max0≤ℎ≤H𝑇,1≤𝑘≤K𝑇. Similarly, maxH𝑇𝑎(ℎ, ˜

ℎ)=max0≤ℎ, ˜

ℎ≤H𝑇𝑎(ℎ, ˜

ℎ), etc. |𝑎|+≡𝑎∨0.

2. Max-correlation with orthonormal transformation

Our test statistic is the maximum of an orthonormal transformed sample covariance. In order to build

intuition, we ﬁrst derive the test statistic under Walsh function and Haar wavelet-based bases. We then

set up a general environment, and present the main results.

In order to reduce notation, assume here 𝜇≡𝐸[𝑋𝑡]=0 is known. In practice this is enforced by

using 𝑋𝑡−¯

𝑋where ¯

𝑋≡1/𝑇Í𝑇

𝑡=1𝑋𝑡. In Hill and Li (2024, Lemmas B.3 and B.3∗) we prove using 𝑋𝑡

−¯

𝑋or 𝑋𝑡−𝜇leads to identical results asymptotically. Thus in proofs of the main results we simply

assume 𝜇=0.

2.1. Walsh functions

The following class of Walsh functions {𝑊𝑖(𝑥)} ≡ {𝑊𝑖(𝑥):𝑖=0,1,2, ...}deﬁne a complete orthonor-

mal basis in L2[0,1). The functions 𝑊𝑖(𝑥)are deﬁned recursively (see, e.g., Ahmed and Rao,1975,

Stoffer,1987,1991,Walsh,1923):

𝑊0(𝑥)=1 for 𝑥∈ [0,1)and 𝑊1(𝑥)=1, 𝑥 ∈ [0, .5)

−1, 𝑥 ∈ [.5,1),

and for any 𝑖=1,2, ...,

𝑊2𝑖(𝑥)=𝑊𝑖(2𝑥), 𝑥 ∈ [0, .5)

(−1)𝑖𝑊𝑖(2𝑥−1), 𝑥 ∈ [.5,1)and 𝑊2𝑖+1(𝑥)=𝑊𝑖(2𝑥), 𝑥 ∈ [0, .5)

(−1)𝑖+1𝑊𝑖(2𝑥−1), 𝑥 ∈ [.5,1).

A bootstrapped test of covariance stationarity 5

In the {−1,1}-valued sequence {𝑊𝑖(𝑥):𝑖=0,1,2, ...},𝑖indexes the number of zero crossings,

yielding a square shaped wave-form. See Figure 1, below, and see Stoffer (1991, Figure 5) and

Jin, Wang and Wang (2015, Figure 1) and their references. The 𝑘𝑡 ℎ discrete Walsh functions used in

this paper are then for 𝑡=1, ..., 𝑇 :

{W𝑘(1), ..., W𝑘(𝑇)} where W𝑘(𝑡)=𝑊𝑘((𝑡−1)/𝑇).

Now deﬁne the covariance coefﬁcient for a covariance stationary time series, 𝛾ℎ≡𝐸[𝑋𝑡𝑋𝑡−ℎ], and

denote the usual (co)variance estimator ˆ𝛾ℎ≡1/𝑇Í𝑇−ℎ

𝑡=1𝑋𝑡𝑋𝑡+ℎ, ℎ ∈N.JWW use {W𝑖(𝑥)} to construct

a set of discrete Walsh covariance transformations: for some integer K ≥ 1,

ˆ𝛾𝑊(𝑘)

ℎ≡1

𝑇

𝑇−ℎ

𝑡=1

𝑋𝑡𝑋𝑡+ℎn1+ (−1)𝑘−1W𝑘(𝑡)o, ℎ =0,1, ..., 𝑇 −1,and 𝑘=1,2, ..., K.(2.1)

As they point out, a sequence of systematic (sub)samples 𝑻𝑊

𝑘:𝑘=1,2, ..., Kin the time domain can

be deﬁned on the basis of Walsh functions:

𝑻𝑊

𝑘≡n𝑡∈𝑇:(−1)𝑘−1W𝑘(𝑡)=1o.

Now let N𝑘be the smallest power of 2 that is at least 𝑘. The ﬁrst systematic sample is the ﬁrst half

of the sample time domain 𝑻𝑊

1={1, ..., [𝑇/2]}; the second is the middle half 𝑻𝑊

2={[𝑇/4],[𝑇/4] +

1, ..., [3𝑇/4]}; the third 𝑻3is the ﬁrst and third time blocks, and so on. Notice 𝑻𝑊

𝑘consists of (𝑘+

1)/2 blocks with at least [𝑇/N𝑘]elements. Thus, when ℎ < 𝑇 /N𝑘then ˆ𝛾𝑊(𝑘)

ℎis just an estimate of 𝛾ℎ

on the 𝑘𝑡 ℎ systematic sample:

ˆ𝛾𝑊(𝑘)

ℎ=1

𝑇

𝑇−ℎ

𝑡=1

𝑋𝑡𝑋𝑡+ℎn1+ (−1)𝑘−1W𝑘(𝑡)o=2

𝑇Õ

𝑡∈𝑻𝑘

𝑋𝑡𝑋𝑡+ℎ.

The condition ℎ < 𝑇/N𝑘holds asymptotically in the Section 4bootstrap setting.

The difference between the 𝑘𝑡 ℎ systematic sample and full sample estimators is:

ˆ𝛾𝑊(𝑘)

ℎ−ˆ𝛾ℎ=(−1)𝑘−11

𝑇

𝑇−ℎ

𝑡=1

𝑋𝑡𝑋𝑡+ℎW𝑘(𝑡).

Notice the {−1,1}-valued nature of W𝑘(𝑡)yields a sub-sample comparison: ˆ𝛾𝑊(𝑘)

ℎ−ˆ𝛾ℎ=

1/𝑇Í𝑡∈𝑻𝑘𝑋𝑡𝑋𝑡+ℎ−1/𝑇Í𝑡≠𝑻𝑘𝑋𝑡𝑋𝑡+ℎ. Our test is based on the maximum |ˆ𝛾𝑊(𝑘)

ℎ−ˆ𝛾ℎ|, in which

case the multiple (−1)𝑘−1is irrelevant. We therefore drop it everywhere. Under the null hypothesis and

mild assumptions this difference is 𝑂𝑝(1/√𝑇)at all lags ℎand for all systematic samples 𝑘. Thus, a

test statistic can be constructed from √𝑇(ˆ𝛾𝑊(𝑘)

ℎ−ˆ𝛾ℎ).

2.2. Haar wavelet functions

Deﬁne the usual Haar wavelet functions 𝜓𝑘,𝑚 (𝑥) ≡ 2𝑘/2𝜓(2𝑘𝑥−𝑚)with 𝑥∈R, where 0 ≤𝑘≤ K𝑇for

some integer sequence {K𝑇}, 0 ≤𝑚≤2𝑘−1, and mother wavelet (Haar,1910):

𝜓(𝑥)=









1, 𝑥 ∈ [0, .5)

−1, 𝑥 ∈ [.5,1)

0 otherwise

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摘要：

arXiv:2210.14086v3[math.ST]21May2024SubmittedtoBernoulliAbootstrappedtestofcovariancestationaritybasedonorthonormaltransformationsJonathanB.Hill1andTianqiLi21Dept.ofEconomics,UniversityofNorthCarolina2Dept.ofEconomics,UniversityofNorthCarolinaWeproposeacovariancestationaritytestforanotherwisedepende...

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