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Econometrics and Statistics

Available online 25 December 2021
Econometrics and Statistics

Partially one-sided semiparametric inference for trending persistent and antipersistent processes

https://doi.org/10.1016/j.ecosta.2021.12.007Get rights and content

Abstract

Hypothesis testing in models allowing for trending processes that are possibly nonstationary and non-Gaussian is considered. Using semiparametric estimators, joint hypothesis testing for these processes is developed, taking into account the one-sided nature of typical hypotheses on the persistence parameter in order to gain power. The results are applicable for a wide class of processes and are easy to implement. They are illustrated with an application to the dynamics of GDP.

Introduction

Testing for persistence is an important subject in time series. It is particularly of interest to macroeconomists to determine how Gross Domestic Product (GDP) and other variables evolve. Due to its potential impact on the choice of economic stabilization policies, this question has generated a huge literature that was largely started by Nelson and Plosser (1982). The current paper considers testing joint hypotheses about the extent of persistence and the possibility of a trend in time series. We use the results of Abadir and Distaso (2007), with the implication here that power can be gained by modifying tests of joint hypotheses to take into account the fact that inference on the persistence parameter is typically one-sided, whereas inference on the remaining components are two-sided. The trend parameters are estimated in the time domain, but we estimate the persistence in the frequency domain using the Fully-Extended Local Whittle (FELW) estimator of Abadir, Distaso, Giraitis, 2007, Abadir, Distaso, Giraitis, 2011 which extends to nonstationarity the classical local Whittle estimator proposed by Künsch (1987) and Robinson (1995). By virtue of the specification being semiparametric, it generates robust inference: it allows for seasonality and other effects to be present at nonzero spectral frequencies and it is valid for a wide class of generating processes that include non-Gaussian ones. It is also easily usable in applied work.

There are precursors to using frequency-domain estimators (including ones obtained via autocorrelation functions) in testing hypotheses about trending persistent series. First, Robinson (1994) introduces such tests that are applied in Gil-Alaña and Robinson (1997). However, the partially one-sided nature of the joint hypotheses is not taken into account in their setup (see the alternative hypothesis in their (28)) and there is power to be gained from doing so. Second, Dolado, Gonzalo, Mayoral, 2008, Dolado, Gonzalo, Mayoral, 2009 allow for trends in the efficient formulation which Lobato and Velasco (2006) introduce as a modification of the original one in Dolado et al. (2002). They generalize the tests of Dickey and Fuller (1979) to allow for fractional persistence. They detrend the series but do not consider joint hypotheses on the trend as well as persistence, which is done by Gil-Alaña and Robinson (1997) and by Dickey and Fuller (1981). Our procedure also differs from Dolado et al. (2002) in the robustness indicated in the previous paragraph when estimating the degree of persistence.

In this paper, p and d denote respectively convergence in probability and in distribution. We write 1A for the indicator of a set A, ν for the integer part of ν, C for a generic constant but

for specific constants. The lag operator is denoted by L, such that Lut=ut1, and the backward difference operator by :=1 L. We write i for the imaginary unit (principal value of 1), in roman typeface to distinguish it from the index i. Consider the processXt=α+βt+ut,t=1,2,...,n,where the sequence {ut} I(d) satisfies the following definition.

Definition 1.1

For d=k+dξ, where kZ is an integer and dξ(1/2,1/2), we say that {ut} is an I(d) process (also denoted by utI(d)) ifkut=ξt,t=1k,2k,,where {ξt} is a second order stationary sequence with spectral densityfξ(λ)=b0|λ|2dξ+o(|λ|2dξ),asλ0where b0>0.

Note that we use the term “stationarity” in a weaker sense than usual, only requiring the leading term of the spectrum to be as in (1.2). Few papers have so far considered such settings with an extended range for d to include regions of nonstationarity and to estimate a time trend, and to conduct joint hypothesis testing, as discussed earlier.

We will assume that the process {ξt} is a linear sequence as follows.

Assumption A.1. {ξt} is a linear sequenceξt=j=0ajεtj,where {aj,j0} are real nonrandom weights, j=0aj2<, and {εj} are i.i.d. variates with zero mean, unit variance, and finite fourth moment Eε04<. Moreover, the spectral density fξ(λ) of {ξt} has the propertyfξ(λ)=|λ|2dξ(b0+b1λ2+o(λ2)),asλ0,for some dξ(1/2,1/2), b0>0, and finite b0,b1. Defining A(λ):=j=0eijλaj, it is also required thatdA(λ)dλ=O(|A(λ)|/λ),asλ0+.

For convenience, we need the following assumption on the true d.

Assumption A.2. {ut} I(d), with d(1/2,3/2), d1/2.

It is technically straightforward to extend our results to all values of d>3/2 that give rise to nonstationarity, as well as to higher-order polynomials. We do not report such extensions, in order to keep the exposition as clear as possible and because the applications’ literature that we just cited requires at most linear trends.

In Section 2, we present the estimators and their basic properties for later use. Section 3 contains the construction of the new tests and their limiting distributions under the null and alternatives. Section 4 illustrates the gains of our approach by means of a simulation study. Section 5 demonstrates the ease of our approach by applying it to the dynamics of GDP. Proofs of the main results are given in Section 6.

Section snippets

The estimators and their properties

This brief section is not new, but it collects results we need from Abadir, Distaso, Giraitis, 2007, Abadir, Distaso, Giraitis, 2011 mainly and lays the ground for the derivations in the following sections. In order to estimate the slope parameter β and the location parameter α of (1.1), we use the standard least squares (LS) estimatorsβ^=t=1n(XtX¯)(tt¯)t=1n(tt¯)2,α^=X¯β^t¯,where X¯=n1t=1nXt and t¯=n1t=1nt=(n+1)/2 are the sample means of the variables. To estimate d, we start by

Testing joint hypotheses

In this section, we discuss testing joint hypotheses in (1.1). Recall that in the case d(1/2,1/2), we have that ut=ξt is stationary with memory parameter d. If d(1/2,3/2), then ut can be written asut=ut1+ξt,t=1,2,...,n,where {ξt} is a stationary I(dξ) process with the memory parameter dξ=d1(1/2,1/2). We assume that {ξt} is a linear process which satisfies Assumption A.1. Note that fractional ARIMA(p,dξ,q) sequences {ξt} with memory parameter dξ(1/2,1/2) satisfy Assumption A.1.

Definition 3.1

Let d0(1

Simulation results

We simulate the following Data Generating Process (DGP)yt=α+βt+ut,t=1,,n,where utI(d), for α=0, β=0,0.05,0.1, d=0.94,0.96,0.98,1 and n=250,500,1000. We consider three different bandwidths for the estimation of d, namely m=n0.65,n0.7,n0.75. The DGP is simulated 10,000 times.

The theory covers both persistent I(d) processes with d(0,1.5) and antipersistent ones with d(0.5,0). To save space, we include simulation results only for testing the null hypothesis H0: d=1 and β=0 against nearby

Empirical illustration: the case of US quarterly GDP

The methods outlined earlier will now be illustrated with the quarterly series of US GDP. This should be viewed as testing for persistence and trend in a context that is more general than Dickey-Fuller tests. It is not an attempt to model GDP, as we will explain that this requires further analysis. Data have been obtained form the Bureau of Economic Analysis and refer to seasonally-adjusted quarterly GDP values, expressed in billions of dollars at 2000 prices. The series is available from the

Proofs

Proof of Theorem 3.1

We assume that H0(d0,β0) is true. Notice that the first claim in (3.3) implies the second claim τdZ12+Z22.

By the Cramér-Wold device, to prove the weak convergence of the finite dimensional distributions (3.3), (τd0,τβ0(d^))d(Z1,Z2), it suffices to show that for any real numbers a1,a2,a1τd0+a2τβ0(d^)da1Z1+a2Z2.First we outline some properties of τd0 and τβ0(d^), used in the proof.

By Theorem 2.1 in Abadir et al. (2011), we have that E(β^β0)2Cn3+2d0. Now, the estimated residuals can be

Acknowledgement.

We would like to the thank the Associate Editor and two Referees for constructive comments on an earlier version of the paper, and Violetta Dalla for expert advice on simulations. We are grateful for the comments of seminar participants at the Bank of Italy, Bilbao, Econometric Society World Congress (London), GREQAM, Liverpool, LSE, Oxford, Queen Mary, Tilburg, Tinbergen Institute, Vilnius Academy of Science, York. This research is supported by the ESRC grants R000239538, RES000230176, and

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