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Unit roots in economic and financial time series: a re-evaluation at the decision-based significance levels.

1. Introduction

2. decision-based level of significance for unit root tests, 2.1. decision-theoretic approach to unit root testing, 2.2. line of enlightened judgement and decision-based significance levels, 2.3. factors affecting the decision-based significance level, 3. calibration rules based on asymptotic local power.

• Model with a constant only ADF: α ^ * = 0.825 − 0.505 p + 0.028 k − 0.025 c − 0.002 X 0 * Phillips–Perron: α ^ * = 0.827 − 0.516 p + 0.029 k − 0.025 c − 0.002 X 0 * DF-GLS: α ^ * = 0.525 − 0.339 p + 0.018 k − 0.019 c + 0.025 X 0 * ERS-P: α ^ * = 0.509 − 0.309 p + 0.017 k − 0.018 c + 0.026 X 0 *
• Model with a constant and a linear trend ADF: α ^ * = 0.933 − 0.655 p + 0.037 k − 0.023 c − 0.001 X 0 * Phillips–Perron: α ^ * = 0.932 − 0.665 p + 0.037 k − 0.023 c − 0.001 X 0 * DF-GLS: α ^ * = 0.802 − 0.546 p + 0.031 k − 0.024 c + 0.017 X 0 * ERS-P: α ^ * = 0.793 − 0.541 p + 0.031 k − 0.023 c + 0.018 X 0 *

4. Re-Evaluation of Past Empirical Results

4.1. extended nelson–plosser data, 4.2. elliott–pesavento data, 4.3. rapach–weber data, 4.4. application of the calibration rules, 4.5. decision-based significance level under a specific loss function, 5. conclusions, supplementary materials, acknowledgments, author contributions, conflicts of interest, appendix a. further sensitivity analyses.

• The real GNP and real per capita GNP are found to be trend-stationary for all c and X 0 * values, for both ADF and DF–GLS tests.
• The employment and money stock are found to be trend-stationary for all c and X 0 * values, for both ADF and DF–GLS tests.
• The nominal GNP is found to be trend-stationary only when c = 2.5. For other c values, it is found to be difference-stationary.
• Other nominal and price variables are found to be difference-stationary for all values of c and X 0 * values.
• The results are nearly the same as those reported in Section 4.1 of the paper, showing little sensitivity to the c and X 0 * values.

• For the ADF test, most of real exchange rates are found to be stationary under a wide range of c values, showing little sensitivity to the X 0 * value. For eight time series, the null hypothesis is rejected for all c values or c ∊ {2.5, 7.5, 12.5}.
• For the DF–GLS test, most of real exchange rates are found to be stationary under a wide range of c values. For ten time series, the null hypothesis is rejected for all c values considered.
• When the DF–GLS test is used, there are cases where the results are sensitive to the choice of X 0 * value, which might be expected from the nature of the DF–GLS test.

• For the ADF test most of real interest rates are found to be stationary under a wide range of c values, showing little sensitivity to the X 0 * value. For six time series, the null hypothesis is rejected for all c values or c ∊ {2.5, 7.5, 12.5}. Two additional time series are found to be stationary when c ∊ {2.5, 7.5}.
• For the DF–GLS test, most of the real exchange rates are found to be stationary under a wide range of c values. For five time series, the null hypothesis is rejected for all c values considered or c ∊ {2.5, 7.5, 12.5}. Three additional time series are found to be stationary when c ∊ {2.5, 7.5}.
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Kim, J.H.; Choi, I. Unit Roots in Economic and Financial Time Series: A Re-Evaluation at the Decision-Based Significance Levels. Econometrics 2017 , 5 , 41. https://doi.org/10.3390/econometrics5030041

Kim JH, Choi I. Unit Roots in Economic and Financial Time Series: A Re-Evaluation at the Decision-Based Significance Levels. Econometrics . 2017; 5(3):41. https://doi.org/10.3390/econometrics5030041

Kim, Jae H., and In Choi. 2017. "Unit Roots in Economic and Financial Time Series: A Re-Evaluation at the Decision-Based Significance Levels" Econometrics 5, no. 3: 41. https://doi.org/10.3390/econometrics5030041

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Article Contents

1. introduction, 2. main results, 3. simulation study, acknowledgement, supplementary material.

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Testing for unit roots based on sample autocovariances

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Jinyuan Chang, Guanghui Cheng, Qiwei Yao, Testing for unit roots based on sample autocovariances, Biometrika , Volume 109, Issue 2, June 2022, Pages 543–550, https://doi.org/10.1093/biomet/asab034

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We propose a new unit-root test for a stationary null hypothesis |$H_0$| against a unit-root alternative |$H_1$|⁠ . Our approach is nonparametric as |$H_0$| assumes only that the process concerned is |$I(0)$|⁠ , without specifying any parametric forms. The new test is based on the fact that the sample autocovariance function converges to the finite population autocovariance function for an |$I(0)$| process, but diverges to infinity for a process with unit roots. Therefore, the new test rejects |$H_0$| for large values of the sample autocovariance function. To address the technical question of how large is large, we split the sample and establish an appropriate normal approximation for the null distribution of the test statistic. The substantial discriminative power of the new test statistic is due to the fact that it takes finite values under |$H_0$| and diverges to infinity under |$H_1$|⁠ . This property allows one to truncate the critical values of the test so that it has asymptotic power 1; it also alleviates the loss of power due to the sample-splitting. The test is implemented in |$\texttt{R}$|⁠ .

Models with unit roots are frequently used for nonstationary time series. The importance of the unit-root concept stems from the fact that many economic, financial, business and social-domain data exhibit segmented trend-like or random wandering phenomena. While the random-walk-like behaviour of stock prices was noticed much earlier, for example by Jules Regnault, a French broker, in 1863 and by Louis Bachelier in his 1900 PhD thesis, the development of statistical inference for unit roots started only in the late 1970s. Nevertheless, the literature on unit-root tests is by now immense and diverse. We review only a selection of important developments below, leading naturally to the new test presented in this paper.

The Dickey–Fuller tests ( Dickey & Fuller, 1979 , 1981 ) deal with Gaussian random walks with independent errors. Efforts to relax the condition of independent Gaussian errors have led to, among others, the augmented Dickey–Fuller tests ( Said & Dickey, 1984 ; Elliott et al., 1996 ), which deal with auto-regressive errors, and the Phillips–Perron test ( Phillips, 1987 ; Phillips & Perron, 1988 ), which estimates the long-run variance of the error process nonparametrically. The augmented Dickey–Fuller tests have been further extended to deal with structural breaks in trend ( Zivot & Andrews, 1992 ), long memory processes ( Robinson, 1994 ), seasonal unit roots ( Chan & Wei, 1988 ; Hylleberg et al., 1990 ), bootstrap unit-root tests ( Paparoditis & Politis, 2005 ), nonstationary volatility ( Cavaliere & Taylor, 2007 ), panel data ( Pesaran, 2007 ) and local stationary processes ( Rho & Shao, 2019 ); see the survey papers by Stock (1994) and Phillips & Xiao (1998) , and the monographs by Hatanaka (1996) and Maddala & Kim (1998) for further references.

The Dickey–Fuller tests and their variants are based on regression of a time series on its first lag, in which the existence of a unit root is postulated as a null hypothesis in the form of the regression coefficient being equal to 1. This null hypothesis is tested against a stationary alternative hypothesis that the regression coefficient is smaller than 1. This setting leads to innate indecisive inference for ascertaining the existence of unit roots, as a statistical test is incapable of accepting a null hypothesis. To place the assertion of unit roots on firmer ground, Kwiatkowski et al. (1992) adopted a different approach: their proposed test considers a stationary null hypothesis against a unit-root alternative. It is based on a plausible representation of possible nonstationary time series in which a unit root is represented as an additive random-walk component. Then, under the null hypothesis, the variance of the random-walk component is zero. The test of Kwiatkowski et al. (1992) is the one-sided Lagrange multiplier test for testing the variance being zero against it being greater than zero.

Despite the many exciting developments mentioned above, testing for the existence of unit roots remains a challenge in time series analysis, as most available methods suffer from lack of accurate size control and low power. In this paper we propose a new test that is based on a radically different idea from existing approaches. Our setting is similar in spirit to that of Kwiatkowski et al. (1992) , in that we test a stationary null hypothesis |$H_0$| against a unit-root alternative |$H_1$|⁠ . However, our approach is nonparametric as |$H_0$| assumes only that the process concerned is |$I(0)$|⁠ , without specifying any parametric forms. The new test is based on the simple fact that under |$H_0$| the sample autocovariance function converges to the finite population autocovariance function, while under |$H_1$| it diverges to infinity. Therefore, we can reject |$H_0$| for large absolute values of the sample autocovariance function. To address the technical question of how large is large, we split the sample and establish an appropriate normal approximation for the null distribution of the test statistic. Our sample autocovariance function-based test statistic offers substantial discriminative power as it takes finite values under |$H_0$| and diverges to infinity under |$H_1$|⁠ . This property allows us to truncate the critical values determined by the normal approximation to ensure that the test has asymptotic power 1; furthermore, it alleviates the loss of power due to the sample-splitting, so that our test outperforms the test of Kwiatkowski et al. (1992) in a power comparison simulation. Another advantage of the new method is that it has remarkable discriminative power, being able to tell the difference between, for example, a random walk and an ar |$(1)$| with autoregressive coefficient close to but still smaller than 1, a case in which most available unit-root tests, including the method of Kwiatkowski et al. (1992) , suffer from weak discriminative power. Admittedly, the new test is technically sophisticated, which we argue is inevitable in order to gain improvement over existing methods. Nevertheless, we have developed an |$\texttt{R}$| ( R Development Core Team, 2022 ) function |$\texttt{ur.test}$| in the package |$\texttt{HDTSA}$| that implements the test in an automatic manner.

2.1. A power-one test

Let |$Y_t$| satisfy ( 2 ) with independent |$\epsilon_t\sim(0,\sigma_{\epsilon}^2)$| and |$\sum_{j=1}^\infty j|\psi_j|<\infty$|⁠ . Write |$a=\sum_{j=0}^\infty\psi_j$| and |$V_{d-1}(t)=F_{d-1}(t)-\int_0^1F_{d-1}(t)\,{\rm d}t$| where |$F_{d-1}(t)$| is the scalar multi-fold integrated Brownian motion defined recursively by |$F_j(t)=\int_0^tF_{j-1}(x)\,{\rm d}x$| for any |$j\geqslant1$|⁠ , with |$F_0(t)$| the standard Brownian motion. For any given integer |$k\geqslant 0$|⁠ , as |$n\rightarrow\infty$| we have that (i) |$n^{-(2d-1)}\hat{\gamma}(k)\rightarrow a^2\sigma_{\epsilon}^2\int_0^1V_{d-1}^2(t)\,{\rm d}t$| in distribution if |$\mu_d=0$|⁠ , and (ii) |$n^{-2d}\hat{\gamma}(k)\rightarrow\phi_{d,k}\mu_d^2$| in probability if |$\mu_d\neq 0$|⁠ , where |$\phi_{d,k}>0$| is a bounded constant depending only on |$d$| and |$k$|⁠ .

Formally, we reject |$H_0$| at the significance level |$\phi \in (0, 1)$| if |$T_n>{\small{\text{cv}}}_\phi$|⁠ , where |${\small{\text{cv}}}_\phi$| is the critical value satisfying |${\mathrm{pr}}_{H_0}(T_n>{\small{\text{cv}}}_\phi)\rightarrow\phi$|⁠ . As we will see in ( 3 ), |$\{\hat{\gamma}_1(k)\}_{k=0}^{K_0}$| are used to determine the critical value |${\small{\text{cv}}}_\phi$|⁠ . One obvious concern with splitting the sample into two halves is loss of testing power. However, the fact that |$T_n$| takes finite values under |$H_0$| and diverges to infinity under |$H_1$| implies that |$T_n$| has strong discriminative power to tell |$H_1$| apart from |$H_0$|⁠ , which is enough to provide more power than, for example, the test of Kwiatkowski et al. (1992) . Our simulation results indicate that the sample-splitting works well even for sample size |$n=80$|⁠ . Under |$H_0$|⁠ , write |$y_{t,k}=2\{(Y_t-\mu)(Y_{t+k}-\mu)-\gamma(k)\}{\rm sgn}(k+t-N-1/2)$|⁠ . For |$\ell\geqslant1$| define |$B_\ell^2=E\{(\sum_{t=1}^\ell Q_t)^2\}$| where |$Q_t=\sum_{k=0}^{K_0}\xi_{t,k}$| with |$\xi_{t,k}=2y_{t,k}\gamma(k)$|⁠ . The following regularity conditions are needed; see the Supplementary Material for a discussion of their validity.

Under |$H_0$|⁠ , |$\max_{1\leqslant t \leqslant n}E(|Y_t|^{2s_1})\leqslant c_1$| for two constants |$s_1\in(2,3]$| and |$c_1>0$|⁠ .

Under |$H_0$|⁠ , |$\{Y_t\}$| is |$\alpha$| -mixing with |$\alpha(\tau)=\sup_{t}\,\sup_{A \in {\mathcal F}_{-\infty}^t, B \in {\mathcal F}_{t+\tau}^{\infty}}|{\mathrm{pr}}(AB)-{\mathrm{pr}}(A){\mathrm{pr}}(B)|\leqslant c_2\tau^{-\beta_1}$| for any |$\tau\geqslant1$|⁠ , where |${\mathcal F}_{-\infty}^t$| and |$\mathcal{F}_{t+\tau}^\infty$| denote the |$\sigma$| -fields generated by |$\{Y_u\}_{u\leqslant t}$| and |$\{Y_u\}_{u\geqslant t+\tau}$|⁠ , respectively, and |$c_2>0$| and |$\beta_1>2(s_1-1)s_1/(s_1-2)^2$| are two constants, with |$s_1$| as specified in Condition 1.

Under |$H_0$|⁠ , there is a constant |$c_3>0$| such that |$B_\ell^2\geqslant c_3\ell$| for any |$\ell\geqslant1$|⁠ .

Let |${\small{\text{cv}}}_\phi$| be defined by ( 3 ) with |${\mathcal T}$| satisfying |${\mathrm{pr}}_{H_0}(\mathcal{T} )\rightarrow1$| and |${\mathrm{pr}}_{H_1}(\mathcal{T}^{\rm c} )\rightarrow1$|⁠ , and suppose that |$\hat{B}_{2N-K_0}/B_{2N-K_0} \rightarrow1$| in probability under |$H_0$| as |$n\rightarrow\infty$|⁠ . Then (i) |${\mathrm{pr}}_{H_0}(T_n>{\small{\text{cv}}}_\phi)\rightarrow\phi$| if the conditions of Theorem 1 hold, and (ii) |${\mathrm{pr}}_{H_1}(T_n>{\small{\text{cv}}}_\phi)\rightarrow1$| if |$Y_t$| satisfies ( 2 ) with independent |$\epsilon_t\sim (0,\sigma_{\epsilon}^2)$| and |$\sum_{j=1}^\infty j|\psi_j|<\infty$|⁠ .

2.2. Determining the event |$\mathcal{T}$| in ( 3 )

To use |$\mathcal{T}$| with finite samples, |$C_*$| must be specified according to the underlying process.

Let |$Y_t\sim I(1)$| satisfy ( 2 ) with independent |$\epsilon_t\sim(0,\sigma_{\epsilon}^2)$| and |$\sum_{j=1}^\infty j|\psi_j|<\infty$|⁠ . Write |$\eta=\sum_{j=0}^{\infty}\psi_j^2+\sum_{j=0}^{\infty}\psi_j\psi_{j+1}$|⁠ . As |$n\rightarrow\infty$|⁠ , we have that (i) |${n}^{-1}R \rightarrow 2a^2\eta^{-1}\int_0^1V_{0}^2(t)\,{\rm d}t$| in distribution if |$\mu_1=0$|⁠ , where |$a$| and |$V_0(t)$| are defined in Proposition 1, and (ii) |$n^{-2}R\rightarrow 6^{-1}\sigma_{\epsilon}^{-2}\eta^{-1}\mu_1^2$| in probability if |$\mu_1\neq 0$|⁠ .

Although the above specification was derived for |$Y_t \sim I(1)$|⁠ , our simulation results indicate that it also works well for |$I(2)$| processes. Testing |$I(0)$| against |$I(d)$| with |$d>1$| is easier than doing so with |$d=1$|⁠ , as the autocovariances are of order at least |$n^{2d-1}$| for |$I(d)$| processes; hence the difference between the values of |$T_n$| under |$H_1$| and those under |$H_0$| increases as |$d$| increases.

2.3. Estimation of |$B_{2N-K_0}^2$|

The kernel function |$\mathcal{K}(\cdot):\mathbb{R}\rightarrow[-1,1]$| is continuously differentiable on |$\mathbb{R}$| and is such that (i) |$\mathcal{K}(0)=1$|⁠ , (ii) |$\mathcal{K}(x)=\mathcal{K}(-x)$| for any |$x\in\mathbb{R}$|⁠ , and (iii) |$\int_{-\infty}^{\infty}|\mathcal{K}(x)|\,{\rm d}x<\infty$|⁠ . Let |$K_*=K_0+2$| satisfy |$K_*^{13}\log K_*=o(n^{1-2/s_2})$| with |$s_2$| as specified in Condition 5. The bandwidth |$b_m\rightarrow\infty$| as |$n\rightarrow\infty$| satisfies |$b_m=o\{n^{1/2-1/s_2}(K_*^5\log K_*)^{-1/2}\}$| and |$K_*^4=o(b_m)$|⁠ .

Under |$H_0$|⁠ , |$\max_{1\leqslant t \leqslant n}E(|Y_t|^{2s_2})\leqslant c_4$| for two constants |$s_2>4$| and |$c_4>0$|⁠ , and the |$\alpha$| -mixing coefficients |$\{\alpha(\tau)\}_{\tau\geqslant1}$| satisfy |$\alpha(\tau)\leqslant c_5\tau^{-\beta_2}$| for two constants |$c_5\,{>}\,0$| and |$\beta_2\,{>}\,\max\{2s_2/(s_2-2), s_2/(s_2-4)\}$|⁠ , where |$\alpha(\tau)$| is as defined in Condition 2.

Suppose that Conditions 4 and 5 hold. Then, as |$n\rightarrow\infty$|⁠ , |$\hat{B}_{2N-K_0}/B_{2N-K_0}\rightarrow1$| in probability under   |$H_0$|⁠ .

2.4. Implementation of the test

Based on § 2.2 and § 2.3 , Algorithm 1 outlines the steps of performing our test, which includes two tuning parameters. The algorithm is implemented in an |$\texttt{R}$| function |$\texttt{ur.test}$| in the package |$\texttt{HDTSA}$| ( Lin et al., 2021 ). To perform the test using function |$\texttt{ur.test}$|⁠ , one merely needs to input the time series |$\{Y_t\}_{t=1}^n$| and the nominal level |$\phi$|⁠ . The package sets the default value |$c_\kappa=0.55$| and returns the five testing results for |$K_0=0, 1, \ldots, 4$|⁠ . One can also set |$(c_\kappa,K_0)$| subjectively. We recommend using |$c_\kappa \in [0.45, 0.65]$| and |$K_0\in\{0,1,2,3,4\}$|⁠ .

To illustrate robustness with respect to the choice of |$(c_\kappa,K_0)$|⁠ , we apply our test to 14 U.S. annual economic time series ( Nelson & Plosser, 1982 ) that are often used for testing unit roots in the literature. The results with |$c_\kappa \in\{0.45, 0.55,0.65\}$| and |$K_0\in\{0,1,2, 3,4\}$| are exactly the same for each of the 14 time series; see the Supplementary Material for details.

Input : Time series |$\{Y_t\}_{t=1}^n$|⁠ , nominal level |$\phi$|⁠ , and two (optional) tuning parameters |$(c_\kappa,K_0)$|⁠ .

Step 1. Compute |$\hat{\gamma}(k)$|⁠ , |$\hat{\gamma}_1(k)$|⁠ , |$\hat{\gamma}_2(k)$| and |$\hat{\gamma}_x(k)$|⁠ . Put |$\hat{\rho}=\hat{\gamma}_x(1)/\hat{\gamma}_x(0)$|⁠ .

Step 2. Call function |$\texttt{lrvar}$| from the |$\texttt{R}$| package |$\texttt{sandwich}$|⁠ , with the default bandwidth of the function, to compute the long-run variances of |$\{\tilde{Q}_t\}$| and |$\{X_t\}$|⁠ , denoted by |$\tilde{V}_{2N-K_0}$| and |$\hat{\sigma}_{\rm L}^2$|⁠ , respectively, where |$\tilde{Q}_t$| is defined in § 2.3 . Put |$\hat{\lambda}=\hat{\gamma}_x(0)/\hat{\sigma}_{\rm L}^2$|⁠ .

Step 3. Calculate the test statistic |$T_n=\sum_{k=0}^{K_0}|\hat{\gamma}_2(k)|^2$| and the critical value |${\small{\text{cv}}}_\phi$| as in ( 3 ) with |$\hat{B}_{2N-K_0}=(2N-K_0)^{1/2}\tilde{V}_{2N-K_0}^{1/2}$| and |$\mathcal{T}$| given in ( 4 ) for |$C_*$| specified in ( 5 ).

Step 4. Reject |$H_0$| if |$T_n>{\small{\text{cv}}}_\phi$|⁠ .

We investigate the finite-sample properties of our test |$T_n$| by simulation with |$K_0\in\{0,1,2,3,4\}$| and |$c_\kappa\in\{0.45, 0.55, 0.65\}$|⁠ . We also consider |$T_n$| with the untruncated critical value |${\small{\text{cv}}}_{\phi,{\rm naive}}$|⁠ , i.e., |$c_\kappa=\infty$| in ( 5 ). Hualde & Robinson (2011) proposed the pseudo maximum likelihood estimator |$\hat{d}$| for the integration order |$d$| in the autoregressive fractionally integrated moving average models that can be used to construct a |$t$| -statistic |$\hat{d}/{\rm sd}(\hat{d})$| for |$H_0: d=0$| versus |$H_1:d\geqslant1$|⁠ . We call the test that rejects |$H_0$| if |$\hat{d}/{\rm sd}(\hat{d})>z_{1-\phi}$| the hr test, where |$z_{1-\phi}$| is the |$(1-\phi)$| -quantile of |$\mathcal{N}(0,1)$|⁠ . For comparison, we include the test of Kwiatkowski et al. (1992) and the hr test in our experiments. We set |$N=40, 70, 100$| and repeat each setting 2000 times. To examine the rejection probability of the tests under |$H_0$|⁠ , we consider the following three models.

Model 1: |$Y_t=\rho Y_{t-1}+\epsilon_t$|⁠ .

Model 2: |$Y_t=\epsilon_t+\phi_1\epsilon_{t-1}+\phi_2\epsilon_{t-2}$|⁠ .

Model 3: |$Y_t-\rho_1Y_{t-1}-\rho_2 Y_{t-2}=\epsilon_t+ 0.5 \epsilon_{t-1}+ 0.3 \epsilon_{t-2}$|⁠ .

To examine the rejection probability of the tests under |$H_1$|⁠ , we consider the following four models.

Model 4: |$\nabla Y_t=Z_t$|⁠ , |$Z_t=\rho Z_{t-1}+\epsilon_t$|⁠ .

Model 5: |$\nabla Y_t=Z_t$|⁠ , |$Z_t=\epsilon_t+\phi_1\epsilon_{t}+\phi_2\epsilon_{t-1}$|⁠ .

Model 6: |$\nabla Y_t=Z_t$|⁠ , |$Z_t-\rho_1Z_{t-1}-\rho_2 Z_{t-2}=\epsilon_t+0.5\epsilon_{t}+0.3\epsilon_{t-1}$|⁠ .

Model 7: |$\nabla^2 Y_t=Z_t$|⁠ , |$Z_t=\epsilon_t+\phi_1\epsilon_{t}+\phi_2\epsilon_{t-1}$|⁠ .

Unless specified otherwise, we always assume that |$\epsilon_t\sim\mathcal{N}(0, \sigma_{\epsilon}^2)$| independently with |$\sigma_\epsilon^2=1$| or 2 and set the nominal level |$\phi$| to |$5\%$|⁠ . The results with different |$(c_\kappa,K_0)$| are similar, indicating once again that our test is robust with respect to the choice of |$(c_\kappa,K_0)$|⁠ . We list the results with |$K_0=0$| and |$\sigma_{\epsilon}^2=1$| in Table 1 , and report other results and the |$\epsilon_t\sim t(2)$| and |$\epsilon_t\sim t(5)$| cases in the Supplementary Material .

Rejection probabilities |$(\%)$| of the proposed test |$T_n$| with |$K_0\,{=}\,0$| and |$c_\kappa\,{=}\,0.45, 0.55, 0.65, \infty$|⁠ , the test of Kwiatkowski et al. (1992) , and the hr test; the nominal level is |$5\%$|

KPSS, the test of Kwiatkowski et al. (1992) ; HR, the test that rejects |$H_0$| if |$\hat{d}/{\rm sd}(\hat{d})>z_{1-\phi}$| where |$z_{1-\phi}$| is the |$(1-\phi)$| -quantile of |$\mathcal{N}(0,1)$|⁠ .

Overall the rejection probabilities of our test under |$H_0$| are close to the nominal level |$\phi=5\%$|⁠ , especially when |$n$| is large, such as |$N=100$|⁠ . The performance of our test is stable across different models with different parameters, different |$K_0$| and different innovation distributions, whereas that of Kwiatkowski et al. ’s test and of the hr test vary and are adequate only in some settings. Table 1 indicates that our test works well for Model 1 with both positive and negative |$\rho$|⁠ , while Kwiatkowski et al. ’s test and the hr test perform poorly when |$\rho<0$| and even worse when |$\rho>0$|⁠ . Kwiatkowski et al. ’s test and the hr test completely fail when |$\rho=0.9$|⁠ , as the rejection probabilities are at least 46.7%. This is due to the fact that when |$\rho$| is close to 1, Kwiatkowski et al. ’s test and the hr test have difficulties distinguishing |$\rho$| from 1, which is a unit root; see also Table 3 of Kwiatkowski et al. (1992) . Our test does not suffer from this closeness to 1, as the order of the magnitude of the autocovariance function matters. Our test and that of Kwiatkowski et al. (1992) work well for Model 2, while the hr test is too conservative. For Model 3, the rejection probabilities of our test and the hr test are close to |$5\%$|⁠ , while Kwiatkowski et al. ’s test does not work as its rejection probabilities range from 16.6% to 26.2%. Our test with |$c_\kappa=\infty$| has no power, which shows that the truncation step for the critical value in ( 3 ) is necessary. The test of Kwiatkowski et al. (1992) has impressive power owing to the fact that it has a tendency to overestimate the rejection probability under |$H_0$|⁠ , leading to inflated power. Nevertheless, our test exhibits greater power in most cases. The hr test has good power for Models 4 and 5, but performs poorly for Model 6. The power-one property of our test is observable in the simulation since the rejection probability tends to 1 as |$N$| increases. Comparing the results of Models 5 and 7, we find that our test displays the power-one property more distinctly as our test statistic has more discriminative power between |$I(2)$| and |$I(0)$| than between |$I(1)$| and |$I(0)$|⁠ .

All authors contributed equally to the paper. We thank the editor, associate editor and referees for their constructive comments. Chang and Cheng were supported by the National Natural Science Foundation of China. Chang was also supported by the Center of Statistical Research and the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics. Yao was supported in part by the U.K. Engineering and Physical Sciences Research Council.

Supplementary Material available at Biometrika online includes all the technical proofs and some additional numerical results.

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Unit Root Tests Are Useful for Selecting Forecasting Models

We study the usefulness of root tests as diagnostic tools for selecting forecasting models. Difference stationary and trend stationary models of economic and financial time series often imply very different predictions, so deciding which model to use is tremendously important for applied forecasters. Forecasters face three choices: always difference the data, never difference, or use a unit-root pretest. We characterize the predictive loss of these strategies for the canonical AR(1) process with trend, focusing on the effects of sample size, forecast horizon, and degree of persistence. We show that pretesting routinely improves forecast accuracy relative to forecasts from models in differences, and we give conditions under which pretesting is likely to improve forecast accuracy relative to forecasts from models in levels.

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A Unit Root Test for an AR(1) Process with AR Errors by Using Random Weighted Bootstrap

• Published: 15 September 2023
• Volume 39 , pages 1834–1854, ( 2023 )

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• Xiao Hui Liu 1 , 2 ,
• Ya Wen Fan 1 , 2 ,
• Yu Zi Liu 1 , 2 &
• Shi Hua Luo 1 , 2  

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A great deal of economic problems are related to detecting the stability of time series data, where the main interest is in the unit root test. In this paper, we consider the unit root testing problem with errors being long-memory processes with the GARCH structure. A new test statistic is developed by using the random weighted bootstrap method. It turns out that the proposed statistic has a chi-squared distribution asymptotically regardless of the process being stationary or nonstationary, and with or without an intercept term. The simulation results show that the statistic has a desired finite sample performance in terms of both size and power. A real data application is also given relying on the inflation rate data of 17 countries.

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We thank one referee and the Associate Editor for their insightful comments which led to many improvements to this paper.

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Xiao Hui Liu, Ya Wen Fan, Yu Zi Liu & Shi Hua Luo

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Xiaohui Liu’s research is supported by the NNSF of China (Grant Nos. 11971208 and 11601197), the Outstanding Youth Fund Project of the Science and Technology Department of Jiangxi Province (Grant No. 20224ACB211003). Yawen Fan’s research is supported by the Science and Technology Research Project of Education Department of Jiangxi Province (Grant No. GJJ200545), the Postgraduate Innovation Project of Jiangxi Province (Grant No. YC2021-B124), and NSSF of China (Grant No. 21BTJ035). Shihua Luo’s research is supported by the National Major Social Science Project of China (Grant No. 21&ZD152), the NNSF of China (Grant No. 61973145) and Natural Science Project of Jiangxi Provincial Department of Science and Technology (Grant No. jxsq2023201048)

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Liu, X.H., Fan, Y.W., Liu, Y.Z. et al. A Unit Root Test for an AR(1) Process with AR Errors by Using Random Weighted Bootstrap. Acta. Math. Sin.-English Ser. 39 , 1834–1854 (2023). https://doi.org/10.1007/s10114-023-1535-x

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Received : 16 October 2021

Revised : 14 June 2022

Accepted : 19 September 2022

Published : 15 September 2023

Issue Date : September 2023

DOI : https://doi.org/10.1007/s10114-023-1535-x

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