2.1 Preliminaries

Let $X=\{X_t\}_{t\ge 0}$ be a real-valued stochastic process. A measurable process $L_X=\{L_X(t,x)\}_{t\ge 0, x\in \mathbb {R}}$ is said to be a local time of X if it satisfies

$$ \begin{align*} \int_0^t h( X_s)ds = \int_{-\infty}^{\infty}h(x) L_X(t, x)dx, \end{align*} $$

for all positive (or bounded) Borel function $h: \mathbb {R}\to \mathbb {R}.$ For a zero mean Gaussian process X with covariance kernel $\mathrm {r}(u, v)=E\big (X_uX_v\big )$ satisfying for all $T>0$

(2.1) $$ \begin{align} \int_0^T\int_0^T [\mathrm{r}(u,u)\mathrm{r}(v,v)-\mathrm{r}^2(u,v)]^{-1/2}dudv<\infty, \end{align} $$

X has a local time $L_X$ which can be represented as

$$ \begin{align*} L_X(t,x) =\frac 1{2\pi}\int_{-\infty}^{\infty}e^{-i\,ux}\int_0^t e^{iuX_s} ds\, du, \end{align*} $$

where the right-hand side is understood in the framework of $L^2$ -theory, that is, as a limit of $L^N(t,x)$ in the sense that

$$ \begin{align*} \lim_{N\to\infty} \mathrm{E}\, |L^N(t,x)-L_X(t,x)|^2=0, \end{align*} $$

where

$$ \begin{align*}L^N(t,x) = \frac 1{2\pi}\int_{-N}^{N}e^{-i\,ux}\int_0^t e^{iuX_s} ds\, du. \end{align*} $$

With this convention, for any bounded Borel function $f(t)$ on $[0,1]$ , $\int _0^1f(u) L_X(du, x)$ is well defined and

(2.2) $$ \begin{align} \int_0^1f(u)L^N(du, x) &=\frac 1{2\pi}\int_{-N}^{N}e^{-i\,tx}\int_0^1f(u) e^{itX_u} du\, dt\nonumber\\ &\to\int_0^1f(u)L_X(du, x), \end{align} $$

in probability, as $N\to \infty $ . Furthermore, if $g(t, x)$ is a Borel function on $ [0,1]\times \mathbb {R}$ satisfying $\int _{-\infty }^{\infty }\sup _{0\le s\le 1}|g(s, x)|dx < \infty $ and $L_X(t, x)$ is continuous, we have

(2.3) $$ \begin{align} c_n \int_0^1 g(t, c_nX_t) dt &=\int_0^1\int_{-\infty}^{\infty}g(t, x)L_X(dt,\, x/c_n) dx \nonumber\\ &\to \int_0^1G(t)L_X(dt, 0),\quad a.s., \end{align} $$

for any $0<c_n\to \infty $ , where $G(t)=\int _{-\infty }^{\infty }g(t, x)dx$ . See, for example, Geman and Horowitz (Reference Geman and Horowitz1976, Reference Geman and Horowitz1980) and also Chapter 2 of Wang (Reference Wang2015).

The linear diffusion $J_{\tau }=\{J_{\tau }(t)=\int _0^t e^{(t-s)\tau }dB_s\}_{t\ge 0}$ in (1.2) is a Gaussian process that satisfies (2.1). Another Gaussian process satisfying (2.1) is fractional Brownian motion $B_H=\{B_H(t)\}_{t\ge 0}$ defined by

$$ \begin{align*} B_H(t)=\kappa_H\, \int_{-\infty}^{t}\big[((t-u)_+)^{H-1/2}-((-u)_+)^{H-1/2}\big]dB_u, \end{align*} $$

where $a_+=\max \{a, 0\}$ , $0<H<1,$ and $\kappa _H>0$ is a constant such that $\mathrm {E}B^2_H(1)=1$ . Both $J_{\tau }$ and $B_H$ have local time processes satisfying (2.3).

2.2 Convergence to Local Time

Let $\{X_{nk}\}_{n\ge 1, k\geq 1}$ be an arbitrary random array defined on $\{\Omega , {\mathcal {F}}, P\}$ for which $X_{n, \lfloor nt \rfloor }\Rightarrow X_t$ on $D[0,1]$ . Note that

$$ \begin{align*} S_n := \frac {c_n}n \sum_{k=1}^n g\Big(\frac kn,\, c_n X_{nk}\Big)=\int_0^1 c_n g(\lfloor nt \rfloor/n, c_n X_{n,\lfloor nt \rfloor}) dt+o_P(1), \end{align*} $$

for any $c_n/n\to 0$ and bounded function $g(t,x)$ . Similar to (2.3), in applications, it is often desirable to establish that

$$ \begin{align*} S_n \to_D \int_0^1 G(t)\, L_X(dt, 0), \end{align*} $$

for certain $0< c_n\to \infty $ . In this regard, a framework was given in Wang and Phillips (Reference Wang and Phillips2009a), allowing for the array $y_{nk}$ to have a certain structure depending on conditional arguments (see also Chapter 2.3 of Wang, Reference Wang2015). Theorem 2.1 of Wang and Phillips (Reference Wang and Phillips2009a) is quite general, so that the main results given by Borodin and Ibragimov (Reference Borodin and Ibragimov1995), Akonom (Reference Akonom1993), Phillips and Park (Reference Phillips and Park1998), and Jeganathan (Reference Jeganathan2004) are all covered, but it is difficult to verify their Assumption 2.3 in applications. To see this, let $d_{nkj}^2=\mathrm {Var} (y_{nk}-y_{nj})$ and ${\mathcal {F}}_{nk}=\sigma (y_{n1},\ldots ,y_{nk}). $ In addition to some regularity conditions on $d_{nkj}$ , Assumption 2.3 of Wang and Phillips (Reference Wang and Phillips2009a) imposed the following condition on the array $y_{nk}$ :

Except for explicit arrays $y_{nk}$ of special structure such as standardized partial sums of linear processes with i.i.d. innovations, the existence of a conditional density can hardly be checked directly or is simply impossible to verify where $y_{nk}$ is a martingale array of the type considered in the present article.

This section provides an alternative to Assumption 2.3 of Wang and Phillips (Reference Wang and Phillips2009a). The new condition imposed on $X_{nk}$ is easy to verify for martingales and many other dependent sequences by using well-developed strong approximation theory. We have the following explicit result.

Theorem 2.1. Suppose that

1. (a) on a richer probability space, there exists a Gaussian process X satisfying (2.1) and a random array $\{X_{nk}^*\}_{1\le k\le n\atop n\geq 1}$ such that, for each $n\ge 1$ , $\{X_{n1}, \ldots ,X_{nn}\}=_d \{X_{n1}^*, \ldots ,X_{nn}^*\},$ and

   (2.4) $$ \begin{align} \sup_{0\le t\le 1}|X_{n, \lfloor nt \rfloor}^*-X_t| =o_P(c_n^{-2}), \end{align} $$

   where $0<c_n\to \infty $ is a sequence of constants satisfying $c_n/n\to 0$ ;

2. (b) for any small $\delta>0$ , there exists two continuous functions $g_{i\delta }(s, x), i=1, 2,$ on $[0,1]\times \mathbb {R}$ satisfying $\sup _{0\le s\le 1}|g_{i\delta }(s,x)|\le C_\delta /(1+|x|^{1+b})$ for some $b>0$ and $C_\delta>0$ depending only on $\delta $ such that $g_{1\delta }(s, x) \le g(s, x) \le g_{2\delta }(s, x)$ and

   $$ \begin{align*} \int_{-\infty}^\infty\sup_{0\le s\le 1}|g_{i\delta}(s, x)-g(s, x)|dx \le \delta. \end{align*} $$

Then, on $D_{\mathbb {R}^{2}}[0, 1],$ we have

(2.5) $$ \begin{align} \big(X_{n, \lfloor nt \rfloor}, S_n \big) \Rightarrow \Big(X_t,\ \int_0^1G(s) L_X(ds, 0)\Big), \end{align} $$

where $G(s)=\int _{-\infty }^{\infty }g(s,x)dx$ .

Remark 2.1. The approximation condition (2.4) can be established using a Skorohod embedding (together with some minor modifications) for various dependent random arrays, including martingales, partial sums of mixing sequences, and nonlinear causal time series. Examples including martingales and partial sums of mixing sequences are given in Sections S1 and S2 of the Supplementary Material. It should be mentioned that Wang and Phillips (Reference Wang and Phillips2009a) and Theorem 2.19 of Wang (Reference Wang2015) established versions of (2.5) suited to nonparametric cointegrating regression for any $0<c_n\to \infty $ satisfying $c_n/n\to 0$ . For the condition imposed on $X_{nk}$ to be useful in applications, there is a trade-off on $c_n$ depending on (2.4) in Theorem 2.1. To illustrate, let $X_{n k}=\frac 1{\sqrt n}\sum _{j=1}^k u_k$ , where $u_k\sim {\mathcal {N}}(0,1)$ are i.i.d. random variables. In this case, there exists a standard Brownian motion $B=\{B_t\}_{t\ge 0}$ so that

$$ \begin{align*}\{X_{n1}, \ldots,X_{nn}\} \ =_d \ \{B_{1/n},\ldots, B_{n/n}\} \ =:\ \{X_{n1}^*, \ldots,X_{nn}^*\}\end{align*} $$

for each $n\ge 1$ . Hence, by noting that $ \sqrt {n}\sup _{0\le t\le 1}|B_{\lfloor nt\rfloor /n}-B_t|\to _P \infty , $ to ensure that (2.4) is true with $X_t$ being replaced by $B_t$ , we have $\sqrt n c_n^{-2}\to \infty $ , and so we require $c_n/n^{1/4}\to 0$ and $c_n\to \infty $ . Such a rate requirement on $c_n$ is insufficient for nonparametric cointegrating regression, as seen in Wang and Phillips (Reference Wang and Phillips2009a, Reference Wang and Phillips2009b) and subsequent research, but it is sufficient for our purpose in this article.

2.3 Extension to More General Settings

We now consider an extension of (2.5) that can be used for many different purposes. Let $v_k$ be a sequence of arbitrary positive random variables and define the sample covariance

$$ \begin{align*} S_{1n} = \frac {c_n}n \sum_{k=1}^n g\Big(\frac kn,\, c_n X_{nk}\Big)\, v_k. \end{align*} $$

We have the following result for the asymptotics of $S_{1n}$ .

Theorem 2.2. In addition to the conditions of Theorem 2.1 , suppose the following hold:

1. (i) X is a Gaussian process satisfying

   $$ \begin{align*} \sup_{|s-t|\le \Delta \atop 0\le t\le 1}|X_t-X_s| = O_P(\Delta^\eta), \quad \text{for some } 0<\eta\le 1; \end{align*} $$

2. (ii) $\sup _{k\ge 1} Ev_k<\infty $ and there exist $A_0\in \mathbb {R}$ and $0<m := m_n \to \infty $ satisfying $m=o(nc_n^{-2/\eta })$ , where $\eta $ is given in (i) so that

   $$ \begin{align*} \max_{0\le j\le n-m} E \, \Big| \frac 1m \sum_{k=j+1}^{j+m} v_k-A_0\Big| =o(c_n^{-1}); \end{align*} $$

3. (iii) $|g(s, x)-g(s', x)| \le C\, |s-s'|/(1+|x|^{1+b})$ on $[0,1]\times \mathbb {R}$ , where $b>0$ is given as in condition (b) of Theorem 2.1 .

Then, for any constant sequence $0<c_n\to \infty $ satisfying $c_n/ n\to 0$ , we have

(2.7) $$ \begin{align} S_{1n} = A_0\, S_n +o_P(1). \end{align} $$

Consequently, on $D_{\mathbb {R}^{2}}[0, 1],$ we have

$$ \begin{align*} \big(X_{n, \lfloor nt \rfloor}, S_{1n} \big) \Rightarrow \Big(X_t,\ A_0\,\int_0^1G(s) L_X(ds, 0)\Big), \end{align*} $$

where $G(s)=\int _{-\infty }^{\infty }g(s,x)dx$ .

Remark 2.4. For a version of (2.7) with $c_n=1$ , we refer to Hu, Kasparis, and Wang (Reference Hu, Kasparis and Wang2021), where the additional condition (i) is not required. When $c_n=1$ , we have

$$ \begin{align*} S_n=\frac {1}n \sum_{k=1}^n g\left(\frac kn,\, X_{nk}\right) \to_D \int_{0}^1 g(t, X_t) dt, \end{align*} $$

providing a different limit distribution from (2.5).

5.1 Selection of the Tuning Parameter $C_0$

To determine the tuning parameter $C_0$ in the truncation rule $b_n = C_0 \log ^{-1} n$ , we employ a cross-validation procedure over a discrete grid within the interval $[\underline {C}_0, \overline {C}_0]$ . To account for the temporal dependence in time-series data, we implement a rolling cross-validation scheme that preserves the ordering of the observations.

Given the time series $\{y_k\}_{k=1}^n$ , we partition the data into K sequential folds. The initial fold is used for model training, and the immediate next fold is used for out-of-sample forecasting. Subsequently, the testing fold is added to the training set, and the procedure continues until the final fold is used. Let $[j]$ denote the index set of the jth fold, for $j = 1, \ldots , K$ . The iterative training–testing proceeds as follows:

• • Training: $[1]$ ; Testing: $[2]$ ;

• • Training: $[1], [2]$ ; Testing: $[3]$ ;

• • $\cdots $

• • Training: $[1], [2], \ldots , [K-1]$ ; Testing: $[K]$ .

In each iteration, we estimate the model by regressing $y_k$ on $y_{k-1}$ on the training data, applying a truncation to the lagged regressor based on the threshold $b_n = C_0 \log ^{-1} n$ , where $C_0$ varies across grid points in $[\underline {C}_0, \overline {C}_0]$ . Using this estimate, out-of-sample predictions are produced for the testing data, and the corresponding mean squared prediction error (MSPE) is calculated. The process is repeated across folds, and the average MSPE (AMSPE) is computed for each candidate $C_0$ . Specifically,

$$ \begin{align*} \text{AMSPE}(C_0) &= \frac{1}{K-1} \sum_{j=2}^{K} \text{MSPE}_j(C_0) \ \text{with} \ \text{MSPE}_j(C_0) = \frac{1}{n_j} \sum_{k \in [j]} \left( y_k - \widehat{y}_k^{(j)}(C_0) \right)^2, \end{align*} $$

where $\widehat {y}_k^{(j)}(C_0)$ denotes the predicted value of $y_k$ using the model trained on the first $(j-1)$ th fold with truncation level determined by $C_0$ , and $n_j$ is the number of observations in the jth testing fold. The optimal tuning parameter $\widehat {C}_0$ is then selected as the value of $C_0$ that minimizes AMSPE over the grid $[\underline {C}_0, \overline {C}_0]$ ,

(5.3) $$ \begin{align} \widehat{C}_0 = \arg\min_{C_0 \in [\underline{C}_0, \overline{C}_0]} \text{AMSPE}(C_0). \end{align} $$

To implement this selection in our simulation study, we consider a sample size of $n = 200$ and perform $10,000$ Monte Carlo replications for each data-generating process (DGP). Figures 1–3 present histograms of the selected $\widehat {C}_0$ across different settings, with normally distributed innovations $\epsilon _k$ having variances $\sigma _{\epsilon }^2 \in \{0.25, 0.5, 1\}$ and localizing parameters $\tau \in \{-0.1, -0.5, -1,0.1,0.5,1\}$ . The search for the optimal tuning parameter is conducted over the interval $[0.2\hat {\sigma }_{1n}, 10\hat {\sigma }_{1n}]$ , where $\hat {\sigma }_{1n}$ denotes an estimate of the standard deviation of $u_k$ , ensuring that the truncation levels considered are appropriate across the simulation designs.

Figure 1

Histogram of choice of $\widehat {C}_0$ using cross-validation when $\epsilon _{k}\sim N(0,0.25)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ .

Figure 1 Long description

The figure consists of six histograms, each with a y-axis labeled Percentage ranging from 0 to 0.09 and an x-axis labeled C sub 0 with values from 0.15 to 6.75. All histograms show a right-skewed distribution with a peak between 0.75 and 1.35.

* Panel a (top-left) shows tau equals minus 0.1. The peak percentage is approximately 0.078.

* Panel d (top-right) shows tau equals 0.1. The peak percentage is approximately 0.075.

* Panel b (middle-left) shows tau equals minus 0.5. The peak percentage is approximately 0.082.

* Panel e (middle-right) shows tau equals 0.5. The peak percentage is approximately 0.078.

* Panel c (bottom-left) shows tau equals minus 1. The peak percentage is approximately 0.083, with a very thin tail approaching zero beyond 4.65.

* Panel f (bottom-right) shows tau equals 1. The peak percentage is approximately 0.072, with a thicker tail that remains above 0.005 through the 6.75 mark.

Across all panels, as the absolute value of tau increases, the distribution becomes slightly more concentrated around the peak for negative tau and more dispersed in the tail for positive tau.

Figure 2

Histogram of choice of $\widehat {C}_0$ using cross-validation when $\epsilon _{k}\sim N(0,0.5)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ .

Figure 2 Long description

The figure consists of six panels, each displaying a histogram with Percentage on the Y-axis and C sub 0 on the X-axis. The X-axis for all panels ranges from 0.2 to 9.8. All histograms exhibit a right-skewed distribution with a peak occurring between C sub 0 values of 1.0 and 1.8.

* Panel a (top-left): tau equals negative 0.1. The peak percentage is approximately 0.07.

* Panel d (top-right): tau equals 0.1. The distribution is similar to panel a with a peak near 0.07.

* Panel b (middle-left): tau equals negative 0.5. The peak percentage is slightly higher, reaching approximately 0.072.

* Panel e (middle-right): tau equals 0.5. The peak is approximately 0.07, with a slightly thicker tail at higher C sub 0 values compared to the left column.

* Panel c (bottom-left): tau equals negative 1. The peak is approximately 0.074.

* Panel f (bottom-right): tau equals 1. The peak is approximately 0.065, showing the most prominent right-hand tail and a slight increase in frequency at the far right edge of the X-axis.

Figure 3

Histogram of choice of $\widehat {C}_0$ using cross-validation when $\epsilon _{k}\sim N(0,1)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ .

Figure 3 Long description

The figure consists of six histograms arranged in two columns and three rows. Each histogram shares identical axes: the Y-axis represents Percentage ranging from 0 to 0.07, and the X-axis represents C sub 0 with values ranging from 0.3 to 13.5.

* Panel a (top-left): tau equals minus 0.1. The distribution peaks sharply at C sub 0 equals 2.1 with a percentage near 0.068, followed by a long right tail that gradually tapers toward zero.

* Panel d (top-right): tau equals 0.1. The distribution is nearly identical to panel a, with a peak around 0.068 at C sub 0 equals 1.8 to 2.1.

* Panel b (middle-left): tau equals minus 0.5. The peak is slightly lower, around 0.066, and the right tail remains thick through C sub 0 equals 6.0.

* Panel e (middle-right): tau equals 0.5. The distribution shows a peak near 0.063. Notably, there is a small uptick in frequency at the far right edge of the X-axis (C sub 0 equals 13.5).

* Panel c (bottom-left): tau equals minus 1. The peak is approximately 0.068 at C sub 0 equals 1.8. The tail decays more smoothly toward the baseline compared to the middle panels.

* Panel f (bottom-right): tau equals 1. The peak is the lowest of all panels, reaching approximately 0.062. This panel shows the most significant terminal bar at C sub 0 equals 13.5, indicating a concentration of values at the upper limit of the cross-validation range.

In addition to the histograms of the selected tuning parameter, Figure 4 plots the cross-validated predicted mean squared error (PMSE) as a function of $C_0$ for a sequence of increasing sample sizes $n \in \{50,100,200,500,1000,2000\}$ , using three-fold cross-validation. Across all designs, the PMSE curves exhibit a well-defined and stable minimum, indicating that the prediction-based criterion yields a practical, identifiable choice of the tuning parameter. Moreover, as the sample size increases, the PMSE curves become increasingly concentrated around a common minimizer, providing empirical evidence that the selected tuning parameter stabilizes and converges as n grows. These results provide supporting finite-sample evidence for the proposed tuning procedure.

Figure 4

Cross-validation: PMSE versus $C_0$ for different sample sizes.

Note: The number of replications is $10,000$ . The figures show the three-fold cross-validated predicted mean squared error (PMSE) as a function of the tuning parameter $C_0$ .

Figure 4 Long description

A multi-panel figure containing six line graphs arranged in two rows and three columns. Each graph plots P M S E on the y-axis against C sub 0 on the x-axis for different sample sizes n.

* Top-left panel: n equals 50, C hat sub 0 equals 3.20. A blue line drops sharply from P M S E of 4.3 at C sub 0 near 0 to approximately 0.8 at C sub 0 equals 2, then plateaus. A vertical dashed line and red asterisk mark the inflection at C sub 0 equals 3.20.

* Top-middle panel: n equals 100, C hat sub 0 equals 3.20. An orange-red line drops from 1.16 to 1.115 and plateaus. Inflection marked at 3.20.

* Top-right panel: n equals 200, C hat sub 0 equals 3.20. A yellow line drops from 3.7 to 1.3 and plateaus. Inflection marked at 3.20.

* Bottom-left panel: n equals 500, C hat sub 0 equals 1.70. A purple line drops from 1.28 to a minimum of 1.165 at C sub 0 equals 1.70, marked by a red asterisk and dashed line, before slightly rising and leveling off at 1.175.

* Bottom-middle panel: n equals 1000, C hat sub 0 equals 3.20. A green line drops from 2.15 to 1.1 and plateaus. Inflection marked at 3.20.

* Bottom-right panel: n equals 2000, C hat sub 0 equals 3.20. A light blue line drops from 1.058 to 1.00 and plateaus. Inflection marked at 3.20.

Across all panels, the x-axis ranges from 0 to 10. The y-axis scales vary to fit the data range of each sample size.

5.2 Finite Sample Comparison of SW and OLS Estimators

To illustrate the finite sample behavior of the OLS and SW estimators, we conduct simulations using the LUR model (1.1) with coefficient $\tau \in \{-1, -0.5, -0.1, 0, 0.1, 0.5, 1\}$ , and innovation process following a GARCH(1,1) specification with variance $\sigma _\varepsilon ^2 =1$ . The sample sizes are $n \in \{50, 100, 200, 500\}$ , and each design is replicated $10,000$ times. For each replication, we compute the OLS and SW estimators and evaluate their finite sample performance using bias and mean absolute error (MAE).

The OLS estimator is well-known to exhibit substantial finite sample bias in LUR settings. For the unit root case ( $\tau = 0$ ), the bias of the OLS estimator is approximately $-{1.72}/{n}$ , as shown in the literature (Phillips and Perron, Reference Phillips and Perron1988). In contrast, the SW estimator employs trimming of extreme values, which reduces the bias while retaining a Gaussian limit.

Table 1 reports the bias and MAE of the two estimators. The OLS estimator displays negative finite sample bias that diminishes slowly with sample size. The bias is significant for small n and in the unit-root case ( $\tau = 0$ ). The SW estimator exhibits smaller finite sample bias, while its MAE is slightly larger than that of the OLS estimator, reflecting the effect of trimming. The MAE differences decrease with the sample size and are limited in magnitude, suggesting that trimming does not have a substantial finite sample cost.

Table 1

Bias and MAE of OLS and SW estimators.

Table 1 Long description

The table is divided into two main vertical sections: Bias and M A E (Mean Absolute Error).

Header Structure:

* The top row lists seven values for tau: -1, -0.5, -0.1, 0, 0.1, 0.5, and 1.

* Below each tau value are two columns for estimators: O L S and S W.

* The leftmost column represents the sample size T, with values 50, 100, 200, and 500.

Bias Section Data Trends:

* For all tau values, Bias is negative.

* As sample size T increases from 50 to 500, the absolute Bias decreases for both O L S and S W. For example, at tau equals 0, O L S bias moves from -0.0352 to -0.0037, and S W bias moves from -0.0300 to -0.0027.

* S W generally shows a smaller absolute bias than O L S across most configurations.

M A E Section Data Trends:

* M A E values decrease significantly as sample size T increases.

* At T equals 50 and tau equals 0, O L S M A E is 0.0455 and S W M A E is 0.0604.

* At T equals 500 and tau equals 0, O L S M A E is 0.0047 and S W M A E is 0.0066.

* Unlike the Bias section, O L S consistently shows lower M A E values compared to S W for all tested sample sizes and tau values.

Note: Entries report Monte Carlo bias and mean absolute error (MAE) for OLS and SW estimators. Number of Monte Carlo replications: $S=10,000$ . Bandwidth parameter: $b_n = \widehat {C}_0 \log ^{-1} n$ , where $\widehat {C}_0$ is selected by cross-validation.

Figure 5 shows kernel density estimates of the sampling distributions of the OLS and SW t-statistics for the unit-root case ( $\!\!\tau =0$ ) with sample sizes $n \in \{100, 200, 500\}$ . The OLS distribution is shifted away from the true parameter. The SW distribution is more centered and approximately symmetric, closely approximating the standard normal distribution, consistent with the theoretical results.

Figure 5

Density estimates of the OLS and SW t-statistics for $\tau =0$ and sample sizes $n=100, 200, 500$ .

Note: The number of replications is $10,000$ . For the SW estimator, we apply the truncation rate $b_n=\widehat {C}_0\log ^{-1}n$ with cross-validated $\widehat {C}_0$ .

Figure 5 Long description

The figure consists of two side-by-side panels, labeled a and b. Both panels share a horizontal x-axis ranging from minus 6 to 6 and a vertical y-axis labeled Density ranging from 0 to 0.5. A vertical dotted line marks the zero point on the x-axis. Each panel contains four curves: three solid lines representing sample sizes n equals 100 in blue, n equals 200 in red, and n equals 500 in yellow, plus a dashed black line representing the standard normal distribution N (0, 1).

* Panel a, titled O L S t-statistic, shows the three solid density curves clustered together, peaking at a density of approximately 0.42 to 0.44. These peaks are shifted to the left of the zero line, centered around minus 0.5. The curves are narrower and taller than the dashed N (0, 1) curve, which peaks at 0.4 at the zero mark.

* Panel b, titled S W t-statistic, shows the three solid density curves much more closely aligned with the dashed N (0, 1) curve. The peaks for n equals 100, 200, and 500 are centered near zero, reaching heights between 0.4 and 0.45. While the S W curves show slight irregularities or small secondary bumps compared to the smooth dashed line, they lack the significant leftward bias seen in the O L S panel.

5.3 Empirical Coverage Probabilities of SW Estimator

Finite sample inference performance of the SW test is studied by examining the empirical coverage probabilities of confidence intervals for $\widehat {\alpha }_{1n}$ for different sample sizes and localizing parameters.

The simulations are based on the LUR model (1.1), where the localizing coefficient $\tau \in \{-1, -0.5, -0.2, -0.1, 0.1, 0.2, 0.5, 1\}$ , and the innovation follows a GARCH(1,1) specification. The innovation variance $\sigma _\varepsilon ^2 \in \{0.25, 0.5, 1\}$ determines the signal-to-noise ratio. We consider sample sizes $n \in \{50, 100, 200, 500\}$ , with $S = 10,000$ Monte Carlo replications for each design.

For each replication, $\alpha $ is estimated using the proposed SW estimator with bandwidth $b_n = C_0 \log ^{-1} n$ , where $C_0$ is chosen by the cross-validation procedure described in Section 5.1. The resulting estimator $\widehat {\alpha }_{1n}$ is defined in (1.3), and its standard error $\widehat {\sigma }_{1n}$ is computed as in Remark 3.1. According to Theorem 3.1, a $(1 - \eta )$ -level confidence interval is constructed as $\mathrm {CI}_{1-\eta } = \big [\, \widehat {\alpha }_{1n} \pm z_{1-\eta /2} \, \widehat {\sigma }_{1n} \, \big ]$ , where $z_{1-\eta /2}$ denotes the $(1-\eta /2)$ quantile of the standard normal distribution. The empirical coverage probability is computed as

(5.4) $$ \begin{align} \mathrm{Coverage\ Rate}= \frac{1}{S} \sum_{s=1}^{S} \mathbb{I}\left\{\alpha \in \mathrm{CI}_{1-\eta}^{(s)}\right\}, \end{align} $$

where $\mathrm {CI}_{1-\eta }^{(s)}$ denotes the confidence interval in the s-th replication. We set $\eta = 0.05$ , so that the nominal coverage level is $95\%$ .

Table 2 reports the empirical coverage probabilities of the 95% confidence intervals for $\widehat {\alpha }_{1n}$ across different combinations of n, $\tau $ , and $\sigma _\epsilon ^2$ . The empirical coverage rates are consistently close to the nominal level across all cases, indicating that the large-sample approximation developed in Theorem 3.1 captures the finite sample distribution of $\widehat {\alpha }_{1n}$ quite well.

Table 2

Empirical coverage probabilities of the 95% confidence intervals for $\widehat {\alpha }_{1n}$ .

Table 2 Long description

The table presents coverage probabilities for alpha hat sub 1 n.

Header Structure:

* The first column is T (sample size).

* The second column is sigma sub varepsilon squared.

* The remaining six columns are grouped under tau, with values: -1, -0.5, -0.1, 0.1, 0.5, and 1.

Data Rows:

* For T = 50:

- sigma sub varepsilon squared = 0.25: Values range from 0.9643 to 0.9686.

- sigma sub varepsilon squared = 0.50: Values range from 0.9647 to 0.9711.

- sigma sub varepsilon squared = 1.00: Values range from 0.9644 to 0.9694.

* For T = 100:

- sigma sub varepsilon squared = 0.25: Values range from 0.9686 to 0.9723.

- sigma sub varepsilon squared = 0.50: Values range from 0.9695 to 0.9731.

- sigma sub varepsilon squared = 1.00: Values range from 0.9690 to 0.9742.

* For T = 200:

- sigma sub varepsilon squared = 0.25: Values range from 0.9711 to 0.9752.

- sigma sub varepsilon squared = 0.50: Values range from 0.9727 to 0.9759.

- sigma sub varepsilon squared = 1.00: Values range from 0.9740 to 0.9771.

* For T = 500:

- sigma sub varepsilon squared = 0.25: Values range from 0.9712 to 0.9751.

- sigma sub varepsilon squared = 0.50: Values range from 0.9709 to 0.9762.

- sigma sub varepsilon squared = 1.00: Values range from 0.9715 to 0.9750.

All values are close to the nominal 95% level, generally increasing slightly as T increases.

Note: Nominal coverage level is 95%. Number of Monte Carlo replications: $S=10,000$ . Bandwidth parameter: $b_n = \widehat {C}_0 \log ^{-1} n$ , where $\widehat {C}_0$ is selected by cross-validation.

5.4 Empirical Size for the SW Local Unit Root Test

The following Monte Carlo simulation is designed to evaluate the finite sample performance of the SW test relative to the ADF test. The DGP follows a near-unit-root AR(1) model, which allows for testing the local null hypothesis $\mathcal {H}_0: \alpha = 1 + \tfrac {\tau }{n}$ . The SW test statistic $T_n$ is constructed according to (4.2) in Theorem 4.1. When testing the local null hypothesis $\mathcal {H}_0$ , which is a moving target, the asymptotic distribution of $T_n$ is derived under $\alpha = 1$ . Consequently, the asymptotic normality of $T_n$ is established in a manner analogous to the classical ADF framework. This construction ensures that the Monte Carlo comparison of the empirical sizes of the SW and ADF tests is both conceptually consistent and theoretically justified.

Specifically, size performance of the SW and ADF tests is evaluated for a nominal size of $5\%$ , sample sizes $n\in \{50,100,200,500\}$ , innovation variance $\sigma _{\epsilon }^2 \in \{0.25, 0.5, 1\}$ , and localizing coefficient under the null $\tau \in \{-0.1,-0.5, -1,0.1,0.5,1\}$ , with $10,000$ replications for each DGP. The simulation design included positive localizing coefficients $\tau>0$ to assess properties of the procedure under the null in the explosive LUR region. The SW test uses the data-driven truncation parameter $\widehat {C}_0$ selected by the cross-validation method described in Section 5.1. For the ADF test, no lagged terms are included due to the mds design in the simulation. The results are shown in Table 3.

Table 3

Empirical size for the local unit root tests when $\epsilon _{k}\sim \mathcal {N}(0,\sigma _{\epsilon }^2)$ .

Table 3 Long description

A data table comparing S W and A D F test results for nominal size 5 percent.

Columns are organized by:

* n: Sample sizes of 50, 100, 200, and 500.

* sigma sub epsilon super 2: Values of 0.25, 0.5, and 1 for each n.

* tau: Six main categories for tau equals negative 1, negative 0.5, negative 0.1, 0.1, 0.5, and 1. Each tau category is split into S W and A D F sub-columns.

Key Data Trends:

* For n equals 50 and tau equals negative 1, empirical sizes range from 0.0789 to 0.0926, showing an upward bias from the 0.05 nominal size.

* As tau increases from negative 1 to 1, the empirical size generally decreases. For example, at n equals 500 and sigma squared equals 0.25, the S W size drops from 0.0700 at tau equals negative 1 to 0.0279 at tau equals 1.

* The A D F test generally shows higher empirical sizes than the S W test across most configurations.

* As sample size n increases, the values tend to stabilize, though they remain sensitive to the tau parameter.

Note: Nominal size 5%. The number of simulations is $10,000$ . For the SW estimator, we apply the truncation rate $b_n=\widehat {C}_0\log ^{-1}n$ with cross-validated $\widehat {C}_0$ . For the ADF test, the lag length $p=0$ .

The results show that the SW test with cross-validated $\widehat {C}_0$ achieves robust size control across different combinations of $n \in \{50, 100, 200, 500\}$ and $\sigma _{\epsilon }^2 \in \{0.25, 0.5, 1\}$ , provided that the localizing coefficient $\tau $ is not too negative. In particular, the empirical rejection rate of the SW test is close to the nominal $5\%$ level across all settings, except when $\tau =-1$ , where the test exhibits a mild degree of size distortion. Notably, size becomes more conservative under the null in the explosive LUR region $\tau>0 $ .

The ADF test also exhibits size distortion when $\tau $ becomes more negative (e.g., $\tau = -0.5$ or $-1$ ), particularly for smaller sample sizes. In comparison, the SW test shows better size control overall and particularly when $\tau $ deviates further from zero in the negative direction, which shows some advantage of SW testing with cross-validation selection of $\widehat {C}_0$ in finite samples. Like the SW test, the ADF test is size conservative in the explosive LUR region, but notably not as conservative as the SW test.

5.5 Local Power of the SW Local Unit Root Test

In the classical ADF framework, the sequence $\alpha = 1 + \tfrac {\tau }{n}$ is typically viewed as a local alternative to the unit root null hypothesis $\mathcal {H}_0:\alpha = 1$ , leading to the conventional notion of local asymptotic power. In contrast, our analysis adopts a broader perspective. We treat $\alpha = 1 + \tfrac {\tau }{n}$ as part of the null hypothesis, thereby considering a far wider null space than the strict unit root case. This formulation is practically motivated, as it reflects the empirical challenge of distinguishing exact unit roots from local-to-unity processes.

Within this framework, we use the term “local power” in a distinct sense, specifically, to refer to the finite sample rejection probabilities computed under sequences of local alternatives $\mathcal {H}_1: \alpha =1+\frac {\tau }{n^{\gamma }}$ with $\gamma \in (0,1]$ , where $\gamma = 1$ corresponds to the empirical size and $\gamma < 1$ captures the local power in our setup. Further, local power performance of the SW and ADF tests was explored using size-corrected power curves for both tests under local alternatives $\mathcal {H}_1$ with values of the localizing exponent $\gamma \in (0,1]$ and $\tau _0>0$ , that is, under stationary LUR alternatives $\mathcal {H}_1$ . For each DGP, $10,000$ replications were employed with sample size $n = 200$ .

Figures 6–8 present the size-corrected power curves of the SW and ADF tests, where the innovations $\epsilon _k \sim _{iid} \mathcal {N}(0,\sigma _{\epsilon }^2)$ , with variances $\sigma _{\epsilon }^2 \in \{0.25, 0.5, 1\}$ , and localizing coefficients $\tau \in \{-0.1,-0.5,-1,0.1,0.5,1\}$ , and $\tau _0=|\tau |>0$ under $\mathcal {H}_1$ . Power performance of the SW test with data-driven truncation $\widehat {C}_0$ selected by cross-validation is compared with the conventional ADF test.

Figure 6

The local power of the SW and ADF unit root tests when $\epsilon _{k}\sim N(0,0.25)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ . The parameter $\tau _0=|\tau |>0$ under $\mathcal {H}_1$ . Empirical critical values are computed under the null to deliver exact 5% rejection rates. For the SW estimator, we apply the truncation rate $b_n=\widehat {C}_0\log ^{-1}n$ with cross-validated $\widehat {C}_0$ . For the ADF test, the lag length $p=0$ .

Figure 6 Long description

A multi-panel display of six line graphs arranged in a two-column by three-row grid. Each graph shares the same axes. The x-axis is labeled Local Alternatives 1 minus tau sub 0 all over n super gamma for gamma in the set 0 to 1. The y-axis ranges from 0 to 1 with a horizontal dashed line at 0.05 representing the rejection rate. Each graph contains two curves: a solid blue line for S W and a dashed orange line for A D F.

* Top-left panel a: tau equals negative 0.1. Both curves start high near 0.9 and 1.0 respectively and decay rapidly toward 0.05 as x increases.

* Top-right panel d: tau equals 0.1. The curves show a nearly identical decay pattern to panel a.

* Middle-left panel b: tau equals negative 0.5. Both curves maintain a plateau near 1.0 until x equals 0.3 before dropping sharply.

* Middle-right panel e: tau equals 0.5. Similar to panel b, the power remains high longer than in the top row before descending.

* Bottom-left panel c: tau equals negative 1. The plateau at 1.0 extends further to x equals 0.45 for A D F and 0.4 for S W before the sharp decline.

* Bottom-right panel f: tau equals 1. The curves show the longest plateau, maintaining maximum power until approximately x equals 0.5 before converging toward the 0.05 baseline.

Figure 7

The local power of the SW and ADF unit root tests when $\epsilon _{k}\sim N(0,0.5)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ . The parameter $\tau _0=|\tau |>0$ under $\mathcal {H}_1$ . Empirical critical values are computed under the null to deliver exact 5% rejection rates. For the SW estimator, we apply the truncation rate $b_n=\widehat {C}_0\log ^{-1}n$ with cross-validated $\widehat {C}_0$ . For the ADF test, the lag length $p=0$ .

Figure 7 Long description

A multi-panel figure containing six line graphs labeled a through f. Each graph shares the same axes. The x-axis is labeled Local Alternatives 1 minus tau sub 0 all over n super gamma for gamma in the interval 0 to 1. The y-axis ranges from 0 to 1. A horizontal dashed line is fixed at 0.05 across all panels. Each panel contains two curves. A solid blue line represents the S W estimator and a dashed orange line represents the A D F test.

* Panel a top left. tau equals minus 0.1. Both curves start high near 1 and decay rapidly toward 0.05 as gamma increases.

* Panel d top right. tau equals 0.1. The curves show a similar rapid decay to panel a.

* Panel b middle left. tau equals minus 0.5. The curves maintain a power of 1 until gamma reaches approximately 0.3 before decaying. The A D F curve stays at 1 longer than the S W curve.

* Panel e middle right. tau equals 0.5. The behavior is nearly identical to panel b with A D F showing slightly higher power in the transition region.

* Panel c bottom left. tau equals minus 1. The plateau at power 1 extends further to gamma equals 0.4 for S W and 0.5 for A D F before the downward slope begins.

* Panel f bottom right. tau equals 1. The curves mirror panel c showing the highest persistence of power at 1 before converging toward the 0.05 baseline at gamma equals 1.

Figure 8

The local power of the SW and ADF unit root tests when $\epsilon _{k}\sim N(0,1)$ .

Note: The sample size $n=200$ . The number of replications is $10,000$ . The parameter $\tau _0=|\tau |>0$ under $\mathcal {H}_1$ . Empirical critical values are computed under the null to deliver exact 5% rejection rates. For the SW estimator, we apply the truncation rate $b_n=\widehat {C}_0\log ^{-1}n$ with cross-validated $\widehat {C}_0$ . For the ADF test, the lag length $p=0$ .

Figure 8 Long description

The figure consists of six panels labeled a through f. Each panel shares the same axes. The x-axis represents Local Alternatives 1 minus tau sub 0 all over n super gamma for gamma in the interval 0 to 1. The y-axis represents power ranging from 0 to 1. A horizontal dashed line at 0.05 indicates the 5% rejection rate.

* Panel a (tau equals minus 0.1) and Panel d (tau equals 0.1) show both S W (solid blue line) and A D F (dashed red line) starting near 1 at x equals 0 and following a steep sigmoidal decline to 0.05 as x approaches 1.

* Panel b (tau equals minus 0.5) and Panel e (tau equals 0.5) show a wider plateau at power 1 for the A D F test compared to the S W test. The A D F curve remains at 1 until x is approximately 0.3, while the S W curve begins its descent earlier at x equals 0.1. Both converge at x equals 0.6.

* Panel c (tau equals minus 1) and Panel f (tau equals 1) show the most significant gap. The A D F test maintains a power of 1 until x equals 0.4, whereas the S W test begins a gradual decline from x equals 0.2.

In all panels, the A D F test generally exhibits higher local power than the S W test in the lower range of the x-axis, with both tests eventually converging to the 0.05 significance level as the local alternative parameter increases.

The size-corrected power function is evaluated over $\gamma \in (0,1]$ . When $\tau _0 = \tau> 0$ , the case $\gamma = 1$ corresponds to the test size, while $\gamma < 1$ represents local alternatives. When $\tau _0 = -\tau> 0$ , the power function for any $\gamma \in (0,1]$ captures the local power performance. To ensure comparability, empirical critical values are computed under the null to deliver exact 5% rejection rates, and these are subsequently used to compute the power under alternatives. The results show that the SW test with cross-validated $\widehat {C}_0$ achieves power levels similar to the ADF test under mild local alternatives. Nonetheless, under more substantial deviations (i.e., smaller values of $\gamma $ ), the SW test exhibits some power deficiency relative to the ADF test. This is no doubt associated with a key property of the test that involves trimming out data, which reduces the effective sample size and, in doing so, thereby power. In some ways, this is an inevitable characteristic of trimming and a well-known implication of such procedures. Power reduction of this type is accentuated in right-sided testing of a UR against explosive alternatives where trimming of extreme observations in the explosive region has a direct impact on the power delivered by such observations. For right-sided tests, an alternative approach would be to formulate testing in a different manner wherein the null hypothesis is mildly explosive paper under which the limit theory is known (Phillips and Magdalinos, Reference Phillips and Magdalinos2009) against the alternative of a UR or stationary LUR. That approach is left as a topic for future research.