Spurious Relationships for Nearly Non-Stationary Series

Abstract

Literature shows that the regression of independent and (nearly) nonstationary time series could result in spurious outcomes. In this paper, we conjecture that under some situations, the regression of two independent and nearly non-stationary series does not have any spurious problem at all. To check whether our conjecture holds, we set up several situations and conduct simulations to justify our conjecture. Our simulations show that under some situations, the chance that the regressions being spurious is very high for all the cases simulated in our paper. Nonetheless, under some other situations, our simulation shows that the rejection rates are much smaller than the 5% level of significance for all the cases simulated in our paper, implying that our conjecture could hold under some situations that regression of two independent and nearly non-stationary series does not have any spurious problem at all.

1. Introduction

Granger and Newbold (1974) and others show that regression of independent (nearly) nonstationary time series could result in spurious outcomes, Pesaran et al. (1999) and others find that a mixed integration of orders; that is, I(0) or I(1), could be cointegrated and the residual is stationary, and Westerlund (2008) documents that many studies commit a Type 1 error by failing to reject the no-cointegration hypothesis. On the other hand, Engle and Granger (1987) establish the relationship between cointegration and error correction models that first suggested in Granger (1981) and develop estimation procedures and tests for the cointegration model. In addition, Phillips (1986) develops an asymptotic theory for regressions of integrated random processes, including the spurious regressions discovered by Granger and Newbold (1974) and the cointegrating regressions developed by Engle and Granger (1987). Entorf (1997) analyses the regression of two independent random walks with drifts and shows that the convergence to pseudo true values applies to the estimation of spurious fixed-effects models. Readers may refer to Ventosa-Santaulária (2009) for an overview of spurious regression.

Is it possible that the regression of two independent and nearly non-stationary series does not have any spurious problem? In this paper, we explore the issue. To explore the problem, we first conjecture that under some situations, regression of two independent and nearly non-stationary series does not have any spurious problem at all. To check whether the conjecture we set holds, we first generate two independent and nearly nonstationary AR(1) processes, $X_t={\alpha }_1{X}_{t-1}+{\epsilon }_t$ and $Y_t={\alpha }_2{Y}_{t-1}+e_t$ with $0.9<|{\alpha }_1|,|{\alpha }_2|<1$. We then regress $Y_t$ on the independent $X_t$ to get $Y_t=\alpha +\beta X_t+u_t$ and check the proportion of rejecting the null hypothesis that the beta ($\beta$) is zero. We first find that under some situations, consistent with the literature, regressing two independent and (nearly) nonstationary time series could be spurious. Nonetheless, we also find that under some other situations, different from the literature, our results show that the rejection rates are much smaller than the 5% level of significance for all the cases simulated in our paper, implying that under some other situations, regressing nearly nonstationary $Y_t$ on independent and nearly nonstationary $X_t$ will not get any spurious problem at all as shown in all the cases being simulated in our paper.

The rest of the paper is organized as follows. In Section 2, we state the basic models for the regression and the regression with a spurious problem. In Section 3, we state our model setup and construct the algorithm for the simulation. In Section 4, we discuss our findings from our simulation and the last section concludes.

2. The Model

In this section, we state the basic models for a simple regression and the regression with a spurious problem. We first state the basic simple regression model.

2.1. Linear Regression Model

In this paper, we consider the following simple regression model:

$$Y_t=\alpha +\beta X_t+u_t,$$

(1)

where $u_t$ is a random component denoted as error term assumed to be independent and identically distributed (iid) with mean 0 and variance ${\sigma }_u^2$, $t=1,\cdots ,T$ in which T is the sample size, $\alpha$ is the intercept parameter, and $\beta$ is the slope parameter.

The most important hypothesis for the simple linear regression stated in (1) is

$$H_0:\beta =0\mathrm{versus}{H}_1:\beta \ne 0,$$

(2)

which is used to detect the linear relationship between two variables Y and X. If the null hypothesis $H_0$ in (2) is true, then for the model in (1), the population mean of $Y_t$ is always equal to $\alpha$ for any value of $X_t$, concluding that $Y_t$ does not depend on the value of $X_t$ and there is no linear relationship between $X_t$ and $Y_t$. On the other hand, if the alternative hypothesis $H_1$ is true, then one could conclude that a change in $X_t$ is associated with a change in $Y_t$ linearly.

To test whether the null hypothesis $H_0$ in (2) is true, one could use the following T test:

$$T=\frac{\widehat{\beta }-\beta }{SE\left(\widehat{\beta }\right)}=\frac{\widehat{\beta }}{SE\left(\widehat{\beta }\right)},$$

(3)

where

$$\widehat{\beta }=\frac{{\sum }_{t=1}^N(X_t-\overline{X})(Y_t-\overline{Y})}{{\sum }_{t=1}^N{(X_t-\overline{X})}^2},$$

(4)

is the estimate of $\beta$ in which $\overline{X}={\sum }_{t=1}^T{X}_t/T$, $\overline{Y}={\sum }_{t=1}^T{Y}_t/T$, and

$$SE\left(\widehat{\beta }\right)=\sqrt{\frac{{\sum }_{t=1}^T\left({\widehat{u}}_t^2\right)/(T-2)}{{\sum }_{t=1}^T{(X_t-\overline{X})}^2}}$$

(5)

is the standard error of the estimate measuring the accuracy of prediction with ${\widehat{u}}_t={(Y_t-\widehat{\alpha }-\widehat{\beta }{X}_t)}^2$. It is well known that the test statistic T follows a t-distribution with $T-2$ degrees of freedom if the null hypothesis $H_0$ is true. We note that Goodness of fit can refer to specific individual factors (or variables) in the regression equation, whereas R-squared refers to a specific numerical value of the entire regression model.

2.2. Spurious Regression

A spurious relationship is a relationship that does not make sense. In this situation, two or more independent variables could appear to be correlated with the effect of an unseen factor (“confounding factor” or “lurking variable”). That is, goodness-of-fit indicators from regression such as $R^2$ are likely to be large, implying a valid fit even if the underlying variables are not truly related.

A variable, say, $Y_t$, is said to be integrated of order d, and denoted as $Y_t=I\left(d\right)$, if it has a stationary, invertible, and stochastic ARMA representation after differencing d times, it is stationary if $d<1$, is non-stationary if $d\ge 1$, and is nearly non-stationary if $d<1$ but close to 1. Granger and Newbold (1974) and others find that if

$$Y_t\sim I\left(1\right)\mathrm{and}{X}_t\sim I\left(1\right),$$

(6)

then regression in (1) could be spurious. It means that even when $Y_t$ and $X_t$ are known to be independent, applying the test statistic T in (3) to an ordinary least square regression could misleadingly indicate a good fit when non-stationary time series data are involved. Granger and Newbold (1974) exhibit an example in which an equation has $R^2=0.997$ and the value of the Durbin–Watson (DW) statistic is $0.093$, implying that when the residuals are strongly autocorrelated in time series regression, the interpretability of the coefficients could be questionable.

Since Granger and Newbold (1974) and others show that regression of independent and nonstationary time series could result in spurious outcomes, it is common to believe that regression of independent and nearly nonstationary time series could also get spurious outcomes. In this paper, we believe that there are some cases in which regression of independent and nearly nonstationary time series may not be spurious. Thus, we set up the following conjecture:

Conjecture 1.

Under some situations, the regression of two independent and nearly non-stationary series does not have any spurious problem at all.

To examine whether the above conjecture could hold in some situations, we will discuss it in the next section.

3. Model Setup and Algorithm

In this section, we first state the model setting of generating two purely independent and nearly nonstationary time series, regressing one of them onto the other, and examining whether the corresponding regression is spurious. We then construct the algorithm for the simulation and discuss our simulation result in the next section.

3.1. Model Setup

We consider the simple linear regression in (1) between two unrelated nearly nonstationary AR(1) series $X_t$ and $Y_t$ such that

$$X_t={\alpha }_1{X}_{t-1}+{\epsilon }_t\mathrm{and}{Y}_t={\alpha }_2{Y}_{t-1}+e_t,\mathrm{with}{\epsilon }_t\stackrel{iid}{\sim }(0,{\sigma }_{\epsilon }^2)\mathrm{and}{e}_t\stackrel{iid}{\sim }(0,{\sigma }_e^2)$$

(7)

in which $0.9<|{\alpha }_i|<1$ $(i=1,2)$. For simplicity, we assume that both ${\epsilon }_t$ and $e_t$ follow:

$$f(a;p)\propto \frac{1}{\sigma }{\left\{1+\frac{a^2}{k{\sigma }^2}\right\}}^{-p}(-\infty <a<\infty ),$$

(8)

where $k=2p-1$ and $p\ge 2$. We note that $E\left(a\right)=0$, $V\left(a\right)={\sigma }^2$, and $t=\sqrt{(k/\nu )}(a/\sigma )$ follows a Student’s t distribution with $\nu =2p-1$ degrees of freedom (df). For $1\le p<2$, k is equated to 1 and in this case, $\sigma$ in (8) is simply a scale parameter. When $\nu =1$, it becomes a Cauchy distribution and when $\nu =\infty$, it becomes a normal distribution. Readers may refer to Pötzelberger (1990), Tiku and Wong (1998), Tiku et al. (1999, 2000), Wong and Bian (2005), Fu and Fu (2015), and others to know more properties of AR(1) series.

To simulate $X_t$ and $Y_t$ properly, without loss of generality, we will consider different factors that could affect the behavior of the time series. First, we consider the distribution of the error terms. We choose a time series that follows the following four different iid error distributions in our study:

Situation 1.

We assume that the distribution of the error terms ${\epsilon }_t$ and $e_t$ defined in (7) follow the following situations:

1. 

a standard normal distribution: that is, both ${\epsilon }_t$ and $e_t$∼ N(0,1);

2. 

a t-distribution with df = 5: that is, both ${\epsilon }_t$ and $e_t$∼ t(5);

3. 

a t-distribution with df = 2: that is, both ${\epsilon }_t$ and $e_t$∼ t(2); and

4. 

a t-distribution with df = 1: that is, both ${\epsilon }_t$ and $e_t$ follow the standard Cauchy distribution.

Second, we vary the lengths of the times series and simulate a time series with the following four different lengths in our study as stated in the following situations:

Situation 2.

We consider that the lengths of the times series $Y_t$ and $X_t$ defined in (7) to be: (i) T = 100; (ii) T = 200; (iii) T = 400; and (iv) T = 800.

After deciding the error distribution and the lengths of the AR(1) processes, we now consider the different values of ${\alpha }_1$ and ${\alpha }_2$. In our model, since both $X_t$ and $Y_t$ are nearly nonstationary, we choose $0.9<|{\alpha }_i|<1$ $(i=1,2)$ and, in particular, we define $A^{-}=\{-0.99,-0.97,-0.95,-0.92,-0.9\}$ and $A^{+}=\{0.99,0.97,0.95,0.92,0.9\}$1 and consider the following values for both ${\alpha }_1$ and ${\alpha }_2$ as stated in Situations 3 and 4:

Situation 3.

We consider that the values of both ${\alpha }_1$ and ${\alpha }_2$ such that ${\alpha }_1\in A^{+}$ and ${\alpha }_2\in A^{-}$.

Situation 4.

We consider that the values of both ${\alpha }_1$ and ${\alpha }_2$ such that ${\alpha }_1\in A^{-}$ and ${\alpha }_2\in A^{+}$.

We note that in this paper, we consider Situations 3 and 4 because when two autoregressive processes in which one is associated to the zero frequency; that is, the AR(1) with a positive coefficient in our paper, and the other is associated to the Nyquist frequency ($\pi$); that is, the AR(1) with a negative coefficient in our paper that has power at frequency $\pi$ and completes a cycle every 2 observations, are independent or even asymptotically orthogonal. Readers may refer to Johansen and Schaumburg (1999), Ghysels and Osborn (2001), and del Barrio Castro et al. (2018, 2019) for more information. Readers may also refer to seasonal unit root tests, see, for example, del Barrio Castro et al. (2012) and Smith et al. (2009), and cointegration for processes integrated at different frequencies, see, for example, del Barrio Castro et al. (2020) with properties that are related to the series we are using in our paper.2

With four different error distributions, four different time series lengths, and the above 50 combinations of ${\alpha }_1$ and ${\alpha }_2$ values as stated in Assumptions 1, 2, 3, and 4, there are in total 800 cases of simulation in our study for the cases when autoregressive coefficients ${\alpha }_1$ and ${\alpha }_2$ have different signs.

Nevertheless, in this paper, we also study the cases when both autoregressive coefficients ${\alpha }_1$ and ${\alpha }_2$ are of the same signs, either positive or negative. Thus, we include the following situations in our study:

Situation 5.

We consider that the values of both ${\alpha }_1$ and ${\alpha }_2$ such that ${\alpha }_1\in A^{+}$ and ${\alpha }_2\in A^{+}$.

Situation 6.

We consider that the values of both ${\alpha }_1$ and ${\alpha }_2$ such that ${\alpha }_1\in A^{-}$ and ${\alpha }_2\in A^{-}$.

3.2. Algorithm

The two series $X_t$ and $Y_t$ are generated from independent error terms, and thus, they are expected not to be related. However, Granger and Newbold (1974) and others have shown that regression of independent nonstationary time series could result in spurious outcomes. In this paper, we believe that it is possible that when regressinng independent and nearly nonstationary $Y_t$ and $X_t$ as shown in Equation (1) may not be spurious under some situations as we stated in Conjecture 1. To check whether Conjecture 1 could hold under some situations, we set the following algorithm for each situation (different error distributions, different time series lengths, different combinations of ${\alpha }_1$ and ${\alpha }_2$) as described in Section 3.1:

Algorithm 1: For each situation (different error distributions, different time series lengths, different combinations of ${\alpha }_1$ and ${\alpha }_2$) as described in Section 3.1, we will conduct the following steps in our simulation:

Simulate 10,000 pairs of $X_t$ and $Y_t$ defined in (7) with coefficients described in Section 3.1. For each pair of simulated $X_t$ and $Y_t$, fit model in (1). Thus, in each subcase, we will obtain 10,000 $\widehat{\beta }$’s and 10,000 corresponding p-values. Plot the distribution of $\widehat{\beta }$’s and record the standard errors and t-statistics of $\widehat{\beta }$. Use the T test defined in Equation (3) to test whether the null hypothesis $H_0$ in (2) hold. $H_0$ is rejected if p-value for the T test is less than 0.05. Calculate the proportion of significant $\widehat{\beta }$’s or proportion of p-values that are less than 0.05 among the 10,000 fitted linear regression models in each subcase. This proportion is denoted as the rejection rate in this paper.

For each situation (different error distributions, different time series lengths, different combinations of ${\alpha }_1$ and ${\alpha }_2$) as described in Section 3.1, we will conduct simulation as described in Algorithm 1 and discuss the results in the next section.

4. Simulation

We follow Algorithm 1 to conduct simulation for each situation (different error distributions, different time series lengths, different combinations of ${\alpha }_1$ and ${\alpha }_2$) as described in Section 3.1. The simulation helps us to examine whether the T statistic as shown in Equation (3) for the model as shown in Equation (1) follow a Student t-distribution. If $X_t$ and $Y_t$ are unrelated, the true null hypothesis that all $\beta$ coefficients are zero should be rejected around 5% of the time at the significance level of 5%. If the T test is good, that is, $\widehat{\beta }$’s follow student t-distribution, the rejection rate should be close to 5%. If the rejection rate is significantly greater than 5%, then we conclude that there exists the spurious problem. In addition, we believe that it is possible that when regressing independent and nearly nonstationary $Y_t$ and $X_t$ as shown in (1) may not be spurious under some situations as we hypothesized in Conjecture 1. To check whether Conjecture 1 could hold under some situations, we discuss it in this section. We first discuss the results of the simulation for the cases when ${\alpha }_1$ and ${\alpha }_2$ are of different signs in the next subsection.

4.1. Simulation for the Cases When ${\alpha }_1$ and ${\alpha }_2$ Are of Different Signs

We first analyze cases as stated in Situation 3 and exhibit the results in

Table A1. Rejection rate, error ∼ N(0,1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.9	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.92	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.95	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.97	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.99	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A2. Rejection rate, error ∼ t(5).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.9	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.92	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.95	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
-0.97	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.99	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A3. Rejection rate, error ∼ t(2).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.0000	0.0001	0.0000	0.0001	0.0001
	0.92	0.0001	0.0000	0.0000	0.0000	0.0001
−0.9	0.95	0.0000	0.0001	0.0000	0.0000	0.0000	0.0000
	0.97	0.0001	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.92	0.95	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0001	0.0000	0.0000	0.0000	0.0000
	0.99	0.0001	0.0000	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0002	0.0000	0.0000	0.0000	0.0001
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.95	0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0001	0.0001	0.0000	0.0000	0.0001
	0.99	0.0000	0.0001	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0001	0.0000	0.0000	0.0000	0.0000
	0.92	0.0001	0.0001	0.0000	0.0000	0.0001
−0.97	0.95	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0003	0.0001	0.0001	0.0000	0.0001
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	0.92	0.0000	0.0000	0.0000	0.0000	0.0000
−0.99	0.95	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

and

Table A4. Rejection Rate, Error ∼ t(1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.0020	0.0005	0.0007	0.0027	0.0015
	0.92	0.0010	0.0006	0.0006	0.0013	0.0009
−0.9	0.95	0.0009	0.0006	0.0002	0.0007	0.0006	0.0008
	0.97	0.0009	0.0007	0.0002	0.0001	0.0005
	0.99	0.0009	0.0001	0.0000	0.0003	0.0003
Caverage		0.0011	0.0005	0.0003	0.0010	0.0008	0.0008
	0.9	0.0008	0.0005	0.0002	0.0006	0.0005
	0.92	0.0012	0.0010	0.0002	0.0006	0.0008
−0.92	0.95	0.0004	0.0004	0.0003	0.0003	0.0004	0.0004
	0.97	0.0005	0.0004	0.0000	0.0001	0.0003
	0.99	0.0005	0.0002	0.0001	0.0000	0.0002
Caverage		0.0007	0.0005	0.0002	0.0003	0.0004	0.0004
	0.9	0.0008	0.0007	0.0002	0.0005	0.0006
	0.92	0.0010	0.0002	0.0001	0.0003	0.0004
−0.95	0.95	0.0011	0.0003	0.0002	0.0001	0.0004	0.0004
	0.97	0.0007	0.0006	0.0000	0.0001	0.0004
	0.99	0.0007	0.0001	0.0000	0.0001	0.0002
Caverage		0.0009	0.0004	0.0001	0.0002	0.0004	0.0004
	0.9	0.0008	0.0006	0.0003	0.0003	0.0005
	0.92	0.0009	0.0007	0.0001	0.0001	0.0005
−0.97	0.95	0.0014	0.0001	0.0001	0.0002	0.0005	0.0004
	0.97	0.0002	0.0002	0.0001	0.0002	0.0002
	0.99	0.0010	0.0002	0.0002	0.0000	0.0004
Caverage		0.0009	0.0004	0.0002	0.0002	0.0004	0.0004
	0.9	0.0011	0.0003	0.0004	0.0000	0.0005
	0.92	0.0005	0.0002	0.0001	0.0000	0.0002
−0.99	0.95	0.0003	0.0003	0.0000	0.0001	0.0002	0.0003
	0.97	0.0009	0.0003	0.0001	0.0000	0.0003
	0.99	0.0005	0.0000	0.0000	0.0000	0.0001
Caverage		0.0007	0.0002	0.0001	0.0000	0.0003	0.0003
Oaverage		0.0008	0.0004	0.0002	0.0003	0.0004	0.0004

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

displaying in Appendix A that report the rejecting frequency of the T test when $({\alpha }_1,{\alpha }_2)\in (A^{+},A^{-})$. From Table A1, Table A2, Table A3 and Table A4, one can observe that when choosing the values of both ${\alpha }_1$ and ${\alpha }_2$ as stated in Situation 3 are from $\left|0.9\right|$ to $\left|0.99\right|$, the rejection rate is about 0.0000 for any n and for any error distribution studied in our paper, except the situation when the error term follows a $t\left(1\right)$ in which the rejection rates are close to 0.0004.

We then analyze the cases as stated in Situation 4 and show the results in

Table A5. Rejection rate, error ∼ N(0,1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.9	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.92	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.95	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.97	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.99	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Oaverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A6. Rejection rate, error ∼ t(5).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.9	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0001	0.0000	0.0000	0.0000	0.0000
0.92	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.95	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.97	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0001	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.99	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Oaverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A7. Rejection rate, error ∼ t(2).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.0000	0.0002	0.0000	0.0001	0.0001
	−0.92	0.0000	0.0000	0.0000	0.0000	0.0000
0.9	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0001	0.0000	0.0000	0.0000
Caverage		0.0000	0.0001	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0001	0.0000	0.0000	0.0001	0.0001
	−0.92	0.0000	0.0001	0.0000	0.0000	0.0000
0.92	−0.95	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0000	0.0000	0.0000	0.0000
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0000	0.0001	0.0000	0.0000	0.0000
0.95	−0.95	0.0002	0.0000	0.0000	0.0000	0.0001	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0000	0.0001	0.0000	0.0001	0.0001
Caverage		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0001	0.0000	0.0000	0.0000	0.0000
	−0.92	0.0002	0.0001	0.0000	0.0000	0.0001
0.97	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0001	0.0001	0.0000	0.0000	0.0001
	−0.99	0.0001	0.0000	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.9	0.0002	0.0001	0.0000	0.0000	0.0001
	−0.92	0.0001	0.0000	0.0000	0.0000	0.0000
0.99	−0.95	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.97	0.0000	0.0000	0.0000	0.0000	0.0000
	−0.99	0.0001	0.0000	0.0000	0.0000	0.0000
Caverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
Oaverage		0.0001	0.0000	0.0000	0.0000	0.0000	0.0000

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

and

Table A8. Rejection rate, error ∼ t(1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.0014	0.0006	0.0009	0.0019	0.0012
	−0.92	0.0010	0.0005	0.0004	0.0016	0.0009
0.9	−0.95	0.0011	0.0003	0.0004	0.0008	0.0007	0.0007
	−0.97	0.0008	0.0006	0.0000	0.0001	0.0004
	−0.99	0.0008	0.0000	0.0001	0.0004	0.0003
Caverage		0.0010	0.0004	0.0004	0.0010	0.0007	0.0007
	−0.9	0.0009	0.0008	0.0002	0.0007	0.0007
	−0.92	0.0012	0.0010	0.0002	0.0007	0.0008
0.92	−0.95	0.0004	0.0003	0.0003	0.0001	0.0003	0.0004
	−0.97	0.0004	0.0005	0.0000	0.0002	0.0003
	−0.99	0.0005	0.0003	0.0001	0.0000	0.0002
Caverage		0.0007	0.0006	0.0002	0.0003	0.0004	0.0004
	−0.9	0.0009	0.0006	0.0002	0.0005	0.0006
	−0.92	0.0017	0.0004	0.0002	0.0001	0.0006
0.95	−0.95	0.0009	0.0003	0.0003	0.0001	0.0004	0.0004
	−0.97	0.0008	0.0004	0.0001	0.0001	0.0004
	−0.99	0.0004	0.0004	0.0000	0.0001	0.0002
Caverage		0.0009	0.0004	0.0002	0.0002	0.0004	0.0004
	−0.9	0.0008	0.0007	0.0005	0.0002	0.0006
	−0.92	0.0009	0.0006	0.0000	0.0001	0.0004
0.97	−0.95	0.0014	0.0004	0.0002	0.0001	0.0005	0.0004
	−0.97	0.0004	0.0002	0.0000	0.0002	0.0002
	−0.99	0.0012	0.0004	0.0001	0.0000	0.0004
Caverage		0.0009	0.0005	0.0002	0.0001	0.0004	0.0004
	−0.9	0.0010	0.0007	0.0002	0.0001	0.0005
	−0.92	0.0008	0.0006	0.0002	0.0000	0.0004
0.99	−0.95	0.0003	0.0002	0.0003	0.0001	0.0002	0.0003
	−0.97	0.0007	0.0004	0.0002	0.0000	0.0003
	−0.99	0.0003	0.0001	0.0002	0.0001	0.0002
Caverage		0.0006	0.0004	0.0002	0.0001	0.0003	0.0003
Oaverage		0.0008	0.0005	0.0002	0.0003	0.0005	0.0005

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

displaying in Appendix B that report the rejecting frequency of the T test when $({\alpha }_1,{\alpha }_2)\in (A^{-},A^{+})$. Similarly, from Table A5, Table A6, Table A7 and Table A8, one can observe that when choosing values of both ${\alpha }_1$ and ${\alpha }_2$ as stated in Situation 4 are between $\left|0.9\right|$ and $\left|0.99\right|$, the rejection rate is zero or close to zero for any n and any error distribution studied in our paper.

Our analysis shows that for all the cases when choosing values for $({\alpha }_1,{\alpha }_2)$ as stated in Situations 3 and 4 and when choosing values of both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, the rejection rates are much smaller than the 5% level of significance, implying that when $({\alpha }_1,{\alpha }_2)$ follow Situations 3 and 4 and when both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, all the corresponding regressions do not encounter any spurious problem for all the cases simulated in our paper, confirming that Conjecture 1 holds. In other words, our analysis shows that when independent $Y_t$ and $X_t$ follow nearly nonstationary AR(1) model and the autoregressive coefficients ${\alpha }_1$ and ${\alpha }_2$ have opposite signs, there is no spurious problem in the regression stated in Equation (1) and Conjecture 1 holds.

4.2. Simulation for the Cases When ${\alpha }_1$ and ${\alpha }_2$ Are of the Same Sign

We turn to examine whether the regression shown in Equation (1) is spurious for the cases when both ${\alpha }_1$ and ${\alpha }_2$ are of the same signs; that is, both ${\alpha }_1$ and ${\alpha }_2$ are positive or both are negative. To do so, we follow Algorithm 1 to conduct simulations for the cases when both ${\alpha }_1$ and ${\alpha }_2$ are positive and both are negative as displayed in Situations 5 and 6 and exhibit the results in

Table A9. Rejection rate, error ∼ N(0,1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.5105	0.5173	0.5191	0.5202	0.5168
	0.92	0.5128	0.5379	0.5446	0.5552	0.5376
0.9	0.95	0.5528	0.5704	0.5800	0.5764	0.5699	0.5669
	0.97	0.5640	0.5887	0.6054	0.6058	0.5910
	0.99	0.5847	0.6203	0.6358	0.6370	0.6195
Caverage		0.5450	0.5669	0.5770	0.5789	0.5669	0.5669
	0.9	0.5193	0.5391	0.5422	0.5445	0.5363
	0.92	0.5392	0.5596	0.5756	0.5744	0.5622
0.92	0.95	0.5633	0.5995	0.6120	0.6102	0.5963	0.5924
	0.97	0.5880	0.6239	0.6368	0.6458	0.6236
	0.99	0.6028	0.6468	0.6584	0.6660	0.6435
Caverage		0.5625	0.5938	0.6050	0.6082	0.5924	0.5924
	0.9	0.5451	0.5805	0.5795	0.5877	0.5732
	0.92	0.5660	0.5970	0.6131	0.6164	0.5981
0.95	0.95	0.6120	0.6391	0.6554	0.6504	0.6392	0.6354
	0.97	0.6252	0.6719	0.6853	0.6889	0.6678
	0.99	0.6486	0.6981	0.7221	0.7248	0.6984
Caverage		0.5994	0.6373	0.6511	0.6536	0.6354	0.6354
	0.9	0.5591	0.5965	0.6113	0.6006	0.5919
	0.92	0.5833	0.6107	0.6367	0.6350	0.6164
0.97	0.95	0.6282	0.6625	0.6897	0.6869	0.6668	0.6623
	0.97	0.6433	0.6984	0.7286	0.7300	0.7001
	0.99	0.6767	0.7288	0.7614	0.7785	0.7364
Caverage		0.6181	0.6594	0.6855	0.6862	0.6623	0.6623
	0.9	0.5775	0.6051	0.6274	0.6385	0.6121
	0.92	0.6106	0.6462	0.6625	0.6677	0.6468
0.99	0.95	0.6454	0.6957	0.7225	0.7240	0.6969	0.6937
	0.97	0.6669	0.7251	0.7632	0.7687	0.7310
	0.99	0.7103	0.7730	0.8071	0.8370	0.7819
Caverage		0.6421	0.6890	0.7165	0.7272	0.6937	0.6937
Oaverage		0.5934	0.6293	0.6470	0.6508	0.6301	0.6301

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A10. Rejection rate, error ∼ t(5).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.5104	0.5152	0.5233	0.5151	0.5160
	0.92	0.5228	0.5420	0.5447	0.5442	0.5384
0.9	0.95	0.5514	0.5588	0.5752	0.5877	0.5683	0.5667
	0.97	0.5608	0.5979	0.6092	0.6065	0.5936
	0.99	0.5805	0.6134	0.6322	0.6435	0.6174
Caverage		0.5452	0.5655	0.5769	0.5794	0.5667	0.5667
	0.9	0.5207	0.5426	0.5490	0.5490	0.5403
	0.92	0.5384	0.5598	0.5697	0.5690	0.5592
0.92	0.95	0.5757	0.5939	0.6042	0.6033	0.5943	0.5922
	0.97	0.5834	0.6336	0.6349	0.6319	0.6210
	0.99	0.6102	0.6454	0.6607	0.6690	0.6463
Caverage		0.5657	0.5951	0.6037	0.6044	0.5922	0.5922
	0.9	0.5539	0.5724	0.5925	0.5902	0.5773
	0.92	0.5729	0.5984	0.6114	0.6120	0.5987
0.95	0.95	0.6088	0.6413	0.6489	0.6484	0.6369	0.6438
	0.97	0.6198	0.6665	0.6827	0.6900	0.6648
	0.99	0.6427	0.7039	0.7159	0.7235	0.6965
Caverage		0.5996	0.6365	0.6503	0.6528	0.6348	0.6438
	0.9	0.5723	0.5962	0.6093	0.6042	0.5955
	0.92	0.5950	0.6304	0.6360	0.6392	0.6252
0.97	0.95	0.6291	0.6753	0.6931	0.6859	0.6709	0.6656
	0.97	0.6448	0.7061	0.7273	0.7295	0.7019
	0.99	0.6669	0.7328	0.7639	0.7739	0.7344
Caverage		0.6216	0.6682	0.6859	0.6865	0.6656	0.6656
	0.9	0.5752	0.6114	0.6252	0.6305	0.6106
	0.92	0.6056	0.6394	0.6601	0.6695	0.6437
0.99	0.95	0.6478	0.7000	0.7093	0.7203	0.6944	0.6941
	0.97	0.6829	0.7336	0.7657	0.7784	0.7402
	0.99	0.7092	0.7747	0.8099	0.8340	0.7820
Caverage		0.6441	0.6918	0.7140	0.7265	0.6941	0.6941
Oaverage		0.5952	0.6314	0.6462	0.6499	0.6307	0.6307

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A11. Rejection rate, error ∼ t(2).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.5075	0.5134	0.5077	0.4965	0.5063
	0.92	0.5198	0.5290	0.5310	0.5337	0.5284
0.9	0.95	0.5502	0.5810	0.5791	0.5798	0.5725	0.5631
	0.97	0.5705	0.5941	0.6000	0.5999	0.5911
	0.99	0.5809	0.6237	0.6312	0.6338	0.6174
Caverage		0.5458	0.5682	0.5698	0.5687	0.5631	0.5631
	0.9	0.5160	0.5321	0.5458	0.5314	0.5313
	0.92	0.5414	0.5557	0.5604	0.5591	0.5542
0.92	0.95	0.5730	0.5941	0.6058	0.6005	0.5934	0.5894
	0.97	0.5976	0.6195	0.6263	0.6381	0.6204
	0.99	0.6202	0.6460	0.6618	0.6627	0.6477
Caverage		0.5696	0.5895	0.6000	0.5984	0.5894	0.5894
	0.9	0.5593	0.5797	0.5696	0.5736	0.5706
	0.92	0.5831	0.5927	0.5959	0.5982	0.5925
0.95	0.95	0.6064	0.6430	0.6475	0.6449	0.6355	0.6341
	0.97	0.6354	0.6709	0.6858	0.6882	0.6701
	0.99	0.6512	0.7045	0.7205	0.7321	0.7021
Caverage		0.6071	0.6382	0.6439	0.6474	0.6341	0.6341
	0.9	0.5736	0.5961	0.5993	0.6070	0.5940
	0.92	0.5908	0.6260	0.6375	0.6308	0.6213
0.97	0.95	0.6407	0.6650	0.6868	0.6908	0.6708	0.6660
	0.97	0.6613	0.7048	0.7284	0.7340	0.7071
	0.99	0.6761	0.7392	0.7588	0.7735	0.7369
Caverage		0.6285	0.6662	0.6822	0.6872	0.6660	0.6660
	0.9	0.5866	0.6290	0.6315	0.6386	0.6214
	0.92	0.6175	0.6508	0.6665	0.6676	0.6506
0.99	0.95	0.6494	0.6986	0.7236	0.7324	0.7010	0.6997
	0.97	0.6891	0.7337	0.7666	0.7798	0.7423
	0.99	0.7113	0.7687	0.8146	0.8380	0.7832
Caverage		0.6508	0.6962	0.7206	0.7313	0.6997	0.6997
Oaverage		0.6004	0.6317	0.6433	0.6466	0.6305	0.6305

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A12. Rejection rate, error ∼ t(1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	0.9	0.4804	0.4400	0.3874	0.3251	0.4082
	0.92	0.5024	0.4703	0.4126	0.3588	0.4360
0.9	0.95	0.5545	0.5287	0.4806	0.4259	0.4974	0.5000
	0.97	0.5883	0.5764	0.5489	0.4802	0.5485
	0.99	0.6014	0.6250	0.6280	0.5858	0.6101
Caverage		0.5454	0.5281	0.4915	0.4352	0.5000	0.5000
	0.9	0.5148	0.4843	0.4174	0.3511	0.4419
	0.92	0.5403	0.5129	0.4579	0.3836	0.4737
0.92	0.95	0.5703	0.5592	0.5233	0.4588	0.5279	0.5344
	0.97	0.6072	0.6253	0.5828	0.5254	0.5852
	0.99	0.6256	0.6597	0.6602	0.6281	0.6434
Caverage		0.5716	0.5683	0.5283	0.4694	0.5344	0.5344
	0.9	0.5466	0.5294	0.4787	0.4128	0.4919
	0.92	0.5742	0.5762	0.5168	0.4573	0.5311
0.95	0.95	0.6191	0.6213	0.5884	0.5311	0.5900	0.5906
	0.97	0.6406	0.6638	0.6519	0.5984	0.6387
	0.99	0.6572	0.7063	0.7291	0.7128	0.7014
Caverage		0.6075	0.6194	0.5930	0.5425	0.5906	0.5906
	0.9	0.5802	0.5792	0.5415	0.4867	0.5469
	0.92	0.6042	0.6231	0.5853	0.5240	0.5842
0.97	0.95	0.6435	0.6701	0.6516	0.6039	0.6423	0.6401
	0.97	0.6524	0.7112	0.7070	0.6761	0.6867
	0.99	0.6907	0.7415	0.7681	0.7625	0.7407
Caverage		0.6342	0.6650	0.6507	0.6106	0.6401	0.6401
	0.9	0.6017	0.6346	0.6252	0.6024	0.6160
	0.92	0.6282	0.6511	0.6643	0.6414	0.6463
0.99	0.95	0.6562	0.6983	0.7158	0.6990	0.6923	0.6951
	0.97	0.6832	0.7425	0.7651	0.7607	0.7379
	0.99	0.7150	0.7673	0.8213	0.8295	0.7833
Caverage		0.6569	0.6988	0.7183	0.7066	0.6951	0.6951
Oaverage		0.6031	0.6159	0.5964	0.5529	0.5921	0.5921

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁺ = {0.99, 0.97, 0.95, 0.92, 0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A13. Rejection rate, error ∼ N(0,1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.5173	0.5196	0.5228	0.5219	0.5204
	−0.92	0.5413	0.5469	0.5444	0.5419	0.5436
−0.9	−0.95	0.5759	0.5852	0.5860	0.5783	0.5814	0.5761
	−0.97	0.6010	0.6006	0.5998	0.6084	0.6025
	−0.99	0.6234	0.6360	0.6361	0.6348	0.6326
Caverage		0.5718	0.5777	0.5778	0.5771	0.5761	0.5761
	−0.9	0.5407	0.5405	0.5435	0.5434	0.5420
	−0.92	0.5637	0.5691	0.5687	0.5730	0.5686
−0.92	−0.95	0.5945	0.6085	0.6101	0.6099	0.6058	0.6034
	−0.97	0.6255	0.6345	0.6330	0.6388	0.6330
	−0.99	0.6600	0.6635	0.6746	0.6724	0.6676
Caverage		0.5969	0.6032	0.6060	0.6075	0.6034	0.6034
	−0.9	0.5701	0.5755	0.5811	0.5755	0.5756
	−0.92	0.5948	0.6122	0.6034	0.6077	0.6045
−0.95	−0.95	0.6445	0.6582	0.6578	0.6568	0.6543	0.6478
	−0.97	0.6730	0.6868	0.6909	0.6867	0.6844
	−0.99	0.7085	0.7217	0.7239	0.7267	0.7202
Caverage		0.6382	0.6509	0.6514	0.6507	0.6478	0.6478
	−0.9	0.6038	0.6032	0.6149	0.6107	0.6082
	−0.92	0.6284	0.6394	0.6403	0.6524	0.6401
−0.97	−0.95	0.6850	0.6862	0.6896	0.6939	0.6887	0.6868
	−0.97	0.7172	0.7305	0.7268	0.7274	0.7255
	−0.99	0.7546	0.7662	0.7781	0.7866	0.7714
Caverage		0.6778	0.6851	0.6899	0.6942	0.6868	0.6868
	−0.9	0.6254	0.6229	0.6409	0.6404	0.6324
	−0.92	0.6588	0.6607	0.6595	0.6741	0.6633
−0.99	−0.95	0.7059	0.7216	0.7328	0.7297	0.7225	0.7237
	−0.97	0.7521	0.7657	0.7790	0.7867	0.7709
	−0.99	0.8010	0.8328	0.8365	0.8480	0.8296
Caverage		0.7086	0.7207	0.7297	0.7358	0.7237	0.7237
Oaverage		0.6387	0.6475	0.6510	0.6530	0.6476	0.6476

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A14. Rejection rate, error ∼ t(5).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.5225	0.5248	0.5206	0.5201	0.5220
	−0.92	0.5422	0.5476	0.5407	0.5536	0.5460
−0.9	−0.95	0.5778	0.5881	0.5766	0.5753	0.5795	0.5783
	−0.97	0.6067	0.6046	0.6102	0.6062	0.6069
	−0.99	0.6292	0.6359	0.6362	0.6473	0.6372
Caverage		0.5757	0.5802	0.5769	0.5805	0.5783	0.5783
	−0.9	0.5404	0.5435	0.5434	0.5410	0.5421
	−0.92	0.5674	0.5709	0.5717	0.5793	0.5723
−0.92	−0.95	0.5939	0.5993	0.5988	0.6146	0.6017	0.6045
	−0.97	0.6354	0.6446	0.6352	0.6476	0.6407
	−0.99	0.6550	0.6670	0.6704	0.6695	0.6655
Caverage		0.5984	0.6051	0.6039	0.6104	0.6045	0.6045
	−0.9	0.5722	0.5734	0.5852	0.5819	0.5782
	−0.92	0.6048	0.6094	0.6097	0.6123	0.6091
−0.95	−0.95	0.6434	0.6499	0.6598	0.6439	0.6493	0.6501
	−0.97	0.6841	0.6888	0.6933	0.6898	0.6890
	−0.99	0.7103	0.7263	0.7365	0.7265	0.7249
Caverage		0.6430	0.6496	0.6569	0.6509	0.6501	0.6501
	−0.9	0.6033	0.6025	0.6084	0.6087	0.6057
	−0.92	0.6363	0.6247	0.6339	0.6396	0.6336
−0.97	−0.95	0.6801	0.6914	0.6917	0.6916	0.6887	0.6845
	−0.97	0.7161	0.7273	0.7294	0.7295	0.7256
	−0.99	0.7502	0.7691	0.7751	0.7811	0.7689
Caverage		0.6772	0.6830	0.6877	0.6901	0.6845	0.6845
	−0.9	0.6334	0.6324	0.6417	0.6349	0.6356
	−0.92	0.6624	0.6638	0.6688	0.6720	0.6668
−0.99	−0.95	0.7057	0.7218	0.7296	0.7299	0.7218	0.7248
	−0.97	0.7579	0.7688	0.7798	0.7776	0.7710
	−0.99	0.7967	0.8320	0.8402	0.8457	0.8287
Caverage		0.7112	0.7238	0.7320	0.7320	0.7248	0.7248
Oaverage		0.6411	0.6483	0.6515	0.6528	0.6484	0.6484

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

,

Table A15. Rejection rate, error ∼ t(2).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.5084	0.5231	0.5100	0.5033	0.5112
	−0.92	0.5475	0.5319	0.5350	0.5316	0.5365
−0.9	−0.95	0.5780	0.5737	0.5793	0.5691	0.5750	0.5721
	−0.97	0.6037	0.6119	0.5992	0.5984	0.6033
	−0.99	0.6182	0.6434	0.6406	0.6354	0.6344
Caverage		0.5712	0.5768	0.5728	0.5676	0.5721	0.5721
	−0.9	0.5371	0.5320	0.5341	0.5235	0.5317
	−0.92	0.5705	0.5649	0.5668	0.5458	0.5620
−0.92	−0.95	0.6131	0.5970	0.6031	0.5987	0.6030	0.6009
	−0.97	0.6299	0.6436	0.6354	0.6268	0.6339
	−0.99	0.6733	0.6692	0.6711	0.6814	0.6738
Caverage		0.6048	0.6013	0.6021	0.5952	0.6009	0.6009
	−0.9	0.5774	0.5781	0.5704	0.5729	0.5747
	−0.92	0.5983	0.6062	0.5938	0.6024	0.6002
−0.95	−0.95	0.6520	0.6551	0.6588	0.6521	0.6545	0.6495
	−0.97	0.6840	0.6888	0.6917	0.6902	0.6887
	−0.99	0.7219	0.7321	0.7377	0.7270	0.7297
Caverage		0.6467	0.6521	0.6505	0.6489	0.6495	0.6495
	−0.9	0.5938	0.6001	0.6034	0.5963	0.5984
	−0.92	0.6337	0.6396	0.6328	0.6212	0.6318
−0.97	−0.95	0.6930	0.6871	0.6899	0.6873	0.6893	0.6846
	−0.97	0.7265	0.7346	0.7349	0.7237	0.7299
	−0.99	0.7635	0.7739	0.7808	0.7757	0.7735
Caverage		0.6821	0.6871	0.6884	0.6808	0.6846	0.6846
	−0.9	0.6359	0.6391	0.6361	0.6334	0.6361
	−0.92	0.6581	0.6756	0.6675	0.6729	0.6685
−0.99	−0.95	0.7182	0.7309	0.7299	0.7198	0.7247	0.7268
	−0.97	0.7606	0.7710	0.7822	0.7704	0.7711
	−0.99	0.8147	0.8337	0.8424	0.8428	0.8334
Caverage		0.7175	0.7301	0.7316	0.7279	0.7268	0.7268
Oaverage		0.6445	0.6495	0.6491	0.6441	0.6468	0.6468

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

and

Table A16. Rejection rate, error ∼ t(1).

${\mathit{\alpha }}_2$	${\mathit{\alpha }}_1$	n = 100	n = 200	n = 400	n = 800	Raverage	Saverage
	−0.9	0.4695	0.4271	0.3674	0.3162	0.3951
	−0.92	0.4981	0.4494	0.3963	0.3422	0.4215
−0.9	−0.95	0.5477	0.5158	0.4538	0.4070	0.4811	0.4882
	−0.97	0.6006	0.5626	0.5176	0.4592	0.5350
	−0.99	0.6476	0.6277	0.5966	0.5613	0.6083
Caverage		0.5527	0.5165	0.4663	0.4172	0.4882	0.4882
	−0.9	0.4982	0.4573	0.4000	0.3320	0.4219
	−0.92	0.5349	0.4850	0.4300	0.3641	0.4535
−0.92	−0.95	0.5857	0.5357	0.4922	0.4330	0.5117	0.5174
	−0.97	0.6268	0.5963	0.5419	0.4870	0.5630
	−0.99	0.6642	0.6678	0.6297	0.5857	0.6369
Caverage		0.5820	0.5484	0.4988	0.4404	0.5174	0.5174
	−0.9	0.5549	0.5045	0.4531	0.3936	0.4765
	−0.92	0.5894	0.5474	0.4904	0.4384	0.5164
−0.95	−0.95	0.6371	0.6096	0.5487	0.5007	0.5740	0.5802
	−0.97	0.6833	0.6588	0.6224	0.5470	0.6279
	−0.99	0.7314	0.7280	0.7055	0.6605	0.7064
Caverage		0.6392	0.6097	0.5640	0.5080	0.5802	0.5802
	−0.9	0.5982	0.5616	0.5063	0.4546	0.5302
	−0.92	0.6262	0.5916	0.5457	0.4965	0.5650
−0.97	−0.95	0.6867	0.6579	0.6199	0.5563	0.6302	0.6343
	−0.97	0.7389	0.7106	0.6681	0.6232	0.6852
	−0.99	0.7851	0.7794	0.7557	0.7235	0.7609
Caverage		0.6870	0.6602	0.6191	0.5708	0.6343	0.6343
	−0.9	0.6437	0.6386	0.5923	0.5597	0.6086
	−0.92	0.6721	0.6643	0.6393	0.6028	0.6446
−0.99	−0.95	0.7295	0.7323	0.7047	0.6644	0.7077	0.7121
	−0.97	0.7847	0.7797	0.7563	0.7301	0.7627
	−0.99	0.8333	0.8511	0.8384	0.8246	0.8369
Caverage		0.7327	0.7332	0.7062	0.6763	0.7121	0.7121
Oaverage		0.6387	0.6136	0.5709	0.5225	0.5864	0.5864

Note: ‘Raverage’ (stands for row average) is the average rejection rate for different values of n for the same case; ‘Caverage’ (stands for column average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} for the same n for the same situation; ‘Saverage’ (stands for situation average) is the average rejection rate for different cases in which α₁ ∈ A⁻ = {−0.99, −0.97, −0.95, −0.92, −0.9} and different n for the same situation and ‘overall average’ in the 3 to 6 columns is the overall average for each n for all situations and all cases while ‘overall average’ in the last two column is the overall average for all the cases in the entire table.

displaying in Appendices Appendix C and Appendix D, respectively.

We first discuss the cases when both ${\alpha }_1$ and ${\alpha }_2$ are positive as stated in Situation 5. Compared with the results in Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7 and Table A8, all of the rejection rates in Table A9, Table A10, Table A11 and Table A12 are significantly higher than 5% and the rejecting frequency of the T test is higher than 49% for any n and any error distribution studied in our paper, except the situation when the error term follows $t\left(1\right)$ in which the rejection rates is higher than 32%. In addition, as n increases, or either ${\alpha }_1$ or ${\alpha }_2$ increases, or as the error distributions are further away from normal distribution, the rejecting rate increases even further.

We turn to discuss the cases when both ${\alpha }_1$ and ${\alpha }_2$ are negative as stated in Situation 6. Similar to the cases when both ${\alpha }_1$ and ${\alpha }_2$ are positive, when both ${\alpha }_1$ and ${\alpha }_2$ are negative, the rejecting frequency of the T test is higher than 50% for any n and for any error distribution studied in our paper, except the situation when the error term follows $t\left(1\right)$ in which the rejection rates is higher than 31%. In addition, Similar to the cases when both ${\alpha }_1$ and ${\alpha }_2$ are positive, as n increases, or either ${\alpha }_1$ or ${\alpha }_2$ increases, or as the error distributions are further away from normal distribution, the rejecting rate increases even further.

Our analysis shows that, different from all the cases when for ${\alpha }_1$ and ${\alpha }_2$ are of different signs, for all the cases when ${\alpha }_1$ and ${\alpha }_2$ are of the same signs, either positive or negative, as stated in Situations 5 and 6, respectively, and when both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, the rejection rate is much higher than the 5% level of significance for all the cases studied in our paper and it could be higher than 49%, implying that when $({\alpha }_1,{\alpha }_2)$ follow Situations 5 and 6 and when both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, the chance that the regressions being spurious is very high for all the cases simulated in our paper, which, in turn, rejects Conjecture 1 for all the cases in Situations 5 and 6.

5. Concluding Remarks

In this paper, we conjecture that under some situations, the regression of two independent and nearly non-stationary series does not have any spurious problem at all. To check whether our conjecture holds, we first generate two independent and nearly nonstationary AR(1) processes, $X_t={\alpha }_1{X}_{t-1}+{\epsilon }_t$ and $Y_t={\alpha }_2{Y}_{t-1}+e_t$ in which $0.9<|{\alpha }_1|,|{\alpha }_2|<1$. We then regress $Y_t$ on independent $X_t$ to get $Y_t=\alpha +\beta X_t+u_t$ and check whether the proportion of rejecting the null hypothesis of the beta ($\beta$) to be zero. We first find that consistent with the literature that supports the hypothesis of regressing two independent and (nearly) nonstationary time series could be spurious, when both ${\alpha }_1$ and ${\alpha }_2$ are of the same signs, either positive or negative, and when the values of both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, the rejection rate is much bigger than the 5% level of significance in all the cases examined in our simulation and it could be higher than 49% in many cases, implying that the chance that the regressions being spurious is very high for all the cases when both ${\alpha }_1$ and ${\alpha }_2$ are of the same signs.

Nonetheless, for all the cases when for ${\alpha }_1$ and ${\alpha }_2$ are of different signs, then different from the literature, our results show that when both $|{\alpha }_1|$ and $|{\alpha }_2|$ are between $0.9$ and $0.99$, the rejection rates are much smaller than the 5% level of significance for all the cases studied in our paper, implying that when ${\alpha }_1$ and ${\alpha }_2$ are of different signs, regressing nearly nonstationary $Y_t$ on independent and nearly nonstationary $X_t$ will not get any spurious problem at all for all the cases being simulated in our paper.

We note that the literature shows that the regression of independent and (nearly) nonstationary time series could result in spurious outcomes. In this paper, we conjecture that under some situations, regression of two independent and nearly non-stationary series does not have any spurious problem at all, and in this paper, we aim to find some situations that our conjecture could hold. In this paper, we find that when $({\alpha }_1,{\alpha }_2)\in (A^{+},A^{-})$ or $(A^{-},A^{+})$, then our conjecture holds. We note that when $({\alpha }_1,{\alpha }_2)\in (A^{+},A^{-})$ or $(A^{-},A^{+})$, our conjecture holds which does not imply that these are only situations that our conjecture holds. There could have other situations that our conjecture could hold. We leave it to future studies to find other situations that our conjecture could hold. The purpose of our paper is to tell readers that when one finds regression of any two or more time series that do not have any spurious problem, this does not necessarily imply that the series are not independent. Thus, academics and practitioners should conduct some proper tests to show whether the series are independent.

Some academics may wonder whether there are some financial or economic time series that exhibit extreme negative autocorrelations. We believe there could have some financial or economic time series exhibit positive autocorrelations and some exhibit negative autocorrelations. We note that the time period used in our paper may not be daily or monthly, it should be set to fit the nature of the time series. It is well-known that stock returns could be overreacted or underreacted, this means that it could be positively auto-correlated or negatively auto-correlated and the true unobserved stock returns are positively auto-correlated or negatively auto-correlated. Whether they are extreme positively auto-correlated or negatively auto-correlated will depend on particular stocks. In addition, as we have mentioned before, when $({\alpha }_1,{\alpha }_2)\in (A^{+},A^{-})$ or $(A^{-},A^{+})$, our conjecture holds which does not imply that these are the only situations that our conjecture could hold. There may have other situations that our conjecture could hold. Some financial or economic time series could follow other situations that yet to be discovered, and thus, the conjecture could be important not only for statistics, but also for economics and finance. We also note that in our paper, we only consider $(A^{+},A^{-})$ to cover nearly non-stationary series but do not cover the situations $A^{+}=1$ and $A^{-}=-1$. We do not cover the situation $A^{+}=1$ because this has been well-studied in the literature. On the other hand, we do not cover the situation $A^{-}=-1$ because this situation, we believe, is of no practice relevance.

We note that as far as we know, this paper is the first paper to discover that under some situations, the regression of two independent and nearly non-stationary series does not have any spurious problem at all. We follow Granger and Newbold (1974) and others to provide simulation results to show our discovery. Academics could follow Phillips (1986), Johansen and Schaumburg (1999), and others to provide formal proof of the finding in our paper to replace Brownian motions by using the OU processes with $exp(c/T)$ to approximate $(1+c/T)$.3 We will leave it to further research to develop the theoretical results to explain the phenomena discovered in this paper. We also note that in this paper, we get very good results by using $0.9<\left|\alpha \right|<1$. One may get good results by using the near-integrated approach. We will leave this to future studies.4 Another problem in our study is that there is a serious problem with under-rejection. Further study could expose this problem and correct the test properly.