A Generalized Residual-Based Test for Fractional Cointegration in Panel Data with Fixed Effects

Abstract

Asymptotic theories for fractional cointegrations have been extensively studied in the context of time series data, with numerous empirical studies and tests having been developed. However, most previously developed testing procedures for fractional cointegration are primarily designed for time series data. This paper proposes a generalized residual-based test for fractionally cointegrated panels with fixed effects. The test’s development is based on a bivariate panel series with the regressor assumed to be fixed across cross-sectional units. The proposed test procedure accommodates any integration order between $[0,1]$, and it is asymptotically normal under the null hypothesis. Monte Carlo experiments demonstrate that the test exhibits better size and power compared to a similar residual-based test across varying sample sizes.

1. Introduction

Numerous studies have explored panel data analysis and cointegration either independently or in tandem. Various methods, including residual-based and spectral-based approaches, have been devised to tackle issues like unit roots, cross-sectional dependence, and heterogeneity. Panel data analysis has garnered considerable attention owing to its wide-ranging applicability across fields such as epidemiology, demography, finance, and economics [1,2,3]. Recently, researchers in engineering and reliability analysis have expanded panel data analysis to encompass count data [4]. These data comprise repeated observations of discrete events over time for each individual or unit. Overall, discussions in panel data encompass topics like developing robust statistical models, devising efficient estimation techniques, and addressing challenges related to data heterogeneity, unit root problems, and correlation structures [5,6,7].

Ref. [8] highlighted the increasing attention given to unit root problems in panel data and the consequent identification of cointegration relationships among variables. The existing panel cointegration techniques were initially designed for balanced panel data with moderate time and cross-sectional units. However, in scenarios involving large time and cross-sectional units with the potential for long memory, conventional panel cointegration tests are inadequate [9,10,11]. The presence of long memory often implies fractional mean reversion, suggesting equilibrium occurs over fractional time periods. Therefore, there is a need to explore fractional cointegration or equilibrium mean reversion within the context of panel data.

There have been numerous fractional cointegration tests developed within the realm of time-series analysis [12]. These tests are typically grouped into two categories: spectral-density-based and residual-based. Ref. [13] developed a residual-based test utilizing the residuals of the multivariate fractional cointegration common-components model with varying memory parameters. Ref. [14] compared semiparametric tests of fractional cointegration, evaluating nine tests for both spectral-density- and residual-based approaches. They found that several methods yield significantly different results when correlated short-run components are present. Moreover, when applied to common-component models rather than triangular systems, these methods exhibit varied power. Notably, there is a significant difference in the power of the tests between the two models.

In empirical studies, Ref. [15] investigated the memory of exchange rates, while ref. [16] explored the dynamics of interest rate futures markets and stock market prices. Both studies identified evidence of fractional cointegration in stock market prices, achieving satisfactory results under the assumption that the observations are $I\left(1\right)$ processes. Subsequently, testing for fractional cointegration was extended to fractionally integrated processes. Ref. [17] proposed a test based on the joint local Whittle estimation of all parameters, which eliminates the possibility of the two underlying series having equal integration orders. In [18], the authors developed a Hausman-type test to detect fractional cointegration, with the additional assumption that the cointegration error is nonstationary. Ref. [19] proposed a Hausman-type test for time series with equal integration orders and no cointegration, which involves determining a bandwidth. In panel settings, Ref. [9] studied a large cross-sectional and time unit heterogeneous panel data model with fixed effects. Their approach allows for cross-sectional dependency, persistency, and fractionally integrated errors. In addition, the methodology provides a general treatment for stationary and nonstationary indicators. Monte Carlo simulation showed that it works effectively in practice. Ref. [20] proposed an extension of the Generalized Method of Moments (GMM) for a fixed-effect, fractionally integrated panel model. Both [9]’s and [20]’s studies assumed that the fixed-effect parameter fizzled out in the long run. The method of [9] is limited in that fractional cointegration is assumed in a panel system if the estimate of mean reversion for the series is greater than that of the residuals [9,21]. This approach is expected to inflate the type I error, as there is an increased possibility of many rejections of mean reversion when there is none [22,23].

Furthermore, Ref. [24] introduced a new panel cointegration test that is robust to nonlinearity, structural breaks, and cross-sectional dependency. The proposed method is a bootstrap panel cointegration test called the Fractional Frequency Flexible Fourier Form for Panel Cointegration Test, and it was empirically illustrated by testing the Feldstein–Horioka paradox for 15 Asian countries. It was discovered that Indonesia, Philippines, Bangladesh, Japan, Thailand, and China are not among the countries that generate cointegration in the cross-section. The study is limited, however, as it only considered country-specific cointegration rather than the panel of interest.

In summary, the findings of the finite sample properties of various fractional cointegration tests for time-series data reviewed by [14] revealed the exemplary applicability of the residual-based methods of [13,25] for stationary systems under the common component model assumption. Also, the study showed that the class of tests with low power under the alternative hypothesis of fractional cointegration are [19,26]. However, the methods of Refs. [17,27,28] are resistant to short-term correlation and are commonly applied due to their simplistic framework. Of all the works conducted on fractional integration and cointegration, none have developed a test particularly for panel data.

Thus, in this paper, a generalized fractional cointegration testing procedure is developed, where both fractionally and non-fractionally cointegrated models are considered such that the observed series with cross-sections and cointegrating error are both fractional and non-fractional processes. In addition, a residual-based testing procedure for generalized fractional cointegration is proposed, and its performance is compared to an existing fractionally cointegrated test when $0\le d\le 1$.

The proposed test involves modifying the residual-based test proposed by [25], which involves two integrated time series $y_t$ and $x_t$, where the observed series are $I\left(d\right)$ processes, the regression residual ${ϵ}_t=y_t-\beta x_t$ is an $I\left(\gamma \right)$, $\gamma$, and d can be real-valued. The test includes traditional cointegration as a special case. Ref. [25] constructed a test statistic which has an asymptotic standard normal distribution under the null hypothesis of no cointegration using a consistent estimate of d and $\gamma$ obtained from $x_t$ and the residual ${ϵ}_t$.

2. Wang et al. (2015) [25] Fractional Cointegration Test

Let $x_t$, $y_t$, where $t=1,2,3,\cdots ,T$, represent two processes, both characterized by $I\left(d\right)$. Ref. [29] demonstrated that for a given scalar $\beta \ne 0$, the linear combination ${\mathbf{ϵ}}_t=y_t-\beta x_t$ also exhibits $I\left(d\right)$ behavior, potentially with ${ϵ}_t$ being $I(d-b)$, where $b>0$. Consequently, given real numbers d and b, the elements of a vector $c_t$ are considered to be cointegrated of order d, b, denoted as $c_t\sim CI(d,b)$, if the following hold:

i.

All elements of $c_t$ follow $I\left(d\right)$.

ii.

There exists a non-zero vector $\alpha$ such that $s_t={\alpha }^{\prime }{c}_t\sim I\left(\gamma \right)=I(d-b)$, where $b>0$.

Here, $\alpha$ and $s_t$ are termed the cointegration vector and error, respectively [30]. Also, it is assumed that $d\ge b$ such that $(d-b)\ge 0$. Therefore, a simple bivariate system of fractionally cointegrated $x_t$ and $y_t$ processes is defined as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_t& =\beta x_t+{(1-L)}^{-\gamma }{ϵ}_{1t}\hfill \\ \hfill x_t& ={(1-L)}^{-d}{ϵ}_{2t}\hfill \end{array}\end{array}$$

(1)

for positive t. The vector ${\mathbf{ϵ}}_t={({ϵ}_{1t},{ϵ}_{2t})}^{\prime }$ now represents a bivariate zero-mean covariance stationary $I\left(0\right)$ process, where $\beta \ne 0$ and $\gamma <d$. In Equation (1), both $x_t$ and $y_t$ are $I\left(d\right)$, and ${ϵ}_{1t}=y_t-\beta x_t$ is $I\left(\gamma \right)$. The lag operator L is defined as $L{y}_t=y_{t-1}$ and the difference operator ${\Delta }^{-d}$ is obtained using ${(1-L)}^d={\sum }_{j=0}^{\infty }\left(\genfrac{}{}{0pt}{}{d}{j}\right){(-1)}^j{L}^j$, where $\left(\genfrac{}{}{0pt}{}{d}{j}\right)=\frac{d!}{j!(d-j)!}$. Unlike the standard $CI(1,1)$ cointegration, the memory parameter d remains unknown in fractionally cointegrated systems and requires estimation. The corresponding hypotheses to assess whether the two processes exhibit fractional cointegration are as follows:

Hypothesis 0 (H0): 

$x_t$ and $y_t$ are not fractionally cointegrated $(d=\gamma )$,

Hypothesis 1 (H1): 

$x_t$ and $y_t$ are fractionally cointegrated $(d>\gamma )$.

The fractional cointegration test proposed by [25] is based on the second component $x_t$ of the process ${\mathit{X}}_t=(y_t,x_t)$ and the associated residuals ${ϵ}_t=y_t-\beta x_t$. Ref. [25] devised a simple t-like test statistic that employs the spectral density of the component $x_t$, denoted as ${\widehat{f}}_{22}=\frac{1}{2\pi T}{\sum }_{t=1}^T{\left({\Delta }^{\widehat{d}}{x}_t\right)}^2$, and the fractional cointegration parameter $\gamma$ of residual ${ϵ}_{2t}$. Thus, the statistic is expressed as follows:

$$F_w=\frac{{\sum }_{t=1}^T{\Delta }^{\widehat{\gamma }}{x}_t}{\sqrt{2\pi T{\widehat{f}}_{22}}}\stackrel{H_0}{\to }N(0,1).$$

(2)

The method requires $d>0.5$ so that a consistent cointegrating vector $\beta$ can be estimated using Ordinary Least Squares (OLS). The first step in the construction of the test is to estimate the cointegration parameter $\beta$ using

$${\widehat{\beta }}_{ols}=\frac{{\sum }_{t=1}^T{x}_t{y}_t}{{\sum }_{t=1}^T{x}_t^2}$$

(3)

and then obtain the residuals ${\widehat{ϵ}}_{1t}=y_t-{\widehat{\beta }}_{ols}{x}_t$. The values $\widehat{d}$ and $\widehat{\gamma }$ are later estimated from the series $x_t$ and ${\widehat{ϵ}}_{1t}$, respectively, using the method of [31]. Correspondingly, the differenced series ${\Delta }^{\widehat{d}}{x}_t$ and ${\Delta }^{\widehat{\gamma }}{x}_t$ are then calculated.

3. Generalized Residual-Based Fractional Cointegration Test for Fixed-Effect Panel Model

Suppose we have a balanced fixed-effect panel model with n being the number of cross-sectional units and T being the time period such that $N=n\times T$ is the total sample size. The model is given as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_{it}& ={\mu }_i+\beta x_{it}+{(1-L)}^{-\gamma }{ϵ}_{1it}\hfill \\ \hfill x_{it}& ={(1-L)}^{-d}{ϵ}_{2it}.\hfill \end{array}\end{array}$$

(4)

where $\beta$ is the cointegration parameter and is assumed to be constant over i cross-sectional units, ${\mu }_i$ is the fixed-effect coefficient for the ith cross-sectional units, $x_t$ and $y_t$ represent simple bivariate processes denoting independent and dependent variables in the panel model, which are fractionally cointegrated if we can establish the following Assumptions:

A₁.

$x_t$ and $y_t$ are both $I\left(d\right)$ with $0\le d\le 1$ and ${ϵ}_{1it}=y_t-\beta x_t$ is $I\left(\gamma \right)$;

A₂.

The vector ${\mathbf{ϵ}}_{it}={({ϵ}_{1it},{ϵ}_{2it})}^{\prime }$ is a bivariate zero mean covariance stationary $I\left(0\right)$ process which is independent across i, $\beta \ne 0$, and $\gamma <d$;

A₃.

The vector ${\mu }_i$ fizzles out in the long-run such that ${\mu }_i=0$ as $N\to \infty$.

The corresponding hypotheses to assess whether the two processes exhibit fractional cointegration are as follows:

Hypothesis 0 (H0): 

$x_{it}$ and $y_{it}$ are not fractionally cointegrated $(d=\gamma )$.

Hypothesis 1 (H1): 

$x_{it}$ and $y_{it}$ are fractionally cointegrated $(d>\gamma )$.

Notice that we can rewrite (4) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_{it}-{\widehat{\mu }}_i& =\beta x_{it}+{(1-L)}^{-\gamma }{ϵ}_{1it}\hfill \\ \hfill x_{it}& ={(1-L)}^{-d}{ϵ}_{2it}.\hfill \end{array}\end{array}$$

(5)

where ${\widehat{\mu }}_i={\overline{y}}_i=T^{-1}{\sum }_{t=1}^T{y}_{it}$ and denotes $z_{it}=y_{it}-{\widehat{\mu }}_i$, such that we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill z_{it}& =\beta x_{it}+{(1-L)}^{-\gamma }{ϵ}_{1it}\hfill \\ \hfill x_{it}& ={(1-L)}^{-d}{ϵ}_{2it}.\hfill \end{array}\end{array}$$

(6)

It is clear that (6) is the demean transformed version of (1), which is reduced to the original time series model in (1) with cross-sectional parameter ${\mu }_i$ factored out. According to [13], when $d\le 0.5$, a consistent $\beta$ in (6) can be estimated using the Tapered Narrow Band Least Square (TNBLS). The TNBLS [13] procedure involves the estimation of a complex-valued taper $q_t$ defined as

$$q_t=\frac{1}{2}\left(1-{exp}^{-i2\pi (t-1/2)T^{-1}}\right),t=1,2,\dots ,T.$$

(7)

The next step involves obtaining the discrete tapered Fourier transform of the series ${\eta }_t$ and the cross-periodogram using

$${\omega }_{\eta ,j}^{\prime }={\left(2\pi \sum _{t=1}^T{\left|q_t^{p-1}\right|}^2\right)}^{-0.5}\sum _{t=1}^T{q}_t^{p-1}{\eta }_t{exp}^{-i{\lambda }_{j}t},$$

(8)

$$I_{\eta \overline{\eta },j}^{\prime }={\omega }_{\eta ,j}^{\prime }{\overline{\omega }}_{\eta ,j}^{\prime }$$

(9)

respectively. The averaged tapered periodogram obtained using m bandwidth is given by

$${\widehat{F}}_{\eta \overline{\eta },j}^{\prime }\left(m\right)=2\pi T^{-1}\sum _{j=1}^m\Re I_{\eta \overline{\eta },j}^{\prime },1\le m\le \frac{T}{2}.$$

(10)

Therefore, the estimate of a consistent long-memory parameter $\beta$ in (4) when $d\le 0.5$ is

$${\widehat{\beta }}_m=\frac{{\widehat{F}}_{xz}^{\prime }\left(m\right)}{{\widehat{F}}_{xx}^{\prime }\left(m\right)}$$

(11)

where $m\ge 1$ is fixed. If instead of keeping m fixed, we substitute $m=T/2$ and avoid differencing and tapering, we obtain the ordinary least squares (OLS) estimator [13].

Ref. [25] established that if $d>0.5$, the OLS estimator is consistent, and otherwise, it is inconsistent. In order to develop a generalized test statistic that is usable for all d’s in the range of $[0,1]$, we developed a piecewise estimator for $\beta$ for the two possible situations, that is, for $d\le 0.5$ and $d>0.5$. Thus, our proposed estimator for the long-memory parameter $\beta$ when $0<d<1$ is denoted by ${\widehat{\beta }}_{mix}$, and it can be estimated using

$${\widehat{\beta }}_{mix}=\left\{\begin{array}{cc}\frac{{\widehat{F}}_{xz}^{\prime }\left(m\right)}{{\widehat{F}}_{xx}^{\prime }\left(m\right)}\hfill & 0<d\le 0.5\hfill \\ \\ \frac{{\sum }_{i,t=1}^N{x}_{it}{z}_{it}}{{\sum }_{i,t=1}^N{x}_{it}^2}\hfill & 0.5<d\le 1\hfill \end{array}\right.$$

(12)

Theorem 1. 

For the fixed-effect fractional cointegrated panel model defined in (4) satisfying A1 and A2, the long-memory parameter can be estimated with (12). Thus, the modified test statistic $M_w=\frac{{\sum }_{i,t=1}^N{\Delta }^{\widehat{\gamma }}{x}_{it}}{\sqrt{2\pi N{\widehat{K}}_{22}}}$, where ${\widehat{K}}_{22}=\frac{1}{2\pi N}{\sum }_{i,t=1}^N{\left({\Delta }^{\widehat{d}}{x}_{it}\right)}^2$, converges; $M_w\stackrel{d}{\to }N(0,1)$ under $H_0$ and diverges under $H_1$.

Proof. 

It is required to show that

$$M_w=\frac{{\sum }_{i,t=1}^N{\Delta }^{\widehat{\gamma }}{x}_{it}}{\sqrt{2\pi N{\widehat{K}}_{22}}}\stackrel{H_0}{\to }N(0,1).$$

(13)

Equation (13) can be rewritten as

$$M_w=\frac{S_N}{{\widehat{k}}_{22}\sqrt{N}}.$$

(14)

where $S_N={\sum }_{i,t=1}^N{\Delta }^{\widehat{\gamma }}{x}_{it}$, ${\widehat{k}}_{22}=\sqrt{2\pi {\widehat{K}}_{22}}$, since $S_N$ is the sum of N independently and identically distributed random variables. Recall that the moment-generating function $Q\left(u\right)=E\left(e^{ux}\right)$ of $S_N$ and correspondingly $M_w$ can be defined as

$$Q_{S_N}\left(u\right)={\left(Q\left(u\right)\right)}^N;$$

$$Q_{M_w}\left(u\right)={\left[Q\left(\frac{u}{{\widehat{k}}_{22}\sqrt{N}}\right)\right]}^N.$$

Now, computing the Taylor’s series expansion of $Q\left(u\right)$ around 0 leads to

$$Q\left(u\right)=Q\left(0\right)+Q^{\prime }\left(0\right)u+\frac{1}{2}{Q}^{″}\left(0\right)u^2+rem=1+\frac{1}{2}{K}_{22}{u}^2+\mathcal{O}\left(u^3\right),$$

since $Q\left(0\right)=E\left(e^0\right)=1$, $Q^{\prime }\left(0\right)=\frac{d}{du}E\left(e^{ux}\right)=E\left(x\right)=0$ (x is assumed to be the differenced $x_{it}$ whose mean is zero under $H_0$), $Q^{″}\left(0\right)=\frac{d^2}{d{u}^2}E\left(e^{ux}\right)=Var\left(x\right)=Var\left({\Delta }^{\widehat{\gamma }}{x}_{it}\right|H_0)=K_{22}$. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Q\left(\frac{u}{{\widehat{k}}_{22}\sqrt{N}}|H_0\right)& =1+\frac{1}{2}{K}_{22}{\left(\frac{u}{k_{22}\sqrt{N}}\right)}^2+\mathcal{O}\left[{\left(\frac{u}{k_{22}\sqrt{N}}\right)}^3\right]\hfill \\ \hfill & =1+\frac{u^2}{2N}+\mathcal{O}\left(\frac{1}{N^{3/2}}\right)\hfill \end{array}\end{array}$$

$$Q_{M_w}\left(u\right|H_0)={\left[1+\frac{u^2}{2N}+\mathcal{O}\left(\frac{1}{N^{3/2}}\right)\right]}^N\stackrel{N\to \infty }{\to }{e}^{u/2}.$$

The moment-generating function of a Gaussian random variable $\varsigma \sim N(0,1)$ with mean 0 and variance 1 is defined as $Q_{\varsigma }\left(u\right)=E\left(e^{u\varsigma }\right)=e^{u/2}$. Thus, $M_w\stackrel{d}{\to }N(0,1)$ under $H_0$. On the other hand, under $H_1$, $E\left(e^{ux}\right)=Var\left(x\right)=Var\left({\Delta }^{\widehat{\gamma }}{x}_{it}\right|H_1)\ne K_{22}$. Let $Var\left({\Delta }^{\widehat{\gamma }}{x}_{it}\right|H_1)=G_{22}$; then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Q\left(\frac{u}{{\widehat{k}}_{22}\sqrt{N}}|H_1\right)& =1+\frac{1}{2}{G}_{22}{\left(\frac{u}{k_{22}\sqrt{N}}\right)}^2+\mathcal{O}\left[{\left(\frac{u}{k_{22}\sqrt{N}}\right)}^3\right]\hfill \\ \hfill & =1+\frac{u^2}{2N}\left(\frac{G_{22}}{{\widehat{K}}_{22}}\right)+\mathcal{O}\left[\frac{1}{N^{3/2}}\left(\frac{G_{33}}{{\widehat{k}}_{22}}\right)\right].\hfill \end{array}\end{array}$$

$$Q_{M_w}\left(u\right|H_1)={\left\{1+\frac{u^2}{2N}\left(\frac{G_{22}}{{\widehat{K}}_{22}}\right)+\mathcal{O}\left[\frac{1}{N^{3/2}}\left(\frac{G_{33}}{{\widehat{k}}_{22}}\right)\right]\right\}}^N\stackrel{N\to \infty }{\to }{e}^{u/2\left(\frac{G_{22}}{{\widehat{K}}_{22}}\right)}.$$

□

4. Simulation Study

We consider the following balanced panel model with fixed effects, where n denotes the count of cross-sectional units and T represents the total time units, resulting in a total sample size of $N=n\times T$. The model is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_{it}& ={\mu }_i+\beta x_{it}+{(1-L)}^{-\gamma }{ϵ}_{1it}\hfill \\ \hfill x_{it}& ={(1-L)}^{-d}{ϵ}_{2it}.\hfill \end{array}\end{array}$$

(15)

where ${\mu }_i=(5,10,15,20,25)$ are the panel intercepts across the units $i=1,2,\dots ,5$. Monte Carlo experiments are conducted to examine the finite sample performance of the tests. Let ${(y_{it},x_{it})}^{\prime }$ be generated from model (1) with $\beta =1$, ${ϵ}_{it}={({ϵ}_{1it},{ϵ}_{2it})}^{\prime }$ being a Gaussian white noise with $E\left({ϵ}_{it}\right)=0$, $Var\left({ϵ}_{1it}\right)=Var\left({ϵ}_{2it}\right)=1$ and $Cov({ϵ}_{1it},{ϵ}_{2it})=\rho$. We consider cases with $\rho =0.0,0.5$ and sample sizes $N=500,1250,2500$ corresponding to $T=100,250,1000$. Similar approaches were used in [14,22,25,32,33,34,35,36,37,38,39].

The test statistic simulation procedure follows the same three-step approach used in [25], which is as follows:

• Step 1: Estimate $\widehat{d}$ using $x_{it}$ by the method of [31].
• Step 2: Compute ${\widehat{K}}_{22}=\frac{1}{2\pi N}{\sum }_{i,t=1}^N{\left({\Delta }^{\widehat{d}}{x}_{it}\right)}^2$.
• Step 3: Compute the estimate of the long-memory parameter using ${\widehat{\beta }}_{mix}$ and use it to estimate ${\widehat{ϵ}}_{1it}=y_{it}-{\widehat{\beta }}_{mix}{x}_{it}$. Again, estimate $\widehat{\gamma }$ using ${ϵ}_{1it}$ by the method used in step 1. Thus, the test statistic $M_w$ is computed. Each statistic is replicated 5000 times so as to estimate the empirical type 1 error rates at $1\%$, $5\%$, and $10\%$.

The empirical type 1 error rates and power are reported in

Table 1. Empirical type I error rate for original [25] test ($F_w$) and proposed modified [25] test ($M_w$) at varying levels of $d=\gamma$, $\rho$, and sample sizes N.

			$\mathit{\alpha }=0.01$			$\mathit{\alpha }=0.05$			$\mathit{\alpha }=0.10$
	$\mathit{d}=\mathbf{\gamma }$	Method/N	500	1250	2500	500	1250	2500	500	1250	2500
	0.3	$F_w$	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.000
		$M_w$	0.015	0.014	0.013	0.066	0.055	0.052	0.118	0.115	0.106
$\rho =0.0$	0.6	$F_w$	0.028	0.018	0.015	0.069	0.063	0.048	0.107	0.094	0.091
		$M_w$	0.026	0.016	0.011	0.079	0.069	0.056	0.145	0.120	0.115
	0.8	$F_w$	0.036	0.027	0.018	0.083	0.073	0.065	0.146	0.125	0.116
		$M_w$	0.015	0.013	0.010	0.068	0.064	0.059	0.128	0.112	0.101
	1	$F_w$	0.026	0.023	0.016	0.076	0.070	0.065	0.122	0.117	0.112
		$M_w$	0.008	0.011	0.013	0.059	0.051	0.051	0.114	0.104	0.101
	0.3	$F_w$	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
		$M_w$	0.012	0.014	0.011	0.059	0.054	0.052	0.123	0.111	0.103
$\rho =0.5$	0.6	$F_w$	0.023	0.017	0.016	0.060	0.054	0.050	0.107	0.091	0.097
		$M_w$	0.020	0.012	0.013	0.084	0.067	0.062	0.148	0.120	0.116
	0.8	$F_w$	0.038	0.022	0.017	0.089	0.070	0.067	0.140	0.126	0.109
		$M_w$	0.014	0.011	0.010	0.060	0.059	0.057	0.127	0.119	0.109
	1	$F_w$	0.030	0.020	0.019	0.076	0.068	0.060	0.123	0.113	0.113
		$M_w$	0.008	0.010	0.010	0.046	0.044	0.043	0.110	0.105	0.102

. In Table 1, it was observed that the original [25] $F_w$ test undersized the nominal size when $d<0.5$, as expected. However, when $d>0.5$, its empirical type 1 error rates compete with the modified test. On the other hand, the modified test’s empirical type 1 error rates are slightly oversized and converge to the nominal size as $N\to \infty$, irrespective of the d values and the correlation values $\rho$. Overall, the empirical type 1 error rates returned by the modified test $M_w$ are relatively closer to the nominal size than the original [25]$F_w$ test. This establishes the validity and applicability of the proposed $M_w$ test for fractionally cointegrated panels.

For the power results in

Table 2. Empirical power for original [25] test ($F_w$) and proposed modified [25] test ($M_w$) at varying levels of d, $\gamma$, $\rho =0.0$, and sample sizes N.

			$\mathit{\alpha }=0.01$			$\mathit{\alpha }=0.05$			$\mathit{\alpha }=0.10$
	$\mathbf{\gamma }$	Method/N	500	1250	2500	500	1250	2500	500	1250	2500
$d=1.0$	$0.9$	$F_w$	0.146	0.169	0.175	0.256	0.258	0.307	0.332	0.362	0.369
		$M_w$	0.169	0.186	0.197	0.286	0.310	0.352	0.390	0.410	0.434
	$0.6$	$F_w$	0.765	0.815	0.877	0.851	0.890	0.919	0.856	0.900	0.910
		$M_w$	0.837	0.860	0.899	0.889	0.907	0.937	0.906	0.919	0.929
	$0.3$	$F_w$	0.913	0.959	0.967	0.944	0.952	0.980	0.949	0.966	0.981
		$M_w$	0.969	0.983	0.984	0.974	0.994	0.994	0.982	0.992	0.990
	$0.0$	$F_w$	0.935	0.961	0.968	0.957	0.969	0.976	0.950	0.971	0.979
		$M_w$	0.993	0.997	1.000	0.995	0.998	1.000	0.999	0.997	1.000
$d=0.9$	$0.6$	$F_w$	0.638	0.694	0.759	0.721	0.747	0.826	0.721	0.747	0.826
		$M_w$	0.737	0.743	0.803	0.795	0.796	0.850	0.795	0.796	0.850
	$0.3$	$F_w$	0.856	0.887	0.939	0.887	0.924	0.952	0.887	0.924	0.952
		$M_w$	0.950	0.971	0.980	0.961	0.977	0.986	0.961	0.977	0.986
$d=0.6$	$0.3$	$F_w$	0.264	0.335	0.421	0.392	0.457	0.530	0.392	0.457	0.530
		$M_w$	0.728	0.739	0.813	0.788	0.793	0.847	0.788	0.793	0.847
	$0.0$	$F_w$	0.389	0.446	0.503	0.518	0.569	0.625	0.518	0.569	0.625
		$M_w$	0.966	0.973	0.977	0.978	0.978	0.979	0.978	0.978	0.979
$d=0.4$	$0.1$	$F_w$	0.005	0.006	0.012	0.011	0.021	0.033	0.055	0.070	0.082
		$M_w$	0.712	0.735	0.803	0.815	0.819	0.868	0.817	0.820	0.870

, we considered $\gamma <d$. The powers of the two tests approach 1 when N increases and when the effect size $\gamma -d$ is large. Again, the powers of the $M_w$ are in most cases closer to 1 than $F_w$. The better performance of $M_w$ observed in Table 1 and Table 2 can be attributed to model adequacy. While $M_w$ was developed using a panel data model assumption, $F_w$ was developed using a time series model. Since the model simulated is a panel one, $M_w$ is expected to be better than $F_w$.

5. A Fractional Cointegration Panel Model for Realized Industry and Market Volatilities in the U.S. Economy

The data used here were drawn from Yahoo Finance and Kenneth French’s Data Library. Five industry portfolios (Cnsmr: Consumer Durables, Nondurables, Wholesale, Retail, and Some Services; Manuf: Manufacturing, Energy, and Utilities; HiTec: Business Equipment, Telephone and Television Transmission; Hlth: Healthcare, Medical Equipment, and Drugs; and Other: Mines, Construction, Building Materials, Transport, Hotels, Bus Services, Entertainment, Finance) in the U.S. economy spanning the time period from 2000 to 2019 $(N_t=240$ months) were extracted from Kenneth French’s Data Library. This dataset was employed to calculate the realized volatility within the industry. The market volatility data were sourced from Yahoo Finance for three composite portfolios (NYSE, NASDAQ, and AMEX). These market portfolios were consolidated to serve as constant input for the realized industry portfolios. The computation of returns followed the method outlined in [9].

We denote $I{V}_{it}$ for $i=1,2,3,4,5$ and $t=1,\dots ,240$ as industry volatility, and $M{V}_{it}$ as market volatility. The corresponding fractional cointegrated panel model is formulated as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I{V}_{it}& ={\mu }_i+\beta M{V}_{it}+{\Delta }^{-\gamma }{ϵ}_{1it}\hfill \\ \hfill M{V}_{it}& ={\Delta }^{-d}{ϵ}_{2it}.\hfill \end{array}\end{array}$$

(16)

We conducted estimation for the fractional cointegration parameters d and $\gamma$ using a bandwidth of ${\eta }_m=0.75$, corresponding to $m={240}^{0.75}=61$ for each industry as well as the pooled industries (panel). It is crucial to examine the equality of d across portfolios to ensure the validity of the pooling. We applied the tests proposed by [19] as cited in [40], yielding estimated results of $(T_{stat}=0.38,p=0.353)$. These findings indicate that the null hypothesis of the equality of d across different portfolios remains valid.

Table 3. Estimates of $\widehat{d}$, $\widehat{\gamma }$, ${\widehat{\beta }}_{mix}$, and $F_w$ and $M_w$ tests of no fractional cointegration for the five industry portfolios and market average.

	Market	Cnsmr	Manuf	HiTec	Hlth	Other	Panel
$\widehat{d}$	0.55	0.55	0.52	0.61	0.46	0.74	0.54
$\widehat{\gamma }$		0.20	0.42	0.87	0.32	0.34	0.52
${\widehat{\beta }}_{mix}$		0.75	0.98	1.11	0.68	1.26	0.96
$SE\left({\widehat{\beta }}_{mix}\right)$		0.016	0.022	0.043	0.028	0.029	0.014
$F_w$		7.54	2.16	0.79	2.80	12.69	1.28
$p(>|F_w\left|\right)$		0.000	0.031	0.432	0.005	0.000	0.201
$M_w$		0.38	0.25	0.23	0.21	0.49	0.12
$p(>|M_w\left|\right)$		0.705	0.799	0.821	0.832	0.626	0.903

presents the estimates of $\widehat{d}$, $\widehat{\gamma }$, ${\widehat{\beta }}_{mix}$, and $F_w$, $M_w$ tests of no fractional cointegration for the five industry portfolios and market average. The ${\widehat{\beta }}_{mix}={\widehat{\beta }}_{ols}$ since $d_{market}=0.55>0.5$, thus the TNBLS approach was not employed here. All the estimates of d’s for both market and industry portfolios are all less than 1 indicating the validity of fractional integration for the U.S. volatilities. Furthermore, the $F_w$ test showed that of the five industry portfolio volatilities, only HiTec is not fractionally cointegrated with market-realized volatility. Also, the $F_w$ fractional panel cointegration test obtained by pooling all industries showed that there is no fractional panel cointegration $(p>0.05)$ for the combined industries against the market. On the other hand, the $M_w$ test showed that all five industry portfolio volatilities are not fractionally cointegrated with market-realized volatility. In addition, the fractional panel cointegration test obtained by adjusting for the fixed effect showed that there is also no fractional panel cointegration for the combined industries against the market. The results of $M_w$ are more reliable compared to $F_w$, as all the individual fractional cointegration tests agree with the overall results obtained for the panel of industries.

6. Conclusions

This paper introduces a generalized residual-based test for a fractionally cointegrated panel model with fixed effects. The test’s development is centered on bivariate panel series $y_{it}$ and $x_{it}$, where $x_{it}$ is assumed to remain fixed across cross-sectional units. Similar to other fractional cointegration tests, both $y_{it}$ and $x_{it}$ are $I\left(d\right)$ and the residual ${ϵ}_{it}=y_{it}-\beta x_{it}$ is $I\left(\gamma \right)$. The proposed test procedure accommodates any values of d and $\gamma$ between $[0,1]$. The modified test, denoted as $M_w$, converges to an asymptotically normal distribution under the null hypothesis and diverges under the alternative. Compared to the test by [25], $M_w$ demonstrates superior size and power across varying sample sizes and simulation conditions. Furthermore, its real-life application to realized industry and market volatilities for the U.S. economy showcases the practicality of the test. However, it is worth noting that the proposed method has limitations; it has not been tested for imbalanced panel data, and it assumes normal distribution of the model’s error.