Hidden panel cointegration

Abdulnasser Hatemi-J

doi:10.1016/j.jksus.2018.07.011

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32 (

); 507-510

doi:

10.1016/j.jksus.2018.07.011

Hidden panel cointegration

Abdulnasser Hatemi-J

UAE University, United Arab Emirates

Received: 2018-3-15, Accepted: 2018-7-9, Published: 2018-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

1 There are also tests for cointegration that allow of structural breaks. See for example Gergory and Hansen (1996) for tests with one unknown break and Hatemi-J (2008) for tests with two unknown breaks.

Abstract

This paper focuses on an important empirical and methodological research question, namely possibly asymmetric and hence nonlinear cointegrating relationships between variables. It extends the Granger and Yoon (2002) method on hidden cointegration for time series data to a panel data framework. Solutions are provided for transforming the panel data variables with deterministic as well as stochastic trend parts into partial cumulative sums for positive and negative components. The transformed data can then be used to test for the long run relationship between the underlying components. The proposed method is applied to a small panel of three Scandinavian countries examining the presence of a long-term relationship between government consumption and economic growth based on quarterly data. First, the standard method that does not allow for asymmetry was implemented. The results do not provide evidence of a long-term relationship between the two variables. However, the results based on the tests suggested in this paper indicate that the underlying variables are indeed related in the long run. Thus, it might be important to separate the impact of positive shocks from the negative ones when the long run relationships between panel data variables are investigated.

Keywords

Asymmetry

Panel data

Cointegration

Testing

Government consumption

Output

Scandinavia

C33

H21

1 Introduction

Since the pioneer work of Granger (1981) cointegration analysis has become an integral part of applied econometrics when the underlying variables are quantities that are measured across time. Based on Granger’s definition, cointegration occurs in a situation in which a linear combination between integrated variables has one unit root less than the integration order of the variables in the model. The variables cointegrate if and only if they have common stochastic trends that cancel each other out. There is a massive literature both theoretically and empirically on cointegration indicating its due importance. Cointegration analysis is important in empirical research in order to avoid spurious results based on a regression model. It is also important for analyzing the long-run relationships between the underlying variables combined with the short-run adjustment mechanism. It is crucial to test for cointegration in order to avoid the spurious regression problem. If the variables have unit roots but not cointegrated then any relationship between them in the level form is spurious according to Granger and Newbold (1974).

Testing for cointegration was initiated by Engle and Granger (1987) via implementing residual based tests for cointegration. More powerful tests were suggested by Phillips (1987) as well as Phillips and Ouliaris (1990).¹ Multivariate tests for cointegration were developed by Johansen (1988, 1991), Johansen and Juselius (1990) as well as Stock and Watson (1988). The idea to conduct tests for unit roots within a panel system originates from Quah (1994). Tests for panel cointegration were suggested by Pedroni (1999, 2004), Kao (1999) and Westerlund (2007), among others.²

In all previous literature on cointegration testing, there was no separation between the impact of negative and positive shocks until Granger and Yoon (2002) introduced the concept of hidden cointegration for time series data. It is hidden in the sense that it illustrates a situation in which the variables in original form do not cointegrate but when the impact of positive shocks is separated from the impact of negative shocks then cointegration is potentially found between the components of the variables.³ The aim of this article is to extend the concept of hidden cointegration to panel data analysis. The suggested tests in this paper are applied to investigate the impact of contractionary as well as expansionary fiscal policy on the economic performance in a panel consisting of Denmark, Norway and Sweden. It should be pointed out that allowing for asymmetry accords well with reality. It is widely agreed in the literature that individuals tend to react more powerfully to negative conditions than to positive ones.

The article continues as follows. Section 2 introduces hidden panel cointegration analysis. Section 3 provides an application. The last section concludes the article. Finally, an appendix at the end of the article provides a simple mathematical example that shows the necessary condition for cointegration in a panel system. It also presents the solution for a variable that has deterministic trend parts. It should be mentioned that Gauss codes are available that can be used for transforming a variable into positive partial sums for positive and negative components (see Hatemi-J, 2014). After transforming the data, tests for panel cointegration can be implemented via a number of software packages that exist in the market.

2 Hidden panel cointegration

Consider the following two variables that are integrated of the first degree, with the resultant solution for each that is found by the recursive approach:⁴ $y_{i, t} = y_{i, t - 1} + e_{i 1, t} = y_{i, 0} + \sum_{j = 1}^{t} e_{i 1, j}$ $x_{i, t} = x_{i, t - 1} + e_{i 2, t} = x_{i, 0} + \sum_{j = 1}^{t} e_{i 2, j}$

For i = 1, …, m. Where m signifies the cross-sectional dimension and e is the disturbance term that is assumed to be a white noise process. The positive and negative shocks for each panel variable are defined as $e_{i 1, t}^{+} : = Max (e_{i 1, t}, 0)$ , $e_{i 2, t}^{+} : = Max (e_{i 2, t}, 0)$ , $e_{i 1, t}^{-} : = Min (e_{i 1, t}, 0)$ and $e_{i 2, t}^{-} : = Min (e_{i 2, t}, 0)$ .⁵ Using these results, the following expressions can be obtained: $y_{i, t}^{+} = y_{i, 0}^{+} + e_{i 1, t}^{+} = y_{i, 0} + \sum_{j = 1}^{t} e_{i 1, j}^{+}$ $x_{i, t}^{+} = x_{i, 0}^{+} + e_{i 2, t}^{+} = x_{i, 0} + \sum_{j = 1}^{t} e_{i 2, j}^{+}$ $y_{i, t}^{-} = y_{i, 0}^{-} + e_{i 1, t}^{-} = y_{i, 0} + \sum_{j = 1}^{t} e_{i 1, j}^{-}$ $x_{i, t}^{-} = x_{i, 0}^{-} + e_{i 2, t}^{-} = x_{i, 0} + \sum_{j = 1}^{t} e_{i 2, j}^{-}$

Assume that our dependent variable is y, and then the two potential panel cointegration equations for the components can be defined as

(1)

y_{i, t}^{+} = α_{i}^{+} + β_{i}^{+} x_{i, t}^{+} + e_{i, t}^{+}

(2)

y_{i, t}^{-} = α_{i}^{-} + β_{i}^{-} x_{i, t}^{-} + e_{i, t}^{-}

The positive cumulative shocks are cointegrated in the panel if $e_{i, t}^{+}$ is stationary. Likewise, the negative cumulative shocks are cointegrated in the panel if $e_{i, t}^{-}$ is stationary.⁶

There is potentially a battery of the tests available in the literature that can be used for testing whether $e_{i, t}^{+}$ or $e_{i, t}^{-}$ is stationary or not. However, the well-known augmented Dickey-Fuller (ADF) test is the simplest one that can be used for this purpose. Assume that we wish to test for cointegration in the panel model (1). Then, the panel ADF test equation is the following:

(3)

e_{i, t}^{+} = ρ^{+} e_{i, t - 1}^{+} + \sum_{l = 1}^{k} γ_{i}^{+} Δ e_{i 1, t - l}^{+} + w_{i, t}^{+}

The optimal lag order l can be determined by minimizing an information criterion. The null hypothesis of no cointegration between the positive components is $ρ^{+} = 1$ . It is also possible to allow for deterministic components such as individual drifts and trends in Eq. (3) if necessary.⁷ Based on the results provided by Kao (1999), the following test statistic can be used to test the null hypothesis of no panel cointegration:

(4)

ADF = \frac{t_{ρ^{+} + \sqrt{6 m} \times \frac{σ_{v}}{2 σ_{0 v}}}}{\sqrt{\frac{σ_{0 v}^{2}}{2 σ_{v}^{2}} + \frac{3 σ_{v}^{2}}{10 σ_{0 v}^{2}}}}

where

t_{ρ^{+}}

is the t-statistic for parameter

ρ^{+}

in Eq. (3). The variance is estimated as

σ_{v}^{2} = σ_{e_{1}^{+}}^{2} - \frac{σ_{e_{1}^{+}, e_{2}^{+}}^{2}}{σ_{e_{2}^{+}}^{2}}

, and the long-run variance is estimated as

σ_{0 v}^{2} = σ_{0 e_{1}^{+}}^{2} - \frac{σ_{0 e_{1}^{+}, e_{2}^{+}}^{2}}{σ_{0 e_{2}^{+}}^{2}}

Let $u_{it} = [\begin{matrix} e_{i 1, t}^{+} \\ e_{i 2, t}^{+} \end{matrix}]$ . The variance-covariance for $u_{it}$ is estimated as $Σ = [\begin{matrix} σ_{e_{1}^{+}}^{2} & σ_{e_{1}^{+}, e_{2}^{+}} \\ σ_{e_{1}^{+}, e_{2}^{+}} & σ_{e_{2}^{+}}^{2} \end{matrix}] = \frac{1}{mT} \sum_{i = 1}^{m} \sum_{t = 1}^{T} u_{it} u_{it}^{'}$

The long-run variance-covariance matrix is estimated via the kernel estimation approach as $Ω = [\begin{matrix} σ_{e_{1}^{+}}^{2} & σ_{e_{1}^{+}, e_{2}^{+}} \\ σ_{e_{1}^{+}, e_{2}^{+}} & σ_{e_{2}^{+}}^{2} \end{matrix}] = \frac{1}{m} \sum_{i = 1}^{m} [\frac{1}{T} \sum_{t = 1}^{T} u_{it} u_{it}^{'} + \frac{1}{T} \sum_{t = 1}^{T} κ (τ / b) \sum_{t = τ + 1}^{T} (u_{it} u_{it - τ}^{'} + u_{it - τ} u_{it}^{'})]$ where $κ$ is representing the kernel function and b is the bandwidth.

The ADF test statistic as presented in Eq. (4) has a standard normal distribution asymptotically. For a proof see Kao (1999). To test for stationarity in the linear combination between negative components as presented in Eq. (2), a similar ADF test can be conducted. Other combinations are also possible.⁸ A Gauss code for constructing partial cumulative sums of positive and negative changes of each variable for each cross sectional unit in the panel is available on request from the author. It should be mentioned that it is also possible to test for hidden panel cointegration using other tests such as seven residual based tests suggested by Pedroni (1999, 2004) as well as the panel version of the multivariate Johansen (1991) test as developed by Maddala and Wu (1999) based on the Fisher (1932) principle of deriving a combined test using the individual test results.

3 An application

The suggested test for hidden panel cointegration is applied to investigating the long-run relationship between government consumption and economic performance in Denmark, Norway and Sweden. Quarterly data is used during the period 1995:Q1-2017:Q4. The source of data is the FRED database that is provided by the Federal Reserve Bank of St. Louis. In order to capture real effects, the variables are expressed at constant prices. The cumulative sums of positive and negative shocks were constructed based on the procedure presented in the previous section. Prior to testing for panel cointegration, panel unit root tests were implemented by using the Im, Pesaran and Shin (2003) test. The results are presented in Table 1, which show that each panel variable has one unit root.

Table 1 The Results of Panel Unit Root Tests.

VARIABLE	H₀: I(1), H₁: I(0)	H₀: I(2), H₁: I(1)
$X$	0.9945	<0.0001
$Y$	0.4222	<0.0001
$X^{+}$	0.6758	<0.0001
$Y^{+}$	0.4457	<0.0001
$X^{-}$	0.4410	<0.0001
$Y^{-}$	0.7116	<0.0001

Notes

The denotation X stands for the government consumption and Y is representing the GDP. The variables are expressed the logarithmic form. The Im, Pesaran and Shin (2003) test is used to test for panel unit root. The p-values are presented.

Given that each variable in the panel has one unit root, it is crucial to conduct tests for panel cointegration. First, we tested for concintegration by using the standard tests and then we implemented the tests that are proposed in this paper. The results of these tests are presented in Table 2, which indicate that there is no cointegration between government consumption and output in the panel sample for these three countries when standard tests are performed. However, when the suggested tests for hidden cointegration are implemented, the results indicate that there is panel cointegration between the underlying components.

Table 2 The Results of Panel Cointegration Tests.

VARIABLES IN THE MODEL	H₀: I(1), H₁: I(0)
$(Y, X)$	−1.0144
$(Y^{+}, X^{+})$	−1.6995*
$(Y^{-}, X^{-})$	−0.5489
$(Y^{-}, X^{+})$	−1.8705*
$(Y^{+}, X^{-})$	−1.6995*

Notes

The null hypothesis of no panel cointegration is rejected at the 5% significance level if the estimated test value is lower than -1.64. * means significant at the 5% significance level.

4 Conclusions

Tests for cointegration are commonly utilized in applied research. The aim of this article is to extend the hidden cointegration tests of time series data as developed by Granger and Yoon (2002) to the hidden cointegration tests of panel data. Panel data combines the time series dimension with the cross-sectional dimension, which results in higher degrees of freedom. It is shown in this paper how partial cumulative sums of positive and negative changes can be constructed for the panel data variables. It is also demonstrated how an augmented Dickey-Fuller test for a panel system can be implemented to test the null hypothesis of no panel cointegration between different components of the underlying variables. A user friendly Gauss algorithm is produced to transform the panel variables into their respective components.

The suggested tests are applied to investigating the impact of government consumption on economic output in a panel of three Scandinavian countries—namely Denmark, Norway and Sweden. The results show that there is no panel cointegration between these variables when the standard tests are applied. However, when the tests suggested in this paper are implemented, it is found that there is indeed a long run relationship between the negative and positive components of the underlying panel data variables. Thus, allowing for asymmetry is crucial in this case in order to avoid misleading empirical inference. It should be pointed out that separating the impact of positive shocks from the negative ones might be informative per se, in order to capture the potential asymmetry that might prevail. Another issue that might be relevant within this context is the issue of cross-section dependence, which has not been addressed in this paper. A potential approach to deal with the cross-section dependency in tests for hidden panel cointegration might be using the results suggested by Pesaran (2006). Furthermore, if the assumption of normality is not fulfilled, the simulation based techniques such as the bootstrapping method suggested by Shin (2015) might be useful for creating critical values that are based on resampling of the underlying data set instead of using the asymptotic critical values.

References

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Engle R.F., Granger C.W.J., . Co-integration and Error Correction: Representation, Estimation and Testing. Econometrica. 1987;55(2):251-276.
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Fisher R.A., . Statistical Methods for Research Workers (12th Edition). Edinburgh: Oliver & Boyd; 1932.
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Granger C.W., Newbold P., . Understanding spurious regressions in econometrics. J. Econometrics. 1974;2:111-120.
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Granger C.W., Yoon G., . Hidden Cointegration. Department of Economics Working Paper. University of California. San Diego; 2002.
Gregory A.W., Hansen B.E., . Residual-based tests for cointegration in models with regime shifts. J. Econometrics. 1996;70(1):99-126.
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Hatemi-J A., . Tests for cointegration with two unknown regime shifts with an application to financial market integration. Empirical Econ.. 2008;35(3):497-505.
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Hatemi-J A., . Asymmetric Causality Tests with an Application. Empirical Econ.. 2012;43(1):1-10.
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Hatemi-J A., . Asymmetric generalized impulse responses with an application in finance. Econ. Model.. 2014;36:18-22.
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Hatemi-J A., El-Khatib Y., . An extension of the asymmetric causality tests for dealing with deterministic trend components. Appl. Econ.. 2016;48(42):4033-4041.
[Google Scholar]
Im K.S., Pesaran M.H., Shin Y., . Testing for Unit Roots in Heterogeneous Panels. J. Econometrics. 2003;115(1):53-74.
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Johansen S., . Statistical Analysis of Cointegration Vectors. J. Econ. Dyn. Control. 1988;12(2–3):231-254.
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Johansen S., . Cointegration and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica. 1991;59(6):1551-1580.
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Johansen S., Juselius K., . Maximum Likelihood Estimation and Inference on Cointegration–with Applications to the Demand for Money. Oxford Bull. Econ. Stat.. 1990;52(2):169-210.
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Kao C., . Spurious regression and residual-based tests for cointegration in panel data. J. Econometrics. 1999;90:1-44.
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Maddala G.S., Wu S., . A Comparative Study of Unit Root Tests with Panel Data and New Simple Test. Oxford Bull. Econ. Stat.. 1999;61:631-652.
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Mohammad, A.A., Mudhir, A.A., 2018. Dynamical approach in studying stability condition of exponential (GARCH) models, J. King Saud Univ.-Sci., forthcoming.
Pedroni P., . Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors. Oxford Bull. Econ. Stat.. 1999;61:653-670.
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Pedroni P., . Panel Cointegration, Asymptotic and Finite Sample Properties of Pooled Time Series tests with an Application to the PPP Hypothesis. Econometric Theory. 2004;20:597-625.
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Pesaran M.H., . Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure. Econometrica. 2006;74(4):967-1012.
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Phillips P.C.B., . Time Series Regression with a Unit Root. Econometrica. 1987;55:277-301.
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Quah D., . Exploiting Cross Section Variation for Unit root Inference in Dynamic Data. Econ. Lett.. 1994;44:9-19.
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Sanaullah A., Hanif M., . A multivariate regression-cum-exponential estimator for population variance vector in two phase sampling. J. King Saud Univ.-Sci.. 2018;30(2):223-228.
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Shin Y., Yu B., Greenwood-Nimmo M., . Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In: Sickles R., Horrace W., eds. Festschrift in Honor of Peter Schmidt: Econometric Methods and Applications. New York: Springer; 2014.
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Appendix

An example regarding panel cointegration

Let us be more explicit about panel cointegration between two panel integrated variables by a simple example. Consider the following two non-stationary panel variables:

(A1)

Z_{i, t} = Z_{i, t - 1} + ε_{i 1, t}

(A2)

W_{i, t} = Z_{i, t - 1} + ε_{i 2, t}

where

ε_{i 1, t}

and

ε_{i 2, t}

are two white noise error terms. Assuming that the initial values are zero, continuous substitutions give the following solutions:

(A3)

Z_{i, t} = \sum_{j = 1}^{t} ε_{i 1, j}

(A4)

W_{i, t} = \sum_{j = 1}^{t - 1} ε_{i 1, j} + ε_{i 2, t}

By taking the first difference of each variable, we obtain

(A5)

Δ Z_{i, t} = Z_{i, t} - Z_{i, t - 1} = \sum_{j = 1}^{t} ε_{i 1, j} - \sum_{j = 1}^{t - 1} ε_{i 1, j} = ε_{i 1, t}

(A6)

Δ W_{i, t} = W_{i, t} - W_{i, t - 1} = \sum_{j = 1}^{t} ε_{i 1, j} + ε_{i 2, t} - (\sum_{j = 1}^{t - 1} ε_{i 1, j} + ε_{i 2, t - 1}) = ε_{i 1, t} + ε_{i 2, t} - ε_{i 2, t - 1}

That is, each panel variable becomes stationary after taking the first difference. Hence, the variables are integrated of the first order, denoted I(1). Now, the question is whether or not these two panel variables are cointegrated. Denote Y_i,t as the difference between the two variables, i.e.

(A7)

Y_{i, t} = W_{i, t} - Z_{i, t} = \sum_{j = 1}^{t} ε_{i 1, j} - (\sum_{j = 1}^{t - 1} ε_{i 1, j} + ε_{i 2, t}) = ε_{i 1, t} - ε_{i 2, t}

This difference is clearly a stationary process. That is, a linear combination of the non-stationary variables is stationary in the panel, which in turn means that the variables cointegrate with (1.0, −1.0) as the cointegrating vector. It should be noted that the variables cointegrate because their stochastic trend cancels each other out. In another words, the panel variables have a common stochastic trend. This is the case in the hidden panel cointegration analysis also. In order to have hidden panel cointegration, there must be at least one common stochastic trend between the cumulative components of the positive or negative shocks in the underlying panel.

Transforming data with deterministic trend parts

It is widely agreed that many economic and financial variables contain deterministic trend in addition to the stochastic trend. In such cases, the following approach can be utilized in order to transform the data into partial cumulative sums for positive and negative components. For example, assume that the data generating process is the following:

(A8)

y_{it} = a_{i} + b_{i} t + y_{it - 1} + u_{it},

where a is the drift and t is the deterministic trend component. By applying the recursive method, the solution to Eq. (A8) is given as the following:

(A9)

y_{ti} = a_{i} t + \frac{t (t + 1)}{2} b_{i} + y_{0 i} + \sum_{j = 1}^{t} u_{ij},

The positive and negative shocks for this panel variable are defined as $u_{i, t}^{+} : = Max (u, 0)$ and $u_{i, t}^{-} : = Min (u_{i, t}, 0)$ and $e_{i 2, t}^{-} : = Min (e_{i 2, t}, 0)$ . It follows that

(A10)

y_{it} = a_{i} + b_{i} t + y_{it - 1} + u_{it} = a_{i} t + \frac{t (t + 1)}{2} b_{i} + y_{0 i} + \sum_{j = 1}^{t} u_{ij}^{+} + \sum_{j = 1}^{t} u_{ij}^{-},

Hence, the positive and negative components can be defined as the following:

(A11)

y_{it}^{+} = \frac{a_{i} t + \frac{t (t + 1)}{2} b_{i} + y_{0 i}}{2} + \sum_{j = 1}^{t} u_{ij}^{+},

and

(A12)

y_{it}^{-} = \frac{a_{i} t + \frac{t (t + 1)}{2} b_{i} + y_{0 i}}{2} + \sum_{j = 1}^{t} u_{ij}^{-} .

Thus, $y_{it} = y_{it}^{+} + y_{it}^{-} .$ . These results can be proved by a proposition provided by Hatemi-J and El-Khatib (2016).

Introduction

Hidden panel cointegration

An application

Conclusions

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[1] AlMuhayfith F.E., Alzaid A.A., . On bivariate Poisson regression models. J. King Saud Univ.-Sci.. 2016;28(2):178-189.
[Google Scholar]

[2] Engle R.F., Granger C.W.J., . Co-integration and Error Correction: Representation, Estimation and Testing. Econometrica. 1987;55(2):251-276.
[Google Scholar]

[3] Fisher R.A., . Statistical Methods for Research Workers (12th Edition). Edinburgh: Oliver & Boyd; 1932.

[4] Granger C.W., . Some Properties of Time Series Data and Their Use in Econometric Model Specification. J. Econometrics. 1981;16:121-130.
[Google Scholar]

[5] Granger C.W., Newbold P., . Understanding spurious regressions in econometrics. J. Econometrics. 1974;2:111-120.
[Google Scholar]

[6] Granger C.W., Yoon G., . Hidden Cointegration. Department of Economics Working Paper. University of California. San Diego; 2002.

[7] Gregory A.W., Hansen B.E., . Residual-based tests for cointegration in models with regime shifts. J. Econometrics. 1996;70(1):99-126.
[Google Scholar]

[8] Hatemi-J A., . Tests for cointegration with two unknown regime shifts with an application to financial market integration. Empirical Econ.. 2008;35(3):497-505.
[Google Scholar]

[9] Hatemi-J A., . Asymmetric Causality Tests with an Application. Empirical Econ.. 2012;43(1):1-10.
[Google Scholar]

[10] Hatemi-J A., . Asymmetric generalized impulse responses with an application in finance. Econ. Model.. 2014;36:18-22.
[Google Scholar]

[11] Hatemi-J A., El-Khatib Y., . An extension of the asymmetric causality tests for dealing with deterministic trend components. Appl. Econ.. 2016;48(42):4033-4041.
[Google Scholar]

[12] Im K.S., Pesaran M.H., Shin Y., . Testing for Unit Roots in Heterogeneous Panels. J. Econometrics. 2003;115(1):53-74.
[Google Scholar]

[13] Johansen S., . Statistical Analysis of Cointegration Vectors. J. Econ. Dyn. Control. 1988;12(2–3):231-254.
[Google Scholar]

[14] Johansen S., . Cointegration and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica. 1991;59(6):1551-1580.
[Google Scholar]

[15] Johansen S., Juselius K., . Maximum Likelihood Estimation and Inference on Cointegration–with Applications to the Demand for Money. Oxford Bull. Econ. Stat.. 1990;52(2):169-210.
[Google Scholar]

[16] Kao C., . Spurious regression and residual-based tests for cointegration in panel data. J. Econometrics. 1999;90:1-44.
[Google Scholar]

[17] Maddala G.S., Wu S., . A Comparative Study of Unit Root Tests with Panel Data and New Simple Test. Oxford Bull. Econ. Stat.. 1999;61:631-652.
[Google Scholar]

[18] Mohammad, A.A., Mudhir, A.A., 2018. Dynamical approach in studying stability condition of exponential (GARCH) models, J. King Saud Univ.-Sci., forthcoming.

[19] Pedroni P., . Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors. Oxford Bull. Econ. Stat.. 1999;61:653-670.
[Google Scholar]

[20] Pedroni P., . Panel Cointegration, Asymptotic and Finite Sample Properties of Pooled Time Series tests with an Application to the PPP Hypothesis. Econometric Theory. 2004;20:597-625.
[Google Scholar]

[21] Pesaran M.H., . Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure. Econometrica. 2006;74(4):967-1012.
[Google Scholar]

[22] Phillips P.C.B., . Time Series Regression with a Unit Root. Econometrica. 1987;55:277-301.
[Google Scholar]

[23] Phillips P.C.B., Ouliaris S., . Asymptotic Properties of Residual Based Tests for Cointegration. Econometrica. 1990;58:165-193.
[Google Scholar]

[24] Quah D., . Exploiting Cross Section Variation for Unit root Inference in Dynamic Data. Econ. Lett.. 1994;44:9-19.
[Google Scholar]

[25] Sanaullah A., Hanif M., . A multivariate regression-cum-exponential estimator for population variance vector in two phase sampling. J. King Saud Univ.-Sci.. 2018;30(2):223-228.
[Google Scholar]

[26] Stock J., Watson M., . Testing for common trends. J. Am. Stat. Assoc.. 1988;83:1097-1107.
[Google Scholar]

[27] Shin D.W., . Stationary bootstrapping for panel cointegration tests under cross-sectional dependence. Statistics J. Theor. Appl. Stat.. 2015;49(1):209-223.
[Google Scholar]

[28] Shin Y., Yu B., Greenwood-Nimmo M., . Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In: Sickles R., Horrace W., eds. Festschrift in Honor of Peter Schmidt: Econometric Methods and Applications. New York: Springer; 2014.
[Google Scholar]

[29] Westerlund J., . Testing for Error Correction in Panel Data. Oxford Bull. Econ. Stat.. 2007;69:709-748.
[Google Scholar]