Citation: Mohammed Yaw Broni, Mosharrof Hosen, Mansur Masih. Does a country’s external debt level affect its Islamic banking sector development? Evidence from Malaysia based on Quantile regression and Markov regime-switching[J]. Quantitative Finance and Economics, 2019, 3(2): 366-389. doi: 10.3934/QFE.2019.2.366
[1] | Minlong Lin, Ke Tang . Selective further learning of hybrid ensemble for class imbalanced increment learning. Big Data and Information Analytics, 2017, 2(1): 1-21. doi: 10.3934/bdia.2017005 |
[2] | Subrata Dasgupta . Disentangling data, information and knowledge. Big Data and Information Analytics, 2016, 1(4): 377-390. doi: 10.3934/bdia.2016016 |
[3] | Qinglei Zhang, Wenying Feng . Detecting Coalition Attacks in Online Advertising: A hybrid data mining approach. Big Data and Information Analytics, 2016, 1(2): 227-245. doi: 10.3934/bdia.2016006 |
[4] | Tieliang Gong, Qian Zhao, Deyu Meng, Zongben Xu . Why Curriculum Learning & Self-paced Learning Work in Big/Noisy Data: A Theoretical Perspective. Big Data and Information Analytics, 2016, 1(1): 111-127. doi: 10.3934/bdia.2016.1.111 |
[5] | Xin Yun, Myung Hwan Chun . The impact of personalized recommendation on purchase intention under the background of big data. Big Data and Information Analytics, 2024, 8(0): 80-108. doi: 10.3934/bdia.2024005 |
[6] | Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen . Big data collection and analysis for manufacturing organisations. Big Data and Information Analytics, 2017, 2(2): 127-139. doi: 10.3934/bdia.2017002 |
[7] | Zhen Mei . Manifold Data Mining Helps Businesses Grow More Effectively. Big Data and Information Analytics, 2016, 1(2): 275-276. doi: 10.3934/bdia.2016009 |
[8] | Ricky Fok, Agnieszka Lasek, Jiye Li, Aijun An . Modeling daily guest count prediction. Big Data and Information Analytics, 2016, 1(4): 299-308. doi: 10.3934/bdia.2016012 |
[9] | M Supriya, AJ Deepa . Machine learning approach on healthcare big data: a review. Big Data and Information Analytics, 2020, 5(1): 58-75. doi: 10.3934/bdia.2020005 |
[10] | Sunmoo Yoon, Maria Patrao, Debbie Schauer, Jose Gutierrez . Prediction Models for Burden of Caregivers Applying Data Mining Techniques. Big Data and Information Analytics, 2017, 2(3): 209-217. doi: 10.3934/bdia.2017014 |
For a continuous risk outcome
Given fixed effects
In this paper, we assume that the risk outcome
y=Φ(a0+a1x1+⋯+akxk+bs), | (1.1) |
where
Given random effect model (1.1), the expected value
We introduce a family of interval distributions based on variable transformations. Probability densities for these distributions are provided (Proposition 2.1). Parameters of model (1.1) can then be estimated by maximum likelihood approaches assuming an interval distribution. In some cases, these parameters get an analytical solution without the needs for a model fitting (Proposition 4.1). We call a model with a random effect, where parameters are estimated by maximum likelihood assuming an interval distribution, an interval distribution model.
In its simplest form, the interval distribution model
The paper is organized as follows: in section 2, we introduce a family of interval distributions. A measure for tail fatness is defined. In section 3, we show examples of interval distributions and investigate their tail behaviours. We propose in section 4 an algorithm for estimating the parameters in model (1.1).
Interval distributions introduced in this section are defined for a risk outcome over a finite open interval
Let
Let
Φ:D→(c0,c1) | (2.1) |
be a transformation with continuous and positive derivatives
Given a continuous random variable
y=Φ(a+bs), | (2.2) |
where we assume that the range of variable
Proposition 2.1. Given
g(y,a,b)=U1/(bU2) | (2.3) |
G(y,a,b)=F[Φ−1(y)−ab]. | (2.4) |
where
U1=f{[Φ−1(y)−a]/b},U2=ϕ[Φ−1(y)] | (2.5) |
Proof. A proof for the case when
G(y,a,b)=P[Φ(a+bs)≤y] |
=P{s≤[Φ−1(y)−a]/b} |
=F{[Φ−1(y)−a]/b}. |
By chain rule and the relationship
∂Φ−1(y)∂y=1ϕ[Φ−1(y)]. | (2.6) |
Taking the derivative of
∂G(y,a,b)∂y=f{[Φ−1(y)−a]/b}bϕ[Φ−1(y)]=U1bU2. |
One can explore into these interval distributions for their shapes, including skewness and modality. For stress testing purposes, we are more interested in tail risk behaviours for these distributions.
Recall that, for a variable X over (−
For a risk outcome over a finite interval
We say that an interval distribution has a fat right tail if the limit
Given
Recall that, for a Beta distribution with parameters
Next, because the derivative of
z=Φ−1(y) | (2.7) |
Then
Lemma 2.2. Given
(ⅰ)
(ⅱ) If
(ⅲ) If
Proof. The first statement follows from the relationship
[g(y,a,b)(y1−y)β]−1/β=[g(y,a,b)]−1/βy1−y=[g(Φ(z),a,b)]−1/βy1−Φ(z). | (2.8) |
By L’Hospital’s rule and taking the derivatives of the numerator and the denominator of (2.8) with respect to
For tail convexity, we say that the right tail of an interval distribution is convex if
Again, write
h\left(z, a, b\right) = \mathrm{log}\left[g\left(\mathrm{\Phi }\left(z\right), a, b\right)\right], | (2.9) |
where
g\left(y, a, b\right) = \mathrm{exp}\left[h\left(z, a, b\right)\right]. | (2.10) |
By (2.9), (2.10), using (2.6) and the relationship
{g}_{y}^{'} = {[h}_{z}^{'}\left(z\right)/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right)]\mathrm{e}\mathrm{x}\mathrm{p}[h({\mathrm{\Phi }}^{-1}\left(y\right), a, b)], \\ {g}_{yy}^{''} = \left[\frac{{h}_{zz}^{''}\left(z\right)}{{{\rm{ \mathsf{ ϕ} }}}^{2}\left(\mathrm{z}\right)}-\frac{{h}_{z}^{'}\left(z\right){{\rm{ \mathsf{ ϕ} }}}_{\mathrm{z}}^{'}\left(z\right)}{{{\rm{ \mathsf{ ϕ} }}}^{3}\left(\mathrm{z}\right)}+\frac{{h}_{\mathrm{z}}^{\mathrm{'}}\left(\mathrm{z}\right){h}_{\mathrm{z}}^{\mathrm{'}}\left(\mathrm{z}\right)}{{{\rm{ \mathsf{ ϕ} }}}^{2}\left(\mathrm{z}\right)}\right]\mathrm{e}\mathrm{x}\mathrm{p}\left[h\right({\mathrm{\Phi }}^{-1}\left(y\right), a, b) ]. | (2.11) |
The following lemma is useful for checking tail convexity, it follows from (2.11).
Lemma 2.3. Suppose
In this section, we focus on the case where
One can explore into a wide list of densities with different choices for
A.
B.
C.
D.D.
Densities for cases A, B, C, and D are given respectively in (3.3) (section 3.1), (A.1), (A.3), and (A5) (Appendix A). Tail behaviour study is summarized in Propositions 3.3, 3.5, and Remark 3.6. Sketches of density plots are provided in Appendix B for distributions A, B, and C.
Using the notations of section 2, we have
By (2.5), we have
\mathrm{log}\left(\frac{{U}_{1}}{{U}_{2}}\right) = \frac{{-z}^{2}+2az-{a}^{2}+{b}^{2}{z}^{2}}{2{b}^{2}} | (3.1) |
= \frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}\text{.} | (3.2) |
Therefore, we have
g\left(\mathrm{y}, a, b\right) = \frac{1}{b}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}\right\}\text{.} | (3.3) |
Again, using the notations of section 2, we have
g\left(y, p, \rho \right) = \sqrt{\frac{1-\rho }{\rho }}\mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{1}{2\rho }{\left[{\sqrt{1-\rho }{\mathrm{\Phi }}^{-1}\left(y\right)-\mathrm{\Phi }}^{-1}\left(p\right)\right]}^{2}+\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\}\text{, } | (3.4) |
where
Proposition 3.1. Density (3.3) is equivalent to (3.4) under the relationships:
a = \frac{{\Phi }^{-1}\left(p\right)}{\sqrt{1-\rho }} \ \ \text{and}\ \ b = \sqrt{\frac{\rho }{1-\rho }}. | (3.5) |
Proof. A similar proof can be found in [19]. By (3.4), we have
g\left(y, p, \rho \right) = \sqrt{\frac{1-\rho }{\rho }}\mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{1-\rho }{2\rho }{\left[{{\mathrm{\Phi }}^{-1}\left(y\right)-\mathrm{\Phi }}^{-1}\left(p\right)/\sqrt{1-\rho }\right]}^{2}+\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\} |
= \frac{1}{b}\mathrm{exp}\left\{-\frac{1}{2}{\left[\frac{{\Phi }^{-1}\left(y\right)-a}{b}\right]}^{2}\right\}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\right\} |
= {U}_{1}/{(bU}_{2}) = g(y, a, b)\text{.} |
The following relationships are implied by (3.5):
\rho = \frac{{b}^{2}}{1{+b}^{2}}, | (3.6) |
a = {\Phi }^{-1}\left(p\right)\sqrt{1+{b}^{2}}\text{.} | (3.7) |
Remark 3.2. The mode of
\frac{\sqrt{1-\rho }}{1-2\rho }{\mathrm{\Phi }}^{-1}\left(p\right) = \frac{\sqrt{1+{b}^{2}}}{1-{b}^{2}}{\mathrm{\Phi }}^{-1}\left(p\right) = \frac{a}{1-{b}^{2}}. |
This means
Proposition 3.3. The following statements hold for
(ⅰ)
(ⅱ)
(ⅲ) If
Proof. For statement (ⅰ), we have
Consider statement (ⅱ). First by (3.3), if
{\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-1/\beta } = {b}^{1/\beta }\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}) | (3.8) |
By taking the derivative of (3.8) with respect to
-\left\{\partial {\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) = \sqrt{2\pi }{b}^{\frac{1}{\beta }}\frac{\left({b}^{2}-1\right)z+a}{\beta {b}^{2}}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}+\frac{{z}^{2}}{2})\text{.} | (3.9) |
Thus
\left\{\partial {\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) = -\sqrt{2\pi }{b}^{\frac{1}{\beta }}\frac{\left({b}^{2}-1\right)z+a}{\beta {b}^{2}}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}+\frac{{z}^{2}}{2})\text{.} | (3.10) |
Thus
For statement (ⅲ), we use Lemma 2.3. By (2.9) and using (3.2), we have
h\left(z, a, b\right) = \mathrm{log}\left(\frac{{U}_{1}}{{bU}_{2}}\right) = \frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}-\mathrm{l}\mathrm{o}\mathrm{g}\left(b\right)\text{.} |
When
Remark 3.4. Assume
li{m}_{z⤍+\infty }-\left\{{\partial \left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) |
is
For these distributions, we again focus on their tail behaviours. A proof for the next proposition can be found in Appendix A.
Proposition 3.5. The following statements hold:
(a) Density
(b) The tailed index of
Remark 3.6. Among distributions A, B, C, and Beta distribution, distribution B gets the highest tailed index of 1, independent of the choices of
In this section, we assume that
First, we consider a simple case, where risk outcome
y = \mathrm{\Phi }\left(v+bs\right), | (4.1) |
where
Given a sample
LL = \sum _{i = 1}^{n}\left\{\mathrm{log}f\left(\frac{{z}_{i}-{v}_{i}}{b}\right)-\mathrm{l}\mathrm{o}\mathrm{g}{\rm{ \mathsf{ ϕ} }}\left({z}_{i}\right)-logb\right\}\text{, } | (4.2) |
where
Recall that the least squares estimators of
SS = \sum _{i = 1}^{n}{({z}_{i}-{v}_{i})}^{2} | (4.3) |
has a closed form solution given by the transpose of
{\rm{X}} = \left\lceil {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {1\;\;{x_{11}} \ldots {x_{k1}}}\\ {1\;\;{x_{12}} \ldots {x_{k2}}} \end{array}}\\ \ldots \\ {1\;\;{x_{1n}} \ldots {x_{kn}}} \end{array}} \right\rceil , {\rm{Z}} = \left\lceil {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{z_1}}\\ {{z_2}} \end{array}}\\ \ldots \\ {{z_n}} \end{array}} \right\rceil . |
The next proposition shows there exists an analytical solution for the parameters of model (4.1).
Proposition 4.1. Given a sample
Proof. Dropping off the constant term from (4.2) and noting
LL = -\frac{1}{2{b}^{2}}\sum _{i = 1}^{n}{({z}_{i}-{v}_{i})}^{2}-nlogb, | (4.4) |
Hence the maximum likelihood estimates
Next, we consider the general case of model (1.1), where the risk outcome
y = \mathrm{\Phi }[v+ws], | (4.5) |
where parameter
(a)
(b)
Given a sample
LL = \sum _{i = 1}^{n}-{\frac{1}{2}[\left({z}_{i}-{v}_{i}\right)}^{2}/{w}_{i}^{2}-{u}_{i}], | (4.6) |
LL = \sum _{i = 1}^{n}\{-\left({z}_{i}-{v}_{i}\right)/{w}_{\mathrm{i}}-2\mathrm{log}[1+\mathrm{e}\mathrm{x}\mathrm{p}[-({z}_{i}-{v}_{i})/{w}_{i}]-{u}_{i}\}, | (4.7) |
Recall that a function is log-concave if its logarithm is concave. If a function is concave, a local maximum is a global maximum, and the function is unimodal. This property is useful for searching maximum likelihood estimates.
Proposition 4.2. The functions (4.6) and (4.7) are concave as a function of
Proof. It is well-known that, if
For (4.7), the linear part
In general, parameters
Algorithm 4.3. Follow the steps below to estimate parameters of model (4.5):
(a) Given
(b) Given
(c) Iterate (a) and (b) until a convergence is reached.
With the interval distributions introduced in this paper, models with a random effect can be fitted for a continuous risk outcome by maximum likelihood approaches assuming an interval distribution. These models provide an alternative regression tool to the Beta regression model and fraction response model, and a tool for tail risk assessment as well.
Authors are very grateful to the third reviewer for many constructive comments. The first author is grateful to Biao Wu for many valuable conversations. Thanks also go to Clovis Sukam for his critical reading for the manuscript.
We would like to thank you for following the instructions above very closely in advance. It will definitely save us lot of time and expedite the process of your paper's publication.
The views expressed in this article are not necessarily those of Royal Bank of Canada and Scotiabank or any of their affiliates. Please direct any comments to Bill Huajian Yang at h_y02@yahoo.ca.
[1] |
Abedifar P, Molyneux P, Tarazi A (2013) Risk in Islamic banking. Rev Financ 17: 2035–2096. doi: 10.1093/rof/rfs041
![]() |
[2] |
Alqahtani F, Mayes DG, Brown K (2016) Economic turmoil and Islamic banking: Evidence from the Gulf Cooperation Council. Pac-Basin Financ J 39: 44–56. doi: 10.1016/j.pacfin.2016.05.017
![]() |
[3] | Basher SA, Kessler LM, Munkin MK (2017) Bank capital and portfolio risk among Islamic banks. Rev Financ Econ 24: 1–9. |
[4] |
Ehrhart H, Minea A, Villieu P (2014) Debt, seigniorage, and the growth laffer curve in developing countries. J Macroecon 42: 199–210. doi: 10.1016/j.jmacro.2014.07.004
![]() |
[5] | Eichengreen B, Rose AK (1998) Staying afloat when the wind shifts: External factors and emerging-market banking crises. National Bureau of Economic Research, No. w6370. |
[6] | Hallak I (2013) Private sector share of external debt and financial stability: Evidence from bank loans. J Int Money Financ 32: 17–41. |
[7] | Hassan MK, Bashir AHM (2003) Determinants of Islamic banking profitability. In 10th ERF annual conference, Morocco, 2–31. |
[8] | Hassan T, Mohamad S, Khaled I. Bader M (2009) Efficiency of conventional versus Islamic banks: evidence from the Middle East. Int J Islamic middle eastern financ manage 2: 46–65. |
[9] |
Hossain AA (2016) Inflationary shocks and real output growth in nine Muslim-majority countries: Implications for Islamic banking and finance. J Asian Econ 45: 56–73. doi: 10.1016/j.asieco.2016.06.004
![]() |
[10] | Koenker R (2005) Quantile regression, Cambridge university press, No. 38. |
[11] | Kumar M, Woo J (2010) Public debt and growth. IMF working papers, 1–47. |
[12] | Krugman P (1988) Financing vs. forgiving a debt overhang. J Dev Econ 29: 253–268. |
[13] | Magd HA, McCoy MP (2014) Islamic Finance Development in the Sultanate of Oman: Barriers and Recommendations. Pro Econ Financ 15: 1619–1631. |
[14] |
Mandilaras A, Bird G (2010) A Markov switching analysis of contagion in the EMS. J Int Money Financ 29: 1062–1075. doi: 10.1016/j.jimonfin.2010.03.001
![]() |
[15] |
Mansoor Khan M, Ishaq Bhatti M (2008) Development in Islamic banking: a financial risk-allocation approach. J Risk Financ 9: 40–51. doi: 10.1108/15265940810842401
![]() |
[16] | Moody J, Levin U, Rehfuss S (1993) Predicting the US index of industrial production. |
[17] |
Naifar N, Hammoudeh S (2016) Do global financial distress and uncertainties impact GCC and global sukuk return dynamics?. Pac-Basin Financ J 39: 57–69. doi: 10.1016/j.pacfin.2016.05.016
![]() |
[18] | Pattillo CA, Poirson H, Ricci LA (2002) External debt and growth. International Monetary Fund, No. 2002–2069. |
[19] | Seguino S (2009) The global economic crisis, its gender implications, and policy responses. Gender Perspectives on the Financial Crisis Panel at the Fifty-Third Session of the Commission on the Status of Women, United Nations, 7. |
[20] | Sukmana R, Ibrahim MH (2017) How Islamic are Islamic banks? A non-linear assessment of Islamic rate–conventional rate relations. Econ Model 64: 443–448. |
[21] |
Trenca I, Petria N, Corovei EA (2015). Impact of Macroeconomic Variables upon the Banking System Liquidity. Pro Econ Financ 32: 1170–1177. doi: 10.1016/S2212-5671(15)01583-X
![]() |