The aim of this paper is to examine exchange rate volatility using GARCH models with a new innovation distribution, the Normal Tempered Stable. We estimated daily exchange rate volatility using different distributions (Normal, Student, NIG) in order to specify the performed model. In addition, a forecasting analysis is performed to check which distribution reveals the best out-of-sample results. We found that the estimated parameters of GARCH-NTS model outperform the GARCH-N and GARCH-t ones for all currencies. Besides, we asserted that GARCH-NTS and EGARCH-NTS are the preferred models in terms of out-of sample forecasting accuracy. Our results indicating the performance of GARCH models with NTS distribution contribute to increase the accuracy of risk measures which is very important for international traders and investors.
Citation: Sahar Charfi, Farouk Mselmi. Modeling exchange rate volatility: application of GARCH models with a Normal Tempered Stable distribution[J]. Quantitative Finance and Economics, 2022, 6(2): 206-222. doi: 10.3934/QFE.2022009
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The aim of this paper is to examine exchange rate volatility using GARCH models with a new innovation distribution, the Normal Tempered Stable. We estimated daily exchange rate volatility using different distributions (Normal, Student, NIG) in order to specify the performed model. In addition, a forecasting analysis is performed to check which distribution reveals the best out-of-sample results. We found that the estimated parameters of GARCH-NTS model outperform the GARCH-N and GARCH-t ones for all currencies. Besides, we asserted that GARCH-NTS and EGARCH-NTS are the preferred models in terms of out-of sample forecasting accuracy. Our results indicating the performance of GARCH models with NTS distribution contribute to increase the accuracy of risk measures which is very important for international traders and investors.
Let p be a prime, f be a polynomial with k variable and Fp=Z/(p) be the finite field, where Z is the integer ring, and let
N(f;p)=#{(x1,x2,⋯,xk)∈Fkp|f(x1,x2,⋯,xk)=0}. |
Many scholars studied the exact formula (including upper bound and lower bound) for N(f;p) for many years, it is one of the main topics in the finite field theory, the most elementary upper bounds was given as follows (see [14])
N(f;p)≤pk−1degf. |
Let ordp denote the p-adic additive valuation normalized such that ordpp=1. The famous Chevalley-Warning theorem shows that ordpN(f;p)>0 if n>degf. Let [x] denote the least integer ≥x and e denote the extension degree of Fq/Fp. Ax (see [2]) showed that
ordpN(f;q)≥e[n−degfdegf]. |
In 1977, S. Chowla et al. (see [7]) investigated a problem about the number of solutions of a equation in finite field Fp as follow,
x31+x32+⋅⋅⋅+x3k≡0, |
where p is a prime with p≡1mod 3 and xi∈Fp, 1≤i≤k.
Let Mk denotes the number of solutions of the above equation. They proved that
M3=p2+d(p−1),M4=p2+6(p2−p),∞∑s=1Msxs=x1−px+x2(p−1)(2+dx)1−3px2−pdx3, |
where d is uniquely determined by 4p=d2+27y2 and d≡1 mod 3.
Myerson [12] extended the result in [2] to the field Fq and first studied the following equation over Fq,
x31+x32+⋅⋅⋅+x3k≡0. |
Recently J. Zhao et al. (see [17]) investigated the following equations over field Fp,
f1=x41+x42+x43,f2=x41+x42+x43+x44. |
And they give exact value of N(f1;p) and N(f2;p). For more general problem about this issue interested reader can see [6,9,10,11].
In this paper, let A(k,p) denotes the number of solutions of the following equation in Fp,
x61+x62+⋅⋅⋅+x6k≡0, |
where p is a prime with p≡1mod 3 and xi∈Fp, 1≤i≤k, and for simplicity, in the rest of this paper, we assume there exists an integer z such that z3≡2mod p, we use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid a lot of complicated calculations. We proved the following:
Theorem 1. For any positive integer k≥1, we have the recurrence formula
A(k+6,p)=5pA(k+4,p)+10dpA(k+3,p)+(46p2+5d2p+dp)A(k+2,p)+(2p2+120dp2+3d3p+d2p+dp)A(k+1,p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)A(k,p)+pk+5−pk+4−(10dp+2d2)pk+3−64pk+5/2−(429+121d+5d2)pk+2−2d2pk+3/2−(3d3+130d2+12d+6)pk+1−d4pk, |
with the initial condition
A(1,p)=1,A(2,p)=4(p−1)+p,A(3,p)=10d(p−1)+p2,A(4,p)=56p(p−1)+10d2(p−1)+p3,A(5,p)=188dp(p−1)+5d3(p−1)+16dC(p)(p−1)+p4,A(6,p)=p5+1400p2(p−1)+(388d2+8d−576)p(p−1)+d2p−d2, |
where d is uniquely determined by 4p=d2+27y2 and d≡1 mod 3, and C(p)=p∑a=1ep(a3).
Remark. Our method is suitable to calculus the number of solutions of the following equation in Fp,
xt1+xt2+⋅⋅⋅+xtk≡0, |
where p satisfied a certain congruence conditions, and t is any nature number.
Our Theorem 2 can be deduced from Theorem 1 and the theory of the Difference equations.
Theorem 2. Let ti (1≤i≤k) be the real root of the below equation with multiplicity si (1≤i≤k) respectively, and ρje±iwj (1≤j≤h) be the complex root of the below equation with multiplicity rj (1≤j≤h) respectively,
x6=5px4+10dpx3+(46p2+5d2p+dp)x2+(2p2+120dp2+3d3p+d2p+dp)x+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p). |
We have
A(n,p)=k∑i=1si∑a=1Ciansi−atni+h∑j=1rj∑b=1Djbnrj−bρnjcosnwj+h∑j=1rj∑b=1Ejbnrj−bρnjsinnwj, |
where Cia,Djb,Ejb, are determined by
A(6,p)=k∑i=1si∑a=1Cia6si−at6i+h∑j=1rj∑b=1Djb6rj−bρ6jcos6wj+h∑j=1rj∑b=1Ejb6rj−bρ6jsin6wj,A(5,p)=k∑i=1si∑a=1Cia5si−at5i+h∑j=1rj∑b=1Djb5rj−bρ5jcos5wj+h∑j=1rj∑b=1Ejb5rj−bρ5jsin5wj,A(4,p)=k∑i=1si∑a=1Cia4si−at4i+h∑j=1rj∑b=1Djb4rj−bρ4jcos4wj+h∑j=1rj∑b=1Ejb4rj−bρ4jsin4wj,A(3,p)=k∑i=1si∑a=1Cia3si−at3i+h∑j=1rj∑b=1Djb3rj−bρ3jcos3wj+h∑j=1rj∑b=1Ejb3rj−bρ3jsin3wj,A(2,p)=k∑i=1si∑a=1Cia2si−at2i+h∑j=1rj∑b=1Djb2rj−bρ2jcos2wj+h∑j=1rj∑b=1Ejb2rj−bρ2jsin2wj,A(1,p)=k∑i=1si∑a=1Ciati+h∑j=1rj∑b=1Djbρjcoswj+h∑j=1rj∑b=1Ejbρjsinwj. | (1.1) |
Before we prove these lemmas, we give some notations, χ2 denotes the second-order character of Fp, χ denotes the third-order character of Fp, ψ denotes the sixth order character of Fp.
ep(x)=e2πixp,τ(χ)=p∑m=1χ(m)ep(m),G(χ,m)=p∑a=1χ(a)ep(am). |
We call G(χ,m) the Gauss sum, and we have the following:
G(χ,m)=τ(χ)¯χ(m),(m,p)=1. | (2.1) |
And also we have
|τ(χ)|=√p, | (2.2) |
where χ is a primitive character of Fp. And let G(m,6;p)=p−1∑a=0ep(ma6). For the property of the exponential sum and the general Gauss sum, interested readers can see [1,4,5,8,13,15].
Lemma 1. Let p be a prime with p≡1mod 3. Then for any third-order character χ of Fp, we have the identity
τ3(χ)+τ3(¯χ)=dp, |
where d is uniquely determined by 4p=d2+27y2 and d≡1mod 3.
Proof. For the proof of this lemma see [3].
Lemma 2. Let χ be a third-order character of Fp with p≡1mod 3, and C(p)=τ(χ)+τ(¯χ), then C(p)=p∑a=1ep(a3).
Proof.
A=τ(χ)+τ(¯χ)=p∑a=1(1+χ(a)+¯χ(a))e(ap)=p∑a=1e(a3p). |
Lemma 3. Let p≡1mod 6, 2≡z3mod p for some z, and let χ be a third-order character of Fp, ψ be a sixth-order character of Fp, then we have the identity
τ(ψ)=τ2(χ)√p. |
Proof. This is Lemma 3 in [16].
Lemma 4. As the definition above, we have the identity
G(m,6;p)=√pχ2(m)+¯x2√pψ(m)+x2√p¯ψ(m)+¯xχ(m)+x¯χ(m), |
where (m,p)=1 and x=τ(χ).
Proof. Firstly we have the identity
1+χ2(m)+χ(m)+¯χ(m)+ψ(m)+¯ψ(m)={6, if m≡a6 mod p;0,otherwise. |
So we have
G(m,6;p)=p−1∑a=0(1+χ2(a)+χ(a)+¯χ(a)+ψ(a)+¯ψ(a))ep(ma)=G(χ2,m)+G(ψ,m)+G(¯ψ,m)+G(χ,m)+G(¯χ,m) |
By (2.1) and Lemma 3, we have
G(m,6;p)=τ(χ2)χ2(m)+τ(¯ψ)ψ(m)+τ(ψ)¯ψ(m)+τ(¯χ)χ(m)+τ(χ)¯χ(m)=√pχ2(m)+¯x2√pψ(m)+x2√p¯ψ(m)+¯xχ(m)+x¯χ(m). | (2.3) |
By (2.3), we complete the proof of our lemma.
Next we let,
Gn(m,6;p)=an+bnχ2(m)+cnψ(m)+dn¯ψ(m)+enχ(m)+fn¯χ(m). | (2.4) |
Then we have following Lemma 5.
Lemma 5. Let an,bn,cn,dn,en,fn are defined as above, then we have that an,bn,cn,dn,en,fn are uniquely determined by n, where n≥1.
Proof. By the orthogonality of characters of Fp, we have
\begin{eqnarray} \sum\limits_{a = 1}^{p-1}\chi\left(a\right) = \left\{ \begin{array}{ll} p-1, & \text{ if}\ χ = χ_0; \\ 0, & \text{otherwise.}\end{array}\right. \end{eqnarray} | (2.5) |
By (2.4) and (2.5) we have
\begin{eqnarray} \sum\limits_{m = 1}^{p-1}G^n(m, 6;p)& = &(p-1)a_n+b_n\sum\limits_{m = 1}^{p-1}\chi_2\left(m\right)+c_n\sum\limits_{m = 1}^{p-1}\psi\left(m\right)+d_n\sum\limits_{m = 1}^{p-1}\overline{\psi}\left(m\right)\\ &&+e_n\sum\limits_{m = 1}^{p-1}\chi\left(m\right)+f_n\sum\limits_{m = 1}^{p-1}\overline{\chi}\left(m\right)\\ & = &(p-1)a_n. \end{eqnarray} |
So we have
\begin{eqnarray} &&a_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}G^n(m, 6;p). \end{eqnarray} | (2.6) |
By the same method, we have
\begin{eqnarray} &&b_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}\chi_2(m)G^n(m, 6;p), \\ &&c_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}\overline{\psi}(m)G^n(m, 6;p), \\ &&d_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}\psi(m)G^n(m, 6;p), \\ &&e_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}\overline{\chi}(m)G^n(m, 6;p), \\ &&f_n = \frac{1}{p-1}\sum\limits_{m = 1}^{p-1}\chi(m)G^n(m, 6;p). \end{eqnarray} |
So now it is easy to see the conclusion of the lemma.
Lemma 6. The sequences \{a_n\} , \{b_n\} , \{c_n\} , \{d_n\} , \{e_n\} , \{f_n\} are defined above, then they satisfied the following recurrence formulae ( n\geq0 ):
\begin{eqnarray} &&a_{n+1} = \sqrt{p}b_n+\frac{\overline{x^2}}{\sqrt{p}}d_n+\frac{x^2}{\sqrt{p}}c_n+xe_n+\overline{x}f_n, \end{eqnarray} | (2.7) |
\begin{eqnarray} &&b_{n+1} = \sqrt{p}a_n+\frac{\overline{x^2}}{\sqrt{p}}e_n+\frac{x^2}{\sqrt{p}}f_n+xd_n+\overline{x}c_n, \end{eqnarray} | (2.8) |
\begin{eqnarray} &&c_{n+1} = \sqrt{p}f_n+\frac{\overline{x^2}}{\sqrt{p}}a_n+\frac{x^2}{\sqrt{p}}e_n+xb_n+\overline{x}d_n, \end{eqnarray} | (2.9) |
\begin{eqnarray} &&d_{n+1} = \sqrt{p}e_n+\frac{\overline{x^2}}{\sqrt{p}}f_n+\frac{x^2}{\sqrt{p}}a_n+xc_n+\overline{x}b_n, \end{eqnarray} | (2.10) |
\begin{eqnarray} &&e_{n+1} = \sqrt{p}d_n+\frac{\overline{x^2}}{\sqrt{p}}c_n+\frac{x^2}{\sqrt{p}}b_n+xf_n+\overline{x}a_n, \end{eqnarray} | (2.11) |
\begin{eqnarray} &&f_{n+1} = \sqrt{p}c_n+\frac{\overline{x^2}}{\sqrt{p}}b_n+\frac{x^2}{\sqrt{p}}d_n+xa_n+\overline{x}e_n, \end{eqnarray} | (2.12) |
with the initial condition
a_0 = 1, b_0 = c_0 = d_0 = e_0 = f_0 = 0. |
Proof. We only prove (2.7), the rest can be proved in the same way. By Lemma 5, we know a_n is unique determined by n . We can compare the coefficient of the equation
G^{n+1}(m, 6;p) = G^{n}(m, 6;p)G(m, 6;p). |
We have
a_{n+1} = \sqrt{p}b_n+\frac{\overline{x^2}}{\sqrt{p}}d_n+\frac{x^2}{\sqrt{p}}c_n+xe_n+\overline{x}f_n. |
So we complete the proof of the lemma.
Lemma 7. Let a_n is defined as above, then we have
\begin{eqnarray} &&a_0 = 1, a_1 = 0, a_2 = 5p, a_3 = 10dp, a_4 = 56p^2+10d^2p, \\&&a_5 = 188dp^2+5d^3p+16dpC(p). \end{eqnarray} |
Proof. By Lemma 4 and after some elementary calculations we have
\begin{eqnarray} G^2(m, 6;p)& = &5p+2dp^{1/2}\chi_2(m)+4p^{1/2}x\psi(m)+4p^{1/2}\overline{x}\overline{\psi}(m)\\ &&+(p^{-1}\overline{x^4}+3x^2)\chi(m)+(p^{-1}x^4+3\overline{x^2})\overline{\chi}(m), \\ G^3(m, 6;p)& = &10dp+(16p^{3/2}+dp^{1/2})\chi_2(m)+(15p\overline{x}+2dx^2+p^{-1}x^5)\chi(m)\\ &&+(15px+2d\overline{x^2}+p^{-1}\overline{x^5})\overline{\chi}(m)\\ &&+(4p^{-1/2}x^4+12p^{1/2}\overline{x^2}+2dp^{1/2}x)\psi(m)\\ &&+(4p^{-1/2}\overline{x^4}+12p^{1/2}x^2+2dp^{1/2}\overline{x})\overline{\psi}(m), \\ G^4(m, 6;p)& = &60p^2+9d^2p+dp+48dp^{3/2}\chi_2(m)\\ &&+(p^{-2}x^8+17\overline{x^4}+46px^2+16dp)\chi(m)\\ &&+(p^{-2}\overline{x^8}+17x^4+46p\overline{x^2}+16dp)\overline{\chi}(m)\\ &&+(56p^{3/2}x+4dp^{-1/2}x^4+12dp^{1/2}\overline{x^2}+8p^{-1/2}\overline{x^5})\psi(m)\\ &&+(56p^{3/2}\overline{x}+4dp^{-1/2}\overline{x^4}+12dp^{1/2}x^2+8p^{-1/2}x^5)\overline{\psi}(m), \\ G^5(m, 6;p)& = &188dp^2+5d^3p+16dpC(p)\\ &&+(52d^2p^{3/2}+208p^{5/2}+16dp^{1/2}(x^2+\overline{x^2}))\chi_2(m)\\ &&+(p^{-2/5}x^{10}+p^{-3/2}\overline{x^8}+4dp^{-1/2}\overline{x^5}+71p^{1/2}x^4\\&&+(46p^{3/2}+16p^{1/2})x^2\\ &&+(129p^{3/2}+10d^2p^{1/2})\overline{x^2}+60dp^{3/2}x+16dp^{3/2})\psi(m)\\ &&+(p^{-2/5}\overline{x^{10}}+p^{-3/2}x^8+4dp^{-1/2}x^5+71p^{1/2}\overline{x^4}\\&&+(46p^{3/2}+16p^{1/2})\overline{x^2}\\ &&+(129p^{3/2}+10d^2p^{1/2})x^2+60dp^{3/2}\overline{x}+16dp^{3/2})\overline{\psi}(m)\\ &&+(8p^{-1}\overline{x^7}+p^{-1}x^7+25x^5+52dpx^2\\&&+(28dp+46p^2)x+16d\overline{x^4}+112p^2\overline{x})\chi(m)\\ &&+(8p^{-1}x^7+p^{-1}\overline{x^7}+25\overline{x^5}+52dp\overline{x^2}\\&&+(28dp+46p^2)\overline{x}+16dx^4+112p^2x)\overline{\chi}(m), \end{eqnarray} |
and comparing the above formulae with (2.6), we have
a_0 = 1, a_1 = 0, a_2 = 5p, a_3 = 10dp, a_4 = 60p^2+9d^2p+dp, a_5 = 188dp^2+5d^3p+16dpC(p).\\ |
Lemma 8. Let a_n , b_n , c_n , d_n , e_n , f_n are defined as above, then we have
\begin{eqnarray} a_6& = &5pa_4+10dpa_3+(46p^2+5d^2p+dp)a_2+(2p^2+120dp^2+3d^3p+d^2p+dp)a_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)a_0\\ b_6& = &5pb_4+10dpb_3+(46p^2+5d^2p+dp)b_2+(2p^2+120dp^2+3d^3p+d^2p+dp)b_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)b_0\\ c_6& = &5pc_4+10dpc_3+(46p^2+5d^2p+dp)c_2+(2p^2+120dp^2+3d^3p+d^2p+dp)c_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)c_0\\ d_6& = &5pd_4+10dpd_3+(46p^2+5d^2p+dp)d_2+(2p^2+120dp^2+3d^3p+d^2p+dp)d_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)d_0\\ e_6& = &5pe_4+10dpe_3+(46p^2+5d^2p+dp)e_2+(2p^2+120dp^2+3d^3p+d^2p+dp)e_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)e_0\\ f_6& = &5pf_4+10dpf_3+(46p^2+5d^2p+dp)f_2+(2p^2+120dp^2+3d^3p+d^2p+dp)f_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}+(129d^2+11d+6)p^2+d^4p)f_0 \end{eqnarray} |
Proof. We only proof the first formula, the rest can be proof in the same way. By Lemma 6, we have
\begin{eqnarray} a_6& = &\sqrt{p}b_5+\frac{\overline{x^2}}{\sqrt{p}}d_5+\frac{x^2}{\sqrt{p}}c_5+xe_5+\overline{x}f_5\\ & = &5pa_4+2dp^{1/2}b_4+4p^{1/2}\overline{x}c_4+4p^{1/2}xd_4\\&&+(3\overline{x^2}+p^{-1}x^4)e_4+(3x^2+p^{-1}\overline{x^4})f_4\\ & = &5pa_4+10dpa_3+(d^2p^{1/2}+12p^{3/2})b_3+(2dp^{1/2}\overline{x}\\&&+8p^{1/2}x^2+p^{-1/2}\overline{x^4})c_3\\ &&+(2dp^{1/2}x+8p^{1/2}\overline{x^2}+p^{-1/2}x^4)d_3+(11px+\overline{x^2}+p^{-1}\overline{x^5})e_3\\ &&+(11p\overline{x}+x^2+p^{-1}x^5)f_3\\ & = &5pa_4+10dpa_3+(46p^2+5d^2p+dp)a_2+(25dp^{3/2}+2p^{3/2})b_2\\ &&+(p^{-3/2}\overline{x^7}+2p^{-1/2}x^5+p^{-1/2}\overline{x^4}+42p^{3/2}\overline{x}+2dp^{1/2}x^2\\&&+(d^2+1)p^{1/2}\overline{x})c_2\\ &&+(p^{-3/2}x^7+2p^{-1/2}\overline{x^5}+p^{-1/2}x^4+42p^{3/2}x+2dp^{1/2}\overline{x^2}\\&&+(d^2+1)p^{1/2}x)d_2\\ &&+(10x^4+(32p+d^2)\overline{x^2}+(4dp+p)x)e_2\\&&+(10\overline{x^4}+(32p+d^2)x^2+(4dp+p)\overline{x})f_2\\ & = &5pa_4+10dpa_3+(46p^2+5d^2p+dp)a_2\\&&+(2p^2+120dp^2+3d^3p+d^2p+dp)a_1\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\&&+(129d^2+11d+6)p^2+d^4p)a_0. \end{eqnarray} |
So we complete the proof of this lemma.
Lemma 9. Let a_n is defined as above, then for any integer n\geq0 , we have
\begin{eqnarray} a_{n+6}& = &5pa_{n+4}+10dpa_{n+3}+(46p^2+5d^2p+dp)a_{n+2}\\&&+(2p^2+120dp^2+3d^3p+d^2p+dp)a_{n+1}\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\&&+(129d^2+11d+6)p^2+d^4p)a_n. \end{eqnarray} |
Proof. By (2.4) and Lemma 8, we have
\begin{eqnarray} G^{6}(m, 6;p)& = &5pG^{4}(m, 6;p)+10dpG^{3}(m, 6;p)+(46p^2+5d^2p+dp)G^{2}(m, 6;p)\\ &&+(2p^2+120dp^2+3d^3p+d^2p+dp)G(m, 6;p)\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\&&+(129d^2+11d+6)p^2+d^4p). \end{eqnarray} |
We multiple G^{n}(m, 6;p) to the both side of the above formula, we have
\begin{eqnarray} G^{n+6}(m, 6;p)& = &5pG^{n+4}(m, 6;p)+10dpG^{n+3}(m, 6;p)\\&&+(46p^2+5d^2p+dp)G^{n+2}(m, 6;p)\\ &&+(2p^2+120dp^2+3d^3p+d^2p+dp)G^{n+1}(m, 6;p)\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\&&+(129d^2+11d+6)p^2+d^4p)G^n(m, 6;p). \end{eqnarray} |
By Lemma 5, we can compare the coefficient of the above equation, we have
\begin{eqnarray} a_{n+6}& = &5pa_{n+4}+10dpa_{n+3}+(46p^2+5d^2p+dp)a_{n+2}\\&&+(2p^2+120dp^2+3d^3p+d^2p+dp)a_{n+1}\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\&&+(129d^2+11d+6)p^2+d^4p)a_n. \end{eqnarray} |
In the formula below, we always let k\geq1 . By the following formula,
\begin{eqnarray} \sum\limits_{a = 0}^{p-1}e_p\left(ma\right) = \left\{ \begin{array}{ll} p, & \text{ if }\ p\mid m; \\ 0, & \text{ otherwise, }\end{array}\right. \end{eqnarray} |
we have
\begin{eqnarray} A(k, p)& = &\frac{1}{p}\sum\limits_{m = 0}^{p-1}\sum\limits_{x_1 = 0, x_2 = 0, \cdot\cdot\cdot, x_k = 0}^{p-1}e_p(m(x_1^6+x_2^6+\cdot\cdot\cdot+x_k^6))\\& = &\frac{1}{p}\sum\limits_{m = 0}^{p-1}G^k(m, 6;p). \end{eqnarray} | (3.1) |
By (8), we have
\begin{eqnarray} A(k, p)& = &\frac{1}{p}\sum\limits_{m = 0}^{p-1}G^k(m, 6;p)\\ & = &\frac{1}{p}(\sum\limits_{m = 1}^{p-1}G^k(m, 6;p)+p^k)\\ & = &\frac{1}{p}((p-1)a_k+p^k) = \frac{p-1}{p}a_k+p^{k-1}. \end{eqnarray} | (3.2) |
So by Lemma 9, we have
\begin{eqnarray} A(k+6, p)-p^{k+5}& = &5p(A(k+4, p)-p^{k+3})+10dp(A(k+3, p)-p^{k+2})\\ &&+(46p^2+5d^2p+dp)(A(k+2, p)-p^{k+1})\\ &&+(2p^2+120dp^2+3d^3p+d^2p+dp)(A(k+1, p)-p^k)\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\ &&+(129d^2+11d+6)p^2+d^4p)(A(k, p)-p^{k-1}). \end{eqnarray} |
So we have
\begin{eqnarray} A(k+6, p)& = &5pA(k+4, p)+10dpA(k+3, p)\\&&+(46p^2+5d^2p+dp)A(k+2, p)\\ &&+(2p^2+120dp^2+3d^3p+d^2p+dp)A(k+1, p)\\ &&+(-4p^5+2d^2p^4+64p^{7/2}+381p^3+2d^2p^{5/2}\\ &&+(129d^2+11d+6)p^2+d^4p)A(k, p)\\ &&+p^{k+5}-p^{k+4}-(10dp+2d^2)p^{k+3}-64p^{k+5/2}\\&&-(429+121d+5d^2)p^{k+2}\\ &&-2d^2p^{k+3/2}-(3d^3+130d^2+12d+6)p^{k+1}-d^4p^k. \end{eqnarray} |
And by Lemma 7 and (3.2), we have the initial conditions
\begin{eqnarray} &&A(1, p) = 1, A(2, p) = 4(p-1)+p, A(3, p) = 10d(p-1)+p^2, \\ &&A(4, p) = 56p(p-1)+10d^2(p-1)+p^3, \\ &&A(5, p) = 188dp(p-1)+5d^3(p-1)+16dC(p)(p-1)+p^4.\\ &&A(6, p) = p^5+1400p^2(p-1)+(388d^2+8d-576)p(p-1)+d^2p-d^2. \end{eqnarray} |
So we complete the proof of the theorem.
The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. And we give an expression of the number of solutions of the above equation by the root of sixth degree polynomial. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.
The author thanks to referees for very important recommendations and warnings which improved the paper.
The author declares that there is no competing interest.
[1] |
Abdullah SM, Siddiqua S, Siddiquee MSH, et al. (2017) Modeling and forecasting exchange rate volatility in Bangladesh using GARCH models: a comparison based on normal and Student's t-error distribution. Financ Innov 3: 18. https://doi.org/10.1186/s40854-017-0071-z doi: 10.1186/s40854-017-0071-z
![]() |
[2] |
Abounoori A, Elmi ZM, Nademi Y (2016) Forecasting Tehran stock exchange volatility; Markov switching GARCH approach. Phys A 445: 264–282. https://doi.org/10.1016/j.physa.2015.10.024 doi: 10.1016/j.physa.2015.10.024
![]() |
[3] |
Atabani Adi A (2019) Modeling exchange rate return volatility of RMB/USD using GARCH family models. J Chinese Econ Bus Stud 17: 169–187. https://doi.org/10.1080/14765284.2019.1600933 doi: 10.1080/14765284.2019.1600933
![]() |
[4] |
Baillie R, Bollerslev T (2002) The message in daily exchange rates: a conditional-variance tale. J Bus Econ Stat 20–1: 60–68. https://doi.org/10.1198/073500102753410390 doi: 10.1198/073500102753410390
![]() |
[5] |
Barndorff-Nielsen OE (1997) Normal inverse Gaussian distributions and stochastic volatility modeling. Scand J Stat 24: 1–13. https://doi.org/10.1111/1467-9469.t01-1-00045 doi: 10.1111/1467-9469.t01-1-00045
![]() |
[6] |
Barndorff-Nielsen OE, Levendorskii SZ (2001) Feller processes of normal inverse Gaussian type. Quant Financ 1: 318–331. https://doi.org/10.1088/1469-7688/1/3/303 doi: 10.1088/1469-7688/1/3/303
![]() |
[7] | Barndorff-Nielsen OE, Schmiegel J (2008) Time change, volatility, and turbulence. Math Control Theory Financ, 29–53. |
[8] | Barndorff-Nielsen OE, Shephard N (2001) Normal Modified Stable Processes. Oxford: Department of Economics, Oxford University Press. https://doi.org/10.1007/978-1-4612-0197-7 |
[9] | Black F (1976) Studies of Stock Price Volatility Changes. Proceedings of the Business and Economics Section of the American Statistical Association, 177–181. |
[10] |
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31–3: 307–327. https://doi.org/10.1016/0304-4076(86)90063-1 doi: 10.1016/0304-4076(86)90063-1
![]() |
[11] |
Calzolari G, Halbleib R, Parrini L (2014) Estimating GARCH-type models with symmetric stable innovations: Indirect inference versus maximum likelihood. Comput Stat Data Anal 76: 158–171. https://doi.org/10.1016/j.csda.2013.07.028 doi: 10.1016/j.csda.2013.07.028
![]() |
[12] |
Cont R (2001) Empirical properties of asset returns: Stylized facts and statistical issues. Quantit Financ 1: 1–14. https://doi.org/10.1080/713665670 doi: 10.1080/713665670
![]() |
[13] |
Danielsson J, De Vries CG (2000) Value-at-risk and extreme returns. Ann Econ Stat 60: 239–270. https://doi.org/10.2307/20076262 doi: 10.2307/20076262
![]() |
[14] |
Engle RF (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50–4: 987–1007. https://doi.org/10.2307/1912773 doi: 10.2307/1912773
![]() |
[15] |
Feng L, Shi Y (2017) Fractionally integrated GARCH model with tempered stable distribution: a simulation study. J Appl Stat 44–16: 2837–2857. https://doi.org/10.1080/02664763.2016.1266310 doi: 10.1080/02664763.2016.1266310
![]() |
[16] |
Fiorani F, Luciano E, Semeraro P (2010) Single and joint default in a structural model with purely discountinuous asset prices. Quantit Financ 10: 249–263. https://doi.org/10.1080/14697680902991965 doi: 10.1080/14697680902991965
![]() |
[17] | Hamilton JD (1994) Time Series Analysis. Princeton University Press, Princeton. https://doi.org/10.1515/9780691218632 |
[18] |
Hirsa A, Madan DB (2003) Pricing american options under variance gamma. J Comput Financ 7: 63–80. https://doi.org/10.21314/JCF.2003.112 doi: 10.21314/JCF.2003.112
![]() |
[19] |
Ho KY, Shi Y, Zhang Z (2013) How does news sentiment impact asset volatility? Evidence from long memory and regime-switching approaches. N Am J Econ Financ 26: 436–456. https://doi.org/10.1016/j.najef.2013.02.015 doi: 10.1016/j.najef.2013.02.015
![]() |
[20] |
Kim YS, Rachev ST, Bianchi L, et al. (2010) Tempered stable and tempered infinitely divisible GARCH models. J Bank Financ 34: 2096–2109. https://doi.org/10.1016/j.jbankfin.2010.01.015 doi: 10.1016/j.jbankfin.2010.01.015
![]() |
[21] |
Letac G, Seshadri V (2005) Exponential stopping and drifted stable processes. Stat Prob Lett 72: 137–143. https://doi.org/10.1016/j.spl.2004.12.007 doi: 10.1016/j.spl.2004.12.007
![]() |
[22] | Lopez JA (2001) Evaluating the predictive accuracy of volatility models. J Forecast 20: 87–109. |
[23] |
Louati M, Masmoudi A, Mselmi F (2020) The normal tempered stable regression model. Commun Stat-Theor M 49: 500–512. https://doi.org/10.1080/03610926.2018.1554121 doi: 10.1080/03610926.2018.1554121
![]() |
[24] |
Madan DB, Carr P, Chang E (1998) The variance gamma process and option pricing. Eur Financ Rev 2: 79–105. https://doi.org/10.1023/A:1009703431535 doi: 10.1023/A:1009703431535
![]() |
[25] |
Madan DB, Seneta E (1990) The Variance Gamma (V.G.) model for share market returns. J Bus 63: 511–524. https://doi.org/10.1086/296519 doi: 10.1086/296519
![]() |
[26] |
Marinelli C, Rachev ST, Roll R (2001) Subordinated exchange rate models: Evidence for heavy tailed distributions and long-range dependence. Math Comput Model 34: 955–1001. https://doi.org/10.1016/S0895-7177(01)00113-3 doi: 10.1016/S0895-7177(01)00113-3
![]() |
[27] |
Nelson DB (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59: 347–370. https://doi.org/10.2307/2938260 doi: 10.2307/2938260
![]() |
[28] | Nolan J (2007) Stable Distributions: Models for Heavy Tailed Data. Brikhauser, Boston. |
[29] |
Podobnik B, Horvatic D, Petersen AM, et al. (2009) Cross-correlations between volume change and price change. Proc Natl Aca 106: 22079–22084. https://doi.org/10.1073/pnas.0911983106 doi: 10.1073/pnas.0911983106
![]() |
[30] |
Rodriguez MJ, Ruiz E (2012) Revisiting Several Popular GARCH Models with Leverage Effect: Differences and Similarities. J Financ Econometrics 10: 637–668. https://doi.org/10.1093/jjfinec/nbs003 doi: 10.1093/jjfinec/nbs003
![]() |
[31] |
Shi Y, Feng L (2016) A discussion on the innovation distribution of the Markov regime-switching GARCH model. Econ Model 53: 278–288. https://doi.org/10.1016/j.econmod.2015.11.018 doi: 10.1016/j.econmod.2015.11.018
![]() |
[32] | Tserakh U, Trusz M (2019) GARCH(1, 1) models with stable residuals. Stud Inf Syst Inf Tech 22: 47–57. |
[33] |
Zhu D, Galbraith JW (2011) Modeling and forecasting expected shortfall with the generalized asymmetric student-t and asymmetric exponential power distributions. J Emp Fin 18: 765–778. https://doi.org/10.1016/j.jempfin.2011.05.006 doi: 10.1016/j.jempfin.2011.05.006
![]() |