Abstract
We develop a test for multidimensional diffusion models with known drift term based on transformation of diffusion matrix. The test in this article is different from the traditional method. We use a onetoone transformation to transform the original model into a unit diffusion. Our approach is not only effective for stationary processes. On the other hand, our test performs well both on size and power.
1. Introduction
A random variable ${X}_{t}$ satisfies the following stochastic differential equation and parametric specification model:
where the random variable ${X}_{t}=\left({X}_{t}^{\left(1\right)},{X}_{t}^{\left(2\right)},\dots ,{X}_{t}^{\left(1\right)}\right)$ is a ddimension state vector on ${S}_{X}\subset {R}^{d}$, drift coefficient $\mu \left({X}_{t}\right)$ and $\mu ({X}_{t},\theta )$ are both $d$dimension vector, diffusion coefficient $\sigma \left({X}_{t}\right)$ and $\sigma ({X}_{t},\theta )$ are both $d\times m$ matrix $m\le d$, and ${W}_{t}$ is a mdimension standard Brownian motion.The focus of this paper is on testing the validity of the parametric specification Mode (2) based on a set of discretely observed data ${\left\{{X}_{t\mathrm{\Delta}}\right\}}_{t=0}^{n}$.
For testing onedimensional diffusion, in a pioneering work, AitSahalia proposed an approach for testing the parametric specification model based on marginal density [1]. The advantage of the test is that the parametric marginal density of most of the diffusion processes is easy to know. But the method has several limitations. Hong and Li, Chen and Gao have an important development after AitSahalia’s work [2, 3]. Separately, the above two articles presents two different methods even though both of the methods are based on TPDF (transition probability distribution density). Chen shows that the method of Hong and Li will excessively accept the null hypothesis [4].
The work in multidimensional case is not going well. Hong and Li's method can be used in multidimensional case. But the result is not very satisfactory. Song develops a martingale approach [5]. “Martingale Problems” is used to transform null hypothesis several times. Then, the multidimensional problem is broken into multiple onedimensional problems. Their method considers all relationship of any two variables, so the difficulty of calculation is geometric growth. This is the socalled dimensional disaster.
Based on Hermite series expansion, AitSahalia makes a breakthrough in giving the closedform of the approximate TPDF for a univariate timehomogeneous diffusion [6]. AitSahalia extends his previous work to the multidimensional diffusions by using Kolmogorov equations [7]. On the basis of AitSahalia, Choi proposes the approximate TPDF of multidimensional timeinhomogeneous diffusion models [8, 9]. In recent years, there are some other works on diffusion model testing [4, 1012].
In this paper, we propose a new test method moved by AitSahalia [7]. For most of diffusion processes, we can transfer diffusion $X$ into a diffusion $Y$ whose diffusion matrix ${\sigma}_{Y}$ is the identity matrix by a onetoone transformation. Then, we test if the diffusion matrix ${\sigma}_{Y}$ is an identity matrix.
Our method has 3 advantages: (a) The condition of the strictly stationary process is not needed. Whether calculating the marginal density or the transitional density, the process needs to be strictly stationary process. While our method is adapted to nonstationary process. Only if the diffusion is a strictly stationary process, we can give the nonparametric estimator of TDPF. But our method does not need to calculate the nonparametric estimator of TDPF. (b) Our test improves the result of size and power. (c) It also works for more complicated cases. The most important improvement is dimensional disaster can be avoided.
The structure of the paper is as follows. In Section 2, we state the setup and assumptions. Section 3 reports the procedure and main results of the test. Simulation studies and empirical application are reported in the next two sections. At last, we conclude the discussion.
2. Setup and assumptions
Assumption 1. ${S}_{X}$ is a product of $m$ intervals with limits ${\underset{\_}{x}}_{i}$ and ${\overline{x}}_{i}$, where possibly ${\underset{\_}{x}}_{i}=\infty $ and ${\overline{x}}_{i}=+\infty $, in which case, the intervals are open at infinite limits.
Assumption 2. $\forall x\in {S}_{X}$, the matrix $a\left(x\right)\equiv \sigma \left(x\right){\sigma}^{\text{'}}\left(x\right)$ is positive definite.
Assumption 3. For $i,j=\mathrm{1,2},\dots ,m$, ${\mu}_{i}\left(x\right)$ and ${\sigma}_{ij}\left(x\right)$ are infinitely differentiable with respect to $x\in {S}_{X}$ and $t\in \left[0,+\infty \right)$.
Assumption 4. There is a constant $K>0$ such that for all $x\in \left[0,+\infty \right)\times {S}_{X}$:
Assumption 3 and Assumption 4 ensure the uniqueness and existence of the solution of the Eq. (1) respectively. Indeed, Assumption 3 implies in particular that the coefficients of the stochastic differential equation are locally Lipschitz under their assumed (once) differentiability, which can be seen by applying the mean value theorem. And Assumption 4 can be relaxed in specific examples; it is not possible to do so in full generality.
3. Approach and test statistics
3.1. Reducible diffusions
Definition 1. If and only if there exists a onetoone transformation of the diffusion $X$ into a diffusion $Y$ whose diffusion matrix ${\sigma}_{Y}$ is the identity matrix, we can say that diffusion $X$ is reducible. There exists an invertible function $\gamma \left(x\right)$ which is infinitely differentiable in $X$ on ${S}_{X}$, such that ${Y}_{t}=\gamma \left({X}_{t}\right)$ satisfies the stochastic differential equation on the domain ${S}_{X}$:
If diffusion is reducible, the change of variable $\gamma $ satisfies $\nabla \gamma \left(x\right)={\sigma}^{1}\left(x\right)$, by Ito’s lemma. Onedimensional diffusion is reducible, by the simple transformation:
where the lower bound of integration is an arbitrary point in the interior of ${S}_{X}$. The differentiability of $\gamma $ ensures that ${\mu}_{Y}$ satisfies assumption 3. This change of variable is a Lamperti transform which plays a critical role in the derivation of closedform Hermite approximations to the transition density of univariate diffusions. In next section, we give the method to solve the case that $1/\sigma \left(u\right)$ cannot be integrated in closed form. For multivariate case, not every diffusion is reducible. It depends on the specification of its σ matrix, in the following way.
Proposition 1. (Necessary and sufficient condition for reducibility). The diffusion $X$ is said to be reducible if and only if:
for each $x$ in ${S}_{X}$ and triplet $x\left(i,j,k\right)=\mathrm{1,2},\dots ,m$ such that $k>j$. If $\sigma $ is nonsingular, then the condition can be expressed as:
Reducibility conditions are only for the matrix $\sigma \left(x\right)$. Under Proposition 1, when $\sigma $ is nonsingular, ${m}^{2}\left(m1\right)/2$ equalities must hold in order for an mdimensional diffusion to be reducible. For example, when $m=2$, only two equalities need to be checked.
3.2. Irreducible diffusions
If the diffusion is irreducible, however, one no longer has the option of transforming $X$ to $Y$. Notice that sample ${\{X}_{1},{X}_{2},...,{X}_{n}\}$ is discrete. Though we can’t give the closedform of transformation for diffusion $X$ directly, we can transfer the sample ${\{X}_{1},{X}_{2},...,{X}_{n}\}$ to sample ${\{Y}_{1},{Y}_{2},...,{Y}_{n}\}$ of a unit diffusion $Y$ discretely. For example, AitSahalia points out that the diffusion is irreducible if $\sigma $ is like case ${\sigma}_{case1}$:
Because ${\sigma}_{11}$ depends on ${x}_{2}$. In practice, we treat the process of variable ${x}_{1}$ and ${x}_{2}$ in ${X}_{t}$ as onedimensional diffusion separately. So ${x}_{2}$ can be seen as a constant even though the ${x}_{2}$ changes every moment. Fortunately, the sample of ${x}_{2}$ is observed so that we can transform the sample of ${x}_{1}$ to the sample of a unit diffusion ${y}_{1}$.
If ${\sigma}_{11}$ just depends on ${x}_{1}$, the diffusion process is reducible. Like ${\sigma}_{case2}$, we treat $b\left({x}_{2}\right)$ in ${\sigma}_{11}$ as constant. The matrix is similar to a diagonal matrix. Then it is like case 1.
Proposition 2. If irreducible diffusion $X$ is diagonalized, we can transform the sample of diffusion $X$ to a sample which belongs to a unit diffusion $Y$.
3.3. Hypothesis and test statistics
With known drift function, the hypothesis is as follows:
By Proposition 1 and Proposition 2, there exists a onetoone transformation of a reducible diffusion to unit diffusion. So we just need to test whether the diffusion function is unit diffusion. We can transfer the reducible diffusion $X$ to unit diffusion $Y$ or the sample of irreducible diffusion $X$ to sample of unit diffusion $Y$. The hypothesis becomes like the following:
where $I$ is a unit matrix.
For the purpose, we need:
(1) Estimate the parameter $\theta $ of diffusion function by the observation sample data set ${\left\{{X}_{t\mathrm{\Delta}}\right\}}_{t=0}^{n}$,
(2) Transfer ${\left\{{X}_{t\mathrm{\Delta}}\right\}}_{t=0}^{n}$ with a unique transformation to sample ${\left\{{Y}_{t\mathrm{\Delta}}\right\}}_{t=0}^{n}$ of a unit diffusion $Y$,
(3) Estimate the diffusion function of $Y$ with ${\left\{{Y}_{t\mathrm{\Delta}}\right\}}_{t=0}^{n}$.
Remark 1. In this paper, ${X}_{i}$ means the $i$th component in $X$. ${X}_{{t}_{i}}$ means the observation at ${t}_{i}$ moment of $X$.
$\frac{n}{T}$(${Y}_{{t}_{i+1}}{Y}_{{t}_{i}}\left)\right({Y}_{{t}_{i+1}}{Y}_{{t}_{i}})\mathrm{\text{'}}$ can be treated as sample of $a\left(y\right)$. So our test statistics is:
Theorem 1. Under Assumptions 14, ${T}_{ii}\to {\chi}^{2}\left(1\right)$ and ${T}_{ij}\to 0$, $i\ne j$, under ${H}_{0}$.
Theorem 2. Under Assumptions 14, ${T}_{ii}\to \infty $ or ${T}_{ij}\to \infty $, $i\ne j$, under ${H}_{1}$.
Theorem 1 and Theorem 2 are applicable to onedimensional situation.
4. Simulation experiment
4.1. Size evaluation
For examining the size of our test for multivariate models, we use a threefactor Vasicek model to generate the sample data. We also do size evaluation with Hong and Li's test and Song’s test for comparison. Following [16], we set:
We use the asymptotic critical values (1.28 and 1.65) at the 10 and 5 % levels as the empirical rejection rates. Reference [8], we compute $T\left(\theta \right)$ over a bandwidth set $\mathcal{H}={\left\{{h}_{k}\right\}}_{k=1}^{J}$. So the results in Table 1 are selected to be the result of the optimal bandwidth at the time of simulation.
Table 1Sizes of Q(j), Song’s test and our T(θ) test
$Q\left(j\right)$  Song  $T\left(\theta \right)$  
${X}_{1t}$  ${X}_{2t}$  ${X}_{3t}$  Total  ${X}_{1t}$  ${X}_{2t}$  ${X}_{3t}$  Total  ${X}_{1t}$  ${X}_{2t}$  ${X}_{3t}$  Total  
5 %  
$n=$250  4.5  3.4  2.9  3.7  4.6  5.1  4.9  6.1  4.3  4.3  3.3  3.4 
$n=$500  4.1  2.9  5.0  4.2  5.6  4.8  4.2  5.7  3.6  3.4  2.7  3.0 
$n=$1000  3.6  3.8  4.8  4.7  3.9  3.8  3.4  5.2  2.9  2.8  2.3  2.4 
10 %  
$n=$250  6.9  5.8  5.9  8.4  8.0  7.9  8.5  9.3  6.5  6.8  7.2  7.4 
$n=$500  7.0  6.5  6.9  8.2  7.7  7.6  7.8  8.6  5.8  6.3  6.6  6.9 
$n=$1000  7.3  7.1  6.7  7.7  7.0  6.5  7.2  7.7  5.3  5.2  6.1  6.2 
Table 1 shows the result of sizes of three tests for three individual and combined generalized residuals separately at the 5 % and 10 % level. Overall, our test has a good performance on sizes at both 5 % and 10 % levels for sample sizes as small as $n=$250 (i.e., about 20 years of monthly data). Our test makes an improvement to the size result.
Then, we consider the test for irreducible case. We use a bivariate Heston Model which is not a stationary process to generate 250, 500 and 1000 random samples. Table 2 shows that our test statistic also works on nonstationary process and has a good result. This is the most important breakthrough of our method. It makes $T\left(\theta \right)$ test can deal with more complex models:
Table 2Sizes of test for Heston model
5 %  10 %  
$n=$250  $n=$500  $n=$1000  $n=$250  $n=$500  $n=$1000  
${X}_{1t}$  5.1  4.9  3.6  7.2  6.7  4.5 
${X}_{2t}$  6.4  5.7  3.9  8.1  6.5  5.0 
Total  4.7  5.1  3.3  7.1  6.1  4.4 
4.2. Power evaluation
Table 3 shows comparison result of $Q\left(j\right)$ our $T\left(\theta \right)$ test. We can see that, for the two models, we can determine which factors cause the error of null hypothesis, and from the whole, it is good to reject the null hypothesis. It will not accept the wrong null hypothesis too much, even though some components may be assumed to be correct. For either the $T\left(\theta \right)$ test or the $Q\left(j\right)$ test, the more the sample data $n$ is, the larger the $n$ is, the more significant the power is. For small samples. The power of the test needs to be improved, especially when $n=\text{250}$. In general, power of our test is not as good as the size. Due to the principle of hypothesis testing, we can only ask for the size of our test to be as good as possible with finite sample.
Table 3Power of Q(j) our T(θ) test (in parentheses)
${X}_{1t}$  ${X}_{2t}$  ${X}_{3t}$  Total  
Model 1  
$n=$250  37.2 (24.2)  6.3 (4.9)  7.7 (3.2)  17.2 (9.4) 
$n=$ 500  63.5 (50.8)  6.7 (4.8)  7.5 (5.2  27.2 (16.3) 
$n=$1000  94.6 (97.4)  6.6 (4.7)  6.6 (4.7)  53.7 (48.3) 
Model 2  
$n=$250  38.4 (26.7)  26.3 (13.7)  6.7 (4.5)  32.2 (18.2) 
$n=$500  74.2 (62.9)  48.1 (37.2)  7.2 (5.3)  65.7 (49.5) 
$n=$1000  95.3 (98.3)  78.5 (72.4)  6.8 (5.9)  93.2 (91.1) 
5. Conclusions
In this article, we have developed a goodness of fit test for multidimensional diffusion models with known drift function. We use TPDF directly to construct our test statistic. Our method is effective for most multidimensional diffusion models. And it has a good performance in empirical applications. Of course, there is something to be improved. Even though our test works for most situations, there are still some cases that have not been covered. Next, we will focus on the test of drift function.
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