Econometric Testing For Financial Continuous Time Models

This dissertation focuses on the methodology theory. It provides some tests for continuous time models based on the theory of empirical processes and has many ad-vantages over the existing methods. For a composite null hypothesis, the empirical process based tests usually have quite complicated null distributions, which depend on the null hypothesis and parameter estimates. Some bootstrap procedures are usually needed to find the critical values. To avoid these complicated procedures, we use the martingale transformation introduced by Khmaladze(1981) to transform the empirical processes so that the effects of null hypothesis and parameter esti-mation can be removed asymptotically. One of the theoretical contribution of this dissertation is that we extend Khmaladze martingale transformation to test theorey for continuous time models. And we provide a complete framework of martingale transformation based tests to test for different component of continuous time mod-els.Four types of tests have been proposed to test for the continuous time models. The major theoretical work and innovations are as follows.First, the existing empirical processes based tests for drift function are either only for simple null hypothesis or not (asymptotically) distribution free for com-posite null hypothesis, some tests for drift function even rely on the specification of volatility function. Due to these facts, we develop two parametric specification tests for drift functions in diffusion models. Based on the kernel estimation of volatility, the tests for drift functions do not impose any restrictions on the functional form of volatility functions. We also provide theoretically the limiting behavior of the two tests under null, and find that they have the same distribution even under different null hypotheses. The following three categories of tests also share this favorable feature.Second, since lots of the existing tests for volatility function involve nonpara-metric smooth technique, their testing results depend on the choice of smooth pa-rameters; although some of them doesn’t involve nonparametric smooth technique, they are not (asymptotically) distribution free; some other tests for volatility func-tion even rely on the specification of drift function. Due to these facts, we develop two parametric specification tests for volatility functions in diffusion models. Simi-larly, the tests of volatility function impose no restriction on the functional form of the drift function. They are consistent against a large class of alternatives and have nontrivial power against a class of root-n local alternatives. Furthermore, unlike existing methods, in the volatility tests we need not select smoothing parameter as we use certain specific features in diffusion functions.Third, the existing joint tests for drift function and volatility function are main-ly based on the marginal density (distribution) or transition density (distribution) function, and the moment conditions; most of them involve the choice of nonpara-metric smooth parameter. Due to these facts, we will propose the first empirical process based joint tests for drift function and volatility function. At first, the asymptotic relation between the empirical processes reflecting drift and volatility respectively is analyzed, and then two joint tests are constructed. The asymptotic behavior of these two joint tests is also given. The proposed joint tests doesn’t involve the choice of nonparametric smooth parameters.Fourth, although there are many forms of jump tests in literature, none of them is based on empirical processes. Due to these facts, we will develop the first jump tests based on empirical processes. The proposed jump tests incorporate the bipower variation estimation, and are robust to model specifications. They are shown to be consistent against the presence of given jumps and have nontrivial power against a class of fourth-root-n local alternatives.In the above four type tests, the joint tests of drift and volatility are based on the approach of Bai and Chen (2008), while the other three type tests are Kolmogorov-Smirnov type and Cramer-von-Mises type. All of them incorporate the method of martingale transformation and are asymptotically distribution-free. Meanwhile, the numerical computation approach of our tests have been pro-vided. Monte Carlo simulations are conducted to examine the finite sample perfor-mance of tests. To get an explicit comparison, we also compute some existing related tests for each category of tests. Monte Carlo simulations indicate that our proposed tests perform well at both size and power in finite samples. Compared with existing tests, our tests make several improvements and are easier to implement.The proposed tests are then applied to empirical analyses in financial markets. First, we analyze the mean-reverting characteristics of SHIBOR using the proposed drift tests, and the volatility characteristics and model dynamics of Treasury bill rate and Eurodollar deposit rate using the proposed volatility tests and joint tests. The results show that the misspecification of short interest model is mainly due to the volatility function misspecification, and drift misspecification is secondary compared to volatility misspecification. Second, we empirically study the jump behavior of stock market using the proposed jump tests. The results show that the probability of jump occurrences in stock index is smaller than that in individual stocks; more jumps can be detected when sampling data more frequently. Compared with the testing results of other jump tests, our proposed jump tests can get more reasonable results. The empirical performances of each compared test is similar to that of Monte Carlo simulation performances, which are also consistent with the theoretical properties of each test.