The difficulty is that lags of the same variables enter in different dimensions of the processes. Here we discuss in detail the case , and results for follow by simply fixing. Generalization for can be done along the same lines; however, this is a lengthy process and is thus omitted. Note that the case is the most relevant for practical purposes. We start with a simple case in which parameters are known, and then we study how the asymptotic distribution changes if parameters are estimated.
We provide analysis under the null, under local and fixed alternatives. We impose assumptions on conditional cdf, the form of parametric family of cdf, dynamics and on the estimator. Assumption 3. The conditional distributions are continuous and strictly increasing in y for all. We first describe the asymptotic behaviour of the process under H 0. Proposition 3. Suppose Assumption 3. Suppose the conditional distribution function is not in the parametric family , i. For any define the conditional cdf and the local alternatives.
We extend weak convergence in Proposition 3. In particular, for some denote. For simplicity, in the following, we assume no information truncation, , omit and write. This requirement can be relaxed, e.
We need the following assumption. We impose two types of restrictions. The first is the smoothness of the distributions with respect to parameters and data. Similar assumptions have been used previously in statistical literature e. Loynes, The second is the convergence in probability of certain averages, which implicitly imposes restrictions on the data dynamics and can be established by means of the uniform law of large numbers ULLN. This part can also be verified directly given a particular model see, e.
Therefore, the following Assumption 3. Conditions similar to Assumption 3. For and also in the i. For , we need an additional condition for h 2. In the i. In a stationary and ergodic case, Assumption 3. The term will appear in the expansion of around and will reflect the parameter estimation effect. Hence, to identify the limit of we need to study the limiting distribution of random vector.
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We make assumptions on the estimation procedure. Under , the estimator admits a linear expansion 3. It will allow us to apply the central limit theorem CLT for random vector. Define and let be a zero mean Gaussian process with covariance function. The following proposition establishes the limiting distribution of our test statistics. Suppose Assumptions 3. Then under H 0 where. The next proposition provides results on asymptotic distribution under the local alternatives. The conditional cdfs are continuous and strictly increasing in y. Then, under where.
Under , are no longer i. Because of this, we have a drift in the asymptotic distribution. Let us now examine this drift more closer. If we fix , we obtain the drift of the process equal. This drift might be zero even if and are different. If they differ only by the conditioning set, then averaging may smooth away this difference. The extreme is in the case of elliptical distribution, where after integrating out one variable from the conditioning set we are still in the same family, so the drift is zero.
As an example, consider testing the AR 1 model against AR 2 , both with standard normal innovations. In other words, In this example,. Assume now that the true DGP is. As before, denote the true distribution and the null distribution with a small abuse of notation because both depend on. We first show that are uniform but not independent , and hence the unconditional expectation of is zero.
Weighted Empirical Processes in Dynamic Nonlinear Models | Hira L. Koul | Springer
Indeed, for some , where in the last equality we use the particular form of and and the property of Gaussian distribution that for chosen parameters. The drift for can be written as , where is The term gives additional power, by preventing the averaging out of dynamic misspecification. In the aforementioned extreme case of dynamic misspecification of elliptical distribution, this term alone delivers the power reported in our Monte Carlo simulations. In terms of , controls the dynamics of , while the other terms control uniformity. Under the alternative, we could also have left 3.
This term does not appear in methods that use projections, as in Bai Now, we discuss the consistency of the test against the fixed alternative. The following limit in probability exists for :. Moreover, under the null. Define the fixed alternative. H 1 : The conditional distribution function of is equal to , which is different from the assumed distribution in the following sense:. Under H 1 , diverges at least for some r , and therefore the KS test is consistent.
For the CvM test, we need at the set of r with a positive measure.
It might be the case that the null is violated, say , but see the example above. In this case, we can distinguish the alternative with but not with. It might also be the case that the null is violated, say , but.
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Tests based on a combination of for different j are consistent against a broader set of alternatives: they check marginals and pairwise structure of , but not serial structure. In theory, we need to aggregate all and , but this approach might not work well in finite samples. We now state the formal condition. Under an alternative, the estimator must converge in probability. Under the null together with Assumption 3.
Then, under H 1 for all sequences of random variables , we have. We show that bootstrap critical values are bounded both under the null and under alternative. Therefore, Proposition 3. We prove that has the same limiting distribution as. The proof is similar to Andrews ; we need to establish an analogue of his 4. The next proposition states that the asymptotic distribution of the test statistics with the bootstrapped data, or denoted shortly under , coincides with the prior result obtained under the null. Then, if ,. Randomness here comes both from the sample and the bootstrap simulations.
In turn, is approximated by when , where the latter is the critical value after B repetitions. Note that we need only , so the analysis holds under the null and under the alternative in the latter case with Assumption 3. For more details, see the discussion on pages — of Andrews In this section, we report the results of a Monte Carlo study to investigate the finite sample performance of the proposed tests. The number of Monte Carlo repetitions is set to To save space, we present here in detail a case of conditional mean misspecification in GARCH models, while other simulation results can be obtained upon request.
We also show how to apply our technique to models of stock exchange indexes. The models are estimated by the ML method. However, they may have no power against other alternatives. Because the goal of the paper is to test the null against a general alternative, parametric tests are not included in the comparison. First, we make evident the ability of our test to detect misspecification in both dynamics and marginal distributions, which are easy to introduce to these models and which are usually tested separately.
Second, the combination of the probability integral transform and residual empirical process is employed by Bai , who applies his tests to GARCH models.
We have discussed above the limitations of univariate residual empirical process tests and our goal is to verify that our approach indeed outperforms them. Thus, we compare the performance of our test with the tests based on a univariate process for the same null hypothesis and the same data.
In order to make tests comparable, we avoid the martingale transform and use bootstrapped versions of all tests. Our methodology applies without modification to all numerous extensions of the basic GARCH model, including models with asymmetry, leverage, higher moments and nonlinear moment dynamics. This example is motivated by the findings in our application for the tests on real data see below.
In Figures 1 b and 1 d , are changed to. Sample sizes are in Figures 1 a and 1 b , and in Figures 1 c and 1 d. The test has power against all alternatives. We also show results for dashed lines with circle markers with on the same plot. They fill monotonically the space from to with closest to. The performance of these tests is decreasing with j. Because the misspecification is in the first lag, most of the power comes from , capturing dependence between and , although in this case there is also dependence in further lags, for instance between and , , etc.
The more tests we aggregate, the less weight is given to the first lag and the less power we have, but we can capture a wider set of alternatives.
The same effect is observed with Ljung—Box tests. In the bottom plots, we repeat the experiment for. FAQ Policy. About this book The role of the weak convergence technique via weighted empirical processes has proved to be very useful in advancing the development of the asymptotic theory of the so called robust inference procedures corresponding to non-smooth score functions from linear models to nonlinear dynamic models in the 's. Show all. Asymptotic Properties of W. Autoregression Pages Koul, Hira L.
Appendix Pages Koul, Hira L. Bibliography Pages Koul, Hira L. Show next xx. Gourieroux, C. Hitczenko, P. Kawczak, J. Weak convergence of a certain class of residual empirical processes, Ph. Thesis, The University of Western Ontario. Google Scholar. Koul, H. Levental, S. A uniform CLT for uniformly bounded families of martingale differences, Journal of Theoretical Probability , 2 , — Pollard, D.
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