#### Degree

Doctor of Philosophy

#### Program

Statistics and Actuarial Sciences

#### Supervisor

Dr. A. Ian McLeod

#### Abstract

Portmanteau test serves an important role in model diagnostics for Box-Jenkins Modelling procedures. A large number of Portmanteau test based on the autocorrelation function are proposed for a general purpose goodness-of-fit test. Since the asymptotic distributions for the statistics has a complicated form which makes it hard to obtain the p-value directly, the gamma approximation is introduced to obtain the p-value. But the approximation will inevitably introduce approximation errors and needs a large number of observations to yield a good approximation. To avoid some pitfalls in the approximation, the Lin-Mcleod Test is further proposed to obtain a numeric solution to this problem based on Monte Carlo Simulation.

In this thesis, we first identify the problem of nuisance parameters for Autoregressive Fractionally Integrated Moving Average Model (ARFIMA model) in the Lin-McLeod Test; the size would be distorted, leading to an inaccurate level of type I error. We solve the problem through a modification of Lin-McLeod Test: Wild Monte Carlo Test, borrowing the idea of Wild Dependent Bootstrapping. In order to validate the algorithm, we derive the asymptotic distribution for the bootstrapped statistics in ARFIMA cases. By perturbing the estimated residuals, the Wild Monte Carlo Test outperforms a wide spread of Portmanteau test for this type of model. It solves the problem of the size underestimation and improves the test power for ARFIMA cases.

Later, we consider the general variance Portmanteau test on Autoregressive Moving Average Model with Generalized Autoregressive Conditional Heteroskedasticity Error (ARMA − GARCH Model), as a special case of weak ARMA model . When we have the null hypothesis of an ARMA − GARCH process, the asymptotic distribution of general variance are derived. With the complicated structure of the asymptotic distribution, the Lin-McLeod Test can serve a better solution to obtain the p-value rather than the gamma approximation. However, the test will still suffer from the size distortion due to the nuisance parameter issues. In this chapter, we mainly derive the asymptotic distribution for general variance Portmanteau tests on the ARMA − GARCH models and propose to use the Wild Monte Carlo Test to reduce the effect of nuisance parameters. The simulation and practical examples show the power of the newly test compared to the results of Francq et al. (2005).

Additionally, we shall consider the data generating process as the ARMA with infinite variance errors. In order to valid the general variance Portmanteau test and Fisher-Gallagher Test under this setup, we use the idea of the autocorrelation of the trimmed time series to construct the modified Portmanteau test and derive the asymptotic distribution for these two kinds of Portmanteau tests on trimmed time series. Still, when we use the Lin-McLeod Test to obtain the p-value, Wild Monte Carlo Test can correct the nuisance parameter distortion for the size and improve the model performance.

Finally, we revisit the cross correlation test for two independent time series. A mistake in Hong (1996) simulation is pointed out and the corrected size for the test is provided later. Then we identify a spurious correlation for the time series with GARCH type errors.

#### Recommended Citation

Xiao, Jinkun, "Advances in Portmanteau Diagnostic Tests" (2016). *Electronic Thesis and Dissertation Repository*. 4100.

https://ir.lib.uwo.ca/etd/4100

#### Included in

Applied Statistics Commons, Longitudinal Data Analysis and Time Series Commons, Statistical Methodology Commons