Portmanteau Tests For Univariate And Multivariate Time Series Models
This thesis examines how the number of available observations of a time series can influence its apparent stationarity as measured by two standard tests, namely the standard Dickey-Fuller (DF) test and the Augmented Dickey-Fuller (ADF) test. The univariate time series case is examined. A stationary time series generated from a first-order autoregressive process with positive or negative values of the parameter 𝜙. Parameters were chosen that ensured that the series were theoretically stationary. The resulting time series produced were examined by, the DF, ADF, DF drift, ADF drift, DF trend and ADF trend tests. Monte Carlo experiments were undertaken using the R program for various values of parameters and different lengths of data, with each simulation repeated 10,000 times. The simulation studies show that the length of time series data affects the stationarity as identified by standard tests. For given values of the parameter 𝜙 of the first-order autoregressive model the minimum length of time series required to ensure the correct identification of the series’ stationarity is presented. Two new portmanteau tests were developed, bases on exponential weights of the residual autocorrelation function and the residual partial autocorrelation function. The asymptotic distributions of the new univariate portmanteau tests were derived. Monte Carlo experiments were used to compare the performance of the two new tests to existing tests. The simulation studies show that one of the new portmanteau tests, which is based on the partial autocorrelation function, is statistically more powerful than previous tests. A new portmanteau test was developed for vector autoregressive moving average models, which is based on exponential weights of the residual covariance matrix. For this new multivariate portmanteau test the asymptotic distribution was derived. This new test was compared with previous tests using Monte Carlo experiments. The simulation study shows that the new multivariate portmanteau test is statistically more powerful than previous tests.
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