Estimation of AR (2) Model with Dependent Errors for Unbounded Stationary and Nonstationary Time Series

In this paper the GLS and the ML estimators, the variance-covariance matrix, the unbiased for the GLS and the ML estimators of parameters of AR (2) model with constant in case of dependent errors have been derived, the simulation results shown that the values of MSE and Thiel’s U in case of unbounded stationary time series for all sample size T are less than the values of MSE and Thiel’s U in case of unbounded nonstationary time series which approved that the results for unbounded stationary times series are better than the results for unbounded nonstationary times series, and the simulation results for unbounded nonstationary time series shown that by using the measurement of MSE the best case among of all cases of nonstationary which gives the smallest values of MSE is case four when the first and the second conditions of stationary conditions for AR (2) model are exists, while by using the measurement of Thiel’s U the best case among of all cases of nonstationary which gives the smallest values of Thiel’s U is case six when the second and the third conditions of stationary conditions for AR (2) model are exists.

Benchmarking Nonstationary Time Series Prediction

The prediction of time series has gained increasingly more attention among researchers since it is a crucial aspect of decision-making activities. Unfortunately, most time series prediction methods assume the property of stationarity, i.e., statistical properties do not change over time. In practice, it is the exception and not the rule in most real datasets. Several transformation methods were designed to treat nonstationarity in time series. In this context, nonstationary time series prediction is challenging since it demands knowledge of both data transformation and prediction methods. Since there are no silver bullets, it leads to exploring a large number of data transformation and prediction method combinations for building prediction setups. However, selecting a prediction setup that is appropriate to a particular time series and application is not a simple task. Benchmarking of different candidate combinations helps this selection. This work contributes by providing a review and experimental analysis of transformation methods and a systematic framework (TSPred) for benchmarking and selecting prediction setups for nonstationary time series. Suitable nonstationary time series transformation methods provided improvements of more than 30% in prediction accuracy for half of the evaluated time series. They improved the prediction by more than 95% for 10% of the time series. The features provided by TSPred are also shown to be competitive regarding prediction accuracy. Furthermore, the adoption of a validation phase during model training enables the selection of suitable transformation methods.

Spurious Relationships for Nearly Non-Stationary Series

Literature shows that the regression of independent and (nearly) nonstationary time series could result in spurious outcomes. In this paper, we conjecture that under some situations, the regression of two independent and nearly non-stationary series does not have any spurious problem at all. To check whether our conjecture holds, we set up several situations and conduct simulations to justify our conjecture. Our simulations show that under some situations, the chance that the regressions being spurious is very high for all the cases simulated in our paper. Nonetheless, under some other situations, our simulation shows that the rejection rates are much smaller than the 5% level of significance for all the cases simulated in our paper, implying that our conjecture could hold under some situations that regression of two independent and nearly non-stationary series does not have any spurious problem at all.