TOWARD A UNIFORM ASYMPTOTIC THEORY FOR MILDLY EXPLOSIVE AUTOREGRESSION

Boosting nonlinear additive autoregressive time series

Thanks to their simplicity and interpretable structure, autoregressive processes are widely used to model time series data. However, many real time series data sets exhibit non-linear patterns, requiring nonlinear modeling. The threshold Auto-Regressive (TAR) process provides a family of non-linear auto-regressive time series models in which the process dynamics are specific step functions of a thresholding variable. While estimation and inference for low-dimensional TAR models have been investigated, high-dimensional TAR models have received less attention. In this article, we develop a new framework for estimating high-dimensional TAR models, and propose two different sparsity-inducing penalties. The first penalty corresponds to a natural extension of classical TAR model to high-dimensional settings, where the same threshold is enforced for all model parameters. Our second penalty develops a more flexible TAR model, where different thresholds are allowed for different auto-regressive coefficients. We show that both penalized estimation strategies can be utilized in a three-step procedure that consistently learns both the thresholds and the corresponding auto-regressive coefficients. However, our theoretical and empirical investigations show that the direct extension of the TAR model is not appropriate for high-dimensional settings and is better suited for moderate dimensions. In contrast, the more flexible extension of the TAR model leads to consistent estimation and superior empirical performance in high dimensions.

A goodness-of-fit test of the errors in nonlinear autoregressive time series models

Strong consistency of the distribution estimator in the nonlinear autoregressive time series

Noise in unspecified, non-linear time series

Noise in unspecified, non-linear time series

While lacking previous knowledge, general users often have difficulty in using nonlinear statistical time series methods (e.g., bilinear model, threshold autoregressive model, smoothing transition autoregressive model, autoregressive conditional heteroscedastic model and generalized autoregressive conditional heteroscedastic model) and spectral analytical approaches. Moreover, standard back-propagation neural networks (BPNNs) using the steepest descent algorithm have the disadvantage that they converge to local optima. To overcome the above limitations, this work develops a self-organizing polynomial neural network (SOPNN) based on the group method of data handling (GMDH) algorithm, which is a statistical learning network. The proposed SOPNN scheme can create an optimal network topology by using an evolutionary process and is easily to be used. Performance of the proposed SOPNN scheme is also measured by using the Mackey-Glass chaotic time series problem. Additionally, numerical results of the proposed SOPNN scheme are compared with those of conventional BPNN, cerebellar model articulation controller NN (CMAC NN), advanced simulated annealing-based BPNN (ASA-BPNN) and artificial immune algorithm-based BPNN (AIA-BPNN). Experimental results indicate that the optimum training and test capabilities of the proposed SOPNN are superior to those of a conventional BPNN. Furthermore, the generalization capability and computational CPU time of the proposed SOPNN scheme are competitive for those of CMAC NN, AIA-BPNN and ASA-BPNN.

Nonlinear time series are time series that are not stable due to a sudden jump. Nonlinear time series often found in financial data. Threshold Autoregressive (TAR) modeling is a time series modeling with a segmented autoregressive (AR)’s model such that among different regimes may have different AR model. This research studied how to obtain the Ordinary Least Square (OLS) estimator for TAR model and examine signification the OLS’s estimator by using t test. This research also studied the other stages of TAR modeling, which are nonlinearity test using Tsay test, TAR model identification by using arranged AR approach and Akaike’s Information Criterion (AIC), and diagnostic test by examining the white noise properties and normality test on the residuals. As an illustration, the TAR modeling was applied on weekly data of rupiah exchange rate against US dollar for period October 4th 2004 to November 7th 2011. The result show that the best TAR model for the data is TAR with threshold value .

The main purpose of this dissertation is to compare the in-sample estimating and out-of-sample forecasting performance of a set of non-linear time series models, i.e., threshold autoregressive models, momentum threshold autoregressive models, exponential autoregressive models, generalized autoregressive models and bilinear models. First, a Monte Carlo simulation is used to study overfitting and forecasting. For AR processes, if the AIC and SBC criteria are used to select models, the possibility of overfitting is very high since the non-linear models and other linear models are very likely to have lower AIC and SBC. However, the MSPE for one-step ahead out-of-sample forecast can be used to identify the true AR processes. For TAR processes, the AIC can be used to identify the TAR-C models. The SBC and MSPE can identify the TAR process only if the difference of the persistence between the two regimes is large enough. Underfitting and misspecification are very likely to happen for a TAR process with small difference of the persistence between the two regimes. However, if we don't know the true AR or TAR process, the MSPE can't select the AR or TAR models in most cases. Thus, none of the AIC, SBC and MSPE can select the AR model for a given AR process with unknown order. For the TAR process, the AIC can consistently identify the TAR-C process and the SBC can identify the TAR-C process only if the difference of the persistency is large enough. Then, a set of linear and non-linear time series models are applied to the term structure of interest rates and the spread of wholesale and retail pork prices in U.S. It is shown that there are non-linear time series models can do better than the conventional ARMA models for both in-sample estimation and out-of-sample forecast. Also, it is very unlikely that the dominance of the non-linear time series models results from overfitting for both the term structure of interest rates and the spread of wholesale and retail pork prices in U.S. Thus, non-linear time series models are very useful for estimating and forecasting the non-linear time series.

This work proposes a method to find the set of the most influential lags and the rule structure of a Takagi---Sugeno---Kang (TSK) fuzzy model for time series applications. The proposed method resembles the techniques that prioritize lags, evaluating the proximity of nearby samples in the input space using the closeness of the corresponding target values. Clusters of samples are generated, and the consistency of the mapping between the predicted variable and the set of candidate past values is evaluated. A TSK model is established, and possible redundancies in the rule base are avoided. The proposed method is evaluated using simulated and real data. Several simulation experiments were conducted for five synthetic nonlinear autoregressive processes, two nonlinear vector autoregressive processes and eight benchmark time series. The results show a competitive performance in the mean square error and a promising ability to find a proper set of lags for a given autoregressive process.

In this paper, a one-step forecasting comparison using a simulated nonlinear autoregressive moving-average time series (NARMA) was conducted between two groups of neural networks. Group I is neural networks that use only autoregressive inputs, while Group II is neural networks that use autoregressive and moving-average (i.e., error feedback) inputs. Simulation results showed that the models in Group II produce more accurate forecasts as compared to the models in Group I. That means, introducing error feedback to neural networks helps in forecasting NARMA time series. Another comparison was conducted between autoregressive moving-average (ARMA) model and neural network models using the simulated NARMA time series. As expected, since it is a nonlinear time series, neural networks show better results as compared to ARMA model.

Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes

Apples constitute the bulk of deciduous fruit produced in South Africa, i.e. in 2000, apples made up the largest percentage of the deciduous fruit crop (43%). From 991/92 to 2002/03 production averaged 574 850 tons per annum with a standard deviation of 43 922 tons. The average distribution of the apple crop between the local market, exports and processing is more or less even. Because of its potential lucrative nature much emphasis in the apple industry is afforded to exports, but relatively little is known about how price transmission takes place on the domestic fresh produce markets (FPMs). Moreover, it is increasingly recognized that the formulation of market-enhancing policies to increase the performance of the local market requires a better understanding of how the market functions. Aggregate market performance is better understood by studying the level of market integration that exists, which in turn is affected by transaction costs in the value chain. Hence, the primary objective of this study was to measure market integration for apples on the South African FPMs to determine the existence of long-run price relationships and spatial market linkages. Specific issues addressed in this study include, (i) determination of the effect of deregulation of the marketing of agricultural products in 1997 on average real market prices, price spread and volatility (risk), (ii) determination of how FPMs where apples are sold are linked and how prices are transmitted across these markets, (iii) determination of the threshold prices beyond which markets adjust and return to equilibrium, and (iv) establish the response of the FPMs to price shocks and how long it takes for shocks to be eliminated. The FPMs included in this study are Johannesburg, Cape Town, Tswhane, Bloemfontein, Port Elizabeth, Durban, Kimberley and Pietermaritzburg. The criteria for selecting the FPMs were based on net market positions (surplus or deficit area), geographical distribution, the volume of trade and the importance of the market to the national apple trade flow. The investigation revealed a statistically significant decline in real prices in six of the eight markets investigated, a statistically significant relation in prices (price spread) between the Johannesburg FPM and five other FPMs, as well as that the price spreads between these markets declined after deregulation, and that the variation in real apple prices declined for five of the eight markets after deregulation. Standard autoregressive (AR) and threshold autoregressive (TAR) error correction models were compared to determine whether transaction cost has significant effects in measuring market integration. Larger adjustment coefficients were found in the TAR model. This is an indication that price adjustments are faster in threshold autoregressive TAR models than in AR models. Also half-life deviations in the TAR model are much smaller than in the AR model. The TAR model requires less time for one-half of the deviation from equilibrium to be eliminated than the standard AR model. Therefore, it is better to use TAR models than AR models because TAR models give a more reliable result. In addition, the parameter estimates of the threshold vector error correction model were analyzed. The results show that bidirectional and unidirectional causality exist between Johannesburg FPM prices and other markets. Regime switching estimates to investigate market integration in the selected markets show that no persistent deviation from equilibrium existed for all but one market pair and no clear evidence was found to support improved market integration after market deregulation in 1997. A nonlinear impulse response function to investigate the impact of positive and negative price shocks in the Johannesburg FPM on other FPMs revealed that it takes about six to twelve months for positive and negative shocks to be completely eliminated in all the markets. Generally, the results obtained confirmed strong market integration in terms of apples for selected FPMs.

There are many parameters which are very difficult to calibrate in the threshold autoregressive prediction model for nonlinear time series. The threshold value, autoregressive coefficients, and the delay time are key parameters in the threshold autoregressive prediction model. To improve prediction precision and reduce the uncertainties in the determination of the above parameters, a new DNA (deoxyribonucleic acid) optimization threshold autoregressive prediction model (DNAOTARPM) is proposed by combining threshold autoregressive method and DNA optimization method. The above optimal parameters are selected by minimizing objective function. Real ice condition time series at Bohai are taken to validate the new method. The prediction results indicate that the new method can choose the above optimal parameters in prediction process. Compared with improved genetic algorithm threshold autoregressive prediction model (IGATARPM) and standard genetic algorithm threshold autoregressive prediction model (SGATARPM), DNAOTARPM has higher precision and faster convergence speed for predicting nonlinear ice condition time series.

Forecasting for nonlinear time series is an important topic in time series analysis. Existing numerical algorithms for multi‐step‐ahead forecasting ignore accuracy checking, alternative Monte Carlo methods are also computationally very demanding and their accuracy is difficult to control too. In this paper a numerical forecasting procedure for nonlinear autoregressive time series models is proposed. The forecasting procedure can be used to obtain approximate m‐step‐ahead predictive probability density functions, predictive distribution functions, predictive mean and variance, etc. for a range of nonlinear autoregressive time series models. Examples in the paper show that the forecasting procedure works very well both in terms of the accuracy of the results and in the ability to deal with different nonlinear autoregressive time series models. Copyright © 2005 John Wiley & Sons, Ltd.