A Powerful Portmanteau Test for Detecting Nonlinearity in Time Series

A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. In this paper, we elaborate on the Toeplitz autocorrelation matrix to the autocorrelation and cross-correlation of residuals and squared residuals block matrix. We derive a new portmanteau test statistic using the log of the determinant of the mth autocorrelations and cross-correlations block matrix. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-squared distributions and can be approximated by a gamma distribution. This test is applied to identify the linearity and nonlinearity dependency of some stationary time series models. It is shown that the convergence of the new test to its asymptotic distribution is reasonable with higher power than other tests in many situations. We demonstrate the efficiency of the proposed test by investigating linear and nonlinear effects in Vodafone Qatar and Nikkei-300 daily returns

L1 methods for shrinkage and correlation

This dissertation explored the idea of L1 norm in solving two statistical problems including multiple linear regression and diagnostic checking in time series. In recent years L1 shrinkage methods have become popular in linear regression as they can achieve simultaneous variable selection and parameter estimation. Their objective functions containing a least squares term and an L1 penalty term which can produce sparse solutions (Fan and Li, 2001). Least absolute shrinkage and selection operator (Lasso) was the first L1 penalized method proposed and has been widely used in practice. But the Lasso estimator has noticeable bias and is inconsistent for variable selection. Zou (2006) proposed adaptive Lasso and proved its oracle properties under some regularity conditions. We investigate the performance of adaptive Lasso by applying it to the problem of multiple undocumented change-point detection in climate. Artificial factors such as relocation of weather stations, recalibration of measurement instruments and city growth can cause abrupt mean shifts in historical temperature data. These changes do not reflect the true atmospheric evolution and unfortunately are often undocumented due to various reasons. It is imperative to locate the occurrence of these abrupt mean shifts so that raw data can be adjusted to only display the true atmosphere evolution. We have built a special linear model which accounts for long-term temperature change (global warming) by linear trend and is featured by p = n (the number of variables equals the number of observations). We apply adaptive Lasso to estimate the underlying sparse model and allow the trend parameter to be unpenalized in the objective function. Bayesian Information Criterion (BIC) and the CM criterion (Caussinus and Mestre, 2004) are used to select the finalized model. Multivariate t simultaneous confidence intervals can post-select the change-points detected by adaptive Lasso to attenuate overestimation. Considering that the oracle properties of adaptive Lasso are obtained under the condition of linear independence between predictor variables, adaptive Lasso should be used with caution since it is not uncommon for real data sets to have multicollinearity. Zou and Hastie (2005) proposed elastic net whose objective function involves both L1 and L2 penalties and claimed its superiority over Lasso in prediction. This procedure can identify a sparse model due to the L1 penalty and can tackle multicollinearity due to the L2 penalty. Although Lasso and elastic net are favored over ordinary least squares and ridge regression because of their functionality of variable selection, in presence of multicollinearity ridge regression can outperform both Lasso and elastic net in prediction. The salient point is that no regression method dominates in all cases (Fan and Li, 2001, Zou, 2006, Zou and Hastie, 2005). One major flaw of both Lasso and elastic net is the unnecessary bias brought by constraining all parameters to be penalized by the same norm. In this dissertation we propose a general and flexible framework for variable selection and estimation in linear regression. Our objective function automatically allows each parameter to be unpenalized, penalized by L1, L2 or both norms based on parameter significance and variable correlation. The resulting estimator not only can identify the correct set of significant variables with a large probability but also has smaller bias for nonzero parameters. Our procedure is a combinatorial optimization problem which can be solved by exhaustive search or genetic algorithm (as a surrogate to computation time). Aimed at a descriptive model, BIC is chosen as the model selection criterion. Another application of the L1 norm considered in this dissertation is portmanteau tests in time series. The first step in time series regression is to determine if significant serial correlation is present. If initial investigations indicate significant serial correlation, the second step is to fit an autoregressive moving average (ARMA) process to parameterize the correlation function. Portmanteau tests are commonly used to detect serial correlation or assess the goodness-of-fit of the ARMA model in these two steps. For small samples the commonly employed Ljung-Box portmanteau test (Ljung and Box, 1978) can have low power. It is beneficial to have a more powerful small sample test for detecting significant correlation. We develop such a test by considering the Cauchy estimator of correlation. While the usual sample correlation is estimated through L2 norm, the Cauchy estimator is based on L1 norm. Asymptotic properties of the test statistic are obtained. The test compares very favorably with the Box-Pierce/Ljung-Box statistics in detecting autoregressive alternatives

Toeplitz Matrix-Based Goodness of Fit Test Statistics for Vector Autoregressive Moving Average Models

In multivariate time series analysis, a Vector Autoregressive Moving Average model would be fitted to the data. Then, diagnostic checking methods are used to assess the goodness of fit for the fitted model. Different multivariate Portmanteau goodness of fit tests had been proposed for multivariate time series analysis. However, these previous tests suffer from low power in many situations, such as small sample sizes. To overcome this, a modified measure of autocorrelation was recently proposed by Fisher and Robbins (2018) using a logarithmic transformation of the determinant of a Toeplitz matrix that contains the multivariate correlations matrices. This new measure of serial correlation improves the power performance of the goodness of fit test statistic while maintaining the same asymptotic distribution under the null hypothesis. In this dissertation, we proposed two Portmanteau test statistics that employ the determinant and the trace of a Toeplitz matrix containing the improved measure of correlation. The asymptotic distributions of each presented test statistic was derived. Also, a simulation study was provided to explore the power performance of the proposed Portmanteau tests. A Monte Carlo method was used to calculate the empirical power in this simulation study. The new trace-based Portmanteau test statistic offered improvements in the power performance over existing tests. On the other hand, the determinant-based test statistic showed good power behavior in the case of moderate and large sample sizes

Monk business: an example of the dynamics of organizations.

In this paper we present a dynamic model of an organization. It is shown that the quality of the members of the organization may cycle and that even if the organization promotes excellency, the organization may end up populated by mediocre agents only.Overlapping generations; Quanty organization;

The changing spatial distribution of economic activity across U.S. counties.

This paper studies the recent trends in the spatial distribution of economic activity in the United States. Using county-level employment data for 13 sector -which cover the entire economy- we apply semi-parametric techniques to estimate how agglometarion and congestion effects have changed between 1972 and 1992. Non-service sectors are found to be spreading out and moving away from centers of high economic activity to areas 20 to 60 kilometers away; service sectors, on the contrary, are increasingly concentrating in areas of high economic activity by attracting jobs from the surrounding 20 kilometers.Economic geography; Spatial externalities; U.S. counties;

Monk business: an example of the dynamics of organizations

In this paper we present a dynamic model of an organization. It is shown that the quality of the members of the organization may cycle and that even if the organization promotes excellency, the organization may end up populated by mediocre agents only

The changing spatial distribution of economic activity across U.S. counties

This paper studies the recent trends in the spatial distribution of economic activity in the United States. Using county-level employment data for 13 sector -which cover the entire economy- we apply semi-parametric techniques to estimate how agglometarion and congestion effects have changed between 1972 and 1992. Non-service sectors are found to be spreading out and moving away from centers of high economic activity to areas 20 to 60 kilometers away; service sectors, on the contrary, are increasingly concentrating in areas of high economic activity by attracting jobs from the surrounding 20 kilometers

Goodness of fit tests and lasso variable selection in time series analysis

This thesis examines various aspects of time series and their applications. In the rst part, we study numerical and asymptotic properties of Box-Pierce family of portmanteau tests. We compare size and power properties of time series model diagnostic tests using their asymptotic c2 distribution and bootstrap distribution (dynamic and fixed design) against various linear and non-linear alternatives. In general, our results show that dynamic bootstrapping provides a better approximation of the distribution underlying these statistics. Moreover, we find that Box-Pierce type tests are powerful against linear alternatives while the CvM due to Escanciano (2006b) test performs better against non linear alternative models. The most challenging scenario for these portmanteau tests is when the process is close to the stationary boundary and value of m, the maximum lag considered in the portmanteau test, is very small. In these situations, the c2 distribution is a poor approximation of the null asymptotic distribution. Katayama (2008) suggested a bias correction term to improve the approximation in these situations. We numerically study Katayama's bias correction in Ljung and Box (1978) test. Our results show that Katayama's correction works well and conrms the results as shown in Katayama (2008). We also provide a number of algorithms for performing the necessary calculations efciently. We notice that the bootstrap automatically does bias correction in Ljung-Box statistic. It motivates us to look at theoretical properties of the dynamic bootstrap in this context. Moreover, noticing the good performance of Katayama's correction, we suggest a bias correction term for the Monti (1994) test on the lines of Katayama's correction. We show that our suggestion improves Monti's statistic in a similar way to what Katayama's suggestion does for Ljung-Box test. We also make a novel suggestion of using the pivotal portmanteau test. Our suggestion is to use two separate values of m, one a large value for the calculation of the information matrix and a smaller choice for diagnostic purposes. This results in a pivotal statistic which automatically corrects the bias correction in Ljung-Box test. Our suggested novel algorithm efciently computes this novel portmanteau test. In the second part, we implement lasso-type shrinkage methods to linear regression and time series models. We look through simulations in various examples to study the oracle properties of these methods via the adaptive lasso due to Zou (2006). We study consistent variable selection by the lasso and adaptive lasso and consider a result in the literature which states that the lasso cannot be consistent in variable selection if a necessary condition does not hold for the model. We notice that lasso methods have nice theoretical properties but it is not very easy to achieve them in practice. The choice of tuning parameter is crucial for these methods. So far there is not any fully explicit way of choosing the appropriate value of tuning parameter, so it is hard to achieve the oracle properties in practice. In our numerical study, we compare the performance of k-fold cross-validation with the BIC method of Wang et al. (2007) for selecting the appropriate value of the tuning parameter. We show that k-fold crossvalidation is not a reliable method for choosing the value of the tuning parameter for consistent variable selection. We also look at ways to implement lasso-type methods time series models. In our numerical results we show that the oracle properties of lasso-type methods can also be achieved for time series models. We derive the necessary condition for consistent variable selection by lasso-type methods in the time series context. We also prove the oracle properties of the adaptive lasso for stationary time series

Persistence and Anti-persistence: Theory and Software

Persistent and anti-persistent time series processes show what is called hyperbolic decay. Such series play an important role in the study of many diverse areas such as geophysics and financial economics. They are also of theoretical interest. Fractional Gaussian noise (FGN) and fractionally-differeneced white noise are two widely known examples of time series models with hyperbolic decay. New closed form expressions are obtained for the spectral density functions of these models. Two lesser known time series models exhibiting hyperbolic decay are introduced and their basic properties are derived. A new algorithm for approximate likelihood estimation of the models using frequency domain methods is derived and implemented in R. The issue of mean estimation and multimodality in time series, particularly in the simple case of one short memory component and one hyperbolic component is discussed. Methods for visualizing bimodal surfaces are discussed. The exact prediction variance is derived for any model that admits an autocovariance function and integrated (inverse-differenced) by integer d. A new software package in R, arfima, for exact simulation, estimation, and forecasting of mixed short-memory and hyperbolic decay time series. This package has a wider functionality and increased reliability over other software that is available in R and elsewhere

Predicting changes in the output of OECD countries: An international network perspective

We use a simple linear regression framework to present evidence, that complex relationships between stock markets and economies may be used to predict changes in the output of 27 OECD countries. We construct new unidirectional return co-exceedance networks to account for complex relationships between stock market returns, and between real economic growths. Although there is heterogeneity between individual country level results, overall our data and analysis provides evidence that topological properties of our networks are useful for in-sample prediction of next quarter changes in the output