The analysis of variance (ANOVA) is a statistical method for assessing the impact of multiple factors and their interactions when there are three or more factors. The method was first developed by R. A. Fisher in the 1910s and since then has been studied extensively by many authors. In the case of i.i.d. data, most literature has focused on the settingwhere the number of groups (a) and the number of observations in each group (n) are small, referred to as fixed-a and -n asymptotics. ANOVA for time series data, commonly referred to as longitudinal or panel data analysis, has been extensively studied in econometrics. In this field, large-a and fixed-n asymptotics or large-a and -n asymptotics are commonly examined, with a primary focus on the regression coefficient. Consequently, results regarding the existence of fixed and random effects of factors have been hardly developed. This monograph aims to present the recent developments related to one- and twoway models mainly for time series data under the framework of fixed-a and large-n asymptotics. Especially,we focus on (i) the testing problems for the existence of fixed and random effects of factors and interactions among factors under various settings, including uncorrelated and correlated groups, fixed and random effects, multi- and high-dimension, parametric and nonparametric spectral densities, and (ii) the local asymptotic normality (LAN) property for one-way models on i.i.d. data. This book is suitable for statisticians and economists as well as psychologists and data analysts. Figure 1 illustrates the relationships between the chapters. In Chapter 1, a historical overview of ANOVA and the fundamentals of time series analysis are provided, along with motivation and concise summary of the content covered in the book. Chapter 2 examines a test for the presence of fixed effects in the one-way model with independent groups. Chapter 3 extends the analysis to high-dimensional settings.Chapters 4 and 5 address correlated groups in one-way and two-waymodels, respectively. Lastly, Chapter 6 explores the log-likelihood ratio process to construct optimal tests in the context of i.i.d. settings.
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