An elementary introduction to mathematical finance

where the concepts are applied. The author could also put these six chapters in an appendix. Graduate students should be familiar with the most of the basic concepts. Full-rank linear models are introduced in Chapter 6; this material is the cornerstone of the whole book. The concepts of the least-squares estimator, its statistical properties, and test statistics are introduced. Some examples of how to use the regression models are provided at the end. This chapter is concise and clear. However, some key topics of linear models are not included, such as the discussion of basic assumptions of linear models, goodness-of-fit, and asymptotic properties of the least-squares estimators. Discussion of such topics should be included. Chapter 7 discusses less than full-rank models, one case of violating basic assumptions of linear models. This chapter introduces the concepts of estimable linear functions, testable hypotheses, and confidence intervals on estimable linear functions of simultaneous inference. The discussion of other topics like heteroscedasticity, serial correlation, and the testing procedures for full-rank hypothesis would be appropriate here. Balanced linear models are covered in Chapter 8, where the key component is the discussion of fixed effect models. Sections 8.5–8.7, covering sufficient statistics and estimation of variance components, should be omitted. Instead, the idea of random effect models, the instrumental variables estimation of random effect models, and generalised method of moments (GMMs) estimation of dynamic panel data models should be discussed in this chapter. Specifically, GMM estimation of dynamic panel models is a very popular topic nowadays, and I am surprised that such an important topic is not covered in this chapter. Later chapters are more specialised. Chapter 9 presents the adequacy of Satterthwaite’s approximation. This is an advanced topic and is of little interest to a general audience, so I think this chapter should be omitted. Chapter 10 discusses unbalanced fixed effect models and Chapter 11 presents unbalanced random and mixed models. These two chapters should be omitted and the author could briefly discuss these topics in Chapter 8. Chapter 12 presents additional topics in linear models. This chapter is quite clear and useful. But I think Sections 12.4 and 12.5 on response surface models are of interest only to advanced researchers. Topics such as structural change and model selection would be very interesting to graduate students. Chapter 13 presents generalised linear models, of interest only to advanced researchers. Topics covered in Sections 13.4–13.7 are very useful to graduate students. This chapter could be combined with Chapter 6 with some details omitted. In summary, this is a very interesting book in the sense that linear models are widely used in many fields, including statistics, economics, finance, and so on. This book can be used as a textbook for first-year or second-year graduate students. The mathematical presentation is thorough and the exercises are quite useful in understanding the concepts of linear models. However, I think the authors may want to cover the more general topics discussed above, so that this book can be useful for readers in many fields.