Translation day01

00-preface.Rmd

@@ -11,7 +11,7 @@
 マクロ数量分析の標準的なツールである Dynare を自分で使ったり, 名前を聞いたことのある学生も多いと思う. 一方で, その裏側でどのような計算が行われているかを知らないという学生も多いのではないかと思う. 理論に無関心でもおおよそ問題が起こらないのは, ひとえに Dynare が優れたインターフェイスを備えたアプリケーションであるということに外ならないのだが, そのブラックボックスを開けて理論を理解しようというのがこの講義の第1の課題である. 
 
 第2の課題は, 読者にコンピュータシミュレーションの基礎的な手法を身につけてもらうことである. 線形合理的期待モデルは近年のマクロ経済の数量分析に大きな役割を果たしているから, 各自の研究プロジェクトに大いに役に立つものと信じている. 
-数学もプログラミングも自分の手を動かし, 頭を悩ませなければ習熟できない. 理論とその実装を行き来しながら総合的な問題解決能力を養ってほしい. この講義では, 主に R言語を用いる.^[R ([https://www.r-project.org/](https://www.r-project.org/)) と RStudio ([https://www.rstudio.com/](https://www.rstudio.com/)) の最新版を各自インストールしてほしい. R言語を選んだ理由は, 計量経済学の学習のために利用したことのある学生が多いだろう考えたからであり, R がシミュレーションのために最適であるとは考えていない. 大変人気のあるプログラミング言語・環境であるから一度は学んでおいて損はないと思う.]
+数学もプログラミングも自分の手を動かし, 頭を悩ませなければ習熟できない. 理論とその実装を行き来しながら総合的な問題解決能力を養ってほしい. この講義では, 主に R言語を用いる.^[[R](https://www.r-project.org/) と [RStudio](https://www.rstudio.com/) の最新版を各自インストールしてほしい. R言語を選んだ理由は, 計量経済学の学習のために利用したことのある学生が多いだろう考えたからであり, R がシミュレーションのために最適であるとは考えていない. 大変人気のあるプログラミング言語・環境であるから一度は学んでおいて損はないと思う.]
 
 最後に, 近年盛んに研究されているマルコフスイッチング合理的期待モデルという非線形モデルを紹介する.  線形モデルのように, (ある種の) 安定性・不安定性に関する必要十分条件が得られるという特徴があり, 理論的に扱いやすい. 金融政策ルールに関するテイラーの条件を緩和できることが知られており, 理論・実証の両面から盛んに研究が進んでいる分野である.
 

01-intro.Rmd

@@ -13,7 +13,7 @@ project_name = 'linear-economic-dynamics'
 $$
 \begin{aligned}
   &\max_{c\in \mathbb{R}^N_+} u(c) \\
-  &\text{subject to}\quad p \cdot c \le I  
+  &\text {subject to}\quad p \cdot c \le I  
 \end{aligned}
 $$
 価格 $p = (p_1, \dots, p_N)$ と所得 $I$ のもとで，効用 $u(c)$ を最大にする消費量の組み合わせ 

Translation/day00_en.Rmd

@@ -0,0 +1,84 @@
+# Preface {#preface .unnumbered}
+
+These notes are based on lectures held at the Graduate School of Economics at Kobe University in the first quarters of 2016 and 2017. I assume as audience graduate students majoring in macroeconomics, macro econometrics and mathematical economics. However, I think that this lecture could be also beneficial for advanced degree students if not bothered by mathematical discussion. At beginning we will start from elementary notions of linear algebra, theory of dynamic system and dynamical system. Since I tried to be as complete as possible, *I think I could also make reference to the book 「Mathematics for economists」 specialized in macroeconomics.* Moreover, since I will hardly use well-known facts of economics, I think that this lecture could be also useful for students of science and engineering who began to be interested in analyzing economic models.
+
+In this lecture, we analyze the linear rational expectation models as a central issue. The main objectives of the lecture are the following three:
+
+- Theoretical analysis of linear rational expectation model
+- Computer simulation
+- Introduction of Markov-switching rational expectations model
+
+I think there are many students who use by themselves or just heard the name "Dynare"", that is a standard tool for macro quantitative analysis. On the other hand, I think that there are many students who do not know how calculations are made on the back side. *[...]* The first part of this lecture is to understand the theory and open the black box of Dynare, that is no other than an application with an excellent interface. 
+
+The second part is to get the students to acquire basic techniques of computer simulation. Since the linear rational expectation models have been playing a major role in quantitative analysis of macroeconomics in recent years, I believe it will be of great use to each research project. Unless you work very hard you cannot master mathematics and programming. 
+I want you to develop comprehensive problem solving skills while coming back and forth between theory and its implementation. In this lecture we will mainly use R language. ^[Please install the last version of [R](https://www.r-project.org/) and [RStudio](https://www.rstudio.com/). The reason why I chose the R language is that I thought that many students had already used it to learn econometrics, *[...]* It is a very popular programming language / environment so I think that it will be very benefical to learn it once.]
+
+Finally, we introduce the Markov-switching rational expectations model, a non linear model that has become very popular in the last years. Like linear models (or some kinds of), there is a characteristic to obtain a necessary and sufficient condition for the stability / instability which is theoretically easy to handle. It is known that Taylor's condition related to monetary policy rules can be relaxed, and research is actively underway from both theory and empirical research. 
+
+
+## Organization {-}
+
+These notes are made in accordance with the system of quarter that consists of 15 lectures (twice a week, 90 minutes × 8 weeks each time).
+In principle, we are going to read through each chapter in order. Since I set up 「exercises」to try to build a comprehensive understanding, I would like you tackle all of them. 
+
+The first few chapters summarize the definitions and results that you learnt in the first year of undergraduate, which are necessary for the analysis of dynamic systems, so it should be a list of things that most of students already know. Therefore, you can skip reading providing that you solve practice problems. However, since each chapter is arranged to be increasingly difficult, I recommend you to read it if you are a little worry. Furthermore, I think that it is not useful to read only the chapters of interest if you do not have a preious knowledge of this field. 
+
+#### \ref@(intro)Chapter: Introduction {-}
+
+- What kind of model do you want to solve?
+- Building a programming environment
+
+#### \ref@(complex numbers)Chapter: Complex numbers
+
+- Review of complex numbers
+- Why are complex numbers necessary? 
+
+#### \ref@(matrix)Chapter: Review of matrices {-}
+
+- Review of matrices
+- Similarity between matrix product and complex numbers product
+
+#### \ref@(eigen)Chapter: Eigenvalues of matrices
+
+- Review of eigenvalues of the matrices
+- Clarify the relationship of complex eigenvalues of linear system
+
+#### Chapter 5: Eigenspace {-}
+
+#### Chapter 6: Stability of Jordan canonical form and linear system {-}
+
+- Jordan canonical form
+- Stability
+
+#### Chapter 7: $\det A \neq 0$ {-}
+
+- @BlanchardKahn1980
+
+#### Chapter 8: Weierstrass canonical form and descriptive system {-}
+
+- $\det A = 0$
+
+#### Chapter 9: General $A$ {-}
+
+- Stock and Watson?
+
+#### Chapter 10: Numerical solution {-}
+
+- Schur analysis
+- QZ analysis
+
+#### Chapter 11: Klein's method {-}
+
+- @Klein2000
+
+#### Chapter 12: Sims' method {-}
+
+#### Chapter 13: Probabilistic system {-}
+
+#### Chapter 14: Lubik-Schorfheide {-}
+
+#### Chapter 15: Markov-switching system {-}
+
+## Acknowledgments {-}
+
+I thank you Professor Ando of Columbia University for his valuable comments.