Update README.md

README.md

@@ -15,22 +15,19 @@ The thesis is a case study on the importance of the **cost of carry model** as w
   - [1.2 Graphs](#1-2)
   - [1.3 Auto-Correlation coefficients](#1-3)
 - [2. Statistical Tests](#2)
-  - [2.1 Augmented Dickey-Fuller unit root tests](#2)
-  - [2.2 Ordinary Leasr Square Regression](#2)
-  - [2.3 Error Correction Models](#2)
-   - [2.3.1 Engle Granger Test and Johansen Test](#2)
-   - [2.3.2 Results for the B&K Regression Model](#2)
-   - [2.3.3 Results for the Vector Error Correction Model](#2)
-    - [2.3.3.1 One Cointegrating Vector](#2)
-    - [2.3.3.2 Three Cointegrating Vector](#2)
+  - [2.1 Ordinary Least Square Regression](#2-1)
+    - [Augmented Dickey-Fuller unit root tests](#2-1-1)
+    - [**Results for OLS**](#2-1-2)
+  - [2.2 Error Correction Models](#2)
+    - [Engle Granger Test and Johansen Test](#2-2-1)
+    - [**Results for the B&K Regression Model**](#2-2-2)
+    - [**Results for the Vector Error Correction Model**](#2-2-3)
+      - [One Cointegrating Vector](#2-2-3-1)
+      - [Three Cointegrating Vector](#2-2-3-2)
 - [3. Comparison of model prediction](#3)
 
-- [I. Overview of Data](#1)
-- [II. Backtesting Philosophy](#2)
-- [III. Backtesting Results](#3)
-- [IV. Return Models](#4)
-- [V. Risk Models](#5)
-- [VI. Future Work](#6)
+### Models
+
 
 ### Keywords
 Cost of carry model; Vector error correction model; Engle-Grangle test; Johansen test
@@ -42,3 +39,49 @@ Cost of carry model; Vector error correction model; Engle-Grangle test; Johansen
   - in the historical period of 2009/12/31 - 2020/07/03
   - 
 ## Project Research
+
+## Project Introduction
+
+### Table of Contents
+- [1. Data](#1)
+  - [1.1 Descriptive statistics](#1-1)
+  - [1.2 Graphs](#1-2)
+  - [1.3 Auto-Correlation coefficients](#1-3)
+- [2. Statistical Tests](#2)
+  - [2.1 Ordinary Least Square Regression](#2-1)
+    - [Augmented Dickey-Fuller unit root tests](#2-1-1)
+    - [**Results for OLS**](#2-1-2)
+  - [2.2 Error Correction Models](#2)
+    - [Engle Granger Test and Johansen Test](#2-2-1)
+    - [**Results for the B&K Regression Model**](#2-2-2)
+    - [**Results for the Vector Error Correction Model**](#2-2-3)
+      - [One Cointegrating Vector](#2-2-3-1)
+      - [Three Cointegrating Vector](#2-2-3-2)
+- [3. Comparison of model prediction](#3)
+
+### Keywords
+Cost of carry model; Brenner and Kroner Model; Error correction model; Engle-Grangle test; Johansen test
+### File
+- carry_model_1204.html: demonstrates thr research results
+### DataSet
+- CU.xlsx: contains the data of SHFE copper 
+  - in four levels: cash price, future price, inventory, interest rate of government bonds.
+  - in the historical period of 2009/12/31 - 2020/07/03
+\newpage
+### Models
+The thesis introduces five models to predict cash price. Two simple models involve only the lagged futures price, as shown in equations $(1)-(2)$. Brenner and Kroner suggested a more complexed model in 1995, as shown in equation $(3)$. Single equation models impose substantial restrictions on the data and so to assess the accuracy of these restrictions a vector error correction model is proposed, see equation $(4)$
+- OLS:  
+$$ DP_t = a_0 + a_1 DF_{t|t-1} + e_t \tag{1} $$
+$$ DP_t = b_0 + b_1 (F_{t|t-1} - P_{t-1}) + e_t \tag{2} $$
+- Brenner and Kroner Model(1995):  
+$$ DP_t = g_0 + g_1 DF_{t|t-1} + g_2 Dr_{t|t-1} + $$
+$$ g_3 DI_{t-1} + g_4 D\sigma_{t-1} + g_5 D\rho(1)_{t-1} + g_6 ECT_{t-2} + e_t \tag{3}$$
+The error correction term, $ECT_{t-2}$ is the residual, $e_{t-1}$ from the regression:
+$$ P_{t-1} = h_0 + h_1 F_{t-1|t-2} + h_2 r_{t-1|t-2} + h_3I_{t-2} + $$ 
+$$ h_4 \sigma_{t-2} + h_5\rho(1)_{t-2} + e_{t-1}  $$
+- Vector Error Correction Model:
+$$ DX_t =  M_1 DX_{t-1} + S^{'}ECT_{t-2}^{*} + L + E_t\tag{4} $$
+$DX_t$ is the change in vector $X$, $M_1$ is the matrix of parameters, $S$ is the vector of speed of adjustment parameters, $L$ is a vector of constant terms and $E_t$ is the vector of residuals. $ECT^*_{t-2}$ are the residuals, $e_t$ from the following regression. 
+$$ P_{t-2} = j_0 + j_1 F_{t-1|t-2} + j_2 r_{t-1|t-2} + j_3I_{t-2} $$ 
+$$ + j_4 \sigma_{t-2} + j_5\rho(1)_{t-2} + e_{t-1} $$
+with $i_i$ and $j_i$ as estimated parameters.