The Elements of Financial Econometrics

Chapter 1 Asset Returns

1.1 Returns

1.1.1 One-period simple returns and gross returns
(1.1) R_t=\frac{P_t-P_{t-1}}{P_{t-1}}
1.1.2 Multiperiod returns
R_t(k)=\frac{P_t-P_{t-k}}{P_{t-k}}
(1.2)\frac{P_t}{P_{t-k}}=\frac{P_t}{P_{t-1}}\frac{P_{t-1}}{P_{t-2}}...\frac{P_{t-k+1}}{P_{t-k}}
(1.3)R_t(k)=\frac{P_t}{P_{t-k}}-1=(R_t+1)(R_{t-1}+1)...(R_{t-k+1}+1)-1
(1.4)R_t(k)\approx R_t+R_{t-1}+...+R_{t-k+1}
1.1.3 Log returns and continuously compounding
(1.5)r_t=logP_t-logP_{t-1}=log(\frac{P_t}{P_{t-1}})=log(1+R_t)
(1.6)r_t(k)=r_t+r_{t-1}+...+r_{t-k+1}
A exp{r_t(k)}=A exp(r_t+r_{t-1}+...+r_{t-k+1})=Ae^{k\overline{r}}
r_t=log(1+R_t)\approx R_t
\lim\limits_{m\to\infty}(1+\frac{r}{m})^m=e^r
C exp(rt)
1.1.4 Adjustment for dividends
R_t=\frac{P_t+D_t}{P_{t-1}}-1, r_t=log(P_t+D_t)-logP_{t-1}
R_t(k)=\frac{(P_t+D_t+...+D_{t-k+1})}{P_{t-k}-1}
r_t(k)=r_t+...+r_{t-k+1}=\sum_{j=0}^{k-1}log(\frac{P_{t-j}+D_{t-j}}{P_{t-j-1}})
1.1.5 Bond yields and prices
(1.7)log(\frac{B_{t+1}}{B_t})=D(r_t-r_{t+1})
1.1.6 Excess returns

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1.2 Behavior of financial return data

(1.8) \hat{r}_k = \frac{1}{T} \sum_{t=1}^{T-k} (r_t - \overline{r})(r_{t+k} - \overline{r})，
其中\overline{r} = \frac{1}{T} \sum_{t=1}^T r_t
1.2.1 样态特征：金融回报
(1.9) f_v(x) = d_v^{-1}\left( 1 + \frac{x^2}{v} \right)^{-\frac{(v+1)}{2}}

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1.3 Efficient markets hypothesis and statistical models for returns

(1.10)r_t=\mu_t+\varepsilon_t,\varepsilon~(0,\sigma_t^2)
(1.11)r_t=\mu+\varepsilon_t,\varepsilon_t\sim WN(0,\sigma^2)
(1.12)E(\varepsilon|r_{t-1},r_{t-2},...)=\sigma_t
cov(\varepsilon_t,\varepsilon_s)=E(\varepsilon_t\varepsilon_s)=E\{E(\varepsilon_t\varepsilon_s|\varepsilon_{t-1},\varepsilon_{t-2},...)\}=E\{\varepsilon_sE(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)\}=0
(1.13)\varepsilon_t=\sigma_t\eta_t
E(\sigma_t|r_{t-1},r_{t-2},...)=\sigma_t
cov(\varepsilon_t,\varepsilon_s)=E(\varepsilon_t\varepsilon_s)=E\{E(\varepsilon_t\varepsilon_s|\varepsilon_{t-1},\varepsilon_{t-2},...)\}=E\{\varepsilon_sE(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)\}=0
E\{(\hat{r}_{t+1}-r_{t+1})^2\}=var(\varepsilon_{t+1})=\sigma^2
E\{(\tilde{r}_{t+1}-r_{t+1})^2\}=E\{(\rho\varepsilon_s-\varepsilon_{t+1})^2\}=(1-\rho^2)\sigma^2<\sigma^2
(1.14)logP_t=\mu+logP_{t-1}+\varepsilon_t
\hat{r}_{t+1}=E(r_{t+1}|r_t,r_{t-1},...)=\mu+E(\varepsilon_{t+1}|\varepsilon_t,\varepsilon_{t-1},...)=\mu

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1.4 Tests related to efficient markets hypothesis

1.4.1 Tests for white noise
\rho_k\equiv Corr(r_t,r_{t-k})=\frac{cov(r_t,r_{t-k})}{\sqrt{var(r_t)var(r_{t-k})}}
(1.15)Q_m=T(T+2)\sum_{j=1}^m\frac{1}{T-j}\hat{\rho}_j^2
1.4.2 Remarks on the Ljung-Box test
1.4.3 Tests for random walks
(1.16)X_t=\mu+\alpha X_{t-1}+\varepsilon_t
(1.17)X_t=\alpha X_{t-1}+\varepsilon_t
(1.18)X_t=\mu+\beta t+\alpha X_{t-1}+\varepsilon_t
(1.19)W=\frac{(\hat\alpha -1)}{SE(\hat\alpha)}
1.4.4 Ljung-Box test and Dickey-Fuller test

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1.5 Appendix: Q-Q plot and Jarque-Bera test

1.5.1 Q-Q plot
(1.20)F^{-1}(\alpha)=max\{x:F(x)\leq\alpha\}
1.5.2 Jarque-Bera test

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1.6 Further reading and software implementation

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Chap 2 Linear Time Series Models

2.1 Stationarity

该协方差函数γ在 lag k 处的值等于变量X_t与X_{t+k}之间的协方差，并且等于期望值E乘以括号内的项。
该相关系数ρ在 lag k 处的值等于变量X_t与X_{t+k}之间的相关性，并由协方差γ在 lag k 处的值除以γ在 lag 0 处的值确定。
这些变量组成的向量的方差矩阵呈现对称结构，在第i行第j列的位置上为γ(|i-j|)，即γ函数随时间间隔递减。
当计算加权和∑a_i X_{t+i} 的方差时，在lag k处的相关性由权重系数a_i和a_j以及γ(i-j)决定。
估计量\hat{\ gamma}(k)通过平均乘积项得到，在计算\hat{\ rho}(k)时则需除以\hat{\ gamma}(0)。

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2.2 Stationary ARMA models

2.2.1 Moving average processes
(2.5)X_t=\mu+\varepsilon_t+a_1\varepsilon_{t-1}+...+a_q\varepsilon_{t-q}
Example 2.1
(2.6)\rho(1)=\frac{a}{1+a^2},\quad\rho(k)=0\,\,\, for\, any\,|k|>1
(2.7)Var(X_t)=E\{(X_t-\mu)^2\}=E\{(\varepsilon_t+a_1\varepsilon_{t-1}+...+a_q\varepsilon_{t-q})^2\}=\sigma^2(1+a_1^2+...+a_q^2)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
2.2.2 Autoregressive processes
(2.15)
(2.16)
**Example 2.2 **
(2.17)
(2.18)
(2.19)
Example 2.3
(2.20)
(2.21)
(2.22)
Example 2.4
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
Example 2.5
2.2.3 Autoregressive and moving average processes
(2.30)
(2.31)
(2.32)
Example 2.6
(2.33)
(2.34)
Example 2.7

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2.3 Nonstationary and long memory ARMA processes

(2.35)
(2.36)