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 = P t − P t − 1 P t − 1 R_t=\frac{P_t-P_{t-1}}{P_{t-1}} Rt​=Pt−1​Pt​−Pt−1​​

1.1.2 Multiperiod returns

R t ( k ) = P t − P t − k P t − k R_t(k)=\frac{P_t-P_{t-k}}{P_{t-k}} Rt​(k)=Pt−k​Pt​−Pt−k​​

(1.2) P t P t − k = P t P t − 1 P t − 1 P t − 2 . . . P t − k + 1 P t − k \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}} Pt−k​Pt​​=Pt−1​Pt​​Pt−2​Pt−1​​...Pt−k​Pt−k+1​​

(1.3) R t ( k ) = P t P t − k − 1 = ( R t + 1 ) ( R t − 1 + 1 ) . . . ( R t − k + 1 + 1 ) − 1 R_t(k)=\frac{P_t}{P_{t-k}}-1=(R_t+1)(R_{t-1}+1)...(R_{t-k+1}+1)-1 Rt​(k)=Pt−k​Pt​​−1=(Rt​+1)(Rt−1​+1)...(Rt−k+1​+1)−1

(1.4) R t ( k ) ≈ R t + R t − 1 + . . . + R t − k + 1 R_t(k)\approx R_t+R_{t-1}+...+R_{t-k+1} Rt​(k)≈Rt​+Rt−1​+...+Rt−k+1​

1.1.3 Log returns and continuously compounding

(1.5) r t = l o g P t − l o g P t − 1 = l o g ( P t P t − 1 ) = l o g ( 1 + R t ) r_t=logP_t-logP_{t-1}=log(\frac{P_t}{P_{t-1}})=log(1+R_t) rt​=logPt​−logPt−1​=log(Pt−1​Pt​​)=log(1+Rt​)

(1.6) r t ( k ) = r t + r t − 1 + . . . + r t − k + 1 r_t(k)=r_t+r_{t-1}+...+r_{t-k+1} rt​(k)=rt​+rt−1​+...+rt−k+1​

A e x p r t ( k ) = A e x p ( r t + r t − 1 + . . . + r t − k + 1 ) = A e k r ‾ A exp{r_t(k)}=A exp(r_t+r_{t-1}+...+r_{t-k+1})=Ae^{k\overline{r}} Aexprt​(k)=Aexp(rt​+rt−1​+...+rt−k+1​)=Aekr

r t = l o g ( 1 + R t ) ≈ R t r_t=log(1+R_t)\approx R_t rt​=log(1+Rt​)≈Rt​

lim ⁡ m → ∞ ( 1 + r m ) m = e r \lim\limits_{m\to\infty}(1+\frac{r}{m})^m=e^r m→∞lim​(1+mr​)m=er

C e x p ( r t ) C exp(rt) Cexp(rt)

1.1.4 Adjustment for dividends

R t = P t + D t P t − 1 − 1 , r t = l o g ( P t + D t ) − l o g P t − 1 R_t=\frac{P_t+D_t}{P_{t-1}}-1, r_t=log(P_t+D_t)-logP_{t-1} Rt​=Pt−1​Pt​+Dt​​−1,rt​=log(Pt​+Dt​)−logPt−1​

R t ( k ) = ( P t + D t + . . . + D t − k + 1 ) P t − k − 1 R_t(k)=\frac{(P_t+D_t+...+D_{t-k+1})}{P_{t-k}-1} Rt​(k)=Pt−k​−1(Pt​+Dt​+...+Dt−k+1​)​

r t ( k ) = r t + . . . + r t − k + 1 = ∑ j = 0 k − 1 l o g ( P t − j + D t − j P t − j − 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}}) rt​(k)=rt​+...+rt−k+1​=j=0∑k−1​log(Pt−j−1​Pt−j​+Dt−j​​)

1.1.5 Bond yields and prices

(1.7) l o g ( B t + 1 B t ) = D ( r t − r t + 1 ) log(\frac{B_{t+1}}{B_t})=D(r_t-r_{t+1}) log(Bt​Bt+1​​)=D(rt​−rt+1​)

1.1.6 Excess returns

1.2 Behavior of financial return data

(1.8) r k ^ = 1 T ∑ t = 1 T − k ( r t − r ‾ ) ( r t + k − r ‾ ) , r ‾ = 1 T ∑ t = 1 T r t \hat{r_k}=\frac{1}{T} \sum_{t=1}^{T-k} (r_t-\overline{r})(r_{t+k}-\overline{r}),\overline{r}=\frac{1}{T}\sum_{t=1}^Tr_t rk​^​=T1​t=1∑T−k​(rt​−r)(rt+k​−r),r=T1​t=1∑T​rt​

1.2.1 Stylized features of financial returns

(1.9) f v ( x ) = d v − 1 ( 1 + x 2 v ) − ( v + 1 ) 2 f_v(x)=d_v^{-1}(1+\frac{x^2}{v})^{-\frac{(v+1)}{2}} fv​(x)=dv−1​(1+vx2​)−2(v+1)​

1.3 Efficient markets hypothesis and statistical models for returns

(1.10) r t = μ t + ε t , ε   ( 0 , σ t 2 ) r_t=\mu_t+\varepsilon_t,\varepsilon~(0,\sigma_t^2) rt​=μt​+εt​,ε (0,σt2​)

(1.11) r t = μ + ε t , ε t ∼ W N ( 0 , σ 2 ) r_t=\mu+\varepsilon_t,\varepsilon_t\sim WN(0,\sigma^2) rt​=μ+εt​,εt​∼WN(0,σ2)

(1.12) E ( ε ∣ r t − 1 , r t − 2 , . . . ) = σ t E(\varepsilon|r_{t-1},r_{t-2},...)=\sigma_t E(ε∣rt−1​,rt−2​,...)=σt​

c o v ( ε t , ε s ) = E ( ε t ε s ) = E { E ( ε t ε s ∣ ε t − 1 , ε t − 2 , . . . ) } = E { ε s E ( ε t ∣ ε t − 1 , ε t − 2 , . . . ) } = 0 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 cov(εt​,εs​)=E(εt​εs​)=E{E(εt​εs​∣εt−1​,εt−2​,...)}=E{εs​E(εt​∣εt−1​,εt−2​,...)}=0

(1.13) ε t = σ t η t \varepsilon_t=\sigma_t\eta_t εt​=σt​ηt​

E ( σ t ∣ r t − 1 , r t − 2 , . . . ) = σ t E(\sigma_t|r_{t-1},r_{t-2},...)=\sigma_t E(σt​∣rt−1​,rt−2​,...)=σt​

c o v ( ε t , ε s ) = E ( ε t ε s ) = E { E ( ε t ε s ∣ ε t − 1 , ε t − 2 , . . . ) } = E { ε s E ( ε t ∣ ε t − 1 , ε t − 2 , . . . ) } = 0 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 cov(εt​,εs​)=E(εt​εs​)=E{E(εt​εs​∣εt−1​,εt−2​,...)}=E{εs​E(εt​∣εt−1​,εt−2​,...)}=0

E { ( r ^ t + 1 − r t + 1 ) 2 } = v a r ( ε t + 1 ) = σ 2 E\{(\hat{r}_{t+1}-r_{t+1})^2\}=var(\varepsilon_{t+1})=\sigma^2 E{(r^t+1​−rt+1​)2}=var(εt+1​)=σ2

E { ( r ~ t + 1 − r t + 1 ) 2 } = E { ( ρ ε s − ε t + 1 ) 2 } = ( 1 − ρ 2 ) σ 2 < σ 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 E{(r~t+1​−rt+1​)2}=E{(ρεs​−εt+1​)2}=(1−ρ2)σ2<σ2

(1.14) l o g P t = μ + l o g P t − 1 + ε t logP_t=\mu+logP_{t-1}+\varepsilon_t logPt​=μ+logPt−1​+εt​

r ^ t + 1 = E ( r t + 1 ∣ r t , r t − 1 , . . . ) = μ + E ( ε t + 1 ∣ ε t , ε t − 1 , . . . ) = μ \hat{r}_{t+1}=E(r_{t+1}|r_t,r_{t-1},...)=\mu+E(\varepsilon_{t+1}|\varepsilon_t,\varepsilon_{t-1},...)=\mu r^t+1​=E(rt+1​∣rt​,rt−1​,...)=μ+E(εt+1​∣εt​,εt−1​,...)=μ

1.4 Tests related to efficient markets hypothesis

1.4.1 Tests for white noise

ρ k ≡ C o r r ( r t , r t − k ) = c o v ( r t , r t − k ) v a r ( r t ) v a r ( r t − k ) \rho_k\equiv Corr(r_t,r_{t-k})=\frac{cov(r_t,r_{t-k})}{\sqrt{var(r_t)var(r_{t-k})}} ρk​≡Corr(rt​,rt−k​)=var(rt​)var(rt−k​)

​cov(rt​,rt−k​)​

(1.15) Q m = T ( T + 2 ) ∑ j = 1 m 1 T − j ρ ^ j 2 Q_m=T(T+2)\sum_{j=1}^m\frac{1}{T-j}\hat{\rho}_j^2 Qm​=T(T+2)j=1∑m​T−j1​ρ^​j2​

1.4.2 Remarks on the Ljung-Box test

1.4.3 Tests for random walks

(1.16) X t = μ + α X t − 1 + ε t X_t=\mu+\alpha X_{t-1}+\varepsilon_t Xt​=μ+αXt−1​+εt​

(1.17) X t = α X t − 1 + ε t X_t=\alpha X_{t-1}+\varepsilon_t Xt​=αXt−1​+εt​

(1.18) X t = μ + β t + α X t − 1 + ε t X_t=\mu+\beta t+\alpha X_{t-1}+\varepsilon_t Xt​=μ+βt+αXt−1​+εt​

(1.19) W = ( α ^ − 1 ) S E ( α ^ ) W=\frac{(\hat\alpha -1)}{SE(\hat\alpha)} W=SE(α^)(α^−1)​

1.4.4 Ljung-Box test and Dickey-Fuller test

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

1.5.1 Q-Q plot

(1.20) F − 1 ( α ) = m a x { x : F ( x ) ≤ α } F^{-1}(\alpha)=max\{x:F(x)\leq\alpha\} F−1(α)=max{x:F(x)≤α}

1.5.2 Jarque-Bera test

1.6 Further reading and software implementation

Chap 2 Linear Time Series Models

2.1 Stationarity

(2.1) γ ( k ) = c o v ( X t , X t + k ) = E { ( X t − μ ) ( X t + k − μ ) } \gamma(k)=cov(X_t,X_{t+k})=E\{(X_t-\mu)(X_{t+k}-\mu)\} γ(k)=cov(Xt​,Xt+k​)=E{(Xt​−μ)(Xt+k​−μ)}

(2.2) ρ ( k ) = C o r r ( X t , X t + k ) = γ ( k ) γ ( 0 ) \rho(k)=Corr(X_t,X_{t+k})=\frac{\gamma(k)}{\gamma(0)} ρ(k)=Corr(Xt​,Xt+k​)=γ(0)γ(k)​

v a r ( X t , . . . , X t + k ) = ( γ ( 0 ) γ ( 1 ) γ ( 2 ) . . . γ ( k − 1 ) γ ( 1 ) γ ( 0 ) γ ( 1 ) . . . γ ( k − 2 ) ⋮ ⋮ ⋮ ⋮ γ ( k − 2 ) γ ( k − 3 ) γ ( k − 4 ) . . . γ ( 1 ) γ ( k − 1 ) γ ( k − 2 ) γ ( k − 3 ) . . . γ ( 0 ) ) var(X_t,...,X_{t+k})=\begin{pmatrix} \gamma(0)&\gamma(1)&\gamma(2)&...&\gamma(k-1)\\ \gamma(1)&\gamma(0)&\gamma(1)&...&\gamma(k-2)\\ \vdots&\vdots&\vdots&&\vdots\\ \gamma(k-2)&\gamma(k-3)&\gamma(k-4)&...&\gamma(1)\\ \gamma(k-1)&\gamma(k-2)&\gamma(k-3)&...&\gamma(0) \end{pmatrix} var(Xt​,...,Xt+k​)=⎝⎜⎜⎜⎜⎜⎛​γ(0)γ(1)⋮γ(k−2)γ(k−1)​γ(1)γ(0)⋮γ(k−3)γ(k−2)​γ(2)γ(1)⋮γ(k−4)γ(k−3)​............​γ(k−1)γ(k−2)⋮γ(1)γ(0)​⎠⎟⎟⎟⎟⎟⎞​

(2.3) v a r ( ∑ i = 1 k a i X t + i ) = ∑ i = 1 k ∑ j = 1 k a i a j c o v ( X t + i , X t + k ) = ∑ i = 1 k ∑ j = 1 k a i a j γ ( i − j ) ≥ 0 var(\sum_{i=1}^ka_iX_{t+i})=\sum_{i=1}^k\sum_{j=1}^ka_ia_jcov(X_{t+i},X_{t+k})=\sum_{i=1}^k\sum_{j=1}^ka_ia_j\gamma(i-j)\geq0 var(i=1∑k​ai​Xt+i​)=i=1∑k​j=1∑k​ai​aj​cov(Xt+i​,Xt+k​)=i=1∑k​j=1∑k​ai​aj​γ(i−j)≥0

(2.4) γ ^ ( k ) = 1 T ∑ t = k + 1 T ( X t − X ‾ ) ( X t − k − X ‾ ) , ρ ^ ( k ) = γ ^ ( k ) γ ^ ( 0 ) \hat\gamma(k)=\frac{1}{T}\sum_{t=k+1}^T(X_t-\overline X)(X_{t-k}-\overline X), \hat\rho(k)=\frac{\hat \gamma(k)}{\hat \gamma(0)} γ^​(k)=T1​t=k+1∑T​(Xt​−X)(Xt−k​−X),ρ^​(k)=γ^​(0)γ^​(k)​

2.2 Stationary ARMA models

2.2.1 Moving average processes

(2.5) X t = μ + ε t + a 1 ε t − 1 + . . . + a q ε t − q X_t=\mu+\varepsilon_t+a_1\varepsilon_{t-1}+...+a_q\varepsilon_{t-q} Xt​=μ+εt​+a1​εt−1​+...+aq​εt−q​

Example 2.1

(2.6) ρ ( 1 ) = a 1 + a 2 , ρ ( k ) = 0     f o r   a n y   ∣ k ∣ > 1 \rho(1)=\frac{a}{1+a^2},\quad\rho(k)=0\,\,\, for\, any\,|k|>1 ρ(1)=1+a2a​,ρ(k)=0forany∣k∣>1

(2.7) V a r ( X t ) = E { ( X t − μ ) 2 } = E { ( ε t + a 1 ε t − 1 + . . . + a q ε t − q ) 2 } = σ 2 ( 1 + a 1 2 + . . . + a q 2 ) 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) Var(Xt​)=E{(Xt​−μ)2}=E{(εt​+a1​εt−1​+...+aq​εt−q​)2}=σ2(1+a12​+...+aq2​)

(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

2.3 Nonstationary and long memory ARMA processes

(2.35)

(2.36)