Here are my notes from the book 'Analysis of Financial Time Series' by Tsay. It's a good introductory book, mostly empirical with decent maths. Someday I will read Hamilton or Green and write that up too.

#### Ch1: Financial Time series and their characteristics

Concerned with asset valuation. The uncertainty in the definition and statistical inference of returns and volatility plays important role.

Asset returns - various definitions. Let $P_t$ be the price of an asset at time index t. Assume no dividends.

1) One-period simple return: $R_t=\frac{P_t}{P_{t-1}}-1$

2) Multi-period simple return: $1+R_t[k]=\Pi_{j=0}^{k-1}(1+R_{t-j})$

3) Continuously compounded or log return: $r_t=ln(1+R_t)=ln(P_t/P_{t-1})=p_t-p_{t-1}$, and $r_t[k] = \sum_{i=0}^{k-1}r_{t-i}$

4) Portfolio return: $R_{p,t}=\sum_{i=1}^{N}w_i R_{it}$ and $r_{p,t}\approx\sum_{i=1}^{N}w_i r_{it}$

5) Dividend payment: $R_t=\frac{P_t+D_t}{P_{t-1}}-1$ and $r_t=ln(P_t+D_t)-ln(P_{t-1})$

6) Excess return: $Z_t=R_t-R_{0t}$ and $z_t=r_t-r_{0t}$

Distributional properties - N assets and T time indices.

Joint distribution:

$$F_{X,Y}(x,y;\theta)=P(X\le x, Y\le y; \theta)=\int_{-\infty}^x \int_{-\infty}^y f_{x,y}(w,z;\theta)dzdw$$

Marginal distributions: marginal distribution of $X$ is obtained by integrating out $Y$.

Conditional distributions:

$$F_{X|Y\le y}(x;\theta) = \frac{P(X\le x, Y \le y ; \theta)}{P(Y \le y; \theta)}$$

$$f_{x,y}(x,y;\theta)=f_{x|y}(x;\theta)\times f_y(y;\theta)$$

Independent variables imply $f_{x,y}(x,y;\theta)=f_x(x;\theta)f_y(y;\theta)$

Moments of a random variable:

$$m_l^{\ast}=E[X^l]=\int_{-\infty}^{\infty}x^lf(x)dx$$

The $l$th central moment of $X$ is defined as

$$m_l=E[(X-\mu_x)^l]=\int_{-\infty}^{\infty}(x-\mu_x)^lf(x)dx$$

Sample central moments:

sample mean: $\hat{\mu_x}=\frac{1}{T}\sum_{t=1}^{T}x_t$

sample variance: $\hat{\sigma_x^2}=\frac{1}{T-1}\sum_{t=1}^{T}(x_t-\hat{mu_x})^2$

sample skewness: $\hat{S_x}=\frac{1}{(T-1)\hat{\sigma}^3_x}\sum_{t=1}^{T}(x_t-\hat{mu_x})^3$, distributed normally with mean 0 and variance $6/T$ (Snedecor and Cochran 1980).

sample kurtosis: $\hat{K_x}=\frac{1}{(T-1)\hat{\sigma}^4_x}\sum_{t=1}^{T}(x_t-\hat{mu_x})^4$, distributed normally with mean 3 and variance $24/T$.

JB normality test: Both $\hat{S}(r)$ and $\hat{K}(r)-3$ are normally distributed and can have individual hypothesis tests for normality. Jarque and Bera (1987) combine the two tests and use the test statistic

$$JB=\frac{\hat{S}^2(r)}{6/T}+\frac{(\hat{K}(r)-3)^2}{24/T}$$

which is asymptotically distributed as a chi-squared random variable with 2 degrees of freedom.

Distribution of returns - The most general model for the

**log returns**${r_{it}; i=1, ...,N; t=1,...,T}$ is its joint distribution funtion:
$$F_r(r_{11},...,r_{N1};r_{12},...,r_{N2};...;r_{1T},...,r_{NT};Y;\theta),$$

where $Y$ is a state vector summarizing the environment in which asset returns are determined and $\theta$ are parameters that uniquely determine the distribution function $F_r(.)$. Many times $Y$ is assumed given and the main concern is the conditional distribution of ${r_{it}}$ given $Y$. Some applications focus on the joint distribution of N returns at a single time index t, i.e. ${r_{1t},...,r_{Nt}}$. Other theories emphasize the dynamic structure of individual asset returns - the uni-variate case - ${r_{i1},...,r_{iT}}$ for a given asset $i$. For the uni-variate case the joint distribution is given as

$$F(r_1, ...,r_T;\theta)=F(r_1;\theta)\Pi_{t=2}^{T}F(r_t|r_{t-1},...,r_{1};\theta).$$

The main concern then is to specify the conditional distribution $F(r_t|r_{t-1},..,r{1};\theta)$. Different distributional specifications lead to different theories. For instance, if the conditional distribution is equal to the marginal distribution it is a random-walk hypothesis. It is customary to treat asset returns as continuous random variables, and use their probability density functions.

$$f(r_1,...,r_T;\theta)=f(r_1;\theta)\Pi_{t=2}^{T}f(r_t|r_{t-1},...,r_{1};\theta).$$

Different distributions on marginals give different models:

Normal distribution: $R_t$ iid and normal with fixed mean and variance - lower bound is wrong, multiperiod returns are not normal due to multiplication, empirical excess kurtosis.

Lognormal distribution: $r_t$ iid and normal with fixed mean $\mu$ and variance$\sigma^2$. The simple returns $R_t$ are then lognormal with:

$$E[R_t]=e^{(\mu+\frac{\sigma^2}{2})}-1 \qquad Var[R_t]=e^{2\mu+\sigma^2}[e^{\sigma^2}-1]$$

summation is also normally distributed, there is no lower bound, but is not consistent with excess kurtosis.

Stable distributions: distribution where a linear combination has the same distribution. Summation and lower bound satisfied but has infinite variance, e.g. Cauchy distribution $f(x)=1/(\pi(1+x^2))$.

Scale mixture of normal distributions: finite mixtures of normal distributions $r_t \sim (1-X)N(\mu,\sigma^2_1)+XN(\mu,\sigma^2_2)$, where $X$ is a Bernoulli random variable and $\sigma^2_2$ is relatively large. Maintain tractability of normal under addition, capture excess kurtosis and finite higher moments, but hard to estimate the mixture parameters. For multivariate returns covariance becomes critical. Everything becomes vectors.

log likelihood function of returns with normal distribution:

$$ln f(r_1,...,r_T;\theta)=ln f(r_1;\theta)-\frac{1}{2}\sum_{t=2}^T(ln(2\pi)+ln(\sigma_t^2)+\frac{(r_t-\mu_t)^2}{\sigma^2_t})$$

This assumes iid of each observation.

Empirical properties of returns:

a) Daily returns have higher kurtosis than monthly returns. For monthly returns market indexes have higher kurtosis than individual stocks.

b) For daily returns market indexes have smaller standard deviation than individual stocks.

c) Empirical returns density is taller, but with fatter tails.

Processes considered: return series, conditional volatility (e.g. clustering), extreme behaviors (frequency, size, impact).

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