Week5 - Descriptive Statistics
5.1 - Covariance Stationarity
- Recall:
- \(E[X_t] = \mu\) indep of t.
- \(var(X_t) = \sigma^2\) indep of t.
- \(cov(X_t,X_{t-j}) = \gamma_j\) indep of t.
- \(cor(X_t,X_{t-j}) = \rho_j\) indep of t.
- In SP500, the capitalization of each company is used to compute the weight for that company in the “SP500 portfolio”: \(w_i = \frac{cap_i}{\sum\limits_j cap_j}\)
- The volativity of SP500 is also called market volatility.
- Descriptive statistics = data summaries (we try to compute some features from the data).
5.2 - Histograms
- We can build blockly histograms or smoothed histograms.
- We can infer the mean, deviation, skewness and kurtosis from the histograms.
5.3 - Sample Statistics
Percentiles
- Sample quantile is often called empirical quantile or percentile.
Quartiles
- q.25 is the first quartile
- q.50 is the second quartile (or median)
- q.75 is the third quartile
- q.75 - q.25 is the interquartile range (IQR)
Historical value-at-risk
- eg. value at risk computed from the data (eg. from empirical quantiles).
Sample Statistics
- We can compute Sample mean, variance, std deviation, skewness, kurtosis and excess kurtosis.