Unit 2 of 14

Probability and financial econometrics

  • 8 lessons
  • Videos 9 to 16

Prices and returns are modeled as random variables. This unit develops distributions, conditional expectation and dependence, then the estimation, regression and time-series tools for financial data.

A change of variables gives the lognormal law and its moments. Conditional expectation on an information partition leads to the tower property and the law of total variance, and Gaussian vectors and copulas separate dependence from the marginals. Likelihood, the law of large numbers and the central limit theorem justify estimates and standard errors under stated conditions. Regression is first a projection, and interpreting it needs identification assumptions. ARMA models, GARCH and the Kalman filter close the unit. General measure-theoretic results are stated, not proved.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Distributions, expectation and dependence

  1. Random variables and distribution transformations

    How a CDF and a density describe the law of a random variable, how a change of variables with its Jacobian gives a new density, and how completing the square yields the lognormal moments.

    Video 9

  2. Expectation conditional expectation and variance

    Why expectation needs integrability, how conditional expectation on an information partition is a random variable, and how the tower property and the law of total variance split variance into within-group and between-group parts.

    Video 10

  3. Multivariate distributions and dependence

    How a joint distribution carries more than its marginals, how covariance gives portfolio variance, why zero correlation need not mean independence, and how copulas and Sklar's theorem separate marginals from dependence.

    Video 11

Estimation and regression

  1. Statistical estimation and likelihood

    How a likelihood function leads to a maximum likelihood estimator, what consistency, the law of large numbers and the central limit theorem require, and how standard errors and confidence intervals follow.

    Video 12

  2. Regression and statistical identification

    How ordinary least squares solves the normal equations as an orthogonal projection, what exogeneity and statistical identification add, and how omitted variable bias and heteroskedasticity change the sandwich covariance.

    Video 13

Time series, volatility and filtering

  1. Stationarity autocovariance and ARMA models

    What weak stationarity and the autocovariance function are, how an AR(1) process gets its causal stationary representation and conditional forecast variance, when an ARMA model is stationary, and what a unit root changes.

    Video 14

  2. Conditional volatility and GARCH

    How ARCH and GARCH models let the conditional variance respond to past shocks, how variance persistence drives variance forecasts, and why strict stationarity differs from a finite unconditional variance.

    Video 15

  3. State space models and the Kalman filter

    How a state-space model links a latent state to noisy observations through a state equation and an observation equation, and how the Kalman filter uses the innovation covariance, the Kalman gain and the posterior covariance.

    Video 16