The topics to be covered under this subject are as follows:

After completion of this topic you will be able to

- Describe and distinguish between continuous and discrete random variables
- Define & distinguish between the probability density function, the cumulative distribution function and the inverse cumulative distribution function
- Calculate the probability of an event given a discrete probability function
- Distinguish between independent and mutually exclusive events
- Define joint probability, describe a probability matrix and calculate joint probabilities using probability matrices
- Define and calculate a conditional probability and distinguish between conditional & unconditional probabilities

After completion of this topic you will be able to

- Interpret and apply the mean, standard deviation and variance of a random variable
- Calculate the mean, standard deviation and variance of a discrete random variables
- Interpret and calculate the expected value of a discrete random variable
- Calculate and interpret the covariance and correlation between two random variables.
- Calculate the mean and variance of sums of variables
- Describe the four central moments of a statistical variable or distribution: mean variance, skewness and kurtosis
- Interpret the skewness and kurtosis of a statistical distribution and interpret the concepts of coskewness and cokurtosis
- Describe and interpret the best linear unbiased estimator

After completion of this topic you will be able to

- Distinguish the key properties among the following distributions: uniform distribution, Bernoulli distribution, Binomial distribution, Poisson distribution, normal distribution, lognormal distribution, Chi-squared distribution, Student's and F-distributions and identify common occurrences of each distribution
- Describe the central limit theorem and the implications it has when combining independent and identically distributed (i.i.d.) random variables
- Describe i.i.d. random variables and the implications of the i.i.d. assumption when combining random variables
- Describe a mixture distribution and explain the creation and characteristics of mixture distributions

After completion of this topic you will be able to

- Describe Bayes' theorem and apply this theorem in the calculation of conditional probabilities
- Compare the Bayesian approach to the frequentist approach
- Apply Bayes' theorem to scenarios with more than two possible outcomes and calculate posterior probabilities

After completion of this topic you will be able to

- Calculate and interpret the sample mean and ample variance
- After completing this reading, you should be able to: Construct and interpret a confidence interval
- Construct an appropriate null and alternative hypothesis and calculate an appropriate test statistic
- Differentiate between a one-tailed and a two-tailed test and identify when to use each test
- Interpret the results of hypothesis tests with a specific level of confidence
- Demonstrate the process of backtesting VaR by calculating the number of exceedances

After completion of this topic you will be able to

- Explain how regression analysis in econometrics measures the relationship between dependent and independent variables
- Interpret a population regression function, regression coefficients, parameters, slope, intercept and the error term
- Interpret a sample regression function, regression coefficients, parameters, slope, intercept and the error term
- Describe the key properties of a linear regression
- Define an ordinary least squares (OLS) regression and calculate the intercept and slope of the regression
- Describe the method and three key assumptions of OLS for estimation of parameters
- Summarize the benefits of using OLS estimators
- Describe the properties of OLS estimators and their sampling distributions and explain the properties of consistent estimators in general
- Interpret the explained sum of squares, the total sum of squares, the residual sum of squares, the standard error of the regression and the regression R
- Interpret the results of an OLS regression

After completion of this topic you will be able to

- Calculate and interpret confidence intervals for regression coefficients
- Interpret the p-value
- Interpret hypothesis tests about regression coefficients
- Evaluate the implications of homoskedasticity and heteroskedasticity
- Determine the conditions under which the OLS is the best linear conditionally unbiased estimator
- Explain the Gauss-Markov Theorem and its limitations and alternatives to the OLS.

After completion of this topic you will be able to

- Define and interpret omitted variable bias and describe the methods for addressing this bias
- Distinguish between single and multiple regression
- Interpret the slope coefficient in a multiple regression
- Describe homoscedasticity and heteroskedasticity in a multiple regression
- Describe the OLS estimator in a multiple regression
- Calculate and interpret measures of fit in multiple regression
- Explain the assumptions of the multiple linear regression model
- Explain the concepts of imperfect and perfect multicollinearity and their implications

After completion of this topic you will be able to

- Construct, apply and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression
- Construct, apply and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a multiple regression
- Interpret the F-statistic
- Interpret tests of a single restriction involving multiple coefficients
- Interpret confidence sets for multiple coefficients
- Identify examples of omitted variable bias in multiple regressions
- Interpret the R2 and adjusted R2 in a multiple regression

After completion of this topic you will be able to

- Describe linear and nonlinear trends
- Describe trend models to estimate and forecast trends
- Compare and evaluate model selection criteria, including mean squared error (MSE), s2, the Akaike information criterion (AIC) and the Schwarz information criterion (SIC
- Explain the necessary conditions for a model selection criterion to demonstrate consistency

After completion of this topic you will be able to

- Describe the sources of seasonality and how to deal with it in time series analysis
- Explain how to use regression analysis to model seasonality
- Explain how to construct an h-step-ahead point forecast

After completion of this topic you will be able to

- Define covariance stationary, autocovariance function, autocorrelation function, partial autocorrelation function and autoregression
- Describe the requirements for a series to be covariance stationary
- Explain the implications of working with models that are not covariance stationary
- Define white noise and describe independent white noise and normal (Gaussian) white noise
- Explain the characteristics of the dynamic structure of white noise
- Explain how a lag operator works
- Describe Wold's theorem
- Define a general linear process
- Relate rational distributed Tags to Wold theorem
- Calculate the sample mean and sample autocorrelation and describe the Box-Pierce Q-statistic and the Ljung-Box Q-Statistic
- Describe sample partial autocorrelation

After completion of this topic you will be able to

- Describe the properties of the first-order moving average (MA(1)) process and distinguish between autoregressive representation and moving average representation.
- Describe the properties of a general finite-order process of order (MA(q)) process
- Describe the properties of the first-order autoregressive (AR(1)) process and define and explain the Yule-Walker equation
- Describe the properties of a general pth order autoregressive (AR(p)) process
- Define and describe the properties of the autoregressive moving average (ARMA) process
- Describe the application of AR and ARMA processes

After completion of this topic you will be able to

- Define and distinguish between volatility, variance rate and implied volatility
- Describe the power law
- Explain how various weighting schemes can be used in estimating volatility
- Apply the exponentially weighted moving average (EWMA) model to estimate volatility
- Describe the generalized autoregressive conditional heteroskedasticity (GARCH (p,q)) model for estimating volatility and its properties
- Calculate volatility using the GARCH(1,1) model
- Explain mean reversion and how it is captured in the GARCH(1,1) model
- Explain the weights in the EWMA and GARCH(1,1) models
- Explain how GARCH models perform in volatility forecasting
- Describe the volatility term structure and the impact of volatility changes

After completion of this topic you will be able to

- Define correlation and covariance and differentiate between correlation and dependence
- Calculate covariance using the EWMA and GARCH(1,1) models
- Apply the consistency condition to covariance
- Describe the procedure of generating samples from a bivariate normal distribution
- Describe properties of correlations between normally distributed variables when using a one-factor model
- Define copula and describe the key properties of copulas and copula correlation
- Describe the Gaussian copula, Student's t-copula, multivariate copula and one factor copula
- Explain tail dependence

After completion of this topic you will be able to

- Describe the basic steps to conduct a Monte Carlo simulation
- Describe ways to reduce Monte Carlo sampling error
- Explain how to use antithetic variate technique to reduce Monte Carlo sampling error
- Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective
- Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them
- Describe the bootstrapping method and its advantage over Monte Carlo simulation
- Describe situations where the bootstrapping method is ineffective
- Describe the pseudo-random number generation method and how a good simulation design alleviates the effects the choice of the seed has on the properties of the generated series
- Describe disadvantages of the simulation approach to financial problem solving