# Quantitative Analysis

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

## 1. Probabilities

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

## 2. Basic Statistics

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

## 3. Distributions

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

## 4. Bayesian Analysis

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

## 5. Hypothesis Testing and Confidence Interval

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

## 6. Linear Regression with One Regressor

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

## 7. Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

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.

## 8. Linear Regression with Multiple Regressor

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

## 9. Hypothesis Tests and Confidence Intervals in Multiple Regression

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

## 10. Modeling and Forecasting Trend

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

## 11. Modeling and Forecasting Seasonality

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

## 12. Characterizing Cycles

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

## 13. Modeling Cycles MA, AR and ARMA Models

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

## 14. Volatility

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

## 15. Correlations and Copulas

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

## 16. Simulation Methods

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