# Greenwicher's WikiFRM - Quantitative Analysis 2017-08-29

[Last Update: Oct 27, 2017]

## The Time Value of Money

### Time Value of Money Concepts and Applications

`compound interest`

or`interest on interest`

- future value
- present value

### Using a Financial Calculator

- learn to solve the TVM (time value of money) problems

### Time Lines

- discounting/compounding
- cash flows occur at the end of the period depicted on the time line
- required rate of return / discount rate / opportunity cost
- noimal risk-free rate = real risk-free rate + expected inflation rate
- required interest rate on a security = noimal risk-free rate + default risk premium + liquidity premium + maturity risk premium

### Present Value of a Single Sum

### Annuities

- ordinary annuity

### Present Value of a Perpetuity

### PV and FV of Uneven Cash Flow Series

### Solving TVM Problems When Compounding Periods are Other Than Annual

## Probabilities

### Random Variables

#### LO 15.1: Describe and distinguish between continuous and discrete random variables.

### Distribution Functions

#### LO 15.2: Define and distinguish between the probability density function, the cumulative distribution function, and the inverse cumulative distribution function.

### Discrete Probability Function

#### LO 15.3: Calculate the probability of an event given a discrete probability function.

### Conditional Probabilities

#### LO 15.6: Define and calculate a conditional probability, and distinguish between conditional and unconditional probabilities.

### Independent and Mutually Exclusive Events

#### LO 15.4: Distinguish between independent and mutually exclusive events.

### Calculating a Joint Probability of Any Number of Independent Events

#### LO 15.5: Define joint probability, describe a probability matrix, and calculate joint probabilities using probability matrices.

### Probability Matrix

## Basic Statistics

### Measure of Central Tendency

#### LO 16.1: Interpret and apply the mean, standard deviation, and variance of a random variable.

#### LO 16.2: Calculate the mean, standard deviation, and variance of a discrete random variable.

- population mean / sample mean

### Expectations

#### LO 16.3: Interpret and calculate the expected value of a discrete random variables.

#### LO 16.4: Calculate the mean and variance of sums of variables.

### Variance and Standard Deviation

### Covariance and Correlation

#### LO 16.5: Calculate and interpret the covariance and correlation between two random variables.

correlation measures the strength of the linear relationship between two random variables

### Interpreting a Scatter Plot

### Moments and Central Moments

- skewness (standardized third central moment): evaluate whether the distribution is symmetric
- kurtosis (standardized fourth central moment): refers to the degree of peakedness or clustering in the data distribution

#### LO 16.6: Describe the four central moments of a statistical variable or distribution: mean, variance, skewness and kurtosis.

### Skewness and Kurtosis

#### LO 16.7: Interpret the skewness and kurtosis of a statistical distribution, and interpret the concepts of coskewness and cokurtosis.

skewness

positively/right skewed distributions: mode < median < mean (for unimodal distribution), positively means more outliner in the positive region

negatively/left skewed distributions: mode > median > mode (for unimodal distribution)

kurtosis

- leptokurtic (more peaked than a normal distribution)
- great probability of an observed value being either close to the mean or far from the mean

- platykurtic (more flatter)
- mesokurtic (same kurtosis)
- excess kurtosis: kurtosis - 3 (relative to a normal distribution)

- leptokurtic (more peaked than a normal distribution)

### Coskewness and Cokurtosis

- coskewness: third cross central moment
- cokurtosis: fourth cross central moment

### The Best Linear Unbiased Estimator

#### LO 16.8: Describe and interpret the best linear unbiased estimator.

- unbiased
- efficient
- consistent
- linear

## Distributions

### Parametric and Nonparametric Distributions

#### LO 17.1: 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 t, and F-distributions, and identify common occurences of each distribution

### The Central Limit Theorem

#### LO 17.2: Describe the central limit theorem and the implications it has when combining independent and identically distributed (i.i.d.) random variables.

#### LO 17.3: Describe the i.i.d. random variables and the implications of the i.i.d. assumption when combining random variables.

### Mixture Distributions

#### LO 17.4: Describe a mixture distribution and explain the creation and characteristics of mixture distributions.

## Bayesian Analysis

### Bayes’ Theorem

#### LO 18.1: Describe Bayes’ theorem and apply this theorem in the calculation of conditional probabilities

### Bayesian Approach v.s. Frequentist Approach

#### LO 18.2: Compare the Bayesian approach to the frequentist approach

### Bayes’ Theorem with Multiple States

#### LO 18.3: Apply Bayes’ theorem to scenarios with more than two possible outcomes and calculate posterior probabilities

## Hypothesis Testing and Confidence Intervals

### Applied Statistics

### Mean and Variance of the Sample Average

#### LO 19.1: Calculate and interpret the sample mean and sample variance

### Population and Sample Mean

### Population and Sample Variance

### Population and Sample Covariance

### Confidence Intervals

#### LO 19.2: Construct and interpret a confidence interval

### Confidence Intervals for a Population Mean: Normal with Unknown Variance

### Confidence Intervals for a Population Mean: Nonnormal with Unknown Variance

### Hypothesis Testing

#### LO 19.3: Construct an appropriate null and alternative hypothesis, and calculate an appropriate test statistic.

### The Null Hypothesis and Alternative Hypothesis

### The Choice of the Null and Alternative Hypotheses

### One-tailed and Two-tailed Tests of Hypotheses

#### LO 19.4: Differentiate between a one-tailed and a two-tailed test and identify when to use each test

### Type I and Type II Errors

- Type I: rejection of the null hypothesis when it is actually true
- Type II: the failure to reject null hypothesis when it is actually false

### The Power of a Test

- power: 1 - P(Type II Errors)

### The Relation Between Confidence Intervals and Hypothesis Tests

### Statistical Significance v.s. Economic Significance

### The p-Value

- p-value: the probability of obtaining a test statistics that would lead to rejection of the null hypothesis

### The t-Test

### The z-Test

#### LO 19.5: Interpret the results of hypothesis tests with a specific level of confidence.

### The Chi-Squared Test

### The F-Test

### Chebyshev’s Inequality

- the percentage of the observations that lie within k std of the mean is at least $1- 1/k^{2}$

### Backtesting

#### LO 19.6: Demonstrate the process of backtesting VaR by calculating the number of exceedances.

## Linear Regression with One Regressor

### Regression Analysis

#### LO 20.1: Explain how regression analysis in econometrics measures the relationship between dependent and independent variables.

### Population Regression Function

#### LO 20.2: Interpret a population regression function, regression coefficients, parameters, slope, interprect, and the error term.

### Sample Regression Function

#### LO 20.3: Interpret a sample regression, regression coefficients, parameters, slope, interprect, and the error term.

### Properties of Regression

#### LO 20.4: Describe the key properties of a linear regression

### Ordinary Least Squares Regression

#### LO 20.5: Define an ordinary least squares (OLS) regression and calculate the intercept and slope of the regression.

### Assumptions Underlying Linear Regression

#### LO 20.6: *Describe the method and three key assumptions of OLS for estimation of parameters.*

### Properties of OLS Estimators

#### LO 20.7: Summarize the benefits of using OLS estimators.

- widely used in practice
- easily understood across multiple fields
- unbiased, consistent, and (under special conditions) efficient

#### LO 20.8: Describe the properties of OLS estimators and their sampling distributions, and explain the properties of consistent estimators in general.

### OLS Regression Results

#### LO 20.9: 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 $R^{2}$.

- the coefficient of determination (interpreted as a percentage of variation in the dependent variable explained by the indepedent variable)
- total sum of squares = explained sum of squares + sum of squared residuals (TSS = ESS + SSR)
- $\sum(Y
*{i} - \bar{Y})^{2} = \sum(\hat{Y} - \bar{Y})^{2} + \sum(Y*{i} - \hat{Y})^{2}$ - $R^{2} = \frac{ESS}{TSS}$

- difference between “correlation coeficient” and “coefficient of determination” (P137)

#### LO 20.10: Interpret the results of an OLS regression.

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

### Regression Coefficient Confidence Intervals

#### LO 21.1: Calculate, and interpret confidence intervals for regression coefficients.

### Regression Coefficient Hypothesis Testing

#### LO 21.3: Interpret hypothesis tests about regression coefficients

#### LO 21.2: Interpret the p-value

`p-value`

: the smallest level of significance for which the null hypothesis can be rejected

### Predicted Values

### Confidence Intervals for Predicted Values

### Dummy Variables

### What is Heteroskedasticity?

#### LO 21.4: *Evaluate the implications of homoskedasticity and heteroskedasticity.*

- homoskedastic (variance of residuals is constant across all observations in the sample)
- heteroskedasticity (opposite of homoskedastic)
- unconditional heteroskedasticity (usually causes no major problems with regression)
- conditional heteroskedasticity (does create significant problems for statistical inference)

### Effect of Heteroskedasticity on Regression Analysis

### Detecting Heteroskedasticity

### Correcting Heteroskedasticity

### The Gauss-Markov Theorem

#### LO 21.5: Determine the conditions under which the OLS is the best linear conditionally unbiased estimator.

#### LO 21.6: Explain the Gauss-Markov Theorem and its limitations, and alternatives to the OLS.

### Small Sample Sizes

#### LO 21.7: Apply and interpret the t-statistic when the sample size is small

## Linear Regression with Multiple Regressors

### Omited Variable Bias

#### LO 22.1: Define and interpret omitted variable bias, and describe the methods for addressing this bias.

### Multiple Regression Basics

#### LO 22.2: Distinguish between single and multiple regression

#### LO 22.5: Describe the OLS estimator in a multiple regression

#### LO 22.3: Interpret the slope coefficient in a multiple regression

#### LO 22.4: Describe homoskedasticity and heteroskedasticity in a multiple regression

### Measures of Fit

#### LO 22.6: Calculate and interpret measures of fit in multiple regression.

### Coefficient of Determination, $R^{2}$

### Adjusted $R^{2}$

- adjusted $R^{2} = 1 - \frac{n-1}{n-k-1} (1 - R^{2})$ is less than or equal to $R^{2}$

### Assumptions of Multiple Regression

#### LO 22.7: Explain the assumptions of the multiple linear regression model.

### Multicollinearity

#### LO 22.8: Explain the concept of imperfect and perfect multicollinearity and their implications.

### Effect of Multicollinearity on Regression Analysis

### Detecting Multicollinearity

- high p-value, high $R^{2}$

### Correcting Multicollinearity

## Hypothesis Tests and Confidence Intervals in Multiple Regression

#### LO 23.1: Construct, apply, and interpret hypothesis tests and confidence intervals for a single coefficient in a multiple regression.

### Hypothesis Testing of Regression Coefficients

### Determining Statistical Significance

### Interpreting p-Values

### Other Tests of the Regression Coefficients

### Confidence Intervals for a Regression Coefficient

### Predicting the Dependent Variable

### Joint Hypothesis Testing

#### LO 23.2: Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a multiple regression

#### LO 23.3: Interpret the F-statistic

#### LO 23.5: Interpret confidence sets for multiple coefficients

### The F-Statistic

`F-statistic`

: always a one-tailed test, calculated as $\frac{ESS/k}{SSR/(n-k-1)}$

### Interpreting Regression Results

### Specification Bias

### $R^{2}$ and Adjusted $R^{2}$

#### LO 23.7: Interpret the $R^{2}$ and adjusted $R^{2}$ in a multiple regression

*Restricted vs. Unrestricted Least Squares Models*

#### LO 23.4: Interpret tests of a single restriction involving multiple coefficients.

### Model Misspecification

#### LO 23.6: Identify examples of omitted variable bias in multiple regressions.

## Modeling and Forecasting Trend

### Linear and Nonlinear Trends

#### LO 24.1: Describe linear and nonlinear trends.

### Linear Trend Models

### Nonlinear Trend Models

### Estimating and Forecasting Trends

#### LO 24.2: Describe trend models to estimate and forecast trends.

### Selecting the Correct Trend Model

### Model Selection Criteria

#### LO 24.3: Compare and evaluate model selection criteria, including mean squard error (MSE), $s^{2}$, the Akaike information criterion (AIC), and the Schwarz information criterion (SIC).

### Mean Squared Error

### The $s^{2}$ measure

- $s^{2}$ measure (unbiased estimate of the MSE): $s^{2} = \frac{\sum
*{t=1}^{T} e*{t}^{2}}{T-k} = \frac{T}{T-k}\frac{\sum*{t=1}^{T} e*{t}^{2}}{T}$ - adjusted $R^{2}$ using the $s^{2}$ estimate

### Akaike and Schwarz Criterion

- Akaike information criterion: $e^{\frac{2k}{T}} \frac{\sum
*{t=1}^{T} e*{t}^{2}}{T}$ - Schwarz information criterion: $T^{\frac{k}{T}} \frac{\sum
*{t=1}^{T}e*{t}^{2}}{T}$

### Evaluating Consistency

#### LO 24.4: Explain the necessary conditions for a model selection criterion to demonstrate consistency

- the most consistent selection criteria with the greatest penalty factor for degrees of freedom is the SIC.

## Modeling and Forecasting Seasonality

### Sources of Seasonality

#### LO 25.1: Describe the sources of seasonality and how to deal with it in time series analysis.

- sources of seasonality
- approaches
- using a seasonally adjusted time series
- regression analysis with seasonal dummy variables

### Modeling Seasonality with Regression Analysis

#### LO 25.2: Explain how to use regression analysis to model seasonality

### Interpreting a Dummy Variable Regression

### Seasonal Series Forecasting

#### LO 25.3: Explain how to construct an h-step-ahead point forecast.

## Characterizing Cycles

### Covariance Stationary

#### LO 26.1: Define covariance stationary, autocovariance function, autocorrelation function, partial autocorrelation function, and autoregression.

#### LO 26.2: Describe the requirements for a series to be covariance stationary.

#### LO 26.3: Explain the implications of working with models that are not covariance stationary.

### White Noise

#### LO 26.4: Define white noise, and describe independent white noise and normal (Gaussian) white noise.

#### LO 26.5: Explain the characteristics of the dynamic structure of white noise.

### Lag Operators

#### LO 26.6: Explain how a lag operator works.

### Wold’s Representation Theorem

#### LO 26.7: Describe Wold’s theorem.

#### LO 26.8: Define a general linear process.

#### LO 26.9: Relate rational distributed lags to Wold’s theorem.

### Estimating the Mean and Autocorrelation Functions

#### Lo 26.10: Calculate the sample mean and sample autocorrelation, and describe the Box-Pierce Q-statistic and the Ljung-Box Q-statistic.

#### LO 26.11: Describe sample partial autocorrelation.

- sample autocorrelation: estimate the degree to which white noise characterizes a series of data
- sample partial autocorrelation: determine whether a time series exhibits white noise
- q-statistics: measure the degree to which autocorrelations vary from zero and whether white noise is present in a dataset
- Box-Pierce q-statistic: reflects the absolute magnitudes of the correlations
- Ljung-Box q-statistic: similar to the above measure

## Modeling Cycles: MA, AR and ARMA Models

### First-order Moving Average Process

#### LO 27.1: Describe the properties of the first-order moving average (MA(1)) process, and distinguish between autoregressive representation and moving average representation.

### MA(q) Process

#### LO 27.2: Describe the properties of a general finite-order process of order q (MA(q)) process.

### First-order Autoregressive Process

#### LO 27.3: Describe the properties of the first-order autoregressive (AR(1)) process, and define and explain the Yule-Walker equation.

### AR(p) Process

#### LO 27.4: Describe the properties of a general $p^{th}$ order autoregressive (AR(p)) process.

### Autoregressive Moving Average Process

#### LO 27.5: Define and describe the properties of the auto regressive moving average (ARMA) process.

### Applications of AR and ARMA Processes

#### LO 27.6: Describe the application of AR and ARMA processes.

## Volatility

### Volatility, Variance, and Implied Volatility

#### LO 28.1: Define and distinguish between volatility, variance rate, and implied volatility.

### The Power Law

#### LO 28.2: Describe the power law.

### Estimating Volatility

#### LO 28.3: Explain how various weighting schemes can be used in estimating volatility.

### The Exponentially Weighted Moving Average Model

#### LO 28.4: Apply the exponentially weighted moving average (EWMA) model to estimate volatility.

#### LO 28.8: Explain the weights in the EWMA and GARCH(1, 1) models.

### The GARCH(1, 1) Model

#### LO 28.5: Describe the generalized autoregressive conditional heteroskedasticity (GARCH(p, q)) model for estimating volatility and its properties.

#### LO 28.6: Calculate volatility using the GARCH(1, 1) model.

### Mean Reversion

#### LO 28.7: Explain mean reversion and how it is captured in the GARCH(1, 1) model.

### Estimation and Performance of GARCH Models

#### LO 28.9: Explain how GARCH models perform in volatility forecasting.

#### LO: 28.10: Describe the volatility term structure and the impact of volatility changes.

## Correlations and Copulas

### Correlation and Covariance

#### LO 29.1: Define correlation and covariance and differentiate between correlation and dependence.

### Covariance using EWMA and GARCH Models

#### LO 29.2: Calculate covariance using the EWMA and GARCH(1, 1) models

### EWMA Model

### GARCH(1, 1) Model

### Evaluating Consistency for Covariances

#### LO 29.3: Apply the consistency condition to covariance.

### Generating Samples

#### LO 29.4: Describe the procedure of generating samples from a bivariate normal distribution.

### Factor Models

#### LO 29.5: Describe properties of correlations between normally distributed variables when using a one-factor model.

### Copulas

#### LO 29.6: Define copula and describe the key properties of copulas and copula correlation

`copula`

: creates a joint probability distribution between two or more variables while maintaining their individual marginal distributions.- copula correlation

### Types of Copulas

#### LO 29.8: Describe the Gaussian copula, Student’s t-copula, multivariate copula, and one factor copula.

### Tail Dependence

#### LO 29.7: Explain tail dependence.

## Simulation Methods

### Monte Carlo Simulation

#### LO 30.1: Describe the basic steps to conduct a Monte Carlo Simulation

### Reducing Monte Carlo Sampling Error

#### LO 30.2: Describe ways to reduce Monte Carlo sampling error.

- increase replications
- antithetic variates
- control variates

### Antithetic Variates

#### LO 30.3: Explain how to use antithetic variate technique to reduce Monte Carlo sampling error

### Control Variates

#### LO 30.4: Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective

### Resulting Sets of Random Numbers

#### LO 30.5: Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them.

- Dickey-Fuller (DF) test: used to determine whether a time series is covariance stationary
- Different Experiments: