Quantitative Methods

Defining Simulation Variables' Distributions

1) Historical Data 2) Cross-sectional (appropriate if peer data is representative of the subject) 3) Analyst subjective estimation

Decision Trees

-Discrete distribution of risk -Good for sequential data -Does not accommodate for correlated variables

Scenario Analysis

-Discrete distribution of risk -Not for sequential data -Accomodates correlated variables

Unit Root

-If value of lag coefficient is equal to one -Time series will follow a random walk process -Not covariance stationary -Eliminate w/ first differencing

Structural Change (Coefficient Instability)

-Indicated by a significant shift in plotted data at a point in time -Divides data into two distinct patterns -Need to run two different models to avoid unreliable results

Covariance stationary

-Requirement for using AR models -Mean, variance, and covariances w/ lagged and leading values do not change over time

Breush-Pagan Test

-Test for heteroskedasity -Correct w/ White-corrected st. errors

Root mean squared error (RMSE)

-Used to compare the accuracy of AR models in forecasting out-of-sample values -Lower RMSE = better predictive power

Standard Error of Estimate (SEE)

-measure of accuracy of predictions -approx. equal to st. dev. of residuals -if relationship between dependent and ind. vars is strong, SEE will be low

Treating Variable Correlation (Simulations)

1) Allow only one variable to vary and compute the other var. 2) Build correlation behavior into simulation

Probabilistic Approaches (Points to Remember)

1) Do not double count risk (i.e. adj. disc. rate AND applying a penalty) 2) Leads to better estimation of input variables 3) Does NOT lead to better estimation of exp. value or better decisions 4) When identifying probabilistic variables, focus attention on few variables that have a large impact

Limitations of Probabilistic Approaches

1) Input data quality 2) Inappropriate specification of statistical distribution 3) Non-stationary distribution 4) Non-stationary correlation

Linear Regression Assumptions

1) Lin. relationship exists between the dependent and independent variables 2) Ind. var. is uncorrelated w/ residuals 3) Exp. value of the residual term is zero 4) Variance of residual term is constant 5) Residual term is independently distributed 6) Residual term is normally distributed

F-stat

MSR/MSE = [RSS/k]/[SSE/(n-k-1)] w/ 1 and n-2 DOF

Simulation

Probabilistic Approach -Continuous distribution of risk -Doesn't matter if sequential data -Accommodates correlated variables

Variance (from ANOVA)

SST/n-1

Time Series Analysis

Step A: Evaluate situation, choose model Step B: Plot data. Check for covariance stationarity Step C: Decide between linear or log-linear model (calc. residuals, check for serial correlation) Step D: If serial correlation, prepare to use AR model by making it covariance stationary

F-test

Test for statistical significance of reg. parameter b1 H0: b1 = 0 Ha: b1 not equal to 0

First Differencing

Used to transform data w/ a unit root into a covariance stationary time series

Durbin Watson Decision Rule

Value of 2 -> no correlation Value less than 2 -> positive correlation Value greater than 2 -> negative correlation

Covariance

measures linear relationship between two random variables

Regression Parameter t-test

t = (b1hat - b1)/sb1

White Corrected Standard Errors Test

t = coefficient/white corrected Check if t < t-critical