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