AGEC 622

1. AGEC 622 • My mission is prepare you for a job in business Have you ever made a price forecast? • How much confidence did you place on your forecast? Was it correct? • We will learn how to make forecasts with confidence intervals • How to use risky forecasts in decision models for business

2. AGEC 622 • My job is not to be easy on people. My job is to make people better. Steve Jobs 2008 • Let’s go do something today rather than dwell on yesterdays mistakes. JWR 2012 • In computer modeling and simulation you learn by making mistakes • But to learn from your mistakes you have to figure out how to correct them

3. Forecasting and Simulation • Forecasters give a point estimate of a variable • Simulation preferred tool for risk analysis • Our goal is to merge these two fields of research and apply them to business decisions • We will use probabilistic forecasting and risk modeling for business decision analysis

4. Materials for this Lecture • Read Chapter 16 of Simulation book • Read Chapters 1 and 2 • Read first half of Chapter 15 on trend forecasting • Read journal article “Including Risk in Economic .. • Readings on the website • Richardson and Mapp • Including Risk in Economic Feasibility Analyses … • Before each class review materials on website • Demo for the days lecture • Overheads for the lecture • Readings in book

5. Simulate a Forecast • Two components to a probabilistic forecast • Deterministic component gives a point forecast Ŷ = a + b1 X + b2 Z • Stochastic component is ẽ and is used as: Ỹ = Ŷ + ẽ Which leads to the complete probabilistic forecast model Ỹ = a + b1 X + b2 Z + ẽ • ẽ makes the deterministic forecast a probabilistic forecast

6. Summarize Probabilistic Forecasting • Simulation provides an easy method for incorporating probabilities and confidence intervals into forecasts • Steps for probabilistic forecasting • Estimate best econometric model to explain trend, seasonal, cyclical, structural variability to get ŶT+i • Residuals (ê) are unexplained variability or risk; an easy way is to assume ê is distributed normal • Simulate risk as ẽ = NORM(0,σe) • Probabilistic forecast is ỸT = ŶT+i + ẽ

7. Introduction to Simulation • The future is risky but it is where we make profits and lose money • Without risk, little or no chance of profit • Simulation is the preferred tool for analyzing the effects of risk for a business decision • Analyze business/investment alternatives • Analyze alternative management strategies • Compare and rank risky decisions • Goal as risk analysts is to “help” decision makers by providing more information than a point forecast

8. Purpose of Simulation • … to estimate distributions that we can not observe and apply them to economic analysis of risky alternatives (strategies) so the decision maker can make better decisions • Profit = (P * Ỹ) – FC – (VC * Ỹ)

9. Major Activities in Simulation Modeling • Estimating parameters for probabilistic forecasts • Ỹt = a + b1Xt + b2 Zt + b3 Ỹt-1 + ẽ • The risk can be simulated with different distributions, e.g. • ẽ = NORMAL (Mean, Std Dev) or • ẽ = BETA (Alpha, Beta, Min, Max) or • ẽ = Empirical (Sorted Values, Cumulative Probabilities Values) or others • Use summary statistics, regression, and forecasting methods to develop the best forecast possible • Make the residuals, ê as small as possible • Estimate parameters (a b1 b2 b3), calculatethe residuals (ê) and specify the distribution for ê • Simulate random values from the distribution (Ỹt‘s) of each risky variable • Validate that simulated values come from their parent distribution • Model development, verification, and validation • Apply the model to analyze risky alternatives and decisions • Statistics and probabilities • Charts and graphs (PDFs, CDFs, StopLight) • Rank risky alternatives (SDRF, SERF)

10. Role of a Forecaster • Analyze historical data series to quantify patterns that describe the data • Extrapolate the pattern into the future for a forecast using quantitative models • In the process, become an expert in the industry so you can identify structural changes before they are observed in the data – incorporate new information into forecasts • In other words, THINK • Look for the unexpected

11. Forecasting Tools • Trend • Linear and non-linear • Multiple Regression • Seasonal Analysis • Moving Average • Cyclical Analysis • Exponential Smoothing • Time Series Analysis

12. Define Data Patterns • A time series is a chronological sequence of observations for a particular variable over fixed intervals of time • Daily • Weekly • Monthly • Quarterly • Annual • Six patterns for time series data (data we work with is time series data because use data generated over time). • Trend • Cycle • Seasonal variability • Structural variability • Irregular variability • Black Swans

13. Patterns in Time Series Data

14. Trend Variation • Trend a general up or down movement in the values of a variable over a historical period • Most economic data contains at least one trend • Increasing, decreasing or flat • Trend represents long-term growth or decay • Trends caused by strong underlying forces, as: • Technological changes • Change in tastes and preferences • Change in income and population • Market competition • Inflation and deflation • Policy changes

15. Simplest Forecast Method • Mean is the simplest forecast method • Deterministic forecast of Mean Ŷ = Ῡ = ∑ Yi / N • Forecast error (or residual) êi = Yi – Ŷ • Standard deviation of the residuals is the measure of the error (risk) for this forecast σe = [(∑(Yi – Ŷ)2/ (N-1)]1/2 • Probabilistic forecast Ỹ = Ŷ + ẽ where ẽ represents the stochastic (risky) residual and is simulated from the êiresuduals

16. Linear Trend Forecast Models • Deterministic trend model ŶT= a + b TT where Tt is time; it is a variable expressed as: T = 1, 2, 3, … or T = 1980, 1981, 1982, … • Estimate parameters for model using OLS • Multiple Regression in Simetar is easy, it does more than estimate a and b • Std Dev residuals & Std Error Prediction (SEP) • When available use SEP as the measure of error (stochastic component) for the probabilistic forecast • Probabilistic forecast of a trend line becomes Ỹt = Ŷt + ẽ Which is rewritten using the Normal Distribution for ẽ Ỹt = Ŷt+ NORM(0, SEPT) where T is the last actual data

17. Non-Linear Trend Forecast Models • Deterministic trend model Ŷt = a + b1Tt+ b2 Tt2+ b3 Tt3 where Tt is time variable is T = 1, 2, 3, … T2 = 1, 4, 9, … T3 = 1, 8, 27, … Estimate parameters for model using OLS • Probabilistic forecast from trend becomes Ỹt = Ŷt+ NORM(0, SEPT)

18. Steps to Develop a Trend Forecast • Plot the data • Identify linear or non-linear trend • Develop T, T2, T3 if necessary • Estimate trend model using OLS • Low R2 is usual • F ratio and t-test will be significant if trend is statistically present • Simulate model using ŶT+iand SEPT assuming Normal distribution of residuals • Report probabilistic forecast

19. Linear Trend Model • F-test, R2, t-test and Prob(t) values • Prediction and Confidence Intervals

20. Non-Linear Trend Regression • Add square and cubic terms to capture the trend up and then the trend down

21. Is a Trend Forecast Enough? • If we have monthly data, the seasonal pattern may overwhelm the trend, so final model will need both trend and seasonal terms (See the Demo for Lecture 1 ‘MSales’ worksheet) • If we have annual data, cyclical or structural variability may overwhelm trend so need a more complex model • Bottom line • Trend is where we start, but we generally need a more complex model

22. Confidence Intervals in Simetar Beyond the historical data you will find: SEP values in column 4 Ŷ values in column 2 Ỹ values in column 1

23. Meaning of the CI and PI • CI is the confidence for the forecast of Ŷ • When we compute the 95% CI for Y by using the sample and calculate an interval of YL to YU we can be 95% confident that the interval contains the true Y0. Because 95% of all CI’s for Y contain Y0 and because we have used one of the CI from this population. • PI is the confidence for the prediction of Ŷ • When we compute the 95% PI we call a prediction interval successful if the observed values (samples from the past) fall in the PI we calculated using the sample. We can be 95% confident that we will be successful.

24. Probabilistic Forecasting Models • Two types of models • Causal or structural models • Univariate (time series) models • Causal (structural) modelsidentify the variables (Xs) that explain the variable (Y) we want to forecast, the residuals are the irregular fluctuations to simulate Ŷ = a + b1 X + b2 Z + ẽ Note: we will be including ẽ in our forecast models • Univariate models forecast using past observations of the same variable • Advantage is you do not have to forecast the structural variables • Disadvantage is no structural equation to test alternative assumptions about policy, management, and structural changes Ŷt = a b1 Yt-1 + b2 Yt-2 + ẽ

25. 20% Return Y 30% Return X Application of Risk in a Decision • Given two investments, X and Y • Both have the same cash outlay • Return for X averages 30% • Return for Y averages 20% • If no risk then invest in alternative X

26. X -20 -10 0 10 20 30 40 50 Y Application of Risk in a Decision • Given two investments, X and Y • Cash outlay same for both X and Y • Return for X averages 30% • Return for Y averages 20% • What if the distributions of returns are known as: • Simulation estimates the distribution of returns for risky alternatives 0 10 20 30 40 50 60 70 80 90

27. Types of Forecasts • There are several types of forecast methods, use best method for problem at hand • Three types of forecasts • Point or deterministic Ŷ = 10.0000 • Range forecast Ŷ = 8.0 to 12.0 • Probabilistic forecast • Forecasts are never perfect