AGEC 622

• 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

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

• Finally we use probabilistic forecasting and risk modeling for business decision analysis

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

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 model of Ỹ = a + b1 X + b2 Z + ẽ

• ẽ makes the deterministic forecast a probabilistic forecast

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Probabilistic Forecast

Summarize Probabilistic Forecasting

• Simulation provides an easy method for incorporating probabilities and confidence intervals into forecasts

• Steps for probabilistic forecasting1. Estimate best econometric model to explain

trend, seasonal, cyclical, structural variability to get ŶT+i

2. Residuals (ê) are unexplained variability or risk, easy way is to assume ê is distributed normal

3. Simulate risk as ẽ = NORM(0,σe)

4. Probabilistic forecast is ỸT = ŶT+i + ẽ

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

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)

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Probabilistic Forecast

Major Activities in Simulation Modeling

• Estimating parameters for probability distributions– Ỹt = a + b1 Xt + b2 Zt + b3 Ỹt-1 + ẽ – The risk can be simulated with different distributions, e.g.

• ẽ = NORMAL (Mean, Std Dv)• ẽ = BETA (Alpha, Beta, Min, Max)• ẽ = Empirical (Sorted Values, Cumulative Probabilities Values)

– 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), calculate the residuals (ê)

and specify the distribution for ê• Simulate random values from the distribution for the

residuals 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

– Statistics and probabilities– Charts and graphs (PDFs, CDFs, StopLight)– Rank risky alternatives (SDRF, SERF)

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

Forecasting Tools• Trend

– Linear and non-linear• Multiple Regression• Seasonal Analysis• Moving Average• Cyclical Analysis• Exponential Smoothing• Time Series Analysis

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

Patterns in Data Series

Trend Seasonal

Cycle Mixed

0 10 20 30 40

years

years

monthsperiods J J J J

Trend Seasonal

Cycle Mixed

0 10 20 30 40

years

years

monthsperiods J J J J

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

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 forecastσe = [(∑(Yi – Ŷ)2/ (N-1)]1/2

• Probabilistic forecast Ỹ = Ŷ + ẽwhere ẽ represents the stochastic (risky) residual and is simulated from the êi

resuduals

Linear Trend Forecast Models

• Deterministic trend modelŶT = a + b TT

where Tt is time 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 becomesỸt = Ŷt + ẽ Which is rewritten using the Normal Distribution for ẽỸt = Ŷt + NORM(0, SEPT) where T is the last actual data

Non-Linear Trend Forecast Models

• Deterministic trend modelŶt = a + b1 Tt + b2 Tt

2 + b3 Tt

3

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 becomesỸt = Ŷt + NORM(0, SEPT)

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+i and SEPT assuming Normal distribution of residuals

• Report probabilistic forecast

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

Non-Linear Trend Regression

• Add square and cubic terms to capture the trend up and then the trend down

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

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

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.

Probabilistic Forecasting Models

• Two types of models– Causal or structural models– Univariate (time series) models

• Causal models identify the variables that explain the variable we want to forecast, the residuals are the irregular fluctuations to simulateŶ = 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

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

20% Return Y 30% Return X

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

-20 -10 0 10 20 30 40 50

X

Y

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