Simetar Download Download three files at http ge

Simetar Download • Download three files at: – http: //ge. tt/3 w 2 lhlr • Before installing Simetar do the following: – Download and install the latest Service Pack for Office 2003 or 2007 if using these systems – We do not run on Excel 2010 yet, install 2007 – If you have a Mac you must have Office and use Parallels to allow you to run Excel • Double click the Simetar. MSI file and use the license code provided in class to install Simetar on your computer

Multiple Regression Forecasts • Materials for this lecture • Demo Lecture 2 Multiple Regression. XLS • Read Chapter 15 Pages 8 -9 • Read all of Chapter 16’s Section 13

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 data generated over time). – – – Trend Cycle Seasonal variability Structural variability Irregular variability Black Swans

Patterns in Data Series

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

Cyclical Variation • Cycle is a recurring up and down movement around a trend • Cycles persist for 2 to 20 years from peak to peak – Business cycle is most notorious – Agriculture examples are cattle and hog cycles • Two components to a cycle – Expansion – Contraction • Cycles caused by – – Changes in tastes and preferences Economic activity (inflation and deflation) Production cycles (animals) Moon and sunspots? • Cycles vary in length and magnitude

Seasonal Variation • Periodic (cyclical) patterns in a time series that complete the cycle within a year • Caused by – Weather, seasons of the year – Production/marketing patterns – Customs and holidays • Agricultural production causes seasonal variation of prices • Holidays cause retail demand sales to vary on a seasonal pattern – – – Thanksgiving – turkey St. Patrick’s Day – corned beef and green beer Easter – ham Winter – cheese Summer – ice cream and gasoline sales Holidays and high temperatures – crushed ice

Structural Variation • Variables you want to forecast are often dependent on other variables Qt. Demand = f( Own Price, Competing Price, Income, Population, Season, Tastes & Preferences, Trend, etc. ) Y = a + b (Time) • Structural models will explain most structural variation in a data series – Even when we build structural models, the forecast is not perfect – A residual remains as the unexplained portion

Irregular Variation • Erratic movements in time series that follow no recognizable regular pattern – Random, white noise, or stochastic movements • Risk is this non-systematic variability in the residuals • This risk leads to Monte Carlo simulation of the risk for our probabilistic forecasts – We recognize risks cannot be forecasted – Incorporate risks into probabilistic forecasts – Provide forecasts with confidence intervals

Black Swans (BSs) • BSs low probability events – An outlier “outside realm of reasonable expectations” – Carries an extreme impact – Human nature causes us to concoct explanations • Black swans are an example of uncertainty – Uncertainty is generated by unknown probability distributions – Risk is generated by known distributions • Recent recession was a BSs – A depression is a BSs – Dramatic increases of grain prices in 2006 and 2007 – Dramtaic increase in cotton price in 2010

Multiple Regression Forecasts • Structural model of the forecast variable is used when suggested by: – – Economic theory Knowledge of the industry Relationship to other variables Economic model is being developed • Examples of forecasting: – – – Planted acres – inputs sales businesses need this Demand for a product – sales and production Price of corn or cattle – feedlots, grain mills, etc. Govt. payments – Congressional Budget Office Exports or trade flows – international ag. business

Multiple Regression Forecasts • Structural model Ŷ = a + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + e Where Xi’s are exogenous variables that explain the variation of Y over the historical period • Estimate parameters (a, bi’s, and SEPe) using multiple regression (or OLS) – OLS is preferred because it minimizes the sum of squared residuals – This is the same as reducing the risk on Ŷ as much as possible, i. e. , minimizing the risk for your forecast

Multiple Regression Model

Steps to Build Multiple Regression Models • Plot the Y variable in search of: trend, seasonal, cyclical and irregular variation • Plot Y vs. each X to see the structural relationship and how X may explain Y; calculate correlation coefficients to Y • Hypothesize the model equation(s) with all likely Xs to explain the Y, based on knowledge of model & theory • Forecasting wheat production, model is Plt Act = f(E(Pricet), Plt Act-1, E(Pth. Cropt), Trend, Yieldt-1) Harvested Act = a + b Plt Act Yieldt = a + b Tt Prodt = Harvested Act * Yieldt • Estimate and re-estimate the model • Make the deterministic forecast • Make the forecast stochastic for a probabilistic forecast

US Planted Wheat Acreage Model Plt Act = f(E(Pricet), Yieldt-1, CRPt, Yearst) • Statistically significant betas for Trend (years variable) and Price • Leave CRP in model because of policy analysis and it has the correct sign • Use Trend (years) over Yieldt-1, Trend masks the effects of Yield

Multiple Regression Forecasts • Specify alternative values for X and forecast the Deterministic Component • Multiply Betas by their respective X’s – Forecast Acres for alternative Prices and CRP – Lagged Yield and Year are constant in scenarios

Multiple Regression Forecasts • Probabilistic forecast uses ŶT+I and SEP or Std Dev and assume a normal distrib. for residuals ỸT+i = ŶT+i + NORM(0, SEPT) or ỸT+i = NORM(ŶT+i , SEPT)

Multiple Regression Forecasts • Present probabilistic forecast as a PDF with 95% Confidence Interval shown here as the bars about the mean in a probability density function (PDF)

Growth Forecasts • Some data display a growth pattern • Easy to forecast with multiple regression • Add T 2 variable to capture the growth or decay of Y variable • Growth function Ŷ = a + b 1 T+ b 2 T 2 Log(Ŷ) = a + b 1 Log(T) Double Log(Ŷ) = a + b 1 T Single Log See Decay Function worksheet for several examples for handling this problem

Multiple Regression Forecasts Single Log Form Log (Yt) = b 0 + b 1 T Double Log Form Log (Yt) = b 0 + b 1 Log (T)

Decay Function Forecasts • Some data display a decay pattern • Forecast them with multiple regression • Add an X variable to capture the growth or decay of forecast variable • Decay function Ŷ = a + b 1(1/T) + b 2(1/T 2)

Forecasting Growth or Decay Patterns • Here is the regression result for estimating a decay function Ŷt = a + b 1 (1/Tt) or Ŷt = a + b 1 (1/Tt) + b 2 (1/Tt 2)

Multiple Regression Forecasts • Examine a structural regression model that contains Trend an X variable Ŷ = a + b 1 T + b 2 Xt does not explain all of the variability, a seasonal or cyclical variability may be present, if so need to remove its effect

Goodness of Fit Measures • Models with high R 2 may not forecast well – If add enough Xs can get high R 2 – R-Bar 2 is preferred as it is not affected by no. Xs • Selecting based on highest R 2 same as using minimum Mean Squared Error MSE =(∑ et 2)/T

Goodness of Fit Measures • R-Bar 2 takes into account the effect of adding Xs where s 2 is the unbiased estimator of the regression residuals and k represents the number of Xs in the model

Goodness of Fit Measures • Akaike Information Criterion (AIC) • Schwarz Information Criterion (SIC) • For T = 100 and k goes from 1 to 25 • The SIC affords the greatest penalty for just adding Xs. • The AIC is second best and the R 2 would be the poorest.

Goodness of Fit Measures • Summary of goodness of fit measures – SIC, AIC, and S 2 are sensitive to both k and T – The S 2 is small and rises slowly as k/T increases – AIC and SIC rise faster as k/T increases – SIC is most sensitive to k/T increases

Goodness of Fit Measures • MSE works best to determine best model for “in sample” forecasting • R 2 does not penalize for adding k’s • R-Bar 2 is based on S 2 so it provides some penalty as k increases • AIC is better then R 2 but SIC results in the most parsimonious models (fewest k’s)