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Data = Truth + Error

Data = Truth + Error. A Paradigm for Any Data. Finding Truth in Forecasting. Smoothing: Truth can be “approximated” by smoothing data. Standard Models: Truth can be “approximated” by “ a regression equation ”. Key Attributes of Standard Models. have simple forms

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Data = Truth + Error

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  1. Data = Truth + Error A Paradigm for Any Data

  2. Finding Truth in Forecasting • Smoothing: Truth can be “approximated” by smoothing data. • Standard Models: Truth can be “approximated” by “aregression equation”

  3. Key Attributes of Standard Models • have simple forms • have enjoyed good track records • software for fitting is “widely” available

  4. Notations • bj: Regression coefficient • Other Greek symbols could be used occasionally. • s: Standard Deviation of Error, e

  5. Standard Trend Models • Specification • Using the model • Estimation of parameters • Interpretation of parameters • Forecasting • Testing if the model is “good”

  6. Linear Trend Models • Linear: Yt = b0 + b1 t + e • Log-linear:ln (Yt) = b0 + b1 t + e efollows White Noise - Random N(0, s)

  7. Interpretation ofb1 • Linear:b1 = Expected Increase of Y • Log-linear: b1 = Expected proportional Increase of Y 100 b1 = Expected % Increase of Y

  8. Estimation of Model Parameters- Least Squares Method • Determine the model parameters so that: Sum (Residual t)2 is minimized. • Eviews: ls

  9. Actual, Fitted & Residual Y Residual t * Yt A trend Curve * * * Fitted: Fit t * * t T Time, t

  10. h Step Ahead Forecast | T • Set e = its expected value, 0 • Assume that parameters are estimated without error • Set t = T+h

  11. Point Forecast • h – step ahead forecast h=1 h=2 Yt Yt * * * * * t 1 T T+1 T+2

  12. Interval Forecast • Set the desired level of confidence, 95%, say. • Interval forecast = point forecast + / - 1.96 SE • SE is an estimate of s, SD of White Noise e

  13. Applications • Performance of funds • Growth of GDP

  14. Trend Models – Two Types • For unbounded data • linear • log-linear • quadratic • log-quadratic • For bounded (S shaped) data • logistic • Gompertz

  15. Unbounded Trend • Linear: Yt = b0 + b1 t + e • Log-linear:ln(Yt)= b0 + b1 t + e • Quadratic: Yt = b0 + b1 t + b2 t2 + e • Log-quadratic: ln(Yt )= b0 + b1 t + b2 t2 + e

  16. g Y = t a b 1+ exp(- t) Bounded S Curves 1. Logistic Curve 2. Gompertz Curve

  17. S - Curves Point of Inflection Y second derivative = 0 Y(ln(a) /b) = g/2 for L Y(ln(a) /b) = g /e for G Concave Up Concave down ln(a)/b Time

  18. S – Growth Model Life Cycle Theory 4 Stages of Technology Life Cycle: 1. Slow growth at the beginning stage 2. Rapid growth 3. Slow growth during the mature stage 4. Decline during the final stage

  19. Nonlinear Regression Using Eviews • Eviews is one of the few statistics packages that provide routines for fitting nonlinear regression models. • You might have to provide initial estimates for parameters for accuracy. • Eviews: param c(1) value c(2) value …

  20. Getting Initial Parameter ValuesLogistic Curve Estimate g from data, and compute Regress the variable on t.

  21. Getting Initial Parameter ValuesGompertz Curve Estimate g from data, and compute Regress the variable on t.

  22. Model Selection Process 1. Timeplot 2. Bounded? No Yes 3. Take a log? No Yes Linear / Quadratic Log - linear Logistic / Gompertz

  23. Applications • MLB average salary • Cardiac operations at a hospital

  24. Recursive Estimation • “Computing is for understanding” • Recursive Estimation • An application of the principle • Experimentation, involving intensive computation

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