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Standard Trend Models

Standard Trend Models. Trend Curves. Purposes of a Trend Curve: 1. Forecasting the long run 2. Estimating the growth rate. Standard Trend Curves. Key Properties: have a simple form have good track records software for fitting is widely available. Types of Standard Trend Curves.

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Standard Trend Models

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  1. Standard Trend Models

  2. Trend Curves • Purposes of a Trend Curve: 1. Forecasting the long run 2. Estimating the growth rate

  3. Standard Trend Curves • Key Properties: • have a simple form • have good track records • software for fitting is widely available

  4. Types of Standard Trend Curves • For unbounded data: • linear • quadratic • exponential • For bounded (S shaped) data: • logistic • Gompertz

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

  6. Two Standard S Curves 1. Logistic Curve 2. Gompertz Curve

  7. S – Curves (Life Cycle Theory) 4 Stages of New 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

  8. 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

  9. Model Selection Process Linear / Quadratic Exponential (linear in log) (standard regression) 1. Timeplot 2. Take a log? No Yes 3. Bounded? No Yes Logistic / Gompertz/ (nonlinear regression)

  10. Nonlinear Least Squares • SPSS is one of the few statistics packages that provide routines for fitting nonlinear regression models. • You have to provide initial estimates for parameters.

  11. Getting Initial Parameter Values- Logistic Curve Estimate g from data, and compute Regress the variable on t.

  12. Getting Initial Parameter Values- Gompertz Curve Estimate g from data, and compute Regress the variable on t.

  13. Durbin-Watson Test

  14. White Noise Residuals • WN (white noise) – uncorrelated • Ex. et~ WN(0, s) (weak WN) • iid – independent and identically distributed • Ex. et ~ iid N(0, s) (strong WN)

  15. Spurious Trend Downward Bias: SE of Coefficient SER Positive Auto- Correlated Residual

  16. Trend Model With Correlated Residual

  17. Durbin Watson Statistic

  18. Some Key Values of DW Stat • E(DW) = 2 if H0 • Table available for DW if H0

  19. DW Test • The Null and Alternative Hypotheses • H0 : r = 0 • H1 : r > 0 -> positive autocorrelated residual

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