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 For unbounded data: –linear –quadratic –exponential For bounded (S shaped) data: –logistic –Gompertz

Unbounded Trend Linear: Y t =  0 +  1 t + e Quadratic: Y t =  0 +  1 t +  2 t 2 + e Log-linear: ln(Y t )=  0 +  1 t + e

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

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 S – Curves (Life Cycle Theory)

S - Curves Point of Inflection Time Y Concave UpConcave down second derivative = 0 ln(  )/  Y(ln(  ) /  for L Y(ln(  ) /  e for G

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

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.

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

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

Durbin-Watson Test

White Noise Residuals WN (white noise) – uncorrelated Ex.  t ~ WN(0,  ) (weak WN) iid – independent and identically distributed Ex.  t ~ iid N(0,  ) (strong WN) 

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

Trend Model With Correlated Residual

Durbin Watson Statistic

Some Key Values of DW Stat E(DW) = 2 if H 0 Table available for DW if H 0

DW Test The Null and Alternative Hypotheses –H 0 :  = 0 –H 1 :  > 0-> positive autocorrelated residual