Course Objectives

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
Time
ln(a)/b
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 gt 0 -gt positive autocorrelated
• residual