Post on 05-Jan-2016
description
Data = Truth + Error
A Paradigm for Any Data
Finding Truth in Forecasting
1. Smoothing: Truth can be “approximated” by
smoothing data.
2. Standard Models: Truth can be “approximated” by
“ a regression equation”
Key Attributes of Standard Models
• have simple forms
• have enjoyed good track records
• software for fitting is “widely” available
Notations
• j: Regression coefficient
– Other Greek symbols could be used occasionally.
• : Standard Deviation of Error,
Standard Trend Models
1. Specification
2. Using the model
• Estimation of parameters
• Interpretation of parameters
• Forecasting
3. Testing if the model is “good”
Linear Trend Models
• Linear: Yt = 0 + 1 t +
• Log-linear: ln (Yt) = 0 + 1 t +
follows White Noise - Random N(0, )
Interpretation of 1
• Linear:1 = Expected Increase of Y
• Log-linear:1 = Expected proportional Increase of Y
100 b1 = Expected % Increase of Y
Estimation of Model Parameters- Least Squares Method
• Determine the model parameters so that:
Sum (Residual t)2 is minimized.
• Eviews: ls
Actual, Fitted & Residual
Time, t
*
*
**
*
*Residual t
t
Yt
Fitted: Fit t
A trend Curve
Y
T
h Step Ahead Forecast | T
• Set = its expected value, 0
• Assume that parameters are estimated without error
• Set t = T+h
h=1 h=2
T+1T+2T
Point Forecast
• h – step ahead forecast
*
*
**
*
t
Yt
Yt
1
Interval Forecast
• Set the desired level of confidence, 95%, say.
• Interval forecast = point forecast + / - 1.96 SE
• SE is an estimate of SD of White Noise
Applications
• Performance of funds
• Growth of GDP
Trend Models – Two Types
• For unbounded data– linear– log-linear– quadratic– log-quadratic
• For bounded (S shaped) data– logistic– Gompertz
Unbounded Trend
• Linear: Yt = 0 + 1 t +
• Log-linear: ln(Yt)= 0 + 1 t +
• Quadratic: Yt = 0 + 1 t + 2 t2 +
• Log-quadratic: ln(Yt )= 0 + 1 t + 2 t2 +
Bounded S Curves
1. Logistic Curve
2. Gompertz Curve
Yt =
1+ exp(-t)
Yt = exp(- exp(-t))
S - Curves Point of Inflection
Time
Y
Concave Up Concave down
second derivative = 0
ln()/
Y(ln() /for L
Y(ln() /e for G
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
S – Growth Model Life Cycle Theory
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 …
Getting Initial Parameter Values
Logistic Curve
ln 1 ln1 exp *t
t
Y tt Y
Estimate from data, and compute
Regress the variable on t.
ln 1tY
Getting Initial Parameter Values
Gompertz Curve
ln ln lnte
tt
Y e tY
Estimate from data, and compute
Regress the variable on t.
ln lntY
Model Selection Process
1. Timeplot
2. Bounded?No
Yes
3. Take a log?No
Yes
Linear / Quadratic
Log - linear
Logistic / Gompertz
Applications
• MLB average salary
• Cardiac operations at a hospital
Recursive Estimation
• “Computing is for understanding”
• Recursive Estimation– An application of the principle– Experimentation, involving intensive
computation