Data = Truth + Error

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