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Chapter 4. Using Regression to Estimate Trends. Trend Models. Linear trend, Quadratic trend Cubic trend Exponential trend. Choosing a trend. Plot the data, choose possible models Use goodness of fit measures to evaluate models Try to Minimize the AIC and SBC Choose a model.

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chapter 4

Chapter 4

Using Regression to

Estimate Trends

trend models
Trend Models
  • Linear trend,
  • Quadratic trend
  • Cubic trend
  • Exponential trend
choosing a trend
Choosing a trend
  • Plot the data, choose possible models
  • Use goodness of fit measures to evaluate models
  • Try to Minimize the AIC and SBC
  • Choose a model
goodness of fit measures
Goodness of Fit Measures
  • Coefficient of Determination or R2
aic and sbc continued
AIC and SBC(continued)
  • Choose the model that minimizes the AIC and SIC
  • Examples
    • choose AIC=3 over AIC=7
    • choose SIC=-7 over SIC=-5
  • The SIC has a larger penalty for extra parameters!
slide9

F-Test

The F-test tests the hypothesis that the coefficients of all explanatory variables are zero. A p-value less than .05 rejects the null and concludes that our model has some value.

testing the slopes
Testing the slopes
  • T-test tests a hypothesis about a coefficient.
  • A common hypothesis of interest is:
steps in a t test
Steps in a T-test
  • 1. Specify the null hypothesis
  • 2. Find the rejection region
  • 3. Calculate the statistic
  • 4. If the test statistic is in the rejection region then reject!
slide12

Figure 5.1 Student-t Distribution

f(t)

()

/2

/2

0

tc

t

-tc

red area = rejection region for 2-sided test

slide13

An Example,n=264

f(t)

.95

.025

.025

0

t

-1.96

1.96

red area = rejection region for 2-sided test

slide14

LS // Dependent Variable is CARSALES

Date: 02/17/98 Time: 13:44

Sample: 1976:01 1997:12

Included observations: 264

Variable Coefficient Std. Error t-Statistic Prob.

C 13.10517 0.311923 42.01413 0.0000

TIME 0.000882 0.005479 0.160947 0.8723

TIME2 2.52E-05 2.02E-05 1.248790 0.2129

R-squared 0.107295 Mean dependent var 13.80292

Adjusted R-squared 0.100454 S.D. dependent var 1.794726

S.E. of regression 1.702197 Akaike info criterion 1.075139

Sum squared resid 756.2412 Schwarz criterion 1.115774

Log likelihood -513.5181 F-statistic 15.68487

Durbin-Watson stat 0.370403 Prob(F-statistic) 0.000000

using our results
Using our results

Plugging in our estimates:

Not in the rejection region, don’t reject!

slide16

P-Value=lined area=.8725

f(t)

.95

.025

.025

0

t

-1.96

1.96

.016

red area = rejection region for 2-sided test

ideas for model building
Ideas for model building
  • F-stat is large, p-value=.000000 implies our model does explain something
  • “Fail to reject” does not imply accept in a t-test
  • Idea, drop one of the variables
slide18

LS // Dependent Variable is CARSALES

Date: 02/17/98 Time: 14:00

Sample: 1976:01 1997:12

Included observations: 264

Variable Coefficient Std. Error t-Statistic Prob.

C 12.81594 0.209155 61.27481 0.0000

TIME 0.007506 0.001376 5.454057 0.0000

R-squared 0.101961 Mean dependent var 13.80292

Adjusted R-squared 0.098533 S.D. dependent var 1.794726

S.E. of regression 1.704014 Akaike info criterion 1.073520

Sum squared resid 760.7597 Schwarz criterion 1.100611

Log likelihood -514.3044 F-statistic 29.74674

Durbin-Watson stat 0.368210 Prob(F-statistic) 0.000000