Using Regression to

1
Chapter 4

• Using Regression to
• Estimate Trends

2
Trend Models

• Linear trend,
• Quadratic trend
• Cubic trend
• Exponential trend

3
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

4
Mean Squared Error
5
Goodness of Fit Measures

• Coefficient of Determination or R2

6
Goodness of Fit Measures

• Adjusted R2

7
AIC and SBC
8
AIC and SBC(continued)

• Choose the model that minimizes the AIC and SIC
• Examples
• choose AIC3 over AIC7
• choose SIC-7 over SIC-5
• The SIC has a larger penalty for extra
  parameters!

9
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.
10
Testing the slopes

• T-test tests a hypothesis about a coefficient.
• A common hypothesis of interest is

11
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!

12
Figure 5.1 Student-t Distribution
f(t)


(???)
?/2
?/2
0
t
tc
-tc
red area rejection region for 2-sided test
13
An Example,n264
f(t)


.95
.025
.025
0
t
-1.96
1.96
red area rejection region for 2-sided test
14
LS // Dependent Variable is CARSALES Date
02/17/98 Time 1344 Sample 197601
199712 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
15
Using our results

• Plugging in our estimates
• Not in the rejection region, dont reject!

16
P-Valuelined area.8725
f(t)


.95
.025
.025
0
t
-1.96
1.96
.016
red area rejection region for 2-sided test
17
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

18
LS // Dependent Variable is CARSALES Date
02/17/98 Time 1400 Sample 197601
199712 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-stati
stic) 0.000000