Transcript Chapter 3

Chapter 4
Using Regression to
Estimate Trends
Trend Models
Linear trend, Y      time  
t
t
Quadratic trend
Yt    1  time  2  time   t
2
Cubic trend
Yt    1  time  2  time  3  time   t
2
Exponential trend
Yt  0 exp( 1time)   t
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
Mean Squared Error
T
MSE 

2
et
t 1
T
et  yt  yˆ t
yt  ˆ0  ˆ1timet
Goodness of Fit Measures
Coefficient of Determination or R2
R  1
2
e
y
2
t
 yt 
2
t
Goodness of Fit Measures
Adjusted R2
R  1
2
e
y
t
2
/(T  k )
 yt  /(T  1)
2
t
AIC and SBC
 T 2
 et 
 t 1  2k
log( AIC )  log


T
T







 T 2
 et 
 t 1  k log(T )
log(SIC )  log


T
T







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!
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.
ˆ  Y  /(k  1)

Y

F
~F
ˆ  /(n  k )

Y

Y

2
t
t
2
t
t
k ,n  k
Testing the slopes
T-test tests a hypothesis about a
coefficient.
A common hypothesis of interest is:
H0 :   0
HA :   0
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!
Figure 5.1 Student-t Distribution
f(t)
()
/2
-tc
0
/2
tc
t
red area = rejection region for 2-sided test
An Example,n=264
f(t)
.025
-1.96
.9
5
0
.025
1.96 t
red area = rejection region for 2-sided test
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
TIME 0.000882
TIME2 2.52E-05
0.311923
0.005479
2.02E-05
42.01413
0.160947
1.248790
0.0000
0.8723
0.2129
R-squared
0.107295
Adjusted R-squared 0.100454
S.E. of regression 1.702197
Sum squared resid 756.2412
Log likelihood
-513.5181
Durbin-Watson stat 0.370403
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
13.80292
1.794726
1.075139
1.115774
15.68487
0.000000
Using our results
Plugging in our estimates:
.000882  0
t
 .1609
.005479
Not in the rejection region, don’t reject!
P-Value=lined area=.8725
f(t)
.025
-1.96
.9
5
0
.025
1.96 t
.016
red area = rejection region for 2-sided test
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
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
TIME
0.209155
0.001376
61.27481
5.454057
0.0000
0.0000
0.101961
0.098533
1.704014
760.7597
-514.3044
0.368210
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
12.81594
0.007506
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
13.80292
1.794726
1.073520
1.100611
29.74674
0.000000