chapter 4 using regression to estimate trends trend models zlinear trend, zquadratic trend zcubic...
TRANSCRIPT
Chapter 4
Using Regression toEstimate Trends
Trend Models
Linear trend, Quadratic trend
Cubic trend
Exponential trend
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tt timetimeY 221
tt timetimetimeY 33
221
tt timeY )exp( 10
Choosing a trend
Plot the data, choose possible models
Use goodness of fit measures to evaluate models
Try to Minimize the AIC and SBCChoose a model
Mean Squared Error
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ttt
T
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timey
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MSE
10
1
2
ˆˆ
ˆ
Goodness of Fit Measures
Coefficient of Determination or R2
2
22 1
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t
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eR
Goodness of Fit Measures
Adjusted R2
)1/(
)/(1 2
22
Tyy
kTeR
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t
AIC and SBC
T
k
T
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AIC
T
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2log)log( 1
2
T
Tk
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SIC
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)log(log)log( 1
2
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.
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Testing the slopes
T-test tests a hypothesis about a coefficient.
A common hypothesis of interest is:
0:
0:0
AH
H
Steps in a T-test
1. Specify the null hypothesis2. Find the rejection region3. Calculate the statistic4. If the test statistic is in the
rejection region then reject!
Figure 5.1 Student-t Distribution
()
t0
f(t)
-tc tc
/2/2
red area = rejection region for 2-sided test
An Example,n=264
.95
t0
f(t)
-1.96 1.96
.025
red area = rejection region for 2-sided test
LS // Dependent Variable is CARSALESDate: 02/17/98 Time: 13:44Sample: 1976:01 1997:12Included observations: 264
Variable Coefficient Std. Error t-Statistic Prob.
C 13.10517 0.311923 42.01413 0.0000TIME 0.000882 0.005479 0.160947 0.8723TIME2 2.52E-05 2.02E-05 1.248790 0.2129
R-squared 0.107295 Mean dependent var 13.80292Adjusted R-squared 0.100454 S.D. dependent var 1.794726S.E. of regression 1.702197 Akaike info criterion 1.075139Sum squared resid 756.2412 Schwarz criterion 1.115774Log likelihood -513.5181 F-statistic 15.68487Durbin-Watson stat 0.370403 Prob(F-statistic) 0.000000
Using our results
Plugging in our estimates:
Not in the rejection region, don’t reject!
1609.005479.
0000882.
t
P-Value=lined area=.8725
.95
t0
f(t)
-1.96 1.96
.025
red area = rejection region for 2-sided test
.016
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 CARSALESDate: 02/17/98 Time: 14:00Sample: 1976:01 1997:12Included observations: 264
Variable Coefficient Std. Error t-Statistic Prob.
C 12.81594 0.209155 61.27481 0.0000TIME 0.007506 0.001376 5.454057 0.0000
R-squared 0.101961 Mean dependent var 13.80292Adjusted R-squared 0.098533 S.D. dependent var 1.794726S.E. of regression 1.704014 Akaike info criterion 1.073520Sum squared resid 760.7597 Schwarz criterion 1.100611Log likelihood -514.3044 F-statistic 29.74674Durbin-Watson stat 0.368210 Prob(F-statistic) 0.000000