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

tt

timey

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MSE

10

1

2

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Goodness of Fit Measures

Coefficient of Determination or R2

2

22 1

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Goodness of Fit Measures

Adjusted R2

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)/(1 2

22

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AIC and SBC

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AIC

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2log)log( 1

2

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SIC

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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:

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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