Chapter 5. Two-Variable Regression: Interval Estimation and Hypothesis Testing

Transcription

1 Chaper 5. Two-Variable Regression: Inerval Esimaion and Hypohesis Tesing

2 Inerval Esimaion: Some Basic Ideas ( ) δ + δ where 0 < Pr < Lower Confidence Upper Confidence Confidence Level Significance Level (Level of Error)

3 Confidence Inervals for Regression Z ( ) Where σ is known and, The area under Normal Curve is: ( ) σ x i µ±σ 68% µ±σ 95% µ±3σ 99.7% where ( ) σ σ ( ) X i X x i

4 Confidence Inervals for Regression However, σ is rarely known, if so: ( ) ( ) σ x i

5 Confidence Inervals for Regression T value follows -disribuion wih n- df. Then, insead of normal disribuion, disribuion is ud for confidence inerval of: ( ) [ ] ) ( : for inerval )% confidence 00( ) ( ) ( Pr ) ( Pr Pr ± +

6 Confidence Inerval for (Consan): [ ] ) ( : for )% confidence inerval 00( ) ( ) ( Pr ± +

7 Confidence Inerval for σ

8 Hypohesis Tesing: General Commens H 0 : he null hypohesis H : alernaive hypohesis I is simply a saemen o be proven by saisical echniques.

9 Hypohesis Tesing: The Confidence Inerval Approach Two Sided or Two Tail Tes H 0 : 0.3 H : 0.3 One Sided or One Tail Tes H 0 : 0.3 H : >0.3

10 Confidence Inerval for

11 Hypohesis Tesing: Tes of Significance Approach Tesing he Significance of Regression Coefficiens: The es ( ) ( ) σ x i ( ) [ ] ) ( : for )% confidence inerval 00( ) ( ) ( Pr ) ( Pr Pr ± +

12 Example: Le s assume as MPC ( df H H 0 : : 8 Then 0.05 :.306 : :, ) : * Two sided or wo ailed hypohesis esing Pr ( )

13 Marginal Propensiy o Consume:

14 One Sided or One Tailed Tes: H H 0 : : > * ( df : ) : ,05.860

15 - es

16 Tesing he Significance of σ : The χ Tes İe: H 0 : σ 85 H : σ 85

17 Rule of Thumb, - If he number of degrees of freedom is 0 or more and if, he level of significance, is a 0,05, hen he null hypohesis 0 can be rejeced if he compued value exceeds in absolue value.

18 The Exac Level of Significance: The p Value I is he probabiliy which measures he obrved or exac level of significance. I is defined also as he lowes significance level a which a null hypohesis can be rejeced.

19 In pracice: and p values If, > or p-value < Then, H 0 is rejeced everyime.

20 Regression Analysis and Analysis of Variance Where, TSS ESS + RSS Componens of TSS is known as he Analysis of Variance (ANOVA) from he regression viewpoin.

21 ANOVA Tes saisic of ANOVA is known as F, which is compued as above. F-es is ud for esing he model fi of regression. This opic is o be considered laer in he following chapers as well.

22 Reporing he Resuls of Regression Analysis: The Ca of MPC

23 Normaliy Tess for Residuals. Chi-Square Goodness of Fi Tes. Jarque-Bera Tes Boh he ess u he residuals, u i and chisquare disribuion.

24 The Chi-Square (χ ) Goodness of Fi Tes Run he regression Obain he residuals, u i Compue sandard deviaion of residuals Rank he residuals and pu hem ino veral groups χ k i ( O E ) i E i i Called Goodness of Fi Measure and follows Chi-Square Disribuion

25 Example o Normaliy Tes H 0 : Residuals are normally disribued H : Residuals are no normally disribued If X is 0.9 and corresponding p value is 0.63, hen he null hypohesis is acceped.

26 Jarque-Bera (JB) Tes of Normaliy JB n S 6 + ( K 3) 4 Where, S reprens skewness and K reprens Kurosis. Since for a normal disribuion, S is zero and K is 3, hen (K-3) is excess kurosis. If he p value is sufficienly low, hen he null hypohesis of normaliy assumpion is rejeced.