Hypothesis Tests and ConfidenceIntervals in Multiple Regression

The future 10-year real earnings growth in the S&P 500(EG10) can be explained by the trailing dividend payout ratio of the stocks in the index (PR) and the yield curve slope(YCS). Test the statistical significance of the independent variable PR in the real earnings growth at 10% significance level. Assume that the number of observation 43.

Coefficient Standard Error

Intercept -12% 1.285%

PR 0.30 0.023

YCS 0.12 0.290

LO 23.1 Statistical significance of a regression coeffiicient

Step 1 : Hypothesis Step 2 : Test statistics Step 3 : Critical value

H : b = 00 PR

H : b ≠ 0a PR

0.30 - 0

0.023

= 13.04 -1.684 +1.684

Outside the range ⸫ reject

10%, two tailed

Dof = 43-2-1= 40

Student’s T- Distribution

Example:

Solution:

https://www.fintreeindia.com/ © 2019 FinTree Education Pvt. Ltd.

FinTree

Watch video with important testable concepts here

P-value

5 ft. 5 ft.3.8 ft. 4.5 ft.4 ft. 6 ft.

Significance level

P-value

FTRFTR FTR

Reject

Reject

P-value is the lowest level of significance at which null hypothesis is rejected

Ÿ P values < Significance Level, Null hypothesis rejected.Ÿ P values > Significance Level, Null hypothesis can’t be rejected.

Interpreting P values :

Two Tail test

One Tail test

Test the null hypothesis that PR is equal to 0.15 versus the alternative that it is not equal to 0.15.Using 1% significance level.

Test the null hypothesis that the intercept term is less than or equal to -15.0% versus the alternative that it is greater than -15.0% using a 5% significance level.

Solution:

Step 1: Ho: PR = 0.15B1

Ha:PR ≠ 0.15B1

Step 2: T-statistics = = 6.5217

Step 3: Since T-statistics is very large therefore, Reject the null hypothesis.

Solution:

Step 1: Ho Bo≤-15% Ha Bo≥-15%

Step 2: t-statistics = = 2.33%

0.30-0.15

0.023

-12-(-15)

1.285

https://www.fintreeindia.com/ © 2019 FinTree Education Pvt. Ltd.

FinTree

Step 3:

The sign of alternate hypothesis can be used to understand direction of rejection area

H : B > -15a 0

+1.684

→ right direction

→ reject on right hand side of critical value

→ right tailed test

2.33 (test stat) ⸫ Reject null

Rejection area

Student’s T- Distribution

Confidence Interval for Regression coefficient

Predicted Value for Dependent Variable

Calculate the 95% confidence interval for the estimated coefficient for the independent variable PR in the real earnings growth example.

Solution: B ± (tc *S.E) = 0.30±2.02*0.023)1 v

= 0.253 to 0.346

Calculate the predicted 10-year real earnings growth for the S&P 500, assuming the payout ratio of the index is 60%.The slope of yield curve is currently 2%.

Solution:

(Eg ) = B +B (PR)+B2(YCS)10 0 1

= -12%+0.3(60%)+0.12(2%) = 6.24%

٨ ٨ ٨

https://www.fintreeindia.com/ © 2019 FinTree Education Pvt. Ltd.

FinTree

Step 1 : Hypothesis Step 2 : Test statistics Step 3 : Critical value

H : B = B = B = B0 1 2 3 4

= B = B = 05 6

H : at least one a

Bj ≠ 0

150

6= 25

Ÿ Mean explained error

=

480-150

53= 6.22

Ÿ Mean unexplained error

=

150

6Ÿ F = = 4.015

2.63

Rejection area

Ÿ Numerator Dof = K = 6

Ÿ Denominator Dof = n – k - 1= 53

(Closet is 60)

⸫ Reject

F test

LO 23.2, 23.3& 23.5

Joint hypothesis testing

Joint Hypothesis ª Joint hypothesis tests two or more coefficients at the same timeª A Robust method for applying joint hypothesis testing, especially when

independent variables are correlated, is known as F- Statistic

ü Test that at least one Independent Variable explains variation of Dependent Variable.

ü Always one-tailed test when testing hypothesis that all regression coefficients are simultaneously zero.(Rejection in tail)

F-Statistic

A regression is run of monthly value stock returns on six Independent variables over 60 months. The total sum of squares is 480 and the sum of squared residuals is 150.Test the null hypothesis at the 2.5% significance level that all the six independent variables are equal to zero.

Example:

F-Table at 2.5%

Ÿ Therefore, we can reject the null hypothesis and conclude that at least one of the six independent variables is significantly different than zero.

https://www.fintreeindia.com/ © 2019 FinTree Education Pvt. Ltd.

FinTree

LO 23.7 2 2 R and adjusted R

When computing both the R2 and the adjusted R2, there are a few pitfalls to acknowledge, which could lead to invalid conclusions

1. If adding an additional independent variable to the regression improves the R2, this variable is not necessary statistically significant.

2 . The R2 measure may be spurious, meaning that the independent variables may show a high R2; however, they are not the exact cause of the movement in the dependent variable.

3. If the R2 is high, we cannot assume that we have found all relevant independent variables. Omitted variables may still exist, which would improve the regression results further.

4. The R2 measure does not provide evidence that the most or least appropriate independent variables have been selected. Many factors go into finding the most robust regression model, including omitted variable analysis, economic theory, and the quality of data being used to generate the model

Restricted vs. Unrestricted Least Square Models

è Imposes value on one or more coefficients to analyze if restriction is significant

2 2 2è R is called restricted R or R r

è Includes both Independent variables.2 2

è Unrestricted R or R r

Restricted -

Unrestricted -

Refers how slope coefficient and other statistics for Independent Variable are different in simple regression compared to the same variables included in multiple regression.

Specification Bias

Approaches:ü Test restriction stated in null.ü Transforms regression and uses null hypothesis as an assumption to simplify

regression model.

LO 23.4 Hypothesis test of single Restriction involving Multiple coefficients

LO 23.6 Model misspecification

Omitted Variable Bias result if,

Ÿ It is a determinant of Dependent variable.

Ÿ It is correlated with at least one Independent variable.

Ÿ Known as Homoskedasticity - only F-Stat

Ÿ Alternative Formula is

Ÿ If error terms aren’t homoskedastic, Heteroskedastic-robust F-Stat is applied.

F-Stat to test restriction is significant or not,

(SSR - SSR )/mur r

SSR /(n - K - 1)ur ur

F =

https://www.fintreeindia.com/ © 2019 FinTree Education Pvt. Ltd.

FinTree