Studentized Versions of the Estimators of the Beta Parameters

Описание: We have been laying some important groundwork over the last several lectures.

We derived several important distributional characteristics of the least-squares estimators, in particular their means, variances, and standard errors.

We then determined the form of the sampling distributions of the least-squares estimators of the beta parameters when the model error terms are Normally distributed. When that is the case, the least-squares estimators are Normally distributed themselves. We also showed that in this situation the standardized versions of the least-squares estimators have a standard Normal distribution. However, because the expressions for the variances and standard errors of the least-squares estimators involve the variance of the model error terms, when that quantity is unknown, which it will be in really all situations in practice, we cannot calculate the exact values of the standardized versions of these estimators.

We then showed how we can ESTIMATE the variance of the model error term. This development led naturally to a two-sided F test about the slope coefficient in the Simple Linear Regression Model.

Using that estimator of the model error term, we then obtained estimators of the variances and standard errors of the beta parameters.

Now that we have estimators of the variances and standard errors of the beta parameters, we can calculate Studentized versions of those estimators and determine their sampling distributions. That is what we are going to discuss in this lecture video.

Once we have done that, we will be able to very easily develop procedures for testing more general hypotheses about the beta parameters, as well as procedures for constructing confidence interval estimates of those parameters.

Here is an outline of the topics covered in this video:

Studentizing Beta_1_Hat
Review the expressions for the Variance and Standard Error of Beta_1_Hat
Obtain estimators of the Variance and Standard Error of Beta_1_Hat
Calculate the Studentized Version of Beta_1_Hat and Identify its Sampling Distribution
Studentizing Beta_0_Hat
Review the expressions for the Variance and Standard Error of Beta_0_Hat
Obtain estimators of the Variance and Standard Error of Beta_0_Hat
Calculate the Studentized Version of Beta_0_Hatand Identify its Sampling Distribution

Next lecture in this series:    • Hypothesis Tests for the Beta Parameters i...  
Previous lecture in this series:    • Estimating the Variances and Standard Erro...  
First lecture in this series:    • Introduction to Linear Regression  

Here is the textbook we are using:
Title: "Introduction to Linear Regression Analysis 5th Edition"
Authors: Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining.
ISBN-13: 978-0470542811
ISBN-10: 0470542810
Link on Amazon: https://www.amazon.com/Introduction-R...

Software Packages Used:
Get access to SAS for free: https://www.sas.com/en_us/software/on...
Minitab: https://www.minitab.com/en-us/

Music Title: Urban Morning
Music Artist: Prigida
Music URL: https://tunetank.com/track/us/urban-m...

You can navigate to a particular section in the lecture by clicking on a timecode below:

Timecodes

0:00 Introduction
1:53 Outline
2:56 Review of the Form of the Estimators of the Variance and Standard Error of Beta_1_hat
5:17 Calculation and Distribution of the Studentized Version of Beta_1_hat
11:07 Review of the Form of the Estimators of the Variance and Standard Error of Beta_0_hat
13:07 Calculation and Distribution of the Studentized Version of Beta_0_hat
15:37 Wrap Up & Funky Music Outro
17:06 Related Lecture Videos