Show explicitly that the following identity holds under a Simple Linear Regression: sum_(i=1)^n r_i mu_i=0 with residuals r_i=y_i-mu_i and mu_i=beta_0+beta_1 x_i

Davirnoilc 2022-11-17 Answered
Show explicitly that the following identity holds under a Simple Linear Regression:
  i = 1 n r i μ i ^ = 0
with residuals r i = y i μ i ^ and μ i ^ = β 0 ^ + β 1 ^ x i .
my steps:
i = 1 n r i μ i ^ = i = 1 n ( y i μ i ^ ) ( β 0 ^ + β 1 ^ x i ) =   i = 1 n ( y i β 0 ^ β 1 ^ x i ) ( β 0 ^ + β 1 ^ x i ) =   i = 1 n ( β 0 ^ y i + β 1 ^ x i y i β 0 ^ 2 β 0 ^ β 1 ^ x i β 0 ^ β 1 ^ x i β 1 ^ 2 x i 2 ) =   i = 1 n ( β 0 ^ y i + β 1 ^ x i y i ( β 0 ^ + β 1 ^ x i ) 2 ) =   i = 1 n ( y i ( β 0 ^ + β 1 ^ x i ) ( β 0 ^ + β 1 ^ x i ) 2 ) =   i = 1 n ( β 0 ^ + β 1 ^ x i ) ( y i β 0 ^ + β 1 ^ x i ) =   i = 1 n μ i ^ r i
how to proceed?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

cismadmec
Answered 2022-11-18 Author has 22 answers
After the third equality, you should use the linearity of the sum and notice that β ^ 0 and β ^ 1 don't depend on the index i. That way, you should get:
i = 1 n r i μ ^ i = β ^ 0 n y ¯ + β ^ 1 i = 1 n x i y i n β ^ 0 2 2 β ^ 0 β ^ 1 n x ¯ β ^ 1 2 i = 1 n x i 2 .
Now, you should use the fact that β ^ 0 = y ¯ β ^ 1 x ¯ and remember the normal equations, namely the second one:
β ^ 0 n x ¯ + β ^ 1 i = 1 n x i 2 i = 1 n x i y i = 0.
The result now directly follows.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-22
Show that: x i e i = 0 and also show that y ^ i e i = 0. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that e i = 0.
asked 2022-10-16
In a linear model, we defined residuals as:
e = y y ^ = ( I H ) y where H is the hat matrix X ( X T X ) 1 X T
and we defined standardized residuals as:
r i = e i s 1 h i i , i = 1 , . . . , n
where s 2 is the usual estimate of σ 2 , v a r ( e i ) = σ 2 h i i , and h i i is the diagonal entry of H at the i t h row and i t h column
Why r i and e i are functions of h i i rather than the whole row h i ?
asked 2022-07-15
Assume a linear regression model: y i = β 0 β 1 x i 1 +...+ β p x i p + ϵ i
C o v ( e , Y ^ )
where: e = the residuals vector
Y ^ = the predicted vector of Y
Use the fact that X T e
asked 2022-08-22
In a simple linear regression Y = X β + ε, residuals are given by ε ^ = M ε, where M = I n P is the annihilator matrix, and P = X ( X T X ) 1 X T is the projection matrix, and X is the design matrix. Assuing that the errors ε are iid normal with mean 0 and standard deviation σ, what is the joint (conditional on X) distribution of the residuals ε ^ ?
asked 2022-10-21
Compute using residuals the integral of the following function over the positively oriented circle | z | = 3
f   ( z ) = e z z 2
My solution: The only singular point of f in | z | 3 is z = 0 (double pole) and its remainder is therefore
Res z = 0 f ( z ) = lim z 0 1 ( 2 1 ) ! ( e z z 2 z 2 ) = lim z 0 e z = 1
Consequently, | z = 3 | f ( z ) = 2 π i Res z = 0 f ( z ) = 2 π i .
this right?
asked 2022-08-16
We've got some data containing two variables, where x is the predictor and y is the response variable. We make a model of the form of:
y = α + β x + ϵ
Then we see that in the residual plot (residuals vs. y ^ ) the variance is increasing as y ^ . We then decide to transform our model to a logarithmic form, i.e.:
l o g ( y ) = α + β x + ϵ
When performing a residual plot analysis, do we plot residuals vs. l o g ( y ) ^ or y ^ ?
asked 2022-09-24
Suppose there is a Quadratic relationship between a predictor, which exhibits a trend over time, and the response but we included only a linear term for that predictor in the linear regression model. Which of the following will happen if we use linear model for this regression model?
Will the diagnostics show autocorrelation in the residuals?
Do you think the residuals will add up to 0?

New questions

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)