derive the general formula for leverage in a regression model using a single binary covariate with
Derive the general formula for leverage in a regression model using a single binary covariate wit...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
3. Consider the multiple linear regression model where Xii, . .. , Xp-i.i are observed covariate values for observation i, and εί udN(0, σ2) (a) What is the interpretation of in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vecto and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and 쓿 in matrix form. (d) Solve = 0 for β, the MLE of the...
Outline the general regression equation for a single index model and, from this, outline the expected return-beta relationship. Explain what is meant by the Security Characteristic Line, making reference to the alpha and beta estimates.
1.Derive the matrix formula for the coefficients of a linear model 2.Derive the matrix of velocity exchange for a couple colliding balls 3.Orthogonal operator versus orthogonal matrix (derive the basic property of an orthonormal matrix from a general definition of an orthogonal transformation). Prove existence of a rotation axis for 3D space
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
3. For a general regression model, show that Tt i=1
3. For a general regression model, show that Tt i=1
Derive the OLS estimator \hat{β}₀ in the regression model yi=β₀+ui. Show all of the steps in your derivation.
1) A regression model that involves a single independent variable is called ________. A) single linear regression B) simple unit regression C) simple linear regression D) individual linear regression
Logistic Regression
In class, we discussed the logistic regression model for binary classification problem. Here, we consider an alternative model. We have a training set {<n, yn) }n where E RD+1 and yn e {0,1}. Like in logistic regression, we will construct a probabilistic model for the probability that yn belongs to class 0 or 1, given en and the model parameters, 0, and 0 (0o,0, ERD+1). More specifically, we model the target Un as: p(yn = 0[xn;00,0) = Cella...
Question 1 Consider the simple regression model (only one covariate): y= BoB1 u Let B1 be the OLS estimator of B1. a) What are the six assumptions needed for B1 to be unbiased, have a simple expression for its variance, and have normal distribution? (3 points) b) Under Assumptions 1-6, derive the distribution of B1 conditional on x\,..., xn. (3 points) In lecture we described how to test the null hypothesis B1 bo against the alternative hypothesis B1 bo, where...