2. This problem explores various aspects of variance estimation in nonparametric re-gression.We have the usual model y = f + e where y is the n-vector of observations, f is then-vector of unknown function values at each z, and e is an n-vector with mean 0 andcovariance matrix Ia. Suppose we estimate f nonparametrically by a linear smoother Sy.Then, by analogy with the linear model, a reasonable estimate of o² is s = ll-Syl. Forthe linear model S is the 'hat' matrix H which maps observations to fitted values. Here S isa smoothing matrix and depends on the smoothing parameter A, but I have suppressed thisdependence for convenience. Unfortunately, while s is unbiased for the standard linearmodel, this is no longer the case for nonparametric smoothers.Verify this by showing that:n-tr(S)E(||u - Syll) = o{n - 2tr(S) + tr(S"s)] + 6" b,where b= f - Sf represents the smoothing bias.Note : This suggests an approximately unbiased alternative estimator, if b-0; see, forexample, Loader's (1999) book, or the book by Ruppert et al. (2003).

In this problem, we delve into the realm of variance estimation in nonparametric regression. We…

Answered: 2. This problem explores various…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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12. The OLS estimators of the coefficients in a multivariate linear regression model will be unbiased and consistent if the following set of conditions holds A. expected value of the error term given the independent variables is zero, the variance of the error term does not depend on the values of the independent variables, the observations (Yi,Xi,1,X₁,2,...,Xik) are i.i.d., large outliers are unlikely B. expected value of the error term given the independent variables is zero, the variance of the error term does not depend on the values of the independent variables, there is no perfect multicollinearity between the independent variables, the observations (Y₁X₁,1X₁,2,...,Xik) are i.i.d. C. expected value of the error term given the independent variables is zero, there is no perfect multicollinearity between the independent variables, the observations (Yi,Xi,1,Xi,2,...,Xik) are i.i.d., large outliers are unlikely D. the variance of the error term does not depend on the values of the…
1) The ratio of the regression mean square to the error mean square is: (a)Usually assumed to be 0.05 (b)The correlation coefficient (c)The covariance (d)The F ratio 2)The coefficient of determination, R2, ... a. Decreases as higher polynomial terms are included in the model b.Can be greater than 1.0 for transformed variables c.Measures the slope of the regression line d.None of these answers is correct 114 3. Assume you have experimental data that could be analyzed using a paired t-test or an unpaired t-test. If there are 22 degrees of freedom for the unpaired t-test, which value below describes the degrees of freedom for the paired t-test?
Which of the following is true of fixed effect estimators A. The fixed effects estimator is equal to the instrumental variable estimator if R^2 is equal to 1. B. The fixed effects estimators are biased if the regression model exhibits multicollinearity. C. The fixed effects estimators have lower variance than the ordinary least squares estimators. D. The fixed effects estimators have large standard errors when R^2 lies close to 0.
Hello, could I please get a solution for A through D
A weight-loss program wants to test how well their program is working. The company selects a simple random sample of 51 individual that have been using their program for 15 months. For each individual person, the company records the individual's weight when they started the program 15 months ago as an x-value. The subject's current weight is recorded as a y-value. Therefore, a data point such as (205, 190) would be for a specific person and it would indicate that the individual started the program weighing 205 pounds and currently weighs 190 pounds. In other words, they lost 15 pounds. When the company performed a regression analysis, they found a correlation coefficient of r = 0.707. This clearly shows there is strong correlation, which got the company excited. However, when they showed their data to a statistics professor, the professor pointed out that correlation was not the right tool to show that their program was effective. Correlation will NOT show whether or not there is…
Suppose that a regression relationship is given by the following:Y = β0 + β1X1 + β2X2 + εIf the simple linear regression of Y on X1 is estimated from a sample of n observations, the resulting slope estimate is generally biased for β1. However, in the special case where the sample correlation between X1 and X2 is 0, this will not be so. In fact, in that case the same estimate results whether or not X2 is included in the regression equation.a. Explain verbally why this statement is true.b. Show algebraically that this statement is true.
Is each of these True or False? (i) The least squares regression line is the line that minimizes the sum of the squares of the residuals. (ii) The least squares regression line must go through as many data points as possible. (iii) If a data point falls on the least squares regression line, then its residual is 0. (iv) The least squares regression line is the line that makes the square of the correlation as small as possible.
5. Indicate if each of the following statements about the simple linear regression model is true or false and explain why. (a). The simple regression line estimated by the method of Least Squared passes through the sample mean of dependent and independent variable. (b). The slope estimator of the Least Squares regression line is free of measure. (c). The sample covariance between the residuals from the least square regression and the explanatory variable is zero. (d). If the values of x more widely spread out, the estimate of the slope coefficient is less concise. (e). Adjusted R-squared will always increase when an additional independent variable is added into the regression.

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