Consider the linear regression modelYt = Bo + B1(x1)e + ..+ Bk(Xx)t + Ut, t 1,... T,where the regressors 1, x1,...,X and errors u satisfy the Gauss-Markov assumptions.Let R? be the (centered) R2 of the regression of xj on all other regressors including aconstant. Show that for the OLS estimators of B1,...Bk it holdsvar ß, =T. Sx(1-R)Remark: The factorRis called the variance inflation factor and is used as an indicator of multicollinearityHint. Use the Frisch-Waugh theorem to partial out all other regressors,

The ideal regression model with various assumptions is given as Description of an ideal dataset…

Answered: Yt = Bo + B1(x1)t + ·…· + Bx(Xx)t + Ue,…

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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Let Bo and ₁ be least square estimators for the simple linear regression model y = Bo+B₁x+e. Prove that E(ŝo) = ßo. Finish the proof based on the following hints. Hint 1: Note that Bo = ÿ – Â₁ñ. - E (B³) = € ( Ñ - B‹ ñ) = 6 (Y) – E(B₁A) - Hint 2: first calculate E(). Note that y = (y₁ + y₂ + ... y2 +yn)/n. In the class we already calculated what is E(y₁), E(y2),..., E(yn), based on these existing results, figure out what is E(ÿ). Yıt 2 € (g) = € ( Y₁+ y₂ + ² X ²) = √( @cy₁+ y₂ + ... ₁₂) есун n Yn) ., n as fixed (not random ..." Hint 3: next we calculate E(₁). Note that if we treat x1, x2, variable), then E(3₁x) = x · E(B₁).
Consider the regression model Y₁ = BX; +u; Y Where ui and X; satisfy the assumptions specified here. Let ẞ denote an estimator of ẞ that is constructed as ẞ = Show that ẞ is a linear function of Y₁, Y2,..., Yn. Show that ẞ is conditionally unbiased. 1. E (YiX1, X2,..., Xn) = == X + +Yn) 2. E(B|×1, X2,..., Xn) = E = B Χ | (X1, X2,..., Xn) = where Y and X are the sample means of Y; and X;, respectively.
Consider a simple linear regression model, Y₁ = Bo + B₁X₁ + ₁ for i=1,2,...,n with the usual random error term (&) assumptions. Suppose that hypotheses Ho: P₁ = 0 against H₁:₁ <0 were tested and Ho: B₁ = 0 was rejected. Assume ₁e₁² = 0. ei i) Briefly explain what the conclusion of the test means for the values of Y given the values of X. -1 ii) Draw an example of a scatter plot with the fitted regression line for the problem mentioned above. iii) Calculate and interpret the correlation coefficient for the problem at hand.
Consider the linear model y=B,+B,x,+B,x,,+B,x+B,+u, You estimate the model y =B,+B,x,+B,x, observations and obtain the OLS residuals . You then estimate the auxiliary regression +u based on 123 31 3" 3i 41 The LM statistic +1 4/ you obtain to test the null hypothesis that H:B,=B,=0 is 20.91. What is the R2 of the auxiliary regression? It is not possible to say O 0.17 O 0.175714 0.177203
Consider the linear regression y, = B, + Bx, + u;, i=1,...,n,n+1,..,n+ p where E(u,) = 0. Is it possible to observe a result that E(B,)+ B,, while E(B) = B,-
17) Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 41 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.9, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 90000 and the sum of squared errors (SSE) is 10000. From this information, what is the number of degrees of freedom for the t-distribution used to compute critical values for hypothesis tests and confidence intervals for the individual…
Example 15.11) The following table shows the marks obtained in two tests by 10 students: Marks in Ist Test (X) 8 8. 10 4 7 Marks in 2nd Test (Y) 8 7 7. 10 5 8. 10 6. (a) Find the least square regression line of Y on X. / 7,
Consider the following population model for household consumption: cons = a + b1 * inc+ b2 * educ+ b3 * hhsize + u where cons is consumption, inc is income, educ is the education level of household head, hhsize is the size of a household. Suppose a researcher estimates the model and gets the predicted value, cons_hat, and then runs a regression of cons_hat on educ, inc, and hhsize. Which of the following choice is correct and please explain why. A) be certain that R^2 = 1 B) be certain that R^2 = 0 C) be certain that R^2 is less than 1 but greater than 0. D) not be certain
Consider the following simple linear regression model, Y; = Po + B₁X₁ + εi, for i=1,2,...,n, where &'s are all independent and normally distributed with E(₁) = 0, and Var(₁) = 0². i) Check whether a statistic Y = Y + B₁ (X₁-X) is an unbiased estimator of the mean of the response variable E(Y) or not. Justify your conclusion.
2
6. Suppose we estimate a linear regression equation Y; = Bo + B1X; + u; by OLS. (a) Show thatE ûi 0, where the û;'s are the regression residuals. i3D1 (b) Suppose we regress X; on û; by OLS, including a constant term in the regression. Show that the estimated coefficient on û; is equal to zero. (c) Suppose we regress Y; on the predicted value Y; by OLS, including a constant term in the regression. Show that the estimated coefficient on Y; is equal to one. (Hint: the regression residual is defined by û; = Y; – Y; where the predicted value Y; = Bo + B,X;. Here, Bo and B1 are the OLS estimators.)
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