Consider the linear regression modely = XBo + u(1)where yis T × 1 with th element yt, X is T × k with tth row x, u is T × 1 withtth element ut, Bo is a k x 1 vector of unknown parameters. Assume that (1) isthe true model for y, X is fixed in repeated samples, rank(X) = k, E[u] = 0 andVar[u]=oIT for some unknown scalar constant o2. Let T = (X'X)-¹X'y thatis, BT is the Ordinary Least Squares (OLS) estimator of 30 based on (1).(a) Show that E[Br] = ßo.(b) Show that Var[ŜT] = o²(X'X)−¹.(c) Let A = = IT - IT(TT)-¹ where is a T x 1 vector of ones. Show that Ais a projection matrix.

Model y=Xβ0+uWe have to show the unbiasness, variance and projection matrix of multiple linear…

Answered: Consider the linear regression model y…

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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Suppose the PRF of two-variable regression is represented in the equation (1) Y₁ =b₁ + b₁X₁ + u₁, u~i.i.d N(0,0²) (1) We use Ordinary Least Square (OLS) method to estimate the coefficients in (1), The OLS estimated SRF is represented in the equation (2) Y₁ =B₁ + B₁X₁ + e, (2) In which , and B, are OLS estimates of b, and b, respectively. (2) Prove Σe, = 0 (3) Prove Sex = 0
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,-
Derive the least squares estimates of a and ß for the centred form of the simple linear regression model given by Yi = a + B(x; – I) + €; i= 1,2,..,n. Check that the estimates do give a minimum in the same way as we saw for the standard form of the simple linear regression model.
Consider a polynomial regression model Y = Bo + BrX + B2X +...+BX + Ui, where E (u₁|X₁, X²,..., X) = 0, observations (Y₁, X;) are independent and identically distributed and 0 < E (Y₂¹) < ∞, 0 < E (X¹¹) < ∞. How is the OLS estimation affected if we estimate a linear regression 1) when the true form of the regression function is quadratic (r = 2) or cubic (r = 3)? (r =
1. Consider two least-squares regressions and y = Xíễ tế y = Xí$i+ XzB2 tê Let R2 and R2 be the R-squared from the two regressions. Show that R22 R2.
1. Simple Linear Regression Estimation: (а) For the model y; B1 + B2x; + ui, define the fitted value ĝ; and residual û;. (b) x to arrive at estimates for B1 and B2? How does OLS take data on the outcome variable y and the independent variable Suppose you have the OLS estimate of the slope coefficient B2 from regressing ax; is equal to (c) y on x. Show that the slope coefficient if you regressed y on x* for x B2/a, where a is some constant. To be clear, I want you to show that B = B2/a, where B2 is the OLS estimate of B2 from the model Yi = B1 + B2x; + ui, and B is the OLS estimate of B; from the model y; = Bi + B5x + U;.
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.)
While shopping for an air conditione, a buyer bonsulted the following table giving a machine's BTUS and the square footage ( that it would cool t' (x) BTUS (y) 5100 5400 6350 6700 7275 7825 8300 8550 9750 9900 150 175 230 250 285315 350 360 435 | 450 (a) Find a linear model for the data. (b) Use the results from part (a) to find the number of BTUS required to cool rooms of 360 ft, 435 r, and 450 (c) Suppose a room measures 215 Use the results from part (a) to decide how many BTUS it requires if the air conditioners are available only with the BTU cholces in the table, which is the best choice? (a) Find a linear model for the data Find the least-aquares regression ine Once you have found the line in ynix+b form, input the values of m and b into the appropriate boxes below A linear model for the data is y (Round to the nearest whole number as needed) %3D
Compute the sum-of-squares error (SSE) by hand for the given set of data and linear model. (7, 7), (8, 8), (9, 10); SSE = y = x - 1
A statistics student is asked to estimate Y = Bo + B1X + E. She calculates the following values: Ex = 280, Σ(x₁ - x)² = 350, Σy, = 600, Σ(y, − y) = 1000 Σ(x,x)(y₁ - y) = -630, n = 20 Which of the following is the sample regression equation? OY--55.2-1.8X + e OY 55.2 +1.8X + e OY-55.2+1.8X + e OY 55.2 1.8X + e

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