Consider a linear model and y₁ = ẞ1 + ẞ2 (1 + α²) + ɛi, i = 1,3,5, ..., 2n - 1 yi = ß₁ + ß2 (1 − α²) + ɛi, i = 2,4,6, ...,2n where a is assumed to be a known constant and ɛi 's are independent and identically distributed N(0, 2) random variables. (i) Let B1 and B2 be the least squares estimators of ẞ₁ and ẞ2, respectively. Find the Var (C1ẞ₁ + C2ẞ2) for any constant C1 and C2 (ii) Find Var(B2 - B1). (iii) Find Var(B2 + ß1). (iv) Comment on the effect of a on the estimability of ẞ2 - ẞ₁ and ẞ2 + ẞ1.

Consider the linear regression model (with deterministic regressors) y = Bi + P2x, + €,, G - (0,0) i.i.d., i = 1.,n. Let ô = (,. BY be the OLS estimator of B = (B. P2) and (x,-x) be another estimator of B. True False Elyı] = Ely.) The estimator , is biased as EIA,) = B (x, - x1) The estimator , is an efficient estimator of Pi as only two observations are needed for its calculation. The estimator B, is a linear estimator of Pi .

Assume that we have the usual Simple Linear Regression model given by: Y₁ = Bo + Bixi +8 Show that in this case: Where Txy R² = rzy is the correlation coefficient between x and y.

Suppose we run a regression of Y on X using the following data points( xy)in a sample: (0, 1,(1,2).(-10, 9). What is the OLS regression line and what is the RSS? A:1+2X, RSS=3 B:1+X, RSS=0 C:2+X, RSS=-1 D:2+X, RSS=0

Consider the following structural model: 31 Bo + B1y2 + B₂²1 +21 Suppose now that there are two exogenous variables excluded from the model: 22 and 23. The assumptions that 2₂ and 23 do not appear in the model and are uncorrelated with the error ₁ are known as The linear combination that is most highly correlated with y2 is given by the reduced form equation for y2: Y2 = πO +1Z1 +T2Z2 + N3Z3+V2 E(v2) = 0, Cov(21, v2) = 0, Cov(22, 2) = 0, and Cov(23, 12) = 0. overidentifying restrictions exclusion restrictions 2SLS rank conditions

Find the least-squares regression line ŷ = bo + bịx through the points (-3, 1), (0, 6), (4, 15), (8, 19), (10, 23). For what value of x is ŷ = 0?

2. We will show a link between regression and the T-test. Suppose we have data of written of the form (Yij, xj), equal to, (Y11, 1), (12, 1), (Yin, 1) (Y21, 1), (22,-1),..., (2n-1) And assume the model, Yij = Bo+B₁xj + Eij, where Eij id N(0.02) (a) Show that Bo Y. : = == +Y=-12. 2n (b) Show for the fitted values, Y₁₁ = Y₁., and Y2j = Y2. for 1jn (c) Show that ²= 82 for the pooled variance t-test (d) Show that the test statistic for the hypothesis test Ho₁ = 0, HA: B00 can be written as, Y₁. - Y2. T And therefore the pooled variance T-test can be written via the SLR equation

y y 90 54 50 53 80 91 35 41 60 48 35 61 60 71 40 56 60 71 55 68 40 47 65 36 55 53 35 11 50 68 60 70 65 57 90 79 50 79 35 59 A data set consist of dependent variable (y) and independent variable (x) as shown above. It is claims that the relationship between the x and y can be modelled through a regression model as follows: ŷ = a+bx where a and b are the estimated values for a and ß (refer to Appendix). (i) Determine the equation of the regression line to predict the y value from the x value. (ii) If the x value is 75, what is the value of y? (iii) Test the hypothesis that a=10 against the alternative a <10. Use a 0.05 level of significance. (iv) Construct a 95% prediction interval for the y with x=35.

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