Question 1: Show the validity of the third equality in the IV representation of the FE estimator: NT βrc = (Xx) xy ΣΣ (xit - Xi) (xit - Xi)" - ") it i=1 t=1 -1 N Τ Ν Τ ΣΣ (x - Xi) x(t) ΣΣ (xit - Xi) yit i=1 t=1 i=1 t=1 || - NT NT ΣΣzx) ΣΣziyit \i=1 t=1 i=1 t=1 -1 N T ΣΣ (xit - Xi) (yit - Ji) i=1 t=1
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- We know that the point estimator o` is an unbiased estimator for the parameter 0 if E(e') = 0.If the estimator is biased (not unbiased), then the difference E(O) – 0 is called the bias of the estimator e and is denoted by Bias(e'). That is,Suppose that @, and ô, are unbiased estimators of the parameter 0 and that V@,) = 15 and V(@2) = 4. What is the relative efficiency of the two estimators?The value of the correlation r = (x,y) for the information would be: n = 8, Sum(x) = 609, Sum(y) = 504, Sum (x)^2 = 47995, Sum (y)^2 = 32954, and Sum (xy) = 395650 Choose answer from the following: a. 0.855 b.0.174
- 1 3 5 7 3 4 6. According to the table, which of the following is the y'(1) value calculated by the forward finite difference Gregory-Newton interpolation relation?Exercise 4. (a) x²(5.7) ~ Gamma(?,?), (b) x²(8) ~ Gamma(?,?), (c) - Gamma(4, 3) ~ x² (?)3.a. Suppose Var(X1) = 2.25, Var(X2) = 2.56, and Cov(X1, X2) = -0.25. Find the correlation p of X1 and X2. b. Suppose E(Y;) = 13. The mean hourly wage W (in dollars) of chemists in the United States from 2000 through 2014 can be modelled by W = 0.903t + 26.08, 0 sts 14 where t represents the year, t = 0 corresponding to 2000. (a) Based from the model, when was the mean hourly wage at least $30, but not more than $32? (b) Use the model to predict when the mean hourly wage will exceed $45 (Larson, 2018).Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.Consider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choiceEvery year, the students at a school are given a musical aptitude test that rates them from 0 (no musical aptitude) to 5 (high musical aptitude). This year's results were: Aptitude Score 0 1 2 3 4 5 Frequency 4 1 1 4 4 1 The mean (T) aptitude score: The median aptitude score: (Please show your answer to 1 decimal place.) (Please separate your answers by ',' in bimodal situation. Enter DNE The mode aptitude score: if there is no mode.) Textbook Measures of CenterLet B₁ and 3₂ denote the OLS estimators of B₁ and 2 in the regression: yi= Bo + Bixi + B2x₁2 + Ui. Let B₁ denote the OLS estimators of B₁ in the regression Vi = Bo + Bixil + Vi. Let 8₁=1(x₁ - x₁)(X₁2 - X₂) / Σ²-₁(x₁₁ — X₁)². a) ₁ is the OLS estimator of a slope parameter from what regression model? b) Show that 1(Vi-Bo - B₁x₁₁ - B2x₁2) (Xi1 — x₁) = 0. c) Prove that B₁ =B1 + Ổ₁ß2. Explain the situations in which omitting x2 from the model will NOT bias the OLS estimate of B₁.This is a problem using the empirical ruleSEE MORE QUESTIONS