Finally consider the regression model with an extra covariate: Yi = Bo + B1.x 1i + B2x2; + €; Let 32=3.13, it's standard error s.e.(B1)=1.12 and assume n=81 Based only on this information, decide if it is useful to include x2i in the regression model. Carefully showing every step of your hypothesis test and explaining your answer. Use signifi- cance level = 05
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- A. B. Consider data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on n = 1,388 births: bwght = 119.772 (0.572) n = 1,388, 0.514 cigs (0.091) R² = 0.0227, where standard errors are shown in parenthesis. What percent of the variation in birth weight is explained by cigs? What is the predicted birth weight when cigs = 0? What about when cigs = 20 (one pack per day)? Comment on the difference.2. Consider a two variable regression model, which satisfies all the Gauss Markov assumptions except that the error variance is proportional to X² i.e.E(u?) = o²X? Y₁ = B₁ + B₂X₁ + Ui How would you obtain the best linear unbiased estimates from the above regression.Consider the regression model Yi=bot Bi Xitui Suppose that you know Bo = 0. Derive the formula for the least squares estimator of B₁. The least squares objective function is O A. n O B. O C. O D. E (Yi-bo-b1Xi) i=1 n Σ (Yi-bo-biXi) i=1 2 n Σ (v₁²-bo-b₁x₁²) i=1 n E (Yi-bo-b+Xi) 3 i=1
- Consider the IV regression model Yi = β0 + β1Xi + β2Wi + ui, where Xi is correlated with ui and Zi is an instrument. Suppose that the first three assumptions in Key Concept (The IV Regression Assumptions) are satisfied. Which IV assumption is not satisfied whena) Zi is independent of (Yi, Xi, Wi)?b) Zi=Wi?c) Wi is1 for all i?d) Zi=Xi?Consider the regression model Yi = β0 + β1Xi + ui.a. Suppose you know that β0 = 0. Derive a formula for the least squaresestimator of β1.b. Suppose you know that β0 = 4. Derive a formula for the least squaresestimator of β1?Consider the regression model Y₁ = BX; +u; Where u; 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. where Y and X are the sample means of Y; and X;, respectively. Show that ẞ is conditionally unbiased. 1. E (Y;|X1, X2,..., Xn) = 1 -B (X ₁ + X 2 + ... + Xn) Х answer these part correctly + ... + Y) +X₂+ ... + Xn) = B 2. E (B|×₁, ×2,..., Xn) = E | (X1, X2,..., Xn) BX; ☑ BX BY
- 1. For a regression model y = XB + u where u is N(0, o?1), y is nx1 matrix, X is nxp matrix, B is px1 matrix and u is nx1 matrix, a. derive the estimators B using the method of least squaresConsider the regression model Yi = b0 + b1X1i + b2X2i + ui. Use approach 2from Section 7.3 to transform the regression so that you can use a t-statistic to testa. b1 = b2.b. b1 + 2b2 = 0.c. b1 + b2 = 1. (Hint: You must redefine the dependent variable in theregression.)Consider the regression model Yi = β0 + β1X1i + β2X2i + β3(X1i * X2i) +ui. a. ΔY>/ΔX1 = β1 + β3X2 (effect of change in X1, holding X2 constant).b. ΔY/ΔX2 = β2 + β3X1 (effect of change in X2, holding X1 constant).c. If X1 changes by ΔX1 and X2 changes by ΔX2, then ΔY =(β1+β3X2)ΔX1 + (β2 + β3X1)ΔX2 + β3ΔX1ΔX2.
- Consider the following regression model where Suppose and are highly (but not perfectly) correlated. Then, a. b. C. d. e. OLS estimators are biased. OLS estimators are not consistent. OLS estimators will have large standard errors. One of,, or the constant should be dropped. cannot be interpreted as the population intercept.Introductory Econometris: A Modern Approach 4th edition, Chapter 17 Problem 1CE: What is the command in R in order to run the "White heteroskedasticity-consistent standard errors & covariance"? In other words, I would like to run the new regression with robust standard errors in it.Using a sample from a population of adults, to estimate the effects of education on health, we run the following regression: hypertension, = a + Beduc; + YX¡ + Ei where hypertension is a dummy variable equals one if a person suffers from hypertension and zero otherwise, educ is years of schooling, and X is a vector of demographic variables such as age, gender, and ethnicity. (a) Show that educ in the regression above is likely to be endogenous and discuss the consequences of this on the OLS estimators. (b) Evaluate whether a government policy that requires children to complete twelve years of schooling is a good instrumental variable for educ.