Consider the following assumptions: 1. ?? = ?(? + ??) (data generating process) 2. E(?? )...
Consider the following assumptions:
1. ?? = ?(? + ??) (data generating process)
2. E(?? ) = 0 for all
3. Var(?? ) = ? 2 for all i
4. Cov(?? , ?? ) for ? ≠ ?
5. ?? ∼ ?????? And suppose you’re interested in generating an estimate for ?.
a. What is the expected value of the sample mean estimator, ?̂= 1 ? ∑?? , under these assumptions? Is ?̂an unbiased estimator for ?? Show all work and state assumptions used.
b. Derive the variance for the sample mean under these assumptions. Show all work and state assumptions used.
c. Given your derivations thus far, and the assumptions listed above, what is the distribution of the sample mean?
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Consider the following assumptions:
1. ?? = ?(? + ??) (data generating process)
2. E(?? )...
1. Select all true statements about sample mean and sample median. A) When the population distribution...
1. Select all true statements about sample mean and sample median. A) When the population distribution is skewed, sample mean is biased but sample median is an unbiased estimator of population mean. B) When the population distribution is symmetric, both mean and sample median are unbiased estimators of population mean. C) Sampling distribution of sample mean has a smaller standard error than sample median when population distribution is normal. D) Both mean and median are unbiased estimators of population mean...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the dis...
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QUESTION 5 Suppose you obtained one sample from the same data generating process as above (Y...
QUESTION 5 Suppose you obtained one sample from the same data generating process as above (Y = 2 + 3x+ u), and your estimate β 1 (the estimated X coefficient) was 2.107. Which of the following best describes whether the estimator is biased? The estimator is biased because the estimate 2.107 is too far below the population parameter of 3 The estimator is not biased because the estimate 2.107 is within 30 percent of the population parameter 3. We do...
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1. Consider a variable y = θ+e where θ is an unknown parameter and e is a random variable with me...
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Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) =...
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) = {pos, li - j| > 1. Let X and S2 be the sample mean and variance, respectively. a. Show that X is a consistent estimator for u. b. Is S2 unbiased for 02? Justify. - c. Show that S2 is asymptotically unbiased for 02.
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Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
QUESTION 5 Suppose you obtained one sample from the same data generating process as above (Y = 2 + 3x+ u), and your estimate β 1 (the estimated X coefficient) was 2.107. Which of the following best describes whether the estimator is biased? The estimator is biased because the estimate 2.107 is too far below the population parameter of 3 The estimator is not biased because the estimate 2.107 is within 30 percent of the population parameter 3. We do...
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Question 2 (0.5 mark) Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4: y = B. +B,X,+B2x2 +Bzx3+u You are interested in estimating the sum of the parameters on Xı and xz; call this 0 = + B2 (1) Show that Ô, = B1 + B2 is an unbiased estimator of , (ii) Find Varê, in terms of Varhi). Var(82), and Corr1. B2).
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
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