Final answer:
In this problem, we are given a population model and asked to find the expected value and variance of Y, derive the formula for the OLS estimator of β1, calculate the mean and variance of β1^, and prove the consistency of β1^ given certain conditions. The expected value of Y can be calculated as β0 + β1E(Xi), while the variance of Y is σ^2. The formula for the OLS estimator β1^ is [Σ(YiXi) + 2nE(Xi)] / Σ(Xi^2), and its mean and variance can be computed using the expected value and variance formulas. Finally, the consistency of β1^ can be proven by applying the Law of Large Numbers, given that ΣXi goes to infinity as n approaches infinity.
Step-by-step explanation:
a) Expected Value:
The expected value of a random variable Y is denoted as E(Y) and represents the average value that Y takes on over many repetitions of the experiment or data collection. For the given population model Yj = β0 + β1Xi + ui, we can calculate the expected value as follows:
E(Yj) = E(β0 + β1Xi + ui)
= E(β0) + E(β1Xi) + E(ui)
= β0 + β1E(Xi) + E(ui)
Since E(ui) = 0 for all i = 1, ..., n, the equation simplifies to:
E(Yj) = β0 + β1E(Xi)
Given that β0 = -2, the expected value becomes:
E(Yj) = -2 + β1E(Xi)
Variance:
The variance of a random variable Y, denoted as Var(Y), measures the spread or dispersion of the values that Y takes on. For the given population model Yj = β0 + β1Xi + ui, we can calculate the variance as follows:
Var(Yj) = Var(β0 + β1Xi + ui)
= Var(β0) + Var(β1Xi) + Var(ui)
Since Var(ui) = σ^2 for all i = 1, ..., n, the equation simplifies to:
Var(Yj) = Var(β1Xi) + σ^2
Given that the error terms ui are independent and the regressors Xi are fixed and positive, Var(β1Xi) = 0 (since β1 and Xi are constants) and the variance becomes:
Var(Yj) = 0 + σ^2
= σ^2
b) OLS Estimator:
The Ordinary Least Squares (OLS) estimator of β1, denoted as β1^, is obtained by minimizing the sum of squared errors (SSE) between the observed values of Y and the predicted values based on the model. The objective function for finding β1^ is given by:
min Σ(Yi - β0 - β1Xi)^2
To derive the formula, we take the derivatives of the objective function with respect to β0 and β1, set them equal to 0, and solve for β1:
d/dβ1 : Σ(Yi - β0 - β1Xi)(-Xi) = 0
Σ(YiXi) - β0Σ(Xi) - β1Σ(Xi^2) = 0
Σ(YiXi) - β0nE(Xi) - β1Σ(Xi^2) = 0
Σ(YiXi) - (-2)nE(Xi) - β1Σ(Xi^2) = 0
Σ(YiXi) + 2nE(Xi) - β1Σ(Xi^2) = 0
Therefore, the formula for β1^ is given by:
β1^ = [Σ(YiXi) + 2nE(Xi)] / Σ(Xi^2)
c) Mean and Variance of β1^:
The mean (expected value) of β1^ can be calculated by taking the expected value of the formula for β1^:
E(β1^) = E([Σ(YiXi) + 2nE(Xi)] / Σ(Xi^2))
Since E(Xi) is a constant, we can take it out of the expectation:
E(β1^) = [Σ(YiE(Xi)) + 2nE(Xi)^2] / Σ(Xi^2)
Similarly, the variance of β1^ can be calculated by taking the variance of the formula for β1^:
Var(β1^) = Var([Σ(YiXi) + 2nE(Xi)] / Σ(Xi^2))
Since the error terms ui are independent, we can treat the regressors Xi as fixed and non-random, which implies that Var(β1Xi) = 0. The variance becomes:
Var(β1^) = Var(2nE(Xi) / Σ(Xi^2))
= [2nE(Xi)]^2 Var(1 / Σ(Xi^2))
d) Consistency of β1^:
To prove the consistency of β1^, we need to show that as the number of observations (n) approaches infinity, β1^ converges to the true population parameter β1. Given that the sum of Xi from i = 1 to n goes to infinity as n approaches infinity (∑Xi → [infinity] as n → [infinity]), we can apply the Law of Large Numbers. According to the Law of Large Numbers, under certain conditions, the sample mean (β1^) will converge to the population mean (β1) as the sample size (n) increases. Therefore, β1^ is consistent.