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Unformatted text preview: around the true value, or whether the mean of the estimator is equal to the true value. ˆ
It means that if we got many different samples, calculating θ for each sample, then ˆ
those θ would tend to be right on average. The distribution centers around the true value of θ . ˆ
Bias: the bias of an estimator θ is defined as E (θ ) − θ . Thus, for an unbiased estimator, bias =0 11 Figures: Prob. distributions of biased vs. unbiased estimators Examples: 1. For any random variable X, the sample average x = 1n
∑ xi is an unbiased n i =1 estimator for the population mean μ . E(x ) = E( 1n
xi ) = ∑ E ( xi ) = ∑ μ = nμ = μ
n i =1
n i =1
n i =1
n The OLS estimator is unbiased or valid under the following four assumptions. Assumption 1: the population model is linear in parameters as Y =α +β X +u One way Assumption 1 can be violated is when the true model is Y = α + β1 X + β 2 X 2 + u , but you estimate a model of the form Y = α + β1 X + u . For example, research shows that income has an inverse‐U shape in age, so if you estimate a simple linear model of income on age, you’ll get a biased slope estimate. Assumption 2: we have a random sample of size n...
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