A find an expression for the value of the vector ˆ

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a.) Find an expression for the value of the vector ˆ β = (ˆ α, ˆ γ ) 0 such that the sum of squared residuals T X t =1 ( y t ˆ α ˆ γx t ) 2 , is as small as possible. (Note: you will receive full credit just for correctly writing down the expression for ˆ β , so if you remember it, you do not have to re-derive it here). b.) Recall the following formulas: sample means: ¯ x = T 1 T X t =1 x t and ¯ y = T 1 T X t =1 y t sample variances: S xx = T 1 T X t =1 ( x t ¯ x ) 2 and S yy = T 1 T X t =1 ( y t ¯ y ) 2 sample covariance: S xy = T 1 T X t =1 ( x t ¯ x )( y t ¯ y ) sample correlation: r xy = S xy p S xx S yy . Find an expression for your value for ˆ γ in part (a) in terms of ¯ x, ¯ y, S xx , S yy , S xy , and r xy . HINT: This calculation and the ones that follow will be easier if you exploit a general property of regressions that include a constant term. c.) The R 2 from the regression is de fi ned as R 2 = 1 P T t =1 ( y t ˆ α ˆ γx t ) 2 P T t =1 ( y t ¯ y ) 2 . Find an expression for R 2 in terms of ¯ x, ¯ y, S xx , S yy , S xy , and r xy . Try to simplify as much as you can. d.) What is the expression for the ordinary OLS F test of the hypothesis ˆ γ = 0? e.) Find the relation between your expression for the F statistic in part (d) and the value of R 2 in part (c).
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–3– 3.) (40 points total) A given researcher is interested in using a cross-section data set to study the e ff ect of a state’s education spending on the earnings of residents of the state. Let s i denote the average education expenditures (as a fraction of state income) for state i over 1990-1999 and let y i denote the average income per person in that state over the same period. The researcher proposes to estimate a regression of the form y i = β 1 + β 2 s i + ε i . The researcher proposes to use the estimated value of β 2 to advise policy-makers as to how much they might expect to be able to increase earning per person in the state if they were to allocate more resources to education. a.) Explain in words the possible source of simultaneous equations bias for this example.
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  • Winter '08
  • Hamilton,J
  • Maximum likelihood, Estimation theory, researcher, Xt, Yt, Sxx Syy

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