Unformatted text preview: ECO2121: Methods of Economic Statistics TA14 – 15 April 2009 「凡事都可行，但不都有益處。 凡事都可行，但不都造就人。 (聖經新約林前 10:23) 」 1. Simple Linear Regression Remember, regression is just conditional expectation. =+ + = 1, ⋯ , We want to know, conditional on X, what is E[YX]. X is nonrandom after conditioning, but Y is random due to ε is random. Assumptions: [  ] = 0 ∀ = 1, ⋯ , ; so [  ] = + (A1): [  ]=σ [  ]= (A2): ∀ = 1, ⋯ , ; so Sometimes we understand it is conditional on X, so we omit the notation “Xi”. The problem is that, β0 and β1 are unknown and therefore to be estimated. squares. (OLS) Different estimators (random variables) are used because of different properties, among many, the OLS is a popular one, which minimizes ∑ , the sum of residual = ∑ ( ∑ − )( − ) ∑ = ∑ (−) = − (−) (−) Expectation: = = ∑ ( − ) [ ] ∑ ( − )( + ) (−) = = ∑ (−) ∑ (−) (−) ∑ (−) ∑ (−) ∑ ( − )( − ) + = 0+ = ∑ (−) ∑ ( −) ∑ (−) ∑ ∑ = 1 − = [ ]− )− = 1 [ ]− = = ( + = + − Variance: = ∑ ∑ = It is also seen that ( , ) must lie in the predicted line by the OLS estimates because taking summation on =+ would give ∑ =+∑ By ∑ =∑ , and divides throughout by n, we get = + . By (A3):
∑ ( ) After estimating β0 and β1, we can “predict ” each Yi from a given Xi, as =+ . We can show that ∑ ̂ = 0 such that ∑ =∑ . = − = (−) (−) = (−) ∑ ( −) = (∑ ( − ) ) (∑ ( − ) ) ∑ (−) × = (∑ ( − ) ) ∑ (−) = [ ]+ −) 1 = 1 [ ]+ ∑ + ∑ ( = + ∑ ( −) . ε ~ N(0, σ2), where σ2 is unknown, or CLT, and +∑ ~
( or test H0: β0 = 0 and H0: β1 = 0 separately. For instance, to test H0: β0 = 0, test statistic =
) ,∑ ( ) . Thus we can construct C.I.’s for β0 and β1, = ~ , + , while for H0: β1 = 0, cases, the degree of freedom is n – 1. For large sample t(n – 1) converges to N(0, 1), thus the test statistic might be written as z0 instead. We shall estimate σ2, by =∑ ̂ ⁄( − 2), (n – 2) is there to adjust the degree of freedom (conceptually difficult – will be taught in later Econometrics course). Sometimes we call RSS = ∑ ̂ . It can be shown that RSS = ∑ ( − )− ∑ ( − )( − ) or ∑ ( −)− ∑ ( − ). ∑ ( ) . For both 2 April 2009 Taylor ...
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This note was uploaded on 11/16/2009 for the course ECO 2121 taught by Professor Professorwen during the Fall '08 term at Al Ahliyya Amman University.
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