TA14a - ECO2121: Methods of Economic Statistics TA14a –...

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Unformatted text preview: ECO2121: Methods of Economic Statistics TA14a – 15 April 2009 This is a supplement to TA14. 1. Example of OLS Estimates Calculations Consider a data set of (Xi, Yi) for i = 1, 2,…,5, where Xi = years of schooling, Yi = weekly income ($), as follows: Xi (years of schooling) Yi (weekly income $) 12 5 14 7 15 7 16 8 18 10 So =− Therefore, we can calculate the following first: ⁄5 = 75/5 = 15 =∑ ⁄5 = 37/5 = 7.4 =∑ ∑ ( − ) = 20 (= (n – 1) × SX2 = 4×5) ∑ ( − )( − ) = ∑ ( − ) = 16 = 16/20 = 0.8, and For a simple linear regression =+ + , the OLS estimators of β0 and β1 are ∑ ( − )( − ) ∑ ( − ) = = ∑ (−) ∑ (−) = 7.4 – 0.8×15 = -4.6 = A plot is made for this regression line + : Using either we need to first calculate ∑ ( − ) = 13.2 (= (n – 1) × SY2 = 4×3.3), thus RSS = 13.2 – 0.8×16 = 0.4 or 13.2 – 0.82 ×20 = 0.4. Hence = 0.4/(5 – 2) = 0.4/3 = 2/15. Hereafter, +∑ ( ) 2) where RSS = ∑ ( − ) − ∑ ( − ). ~(0, ), but σ2 is unknown and therefore to be estimated by ∑ ( − )( − ) or ∑ = RSS/(n – ( −)− = (2/15) × (1/5 + 152/20) = 229/150 = 1.5267. =∑ ( ) = (2/15)/20 = 1/150, and = You are expected to do similar calculations without a computer. 2. Some Notations In the textbook, sometimes we write SSXX = ∑ ( − ) , SSYY = ∑ ( − ) , and SSXY = ∑ ( − )( − ), where SS denotes “Sum of Squares of ”. Hence = SSXY/SSXX. ∑ Sometimes we let =∑ ( ( ) ) , the nonrandom “weights”, so , which is linear in Yi, and we call a linear estimator in this sense. = Taylor 8 April 2009 ...
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