11+Regression

Parameters i are obtained if we minimize the sum of

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Unformatted text preview: own), xi are variables measured without error, ε - is a random error component, the sum k Ey = βi xi = β0 + β1 x1 + · · · + βk xk i=0 is the deterministic portion of the model, βi determines the contribution of the independent variable xi (i = 1 : k ). For a sample of size n we may rewrite this model more detail y1 = β0 + β1 x11 + · · · + βk x1k + ε1 y2 = β0 + β1 x21 + · · · + βk x2k + ε2 ................................. yn = β0 + β1 xn1 + · · · + βk xnk + εn As before we make assumptions about the random error ε 1. The mean of the probability distribution of ε equals to 0. 2.The variance σ 2 of the probability distribution of ε is constant for all x1 , ..., xk . 3.The probability distribution of ε is normal. 4. The random values of ε are independent. Estimation of σ 2 . The estimator of σ 2 is σ 2 = s2 = ˆ where SSE , n−k−1 n n ε2 i SSE = (yi − yi )2 , ˆ = i=1 i=1 ˆ ˆ ˆ yi = β0 + β1 xi1 + · · · + βk xik ˆ (i = 1 : n), ˆ coefficients βj (j = 1 : k ) minimize the error SSE , ˆ ˆ¯ β 0 = y − β 1 x1 − · · · − β k xk ¯ ˆ¯ 10 A. Zhensykbaev and Regression ˆ yi = y + β1 (xi1 − x1 ) + · · · + βk (xik − xk ) ˆ ¯ˆ ¯ ¯ (i = 1 : n), In the case k = 2 SSx2 x2 SSx1 y − SSx1 x2 SSx2 y ˆ β1 = , 2 SSx1 x1 SSx2 x2 − SSx1 x2 SSx1 x1 SSx2 y − SSx1 x2 SSx1 y ˆ β2 = , 2 SSx1 x1 SSx2 x2 − SSx1 x2 ˆ ˆ¯ β 0 = y − β 1 x1 − β 2 x2 , ¯ ˆ¯ ˆ yi = y + β1 (xi1 − x1 ) + β2 (xi2 − x2 ) ˆ ¯ˆ ¯ ¯ (i = 1 : n), where as usually 1 xi = ¯ n n 1 y= ¯ n xνi , ν =1 n SSxi xj = n yν , ν =1 n (xνi − xi )(xνj − xj ), ¯ ¯ SSxi y = (xνi − xi )yν ¯ ν =1 (i, j = 1, 2). ν =1 Inferences about the β parameters. Parameters βi are obtained if we minimize the sum of squares of errors SSE . Inferences about the individual parameters βi are obtained using either test hypotheses or a confidence interval. We consider the case k = 2. Hypothesis about βi . The null hypothesis is H0 = {βi = 0}. The test statistic is a t-statistic ˆ βi t= , sβi ˆ where √ sβi = σ cii , ˆ ˆ (i = 1, 2), 1 σ= ˆ n−3 n 2 11 (yν − yν )2 , ˆ ν =1 A. Zhensykbaev c11 = Regression SSx2...
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