<![CDATA[A heteroskedastic semiparametric regression model consists of two main components, i.e. parametric component and nonparametric component. The model assumes that any data (xÌ° i′ , t i , y i ) follows y i = xÌ° i′ β̰+ f(t i ) + σ i ε i , where i = 1,2, … , n , xÌ° i′ = (1, x i1 , x i2 , … , x ir ) and t i is the predictor variable. Parameter vector β̰ = (β 1 , β 2 , … , β r ) ′ ∈ â„œ r is unknown and f(t i ) is also unknown and is assumed to be in interval of C[0,Ï€] . Random error ε i is independent on zero mean and varianceσ 2 . Estimation of the heteroskedastic semiparametric regression model was conducted to evaluate the parametric and nonparametric components. The nonparametric component f(t i ) regression was approximated by Fourier series F(t) = bt + 12 α 0 + ∑ α k ð‘ ð‘œð‘ kt Kk=1 . The estimation was obtained by means of Weighted Penalized Least Square (WPLS): min f∈C(0,Ï€) {n −1 (y̰− Xβ̰−fÌ°) ′ W −1 (y̰− Xβ̰− fÌ°) + λ ∫ 2Ï€ [f ′′ (t)] 2 dt Ï€0 } . The WPLS solution provided nonparametric component fÌ‚Ì° λ (t) = M(λ)yÌ° ∗ for a matrix M(λ) and parametric component β̰̂ = [X ′ T(λ)X] −1 X ′ T(λ)yÌ°]]>
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