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In order to simplify the computation, it is practical to take the logarithm of equation (7), which yields the following log-likelihood function: (8) log L (θ, σ 2 ) = − M − 1 2 log (2 π σ 2 ) − 1 2 σ 2 ∑ k = 1 M − 1 [ x [ t k ] − φ [ t k ] ⋅ θ ] 2 In equation (8), x [ t k ] and φ[ t k ] are the kth elements of X and Φ respectively.
In order to simplify the computation, it is practical to take the logarithm of equation (8), which yields the following log-likelihood function: (9) log L (θ, σ 2 ) = − m − 1 2 log (2 π σ 2 ) − 1 2 σ 2 ∑ k = 1 m − 1 [ y [ t + k + 1 ] − ϕ [ t + k ] ⋅ θ ] 2 where y [ t + k] and ϕ[ t + k] are the k-th elements of Y and Φ in (6), respectively.
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