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49Dependence CaseWhen |β| < 1, then the following statements hold (Hamilton, 1994).E[Y i ] = C/(1 − β) (4.8)V AR[Y i ] = σ 2 Y = σ2 ɛ /(1 − β2 ) (4.9)E[Ȳ ] = E[Y i] = C/(1 − β) (4.10)V AR(Ȳ ) =( ) ( )1 + β σ2ɛ1 − β N(4.11)We are able to observe the Y i−1 value preceding each Y i for every observationexcept the first. Not surprisingly, the contribution of the first observation to theoverall loglikelihood is different than that of the other observations, and this makesthe loglikelihood equation more difficult to analyze. Since the proportion of thecontribution of Y 1 to the loglikelihood becomes small as N increases, we chooseto condition on its value, which has the effect of simplifying the formula for theloglikelihood. Furthermore, estimates based on this conditional loglikelihood areconsistent and asymptotically equal to estimates based on the exact loglikelihood.The exact loglikelihood isL(Y |M) = − 1 2 log(2π) − 1 2 log(σ2 ɛ /(1 − β 2 )) − (Y 1 − C/(1 − β)) 2− N − 12log(2π) − N − 1 log(σɛ 2 ) − 122σɛ22σɛ 2 /(1 − β 2 )N∑(Y i − C − βY i−1 (4.12) ) 2The first three terms in equation 4.12 can be thought of as the contribution ofY 1 , with the remaining terms resulting from Y 2 ...Y N . Conditioning on Y 1 we obtaini=2