1. Given two observations x 1 and x 2 from the causal AR(1) process satisfying X t = φX t −1 + Z t , { Z t } ∼ WN(0 ,σ 2 ) , and assuming that | x 1 | ≠ | x 2 |, find the maximum likelihood estimates...

1. Given two observationsx
₁ andx
₂ from the causal AR(1) process satisfying

X_t
=φX_t
−1 +Z_t,{Z_t
} ∼ WN(0,σ
²),

and assuming that |x
₁| ≠ |x
₂|, find the maximum likelihood estimates ofφandσ
².

2. Derive a cubic equation for the maximum likelihood estimate of the coefficientφof a causal AR(1) process based on the observationsX
₁
, . . . , X_n