Autoregressive Processes of Order 1
Formal Definition of (𝐵)−1
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
• Substitute 𝐵 for 𝑧 to formally define (𝐵)−1
• Let 𝜖𝑖= 0 for 𝑖 ≤ 𝑠
• Key equation: 𝑋 = (𝐵)−1𝜖, or more explicitly:
𝑋𝑛= ∑ ∑ 𝑟−𝑠
|𝑠|=𝑟𝑘≥0 𝜖𝑛−𝑘
Convergence of 𝑋𝑛
• When all roots 𝑟𝑖 are outside the unit circle:
∑|𝑟−𝑠|
|𝑠|=𝑘 ≤(1+𝜂)−𝑘 (𝑘+𝑝−1
𝑝−1 )∼(1+𝜂)−𝑘 𝑘𝑝
(𝑝−1)! as 𝑘 → ∞
• For weak* convergence of 𝑋𝑛, a sufficient condition on 𝜖𝑖 is:
∫𝑃{|𝜖𝑖|> 𝑡}
𝑡
∞
1𝑑𝑡 < ∞
• If 𝜖𝑖 is integrable, 𝑋𝑛 converges in 𝐿1 as well
Behavior with Roots Inside Unit Disk
• If 𝑟1 is the unique root with smallest modulus and |𝑟1|< 1:
– Distribution of 𝑟1
𝑛𝑋𝑛 converges weakly* to a non-degenerate limit
– 𝑋𝑛 explodes at exponential rate
• If smallest root is not unique, 𝑋𝑛 still explodes, essentially at an exponential rate
Classical Theory vs. Finite Sample Behavior
• Classical theory focuses on location of roots of 𝜙 with respect to unit disk
• Only considers behavior as time 𝑛 → ∞
• Key insight: This is only part of the overall behavior of these processes
Autoregressive Process of Order 1
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝜖𝑖
• Non-explosive if |𝑎|< 1
Autoregressive Process of Order 2
• Defined as: 𝑋𝑖= 𝑎𝑋𝑖−1 +𝑏𝑋𝑖−2 +𝜖𝑖
• Non-explosive conditions:
– If 𝑎2+4𝑏 < 0: −𝑏 < 1
– If 𝑎2+4𝑏 ≥ 0: 𝑏+𝑎 < 1
Tail Behavior of Autocovariances (Order 1)
• Define matrix 𝐴 such that 𝑋 = 𝐴𝜖
• Empirical covariance: 𝑛𝛾𝑛(𝑘)= ⟨𝐴𝑇𝐵𝑘𝐴𝜖,𝜖⟩
• Tail behavior depends on value of 𝑎 and parity of 𝑘
• Key result: Theorem 11.2.1 gives equivalent expressions for 𝑃{𝑛𝛾𝑛(𝑘)> 𝑡} as 𝑡 →
∞
• Important observation: Odd and even autocovariances exhibit very different
decays when 𝑎 is negative
