Analyzing Aggregated AR(1) Processes Jon Gunnip Supervisory - PowerPoint PPT Presentation

Analyzing Aggregated AR(1) Processes Jon Gunnip Supervisory Committee Professor Lajos Horv´ ath (Committee Chair) Professor Davar Koshnevisan Professor Paul Roberts Analyzing Aggregated AR(1) Processes – p.1

What is an AR(1) Process? Let ǫ i be i.i.d. with Eǫ = 0 , Var ǫ = σ 2 . For some constant ρ , −∞ < ρ < ∞ , and for all i ∈ Z , let X i = ρX i − 1 + ǫ i . This is an autoregressive process of order 1. Analyzing Aggregated AR(1) Processes – p.2

AR(1) Process Example ρ = . 5 , ǫ ∼ N (0 , 1) Analyzing Aggregated AR(1) Processes – p.3

Aggregated AR(1) Processes • What if a financial statistic for N companies each followed an AR(1) process? X ( j ) = ρ ( j ) X ( j ) i − 1 + ǫ ( j ) 1 ≤ j ≤ N, 1 ≤ i ≤ ∞ i , i Analyzing Aggregated AR(1) Processes – p.4

Aggregated AR(1) Processes • What if a financial statistic for N companies each followed an AR(1) process? X ( j ) = ρ ( j ) X ( j ) i − 1 + ǫ ( j ) 1 ≤ j ≤ N, 1 ≤ i ≤ ∞ i , i • Only summary statistics might be reported: j =1 X ( j ) � N Y i = 1 1 ≤ i ≤ n i , N Analyzing Aggregated AR(1) Processes – p.4

Aggregated AR(1) Processes • What if a financial statistic for N companies each followed an AR(1) process? X ( j ) = ρ ( j ) X ( j ) i − 1 + ǫ ( j ) 1 ≤ j ≤ N, 1 ≤ i ≤ ∞ i , i • Only summary statistics might be reported: j =1 X ( j ) � N Y i = 1 1 ≤ i ≤ n i , N • Is it plausible to consider Y 1 , ..., Y n as an AR(1) process? Y i = ρ ∗ Y i − 1 + ǫ ∗ 1 ≤ i ≤ n i , Analyzing Aggregated AR(1) Processes – p.4

Agenda • Discuss some elementary facts about AR(1) processes Analyzing Aggregated AR(1) Processes – p.5

Agenda • Discuss some elementary facts about AR(1) processes • Derive an estimator ˆ ρ for ρ in an AR(1) process and analyze the distribution of √ n (ˆ ρ − ρ ) Analyzing Aggregated AR(1) Processes – p.5

Agenda • Discuss some elementary facts about AR(1) processes • Derive an estimator ˆ ρ for ρ in an AR(1) process and analyze the distribution of √ n (ˆ ρ − ρ ) • Consider aggregations of AR(1) processes where ρ is a random variable and use simulations to test two estimators for Eρ that rely on the aggregated data Analyzing Aggregated AR(1) Processes – p.5

Elementary Facts about AR(1) Processes Analyzing Aggregated AR(1) Processes – p.6

Stationary, Predictable Solutions • A solution to an AR(1) process is weakly stationary if EX i is independent of i and Cov ( X i + h , X i ) is independent of i for each integer h Analyzing Aggregated AR(1) Processes – p.7

Stationary, Predictable Solutions • A solution to an AR(1) process is weakly stationary if EX i is independent of i and Cov ( X i + h , X i ) is independent of i for each integer h • A solution is predictable if X i is a function of ǫ i , ǫ i − 1 , . . . Analyzing Aggregated AR(1) Processes – p.7

Solutions to AR(1) Processes • An AR(1) process has a unique, stationary, predictable solution if and only if | ρ | < 1 Analyzing Aggregated AR(1) Processes – p.8

Solutions to AR(1) Processes • An AR(1) process has a unique, stationary, predictable solution if and only if | ρ | < 1 • Assume | ρ | < 1 . Using X i − i = ρX i − 2 + ǫ i − 1 , recursively expand X i = ρX i − 1 + ǫ i to get Y i = � ∞ k =0 ρ k ǫ i − k = ǫ i + ρǫ i − 1 + ρ 2 ǫ i − 2 + . . . as solution Analyzing Aggregated AR(1) Processes – p.8

Solutions to AR(1) Processes • An AR(1) process has a unique, stationary, predictable solution if and only if | ρ | < 1 • Assume | ρ | < 1 . Using X i − i = ρX i − 2 + ǫ i − 1 , recursively expand X i = ρX i − 1 + ǫ i to get Y i = � ∞ k =0 ρ k ǫ i − k = ǫ i + ρǫ i − 1 + ρ 2 ǫ i − 2 + . . . as solution • Solution is predictable. It is also defined with probability 1 and that it satisfies X i = ρX i − 1 + ǫ i Analyzing Aggregated AR(1) Processes – p.8

Solutions to AR(1) Processes (cont’d) • Mean function is µ Y ( i ) = 0 and covariance function is γ Y ( h ) = ρ − h � � σ 2 so solution is 1 − ρ 2 stationary Analyzing Aggregated AR(1) Processes – p.9

Solutions to AR(1) Processes (cont’d) • Mean function is µ Y ( i ) = 0 and covariance function is γ Y ( h ) = ρ − h � � σ 2 so solution is 1 − ρ 2 stationary • For | ρ | > 1 , there is a unique, stationary, non-predictable solution Analyzing Aggregated AR(1) Processes – p.9

Solutions to AR(1) Processes - Conclusion • Since we have a unique, stationary, predictable solution if and only if | ρ | < 1 , we assume | ρ | < 1 throughout the rest of the presentation Analyzing Aggregated AR(1) Processes – p.10

Solutions to AR(1) Processes - Conclusion • Since we have a unique, stationary, predictable solution if and only if | ρ | < 1 , we assume | ρ | < 1 throughout the rest of the presentation • Next step is to have a way to estimate ρ given data X 1 , ..., X n from an AR(1) process Analyzing Aggregated AR(1) Processes – p.10

Deriving ˆ ρ and Analyzing √ n (ˆ ρ − ρ ) Analyzing Aggregated AR(1) Processes – p.11

Estimating ρ in AR(1) Process • Least squares estimation: using ǫ k = X k − ρX k − 1 and minimizing k =2 ( X k − ρX k − 1 ) 2 yields � n n � X k X k − 1 k =2 ρ = ˆ n � X 2 k − 1 k =2 Analyzing Aggregated AR(1) Processes – p.12

Estimating ρ in AR(1) Process • Least squares estimation: using ǫ k = X k − ρX k − 1 and minimizing k =2 ( X k − ρX k − 1 ) 2 yields � n n � X k X k − 1 k =2 ρ = ˆ n � X 2 k − 1 k =2 • Agrees with maximum liklihood estimator for ǫ ∼ N (0 , σ 2 ) Analyzing Aggregated AR(1) Processes – p.12

Properties of √ n (ˆ ρ − ρ ) • By substituting ρX k − 1 + ǫ k for X k in ˆ ρ we derive n n � � X k − 1 ǫ k X k − 1 ǫ k k =2 k =2 ρ − ρ = ˆ ≈ n nEX 2 0 � X 2 k − 1 k =2 Analyzing Aggregated AR(1) Processes – p.13

Properties of √ n (ˆ ρ − ρ ) • By substituting ρX k − 1 + ǫ k for X k in ˆ ρ we derive n n � � X k − 1 ǫ k X k − 1 ǫ k k =2 k =2 ρ − ρ = ˆ ≈ n nEX 2 0 � X 2 k − 1 k =2 • Predictability of X k implies EX k − 1 ǫ k = 0 Analyzing Aggregated AR(1) Processes – p.13

Properties of √ n (ˆ ρ − ρ ) • By substituting ρX k − 1 + ǫ k for X k in ˆ ρ we derive n n � � X k − 1 ǫ k X k − 1 ǫ k k =2 k =2 ρ − ρ = ˆ ≈ n nEX 2 0 � X 2 k − 1 k =2 • Predictability of X k implies EX k − 1 ǫ k = 0 • Thus, E √ n (ˆ ρ − ρ ) ≈ 0 Analyzing Aggregated AR(1) Processes – p.13

Properties of √ n (ˆ ρ − ρ ) (cont’d) • Similarly we can show σ 2 Var √ n (ˆ ρ − ρ ) ≈ EX 2 0 Analyzing Aggregated AR(1) Processes – p.14

Properties of √ n (ˆ ρ − ρ ) (cont’d) • Similarly we can show σ 2 Var √ n (ˆ ρ − ρ ) ≈ EX 2 0 • If √ n (ˆ ρ − ρ ) is normally distributed we would σ 2 expect it to be approximately N (0 , 0 ) EX 2 Analyzing Aggregated AR(1) Processes – p.14

Properties of √ n (ˆ ρ − ρ ) (cont’d) • Similarly we can show σ 2 Var √ n (ˆ ρ − ρ ) ≈ EX 2 0 • If √ n (ˆ ρ − ρ ) is normally distributed we would σ 2 expect it to be approximately N (0 , 0 ) EX 2 • We examine this proposition through simulations using several combinations of ρ and ǫ Analyzing Aggregated AR(1) Processes – p.14

Properties of √ n (ˆ ρ − ρ ) (cont’d) ρ = . 1 , ǫ ∼ N (0 , 1) Analyzing Aggregated AR(1) Processes – p.15

Properties of √ n (ˆ ρ − ρ ) (cont’d) ρ = . 5 , ǫ ∼ N (0 , 1) Analyzing Aggregated AR(1) Processes – p.16

Properties of √ n (ˆ ρ − ρ ) (cont’d) ρ = . 9 , ǫ ∼ N (0 , 1) Analyzing Aggregated AR(1) Processes – p.17

Properties of √ n (ˆ ρ − ρ ) (cont’d) ρ = . 99 , ǫ ∼ N (0 , 1) Analyzing Aggregated AR(1) Processes – p.18

Properties of √ n (ˆ ρ − ρ ) (cont’d) f ( x ) = 1 2 e −| x | ρ = . 5 , ǫ ∼ DE (1 , 0) Analyzing Aggregated AR(1) Processes – p.19

Properties of √ n (ˆ ρ − ρ ) (cont’d) 1 ρ = . 5 , ǫ ∼ CAU (1 , 0) f ( x ) = π (1+ x 2 ) Analyzing Aggregated AR(1) Processes – p.20

Properties of √ n (ˆ ρ − ρ ) - Conclusion • When ǫ is distributed as N (0 , 1) or DE (1 , 0) , √ n (ˆ ρ − ρ ) is distributed approximately as σ 2 N (0 , 0 ) EX 2 Analyzing Aggregated AR(1) Processes – p.21

Properties of √ n (ˆ ρ − ρ ) - Conclusion • When ǫ is distributed as N (0 , 1) or DE (1 , 0) , √ n (ˆ ρ − ρ ) is distributed approximately as σ 2 N (0 , 0 ) EX 2 • We can create confidence intervals or do hypothesis testing for ρ under these circumstances Analyzing Aggregated AR(1) Processes – p.21

Properties of √ n (ˆ ρ − ρ ) - Conclusion • When ǫ is distributed as N (0 , 1) or DE (1 , 0) , √ n (ˆ ρ − ρ ) is distributed approximately as σ 2 N (0 , 0 ) EX 2 • We can create confidence intervals or do hypothesis testing for ρ under these circumstances • We will use ǫ distributed as N (0 , 1) and DE (1 , 0) for our simulations of aggregrated AR(1) processes Analyzing Aggregated AR(1) Processes – p.21

Aggregated AR(1) Processes Analyzing Aggregated AR(1) Processes – p.22