Analyzing Aggregated AR(1) Processes

Transcription

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

2 What is an AR(1) Process? Let with ǫ i be i.i.d. Eǫ = 0, Varǫ = 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

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

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

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

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

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

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

9 Agenda Discuss some elementary facts about AR(1) processes Derive an estimator ˆρ for ρ in an AR(1) process and analyze the distribution of n(ˆρ 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

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

11 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

12 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

13 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

14 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 ρk ǫ i k = ǫ i + ρǫ i 1 + ρ 2 ǫ i as solution Analyzing Aggregated AR(1) Processes p.8

15 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 ρk ǫ i k = ǫ i + ρǫ i 1 + ρ 2 ǫ i 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

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

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

18 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

19 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

20 Deriving ˆρ and Analyzing n(ˆρ ρ) Analyzing Aggregated AR(1) Processes p.11

21 Estimating ρ in AR(1) Process Least squares estimation: using ǫ k = X k ρx k 1 and minimizing n k=2 (X k ρx k 1 ) 2 yields ˆρ = n X k X k 1 k=2 n k=2 X 2 k 1 Analyzing Aggregated AR(1) Processes p.12

22 Estimating ρ in AR(1) Process Least squares estimation: using ǫ k = X k ρx k 1 and minimizing n k=2 (X k ρx k 1 ) 2 yields ˆρ = n X k X k 1 k=2 n k=2 X 2 k 1 Agrees with maximum liklihood estimator for ǫ N(0,σ 2 ) Analyzing Aggregated AR(1) Processes p.12

23 Properties of n(ˆρ ρ) By substituting ρx k 1 + ǫ k for X k in ˆρ we derive n n X k 1 ǫ k X k 1 ǫ k ˆρ ρ = k=2 n k=2 X 2 k 1 k=2 nex 2 0 Analyzing Aggregated AR(1) Processes p.13

24 Properties of n(ˆρ ρ) By substituting ρx k 1 + ǫ k for X k in ˆρ we derive n n X k 1 ǫ k X k 1 ǫ k ˆρ ρ = k=2 n k=2 X 2 k 1 k=2 nex 2 0 Predictability of X k implies EX k 1 ǫ k = 0 Analyzing Aggregated AR(1) Processes p.13

25 Properties of n(ˆρ ρ) By substituting ρx k 1 + ǫ k for X k in ˆρ we derive n n X k 1 ǫ k X k 1 ǫ k ˆρ ρ = k=2 n k=2 X 2 k 1 k=2 nex 2 0 Predictability of X k implies EX k 1 ǫ k = 0 Thus, E n(ˆρ ρ) 0 Analyzing Aggregated AR(1) Processes p.13

26 Properties of n(ˆρ ρ) (cont d) Similarly we can show Var n(ˆρ ρ) σ2 EX 2 0 Analyzing Aggregated AR(1) Processes p.14

27 Properties of n(ˆρ ρ) (cont d) Similarly we can show Var n(ˆρ ρ) σ2 EX 2 0 If n(ˆρ ρ) is normally distributed we would expect it to be approximately N(0, σ 2 EX 2 0 ) Analyzing Aggregated AR(1) Processes p.14

28 Properties of n(ˆρ ρ) (cont d) Similarly we can show Var n(ˆρ ρ) σ2 EX 2 0 If n(ˆρ ρ) is normally distributed we would expect it to be approximately N(0, σ 2 EX 2 0 ) We examine this proposition through simulations using several combinations of ρ and ǫ Analyzing Aggregated AR(1) Processes p.14

29 Properties of n(ˆρ ρ) (cont d) ρ =.1, ǫ N(0, 1) Analyzing Aggregated AR(1) Processes p.15

30 Properties of n(ˆρ ρ) (cont d) ρ =.5, ǫ N(0, 1) Analyzing Aggregated AR(1) Processes p.16

31 Properties of n(ˆρ ρ) (cont d) ρ =.9, ǫ N(0, 1) Analyzing Aggregated AR(1) Processes p.17

32 Properties of n(ˆρ ρ) (cont d) ρ =.99, ǫ N(0, 1) Analyzing Aggregated AR(1) Processes p.18

33 Properties of n(ˆρ ρ) (cont d) ρ =.5, ǫ DE(1, 0) f(x) = 1 2 e x Analyzing Aggregated AR(1) Processes p.19

34 Properties of n(ˆρ ρ) (cont d) ρ =.5, ǫ CAU(1, 0) f(x) = 1 π(1+x 2 ) Analyzing Aggregated AR(1) Processes p.20

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

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

37 Properties of n(ˆρ ρ) - Conclusion When ǫ is distributed as N(0, 1) or DE(1, 0), n(ˆρ ρ) is distributed approximately as N(0, σ 2 EX 2 0 ) 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

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

39 Aggregated AR(1) Processes Assume a financial statistic for each of N companies follows an AR(1) process with ρ (j) randomly distributed X (j) i = ρ (j) X (j) i 1 + ǫ(j) i, 1 j N, 1 i Analyzing Aggregated AR(1) Processes p.23

40 Aggregated AR(1) Processes Assume a financial statistic for each of N companies follows an AR(1) process with ρ (j) randomly distributed X (j) i = ρ (j) X (j) i 1 + ǫ(j) i, 1 j N, 1 i We only have summary statistics and we want to estimate Eρ Y i = 1 N N j=1 X (j) i, 1 i n Analyzing Aggregated AR(1) Processes p.23

41 Using ˆρ LSE to estimate Eρ If we consider Y 1,...,Y n as an AR(1) process, Y i = ρ Y i 1 + ǫ i, 1 i n, our initial estimator for Eρ would be the least squares estimator ˆρ LSE = n i=2 n i=2 Y i Y i 1 Y 2 i 1 Analyzing Aggregated AR(1) Processes p.24

42 Using ˆρ LSE to est. Eρ - Granger and Morris Important results in the aggregation of stationary time series were achieved by Granger and Morris (1976). In particular, they showed that the sum of two AR(1) processes is not an AR(1) process, but is a more complex autoregressive moving average (ARMA) process. Analyzing Aggregated AR(1) Processes p.25

43 Using ˆρ LSE to est. Eρ - Horváth and Leipus Results of Horváth and Leipus (2005) Analyzing Aggregated AR(1) Processes p.26

44 Using ˆρ LSE to est. Eρ - Horváth and Leipus Results of Horváth and Leipus (2005) If P(0 < ρ < 1) = 1 or P( 1 < ρ < 0) = 1, ˆρ LSE converges in probability as the number of companies or time periods increase but the limit is not Eρ. Analyzing Aggregated AR(1) Processes p.26

45 Using ˆρ LSE to est. Eρ - Horváth and Leipus Results of Horváth and Leipus (2005) If P(0 < ρ < 1) = 1 or P( 1 < ρ < 0) = 1, ˆρ LSE converges in probability as the number of companies or time periods increase but the limit is not Eρ. If the distribution of ρ is symmetric around 0, ˆρ LSE converges to Eρ in probability as the number of companies or time periods goes to infinity. Analyzing Aggregated AR(1) Processes p.26

46 Using ˆρ LSE to estimate Eρ (cont d) We run simulations to test ˆρ LSE for ρ U(0, 1),ρ U( 1, 1) ǫ N(0, 1),ǫ DE(1, 0) N {100, 200, 300} n {10, 20, 30, 40, 50} Analyzing Aggregated AR(1) Processes p.27

47 Simulation Results for ρ U(0, 1) Analyzing Aggregated AR(1) Processes p.28

48 Simulation Results for ρ U( 1, 1) Analyzing Aggregated AR(1) Processes p.29

49 Using ˆρ MLSE to estimate Eρ Horváth and Leipus suggest the following modification of ˆρ LSE which should converge to Eρ for ρ U(0, 1) ˆρ MLSE = n i=2 Y i Y i 1 n i=1 Y 2 i n i=4 n i=3 Y i Y i 3 Y i Y i 2 Analyzing Aggregated AR(1) Processes p.30

50 Using ˆρ MLSE to estimate Eρ Horváth and Leipus suggest the following modification of ˆρ LSE which should converge to Eρ for ρ U(0, 1) ˆρ MLSE = n i=2 Y i Y i 1 n i=1 Y 2 i n i=4 n i=3 Y i Y i 3 Y i Y i 2 We repeat previous simulations using ˆρ MLSE instead of ˆρ LSE Analyzing Aggregated AR(1) Processes p.30

51 Simulation Results for ρ U(0, 1) Analyzing Aggregated AR(1) Processes p.31

52 Simulation Results for ρ U( 1, 1) Analyzing Aggregated AR(1) Processes p.32

53 Aggregated AR(1) Processes - Final Thoughts Did not fully replicate results of Horváth and Leipus with ˆρ MLSE for ρ U(0, 1) Analyzing Aggregated AR(1) Processes p.33

54 Aggregated AR(1) Processes - Final Thoughts Did not fully replicate results of Horváth and Leipus with ˆρ MLSE for ρ U(0, 1) Examined limited range of values for the number of companies and time periods Analyzing Aggregated AR(1) Processes p.33

55 Aggregated AR(1) Processes - Final Thoughts Did not fully replicate results of Horváth and Leipus with ˆρ MLSE for ρ U(0, 1) Examined limited range of values for the number of companies and time periods The AR(1) processes for each company were started with an initial value of 0 Resulted in non-stationary process? Throwing away a number of initial values in the series might have overcome this issue Analyzing Aggregated AR(1) Processes p.33

56 Questions? Analyzing Aggregated AR(1) Processes p.34

57 Thank You! Analyzing Aggregated AR(1) Processes p.35