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 ǫi be i.i.d. with Eǫ = 0, Varǫ = σ2. For some constant ρ, −∞ < ρ < ∞, and for all i ∈ Z, let Xi = ρXi−1 + ǫi. 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)

i

= ρ(j)X(j)

i−1 + ǫ(j) i ,

1 ≤ j ≤ N, 1 ≤ 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)

i

= ρ(j)X(j)

i−1 + ǫ(j) i ,

1 ≤ j ≤ N, 1 ≤ i ≤ ∞

• Only summary statistics might be reported:

Yi = 1

N

N

j=1 X(j) i ,

1 ≤ 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)

i

= ρ(j)X(j)

i−1 + ǫ(j) i ,

1 ≤ j ≤ N, 1 ≤ i ≤ ∞

• Only summary statistics might be reported:

Yi = 1

N

N

j=1 X(j) i ,

1 ≤ i ≤ n

• Is it plausible to consider Y1, ..., Yn as an

AR(1) process? Yi = ρ∗Yi−1 + ǫ∗

i,

1 ≤ i ≤ n

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 EXi is independent of i and Cov(Xi+h, Xi) 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 EXi is independent of i and Cov(Xi+h, Xi) is independent of i for each integer h

• A solution is predictable if Xi is a function of

ǫi, ǫi−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 Xi−i = ρXi−2 + ǫi−1,

recursively expand Xi = ρXi−1 + ǫi to get Yi = ∞

k=0 ρkǫi−k = ǫi + ρǫi−1 + ρ2ǫi−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 Xi−i = ρXi−2 + ǫi−1,

recursively expand Xi = ρXi−1 + ǫi to get Yi = ∞

k=0 ρkǫi−k = ǫi + ρǫi−1 + ρ2ǫi−2 + . . . as

solution

• Solution is predictable. It is also defined with

probability 1 and that it satisfies Xi = ρXi−1 + ǫi

Analyzing Aggregated AR(1) Processes – p.8

Solutions to AR(1) Processes (cont’d)

• Mean function is µY (i) = 0 and covariance

function is γY (h) = ρ−h

σ2 1−ρ2

• so solution is

stationary

Analyzing Aggregated AR(1) Processes – p.9

Solutions to AR(1) Processes (cont’d)

• Mean function is µY (i) = 0 and covariance

function is γY (h) = ρ−h

σ2 1−ρ2

• so solution is

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 X1, ..., Xn 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

ǫk = Xk − ρXk−1 and minimizing n

k=2(Xk − ρXk−1)2 yields

ˆ ρ =

n

• k=2

XkXk−1

n

• k=2

X2

k−1

Analyzing Aggregated AR(1) Processes – p.12

Estimating ρ in AR(1) Process

• Least squares estimation: using

ǫk = Xk − ρXk−1 and minimizing n

k=2(Xk − ρXk−1)2 yields

ˆ ρ =

n

• k=2

XkXk−1

n

• k=2

X2

k−1

• Agrees with maximum liklihood estimator for

ǫ ∼ N(0, σ2)

Analyzing Aggregated AR(1) Processes – p.12

Properties of √n(ˆ ρ − ρ)

• By substituting ρXk−1 + ǫk for Xk in ˆ

ρ we derive ˆ ρ − ρ =

n

• k=2

Xk−1ǫk

n

• k=2

X2

k−1

≈

n

• k=2

Xk−1ǫk nEX2

Analyzing Aggregated AR(1) Processes – p.13

Properties of √n(ˆ ρ − ρ)

• By substituting ρXk−1 + ǫk for Xk in ˆ

ρ we derive ˆ ρ − ρ =

n

• k=2

Xk−1ǫk

n

• k=2

X2

k−1

≈

n

• k=2

Xk−1ǫk nEX2

• Predictability of Xk implies EXk−1ǫk = 0

Analyzing Aggregated AR(1) Processes – p.13

Properties of √n(ˆ ρ − ρ)

• By substituting ρXk−1 + ǫk for Xk in ˆ

ρ we derive ˆ ρ − ρ =

n

• k=2

Xk−1ǫk

n

• k=2

X2

k−1

≈

n

• k=2

Xk−1ǫk nEX2

• Predictability of Xk implies EXk−1ǫk = 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

Var√n(ˆ ρ − ρ) ≈ σ2 EX2

Analyzing Aggregated AR(1) Processes – p.14

Properties of √n(ˆ ρ − ρ) (cont’d)

• Similarly we can show

Var√n(ˆ ρ − ρ) ≈ σ2 EX2

• If √n(ˆ

ρ − ρ) is normally distributed we would expect it to be approximately N(0,

σ2 EX2

0 )

Analyzing Aggregated AR(1) Processes – p.14

Properties of √n(ˆ ρ − ρ) (cont’d)

• Similarly we can show

Var√n(ˆ ρ − ρ) ≈ σ2 EX2

• If √n(ˆ

ρ − ρ) is normally distributed we would expect it to be approximately N(0,

σ2 EX2

0 )

• 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)

ρ = .5, ǫ ∼ DE(1, 0) f(x) = 1

2e−|x|

Analyzing Aggregated AR(1) Processes – p.19

Properties of √n(ˆ ρ − ρ) (cont’d)

ρ = .5, ǫ ∼ CAU(1, 0) f(x) =

1 π(1+x2)

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 N(0,

σ2 EX2

0 )

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 N(0,

σ2 EX2

0 )

• 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 N(0,

σ2 EX2

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

Aggregated AR(1) Processes

Analyzing Aggregated AR(1) Processes – p.22

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

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 Yi = 1 N

N

• j=1

X(j)

i ,

1 ≤ i ≤ n

Analyzing Aggregated AR(1) Processes – p.23

Using ˆ ρ∗

LSE to estimate Eρ If we consider Y1, ..., Yn as an AR(1) process, Yi = ρ∗Yi−1 + ǫ∗

i,

1 ≤ i ≤ n,

• ur initial estimator for Eρ would be the least

squares estimator ˆ ρ∗

LSE = n

• i=2

YiYi−1

n

• i=2

Y 2

i−1

Analyzing Aggregated AR(1) Processes – p.24

Using ˆ ρ∗

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

Using ˆ ρ∗

LSE to est. Eρ - Horváth and Leipus

Results of Horváth and Leipus (2005)

Analyzing Aggregated AR(1) Processes – p.26

Using ˆ ρ∗

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,

ˆ ρ∗

LSE converges in probability as the number

• f companies or time periods increase but the

limit is not Eρ.

Analyzing Aggregated AR(1) Processes – p.26

Using ˆ ρ∗

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,

ˆ ρ∗

LSE converges in probability as the number

• f companies or time periods increase but the

limit is not Eρ.

• If the distribution of ρ is symmetric around 0,

ˆ ρ∗

LSE converges to Eρ in probability as the

number of companies or time periods goes to infinity.

Analyzing Aggregated AR(1) Processes – p.26

Using ˆ ρ∗

LSE to estimate Eρ (cont’d) We run simulations to test ˆ ρ∗

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

Simulation Results for ρ ∼ U(0, 1)

Analyzing Aggregated AR(1) Processes – p.28

Simulation Results for ρ ∼ U(−1, 1)

Analyzing Aggregated AR(1) Processes – p.29

Using ˆ ρ∗

MLSE to estimate Eρ

• Horváth and Leipus suggest the following

modification of ˆ ρ∗

LSE which should converge to

Eρ for ρ ∼ U(0, 1) ˆ ρ∗

MLSE = n

• i=2

YiYi−1 −

n

• i=4

YiYi−3

n

• i=1

Y 2

i − n

• i=3

YiYi−2

Analyzing Aggregated AR(1) Processes – p.30

Using ˆ ρ∗

MLSE to estimate Eρ

• Horváth and Leipus suggest the following

modification of ˆ ρ∗

LSE which should converge to

Eρ for ρ ∼ U(0, 1) ˆ ρ∗

MLSE = n

• i=2

YiYi−1 −

n

• i=4

YiYi−3

n

• i=1

Y 2

i − n

• i=3

YiYi−2

• We repeat previous simulations using ˆ

ρ∗

MLSE

instead of ˆ ρ∗

LSE

Analyzing Aggregated AR(1) Processes – p.30

Simulation Results for ρ ∼ U(0, 1)

Analyzing Aggregated AR(1) Processes – p.31

Simulation Results for ρ ∼ U(−1, 1)

Analyzing Aggregated AR(1) Processes – p.32

Aggregated AR(1) Processes - Final Thoughts

Did not fully replicate results of Horváth and Leipus with ˆ ρ∗

MLSE for ρ ∼ U(0, 1)

Analyzing Aggregated AR(1) Processes – p.33

Aggregated AR(1) Processes - Final Thoughts

Did not fully replicate results of Horváth and Leipus with ˆ ρ∗

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

Aggregated AR(1) Processes - Final Thoughts

Did not fully replicate results of Horváth and Leipus with ˆ ρ∗

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

Questions?

Analyzing Aggregated AR(1) Processes – p.34

Thank You!

Analyzing Aggregated AR(1) Processes – p.35