B = Y Z Z Z where y z 1 1 y z - - PowerPoint PPT Presentation

ARMAX Models

• Vector (multivariate) regression:

– output vector

yt =

    

yt,1 yt,2 . . . yt,k

    

– input vector

zt =

    

zt,1 zt,2 . . . zt,r

    

1

• Regression equation:

yt,i = βi,1zt,1 + βi,2zt,2 + · · · + βi,rzt,r + wt,i

• r in vector form

yt = Bzt + wt.

• Here {wt} is multivariate white noise:

E(wt) = 0, cov

• wt+h, wt
• =

  

Σw,

h = 0

0,

h = 0.

2

• Given observations for t = 1, 2, . . . , n, the least squares es-

timator of B, also the maximum likelihood estimator when {wt} is Gaussian white noise, is ˆ

B = Y′Z

• Z′Z

−1 ,

where

Y =

    

y′

1

y′

2

. . .

y′

n

     and Z =     

z′

1

z′

2

. . .

z′

n

    

• ML estimate of Σw (replace n with (n − r) for unbiased):

ˆ

Σw = 1

n

n

• t=1
• yt − ˆ

Bzt yt − ˆ Bzt

′ .

3

• Information criteria:

– Akaike: AIC = ln

• ˆ

Σw

• + 2

n

• kr + k(k + 1)

2

• ;

– Schwarz: SIC = ln

• ˆ

Σw

• + ln n

n

• kr + k(k + 1)

2

• ,

– Bias-corrected AIC (incorrect in Shumway & Stoffer): AICc = ln

• ˆ

Σw

• +

2 n − k − r − 1

• kr + k(k + 1)

2

• .

4

Vector Autoregression

• E.g., VAR(1):

xt = α + Φxt−1 + wt.

• Here Φ is a k × k coefficient matrix, and {wr} is Gaussian

multivariate white noise.

• This resembles the vector regression equation, with:

yt = xt, B =

• α Φ
• ,

zt =

• 1

xt−1

• .

5

• Observe x0, x1, . . . , xn, and condition on x0.
• Maximum conditional likelihood estimators of B and Σw are

same as for ordinary vector regression.

• VAR(p) is similar, but we must condition on the first p ob-

servations.

• Full likelihood = conditional likelihood × likelihood derived

from marginal distribution of first p observations, and is dif- ficult to use.

6

Example: 1-year, 5-year, and 10-year weekly interest rates

• Data from http://research.stlouisfed.org/fred2/series/WGS1YR/,

etc.

a = read.csv("WGS1YR.csv"); WGS1YR = ts(a[,2]); a = read.csv("WGS5YR.csv"); WGS5YR = ts(a[,2]); a = read.csv("WGS10YR.csv"); WGS10YR = ts(a[,2]); a = cbind(WGS1YR, WGS5YR, WGS10YR); plot(a); plot(diff(a));

7

• Use the dse package to fit VAR(1) and VAR(2) models to

differences:

library(dse); b = TSdata(output = diff(a)); b1 = estVARXls(b, max.lag = 1); cat("VAR(1)\n print method:\n"); print(b1); cat("\n summary method:\n"); print(summary(b1)); b2 = estVARXls(b, max.lag = 2); cat("\nVAR(2)\n print method:\n"); print(b2); cat("\n summary method:\n"); print(summary(b2));

8

VAR(1) print method:

• neg. log likelihood= -7188.785

A(L) = 1-1.014698L1 0+0.05794167L1 0-0.04292339L1 0-0.02482398L1 1-0.9224325L1 0-0.05304638L1 0-0.0144053L1 0+0.03872528L1 1-1.024605L1 B(L) = 1 1 1 summary method:

• neg. log likelihood = -7188.785

sample length = 2448 WGS1YR y.WGS5YR WGS10YR RMSE 0.2005654 0.1713752 0.1563661 ARMA: model estimated by estVARXls inputs :

• utputs:

WGS1YR y.WGS5YR WGS10YR

9

input dimension =

• utput dimension =

3

• rder A =

1

• rder B =
• rder C =

9 actual parameters 6 non-zero constants trend not estimated. VAR(2) print method:

• neg. log likelihood= -7414.944

A(L) = 1-1.329215L1+0.3221239L2 0+0.1030711L1-0.05850615L2 0-0.1539836L1+0.1172694L2 0-0.07336772L1+0.05027099L2 1-1.117284L1+0.1974304L2 0-0.1148573L1+0.05777105L2 0+0.0002002881L1-0.01317073L2 0-0.02287398L1+0.06233586L2 1-1.252808L1+0.2262312L2 B(L) = 1 1 1 summary method:

• neg. log likelihood = -7414.944

sample length = 2448

WGS1YR y.WGS5YR WGS10YR RMSE 0.1910442 0.1666275 0.1534016 ARMA: model estimated by estVARXls inputs :

• utputs:

WGS1YR y.WGS5YR WGS10YR input dimension =

• utput dimension =

3

• rder A =

2

• rder B =
• rder C =

18 actual parameters 6 non-zero constants trend not estimated.

• AIC is smaller (more negative) for VAR(2), but SIC is smaller

for VAR(1).

• For VAR(1),

ˆ

Φ1 =

  

0.3288773 −0.08581201 0.136938 0.06575108 0.1534516 0.08875425 −0.004959931 0.04152504 0.2406055

  

• Largest off-diagonal elements are (1,3) and (2,3), suggesting

that changes in the 10-year rate are followed, one week later, by changes in the same direction in the 1-year and 5-year rates.

10