autoUniAr

Automatic selection and fitting of a univariate autoregressive time series model. The lag for the model is automatically selected using Akaike’s information criterion (AIC). Estimates of the autoregressive parameters for the model with minimum AIC are calculated using method of moments, method of least squares, or maximum likelihood.

Synopsis

autoUniAr (z, maxLag, p)

Required Arguments

float z[] (Input)

Array of length nObs containing the stationary time series.

int maxLag (Input)

Maximum number of autoregressive parameters requested. It is required that 1≤ maxLag ≤ nObs/2.

int p (Output)

Number of autoregressive parameters in the model with minimum AIC.

Return Value

Vector of length 1+ maxLag containing the estimates for the constant and the autoregressive parameters in the model with minimum AIC. The estimates are located in the first 1+ p locations of this array.

Optional Arguments

printLevel (Input)

Printing option:

printLevel	Action
0	No Printing.
1	Prints final results only.
2	Prints intermediate and final results.

Default: printLevel = 0.

maxIterations (Input)Maximum number of estimation iterations.

Default: maxIterations = 300

method, int (Input)

Estimation method option:

method	Action
0	Method of moments.
1	Method of least squares realized through Householder transformations.
2	Maximum likelihood

Default: method = 1.

varNoise (Output)

Estimate of innovation variance.

aic (Output)

Minimum AIC.

meanEstimate (Input/Output)

Estimate of the mean of the time series z. On return, meanEstimate contains an update of the mean.

Default: Time series z is centered about its sample mean.

Description

Function autoUniAr automatically selects the order of the AR model that best fits the data and then computes the AR coefficients. The algorithm used in autoUniAr is derived from the work of Akaike, H., et. al (1979) and Kitagawa and Akaike (1978). This code was adapted from the UNIMAR procedure published as part of the TIMSAC-78 Library.

The best fit AR model is determined by successively fitting AR models with 0, 1, 2, …, maxLag autoregressive coefficients. For each model, Akaike’s Information Criterion (AIC) is calculated based on the formula

\[\mathit{AIC} = -2 \ln (\mathit{likelihood}) + 2(\mathtt{p} + 1)\]

Function autoUniAr uses the approximation to this formula developed by Ozaki and Oda (1979),

\[\mathit{AIC} = (\mathrm{nObs} - \mathrm{maxlag}) \ln \left(\hat{\sigma}^2\right) + 2(p+1) + (\mathrm{nObs} - \mathrm{maxlag}) (\ln (2 \pi) + 1),\]

where \(\hat{\sigma}^2\) is an estimate of the residual variance of the series, commonly known in time series analysis as the innovation variance. By dropping the constant

\[(\mathrm{nObs} - \mathrm{maxlog}) (\ln (2\pi) + 1),\]

the calculation is simplified to

\[\mathit{AIC} = (\mathrm{nObs} - \mathrm{maxlag}) \ln \left(\hat{\sigma}^2\right) + 2(p+1)\]

The best fit model is the model with minimum AIC. If the number of parameters in this model is equal to the highest order autoregressive model fitted, i.e., p=maxLag, then a model with smaller AIC might exist for larger values of maxLag. In this case, increasing maxLag to explore AR models with additional autoregressive parameters might be warranted.

If method = 0, estimates of the autoregressive coefficients for the model with minimum AIC are calculated using method of moments. If method =1, the coefficients are determined by the method of least squares applied in the form described by Kitagawa and Akaike (1978). Otherwise, if method =2, the coefficients are estimated using maximum likelihood.

Example

Consider the Wolfer Sunspot data (Anderson 1971, p. 660) consisting of the number of sunspots observed each year from 1770 through 1869. In this example, autoUniAr found the minimum AIC fit is an autoregressive model with 3 lags:

\[\tilde{w}_t = \phi_1 \tilde{w}_{t-1} + \phi_2 \tilde{w}_{t-2} + \phi_3 \tilde{w}_{t-3} + a_t,\]

where

\[\tilde{w}_t := w_t - \mu,\]

μ the sample mean of the time series \(\left\{ w_t \right\}\). Defining the overall constant \(\phi_0\) by \(\phi_0 :=\mu\left( 1-\textstyle\sum_{i=1}^{3} \phi_i \right)\), we obtain the following equivalent representation:

\[w_t = \phi_0 + \phi_1 w_{t-1} + \phi_2 w_{t-2} + ]phi_3 w_{t-3} + a_t.\]

The example computes estimates for \(\phi_0,\phi_1,\phi_2,\phi_3\) for each of the three parameter estimation methods available.

from __future__ import print_function
from numpy import *
from pyimsl.stat.autoUniAr import autoUniAr
from pyimsl.stat.dataSets import dataSets
from pyimsl.stat.writeMatrix import writeMatrix

maxlag = 20
n_obs = 100
p = []
aic = []
avar = []
z = empty(100)

# get wolfer sunspot data
w = dataSets(2)
for i in range(0, n_obs):
    z[i] = w[21 + i][1]

# Compute AR parameters for minimum AIC by method of moments
print("AIC Automatic Order selection")
print("AR coefficients estimated using method of moments")

parameters = autoUniAr(z, maxlag, p,
                       varNoise=avar,
                       method=0,
                       aic=aic)

print("Order selected: %d" % p[0])
print("AIC =  %11.4f, Variance = %11.4f" % (aic[0], avar[0]))
print("Constant estimate is %11.4f." % parameters[0])
writeMatrix("Final AR coefficients estimated by method of moments",
            parameters[1:p[0] + 1], column=True)

# Compute AR parameters for minimum AIC by method of least squares
print("\nAIC Automatic Order selection")
print("AR coefficients estimated using method of least squares")

ar = autoUniAr(z, maxlag, p,
               varNoise=avar,
               method=1,
               aic=aic)

print("Order selected: %d\n" % p[0])
print("AIC =  %11.4f, Variance = %11.4f" % (aic[0], avar[0]))
print("Constant estimate is %11.4f." % ar[0])
writeMatrix("Final AR coefficients estimated by method of least squares",
            ar[1:p[0] + 1], column=True)

# Compute AR parameters for minimum AIC by maximum likelihood estimation
print("\nAIC Automatic Order selection")
print("AR coefficients estimated using maximum likelihood")

ar = autoUniAr(z, maxlag, p,
               varNoise=avar,
               method=2,
               aic=aic)

print("Order selected: %d" % p[0])
print("AIC =  %11.4f, Variance = %11.4f" % (aic[0], avar[0]))
print("Constant estimate is %11.4f." % ar[0])
writeMatrix("Final AR coefficients estimated by maximum likelihood",
            ar[1:p[0] + 1], column=True)

Output

AIC Automatic Order selection
AR coefficients estimated using method of moments
Order selected: 3
AIC =     633.0114, Variance =    287.2699
Constant estimate is     13.7098.

AIC Automatic Order selection
AR coefficients estimated using method of least squares
Order selected: 3

AIC =     633.0114, Variance =    144.7149
Constant estimate is      9.8934.

AIC Automatic Order selection
AR coefficients estimated using maximum likelihood
Order selected: 3
AIC =     633.0114, Variance =    218.8486
Constant estimate is     11.4218.
 
Final AR coefficients estimated by method of moments
                   1        1.368
                   2       -0.738
                   3        0.078
 
Final AR coefficients estimated by method of least squares
                      1        1.604
                      2       -1.024
                      3        0.209
 
Final AR coefficients estimated by maximum likelihood
                   1        1.551
                   2       -0.999
                   3        0.204