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### IMSL_AUTOCORRELATION

IMSL_AUTOCORRELATION

The IMSL_AUTOCORRELATION function computes the sample autocorrelation function of a stationary time series.

## Syntax

Result = IMSL_AUTOCORRELATION(x, lagmax [, ACV=variable] [ /DOUBLE] [, SE_OPTION=value] [, SEAC=variable] [, XMEAN_IN=value] [, XMEAN_OUT=variable])

## Return Value

One-dimensional array of length lagmax + 1 containing the auto-correlations of the time series x. The 0-th element of this array is 1. The k-th element of this array contains the autocorrelation of lag k where k = 1, ..., lagmax.

## Arguments

#### lagmax

Scalar integer containing the maximum lag of autocovariance, auto-correlations, and standard errors of auto-correlations to be computed. lagmax must be greater than or equal to 1 and less than N_ELEMENTS(x).

#### x

One-dimensional array containing the time series. N_ELEMENTS(x) must be greater than or equal to 2.

## Keywords

#### ACV

Named variable into which an array of length lagmax + 1 containing the variance and auto-covariances of the time series x is stored. The 0-th element of this array is the variance of the time series x. The k-th element contains the autocovariance of lag k where k = 1, ..., lagmax.

#### DOUBLE

If present and nonzero, double precision is used.

#### SE_OPTION

Method of computation for standard errors of the auto-correlations. Keywords Se_Option and Seac must be used together.

• 1—Compute the standard errors of autocorrelation using Barlett's formula.
•

• 2—Compute the standard errors of autocorrelation using Moran's formula.

#### SEAC

Named variable into which an array of length lagmax containing the standard errors of the auto-correlations of the time series x is stored. Keywords Seac and Se_Option must be used together.

#### XMEAN_IN

The estimate of the mean of the time series x.

#### XMEAN_OUT

Named variable into which the estimate of the mean of the time series x is stored.

## Discussion

The IMSL_AUTOCORRELATION function estimates the autocorrelation function of a stationary time series given a sample of n = N_ELEMENTS(x) observations {Xt} for t = 1, 2, ..., n.

Let:

be the estimate of the mean m of the time series {Xt} where:

The autocovariance function s(k) is estimated by:

where K = lagmax. Note that:

is an estimate of the sample variance. The autocorrelation function r(k) is estimated by:

Note that:

by definition.

The standard errors of the sample auto-correlations may be optionally computed according to the keyword Se_Option for the output keyword Seac. One method (Bartlett 1946) is based on a general asymptotic expression for the variance of the sample autocorrelation coefficient of a stationary time series with independent, identically distributed normal errors. The theoretical formula is:

where:

assumes m is unknown. For computational purposes, the auto-correlations r(k) are replaced by their estimates:

for |k| £ K, and the limits of summation are bounded because of the assumption that r(k) = 0 for all k such that |k| > K.

A second method (Moran 1947) utilizes an exact formula for the variance of the sample autocorrelation coefficient of a random process with independent, identically distributed normal errors. The theoretical formula is:

where m is assumed to be equal to zero. Note that this formula does not depend on the autocorrelation function.

## Example

Consider the Wolfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. The IMSL_AUTOCORRELATION function computes the estimated auto-covariances, estimated auto-correlations, and estimated standard errors of the auto-correlations.

`.RUN  PRO print_results, xm, acv, result, seac     PRINT, ';Mean =', xm     PRINT, ';Variance =', acv(0)     PRINT, ';      Lag       ACV          AC         SEAC'     PRINT, ';       0', acv(0), result(0)     FOR j  =  1, 20 DO \$        PRINT, j, acv(j), result(j), seac(j - 1)  END    lagmax = 20  data = IMSL_STATDATA(2)  x = data(21:120,1)  result = IMSL_AUTOCORRELATION(x, lagmax, ACV = acv, \$     SE_OPTION = 1, SEAC = seac, XMEAN_OUT = xm)  print_results, xm, acv, result, seac    Mean =      46.9760  Variance =      1382.91     Lag       ACV          AC         SEAC     0      1382.91      1.00000     1      1115.03     0.806293    0.0347834     2      592.004     0.428087    0.0962420     3      95.2974    0.0689109     0.156783     4     -235.952    -0.170620     0.205767     5     -370.011    -0.267560     0.230956     6     -294.255    -0.212780     0.228995     7     -60.4423   -0.0437067     0.208622     8      227.633     0.164604     0.178476     9      458.381     0.331462     0.145727     10      567.841     0.410613     0.134406     11      546.122     0.394908     0.150676     12      398.937     0.288477     0.174348     13      197.757     0.143001     0.190619     14      26.8911    0.0194453     0.195490     15     -77.2807   -0.0558828     0.195893     16     -143.733    -0.103935     0.196285     17     -202.048    -0.146104     0.196021     18     -245.372    -0.177432     0.198716     19     -230.816    -0.166906     0.205359     20     -142.879    -0.103318     0.209387  `

## Version History

 6.4 Introduced

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