IRIS Toolbox Reference Manual

## acf

Autocovariance and autocorrelation functions for model variables

#### Syntax

    [C,R,List] = acf(M,...)

#### Input arguments

- `M` [ model ] - Solved model object for which the ACF will be computed.

#### Output arguments

- `C` [ namedmat | numeric ] - Auto/cross-covariance matrices.

- `R` [ namedmat | numeric ] - Auto/cross-correlation matrices.

- `List` [ cellstr ] - List of variables in rows and columns of `C` and `R`.

#### Options

- `'applyTo='` [ cellstr | char | `Inf` ] - List of variables to which the `'filter='` will be applied; `Inf` means all variables.

- `'contributions='` [ `true` | `false` ] - If `true` the contributions of individual shocks to ACFs will be computed and stored in the 5th dimension of the `C` and `R` matrices.

- `'filter='` [ char | empty ] - Linear filter that is applied to variables specified by 'applyto'.

- `'nFreq='` [ numeric | `256` ] - Number of equally spaced frequencies over which the filter in the option `'filter='` is numerically integrated.

- `'order='` [ numeric | `0` ] - Order up to which ACF will be computed.

- `'output='` [ `'namedmat'` | `'numeric'` ] - Return matrices `C` and `R` as either namedmat objects (matrices with named rows and columns) or plain numeric arrays; if the option `'select='` is used, `'output='` is always a namedmat object.

- `'select='` [ cellstr | `Inf` ] - Return ACF for selected variables only; `Inf` means all variables.

#### Description

`C` and `R` are both N-by-N-by-(P+1)-by-Alt matrices, where N is the number of measurement and transition variables (including auxiliary lags and leads in the state space vector), P is the order up to which the ACF is computed (controlled by the option `'order='`), and Alt is the number of alternative parameterisations in the input model object, `M`. If `'contributions=' true`, the size of the two matrices is N-by-N-by-(P+1)-by-E-Alt, where E is the number of measurement and transition shocks in the model.

##### ACF with linear filters

You can use the option `'filter='` to get the ACF for variables as though they were filtered through a linear filter. You can specify the filter in both the time domain (such as first-difference filter, or Hodrick-Prescott) and the frequncy domain (such as a band of certain frequncies or periodicities). The filter is a text string in which you can use the following references:

- `'L'`, the lag operator, which will be replaced with `exp(-1i*freq)`;
- `'per'`, the periodicity;
- `'freq'`, the frequency.

#### Example

A first-difference filter (i.e. computes the ACF for the first differences of the respective variables):

    [C,R] = acf(m,'filter=','1-L')

#### Example

The cyclical component of the Hodrick-Prescott filter with the smoothing parameter, $lambda$, 1,600. The formula for the filter follows from the classical Wiener-Kolmogorov signal extraction theory,

$$w(L) = \frac{\lambda}{\lambda + \frac{1}{ | (1-L)(1-L) | ^2}}$$

    [C,R] = acf(m,'filter','1600/(1600 + 1/abs((1-L)^2)^2)')

#### Example

A band-pass filter with user-specified lower and upper bands. The band-pass filters can be defined either in frequencies or periodicities; the latter is usually more convenient. The following is a filter which retains periodicities between 4 and 40 periods (this would be between 1 and 10 years in a quarterly model),

    [C,R] = acf(m,'filter','per >= 4 & per <= 40')

IRIS Toolbox. Copyright © 2007—2013 Jaromir Benes.