[C,R,List] = acf(M,...)
M [ model ] - Solved model object for which the ACF will be computed.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.
'applyTo=' [ cellstr | char | @all ] - List of variables to which the 'filter=' will be applied; @all 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.
'matrixFmt=' [ 'namedmat' | 'plain' ] - Return matrices C and R as either namedmat objects (i.e. matrices with named rows and columns) or plain numeric arrays.
'select=' [ @all | char | cellstr ] - Return ACF for selected variables only; @all means all variables.
C and R are both N-by-N-by-(P+1)-by-NAlt 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 NAlt 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-by-NAlt, where E is the number of all shocks (measurement and transition combined) in the model.
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.A first-difference filter (i.e. computes the ACF for the first differences of the respective variables):
[C,R] = acf(m,'filter=','1-L')
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)')
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')