ecmnfish

Fisher information matrix

Syntax

Fisher = ecmnfish(Data,Covariance)

Fisher = ecmnfish(___,InvCovar,MatrixType)

Description

Fisher = ecmnfish(Data,Covariance) computes an NUMPARAMS-by-NUMPARAMS Fisher information matrix based on the current maximum likelihood parameter estimates.

Use ecmnfish after estimating the mean and covariance of Data with ecmnmle.

Fisher = ecmnfish(___,InvCovar,MatrixType) adds optional arguments for InvCovar and MatrixType.

Examples

Compute Fisher Information Matrix Based on Parameter Estimates for Data

This example shows how to compute the Fisher information matrix based on parameter estimates for Data for five years of daily total returns for 12 computer technology stocks, with six hardware and six software companies

load ecmtechdemo.mat

The time period for this data extends from April 19, 2000 to April 18, 2005. The sixth stock in Assets is Google (GOOG), which started trading on August 19, 2004. So, all returns before August 20, 2004 are missing and represented as NaNs. Also, Amazon (AMZN) had a few days with missing values scattered throughout the past five years.

[ECMMean, ECMCovar] = ecmnmle(Data)

ECMMean = 12×1

    0.0008
    0.0008
   -0.0005
    0.0002
    0.0011
    0.0038
   -0.0003
   -0.0000
   -0.0003
   -0.0000
   -0.0003
    0.0004
      ⋮

ECMCovar = 12×12

    0.0012    0.0005    0.0006    0.0005    0.0005    0.0003    0.0005    0.0003    0.0006    0.0003    0.0005    0.0006
    0.0005    0.0024    0.0007    0.0006    0.0010    0.0004    0.0005    0.0003    0.0006    0.0004    0.0006    0.0012
    0.0006    0.0007    0.0013    0.0007    0.0007    0.0003    0.0006    0.0004    0.0008    0.0005    0.0008    0.0008
    0.0005    0.0006    0.0007    0.0009    0.0006    0.0002    0.0005    0.0003    0.0007    0.0004    0.0005    0.0007
    0.0005    0.0010    0.0007    0.0006    0.0016    0.0006    0.0005    0.0003    0.0006    0.0004    0.0007    0.0011
    0.0003    0.0004    0.0003    0.0002    0.0006    0.0022    0.0001    0.0002    0.0002    0.0001    0.0003    0.0016
    0.0005    0.0005    0.0006    0.0005    0.0005    0.0001    0.0009    0.0003    0.0005    0.0004    0.0005    0.0006
    0.0003    0.0003    0.0004    0.0003    0.0003    0.0002    0.0003    0.0005    0.0004    0.0003    0.0004    0.0004
    0.0006    0.0006    0.0008    0.0007    0.0006    0.0002    0.0005    0.0004    0.0011    0.0005    0.0007    0.0007
    0.0003    0.0004    0.0005    0.0004    0.0004    0.0001    0.0004    0.0003    0.0005    0.0006    0.0004    0.0005
    0.0005    0.0006    0.0008    0.0005    0.0007    0.0003    0.0005    0.0004    0.0007    0.0004    0.0013    0.0007
    0.0006    0.0012    0.0008    0.0007    0.0011    0.0016    0.0006    0.0004    0.0007    0.0005    0.0007    0.0020
      ⋮

To evaluate the negative log-likelihood function for ecmnmle, use ecmnfish based on the current maximum likelihood parameter estimates for ECMCovar.

Fisher = ecmnfish(Data,ECMCovar)

Fisher = 90×90
107 ×

    0.0001    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.0000    0.0001   -0.0000    0.0000   -0.0000    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000   -0.0001         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000    0.0002   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000    0.0003   -0.0000    0.0000   -0.0000   -0.0000   -0.0001   -0.0001    0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000   -0.0000    0.0001   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0001   -0.0000    0.0000   -0.0000    0.0002    0.0000   -0.0000    0.0000    0.0001    0.0000   -0.0002         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000   -0.0000   -0.0000    0.0000    0.0002   -0.0001   -0.0000    0.0000   -0.0000   -0.0001         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0001    0.0004   -0.0000   -0.0001   -0.0000    0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000   -0.0001    0.0000    0.0000   -0.0000   -0.0000    0.0002   -0.0001   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.0000    0.0000   -0.0000   -0.0001   -0.0000    0.0001    0.0000   -0.0001   -0.0001    0.0004   -0.0000   -0.0001         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000    0.0000   -0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0001   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0001   -0.0000   -0.0000   -0.0000   -0.0002   -0.0001    0.0000   -0.0000   -0.0001   -0.0000    0.0004         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0         0         0    0.0756    0.0007    0.0000   -0.0036   -0.0000    0.0000   -0.0395   -0.0002    0.0009    0.0052   -0.0084   -0.0000    0.0002    0.0022    0.0002   -0.0030   -0.0000    0.0001    0.0008    0.0002    0.0000   -0.0194   -0.0001    0.0005    0.0051    0.0011    0.0004    0.0012   -0.0137   -0.0001    0.0003    0.0036    0.0008    0.0003    0.0018    0.0006   -0.0317   -0.0002
         0         0         0         0         0         0         0         0         0         0         0         0    0.0007    0.0990    0.0005   -0.0122   -0.0024    0.0003    0.0045   -0.0258    0.0031   -0.0012   -0.0267   -0.0057    0.0013    0.0067    0.0015    0.0665   -0.0017   -0.0013   -0.0175   -0.0032   -0.0013    0.0069   -0.0127    0.0014   -0.0024    0.0030   -0.0087   -0.0009    0.0008   -0.0089    0.0011   -0.0006    0.0024   -0.0060   -0.0007   -0.0001    0.0015   -0.0208
         0         0         0         0         0         0         0         0         0         0         0         0    0.0000    0.0005    0.0324   -0.0001   -0.0080    0.0005    0.0000    0.0031   -0.0004    0.0001   -0.0001   -0.0175    0.0021   -0.0008    0.0024    0.0003    0.0435   -0.0054    0.0021   -0.0117    0.0146    0.0000    0.0046   -0.0006    0.0002   -0.0012    0.0031    0.0002    0.0000    0.0005   -0.0001    0.0000   -0.0001    0.0004    0.0000    0.0000    0.0000    0.0011
      ⋮

Input Arguments

Data — Data

matrix

Data, specified as an NUMSAMPLES-by-NUMSERIES matrix with NUMSAMPLES samples of a NUMSERIES-dimensional random vector. Missing values are indicated by NaNs.

Data Types: double

Covariance — Maximum likelihood parameter estimates for covariance of Data

matrix

Maximum likelihood parameter estimates for the covariance of the Data using the ECM algorithm, specified as a NUMSERIES-by-NUMSERIES matrix.

InvCovar — Cholesky decomposition of covariance matrix

[ ] (default) | matrix

(Optional) Inverse of covariance matrix, specified as a matrix using inv as:

inv(Covariance)

Data Types: double

MatrixType — Matrix format

'full' (default) | character vector

(Optional) Matrix format, specified as a character vector with a value of:

• 'full' — Computes the full Fisher information matrix.

• 'meanonly' — Computes only the components of the Fisher information matrix associated with the mean.

Data Types: char

Output Arguments

Fisher — Fisher information matrix

matrix

Fisher information matrix, returned as an NUMPARAMSNUMPARAMS matrix based on current parameter estimates, where NUMPARAMS = NUMSERIES * (NUMSERIES + 3)/2 if the MatrixFormat = 'full'. If the MatrixFormat = 'meanonly', then the NUMPARAMS = NUMSERIES.

More About

Fisher Information Matrix

The Fisher information matrix is a key concept in statistics and information theory that measures the amount of information that an observable random variable carries about an unknown parameter upon which the probability of the random variable depends.

The Fisher information matrix is used to evaluate the efficiency of estimators and to derive confidence intervals.

Version History

Introduced before R2006a