Matlab: Calculating inverse of covariance matrix for time series model -


<code>x(n) = \sum_{j=1}^p a_jx(n-j) + u(n)</code>

is univariate autoregressive ar model of order p = 2 , data samples n excited u gaussian 0 mean noise , variance sigma_u^2.

i aware of function cov(x) how use pxp covariance matrix c(n) , inverse, s(n) = c(n) coefficients of ar model. mathematical formula inverse covariance matrix, s(n) expressed

invcovformula

where a_1 , a_2 lower triangular toeplitz matrices.

inversecov

can directly pinv(cov(coefficients))? unsure how pass arguments ar coefficients function.

how implement formula? thank help

your question entirely ill posed, others have noted, encourage reading , think what want do first. nonetheless, can provide guidance, has more straightening out concepts programming. based on question , comments, question should re-phrased as, "how 1 estimate covariance matrix parameters of ar(n) model in parameters obtained via maximum likelihood estimation?" in case can answer follows.

  1. for sake of discussion let's first generate test data. 
>> rng(0); >> n = 10000; >> x = rand(n,2);
  1. first obtain initial conditions smartly running ols. how people estimate autoregressions anyway (nearly entire literature on vector autoregressions) , ols , mle asymptotically equivalent under conditions. include column of zeros if want include constant (which want if working non-demeaned data). in output b parameter estiamte vector , c corresponding vector of standard errors. 
>> x = [x,ones(n,1)]; >> [b,c]=lscov(x,y)  b =      4.9825     9.9501    20.0227   c =      0.0347     0.0345     0.0266
  1. before can maximum likelihood need likelihood first. can construct 1 composition of couple of simple anonymous functions. need construct initial conditions vector includes standard deviation of prediction error because 1 of things going estimate using mle. in case going assume distributed error, , base example on that, didn't , of course choose (not having distributed error typically motivation not doing ols). also, build negative log likelihood since using minimization routines. 
>> err = @(b) y - x*b  err =       @(b)y-x*b  >> std(err(b))  ans =      0.9998  >> b0 = [b; std(err(b))]; >> nll = @(b) (n/2)*log(2*pi*b(end)^2) + (1/(2*b(end)^2))*sum(err(b(1:end-1)).^2);
  1. next use type of minimization routine (e.g., fminsearch, fminunc, etc.) mle. 
>> bmle = fminsearch(nll,b0)  bmle =      4.9825     9.9501    20.0227     0.9997

unsurprisingly, estimates identical obtained under ols, expect when errors distributed, , why people opt doing ols unless there compelling reason think errors non-normal. covariance matrix can estimated outer product of scores, particularly simple expression in normal case.

>> inv(x'*x/bmle(end))  ans =      0.0012    0.0000   -0.0006     0.0000    0.0012   -0.0006    -0.0006   -0.0006    0.0007

finally, standard errors match obtained in least squares case. 

>> sqrt(diag(inv(x'*x/bmle(end))))  ans =      0.0347     0.0345     0.0266

edit: sorry, realized test data cross sectional instead of time series data. fix time. methodology estimating models stays same.


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