| 23 | | Following these steps for each Metropolis-Hastings step we can generate a candidate of vector [[LatexEquation(\beta' $$)]] calculating simultaneously its exact density: |
| 24 | | 1. Since this system has been previously decomposed is very fast to generate a pre-candidate vector [[LatexEquation(\beta' $$)]] matching it |
| 25 | | 1. The corresponding ARIMA noise is simply [[BR]][[LatexEquation(z' = Y' - X' \beta' $$)]] |
| 26 | | 1. By means of Almagro method it's posible to calculate residuals [[LatexEquation(e' $$)]] and initial values [[LatexEquation(u' $$)]] that solve difference equation [[BR]] [[LatexEquation(e'_t = \frac{\phi'\left(B\right)}{\theta'\left(B\right)} z'_t $$)]] |
| 27 | | 1. Let be the standarized multinormal residuals [[BR]] [[LatexEquation( \epsilon' =\frac{1}{\sigma'}e' $$)]] [[BR]] |
| 28 | | 1. Since [[LatexEquation(\beta' $$)]] are determined by [[LatexEquation(\epsilon' $$)]] by means of a full range linear equation, their densities have constant ratio, and logarithm of density of candidate vector can be calculated directly as [[BR]] [[LatexEquation(cte -\frac{T}{2}\log\left(2\pi\right)-\frac{1}{2}\sum_{t=1}^{T}\epsilon'_{t}^{2}$$)]] |
| 29 | | 1. Then we can propose residuals and initial values for current system as [[BR]][[LatexEquation(e^{*} = \frac{\sigma}{\sigma'}e' $$)]] [[BR]] [[LatexEquation(u^{*} = \frac{\sigma}{\sigma'}u' $$)]] |
| 30 | | 1. ARIMA noise for current system becomes simply [[BR]][[LatexEquation(z^{*}_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e^{*}_t $$)]] |
| 31 | | 1. In order to get the refined candidate, we will take minimum residuals solution of sparse linear system [[BR]][[LatexEquation(X \beta = Y-z^{*} $$)]] |
| 32 | | 1. In order to calculate its exact density, it will be obtained the ARIMA noise [[BR]][[LatexEquation(z = Y - X \beta $$)]] [[BR]] to get standarized residuals from ARIMA equations. |
| 33 | | 1. If resulting vector doesn't match constraining inequations [[BR]] [[LatexEquation(A \beta <= a $$)]][[BR]] density will be toggled to [[LatexEquation(-\infty $$)]] in order to force rejection. |
| | 23 | Following these steps for each Metropolis-Hastings step we can generate a candidate of vector [[LatexEquation(\beta $$)]] calculating simultaneously its exact density: |
| | 24 | 1. Since this system has been previously decomposed is very fast to generate a candidate vector and its density [[LatexEquation( \log\left(Q\left(\beta\right)\right) $$)]], that is not depending on previous state. |
| | 25 | 1. In order to calculate the density for current system we will get corresponding ARIMA noise [[BR]][[LatexEquation(z = Y - X \beta $$)]] |
| | 26 | 1. By means of Levinson or Almagro method of ARMA evaluation it's posible to calculate in a very fast way differential equation [[BR]][[LatexEquation(\phi\left(B\right) z_t = \theta\left(B\right) e_t $$)]][[BR]] getting also residuals likelihood which logarithm is, but a constant, the exact density [[LatexEquation( \log\left(P\left(\beta\right)\right) $$)]] |
