| logLik.varlse {bvhar} | R Documentation |
Extract Log-Likelihood of Multivariate Time Series Model
Description
Compute log-likelihood function value of VAR(p), VHAR, BVAR(p), and BVHAR
Usage
## S3 method for class 'varlse'
logLik(object, ...)
## S3 method for class 'vharlse'
logLik(object, ...)
## S3 method for class 'bvarmn'
logLik(object, ...)
## S3 method for class 'bvarflat'
logLik(object, ...)
## S3 method for class 'bvharmn'
logLik(object, ...)
Arguments
object |
Model fit |
... |
not used |
Details
Consider the response matrix Y_0.
Let n be the total number of sample,
let m be the dimension of the time series,
let p be the order of the model,
and let s = n - p.
Likelihood of VAR(p) has
Y_0 \mid B, \Sigma_e \sim MN(X_0 B, I_s, \Sigma_e)
where X_0 is the design matrix,
and MN is matrix normal distribution.
Then log-likelihood of vector autoregressive model family is specified by
\log p(Y_0 \mid B, \Sigma_e) = - \frac{sm}{2} \log 2\pi - \frac{s}{2} \log \det \Sigma_e - \frac{1}{2} tr( (Y_0 - X_0 B) \Sigma_e^{-1} (Y_0 - X_0 B)^T )
In addition, recall that the OLS estimator for the matrix coefficient matrix is the same as MLE under the Gaussian assumption.
MLE for \Sigma_e has different denominator, s.
\hat{B} = \hat{B}^{LS} = \hat{B}^{ML} = (X_0^T X_0)^{-1} X_0^T Y_0
\hat\Sigma_e = \frac{1}{s - k} (Y_0 - X_0 \hat{B})^T (Y_0 - X_0 \hat{B})
\tilde\Sigma_e = \frac{1}{s} (Y_0 - X_0 \hat{B})^T (Y_0 - X_0 \hat{B}) = \frac{s - k}{s} \hat\Sigma_e
In case of VHAR, just consider the linear relationship.
While frequentist models use OLS and MLE for coefficient and covariance matrices, Bayesian models implement posterior means.
Value
A logLik object.
References
Lütkepohl, H. (2007). New Introduction to Multiple Time Series Analysis. Springer Publishing.
Corsi, F. (2008). A Simple Approximate Long-Memory Model of Realized Volatility. Journal of Financial Econometrics, 7(2), 174–196.
Bańbura, M., Giannone, D., & Reichlin, L. (2010). Large Bayesian vector auto regressions. Journal of Applied Econometrics, 25(1).
Litterman, R. B. (1986). Forecasting with Bayesian Vector Autoregressions: Five Years of Experience. Journal of Business & Economic Statistics, 4(1), 25.
Ghosh, S., Khare, K., & Michailidis, G. (2018). High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models. Journal of the American Statistical Association, 114(526).