var_lm: Fitting Vector Autoregressive Model of Order p Model

In bvhar: Bayesian Vector Heterogeneous Autoregressive Modeling

Fitting Vector Autoregressive Model of Order p Model

Description

This function fits VAR(p) using OLS method.

Usage

var_lm(
  y,
  p = 1,
  exogen = NULL,
  s = 0,
  include_mean = TRUE,
  method = c("nor", "chol", "qr")
)

## S3 method for class 'varlse'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

## S3 method for class 'varlse'
logLik(object, ...)

## S3 method for class 'varlse'
AIC(object, ...)

## S3 method for class 'varlse'
BIC(object, ...)

is.varlse(x)

is.bvharmod(x)

## S3 method for class 'varlse'
knit_print(x, ...)

Arguments

y	Time series data of which columns indicate the variables
p	Lag of VAR (Default: 1)
exogen	Exogenous variables
s	Lag of exogeneous variables in VARX(p, s). By default, s = 0.
include_mean	Add constant term (Default: TRUE) or not (FALSE)
method	Method to solve linear equation system. (nor: normal equation (default), chol: Cholesky, and qr: HouseholderQR)
x	Any object
digits	digit option to print
...	not used
object	A varlse object

Details

This package specifies VAR(p) model as

Y_{t} = A_1 Y_{t - 1} + \cdots + A_p Y_{t - p} + c + \epsilon_t

If include_type = TRUE, there is constant term. The function estimates every coefficient matrix.

Consider the response matrix Y_0. Let T 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 n = T - 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{nm}{2} \log 2\pi - \frac{n}{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, n.

\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

Let \tilde{\Sigma}_e be the MLE and let \hat{\Sigma}_e be the unbiased estimator (covmat) for \Sigma_e. Note that

\tilde{\Sigma}_e = \frac{n - k}{n} \hat{\Sigma}_e

Then

AIC(p) = \log \det \Sigma_e + \frac{2}{n}(\text{number of freely estimated parameters})

where the number of freely estimated parameters is mk, i.e. pm^2 or pm^2 + m.

Let \tilde{\Sigma}_e be the MLE and let \hat{\Sigma}_e be the unbiased estimator (covmat) for \Sigma_e. Note that

\tilde{\Sigma}_e = \frac{n - k}{T} \hat{\Sigma}_e

Then

BIC(p) = \log \det \Sigma_e + \frac{\log n}{n}(\text{number of freely estimated parameters})

where the number of freely estimated parameters is pm^2.

Value

var_lm() returns an object named varlse class. It is a list with the following components:

coefficients

Coefficient Matrix

fitted.values

Fitted response values

residuals

Residuals

covmat

LS estimate for covariance matrix

df

Numer of Coefficients

p

Lag of VAR

m

Dimension of the data

obs

Sample size used when training = totobs - p

totobs

Total number of the observation

call

Matched call

process

Process: VAR

type

include constant term (const) or not (none)

design

Design matrix

y

Raw input

y0

Multivariate response matrix

method

Solving method

call

Matched call

It is also a bvharmod class.

References

Lütkepohl, H. (2007). New Introduction to Multiple Time Series Analysis. Springer Publishing.

Akaike, H. (1969). Fitting autoregressive models for prediction. Ann Inst Stat Math 21, 243-247.

Akaike, H. (1971). Autoregressive model fitting for control. Ann Inst Stat Math 23, 163-180.

Akaike H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, vol. 19, no. 6, pp. 716-723.

Akaike H. (1998). Information Theory and an Extension of the Maximum Likelihood Principle. In: Parzen E., Tanabe K., Kitagawa G. (eds) Selected Papers of Hirotugu Akaike. Springer Series in Statistics (Perspectives in Statistics). Springer, New York, NY.

Gideon Schwarz. (1978). Estimating the Dimension of a Model. Ann. Statist. 6 (2) 461 - 464.

See Also

• summary.varlse() to summarize VAR model

Examples

# Perform the function using etf_vix dataset
fit <- var_lm(y = etf_vix, p = 2)
class(fit)
str(fit)

# Extract coef, fitted values, and residuals
coef(fit)
head(residuals(fit))
head(fitted(fit))

bvhar documentation built on June 25, 2025, 9:07 a.m.