Fitting Vector Autoregressive Model of Order p Model
Source:R/varlse.R, R/print-varlse.R, R/criteria.R, and 1 more
var_lm.RdThis function fits VAR(p) using OLS method.
Usage
var_lm(y, p = 1, 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)
- include_mean
Add constant term (Default:
TRUE) or not (FALSE)- method
Method to solve linear equation system. (
nor: normal equation (default),chol: Cholesky, andqr: HouseholderQR)- x
A
varlseobject- digits
digit option to print
- ...
not used
- object
A
varlseobject
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.
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 \(pm^2\).
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)
#> [1] "varlse" "olsmod" "bvharmod"
str(fit)
#> List of 16
#> $ coefficients : num [1:19, 1:9] 0.9588 0.0313 -0.0235 -0.0944 -0.0048 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:19] "GVZCLS_1" "OVXCLS_1" "VXFXICLS_1" "VXEEMCLS_1" ...
#> .. ..$ : chr [1:9] "GVZCLS" "OVXCLS" "VXFXICLS" "VXEEMCLS" ...
#> $ fitted.values: num [1:903, 1:9] 21.3 22.2 21.5 21.1 21.3 ...
#> $ residuals : num [1:903, 1:9] 1.008 -0.639 -0.284 0.298 0.308 ...
#> $ covmat : num [1:9, 1:9] 1.113 0.373 0.304 0.445 1.366 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : chr [1:9] "GVZCLS" "OVXCLS" "VXFXICLS" "VXEEMCLS" ...
#> .. ..$ : chr [1:9] "GVZCLS" "OVXCLS" "VXFXICLS" "VXEEMCLS" ...
#> $ df : int 19
#> $ m : int 9
#> $ obs : int 903
#> $ y0 : num [1:903, 1:9] 22.3 21.6 21.2 21.4 21.6 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:9] "GVZCLS" "OVXCLS" "VXFXICLS" "VXEEMCLS" ...
#> $ p : int 2
#> $ totobs : num 905
#> $ process : chr "VAR"
#> $ type : chr "const"
#> $ design : num [1:903, 1:19] 21.5 22.3 21.6 21.2 21.4 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:19] "GVZCLS_1" "OVXCLS_1" "VXFXICLS_1" "VXEEMCLS_1" ...
#> $ y : num [1:905, 1:9] 21.5 21.5 22.3 21.6 21.2 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:9] "GVZCLS" "OVXCLS" "VXFXICLS" "VXEEMCLS" ...
#> $ method : chr "nor"
#> $ call : language var_lm(y = etf_vix, p = 2)
#> - attr(*, "class")= chr [1:3] "varlse" "olsmod" "bvharmod"
# Extract coef, fitted values, and residuals
coef(fit)
#> GVZCLS OVXCLS VXFXICLS VXEEMCLS VXSLVCLS
#> GVZCLS_1 0.958839561 -0.062827385 -0.019710690 0.01257027 0.291868669
#> OVXCLS_1 0.031306519 0.946200622 -0.006632872 -0.03089072 0.005701801
#> VXFXICLS_1 -0.023546344 -0.052053167 0.853672620 -0.06031077 -0.096749298
#> VXEEMCLS_1 -0.094354618 -0.073106564 0.074752695 0.86705288 -0.048927384
#> VXSLVCLS_1 -0.004803889 0.011100124 -0.028180528 -0.00787754 0.741225547
#> EVZCLS_1 0.125516130 -0.138276981 0.077667943 -0.05620528 0.213417413
#> VXXLECLS_1 0.077038206 0.116602536 0.049019313 0.19068844 0.123664057
#> VXGDXCLS_1 -0.041364201 0.022893379 -0.012790918 -0.02444514 -0.051351386
#> VXEWZCLS_1 0.032078312 0.081017743 0.002689773 0.02421098 0.011793646
#> GVZCLS_2 -0.104834812 -0.017750378 -0.063640299 -0.06563731 -0.284901251
#> OVXCLS_2 -0.057264753 0.000198619 -0.012230807 0.01529842 -0.021755404
#> VXFXICLS_2 -0.010770464 0.007308545 0.070962129 0.06532811 0.035457057
#> VXEEMCLS_2 0.109941832 0.031515218 -0.087805109 0.05027743 0.106490192
#> VXSLVCLS_2 0.031836190 0.026261295 0.084884303 0.07330380 0.165164782
#> EVZCLS_2 -0.039927065 0.240594601 0.018595199 0.07171981 -0.095041473
#> VXXLECLS_2 -0.063856767 -0.045562528 -0.052978615 -0.18149299 -0.106494854
#> VXGDXCLS_2 0.084605713 0.007572189 0.029414253 0.02242848 0.085005069
#> VXEWZCLS_2 -0.033916295 -0.071267512 0.007601670 -0.02085634 -0.019106806
#> const 0.484435224 0.071858917 0.764593553 0.73643928 0.993267584
#> EVZCLS VXXLECLS VXGDXCLS VXEWZCLS
#> GVZCLS_1 -0.0070051856 0.010258469 0.186062079 0.016397003
#> OVXCLS_1 0.0057743474 -0.055730279 -0.008620592 -0.017425642
#> VXFXICLS_1 -0.0013000787 -0.071102522 -0.019267681 -0.093669606
#> VXEEMCLS_1 0.0082641013 0.043573682 0.052515700 0.111449434
#> VXSLVCLS_1 0.0048970514 -0.010365439 -0.037485186 -0.035578884
#> EVZCLS_1 0.9530755788 -0.159179164 -0.029104883 -0.109896535
#> VXXLECLS_1 0.0073924911 1.090977549 0.173739155 0.097903592
#> VXGDXCLS_1 -0.0170045623 -0.002534079 0.717661085 0.014081721
#> VXEWZCLS_1 0.0295958720 0.039727624 0.029208637 0.919861136
#> GVZCLS_2 0.0011140822 -0.064661782 -0.129207944 -0.025690608
#> OVXCLS_2 -0.0013553534 0.045540919 0.007661278 0.007209753
#> VXFXICLS_2 0.0193175978 0.067148832 0.057541033 0.120471274
#> VXEEMCLS_2 -0.0273314378 -0.088654468 -0.128280935 -0.177756749
#> VXSLVCLS_2 0.0074048747 0.057079506 0.076151265 0.072552465
#> EVZCLS_2 -0.0026576824 0.171823659 -0.062484101 0.077971334
#> VXXLECLS_2 -0.0007146444 -0.109371143 -0.146653392 -0.060460831
#> VXGDXCLS_2 0.0111059419 0.007634580 0.222052920 -0.026452775
#> VXEWZCLS_2 -0.0304841577 -0.030496662 -0.018978557 0.055930225
#> const 0.1341878224 0.742494021 0.705738212 0.796582789
head(residuals(fit))
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1.0084235 -0.01665024 -0.1189584 0.18259423 1.0345739 0.54161189
#> [2,] -0.6393271 1.07781122 -0.7889172 0.06023516 -0.2598844 -0.30429270
#> [3,] -0.2841251 -1.13853760 0.8285284 1.09281272 1.3066535 0.53399638
#> [4,] 0.2975186 -0.89689641 -0.5797484 -0.31164787 1.3474286 -0.09610840
#> [5,] 0.3075188 -0.98987446 -0.6230728 -0.46134610 1.2499047 0.01799667
#> [6,] -0.4052045 -1.62792068 -0.9718923 -2.26508501 -1.1137253 -0.25997035
#> [,7] [,8] [,9]
#> [1,] 0.97639374 0.3811097 -0.2513162
#> [2,] 0.08960441 -0.6696317 0.4110768
#> [3,] 0.24450709 -0.9864375 1.7865058
#> [4,] -0.20575733 0.6892149 -0.3931113
#> [5,] -0.21831258 1.3123282 -0.5570114
#> [6,] -0.18371357 -1.1068942 -1.1856081
head(fitted(fit))
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 21.33158 35.53665 29.17896 29.63741 42.47543 12.57839 26.67361 33.37889
#> [2,] 22.23933 35.51219 29.24892 29.94976 43.08988 13.07429 27.55040 34.16963
#> [3,] 21.48413 36.75854 28.71147 30.03719 42.17335 12.77600 27.55549 33.79644
#> [4,] 21.10248 35.69190 29.68475 30.97165 42.64257 13.32611 27.66076 33.26579
#> [5,] 21.29248 34.95987 29.29307 30.65135 43.25010 13.13200 27.32831 33.78767
#> [6,] 21.54520 34.21792 28.93189 30.24509 43.75373 13.04997 27.03371 34.90689
#> [,9]
#> [1,] 30.43132
#> [2,] 30.33892
#> [3,] 30.91349
#> [4,] 32.54811
#> [5,] 32.16701
#> [6,] 31.65561