Bose, E., Hravnak, M., & Sereika, S. M. (2017). Vector autoregressive (VAR) models and Granger causality in time
series analysis in nursing research: Dynamic changes among vital signs prior to cardiorespiratory instability events
as an example. Nursing Research, 66(1). Supplemental Digital Content 1.
Exemplar Model
Illustration of a VAR model with 12 lags for HR as the dependent variable for one patient. VAR model is
constructed beginning with the first lag up to the lag order specified by the lag order selection criteria. This
includes HR from lag 1 (HR (-1)) up to lag 12, represented as HR (-12), RR up to lag 12, represented as RR(-12) and
SpO2 up to lag 12, represented as SPO2(-12) with coefficients present for each of the terms along with a constant
term represented as C(1,37).
General Form of the VAR (p) Model
Consider a given sample time-period with t = 1, 2, ….T, and a variable Y measured over
time. Then Yt = (Y1,t , Y2,t , Y3,t,, …..Yk,t) is a k x 1 (k ԑ N) vector of time-series variables, a p-lag
VAR(p) model is denoted as
where the p-time periods back observation
is called the pth lag of Y, c is a k x 1 vector of
constants, Ai is a time invariant k x k matrix and et is a k x 1 vector of error terms.
A general VAR (p) with k variables and T+1 observations Y0 to YT can be written as
where
Bose, E., Hravnak, M., & Sereika, S. M. (2017). Vector autoregressive (VAR) models and Granger causality in time
series analysis in nursing research: Dynamic changes among vital signs prior to cardiorespiratory instability events
as an example. Nursing Research, 66(1). Supplemental Digital Content 1.
and
and
The coefficient matrix B can be obtained by solving using ordinary least squares (OLS) estimation of Y
BZ.
A detailed explanation of this procedure is in Lütkepohl (2005). Since within the VAR (p), each
equation has the same explanatory variables, each equation may be estimated separately by
ordinary least squares. Recommended sources for information about estimation in VAR
modeling are Lütkepohl (2005) and Hamilton (1994).
Residual Autocorrelation
To illustrate, for a linear regression equation such as
where the residuals follow an autoregressive scheme, ut can be expressed as
For the LM test, Equation 7 is first fitted using OLS to obtain a set of sample residuals
and
then an auxiliary model is fitted with
The R2 statistic is calculated for this model and asymptotic approximation used for the
distribution of the test statistic. The null hypothesis is H0 :
that is there is no
autocorrelation and each of the observations are independent of each other versus the alternative
H1 :
for at least one i ε {1,……p}meaning that autocorrelation exists and that future
observations are affected by past values. The LM test may be viewed as a test for zero coefficient
Bose, E., Hravnak, M., & Sereika, S. M. (2017). Vector autoregressive (VAR) models and Granger causality in time
series analysis in nursing research: Dynamic changes among vital signs prior to cardiorespiratory instability events
as an example. Nursing Research, 66(1). Supplemental Digital Content 1.
matrices in (9) for the residuals. A more precise form of the statistic is available in Lütkepohl
(2005, Section 4.4.4).
Stability
Given an nth degree polynomial,
if ri represents the root, then P(ri) = 0. The roots can be determined by computing the
eigenvalues (λi) of the matrix
and taking ri = 1/ λi. If no root lies outside the unit circle as shown in Figure 3 for one patient
(#1), then VAR system satisfies the stability condition, indicating model stability.
Granger Causality
To illustrate mathematically, if x and y represent two stationary time-series, the univariate
autoregression of y is
where
m ԑ N represents the regression coefficients of the autoregressive equation above.
Equation SDC11 is augmented by then including lagged values of x as follows
where p is the shortest lag length of x and q the longest lag length of x.
The null hypothesis for GC is that no explanatory power is added by the predictors when an Ftest (and subsequently a χ2) is conducted on Equation SDC12.
Bose, E., Hravnak, M., & Sereika, S. M. (2017). Vector autoregressive (VAR) models and Granger causality in time
series analysis in nursing research: Dynamic changes among vital signs prior to cardiorespiratory instability events
as an example. Nursing Research, 66(1). Supplemental Digital Content 1.
References in SDC
Hamilton, J. D. (1994). Time series analysis. Princeton, NJ: Princeton University Press.
Lütkepohl, H. (2005). New introduction to multiple time series analysis. Berlin, Germany:
Springer Berlin Heidelberg.