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I am trying to run VAR on the first differences of XLE and Brent futures. Prior to this I have already tested that the series in levels are I(1) and are not cointegrated.

Below are my results using VAR in the VARS package in R and I'm not really sure what I should be looking at - could someone shed some light on the interpretation please? Many thanks!

> Diff1VAR<-VAR(data.frame(XLE.Diff1,Brent.Diff1),type=c("const"),p=2)
> summary(Diff1VAR)

VAR Estimation Results:
========================= 
Endogenous variables: XLE.Diff1, Brent.Diff1 
Deterministic variables: const 
Sample size: 175 
Log Likelihood: 470.915 
Roots of the characteristic polynomial:
0.3968 0.3968 0.3812 0.3514
Call:
VAR(y = data.frame(XLE.Diff1, Brent.Diff1), p = 2, type = c("const"))


Estimation results for equation XLE.Diff1: 
========================================== 
XLE.Diff1 = XLE.Diff1.l1 + Brent.Diff1.l1 + XLE.Diff1.l2 + Brent.Diff1.l2 + const 

                Estimate Std. Error t value Pr(>|t|)
XLE.Diff1.l1    0.024898   0.101310   0.246    0.806
Brent.Diff1.l1 -0.043074   0.073885  -0.583    0.561
XLE.Diff1.l2    0.087136   0.100689   0.865    0.388
Brent.Diff1.l2  0.045034   0.070635   0.638    0.525
const           0.005842   0.004818   1.212    0.227


Residual standard error: 0.06313 on 170 degrees of freedom
Multiple R-Squared: 0.01663,    Adjusted R-squared: -0.006507 
F-statistic: 0.7188 on 4 and 170 DF,  p-value: 0.5802 


Estimation results for equation Brent.Diff1: 
============================================ 
Brent.Diff1 = XLE.Diff1.l1 + Brent.Diff1.l1 + XLE.Diff1.l2 + Brent.Diff1.l2 + const 

                Estimate Std. Error t value Pr(>|t|)  
XLE.Diff1.l1    0.254914   0.137533   1.853   0.0655 .
Brent.Diff1.l1  0.069256   0.100301   0.690   0.4908  
XLE.Diff1.l2    0.275127   0.136690   2.013   0.0457 *
Brent.Diff1.l2 -0.099845   0.095890  -1.041   0.2992  
const           0.005070   0.006541   0.775   0.4394  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Residual standard error: 0.0857 on 170 degrees of freedom
Multiple R-Squared: 0.07946,    Adjusted R-squared: 0.0578 
F-statistic: 3.669 on 4 and 170 DF,  p-value: 0.006809 



Covariance matrix of residuals:
            XLE.Diff1 Brent.Diff1
XLE.Diff1    0.003985    0.003544
Brent.Diff1  0.003544    0.007344

Correlation matrix of residuals:
            XLE.Diff1 Brent.Diff1
XLE.Diff1      1.0000      0.6551
Brent.Diff1    0.6551      1.0000
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  • $\begingroup$ What are you looking for? Are you conducting a descriptive / explanatory / predictive study? $\endgroup$ – Richard Hardy Nov 17 '15 at 10:49
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Recall that VAR is a reduced form model and as such its coefficients are difficult to interpret. You could refer to impulse-response functions (IRF) and forecast error variance decomposition (FEVD) to see how the variables develop and affect each other. You could test for Granger causality as well.

Note that the model's $R^2$ is very low (adjusted $R^2$ is even negative) and the $F$-statistic is not statistically significant. Hence, a model including only an intercept in each equation could be superior to your current model. This should not come as a big surprise since oil price (or changes in oil price, as in this model) is notoriously difficult to predict. I would be more surprised if a VAR model was successful here.

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  • $\begingroup$ Thanks @Richard, I am trying to conduct a predictive study. What do you mean by a model that includes only an intercept in each equation, and how do I do that? $\endgroup$ – ElizaTYX Nov 19 '15 at 3:28
  • $\begingroup$ I mean $y_t=\beta_0+\varepsilon_t$, the simplest possible model. $\endgroup$ – Richard Hardy Nov 19 '15 at 6:03
  • $\begingroup$ Hi @Richard, i'm sorry if my questions sound really silly but what are we looking for here in a VAR result? My understanding is that we are looking for how the variables interact, so I should look at the correlation scores between XLE.diff1 Brent.diff1, assuming that the F-stat is statistically significant and R-squared is high. Are these the only results that should be of concern when i look at the results? Also, do you know of any VAR or VECM walkthroughs online that I can read more on? Very much appreciated! $\endgroup$ – ElizaTYX Nov 19 '15 at 7:05
  • $\begingroup$ Correlation scores? What is that? Also, what you look for in the model depends almost entirely on what you intend to use the model for. Since you mentioned your goal is prediction, you could test for Granger causality that is intimately related to prediction. It would show whether one variable is useful when predicting the other. You could also conduct a forecasting exercise by having a rolling window within your original sample, specifying and estimating your model there, and checking how well its forecasts match the actual realized values. Done properly that would be an honest evaluation. $\endgroup$ – Richard Hardy Nov 19 '15 at 7:19

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