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Hello,

is someone familiar with vector autoregressive models (VAR-models) and forecast error variance decomposition (FEVD) and can help me with following issue??

I have set up two bivariate VAR-models with lag length 1.
The variables are:
XA = daily time series of trading volume on stock exchange in country A
YA = daily time series of stock market volatility in country A
XB = daily time series of trading volume on stock exchange in country B
YB = daily time series of stock market volatility in country B
(b = VAR-coefficients)
(a = constant)
(e = error term)

The bivariate VAR-models:
VAR(I):
XA(t) = a(a1) + b(a1,1)*XA(t-1) + b(a1,2)*YA(t-1) + e(a1,t)
YA(t) = a(a2) + b(a2,1)*XA(t-1) + b(a2,2)*YA(t-1) + e(a2,t)

VAR(II):
XB(t) = a(b1) + b(b1,1)*XB(t-1) + b(b1,2)*YB(t-1) + e(b1,t)
YB(t) = a(b2) + b(b2,1)*XB(t-1) + b(b2,2)*YB(t-1) + e(b2,t)

Now we conduct forecast error variance decomposition on the two VAR-models. For forecast step 20, the FEVD returns the following values:

VAR(I): XA: 80% on XA / 20% on YA YA: 10% on XA / 90% on YA

VAR(II): XB: 60% on XB / 40% on YB YB: 5% on XB / 95% on YB

I interpreted the results as follows:
(i) Over a horizon of 20 days, 20% of the movement of XA can be explained by the movement of YA (following the VAR-model)
(ii) Over a horizon of 20 days, YA does a better job in explaining the movement of XA than XA does in explaining changes in YA
(iii) In country B, volatility (YB) has a bigger influence on trading volume (XB) than volatility (YA) on trading volume (XA) in country A over a horizon of 20 days

I am not completely sure if I got the FEVD methodology right, so could someone please help me on this issue with a short feedback if my interpretation of the results in right?
Thank you very much in advance!

Best, Jost

First of all, make sure that you have tried alternative (longer) horizons. You should see that variance decomposition is actually converged.
Try also changing the order of variables, since results may dramatically change if the correlation between the error terms are significantly high.
Having said that, results of the second VAR model would suggest that YB behaves more like an exogenous variable.

Thank you very much for the useful comments.
actually, I have set up several models with varying time horizons and converging results. I also tried different lag length, order of variables etc. for robustness checks.
However, I am now mainly interested in how to interpret the results of FEVD qualitatively. So the VAR-model stated above is an example with how I would interpret the FEVD results with my knowledge about the methodology. I just want to be sure that these statements are supported by the results and not tee progressively interpreted or even contrary to the results.

I can give another example in a less abstract sense:

I want to analyze the interrelation of daily stock market returns [return] and daily investor mood [mood], proxied by a measure of sentiment retrieved from Twitter-posts for countries A and B.

VAR(I) for country A:
[mood_A](t) = a(a1) + b(a1,1)*[mood_A](t-1) + + b(a1,2)*[return_A](t-1) + e(a1,t)
[return_A](t) = a(a2) + b(a2,1)*[mood_A](t-1) + b(a2,2)*[return_A](t-1) + e(a2,t)

VAR(II) for country B:
[mood_B](t) = a(b1) + b(b1,1)*[mood_B](t-1) + b(b1,2)*[return_B](t-1) + e(b1,t)
[return_B](t) = a(b2) + b(b2,1)*[mood_B](t-1) + b(b2,2)*[return_B](t-1) + e(b2,t)

Now I conduct forecast error variance decomposition on the two VAR-models. For forecast step 100, the FEVD returns the following values:

VAR(I):
[mood_A]: 85% on [mood_A] / 15% on [return_A]
[return_A]: 5% on [mood_A] / 95% on [return_A]

VAR(II):
[mood_B]: 65% on [mood_B] / 35% on [return_B]
[return_B]: 5% on [mood_B] / 95% on [return_B]

My interpretation:
(i) for a forecast horizon of 100 days, 15% of the movement in investor mood in country A [mood_A] can be explained by changes in stock market returns [return_A].
(ii) for a forecast horizon of 100 days, market return [return_A] does a better job in explaining the movement of investor mood [mood_A] than investor mood [mood_A] does in explaining changes in market return [return_A] .
(iii) In country B, stock market return [return_B] has a bigger influence on investor mood [mood_B] than stock market return [return_A] has on investor mood [mood_A] in country A over a horizon of 20 days.

-> given that I can economically argue that ordering the variables in the I we do (first market return than investor mood), is it possible to interpret these results the way I do?

Yes, I believe so. FEVD measures the relative importance of each random innovations/shocks. Therefore I would prefer to use these terms instead of "changes". And would call the impact of endogenous variables as the major driving factors, if they were not "negligible". Yet, these are merely subjective choices of qualitative expressions...