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I have monthly financial time-series data from 2011-present of four stock market indices. I conducted various stationarity tests and found that the series are I(1) processes (stationary only in first differences). Then, testing for cointegration with the Johanssen, I find that the series are NOT cointegrated....

Do I use VAR in levels or in differences? I am getting conflicting answers in my research.

Richard Hardy
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DGD_987
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  • Could you point to a source that suggests not to difference in such a setting? – Richard Hardy May 11 '21 at 11:54
  • I think they also deal with this stuff at quant.stackexchange – Dom Fomello May 11 '21 at 11:57
  • @DomFomello, likely so, though the database of such questions is larger on Cross Validated, so it makes sense to ask here. – Richard Hardy May 11 '21 at 12:00
  • @RichardHardy https://stats.stackexchange.com/questions/232573/var-in-levels-for-cointegrated-data ------- SimonCW says "You can also estimate VAR in levels of non-stationary variables when there is no cointegration present!" – DGD_987 May 11 '21 at 12:07
  • @DGD_987, see my comment under his answer. I can recommend Ashley & Verbrugge (2009) from that answer. Clearly written with clearly summarized results. – Richard Hardy May 11 '21 at 13:10
  • @RichardHardy Thanks for the info. I have a few more questions if you don't mind: - given I am estimating a VAR in differences, should I also select the optimal lag (e.g. with varsoc in stata) using the differenced variables? - what would it mean to 'specify' different VAR models? I don't understand what things could be 'changed around'. E.g. if I have four stock market indices, is there more than one VAR combination? – DGD_987 May 11 '21 at 14:12
  • @DGD_987, yes, I would select the lags using the selected specification (in your case, in differences). To "specify" may mean to choose (1) transformation such as differencing, if any, (2) lag length, (3) presence/absence of trend and/or intercept. – Richard Hardy May 11 '21 at 14:17
  • @RichardHardy Hi, just another query. If I am asked to estimate/forecast volatility, should I assume that since the series are I(1) that I should therefore estimate a GARCH with the variables first differenced? As I understand, VAR won't capture volatility. Thanks – DGD_987 May 12 '21 at 15:58
  • @DGD_987, GARCH is about the conditional variance of the residuals of the conditional mean model. Therefore, it does not matter what conditional mean model you have as long as you are satisfied with it. – Richard Hardy May 12 '21 at 16:05

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You should go for differencing. Without it, you would have I(1) variables that are not cointegrated on the left and the right hand sides of each equation, causing one side to diverge from the other unless (1) the slope coefficients of own lags sum to one and (2) the slope coefficients of lags of other variables sum to zero for each variable (assuming there is no cointegration between subgroups of variables). E.g. take one equation of a bivariate VAR(1) model for variables $(y,x)$: $$ y_{t}=\beta_0+\beta_1 y_{t-1}+\beta_2 x_{t-1}+\varepsilon_t $$ where $y_t$ and $x_t$ do not share a common trend and thus are diverging. The left hand side of the equation would diverge from the right hand side unless $\beta_1=1,\beta_2=0$, provided that the residual $\varepsilon_t$ is required to be I(0). Proceeding with such a model would make little sense.

After differencing, you will be dealing with stationary series, and the problem will be solved.

The above presumes the variables are truly I(1). If there is some doubt about that and you are interested in impulse responses, see Ashley & Verbrugge (2009) for the idea of using VAR in levels augmented with lags of first differences.

Richard Hardy
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