# Do low $R^2$ values mean that my vector autoregressive model is bad?

I have a VAR model, which shows very low $R^2$ values (below 0.05). Does this mean that my model is very bad in explaining?

• Can you describe your data a little more? – cardinal Jul 18 '12 at 14:44
• Thank you very much for your answers! I will describe my model a little bit better: I am actually trying to see if there is a causality (Granger) between two variables and the yield on certain european 6month T-Bill yields. The first is the spread between 6month EURIBOR and EONIAswap rates. The second is the spread between all-rated 10-year european government debt and AAA-rated 10-year european government debt. – Johannes Jul 18 '12 at 17:35
• What is the "sampling rate" to accumulate your 682 observations? – cardinal Jul 18 '12 at 17:37
• its daily data (business days, i.e. 250 per year) – Johannes Jul 18 '12 at 17:39

## 2 Answers

The low $R^2$ is a result of your choice of using differences in interest rates. However, $R^2$ is not a particularly indicative of whether you have a good model or not. Estimating a VAR model on interest rate levels rather than differences would give you a much higher $R^2$, but the higher $R^2$ is not strictly evidence of having a better model (though the model is better because it incorporates the mean-reversion and cointegration effects that your VAR model in differences ignores). In economic and financial data it is common to see approximately AR(1) data with high $R^2$ values and then when differencing the $R^2$ value declines. Sometimes it is appropriate to difference, but when there is mean-reversion (or cointegration), then not accounting for those effects can lead to bad forecasts. An error-correction model would provide a more accurate $R^2$ than a VAR in levels would and would provide the same forecasts.

Also, rather than imposing a spread in the model, it is often better to let the statistical model do the hard work of determining the appropriate relationship. The cointegrating vectors from an ECM are better to use than typical spreads. Just using the spreads is kind of like you imposing your own prior on the cointegrating vectors (but ignoring the rest of the Bayesian framework).

This might not be appropriate for this form, but it looks like you are using VAR(1) with interest rate data. Interest rate data are often cointegraded. Make you sure you test for cointegration. I think you are doing something w/ the yield curve but from what you posted the model is not clear. I would take a deeper look at the yield curve literature and see how people model these types of interest rates in a VAR framework. Also, you would probably want to include libor or the fed funds rate. In general, it is often most interesting to use credit spreads in these types of regressions, e.g. BAA - AAA

To answer your main question: no, it's bad if a model has a low $R^2$, especially if it's based on a theoretical construct. You need to realize, however, that there may be other regressors that explain the dependent variable and be careful about omitted variable bias (e.g. inflation expectations in your model)