Have you ever encountered a situation where the multivariate R^2 is higher than the R^2 of any individual variable? It sounds counterintuitive, right? I mean, shouldn’t the overall R^2 be a weighted average of the individual R^2 values?
A Reddit user, fuckosta, recently asked this very question after fitting a harmonic regression model to a set of time series data. They calculated the R^2 for each individual time series and the overall R^2 using the observations and fitted values as matrices. To their surprise, the overall R^2 was significantly higher than those of the individual time series.
So, what’s going on here? Is this a flaw in their approach, or is there something more subtle at play?
## The Multivariate R^2 Conundrum
When we calculate the R^2 for each individual variable, we’re essentially measuring how well that variable explains the variation in the response variable. In a multivariate setting, however, the R^2 is calculated using the entire set of predictors. This is where things can get interesting.
In some cases, the relationships between the predictors can lead to an overall R^2 that’s higher than any individual R^2 value. This doesn’t necessarily mean that there’s a flaw in the approach; it might simply be a reflection of the complex interactions between the variables.
## Understanding the Differences
To make sense of this phenomenon, let’s consider a few possible explanations:
– **Interaction effects**: The relationships between the predictors can lead to interaction effects that increase the overall R^2. These interactions might not be captured by individual R^2 values, which only consider the relationship between a single predictor and the response variable.
– **Correlated predictors**: When the predictors are correlated, the multivariate R^2 can be higher than the individual R^2 values. This is because the correlated predictors can together explain more variation in the response variable than they would individually.
– **Model specification**: The choice of model specification can also impact the R^2 values. For example, if the harmonic regression model is well-suited to the data, it might capture more variation than individual linear regression models.
## Takeaway
The next time you encounter a multivariate R^2 that’s higher than expected, don’t panic! Instead, take a closer look at the relationships between the predictors and consider the possible explanations mentioned above. It might just be a sign that your model is doing a better job of capturing the underlying patterns in the data.
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*Further reading: [Multiple Regression](https://en.wikipedia.org/wiki/Multiple_regression)*