Hi all, I am looking for a test that will tell me whether a hedge fund return has a linear relationship with market index return. I am working with return data series from 100 over hedge funds and comparing them with Morgan Stanley family of indices. Is there any MATLAB functions that do this? Any advice much appreciated!!P.S. I understand that hedge funds returns would not be linear in the first place due to the type of trading strategies they adopt (eg. options and short selling and leverage etc). I came across a linearity test by Tsay entitled "Testing and Modeling Threshold Autoregressive Processes" in the Journal of the American Statistical Association in 1989. Is such a test already available in standard MATLAB package?

The standard test for linearity (or indeed any model misspecification) is the Ramsey RESET test. I am not sure if this can be done automatically in Matlab, but any statistical package, such as Eviews, will carry it out. As shown below it's easy enough to implement manually:The test is as follows:original regression:y = a + bxRamsey RESET test (to order 2) augments the original regression with a dependant variable squared and cubed term:y = a + bx + c(yhat^2) + d(yhat^3)The joint null hypothesis is then tested using either likelihood ratios or F-tests:H0 : c = d = 0If you cannot reject the null, the original regression is well specified + the relationship can be assumed to be linear.I recently read a paper from UBS analysing hedge fund returns as a market index augmented with coskewness and cokurtosis (the 4 moment CAPM). Leave me a post here if that paper would interest you. Edit = adding in yhat rather than y

Using the R-squared is not the best approach. As more and more explanatory variables are added (even if they are obviously unrelated by standard logic), the R-Squared tends to increase, this is related to the spurious regression problem. So if you are comparing a linear with a quadratic relationship, the quadratic (y = a + bx +cx^2) will almost always have a higher R-squared regardless of the population relationship. Some of this error can be removed by using a goodness of fit measure which takes into account the number of explanatory variables, such as adjusted R-Squared, Akaike IC or Schwarz Bayesian Criterion.Moreover it is difficult to draw robust statistical inferences from goodness of fit ratios, whereas the Ramsey-RESET test allows easy hypothesis testing.Hope that's a decent explanation.

I would say something stronger, R^2 is almost useless for testing linearity. The integers from 0 to 10 have an R^2 of 0.93 with their squares, and that is not linear at all. As long as a function is monotonic and does not have large outliers, it will have a high R^2.I don't like general-purpose tests for linearity (and almost anything else). You should always test for something of practical meaning. There's a reason you want to know if hedge fund returns are linear. Probably it's because you're worried about negative skew. In that case, test for negative skew directly.And don't test in the data, consider a model for the returns that will have negative skew, and determine what you will see in the data. This is a crucial distinction. For example, consider someone playing a martingale strategy in roulette (double the bet after every loss). You will only observe the negative skew after a disaster, and then it's too late. Instead you will observe abnormally steady returns given the variance of individual bets.