Forums > Pricing & Modelling > Relative Value using Regression vs ARIMA

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 Mistro Total Posts: 22 Joined: Aug 2018
 Posted: 2018-12-09 10:10 Hi everyone, I am building a model for relative value equity options trading and could use some help with some of the statistical aspects. The two assets under the microscope are VNQ and SPY. Using OLS we get a very nice relationship when we take the log of both their volatility. log(Spy_vol) ~ log(VNQ_vol). The problem is, the auto correlation of the residuals is very high.I am looking at ARIMA models (generalized least squares did not seem to have a better fit), but I am not sure on how I can apply it to this data set or if there are other alternatives. Below is the OLS regression and the AC men lie, women lie, numbers don't
 day1pnl Total Posts: 54 Joined: Jun 2017
 Mistro Total Posts: 22 Joined: Aug 2018
 Posted: 2018-12-09 22:43 It is the 20 day yang.zhang vol. Here is the non- log plot for a better understanding. men lie, women lie, numbers don't
 Mistro Total Posts: 22 Joined: Aug 2018
 Posted: 2018-12-09 23:48 Thanks day1 for your in depth response. The problem I have is, the residuals are correlated. This means the regression is not BLUE no matter what I incorporate into the model. One of my models gets a R^2 of .89, but it still has correlated errors. Are you saying I should take the daily returns rather than the realized vol over n amount of periods? Do you personally use linear regression for time series data? men lie, women lie, numbers don't
 schmitty Total Posts: 62 Joined: Jun 2006
 Posted: 2018-12-10 00:23 I suspect that your main problem is that you have overlapping data. If each data point represents a single day, then you have 19 days overlap in your YZ estimator with the previous day's value. Throw out your current model. Start with a model on a non-overlapped subset (current data sampled every 20 days) of your data. It will still have autocorrelation in both your predictor and response, but nothing like the 0.97 figure at lag 1 in the acf of the residuals like you have now. OLS as you have it set up in your current model is inherently misspecified here. Try fitting a bivariate VAR (R package vars and several other packages) model to the data then report back here with the acf and pacf of the resisuals of that model.
 nikol Total Posts: 853 Joined: Jun 2005
 Posted: 2018-12-10 00:58 I would approach it this way:- take daily rates, r, over calibration period (from t= -N to 0) for each series, j.- calibrate ARIMA to each series, such that you get ARIMA(param_j)- check (ARIMA)residuals from each calibration: res (from t=-N to 0 for j)- check marginal distributions of these residuals and cova matrix- apply PIT (prob integral transformation) to each of res_j and check cova matrix again. PIT is empirical CDF^{-1}(x). - try to model (ARIMA)residuals with something like GARCH or else. The result of that will be (GARCH)residuals.When to stop this research?The moment, when you (ARIMA-GARCH-xyz)residuals becoming Normal you can trust you have a good model.How to use it?- Simulate Correlated Normal random core.- Apply PIT or GARCH to model ARIMA-residuals- Apply ARIMA to model daily process of returns- integrate your process to the horizon necessary (10 days, 1 months, 1 year etc etc). It is like "Russian doll" (Matreshka) - you open layers till the core of normal correlated sample, then you have to re-assemble reality back to the space of returns.
 day1pnl Total Posts: 54 Joined: Jun 2017
 Posted: 2018-12-10 06:35 You cant really use R^2 for anything.> Are you saying I should take the daily returns rather than the realized vol over n amount of periods? Do you personally use linear regression for time series data?No no, just that realized vol (SSD)^(1/2) is a non-linear transformation of the former, which in turn is described by “CAPM”. I personally dont much time series at all - but if you need to find the beta ratio in which its appropriate to compose your portfolio then regression is correct. Still not sure exactly what you want to achieve with your model except its “for trading”?On Garch-ARIMA: have yet to see a economic time series model that was not hysterically overfitted or could actually predict anything, and u open a different can of worms there in terms of T0 distribution for your data imo.. but im sure some people can actually make it workAnyways just my 2c, hope it helps
 nikol Total Posts: 853 Joined: Jun 2005
 Posted: 2018-12-10 22:21 time series model that was not hysterically overfitted or could actually predict anythingTotally agree. I descried exemplar framework, where the modeler can add/remove components at his/her taste or reasoning.At least what I can say for sure is that the distributions of returns shown on plots cannot be correlated directly in the model without proper transformation. If you do correlate them like shown in the plot, you get into trouble.
 rickyvic Total Posts: 191 Joined: Jul 2013
 Posted: 2019-04-01 18:11 Hey without getting into too much detail, your residual has a unit root, clearly.Meaning that the distribution explodes (non stationary) so no statistical model applies.Check for cointegration and you'll see.I don't think you can do much in linear space and for sure if you look at the level.Try to diff once and see what else you can do, but I wouldnt waste much time on it. "amicus Plato sed magis amica Veritas"
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