# Time series price prediction and linear regression: using high/low rather than last quotes price

Discrete time series regression models, like ARIMA, are usually built around the assumption that we only have 1 available price for each period t, which I will call the Close.

In reality asset time series (bid, ask) are continuos processes and we might have more data than just the Close for each interval. Let's say we have, for each period, an Open, High, Low and Close price. t1(Low) will therefore be the lowest asset price during t1.

Now let's also assume we want to build a mean reversion model. We have reasons to believe that some of our predictors have significant negative correlation to the next period price, although we can't be confident that mean reversion will materialize precisely at t1(Close). If we want to make R-squared more meaningful, does it make sense to use the next period Close as the DV? Shouldn't we use the Low/High? That is, if the asset price at interval t0 as travelled from an Open of 10 to a Close of 20, we will want to regress t0 IVs against t1(Low), not t1(Close).

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Please edit your question for clarity. You say you are predicting prices but then define Y as a return in T1. Also you say you are predicting price at "any time" -- not sure that what means. Maybe you are trying to build a model that achieves some target return over the course of a horizon -- not clear from your question. – Ram Ahluwalia Dec 18 '11 at 15:32
I am, if anything, even more confused as to what you are asking after your most recent edits. At least, I am confused as to what you are proposing. I feel your initial goal of "maximizing prediction value" may be misguided. A model that overly depends on the specific regression you run is not very robust. Also, if I understand you right, rounding your dependent variable, particularly if you are trying to predict prices or returns, is rarely a good idea. Better to use robust regression, if outliers are a problem. – Tal Fishman Dec 18 '11 at 18:59
Alright, let's make it simple and I will walk you through: in time series analysis, be it ARIMA or any other forecasting model, what is the generally used observed value Y in the regression? If the answer is the last available differenced price at time t1 for the continuous process, why is that? Why not to regress against the differenced high or low price for the period t0-t1? And why to close the question anyway, other people might provide more specific critique than general "confused" comments. – Robert Kubrick Dec 18 '11 at 21:29
@RobertKubrick Closing need not be permanent. Closing only says that, as it stands right now, at least 5 active users believe your question is not clear. If you edit your question to be clearer, then it will be re-opened. If you wish, you could make your last comment, "Why not regress against the differenced high or low price?" your entire question, and perhaps that is good enough. – Tal Fishman Dec 18 '11 at 22:42
@RobertKubrick following your latest edits, is your question now "shouldn't we use high/low?" Please make it clear that that is your ultimate question. As it stands, the question still seems a bit overly chatty, more like giving your opinion than asking a question. – Tal Fishman Dec 19 '11 at 13:51

High and low prices are frequently used in many contexts, such as estimating volatility. See, for example, the Garman-Klass and Yang-Zhang estimators. Brandt and Kinlay provide a nice summary of some of these estimators.

However, it sounds like you are more interested in using high/low information for evaluating whether mean reversion has taken place. In that case, I would counsel against using the range to measure whether the predicted mean reversion materialized. Having worked with real financial time series, I can tell you that the high/low quotes are much less reliable. The high or low of a given period is often only the active price for a very short period, and may not be tradeable, particularly in large size and/or without expensive high-frequency execution (for which you should really be using tick data, not OHLC data, anyhow).

Instead, I would recommend that you use the VWAP over the interval (calculated from actual ticks) as a slightly more meaningful alternative to the close alone. I've seen some research that approximates VWAP over short horizons for typical equities from OHLC data as

$$\frac{O+C+\frac{H+L}2}3$$

If OHLC data is all you have, this may be your best bet.

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One problem I have with VWAP data is that for some intervals (1 minute in my case) the asset hasn't traded at all. I could discard those observation from the study, but I actually see quotes mean reversion as a validation of the hypothesis. – Robert Kubrick Dec 19 '11 at 17:36
@RobertKubrick If there are no trades sometimes in 1-minute intervals for this asset, then you should really be using longer intervals. Quotes, btw, are very easy to manipulate. You will not know for sure about those quotes until you try to trade on them. – Tal Fishman Dec 19 '11 at 18:47
@TalFishman Could you supply a reference for the approximate VWAP formula you give in this answer? – babelproofreader Dec 20 '11 at 21:39
@babelproofreader It was proprietary research. I'd suggest doing your own research before using it. – Tal Fishman Dec 20 '11 at 22:40