Library
Augmented Dickey–Fuller (ADF) mean reversion testThe augmented Dickey-Fuller test (ADF) is a statistical test for the tendency of a price series sample to mean revert .
The current price of a mean-reverting series may tell us something about the next move (as opposed, for example, to a geometric Brownian motion). Thus, the ADF test allows us to spot market inefficiencies and potentially exploit this information in a trading strategy.
Mathematically, the mean reversion property means that the price change in the next time period is proportional to the difference between the average price and the current price. The purpose of the ADF test is to check if this proportionality constant is zero. Accordingly, the ADF test statistic is defined as the estimated proportionality constant divided by the corresponding standard error.
In this script, the ADF test is applied in a rolling window with a user-defined lookback length. The calculated values of the ADF test statistic are plotted as a time series. The more negative the test statistic, the stronger the rejection of the hypothesis that there is no mean reversion. If the calculated test statistic is less than the critical value calculated at a certain confidence level (90%, 95%, or 99%), then the hypothesis of a mean reversion is accepted (strictly speaking, the opposite hypothesis is rejected).
Input parameters:
Source - The source of the time series being tested.
Length - The number of points in the rolling lookback window. The larger sample length makes the ADF test results more reliable.
Maximum lag - The maximum lag included in the test, that defines the order of an autoregressive process being implied in the model. Generally, a non-zero lag allows taking into account the serial correlation of price changes. When dealing with price data, a good starting point is lag 0 or lag 1.
Confidence level - The probability level at which the critical value of the ADF test statistic is calculated. If the test statistic is below the critical value, it is concluded that the sample of the price series is mean-reverting. Confidence level is calculated based on MacKinnon (2010) .
Show Infobox - If True, the results calculated for the last price bar are displayed in a table on the left.
More formal background:
Formally, the ADF test is a test for a unit root in an autoregressive process. The model implemented in this script involves a non-zero constant and zero time trend. The zero lag corresponds to the simple case of the AR(1) process, while higher order autoregressive processes AR(p) can be approached by setting the maximum lag of p. The null hypothesis is that there is a unit root, with the alternative that there is no unit root. The presence of unit roots in an autoregressive time series is characteristic for a non-stationary process. Thus, if there is no unit root, the time series sample can be concluded to be stationary, i.e., manifesting the mean-reverting property.
A few more comments:
It should be noted that the ADF test tells us only about the properties of the price series now and in the past. It does not directly say whether the mean-reverting behavior will retain in the future.
The ADF test results don't directly reveal the direction of the next price move. It only tells wether or not a mean-reverting trading strategy can be potentially applicable at the given moment of time.
The ADF test is related to another statistical test, the Hurst exponent. The latter is available on PulseWire as implemented by balipour , QuantNomad and DonovanWall .
The ADF test statistics is a negative number. However, it can take positive values, which usually corresponds to trending markets (even though there is no statistical test for this case).
Rigorously, the hypothesis about the mean reversion is accepted at a given confidence level when the value of the test statistic is below the critical value. However, for practical trading applications, the values which are low enough - but still a bit higher than the critical one - can be still used in making decisions.
Examples:
The VIX volatility index is known to exhibit mean reversion properties (volatility spikes tend to fade out quickly). Accordingly, the statistics of the ADF test tend to stay below the critical value of 90% for long time periods.
The opposite case is presented by BTCUSD. During the same time range, the bitcoin price showed strong momentum - the moves away from the mean did not follow by the counter-move immediately, even vice versa. This is reflected by the ADF test statistic that consistently stayed above the critical value (and even above 0). Thus, using a mean reversion strategy would likely lead to losses.
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Pythagorean Moving Averages (and more)When you think of the question "take the mean of this dataset", you'd normally think of using the arithmetic mean because usually the norm is equal to 1; however, there are an infinite number of other types of means depending on the function norm (p).
Pythagoras' is credited for the main types of means: his harmonic mean, his geometric mean, and his arithmetic mean:
Harmonic Average (p = -1):
- Take the reciprocal of all the numbers in the dataset, add them all together, divide by the amount of numbers added together, then take the reciprocal of the final answer.
Geometric Average (p = 0):
- Multiply all the numbers in the dataset, then take the nth root where n is equal to the amount of number you multiplied together.
Arithmetic Mean (p = 1):
- Add all the numbers in the dataset, then divide by the amount of numbers you added by.
A couple other means included in this script were the quadratic mean (p = 2) and the cubic mean (p = 3).
Quadratic Mean (p = 2):
- Square every number in the dataset, then divide by the amount of numbers your added by, then take the square root.
Cubic Mean (p = 3):
- Cube every number in the dataset, then divide by the amount of numbers you added by, then take the cube root.
There are an infinite number of means for every scenario of p, but they begin to follow a pattern after p = 3.
Read more:
www.cs.uni.edu
en.wikipedia.org
en.wikipedia.org
Note : I added the functions for the quadratic mean and cubic mean, but since market charts don't have those types of graphs, the functions don't usually work. It's the same reason why sometimes you'll see the harmonic average not working.
Disclaimer : This is not financial or mathematical advice, please look for someone certified before making any decisions. Indicator
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Mean recursion envelopeFree for public consumption
There is very little original here, the idea is discussed in the underground traders alliance, (google em), and was apparently the basis of what was at one time myfxbooks most profitable strategy.
I can't find the original video that was floating around on youtube, but if i find it again, i'll link it here.
This is bascially just the TV default envelope code copied and modified.
The idea is to have an envelope based on a low length, exponential basis. Then to manually "tune" the percent input so that the envelopes engulf most bars. Whenever price goes outside the envelopes (especially at key levels), look for a change to enter a reversion back to the ema.
This manual tuning when switching between time-frames and symbols of the percentage input, becomes arduous.
Instead this script uses the TV envelope code, but gets a setting based on the average of true range and "autotunes" with this.
Anything that protrudes beyond that level, especially at key levels, is likely to revert back to the ema. Bear in mind, a run away trend will also push past the envelopes and continue running for several (3-5) bars so, use it mindfully and thoughtfully with all the usual cautions about risk management.
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Spread by//Every spread & central tendency measure in 1 script with comfortable visualization, including scrips's status line.
Spread measures:
- Standard deviation (for most cases);
- Average deviation (if there are extreme values);
- GstDev - Geometric Standard Deviation (exclusively for Geometric Mean);
- HstDev - Harmonic Deviation (exclusively for Harmonic Mean).
These modified functions will calculate everything right, they will take source, length, AND basis of your choice, unlike the ones from TW.
Central tendency measures:
- Mean (if everything's cool & equal);
- Median (values clustering towards low/high part of the rolling window);
- Trimean (3/more distinguishable clusters of data);
- Midhinhe (2 distinguishable clusters of data);
- Geometric Mean ( |low.. ... ... .. .... ... . . . . . . . . . . . .high| this kinda data); <- Exp law
- Harmonic Mean { |low. . . . . . . . . . . . . . .. . . .high| kinda data). <- Reciprocal law
Listen:
1) Don't hesitate using Standard Deviation with non-mean, like "Midhinge Standard Devition", despite what ol' stats gurus gonna say, it works when it's appropriate;
2) Don't check log space while using Geometric Mean & Geometric Standard Deviation, these 2 implement log stuff by design, I mean unless u wanna make it double xd
3) You can use this script, modify it how you want, ask me questions whatever, just make money using it;
4) Use Midrange & Midpoints in tandem when data follows ~addition law (like this . . . . . . . . . . . . . . . . . . . . .). <- just addition law
Look at the data, choose spread measure first, then choose central tendency measure, not vice versa.
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Ain't gonna place ® sign on standard deviations like one B guy did in 1980s lmao, but if your wanna use Harmonic Deviations in science/write about/cite it/whatever, pls give me a lil credit at least, I've never seen it anywhere and unfortunately had to develop it by myself. it's useful when your data develops by reciprocals law (opposite to exponential).
Peace TW Indicator
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