How Price and Volume Relate: Independence, Dependence, and Testing
Article Quant Q&A · Author: SiXUlm
Summary
The discussion interprets the claim that volume is independent of price as a distinction between the data series: price-based indicators transform prices, while volume is a separate measurement. Price can rise, fall, or remain broadly flat on high or low volume. That distinction does not establish statistical independence; the relationship between price changes and trading activity remains an empirical question and can be relevant to interpreting market sentiment.
One answer describes using mutual information to assess dependence, noting that zero mutual information corresponds to independence. It reports a nonzero estimate from a discretized sample of IBM daily closing prices and volume, but provides no broader validation or evidence that the result generalizes to all securities. Another answer discusses research suggesting that price changes per trade and trade counts over clock time can have different distributional behavior, potentially linking trading activity with the magnitude, though not necessarily the direction, of price movement. The discussion also mentions Granger causality as a separate test, with stationarity as a requirement. These methods address different questions and the examples do not establish universal causal relationships.
Key ideas
- Volume is a separate market measurement from price, but that does not mean the two are statistically independent.
- Price can rise, fall, or move sideways under different volume conditions.
- Mutual information can be used to assess statistical dependence; a sample IBM estimate is reported as nonzero.
- The IBM example does not establish the same relationship for every security.
- Granger causality is mentioned as a distinct analysis that requires stationary time series.
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Full text
# Why is volume a totally independent variable from price? # Why is volume a totally independent variable from price? In Martin Pring's book "Technical analysis explained", when talking about volume, he asserts that it is "a totally independent variable from price" Why is this? ## Answer by GoneAsync (score 6, accepted) https://quant.stackexchange.com/a/17846 Pring was (probably) simply referring to the fact that most indicators are function of price -- lots of different ways to twist and contort prices to define trends, reversal points, etc. Volume is another parameter entirely, as it doesn't depend on price; the market or share price can have an up day on average, high, or low volume, it can have a down day on average, high, or low volume, it can have a sideways day on average, high, or low volume. Whether price and volume are parameterized on a common factor, as you suggest, is certainly a valid question, and indeed what a lot of technical analysis is based on -- trying to divine crowd sentiment and intentions from the combination of price and volume. More informed, certainly more thorough, insight was given by Mandelbrot. He noted that price change per market time (i.e. per trade, although some use volume, but I'm not sure how thoroughly tested this is) tends to follow a normal distribution (Gaussian, although it should never be assumed so), while the number of trades per clock time tends to follow fat-tailed distributions, the result being very non-Gaussian distribution of prices as is normally assumed for simplicity. This does suggest some coupling of price movement (but not direction) with number of trades, and possibly volume. References: Mandelbrot, The (Mis)Behaviour of Markets Mandelbrot, Fractals and Scaling in Finance Kobeissi, Multifractal Financial Markets ## Answer by Barnaby (score 0) https://quant.stackexchange.com/a/17852 So lets test if volume and price are independent. We can estimate this by using mutual information. When x and y are independent, their mutual information is 0 so measures of mutual information entropy should be zero. Lets calculate this for the IBM case with 10 years of daily closing price and data volumen. library(infotheo) ``` a<-c(186.76,186.22,183.9,182.21,185.81,186.81,184.71,187.9,186.67,188.25,195.04,192.62,189.83,190.45,192.49,194.5,193.55,192.69,191.77,194.52,193.29,196.64,195.68,195.19,197.77,197.02,196.4,190.01,192.27,192.15,191.73,190.22,189.63,193.14,195.11,196.47,193.53,191.44,191.26,190.03,189.3,188.91,190.08,192.57,192.19,188.72,186.46,187.06,187,184.89,186.39,185.68,185.94,184.78,183.08,183.76,184.36,185.69,184.37,184.51,185.98,186.37,186.22,184.29,182.25,181.22,182.56,182.35,182.26,183.6,182.82,181.55,182.14,180.88,180.72,180.37,181.71,181.27,186.35,188.39,188.53,188.04,187.22,188.42,187.7,188,189.86,188.49,192.36,192.49,192.5,190.85,194.09,193.63,195.24,194.4,195.78,194.57,194,191.67,189.15,189.64,187.1,185.97,184.3,186.63,187.47,187.34,187.95,187.88,187.38,189.36,190.07,190.1,191.23,190.41,191.16,192.99,192.25,192,192.3,191.56,191.95,190.68,191.2,190.14,189.99,191.54,191.72,191.28,191.81,192.96,192.8,193.75,194,193.11,191.62,192.31,189.01,190.06,189.64,189.83,187.17,186.91,188.67,189.04,185.71,189.36,186.42,185.93,183.52,183.8,181.75,179.84,182.05,169.1,163.23,161.79,162.18,162.08,161.87,163.6,163.46,164.35,164.4,164.36,162.65,161.82,161.46,162.07,163.49,163.3,161.92,162.79,164.16,164.16,161.89,161.43,160.64,160.92,162.15,161.76,161.95,162.17,161.54,162.67,164.52,164.05,163.27,161.86,162.99,160.51,161.07,155.38,153.06,151.41,151.93,157.68,158.51,161.44,162.24,161.82,162.34,160.51,160.05,160.44,162.06,159.51,156.07,155.05,158.42,159.11,156.44,156.81,155.8,154.57,157.14,156.95,152.09,155.39,155.87,156.36,153.67,151.55,155.48,153.31,154.66,158.47,156.96,157.91,156.72,155.75,158.56,158.2,158.52,160.4,160.96,162.19,163.89,163.65,162.91,164.83,162.81,160.87,161.94,160.48,161.03,159.42,161.18,158.5,160.77,157.81,156.8,157.98,154.28,157.08,156.96,159.81,159.81,162.88,164.63,163,159.2,160.59,160.4,162.67,160.5,159.18,160.45,162.04,162.07,161.85,162.34,162.86,162.38,162.3,164.13,163.13,160.67,166.16,164.26,165.36,170.24,169.78,170.73,173.92,174.4,171.29,173.67,173.97,173.08,170.05,170.99,172.68,171.12,170.55,172.28,174.05,173.26) b<-c(4229458,4833043,4912569,5479729,4938122,3959058,4319687,5059611,12535240,5738487,10902080,6851710,6749697,5193721,8537265,5394135,4923642,4074541,6089856,6740548,5112029,5480087,8417865,4834989,5431064,5351281,8467967,11248880,5419311,3878576,4735761,4407111,3871007,5632139,4626270,4206079,3673954,3049853,2222041,3186237,3817452,2431943,2261646,3777411,2226577,5251499,4202156,3946049,3270670,3023761,2984907,2215309,2562907,4793716,3721398,2759903,4620386,3200540,2515006,2375817,2852248,3296692,2728403,4138348,4061486,4425086,2773423,3538718,2445357,3925158,3551005,10686760,3231689,3875358,2762820,3258470,4575380,4223820,6643078,5092954,2422363,2958707,3135115,3309559,3177816,2402540,4501073,4858779,7811181,8827453,8165628,4154446,4851325,3584165,3503667,3376430,3242107,3264124,3943779,4207339,5178022,2125035,3307945,3847041,2708590,2781486,2527208,1858635,1794941,1929422,2814815,2418687,2038470,2177642,2496475,1940334,1723557,3158767,2190185,1503582,2909422,2679529,1822765,2864944,2260202,2524233,2390391,2763994,2297955,2900953,2456411,2561528,3126159,2963346,8847306,3288454,3300715,3082583,4151355,2493862,2336317,2870283,3705432,2281749,3071521,2099525,2990954,2982451,2625359,5090174,3596738,3924621,6895843,5578469,4350238,23393800,20947830,11084830,7599154,6652126,4989112,7894973,4738734,3895869,5817961,4683808,4241264,4102438,4067487,3491874,4956409,3534400,3377542,3239343,4974605,4795042,5410056,3801179,4182186,4076193,6615896,4062338,3966022,2405512,4165523,3465647,6432090,3860288,3013455,2851383,3865355,4081495,3987625,8603486,6489279,6781567,5135799,7289467,8863708,4671176,4044120,1869977,1912278,3328597,2820875,4007568,5521275,4877949,6140885,4694891,4240432,4485123,4185540,4376654,4679570,4251472,5755866,8368458,11891690,6118450,4830128,7887907,5659328,4493190,8312100,6563442,4711986,5536804,3676106,5253628,3256440,2981444,4440579,3626629,3331904,3706909,2892929,4358252,3504269,3346955,2712218,4048109,4003088,7107766,4403866,5910972,3234006,3633335,3787426,4538724,5193874,4600260,5709280,4566291,6057170,3749638,3311792,5737794,3871119,9001842,5929727,4331090,5428222,4391017,3477801,4137986,4254808,3694519,4671478,3465581,3147566,2521583,2263480,2511035,3868071,2716418,3495848,3130813,4308880,9572126,9681557,4022236,8226707,3887859,3166588,5814978,4515143,5058348,3311743,4023794,3593415,3610295,2472627,3092479,2657123,2954387,2457170,2438669,2916087) c<-data.frame(a,b) d<-discretize( c, disc="equalfreq", nbins=NROW(c)^(1/3) ) mutinformation(d[,1],d[,2], method= "shrink") 0.09123699 ``` You can see that the two variables are not totally independent in the case of IBM and I guess this is the same case for all securities. You can calculate if there is causality of one time series causing the other and viceversa via granger causality, however this needs the data to be stationary.
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