# Correlation - what is such simple words

## What is a correlation and what does correlate - simple words about the complex

January 22, 2021.

Hello, dear blog readers KtonanovenKogo.ru. When some people hear the word "correlation", then often just fall into a stupor. It is clear: a terrible term from the world of higher mathematics and statistics.

Immediately see dull graphs, multi-storey formulas, when looking at which you want to score and cry. In fact, everything is much easier.

Having spent a few minutes to read this article, you will learn what a correlation is and how to use it in everyday life.

## Correlation definition - what is it

Simple words, correlation - This is interconnection two or several random parameters. When one value grows or decreases, the other also changes.

Explaining example: There is a correlation between the air temperature and the intake of ice cream. The hotter the weather, the more cold delicacy people buy people. And vice versa.

Such patterns are established by studying large volumes of statistical data. We collect information about ice cream consumption over several years and information about temperature fluctuations in the same period. And then comparing and looking for addiction.

Correlate - it means be interrelated with something. There is a positive and negative correlation.

With a positive than one parameter, the more and the other. For example, the larger than the waste of the farmer on fertilizers, the more abundant harvest. With reverse correlation, the growth of one value is accompanied by a decrease in the other. The higher the building, the worse it is opposed to earthquakes.

## Correlation is a relationship without guarantees

Consider an example of direct correlation: the higher the level of human well-being, the greater its life expectancy. Provided people feed on high-quality food and receive medical care in a timely manner. Unlike the poor.

However, it is impossible to say confidently that a certain oligarch will live longer than this beggar.

This is only a statistical probability that may not work for one particular case. This correlation differs from linear dependence, where the outcome is known with a 100 percent probability.

But if we take a sample from hundreds of thousands of rich and the same number of poor, compare their life expectancy, then the general trend will be correct.

## Correlation coefficient

This is a number that is indicated as "R". It is in the interval from -1 to 1. Reflects the strength and pole of the interconnection of values. Let's look at the example:

The value of the coefficient | What is the correlation? | What does it say about? |
---|---|---|

R = 1. | Strong positive correlation | People who eat blueberries have sharp eyesight. Eat blueberries! |

R is less than 0.5 | Weak positive correlation | Some people who like blueberries have sharp eyesight. But it is not exactly. In short, nothing is understandable yet. But it is better to eat blueberries just in case. |

R = 0. | No correlation | Blueberries and vision are not connected. |

R is less -0.5 | Weak negative correlation | There are cases of impairment of vision due to blueberries. Do not risk. |

R = -1. | Strong negative correlation | Almost everyone who has eaten blueberries, darken. Burst blueberries! |

The magnitude of the correlation coefficient is calculated by the formula:

If it suddenly darkened in the eyes and an irresistible desire to close an article (Humanitarian syndrome), that is, the option easier. Microsoft Exel Everything will perform with the help of the "Cornel" function. This is done like this:

Judging by the calculations, the human growth practically does not affect the level of salary.

## Real reasons for correlation and possible hypotheses

The dollar rate and the cost of oil correlate negatively. We can hypothesize: the rise in prices for ferrous gold causes the value of the US currency. But why is it going on? Where did the connection between these phenomena come from?

Determination of the cause of the correlation is a very difficult task. Thousands of various factors are intertwined, some of which is hidden.

Perhaps the fact is that the United States is the largest consumer of oil in the world. Every day they import about 7.2 million barrels. Reducing the price of ferrous gold is good for the American economy, because it allows you to spend less money. Consequently, the dollar is growing.

Correlation provides the ability Make output From statistical data.

For example, we found out that there is a negative relationship between personnel income and its efficiency. Our hypothesis: "Lazy and loafers get more than responsible employees." Then we will revise the motivation system and get rid of useless people.

The hypothesis is only a statistical output, assumption. It may well be erroneous.

According to statistics, the more firefighters participate in fire extinguishing, the more significant damage. What hypothesis can do from here? Firefighters bring harm, let's cut them! But if you figure it out, the real cause of damage is the fire. And an increase in the number of persons involved in its extinguishing is a consequence of a fire scale.

Our universe is infinite, which means you can always find several variables that will be correlated with each other, despite the complete absence of causal relationships. Even the most violent imagination will not be able to explain that it combines cheese and a killer blanket:

For more information on this topic, see the video:

## How, with the help of the correlation, people become richer

The main rule of any investor: Do not put all the eggs in one basket. Attachments are recommended to diversify (what is it?) - Distribute. Therefore, people buy shares not one company, but a dozen of different, forming investment portfolios. If some firm quotes fall, then the remaining nine will be able to play the fall or at least reduce damages.

But this is in theory, and in practice, all spoils the correlation. The problem is that the cost of shares of various companies within the industry or even the whole country can be strongly correlated. The problems of a huge corporation provoke a panic on the market, reduce the cost of other assets, at first glance, not related to each other. In 2008, there was a collapse of Lehman Brothers, which caused a chain reaction and collapse in world markets.

Therefore, when investing, you need to try to choose directions that not related to each other (R is striving for 0).

For example, a pair of "Gold - US Bonds" = -0.13. If you collect a briefcase from completely independent parts, the risks of financial losses will be reduced.

Territorial approximation of assets to each other enhances the correlation. So, it is necessary to consider options at different points of the world, as much as possible from each other.

In life, this principle is also valid. If your skills and knowledge allow the programmer, a taxi driver, a plumber and a journalist - you are well protected from the risk of unemployment.

## Memo

- Correlation is a ratio, interdependence of several variables.
- Communication is positive and negative.
- The correlation coefficient determines the degree of interdependence of one variable from the other.
- Based on the correlation, people push hypotheses (often erroneous).
- The true reason for the correlation is sometimes hidden under a variety of factors and external forces.
- There is a false correlation dependence.
- Staying eggs to baskets, remember that they should not be correlated with each other.

Good luck to you! Seeing fast meetings on the pages of KtonanovenKogo.ru

Many novice traders have heard about trade strategies based on the correlation of various assets that have certain relationships in their price dynamics, which, in fact, allows traders to make profitable transactions (for example, trade on the basis of correlation of currency pairs). The correlation of assets is traditionally associated with short-term trade methods, but the concept of correlation is very actively used and portfolio managers, which makes this phenomenon very significant for successful trading.

**Definition and Correlation Logic **

It can be said that the correlation is a statistical measure of interaction between two random variables. And if we talk about correlation in trading, then two trading assets. Those. If two of any asset show synchronous cost dynamics, then we can say that these assets show direct correlation or their correlation coefficient is approximately equal to one.

If the assets show the opposite dynamics of the price change, then we can conclude that they show the reverse correlation, which in this case will be approximately equal to minus one.

But, of course, not all assets show this kind of correlation, and even the assets with direct and reverse correlation can sometimes begin to live their own lives and show a completely different dynamics with a correlating asset. Those. The correlation value may be okolonul (when assets show an unrelated nature of price changes), with the correlation can periodically how to increase (when assets begin to show some similarity in the dynamics) and decrease, leve into a negative value (show the opposite speaker).

**Correlation in short-term trade **

One of the first scalper strategies were trading systems based on correlation. Thus, the Futures on the RTS index repeated the futures dynamics of E-MINI SNP, and this repetition was implemented with some time delay, which allowed scalperas to make transactions in the direction of the price impulse of the American index with a high probability of obtaining a profit. For this kind of advanced correlation on the Trader Slanga, there is a term "guide" - that is, an asset that its impulse dynamics predetermines the dynamics of the "slave" correlating tool. Indeed, scalpers, watching the direction of SNP dynamics during the opening of American trading, when leaving important statistics and, when testing important levels, the SNP made transactions of a similar foucher index on the RTS index and earned with such a straightforward way. Yes, sometimes our futures began to move in the opposite direction, but in this case the scalperst closed their transactions, observing the risk management.

Also in short trade used the correlation of Gazprom and Sberbank shares to make transactions with the Futures on the RTS index. The fact is that the main "weight" in the RTS index is just have such heavyweights as Gazprom and Sberbank. And if both of these asset in the intraday frames began to show the synchronicity of the speakers, the scalpersmen made transactions of a similar direction, realizing that the RTS index and futures on the RTS index would most likely repeat this movement of such an integrated "guide". Moreover, if the trader is in a profitable transaction and begins to observe that synchronously moving assets begin to show already different dynamics, then there is a complete or partial fixation of the transaction profits.

However, it is worth understanding that market conditions are subject to change, and previously acting correlations may violate. But new ones can also form - it all depends on the focus of market attention.

**Correlation when building securities portfolio **

In various analytical reviews, we often can meet the phrase like "with such an index values of the index, you should recruit paper into the portfolio, and with such something - fix profits." If you look at this phrase from the point of view of correlation, you can make the following conclusions. At the Moscow Stock Exchange, about three hundred shares is traded, of which the order of fifty are indoors, and the main heavyweights in the structure of the index are not so much. Nevertheless, the index is a barometer of the Russian stock market, so we can say that if the index shows a strong increase in dynamics, it is most likely to differ to a different degree to a sufficiently large number of securities with a different value of the correlation to the index.

On the contrary, if the index decreases, the wide front of the shares will be much heavier to grow against the dynamics of the index. Thus, it can be said that the main portfolio acquisitions must be made from meaningful support in the index, when the index itself begins to repel up from the support achieved. Also, if we consider that the likelihood of its decline increases from index resistances, in such a situation, part of the profits on previously open positions can be fixed.

Naturally, various papers will have a different correlation with an index. A number of papers will show a very significant correlation approximate to one, a number of papers can show even the reverse correlation, but most of the papers will show the separation correlation. But in the case of a powerful growth of the index, most likely, part of the growth will be able to host on such papers.

It is also worth noting that within the paper industry can show a slightly increased correlation, but not always. So, if, for example, oil grows in price, then we can assume the growth of the oil and gas sector. If we see the rapid growth of the heavyweight industry, we can assume a certain growth and a certain number of other branch securities. This will be an example of direct correlation.

The opposite situations are found when, with the growth of one branch of the paper, any other industry show a decrease. So, if, for example, the US dollar growth can grow paper exporters, which at the same time will be able to get a high ruble revenue when converting currency, then paper of domestic demand, on the contrary, can show the downward dynamics, since the purchasing power of the population with the increase in the cost of the currency will decrease.

**Output **

Understanding market correlations is necessary for successful trading, both in terms of the implementation of short-term trade and to build a securities portfolio. Understanding the correlation on the stock exchange allows you to look at the market not as fraginal securities, but as an integrated structure showing holistic development. Namely, a comprehensive understanding of market processes already displays a trader to a new level of understanding of trade.

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Correlation is a similarity or relationship between two things, people or ideas. Means similarity or equivalence that exist between two hypotheses, situations or things.

In the field of statistics and mathematics, the correlation refers to the measure between variables (two or more) associated with each other.

The word correlation is a nouns of the female genus, it happened from the Latin CorrelAtiōne ("CUM" (at the same time) + "Relatio" (ratio)) is read as "correlation" and means "ratio" or "relationship".

The word "correlation" can be replaced by synonyms, such as: communication, dependence, relationship, relationship, interdependence and interconnectivity.

## Correlation analysis

The purpose of the correlation coefficient is to determine the intensity of the relation that exists between well-known data sets or other known information.

The value of the correlation coefficient may vary from -1 to 1, and the result determines whether the correlation is negative or positive.

To interpret the coefficient, it is necessary to know that 1 means that the correlation between variables is full positive, and -1 means that it is complete negative. If the coefficient is 0, then the variables do not depend on each other.

## Pearson correlation coefficient (PEARSON)

In statistics, the Pearson correlation coefficient (R-perester), which is also called the Pearson Product Correlation Coefficient (or PPMCC, or PCC), measures the relationship between two variables in the same metric scale.

### Calculation of the Pearson correlation coefficient

#### Method 1) Calculation of the Pearson correlation coefficient using covariance and standard deviation

Where:

This is covaria

This is the standard deviation of the variable x,

This is the standard deviation of the variable Y.

In this case, the calculation includes first searching for covariance between variables and the standard deviation of each of them.

Then you need to divide the covariance to multiply from two standard deviations - do a fraction and put covariance from above, and multiplication of two standard deviations from below.

Often, these tasks already have either standard variables deviations, or covariance between them, it remains only to apply the formula.

#### Method 2) calculation of the Pearson correlation coefficient with source data (without covariance or standard deviation)

With this method, the easiest formula looks like this:

For example, if we assume that we have data with n = 6 by observations of two variables: the level of glucose (y) and age (X). For example, this is the statistics of six people, from which we know their age and level of glucose. In the following table, you will see this data: at the first person to which 43 years old, the level of glucose 99, in the second, which is 21 years old, the level of glucose 65, in the third, which is 25 years old, glucose 79 and so on. The calculation should be performed by the following steps.

Step 1: Fill out a table as follows: Make existing data I, X, Y and add empty columns for XY, X², Y².

Step 2: Multiply X and Y to fill the "XY" column. For example, in the first line it will be x1y1 = 43 × 99 = 4257.

Step 3: Take the value of the column X and build it in a square, write the result in the X² column. For example, in the first row in our table will be x12 = 43 × 43 = 1849.

Step 4: Do the same as in step 3, but now use the Y column and write down your calculations in the Y² column. For example, in the first row in our table will be Y12 = 99 × 99 = 9801.

Step 5: Make the amount of each of the columns and put the result below, to each column. For example, the sum of the column age x is 43 + 21 + 25 + 42 + 57 + 59 = 247.

Step 6: Use the correlation coefficient formula.

The range of correlation coefficient from -1 to 1. Our result is 0.5298 or 52.98%. This means that variables have a moderate positive correlation.

Those. The age and glucose level depend on each other (since the coefficient of 0.5298 is far from 0), but not very strong (since the coefficient is still very far from 1). And positive, since the coefficient is greater than 0, this means that glucose and age rise together, and not vice versa (i.e., the higher the age, the higher the level of glucose).

## Spearman correlation coefficient

In statistics there is also a correlation coefficient of a spirit, which is named after the statistics of Charles Edward Spearman (Spearman).

The purpose of this coefficient is to measure the intensity of the ratio between the two variables, regardless of whether they are linear or not.

The correlation of the spirit is used to estimate whether the intensity of the relationship between the two analyzed variables is measured by a monotonous function (a mathematical function that retains or inverts the ratio of the initial sequence).

### How to count the correlation coefficient of spirit

The calculation of the correlation coefficient of the spirit is already slightly different from the previous one. To do this, you need to organize available data to the following table.

1. You must have two pairs of data corresponding to each other. You must make them in this table. For example, the Restaurant's Directorate wants to know if there is a connection between the number of orders of water bottles and the number of dessert orders. The director took at random data of 4 tables. Thus, it turned out two pairs of data: where "Data A" is orders of desserts, and "Data B" - water orders (i.e., the first table ordered 7 desserts and 8 bottles of water, the second - 6 desserts and 3 bottles With water, etc.):

2. In the "Ranking A" column, we will classify observations that are in "Data A", increasing: "1" is the lowest value in the column and N (total observation) - the highest value in the "Data A" column . In our example, this is:

3. Make the same positioning (observation classification) for the second column "Data B", writing it in the "Ranking B" column.

4. In the column "D", consider the difference between the two last columns-ranking (A - B). You do not need to consider here (in the next step you will find out why).

5. ENOUNG TO THE WORLD ENERGY Each of the values obtained in the "D" column.

6. Make the amount of all the data that you have turned in the "D2" column. It will be ΣD². In our example σd² = 0 + 1 + 0 + 1 = 2.

7. Now we use the formula of the Spearman.

In our case, n = 4, we see this by the number of pairs of data (corresponds to the number of observations).

8. Finally, replace the data in the formula.

Our result is 0.8 or 80%. This means that variables have a positive correlation.

That is, orders of water bottles and orders of desserts by customers of this restaurant depend on each other (since the coefficient is 0.8 distale from 0), but not completely (since the coefficient is very close to 1, but not equal to 1) . And positive, since the coefficient is greater than 0, this means that the amount of water and the number of desserts increase together, and not vice versa (i.e., the higher the amount of water consumed, the higher the number of desserts consumed).

## Linear regression

This formula used to estimate the possible value of the variable (y) when the values of other variables are known (X).

The value "X" is an independent variable or predictor, and the "Y" dependent variable (also variable response) or the answer to the specified question.

Linear regression is used to verify how the "Y" value can vary depending on the variable "x". Direct, containing values of checking this variation, is called linear regression line.

If the relationship is between the dependent variable ("y") and an independent variable (x "), the regression will be called a simple linear regression.

### Simple linear regression

Yi = β0 + β1xi + εi

Where:

β0 - shift (length of the segment cut off on the coordinate axis direct y)

β1 - Tilt straight y,

εi-random error of the variable Y in the I-M observation.

See also logarithm and standard deviation values.

However, we are not alone. Almost any stock market in the world is closely associated with the American stock market and reacts primarily on what is happening there. And here in addition to the fundamental reasons for the interaction of markets **Capital **It also has a strong impact of automatic trading tools. This manifests itself especially clearly at the micro level (ticks). Each tick motion of the S & P500 index is immediately responds to the corresponding change in the indexes of FTSE, DAX, MICEX, BOVESPA. Such correlation exists everywhere and is the basis for making traders decisions.

Next, there are several charts that shown how the S & P500 index interacts the index **RTS **and oil prices. These graphs show the change of S & P500, index *RTS *and oil prices as a percentage of the reference point specified on the schedule.

The figure highlighted the situation in March, when the RTS index went beyond black gold, and not behind the S & P500 index. It was a period of exacerbation of the situation in North Africa and in the Middle East. Increasing oil prices have negatively affected the US stock market, but at the same time led to the rally on the Russian stock market. Pay attention to another fact: the reversal on the Russian stock market almost always takes place a little earlier than the prices of oil do. The following graph shows the same correlations since the speech **Ben Bernenice **In Jacksonholl, where he announced the upcoming QE2 program.

As we see, almost to the new year S & P500, the RTS index and oil moved almost synchronously. In January - February, seasonal correction in black gold happened, but the Russian market continued to grow with America, mastering *money *which usually allocate investment funds at the beginning of the year. The following chart shows the same correlations since the peak of the American stock market in 2007. An impressive parabolic rally in black gold was still uneasyl by the Russian stock market.

This graph draws the stability of the spread between oil prices and the RTS index. The following chart shows us the correlation from January 2004. Investing in the US stock market for this period did not bring any profit.

Finally, the most impressive schedule from this series: from the beginning of 2000.

As we see, while Oil and the RTS index issued a very strong increase in this period, 450% and 1500%, respectively, the American stock market during this time almost did not leave the negative zone. Undoubtedly, there are other factors that influence the Russian stock market. For example, the ruble exchange rate. Strengthening the ruble rate leads to the influx of money to the Russian market. Increase rate **Additional capital investment **It leads to an increase in the ruble and accordingly contribute to the growth of the Russian market (usually it is played in advance by insiders).

When the dollar is getting cheaper relative to the ruble, then, if we assume that the prices of assets in rubles remain unchanged, therefore they should be expening about the dollar and other currencies. Perhaps the dependence of the Russian market from oil prices expresses the relationship between the market with a change in the national currency rate with some correlastic coefficient. Therefore, although there is also a certain correlation here, to identify the interaction of the RTS index with a ruble course or some other currency makes no sense.

Correlation (CORRELATION) is

In short: you can draw the following conclusions: the interaction of the Russian stock market with the S & P500 index reflects the Global Market Sentiment in relation to the stock markets in general; Interaction with oil prices reflect both the traditional predominance in the Russian indices of the oil and gas sector, and most of the relationship with changes in currency exchange rates.

There are other correlations that need to be considered when investing in the Russian stock market: for example, the interaction of the Russian market with an influx / outflow of foreign *Capital *.

### Correlation of securities

Between profitability **valuable papers **A functional dependence may be observed. This means that there is a strict rule that binds the values of their return. The simplest is a linear dependence.

In the financial market relationship between profitability *valuable papers *It is often not functional, i.e. not tough. In this case, one paper profitability value may correspond to different values of other paper yields. Thus, there is no strict law that would associate the values of their return. The dependence of this kind is called stochastic or probabilistic, or statistical. This means that when changing the yield of one paper, you can only talk about what other types of yield can take another paper and with what probability. This state of affairs is explained by the existence of a large number of factors affecting the profitability of specific assets and the fact that all of them is difficult to consider.

When forming a portfolio, the degree of relationship between the yields of two securities can be determined using such indicators as covariance and correlation coefficient.

Covariance talks about the degree of dependence of two random variables. It can take positive, negative values and equal to zero. If the covariance is positive, this suggests that when changing the value of one variable, the other has a tendency to change in the same direction. So, with a positive covariance of the returns of two papers with increasing first paper yields **yield **The second will also grow. When falling first paper yields *yield *The second will also decrease.

With a negative covariance, variables tend to change in opposite directions. In this case, the growth of the yield of first paper will be accompanied by a drop in the yield of second paper, and vice versa. The greater the value of covariance, the stronger the relationship between variables. If the covariance is zero, there is no relationship between the variables.

The correlation coefficient characterizes the degree of tightness of the linear dependence of two variables and is a dimensionless value. The trend towards the linear dependence of two variables can have a more or less pronounced character. Therefore, the values of the coefficient vary in the range from -1 to +1. If the coefficient is +1, there is a positive functional dependence between the profitability of two papers. If the correlation coefficient is positive, but less than +1, there is also a dependence between the profitability of two papers, but less strict.

If the correlation coefficient is -1, there is a negative functional dependence between paper profitability. When the correlation factor is equal to zero, there is no relationship between the variables.

### Correlation of investments

Many Lie **Investors **- Participants in our forum adjust their set of tools using *Diversification *and correlation. I think not many. If the concept of diversification is familiar to most at least at the proverb level: "Do not keep all the eggs in one basket." The notion of the correlation of assets, for example, I found quite recently.

Compilation of the diversification of the investment portfolio from assets with uncorrelated results reduces the risk, because at the time the profit on one asset falls to another, it probably grows. When trying to build a diversified investment portfolio from assets with a pronounced negative correlation, we can get an unexpected and very useful effect for us. The total yield of the investment portfolio may be higher than the profitability of individual assets, and, accordingly, the risk may be lower than the risk of other assets.

What does the US Stock Exchange data on the correlation dependence between different groups of assets for 1926-2009 are talking about 1926-2009: mutual correlation between shares of small enterprises and shares of large enterprises - (+0.79). This is a rather high correlation. Although not 1. Still, large stocks and small shares behave somewhat differently. Between the shares and bonds, the correlation is already close to zero.

Correlations between shares and short-term bonds and treasury bills are also close to zero and even somewhat negative.

Bonds with each other correlated high enough. Long-term short-term bonds have a correlation of 0.8 - 0.9.

Long-term bonds with treasury **bills **On the contrary - a sharp decrease in correlation.

Separately, USA, Canada, Japan and United Kingdom, separate Europe, Asian Region and Pacific: The correlation between closely lying regions is quite high. Between Asia and the Pacific, the correlation is about 0.92. There is also a fairly high correlation between Canada and the United States. But the farther from each other the regions are, the lower between them the correlation. Even in Japan with England or Japan with Canada and the US, the correlation is less than 0.5. Important! If you wish to reduce the risk of an investment portfolio, we can include shares from different parts of the world.

Correlation between the MICEX index, two PIPs of the Troika dialogue, gold, silver, dollar, euro and Moscow real estate: correlation between the shares index and the stock fund, of course high. Correlation between shares and bonds somewhere at the level of 0.5. Between securities and gold, the correlation is close to zero (even a little negative). The correlation between gold and silver is high. Therefore, try to include in your investment portfolio and gold and silver does not make sense.

The correlation between the dollar and the euro and between the shares and bonds is again zero or even negative. The correlation between housing and the MICEX index in the Russian Federation is even negative (at minus 0.17-0.18). What, by the way, is rather not typical of world standards.

Conclusions: Without the correct diversification of assets, taking into account their mutual correlation, it is impossible to form an effective investment portfolio that will allow you to multiply your capital or, in any case, save it.

### Correlation of the dollar and oil prices and inverse proportionality

Fundamental factors are the basis of trade in the foreign exchange market, they allow you to establish a relationship **exchange rates **With those or other events. This article will talk about the correlation of such an indicator as the price of oil with a course of the United States dollar. The America's economy is one of the most energy-dependent economies in the world. The United States of America consumes just a huge amount of petroleum products, so the increase in the price of crude oil simply cannot but affect the course of the national currency.

The reason for this connection lies quite deeply, but the changes occur literally immediately, as the market is inclined to respond to the fundamental changes on the basis of psychological factors. When considering the influence of oil prices on the dollar rate, there is a rather unambiguous situation because the United States is one of the largest oil producers, at the same time actually act as the largest acquisitioner of this type of raw materials.

According to statistical data **Economy of America **There are not enough of its own reserves of petroleum products to ensure the needs of all production, while part of the mined black gold within the country goes to *export *. For this reason, America is forced to purchase about 9 billion annually. **barrels **Black gold, which is significantly displayed at an increase in the value of American goods both within the country and in foreign markets.

And the increase in the cost of goods, as it is known to always lead to negative consequences for the national currency. In addition, the negative impact on the exchange rate of the American dollar also has the fact that for the purchase of black gold companies have to buy other foreign currencies, as exporters do not always agree to settlements in the United States dollars. For example, a number of Arab countries have not yet been fully transferred in the calculations for oil on the euro. As a result of these two factors, we see the following picture, the price of oil increases, as a result increases **sentence **The United States of America's United States in the Forex market, as a result of its course goes down.

At the same time, when the price of oil falls, there is an inverse situation, the United States of America's dollar is actively growing in relation to such currencies as euro, the Canadian dollar and some other currencies. This dependency can be quite successfully used in the game on the Forex currency exchange, for trading the most optimal choice will be a pair of the US dollar, the US-Canadian dollar, since it is on this instrument that the greatest volatility will be observed. If possible, you can use such a currency pair as a USD / RUR, it will react similarly to the previous tool.

Purchase warbeds are open in the case of the rise in price of black gold, selling orders - in case of falling prices for ferrous gold. Also, the inverse proportionality is also tracked, while strengthening the American dollar, the price of petroleum products and crude oil is noticeably falling, this property can be used when trading on raw materials.

### Correlation of the ruble course and oil prices

О *War *In Syria they say all who trades black gold. Black gold brand Brent was in the range of 100-110 dollars per *barrel *. But on the likelihood of overthrow the Americans another government **Futures on oil **Quickly rose to 117 dollars. Then it was logical **correction **, and now **Brent. **About 115 dollars traded.

How did the ruble behave? Very often, Analystov can be heard: " **ruble **Rose on the backdrop of oil growth, "or" the growth of the dollar is associated with the fall in oil prices. " Is there a correlation of the dollar to ruble and oil prices? Is this a correlation now? This year? The dollar rate to the ruble correlated with oil prices until July, and in July *Brent. *went up, and *ruble *- not. Why did it happen?

There are several reasons here. First, the budget that depends on the prices of oil and the dollar rate to the ruble. This budget is the higher dollar rate and oil prices, the better. Secondly, not only oil workers want to see a weaker ruble. The reporting of many exporters are "asking" a more profitable course for them. Thirdly, the outflow of capital has not gone anywhere. The outflow goes and goes big. Fourth, in the dollar exchange rate, the Bank of Russia's banks for the Ministry of Finance were still laid. Fifth, the dollar is now growing in relation to all the "weak" currency of the ruble type (Brazilian Rial or Indian Rupee).

Addiction **budget **The Russian Federation from raw material exports has already become a parable in languages. The federal budget is 45 percent is filled with earnings from the sale of black gold and petroleum products. Approximately half of the Black Gold mined in the Russian Federation (246 million tons) go abroad, and the second half is recycled on Russian refineries. Calculations for black gold with importers are carried out in dollars. As a result, currency revenues from the sale of black gold and determine the ruble exchange rate on the dollar. The greater the price of black gold, the more dollar income, the greater the international foreign exchange market, the Forex comes with dollars, the stronger ruble. And vice versa.

Sergey Guriev, the most successful and figurative definition of the cost of the ruble gave the Rubble, the rector of the Russian Economic School: "The Russian ruble is a paper version of black gold. What oil, such a ruble. We decided to check with which accuracy the quotes of the ruble and dollar coincide with each other. The graphs of the Barrel - Ruble ratio for a biennium, which includes the peak of oil prices $ 146 per barrel, which came in 2008, and the decline in prices up to $ 40 per barrel in the winter 2008-2009, is shown in the chart.

The degree of compliance of the cost of the ruble to the price of black gold can be characterized by the correlation coefficient establishing the statistical relationship of these quantities. The correlation coefficient (usually use the Pearson coefficient) can take values from minus units to one. For independent processes (values), the correlation coefficient takes a value close to zero. And, on the contrary, for functionally dependent processes, this coefficient is approaching a unit or minus unit, depending on the heated or oncoming nature of the movement of the values under study.

In our case, the correlation coefficient, calculated on the period of one year (from February 1, 2009 until February 1, 2010) is the value equal to the module 0.935. This is a very high degree of compliance of the cost of the ruble and oil prices: from the point of view of mathematical statistics, the functional connection exists. We will construct the simplest mathematical model of the ruble of the ruble on the dollar, involving a linear dependence of one from the other. The green line on the chart displays the modeling behavior of the ruble.

No need to know the word "correlation" to evaluate such a visual result. Missing with the model during the maximum oil prices when the ruble was strengthened and became a brake for domestic exporters, due to ruble interventions **Central Bank **In the foreign exchange forex market to deter strengthen the ruble rate. And vice versa - intense dollar **Interventions **With the weakening of the ruble during the period of failure prices for oil.

The model allows you to evaluate the future ruble exchange rate, for example, for the price of Barrel 90 dollars, the ruble rate can rise to 27 rubles / dollars, and at the price of Barrel 50 dollars can drop to 35 rubles / dollars. It should be recognized, a specific model does not take into account many factors, including, as already shown, the intervention *central bank *But, nevertheless illustrates the general principle.

The question arises how long will the tough bond "barrel-ruble"? Answer: Until the structure of Russian exportation or settlement currency on oil contracts should be changed.

### Correlation of prices for oil and GDP of Russia

In Ov *work *Manager I constantly use different performance indicators (KPI). I was interested in a kind of KPI macroeconomic level. Earlier, I told about how corruption is the level of corruption in the Russian Federation and the countries of the world according to the estimates of the Center for Anti-Corruption Research and Initiatives Transparency International. Then I considered the dynamics of another macroeconomic indicator - the rating of economic freedom formed by the American Research Center "The Heritage Foundation" and the newspaper The Wall Street Journal. And finally, presented tax burden in the countries of the world (Tax Misery) published by Forbes magazine.

Recently, due to the fall in oil prices, they spoke about possible problems with execution **Budget Countries **. And I was interested in the question, how closely the prices of oil with the macroeconomic indicators of the domestic economy are correlated !?

There are many different types of oil prices, and the data that I refer to, not the most common ... But, how they are represented as fully and convenient, allows you to analyze them from different sides. Despite the fact that the correlation between the various types of oil prices, in my opinion, is complete. Often, problems in the economy of the country are associated with the name of Yeltsin, and the successes - Putin. At first glance, the dependence is unambiguous, but, as the subsequent analysis will show, superficial.

The correlation of oil prices and the size of the Russian GDP of the Russian Federation simply struck me. Calculating the correlation coefficient, I realized what the expression "on the oil needle" means. If 97% of the speakers **GDP **The Russian Federation is associated with the price of oil, what remains for other factors!? Are they playing, at least some role!?

Do not think that such a high correlation is characteristic of all macroeconomic indicators. So the dollar rate shows only a 50% correlation with the cost of black gold. That is, only half of the changes in the dollar's course can be explained by the global oil market environment.

*GDP *The United States also demonstrates a very moderate correlation with oil prices. Although in the US, the relationship is also very close.

### Correlation in psychology

The concept of illusory correlation. Illusory Correlation (Illusory Correlation) is a psychological phenomenon that is observed with almost all people, just as almost all people are subject to the illusions of Muller Lyer and other optical illusions.

Perhaps the phenomenon of the illusory correlation will be easier to understand if you call it the words "illusion of communication", and the essence of the illusory correlation is that a person for one reason or another sees the relationship between the parameters, properties, the phenomena, which is not really not. Usually, the illusory correlation is observed in the pair "Property - a sign of the presence of this property." For example, if a person believes that the color of the hair can talk about the degree of mental human development, and the hair rigidity is about the rigidity of character, then it is just about the illusory correlation. In fact, it is clear that there is no connection between the color of the hair and the intelligence or between the rigidity of the hair and there is no character.

The experimental phenomenon of illusory correlation was first investigated by Lauren Chepman (by the way, it is a single-fampot of our famous, although the failed agent illegal Anna Chapman) back in 1967. And it was this researcher that the term "illusory correlation" itself introduced. The study was conducted so. The tests for a certain time were presented (projected on the screen) of a couple of words, for example, "bacon - eggs". The couples were compiled as follows: one of the following four words was the left word: bacon, lion, buds, boat, and right - one of the following three words: eggs, tiger, notebook.

Thus, 12 pairs of words were presented to the test: "Bacon - Eggs", "Bacon - Tiger", "Bacon - Notebook", etc. Moreover, these couples were imposed many times and alternated in random order, but each pair was presented an equal number of times.

Then the subjects asked to estimate the frequency of each pair of words. And this is the key point of the experiment. Despite the fact that objectively the frequency of presentation of each pair of words was the same, higher subjects declared the frequency of the presentation of words of words having, by expressing the author of the Experiment of the "Strong Verbal Association". These are the following pairs of words: "Bacon - Eggs" ( **association **on adjacentness) and "Lev - Tiger" ( *association *in similarity).

Thus, the tests had illusory ideas that the word "bacon" is more closely associated with the word "eggs", and the word "lion" with the word "tiger" than other words with each other. Let me remind you that in fact each of the 12 pairs of words have been imposed equal to the number of times.

So, with the illusory correlation, a person, as they say, confuses God's gift with the scrambledness: sees the connection where it really is not.

Illusory correlation and projective tests. Investigated Lauren Chepman (along with his wife Jin Chepman) and the role of illusory correlations in determining the nature of a person with the help of so-called projective tests. Such projective tests were studied as a "drawing of a person" and "Test Rorshah".

At the same time, Chepman's spouses were interested in why psychologists continue to use projective tests, although their insolvency (bankruptcy) was repeatedly shown in scientific research (bankruptcy) as a psychodiagnostic tool, i.e. The lack of communication between the proposed developers of these tests by keys and interpretations with the psychological characteristics of the test individuals. Chepmans suggested that such persistence in the use of non-valid tests is due to the phenomenon of illusory correlation, which is subject to psychologists (like all people).

Before proceeding to the description of the actually experiments, it is necessary to say a few words about projective tests.

Projective tests are based on the assumption that when interpreting non-delayed visual incentives (blots) or when performing an indefinite task (to draw a person), the subject allegedly will definitely refer their character traits. For example, the developer of the test "Drawing of man" Karen Makhovener argued that the paralyonal (suspicious) entity when drawing a person's painting a special emphasis would give his eyes concerned about his masculine - draws a muscular person, concerned about his own intelligence - draws a big head, etc. In the keys to the test of Rorshach It is approved, for example, that if a person has homosexual inclinations, then in the blots he will see: buttocks, rear pass, genitals, women's clothing, people of indefinite sex, people with signs of both sexes.

I think the reader easily noticed that the links described above between the signs and the features of the character are purely associative and are based on domestic, everyday, trivial ideas. Indeed, why would a person with doubts about his masculinity and not draw muscular people, and homosexuals - not to see the rear aisles in the blots? But in fact there is no connection here.

And Chepmans experimentally showed that such illusory correlations in the interpretation of mentioned-projective tests are subject to both professional psychologists, and people who do not have any attitude towards psychology.

The experiment scheme was somewhat similar to the experimental scheme to identify illusory correlations, which we looked higher. The subjects were offered drawings of a person, made as patients of a psychiatric clinic and healthy people, and the corresponding psychological characteristics. For example, a characteristic "concerned about the level of its intelligent" was attached to the figure of a person with a large head. At the same time, pay attention (!), Some and the same psychological characteristics were attached to different drawings. For example, the characteristic "refers to people with distrust and suspicion" was attached to both drawings with a pronounced accent in their eyes and to drawings that do not have any features of the eye image. Moreover, such combinations were, as in the experiment already considered, the same amount.

The subjects were asked to establish the connection between the features of the drawings and the psychological characteristics of the authors of these drawings. And as a reader must have already guessed, the subjects demonstrated the illusory correlation: for example, they argued that such a character trait as suspicionally combined with a pronounced accent in the eyes. Moreover: the same picture was observed in the next series of experimeth, in which these two characteristics (pronounced eyes and suspicion) did not meet together at all!

A similar way was carried out with experiment with spots of Rorshah. Interpretations are attached to the spots, formulated by persons who passed psychodiagnostics, and the psychological characteristics of these people. For example, the interpretation of the "rear pass" equal to the number of times coincided with each of the following four psychological characteristics: it exists a sexual attraction to other men; He believes that those surrounding consistent around him; He is experiencing sadness and depression for a long time; He is experiencing a strong sense of own inferiority.

As in the previous experiment, the subjects again demonstrated the phenomenon of the illusory correlation, the lines of the "rear pass" with a psychological characteristic "he shows sexual attraction to other men."

Illusory correlation in our lives. Of course, illusory correlations distort our perception with you not only in laboratories. For example, it is the phenomenon of illusory correlation that largely determines the formation of stereotypes with respect to those or other peoples or social sections.

Many Lzhenauki (especially Lzhenayuki about the soul), in particular, physiognomic, socionics, grapheus, typology of criminals Cesare Lombroso, Francology, Frams of B. Chigyr about the fact that the person's name determines its character, as well as obviously occult teachings, Such as chiromantia. Many aspects of psychological occultism are also rooted in illusory correlations. On illusory correlations, many representations of modern psychoanalysis and other types of psychotherapy are based on illusory correlations (for example, when the cough is declared by the manifestation of a secret desire to say the nastiness, and the pain in the back - the manifestation of a severe psychological oshche, which man dated).

### Correlation in everyday life

Strengthening interest in psychological science to the potential of correlation analysis is due to the whole nearby. First, it becomes allowed to study a wide range of variables, the experimental check of which is difficult or impossible. After all, for ethical considerations, for example, it is impossible to conduct experimental studies of suicides, drug addiction, destructive parental influences, the influence of authoritarian sects. Secondly, it is possible to obtain a short time of valuable data on large quantities of the studied persons. Thirdly, it is known that many phenomena change their specificity during strict laboratory experiments. A correlation analysis provides a researcher with the ability to operate with information obtained in conditions as close as possible to real. Fourth, the implementation of the statistical study of the dynamics of one or another dependence often creates the prerequisites for reliable prediction of psychological processes and phenomena.

However, it should be borne in mind that the use of the correlation method is connected with very significant principled limitations.

So, it is known that the variables can be correlated and in the absence of a causal relationship between themselves.

It is sometimes possible due to the action of random causes, with the inhomogeneity of the sample, due to the inadequacy of the research tool for the tasks. Such a false correlation is able to become, say, "proof" that women disciplined men, teenagers from incomplete families are more inclined to offenses, the extroverts are aggressively introverts, etc.

It is necessary to remember: the presence of correlation is not an indicator of severity and the direction of causal relations.

In other words, by setting the correlation of variables, we can judge not about determinants and derivatives, but just how closely the changes in variables are interrelated and how one of them responds to the dynamics of the other.

Correlation (CORRELATION) is

Not with all the problems can be cope with the experimental method. There are many situations where the researcher cannot control which subjects fall into certain conditions. For example, if you need to check the hypothesis that people with anorexia are more sensitive to changes in taste than people with normal weight, then we cannot assemble a group of subjects with normal weight and demand that half of them have anorexia! In fact, we will have to take away people already suffering from anorexia, and those who have the weight in the norm, and check whether they also differ in flavor sensitivity. Generally speaking, you can use the correlation method to determine if some variable is connected, which we cannot control, with another variable you are interested in, or, in other words, they correlate with each other.

In the above example, variable weight has only two values - normal and anorexic. More often it happens that each of the variables can take many values, and then it is necessary to determine how much the values of one and the other variable correlate between themselves. It may determine this statistical parameter called the correlation coefficient and the letter R. The correlation coefficient allows us to evaluate how connected two variables are, and is expressed by the number from -1 to +1. Zero means lack of communication; Complete connection is expressed by one (+1, if the ratio is positive, and -1, if it is negative). As R from 0 to 1 increases, the communication force increases.

Dispelling graphs illustrating correlation. These hypothetical data belongs to 10 patients, each of which has some damage to the brain sites responsible, as far as known, for the recognition of individuals. In the figure, patients are arranged along the horizontal, respectively, the volume of brain damage, and the most left point shows the patient with the smallest damage (10%), and the most right point shows the patient with the greatest damage (55%). Each point on the graph reflects an indicator for a separate patient in the test of the recognition of persons. The correlation is positive and equal to 0.90. The figure shows the same data, but now they show the share of the correct answers, and not mistakes. Here the correlation is negative, equal to -0.90. In the figure, patient's successes in the recognition test are displayed depending on their growth. Here the correlation is zero.

The essence of the correlation coefficient can be illustrated by the example of the graphical representation of the data of the hypothetical research. As shown in the figure, patients are involved in the study, which are known in advance that they are damaged by the brain, and it caused varying degrees of difficulty in recognizing persons (transcopaging). It is necessary to find out whether the difficulty is increasing, or an error of recognizing persons, with an increase in the percentage of damaged cerebral tissue. Each point on the graph shows the result for a separate patient when testing it to recognize persons. For example, a patient with 10% damage was error in the test recognition test in 15% of cases, and a patient with 55% damage made errors in 95% of cases. If the error of the recognition of persons was constantly increased with an increase in the percentage of brain damage, the points on the chart would be all the time above when moving from left to right; If they were placed on the diagonal of the figure, the correlation coefficient would be R = 1.0. However, several points are located along different directions of this line, so the correlation is about 90%. Correlation 90% means a very strong connection between the volume of the damaged brain and the errors of the recognition of individuals. The correlation in the figure is positive, since greater brain damage causes more mistakes.

If, instead of errors, we decided to display the proportion of the correct answers in the recognition test, they would receive a schedule depicted by Narusunk. Here, the correlation is negative (equal to approximately -0.90), since with an increase in brain damage, the proportion of correct answers is reduced. Diagonal in the picture is just an inverse version of the one that is in the previous figure.

Correlation (CORRELATION) is

Finally, turn to the graphics in the picture. Here is the proportion of patient errors in the test to recognize persons depending on their growth. Of course, there is no reason to believe that the share of recognized persons is associated with the growth of the patient, and the schedule confirms it. When moving from left to right, the point does not show the agreed movement, nor up, and scattered around the horizontal line. The correlation is zero.

The correlation is positive (+) and negative (-). The correlation sign shows whether two variables are associated with positive correlation (the value of both variables is growing or decreased at the same time) or negative correlation (one variable grows with a decrease in the other). Suppose, for example, that the number of classes of classes by the student has a correlation -0.40 with points at the end of the semester (the more passes, the less points). On the other hand, the correlation between the points received and the number of visited classes will be +0.40. The connection strength is the same, but it depends on whether we consider the missed or visited classes.

As the connection of two variables, R increases from 0 to 1. To present it better, consider several known positive correlation coefficients: the correlation coefficient between points obtained in the first year of study in college, and points received in the second year is about 0, 75, the correlation between the gestures of the intellect at the age of 7 years and during re-testing at 18 years is approximately 0.70, the correlation between the growth of one of the parents and the growth of the child in adulthood is about 0.50, the correlation between the test results for the ability The training obtained in school and college is approximately 0.40, the correlation between points obtained by individuals in the blank tests, and the judgment of the psychologist expert on their personal qualities is about 0.25.

In psychological studies, the correlation coefficient is 0.60 and higher is considered rather high. The correlation in the range from 0.20 to 0.60 has practical and theoretical value and is useful when extending predictions. Correlation from 0 to 0.20 should be treated carefully, when the predictions extending its benefits are minimal.

Correlation (CORRELATION) is

Tests. A familiar example of using the correlation method is tests for measuring certain abilities, achievements and other psychological qualities. When testing a group of people who differ in some quality (for example, mathematical abilities, dexterity of hands or aggressiveness), there is some standard situation. Then you can calculate the correlation between changes in the indicators of this test and the change in another variable. For example, it is possible to establish a correlation between the indicators of a group of students in the test of mathematical abilities and their mathematics estimates with further training in college; If the correlation is significant, then on the basis of the results of this test, you can decide who from the new set of students can be translated into a group with increased requirements.

Testing is an important tool of psychological research. It allows psychologists to receive a large amount of data on people with a minimum separation of them from everyday affairs and without the use of complex laboratory equipment. Test construction includes many steps that we will consider in detail in subsequent chapters.

Correlation and causal relationships. There is an important difference between experimental and correlation studies. As a rule, in an experimental study, systematically manipulates one variable (independent) to determine its causal impact on some other variables (dependent). Such causal relationships cannot be derived from correlation research. An erroneous understanding of the correlation as a causal relationship can be illustrated in the following examples. There may be a correlation between the softness of the asphalt on the streets of the city and the number of solar strikes that happened in the day, but it does not follow from here that the softened asphalt highlights some poison that leads people to the hospital bed. In fact, the change in both of these variables is the softness of the asphalt and the number of solar strikes - is caused by the third factor - solar heat. Another simple example is a high positive correlation between a large number of storks, nesting in French villages, and a high birth rate registered in the same place. We will provide inventive readers to guess the possible reasons for such a correlation, without resorting to the postulation of the causal relationship between storks and babies. These examples serve as sufficient caution from understanding the correlation as a causal relationship. If there is a correlation between two variables, the change in one can cause changes to another, but without special experiments, this conclusion will be unjustified.

### Sources and links

ru.wikipedia.org - Wikipedia's free encyclopedia

bank24.ru - 24-hour bank of the Russian Federation

dic.academic.ru - portal of dictionaries and ecyclopedia

Statsoft.ru - Electronic Statistics Textbook

Superscalper.ru - Skalping Service for Forts and NYSE

Machinelearning.Ru - Information and analytical resource of intelligent data analysis

Uchebnik.biz - Student Library of Humanitarian Direction

TESTENT.RU - Educational site of Kazakhstan

FDVLADIMIR.ru - Financial House "Vladimir" - Broker in the securities market

Lib.Qrz.ru - electronic library of technical orientation

Uchimatchast.ru - website on applied mathematics

Cito-web.yspu.org - Yaroslavl State Pedagogical University

math.sessr.ru - Online Calculator of Mathematical and Economic Values

Stathelp.ru - statistical help, statistics news

Gaap.ru - Theory and Practice of Financial Accounting

GOLDENFRONT.RU - Site of investment in gold

Newsland.com - News in the Russian Federation and the World

*Encyclopedia Investor . 2013. .***Synonyms **:

### Watch what is a "correlation" in other dictionaries:

**correlation**- Correlation (p. 325) (from Late. Correlatio ratio) The term used in various areas of knowledge, including in psychology, to designate the mutual relationship, compliance with concepts and phenomena. Most psychological ... ... Big psychological encyclopedia**CORRELATION**- [Lat. CorrelAtio] Mutual relationship, ratio of objects or concepts. Dictionary of foreign words. Komlev N.G., 2006. Correlation Novolatinsk. from Relata. Mutual attitude, for example, existing between the guardian and swelling. Explanation 25000 ... ... Dictionary of foreign words of the Russian language**CORRELATION**- (Correlation) The degree of relationship between two variables. The linear correlation between two variables x and y is determined by the sign and the value of σi (xi μx) (yi μy), where μx and μy the average value of x and y. Between two variables there is a positive ... ... Economic Dictionary**correlation**- ratio, correlation, relationship, interdependence, interdependence, interconnectivity of Russian synonyms. Correlation of land., Number of synonyms: 8 • Autocorrelation (1) ... Synonym dictionary**CORRELATION**- (from Franz. Correlation ratio) in statistics is understood as a relationship between the studied statistical values, ranks and groups; To determine the presence or absence of K. Statistics enjoys a special method. K. method applied ... ... Big medical encyclopedia**correlation**- - - [http://www.rfcmd.ru/glossword/1.8/index.php?a=index d = 23] correlation The value that characterizes the mutual dependence of two random variables x and y is indifferent whether it is determined by some causative bond or Just random ... ... Technical translator directory**Correlation**- The relationship of two or more values, in which changes in one or more of them lead to a change in other or others. K. It is considered simple when it comes to relationships between two quantities or variables (for example, between ... ... Business Terms Dictionary**CORRELATION**- In mathematical statistics, probabilistic or statistical dependence. In contrast to the functional dependence, the correlation occurs when the dependence of one of the signs from another is complicated by the presence of a number of random factors ... Big Encyclopedic Dictionary**CORRELATION**- (from lat. CorrelAlatio ratio) 1) in logic - the ratio between the two is the same in the form of connections. If, thanks to a regular change in the structure, one relationship becomes isomorphic (equal in form) another, then this is the relationship of both connections ... ... Philosophical encyclopedia**correlation**- And, g. Corrélation f., it. Korrelation & LT; Lat. Correlatio ratio. First noted in the dictionary of Having 1894 ES. Mutual relationship, the ratio of objects or concepts. The law of correlation. Functional correlation. Bass 1. Growing unemployment and ... ... Historical Dictionary of Gallicalism Russian Language**Correlation**- [CorRelation] The value characterizing the mutual dependence of two random variables X and Y is indifferent whether it is determined by some causal bond or just a random coincidence (false correlation). In order to determine this ... ... Economics and Mathematical Dictionary

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