# R-squared

R-squared is a statistical measure of how well a statistical model (it may be just a line) fits the data. It is also known as the coefficient of determination, and denoted as R2 or r2. To be more specific, R-squared is the percentage of the variability of the data that is explained by the model (i.e., it is the explained variation divided by the total variation). In simple words, R2 shows how much of the changes in one thing (like the prices of gold stocks) can be explained by changes in another thing (like the price of gold). R-squared is always between 0 and 100 percent, where the higher the-R squared, the better the model fits the data. An R2 of 1 indicates that the model (it may be a simple regression line) perfectly fits the data, while an R2 of 0 means that the model does not fit the data at all.

In case of a typical linear regression, R-squared is simply the square of the correlation coefficient, which is a measure of the linear correlation between two variables. Therefore, it may be a useful measure of how much of the changes in the price of gold can be explained by changes in other variables.

## R-squared in Gold Market

The R2 may be used to determine the strength of the relationship between gold and other variables (like the USD Index, the general stock market). The Correlation Matrix shows correlation coefficients among several markets (including precious metals) in different time frames. To get the R-squared, one only needs to square the correlation coefficient (assuming that the correlation is linear).

Let’s assume that one invests in silver using a simple linear regression under the assumption that the price of silver is driven by the price of gold. If the correlation between gold and silver prices in the very long-term (1,500 days) is 0.9 (according to the Correlation Matrix, as of March 29, 2016), it means that the changes in the price of silver may be explained in 81 percent by the changes in the price of gold. It implies that investing in silver looking only at gold is simple, but quite effective investment strategy, as only 19 percent of the variability of silver prices is not explained by the model.

Let’s analyze another example. Some investors believe that the price of gold is driven by the stock market. However, the simple linear regression model using S&P as an explanatory variable would bring mediocre results, at least in the medium term. In such a time frame (250 days), the correlation between gold and S&P was only 0.02 (according to the Correlation Matrix, as of March 29, 2016). It means that the variation in the S&P does not explain the changes in the price of gold practically at all (the R2 is 0.0004).

To sum up, R-squared is a statistical measure of how well a statistical model fits the data, which may be a useful investment tool in the gold market. However, investors should be aware of its limitations. First, correlation does not mean causation (the number by itself doesn’t say if the price of gold stocks is determined by the price of gold or the other way around – we know that based on the common sense). Second, simple models assume normal distributions of returns and a linear relationship, while the markets, including the precious metals market, are dominated by non-linear relations. Third, the value of the correlation coefficient and R2 depends on the time frame. The same assets (i.e., gold and S&P) may be simultaneously correlated and not correlated, depending on the chosen time frame. Having said that, we can use correlation and determination analysis to find important connections in the precious metals markets and use them in connection with other types of analysis, which would increase the efficiency of the latter.

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