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Intermediate
R-Squared is a measure of the strength of the relationship between two data sets, often called variables. With investments it is used for statistical interpretations, most commonly for single-variable linear regressions.
The R refers to Pearson's R, sometimes just r, from correlation and is calculated by squaring the correlation. between a pair of securities. This transforms the correlation scale, of -1 to +1, to an easily interpreted scale of 0 to 1. On the one end, 0 indicates no relationship and 1 indicates a perfect linear relationship between the variables.
Synonym: coefficient of determination, r2, R Squared
For context, since the range is from 0 to 1, or 0.01 to 1.00, or 0% to 100%, it's easy to assume the pattern is linear, when it isn't. Recall R-Squared comes from squaring correlation, so the pattern is curved. So the way we naturally interpret the percentile scale doesn't exactly work the same here.
Also, the interpretation depends on which branch of the sciences you're analyzing. In the hard sciences or earth sciences high R-Squared measures may be above 0.90, whereas for the social sciences like economics, an R-Squared of 0.70 may represent a high reading.
Doc: Who remembers why it is important to
review R-Squared with beta.
Mia: Even a random shotgun pattern has a
line of best fit.
This video can be accessed in a new window or App , at the YouTube Channel or from below.
R-Squared definition for investment modeling (4:31)
The script includes two sections where we visualize and demonstrate the concept of R-squared.
We're sitting right here in Excel and this is a snippet from our boot camp course.
This is one depiction of R-squared, from a scatterplot of returns for one stock Merck versus a basket of stocks, the Market, for 60 periods. Think about each dot here as returns for the Market on the x-axis and Merck on the y-axis, for each month.
So if stocks exhibit co-movements, as they appear to here, then a pattern will look linear. A random shotgun pattern would have low R-squared.
Once we square correlation, the direction of the line goes away, and we can only interpret the goodness of fit of the line versus the data points.
The best way to understand R-squared is with an example, so let's head there now.
We'll walk all the way through a calculation for two stocks, Microsoft and eBay.
We have six monthly returns for each stock from April to September 2003. Column F is the return on Microsoft, eBay is in column G.
Next we compute the average of each, here 2.38% and 3.98%. Then we move those over to columns H and I.
In column J take the return minus this average which gives us 3.24%. That's 5.62% minus 2.38%. For eBay it is 8.91% minus 3.98% or 4.93%. Carry that formula down, and let's call these demeaned returns.
Next, in column L, multiply these together. As you notice, when the stocks move together, like in April, the product is negative. And when they move in opposite directions, the product is negative.
Next, using the =SUM function, add up the products to get -0.0037 for the pair of stocks. Next, divide by 6 observations for a covariance of -0.0006 for the pair of stocks. This isn't interpretable as the units are returns-squared, so we translate to correlation by dividing by the product of the two standard deviations.
So to interpret, the R-squared explains what percentage of the total errors is explained by the x-variable.
Click box for answer.
True. The first part is correlation.
0.25. R-Squared are positive numbers.
Still unclear on R-squared? Leave a question in the comments section on YouTube or check out the Quant 101 Series, specifically How to interpret Correlation and R-squared.
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