I sill don’t fully understand what the difference is between these. Can someone help me understand this statement: “Suppose we conclude that market cap is not linearly related to P/E but the natural log of market cap is linearly related to P/E” When you take a natural log of something like market cap, how does that make it linearly related to P/E vs how it was in its normal state?
i’m not positive but I believe it goes like this. One of the major assumptions in regression analysis is that there is a linear relationship between independent and dependent variables. If this assumption is violated then your regression is bollocks. This is not an issue when variables grow by constant UNITS, but when they grow by constant RATES (i.e. exponential growth) they are no longer linear but curved. By taking the natural log of variables with this issue you can turn them into a linear function. In the example of market cap, if you assume that a company with a market cap of $100 (just to make numbers small) grows at a RATE of 5% for 10 years you get a market cap that grows as follows. 100 150 225 337.5 506.25 759.375 1139.0625 1708.59375 2562.890625 3844.335938 If you plot this out you will see that this is not a straight line but curved. But if you take the natural log of these numbers you get points that plot as follows, which when graphed does form a straight line and allows the variables to usuable for regression analysis. 4.605170186 5.010635294 5.416100402 5.82156551 6.227030618 6.632495727 7.037960835 7.443425943 7.848891051 8.254356159 I hope this helps.
correction I just realized that they were grown at 50% rather than 5% but this doesnt change the underlying principle only the numbers
Good explanation cfakid
thanks cfakid