Can you tell which part of a log is closer to the top or which part is closer to the root???
Answers
Explanation:
Economists (like me) love the log transformation. We especially love it in regression models, like this:
lnYi=β1+β2lnXi+ϵi
Why do we love it so much? Here is the list of reasons I give students when I lecture on it:
It respects the positivity of Y. Many times in real-world applications in economics and elsewhere, Y is, by nature, a positive number. It might be a price, a tax rate, a quantity produced, a cost of production, spending on some category of goods, etc. The predicted values from an untransformed linear regression may be negative. The predicted values from a log-transformed regression can never be negative. They are Yˆj=exp(β1+β2lnXj)⋅1N∑exp(ei) (See an earlier answer of mine for derivation).
The log-log functional form is surprisingly flexible. Notice:
lnYiYiYi=β1+β2lnXi+ϵi=exp(β1+β2lnXi)⋅exp(ϵi)=(Xi)β2exp(β1)⋅exp(ϵi)
Which gives us:Loving the log-log functional forms That's a lot of different shapes. A line (whose slope would be determined by exp(β1), so which can have any positive slope), a hyperbola, a parabola, and a "square-root-like" shape. I've drawn it with β1=0 and ϵ=0, but in a real application neither of these would be true, so that the slope and the height of the curves at X=1 would be controlled by those rather than set at 1.
As TrynnaDoStat mentions, the log-log form "draws in" big values which often makes the data easier to look at and sometimes normalizes the variance across observations.
The coefficient β2 is interpreted as an elasticity. It is the percentage increase in Y from a one percent increase in X.
If X is a dummy variable, you include it without logging it. In this case, β2 is the percent difference in Y between the X=1 category and the X=0 category.
If X is time, again you include it without logging it, typically. In this case, β2 is the growth rate in Y---measured in whatever time units X is measured in. If X is years, then the coefficient is annual growth rate in Y, for example.
The slope coefficient, β2, becomes scale-invariant. This means, on the one hand, that it has no units, and, on the other hand, that if you re-scale (i.e. change the units of) X or Y, it will have absolutely no effect on the estimated value of β2. Well, at least with OLS and other related estimators.
If your data are log-normally distributed, then the log transformation makes them normally distributed. Normally distributed data have lots going for them.
Answer:
Throw the log into water. The part that was near the roots will sink and the part that was near to top would float.
Explanation: