WEBVTT
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So what is log.
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It's not this big block of wood that people carry,
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at least in our context it is not.
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But we just saw that log is this transformation that we do that's gonna give us,
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pay us some dividends when we calculate elasticity.
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We saw that if you use a log log model the coefficient directly gives us the elasticity.
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Now let's see why that is the case.
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So in the table here I have world population from year one to year 2000.
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And here is a population in millions.
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We went from 170 million to 6080 million in 2000.
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On this column here,
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we have log of population.
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So log of 170 is equal to 5.14.
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So what log does is that it allows us to compute change.
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Right, rate at which something changes.
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That can be seen in this chart right here.
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What I have plotted is on the left side,
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I have population in millions and
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these blue dots plot the population in millions over time.
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So you can see that that is a nice hockey stick in world population.
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So around this time period here between 100,
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1500 and 2000, world population actually took off.
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You get a nice hockey stick.
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There was something happening there.
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So what was it that was happening.
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Maybe the log transformation will help us understand.
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So if you see in the log transformation out here,
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this plots log of population in millions.
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So what you see here is something happened from around between 1500 and 2000.
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But even the rate of change in world population increased.
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That is the world population started growing at a faster rate between 1500 and 2000.
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So this is the kind of conceptual value that the log transformation provides,
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when you convert something into a log.
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Now what does that mean for regression and elasticity?
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So the key thing here is,
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the first difference of natural log gives you percent change.
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So what I have plotted in the graph here is in the blue line,
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it is percent change in dollars
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spent and the green is difference in log of dollars spent.
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OK. So if you see here, it's not if,
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you are not able to make a difference between the blue and the green line here.
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That really tells you that percent change in dollar
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spent is the same as difference in log dollars spent.
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So if you take a log of dollar spent and take
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the difference between two values of log of dollar spent,
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that is equal into percent change in dollars spent.
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And why is that important for elasticity.
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Let's think about this.
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Elasticity is percent change in sales for a percent change in price.
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Now what is that thing?
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That thing is pretty much the coefficient of price in a regression
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of log of sales on log of price and that is because the coefficient tells you,
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difference in y over difference in x.
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Now difference in log is percent change.
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So if you take the coefficient off a log log model it gives you
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percent change in y for a person change in x.
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Let's look at the numbers here.
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So let's recap. Elasticity can be obtained from Log/Log models.
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If you regress log of dollars spent on log of number of promotions.
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The coefficient here, 0.317 gives you the elasticity of sales to number of promotions.
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So 0.317 here is change in Log spent,
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when log number of promotions increases by one unit
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or it is equal into percent change in dollar
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spent without that transformation what
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percent change in number of promo.
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Which is equal to the elasticity of promotion.
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Right.
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So what I have here is basically showing that the coefficient gives you this one.
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When log spent, what is log spent when promotion is zero. It's 2.2.
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Log of dollars spent when promotion is one it's 2.553.
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So if you take 2.553 minus 2.236.
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That is 2.553 - 2.236. It gives you 0.317.
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So I hope you are able to understand the value of
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the log transformation when you're calculating price elasticity.
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This is a very important concept that would be
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very useful at least in marketing mix models.