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Linear Regression - level dummies, another example

5 important questions on Linear Regression - level dummies, another example

Does Province play a role?LogHPi=........................B13SMi+B14YOi+ei

Either include dummy Di^j (until j, j runs from 1 to 12) so
B1Di^1+B2Di^2......B12Di^12
but we can use summation to write down neater.
12Sumj=1BjDi^j
remember each Provence has its own beta. This is the sum of beta 1 times the dummy for south holland + the beta 2 times the dummy for north holland etc.

The second way to write this down:
Does Province play a role?LogHPi=........................B13SMi+B14YOi+ei

If you want to include a constant in the regression, but then we can't put all 12 dummies there. We'd have to leave 1 out to avoid perfect multicollinearity. Start by 2=> alpha+12Sumj=2BjDi^j+B13 etc..

So suppose you agree that j=1 was south holland, then now this alpha has immediately the interpretation that that's the level of the log house price if we would have no SM (SM=0) and Yoi=0), corresponding to housing price in south holland. (so interpretation is nonsense but alpha belongs now to south holland).

What's now the interpretation of b2 if we write it the second way, so with a constant (suppose we agreed on that j=2 corresponds with North Holland),

That has a logical interpretation, that means its the top up of b2*100% compared to the reference category (housing price of south holland), ceteris paribus.

Always, with the second way of using a constant with inclusion of level dummies

Check what's the reference, the constant and the beta is always the top up wrt the reference category.

One example
- 3 dummy variables, now industries
So we have certain companies and for each company I, we have yi=avg CEO salary and we have certain companies which we can allocate into banks, High-Tech and Other (B,i, HT,i and O,i are then the dummies). Research question: Is the avg ceo salary different between industries.

Yi = B0+b1B,i+B2HT,i+e,i
B0=reference category = other

So that means now (we don't have any other x-variables)
b0=avg salary other industry
b1=top up if you're in a bank (on other), so more (if pos) than for other industry.

Similarly, in y,i=B0+B1B,i+B2O,i+e,i, b0 is reference category HT. Which is the avg salary of HT. B2 is the top up or extra (if pos) salary of ceo from Other industry, on top of avg ceo salary the HT.

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Linear Regression

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Linear Regression - level dummies, another example

5 important questions on Linear Regression - level dummies, another example

Does Province play a role?LogHPi=........................B13SMi+B14YOi+ei

The second way to write this down:Does Province play a role?LogHPi=........................B13SMi+B14YOi+ei

What's now the interpretation of b2 if we write it the second way, so with a constant (suppose we agreed on that j=2 corresponds with North Holland),

Always, with the second way of using a constant with inclusion of level dummies

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The second way to write this down:
Does Province play a role?LogHPi=........................B13SMi+B14YOi+ei