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Okay, so now,
let's talk about how we're gonna interpret the coefficients of our fitted line.
And let's start by talking about W-hat 0, our estimated intercept.
So, that's this point right here, this green star, and what does that represent?
Well, it's the predicted value of a house with 0 square feet.
So, we can think of that just as land,
because we have that Y-hat is = W-hat 0 when X=0.
So, when there's no square feet, we have that our predicted
house value is exactly equal to W-hat 0.
But I'll say that in general, this is not normally that meaningful.
In many scenarios, you don't really think of having no input and
having the output have some meaningful value.
And actually in our case, if you go back and look at the number,
we had that our intercept was -44,850.
So, it's saying that land you should actually as the seller be giving
the buyer money, to take that land off your hands.
But remember that our dataset's really noisy,
it's not a perfect model of what's going on.
So, of course, we have to think about interpreting our parameters
in the context of what we've included in our model.
Okay, we'll talk more about that in the next module.
But, I also want to talk about W-hat 1 which is our estimated slope.
And that tends to have a bit more meaning associated with it.
So, the estimated slope, I'm highlighting with this green triangle here,
saying that per unit change, in our inputs, so for
1 unit, so 1 square foot,
what's our predicted change in the value of the house.
So, for example, let's say I'm looking at estimating
the value of a house with 1001 square feet.
And I want to look at the difference relative to the estimated value of a house
with just 1000 square feet.
Well, what is this, this is equal to, if I look at the first term
estimating a house with 1,001 square feet, how would I estimate that?
I would take W-hat 0,
+ W-hat 1 * 1,001 square feet.
And then, I'm gonna subtract for
the second term I have it's estimated value is again
W-hat 0 + W-hat 1 * 1,000 square feet.
And these W-hat 0's are gonna cancel.
And the difference between these numbers is just 1.
So, this will just be W-hat 1.
So again,
W-hat 1 represents
the predicted change
in the output per unit
change in the input.
Okay, so that's how we're going to interpret W-hat 1.
But one thing I want to make very,
very clear is that the magnitude of this slope,
depends both on the units of our input, and on the units of our output.
So, in this case, the slope, the units of slope,
are dollars per square feet.
And so, if I gave you a house that was measured in some other unit,
then this coefficient would no longer be appropriate for that.
So, let's make this a little bit more explicit.
So now that we know that there are units attached to these numbers,
I'm gonna put these in, so our intercept has units just dollars.
So, let me be clear what I'm actually highlighting here.
The intercept has units dollars, and the slope has units dollars per square feet.
And when I went through and
did my calculation of predicting the value of a house with a given square feet,
if I actually put in these units, I see that everything works out.
Here I have dollars, here I have dollars per square feet, and
then I'm multiplying by square feet.
So when I multiply these two things the resulting units of
this is just gonna be dollars, so the end result is something in terms of dollars.
That's great.
And likewise, when I use the equation in reverse, I have dollars,
dollars divided by dollars per square feet,
and I end up with something that's in square feet.
So, I'll just say units work out,
which is great.
But what if the house was measured in square meters instead of square feet,
well, clearly I can't just plug into that equation, and
likewise what if the price was measured in Chinese Yuan instead of in US Dollars.
Again, can't just plug into that equation, so when I think about magnitude,
it only has meaning in terms of the units that I use for my input and output.
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