# Ridiculous manipulation question

1. May 16, 2010

### operationsres

ECONOMETRICS CALCULUS.

1. The PRECEDING question and solution

Differentiate with respect to b1

My solution:
df(b0,b1)/db1 = 2Σ[(yi-b0-b1xi)*(-xi)]

2. the ACTUAL question

By setting d,f(b0,b1)/d,b1 = 0, show that you obtain .
(Hint: you will have to substitute b0 = yBAR - b1(xBAR), and use the results from the previous question.)

3. Some properties that we can assume while answering the question.

4. The attempt at a solution (I can't do it)

Letting 2Σ[(yi-b0-b1xi)*(-xi)] = 0
0= Σ[(yi-b0-b1xi)*(-xi)]
0= Σ(-yi xi)+Σb0xi+ Σb1xi ^2
0= -Σ (yi xi)+b0 Σxi+b1Σxi^2
b1Σxi^2=Σyi Σxi – (yBAR-b1xBAR)Σxi ............... (Re-arranging and substituting for b0)
b1Σxi=Σyi -yBAR+b1xBAR ............................ (dividing both sides by Σxi)
b1Σxi – b1xb = Σyi – yBAR
b1(Σxi – xb)=Σyi – yBAR
b1= (Σyi – yb)/(Σxi – xBAR)
= (Σyi – Σyi /n)(Σxi – Σxi/n)

Last edited: May 16, 2010
2. May 16, 2010

### Staff: Mentor

The line above should be an equation. It should NOT start with =. The equation you're starting with says that
$$\frac{\partial f}{\partial b_1} = 0$$
Each line after the first one should be an equation. Ultimately you should get an equation that starts with b1.

3. May 16, 2010

### operationsres

That's just me being unformal though.

2=x
=y

is just 2=x=y.

I will formalise it when I figure out the answer.

Do you know where my actually working went wrong?

4. May 16, 2010

### Staff: Mentor

If 2 = x, and x = y, then you can say 2 = y or 2 = x = y. This is different from what I'm talking about, and it's not just a matter of being informal vs. being formal. It's the difference between solving an equation and simplifying an expression, which are two entirely different operations.

and solve for b1, using the properties of summations.

Each step should be an equation!

5. May 16, 2010

### gabbagabbahey

Hi operationres, it may seem that Mark is being nit-picky, but his comments are good advice. Even though you aren't handing in your forum post for marks, it is still in your best interests to make your workings as clear as possible. Doing so makes it easier for us to see where you are going wrong, and also gives you practice in writing things in a logical, mathematically correct manner (practice at this helps make things go quicker on your exams ).

Here it looks like you are claiming that

$$\sum_{i=1}^nx_iy_i=\left(\sum_{i=1}^nx_i\right)\left(\sum_{i=1}^ny_i\right)$$

Try plugging in some actual numbers to convince yourself that this is not correct.

You also seem to think that

$$\frac{\sum_{i=1}^n(x_i)^2}{\sum_{i=1}^nx_i}=\sum_{i=1}^nx_i$$

Which again isn't true (plug in some actual numbers to convince yourself of this).