Question

3. Suppose the following data has been obtained for the linear model Y Bo+ x 14 2 1 4 0 22 (a) Find the OLS estimators βο and A using the non-matrix method. (b) Find the OLS estimators using the matrix method. (c) Find the coefficient of determination. (d) Find the standard error of

a,b,c,d

Answer 1

Solution:

Given the data, we know that number of observations, n = 4 and

Sum(xi) = 1 + (-4) + 2 + 1 = 0 ; so, x(bar) = sum(xi)/n = 0/4 = 0

Sum(yi) = 4 + 0 + 2 + (-2) = 4 ; so, y(bar) = sum(yi)/n = 4/4 = 1

(a) Denoting estimated betas by b for ease of writing, so finding b0 and b1

b1 = sum[(xi - x(bar))(yi - y(bar))]/sum[xi - x(bar)]²

Now, sum[(xi-x(bar))(yi-y(bar))] = [((1-0)*(4-1))+((-4-0)*(0-1))+((2-0)*(2-1))+((1-0)*(-2-1))]

= 3 + 4 + 2 + (-3) = 6

And, sum[xi - x(bar)]² = [(1-0)²+(-4-0)²+(2-0)²+(1-0)²] = 1 + 16 + 4 + 1 = 22

So, b1 = 6/22 = 0.273 (approx)

b0 = y(bar) - b1*x(bar)

b0 = 1 - (6/22)*0 = 1

So, the regression line is y^ = 1 + 0.273x

(b) Using the matrix method, the model can be written as:

Y = XB + e where Y is column sector of dependent variable and so on. (To solve this you should be very clear with basic matrices properties and formulae). So,

Y =
402-2
; X =
4 1
, where column of 1 is for intercept term estimation. B =
bl
and finally, e =
1234

Then, we are required to find the B vector. Now, required formula is:

B = (X'X)^-1(X'Y)

X' =

1-4 2 1

So, using simple matrix multiplication, we find

X'X =

1*1+(-4) * 1 + 2 * 1 + 1 * 1 1*1+(-4) * (一4) + 2 * 2 + 1 * 1

X'X =
111 1(-4)2 +1 1-4) +2+1 116 +4+1
=
0 2 40

Then, (X'X)^-1 = adjoint(X'X)/|X'X|

|X'X| = 4*22 - 0*0 = 88

Adjoint(X'X) = (coming out to be same as (X'X)

0 2 40

So, (X'X)^-1 = (1/88)*

0 2 40

And, X'Y =