Question

Suppose we assume the usual linear relationship yibo byai e for n pairs of observations (xi, yi), (x2,32),. ,(En, yn). Now, show what we would calculate for the coefficients bo and bi under the following criteria. a. (8 pts) Suppose we insist that the average error is zero: e-0. What condition must b0 satisfy in order for that relationship to hold? b. (12 pts) Now further suppose that we insist upon the correlation between the inputs and the errors being zero: relationship to hold? a, what condition must b satisfy in order for that rze 0. Using the bo you found in part

Answer 1

To establish the linear regression model,

Y_i = b₀ + b₁ X_i + e_i

We will apply the technique of Method Of Least Squares (MLS) to estimate the parameters b₀ and b₁.

This method consists of minimizing Error sum of squares given by

E = Σe_i² = Σ(Y_i - b₀ - b₁ X_i)²

On differentiating with respect to the parameters b₀ and b₁ we get two normal equations and solving them we can see that estimates of parameters b₀ and b₁ are as follows:

b₁ = Cov(X,Y)/Var(X) and b₀ = Ybar – b₁ Xbar.

Thus for the sake of calculations we should prepare the following table:

i	Xi	Yi	Xi2	Yi2	Xi* Yi
1
2
.
.
n
Total	ΣX	ΣY	ΣX2	ΣY2	ΣXY

After getting these totals in table it is easy to estimate both the parameters.

In order to hold this relation the condition b₀ is it should be NON ZERO.

Condition on b₁ is also same that it must be NON ZERO.

Here we must note that the relation is linear so that graphically it represents a straight line, where b₀ is interpreted as intercept on Y axis or it is the value of Y in the absence of X. Similarly b₁ is interpreted as slope of the line, which is rate of change in Y per unit change in X.