Question

Suppose you are interested in estimating the relationship between edu (number of years of university education) and the inc (annual income measured in ten thousand) and you run the following regression: ??? = ?? + ?? ??? + ?

4.A Suppose ?? = ???? , ?? = ???? , ??? ? = 8. Further, if you know, ?̅ = 3.2125, ?̅= 25.875, ∑ (?? − ?̅ ? ?=1 ) (?? − ?̅) = 5.8125 and ∑ (?? − ?̅) 2 = 56.875 ? ?=1 , calculate the value of the parameter ?2.

4.B Based on the information in 4.A and value of ?2 you computed, calculate the value of the intercept ?1. Please show all the calculation step by step.

4.C According to your parameter estimates, what is the predicted value of inc when edu = 20?

4.D If the sum of the residuals of squares, ∑ ?? 2 = 0.2247 ? ?=1 and the total sum of squares, ∑ (?? − ?̅) 2 = 1.0288 ? ?=1 , can you calculate the ? 2 ? Please show the calculation and make sure to interpret the result.

Answer 1

The model to be estimated is given as:

A.

If we need to calculate the value of the parameter , we need to find the OLS estimate of that. We represent the OLS estimate of the parameter as

OLS estimation is done by minimising the sum of sqaures of the estimated error terms. This yields,

We are given the value of the numerator as  ∑ (?? − ?̅ ) (?? − ?̅) = 5.8125

and the denominator as ∑ (?? − ?̅) 2 = 56.875

Hence, we can write :

or,

.................................................................

B.

The value of the intercept is obtained by the following formula:

We are already given from the problem, ?̅ = 3.2125, ?̅= 25.875 and we have also obtained

Hence, we substitute these values to get:

or,

or,

..........................................................

C.

Hence after we have predicted the intercept and the slope parameters, the predicted model will look like:

When edu=20, we put this value in the above equation to get the predicted value of inc

This means,

or,

or,

............................................................

D.

We know that :

Residual Sum of Sqaures (RSS) + Explained Sum of Sqaures (ESS) = Total Sum of Sqaures. (TSS)

And,

or,

(as ESS = TSS - RSS)

or,

Now, we are given

residuals of squares (RSS)= ∑ ?? 2 = 0.2247

and,

total sum of squares (TSS)= ∑ (?? − ?̅) 2 = 1.0288

Hence,

or,

or,

This result means that on an average, the model is able to capture 0.7816 proportion (approx) or 78.16% of the variabilities in the dependent variable through the variabilities of the independent variable. Hence, we can say that the model is a good fit.