Regression without any regressor Suppose you are given the m

Regression without any regressor. Suppose you are given the model Y_i = beta_1 + U_i. Use OLS to find the estimator of beta_1. What is its variance and the RSS (residual sum of squares)? Does the estimated beta_1 make intuitive sense? Now consider the standard two-variable model we have been discussing in class. Is it worth adding the X variable to the model? If not, why bother with regression analysis?

Solution

The error term has to be minimised.

So,

( Yi - b)² has to be minimised with respect to \'b\' by partial differentiation

Thus,

we get:

2Yi = 2b

or Yi = b.

thus, the estimated value of the b is \'b\' itself since the variance and the residual sum of squares obtained when each Yi is plugged with the value \'b\' will come out to be 0.

Hence, \'b\' does make an intuitive sense because in absence of a regressor, the output function is dependent on a constant function that does not change.

(I am not sure about the model you have been discussing in your class) However, it is always useful to add a variable that helps in precisely predicting the value of the output given changes in the input.

Hope this helps. Ask if you have doubts.