ECON301_Handout_03_1213_02

Although we have a sample since we do not know the

Info iconThis preview shows pages 18–20. Sign up to view the full content.

View Full Document Right Arrow Icon
’s. Although we have a sample since we do not know the Population Regression Function (PRF), the corresponding disturbances are still unobservable. If the disturbances were observable, an unbiased estimator of 2 would be obtained as (Erlat, 1992, pp.32-33): 2 2 1 1 T t t u s T However as it has emphasized above, t u ’s are unobservable and we need sample counterparts for them, which are called residuals ( ˆ t u ). Thus, we will substitute ˆ t u ’s for t u ’s and we will calculate the variance of these ˆ t u ’s. However, this is a bit problematic. Note that ˆ t u ’s are OLS residuals . Hence, they must obey the OLS estimator rules of: (1) 1 ˆ 0 T t t u , and (2) 1 ˆ 0 T tt t uX Hence, to call a series of observations as OLS residuals, these two conditions must be met. Now, the question here is: how many of ˆ t u ’s are random or how many ˆ t u are free to vary ? This is an important question since in statistics, the variance is calculated by dividing the sum of squared deviations from mean to the number of observations that are free to vary . To digest this crucial issue let us give an example. Suppose we have a sample size of 5 (i.e., T=5):
Background image of page 18

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 19 t=1 t=2 t=3 t=4 t=5 1 ˆ u may happen randomly 2 ˆ u may happen randomly 3 ˆ u may happen randomly If 4 ˆ u happen randomly, there is no guarantee that 1 ˆ 0 T tt t uX holds. If 5 ˆ u happen randomly, there is no guarantee that 1 ˆ 0 T t t u holds. As can be seen from this example that the number of values of ˆ t u that are free to vary is not 5 but instead 3! In other words, in this example we are free to choose the first three numbers at random, but the fourth and fifth must be chosen so that 1 ˆ 0 T t and 1 ˆ 0 T t t u conditions hold, thus the number of values of ˆ t u that are free to vary is not 5 but instead 3. We call the number of observations that are free to vary as degrees of freedom 2 . For a regression model with k independent variables and 1 intercept term, we can generalize the degrees of freedom as df=T-k-1. Hence, an unbiased estimator of 2 can be calculated as: 2 2 1 ˆˆ () ˆ 1 T t uu Tk  Note that ˆ 0 t u since 1 ˆ 0 T t t u condition of OLS estimation. Hence, the unbiased estimator of 2 can be rewritten as: 2 The concept of degrees of freedom is central to the principle of estimating statistics of populations from samples of them.
Background image of page 19
ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 20 2 2 1 ˆ ˆ 1 T t t u Tk   where ˆ t u is the OLS residual, and it is given by ˆ ˆ t t t u Y Y  .
Background image of page 20
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page18 / 20

Although we have a sample since we do not know the...

This preview shows document pages 18 - 20. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online