1. 1 Research Method Lecture 4 (Ch5) OLS Asymptotics ©

2. OLS Asymptoticc 4OLS asymptotics are the analyses of OLS properties when the sample size (n) increases to infinity. We will talk about the concept of (i) consistency and (ii) asymptotic normality. 2

3. Consistency 4Consistency is a similar concept as the unbiasedness. Unbiasedness: Given the sample size n, the expected value of the estimator is equal to the true value β j. Consistency: The estimator approaches to the true value β j as the sample size n increases to infinity. 3

4. Why we need the concept of consistency? 4Often, unbiasedness is difficult to achieve. 4But consistency is easier to achieve under less strict conditions. 4Econometrician consider that consistency is the minimum requirement for any estimators. 4

5. Theorem 5.1: Consistency of OLS Under assumptions MLR.1 through MLR.4, OLS estimators is consistent for β j for j=0,1,…,k. That is: 5 Proof: See front board

6. 4Consistency can be achieved under less strict assumptions, given below. Assumptions MLR.4’ E(u)=0 and cov(xj,u)=0 for j=0,1,2,…,k 6

7. Asymptotic normality 4In the previous handout, we assumed that the error term is normal (MLR.6) in order to do the hypothesis testing. 4But in many cases, normality assumption is not appropriate. 4We want to conduct hypothesis testings while making no assumption about the distribution of the error term. 4Asymptotic normality result (in the next slide) will shows that using t-test is just fine for any type of distribution. 7

8. 4Theorem 5.2 Asymptotic normality Under Gauss-Markov Assumptions (MLR.1 through MLR.5), the distribution of the following will approach to N(0,1) as sample size increases to infinity. That is: 8 Or an equivalent notation is: Proof: See front board

9. 4Theorem 5.2 tells us that, even if we do not know the distribution of the error term u, we can use the usual t-test in a usual way to conduct hypothesis testing. 9

10. Lagrange Multiplier Statistic (or nR 2 -statistic) 4Remember that F test relies on the normality assumption about u. 4There is a test of the exclusion restrictions that does not need the normality assumption. 4This uses LM-statistic (or often called n-R- squared statistic) 4This is a test of exclusion restrictions. 10

11. 4I explain the procedure by using the following example Y= β 0 +β 1 x 1 +β 2 x 2 +β 3 x 3 +β 4 x 4 +u --------------(1) H 0 : β 2 =0, β 4 =0 H 1 : H 0 is not true Next slide shows the procedure 11

12. 4The procedure (i)Regress the restricted model. That is, Y= β 0 +β 1 x 1 +β 2 x 2 +u. Then, get the residual,. (ii)Regress on all the independent variables. That is. Then compute R-squared. Call this R u 2. (iii)Compute LM=n R u 2. The asymptotic distribution of LM-stat is chi-squared distribution with df equal to number of equations in H 0. That is 12 # equations in H 0. In this example q=2.

13. (iv) Set the significance level. This is usually set at 0.05. (v) Find the cutoff point such that P( χ 2 q >c)=. (vi) Reject if LM is greater than the cutoff number. This is illustrated in the next slide. 13

14. 14 c 1- The density of χ 2 q Rejection region The cutoff points can be found in the table in the next slide.

15. Copyright © 2009 South- Western/Cengage Learning15

16. Example 4Using crime1.dta, consider the following model. Narr86=β 0 +β 1 pcnv+β 2 avgsen+β 3 tottime+β 4 ptime86+ β 5 qemp86+u Narr86: the number of time a man was arrested until 1986 Pcnv: proportion of prior arrests leading to conviction Avgsen: average sentence served from past conviction Tottime: total time the man has spent in prison prior to 1986 Ptime86: month spent in prison in 1986 Qemp86:number of quarters in 1986 during which the man was legally employed. 16

17. 4Test if avgsen and tottime have no effect on narr86 once the other factors have been controlled for. That is test the following hypothesis. (Use LM statistic instead of F-test) H 0 : β 2 =0,β 3 =0 H 1 : H 0 is not true. 17