政治大學中山所共同選修 課程名稱： 社會科學計量方法與統計－計量方法 Methodology of Social Sciences 授課內容： Inference in the Simple Regression Model 日期： 2003 年 10 月日

政治大學 中山所共同選修 黃智聰 b~N(β, σ b 2 ) Z= ~N(0,1) Chi-square random variable arise when standard normal, N(0,1), random variables are squared. If Z~N(0,1) and V~x 2 (m) and if Z and V are independent, then m is the degree of freedom i=1,2 we estimated variance unknown

政治大學 中山所共同選修 黃智聰 P(t ≧ t c )=P(t ≦ t c )=α/2 P(- t c ≦ t ≦ t c )=1-α P[- t c ≦ ≦ t c ]=1-α P[b 2 - t c Se (b 2 ) ≦ β 2 ≦ b 2 + t c Se(b 2 ) ]=1-α The interval endpoints, and both b 2 and Se(b 2 ) are random variables, since their values are not known until a sample of data is drawn. b 2- β 2 Se( b 2 )

政治大學 中山所共同選修 黃智聰 b 2 ± t c Se(b 2 ) is call a (1- α) ×100% interval estimate of β 2 or called a (1- α) ×100% confidence interval. (1- α) ×100% of all the interval constructed would contain the true parameter β 2. This we known before any data are actually collected. If the interval is [0.0666,0.1900], is β 2 in the interval? We don ’ t known, and we will never known!! We just know 95% of all the interval estimates constructed using this procedure will contain the true parameter.

政治大學 中山所共同選修 黃智聰 5.2 Hypothesis Testing Components of Hypothesis Tests 1.A Null hypothesis, H 0 2.An alternative hypothesis, H 1 3.A test statistic 4.A rejection region

政治大學 中山所共同選修 黃智聰 1. H 0 : β 2 =c, c is a constant, and is an important value in the context of special regression model. 2. H 1 : β 2 ≠ c H 1 : β 2 ＞ c b/c theoretically, β 2 can not be negative H 1 : β 2 ＜ c when there is no chance that β 2 ＞ c

政治大學 中山所共同選修 黃智聰 3.The test statistic Ex: H 0 : β 2 =c, β 1 ≠ c Therefore don ’ t have standard normal distribution and the formation of a t random variable. b 2 -c Var(b 2 )

政治大學 中山所共同選修 黃智聰 The rejection Region The rejection region is the range of values of the test statistic that leads to rejection of the null hypothesis. ie: when the null hypothesis is true, are unlikely and have low probability. Two-tailed Test If the value of the test statistic falls in the rejection region, either tail the t-distribution, then we reject the null hypothesis and accept the alternative. Avoid sampling that we accept the null hypothesis instead of saying we fail to reject the null hypothesis

政治大學 中山所共同選修 黃智聰 Format for Testing Hypothesis 1.Determine the null and alternative hypothesis 2.Specify the test statistic and its distribution if the null hypothesis is true. 3.Select α and determine the rejection region 4.Calculate the sample values of the test statistic 5.State your conclusion

政治大學 中山所共同選修 黃智聰 Type I and Type II errors The P-value of A Hypothesis Test T= If P ＜ α then the test procedure leads to rejection of the null hypothesis A significance Test in the Food Expenditure Model A statistically significant relationship exists b/w x and y. If α more likely to reject H 0 How to choose α 0.1, 0.05, 0.01

政治大學 中山所共同選修 黃智聰 One-tailed Test H 0 : β k =c H 1 : β k ＜ c or β k ＞ c 電腦是以 One-tailed 來算 因為 if two-tailed 算出 P=0.08 one-tailed P=0.04 所以在 two-tailed at α=0.05 時 reject H 0 But one-tailed can ’ t reject H 0