Methodology of Social Sciences: Inference in the Simple Regression Model

1. 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

2. unknown unknown • b~N(β, σb2 ) 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~x2 (m) and if Z and V are independent, then m is the degree of freedom i=1,2 we estimated variance 政治大學 中山所共同選修 黃智聰

3. P(t≧tc )=P(t≦ tc)=α/2 • P(- tc ≦t ≦ tc)=1-α • P[- tc ≦ ≦ tc]=1-α • P[b2- tcSe (b2) ≦β2 ≦ b2+ tcSe(b2) ]=1-α • The interval endpoints, and both b2 and Se(b2) are random variables, since their values are not known until a sample of data is drawn. b2- β 2 Se(b2) 政治大學 中山所共同選修 黃智聰

4. b2± tcSe(b2) is call a (1- α) ×100% interval estimate of β2 or called a (1- α) ×100% confidence interval. • (1- α) ×100% of all the interval constructed would contain the true parameter β2 . This we known before any data are actually collected. • If the interval is [0.0666,0.1900], is β2 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. 5.2 Hypothesis Testing • Components of Hypothesis Tests • 1.A Null hypothesis, H0 • 2.An alternative hypothesis, H1 • 3.A test statistic • 4.A rejection region 政治大學 中山所共同選修 黃智聰

6. 1. H0:β2=c, c is a constant, and is an important value in the context of special regression model. • 2. H1:β2≠c • H1:β2＞c b/c theoretically, β2 can not be negative • H1:β 2＜c when there is no chance that β2＞c 政治大學 中山所共同選修 黃智聰

7. 3.The test statistic • Ex: H0:β2=c, β1≠c • Therefore don’t have standard normal distribution and the formation of a t random variable. b2-c Var(b2) 政治大學 中山所共同選修 黃智聰

8. 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

9. 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

10. 5.2.6 Type I and Type II errors • 5.2.7 The P-value of A Hypothesis Test • T=0.9263 • If P＜α then the test procedure leads to rejection of the null hypothesis • 5.2.8 A significance Test in the Food Expenditure Model • A statistically significant relationship exists b/w x and y. • If α more likely to reject H0 • How to choose α 0.1, 0.05, 0.01 政治大學 中山所共同選修 黃智聰

11. 5.2.10 One-tailed Test • H0:βk=c H1:βk＜c or βk＞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 H0 • But one-tailed can’t reject H0 政治大學 中山所共同選修 黃智聰