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： Properties of the Least Square Estimator 日期：2003年10月16日,4.1 b1 and b2 are random variable b/c their value depend on the random variable y whose values are not known until the sample is collected. After the data are collected, the least squares estimates are calculated numbers without statistical properties b/c there are nonrandom quantities.,4.2 The least squares estimators b1 and b2 are random variable and have probability distributions that we can study prior to the of any data. determine the means and variances of the least squares estimators b1 , b2 .,true value, unknown,unbiased,estimator,and,Two assumptions!,If then b2 is not unbiased! et contains other factors yt affecting that are omitted from the economic model. If we have omitted anything that is important, the we would expert that and . b2 is unbiased does not imply anything about what might happen is just are sample.,Note: An individual estimate b2 may be near to, or far from 2 . But Also, we can calculate var(b2) and var(b2) Same mean value but different variances Sampling Precision The lower the variance of an estimator, the greater the sampling precision of an precision. 1.if var(y) var(b2) less precision,2. if var(x) var(b2) more precision 3. if T var(b2) high precision 4. X2 var (b2) loss precision 5. X 0 Cov(b1,b2) 0,X,b2 but b1,X,If X 0,4.4 The Probability Distributions of the Least Squares Estimates If eN(0, 2) then b1N(1, b12) b2N(2, b22) What if b1 , b2 N If SR1SR5 hold, and if the sample size T is sufficiently large, then the least squares estimators have a distribution that approximates the normal distribution. Ex: Food Expenditure Data,