北大暑期课程《回归分析》(Linear Regression Analysis)讲义pku4.doc

Class 4: Inference in multiple regression. I. The Logic of Statistical Inference The logic of statistical inference is simple: we would like to make inferences about a population from what we observe from the sample that has been drawn randomly from the population. The samples characteristics are called point estimates. It is almost certain that the samples characteristics are somewhat different from the populations characteristics. But because the sample was drawn randomly from the population, the samples characteristics cannot be very different from the populations characteristics. What do I mean by very different? To answer this question, we need a distance measure (or dispersion measure), called the standard deviation of the statistic. To summarize, statistical inferences consist of two steps: (1) Point estimates (sample statistics) (2) Standard deviations of the point estimates (dispersion of sample statistics in a hypothetical sampling distribution). II. Review: For a sample of fixed size is the dependent variable; contains independent variables. We can write the model in the following way: Under certain assumptions, the LS estimator As certain desirable properties: A1 = unbiasedness and consistency A1 + A2 = BLUE, with A1 + A2 + A3 =BUE (even for small samples) III. The Central Limit Theorem Statement: The mean of iid random variables (with mean of , and variance of ) approaches a normal distribution as the number of random variables increases. The property belongs to the statistic -- sample mean in the sampling distribution of all sample means, even though the random variables themselves are not normally distributed. You can never check this normality because you can only have one sample statistic. In regression analysis, we do not assume that ? is normally distributed if we have a large sample, because all estimated parameters approach to normal distributions. Why: all LS estimates are linear function of ? (proved last time).