# notes04 - Statistical Inference ECON 399 Neil Hepburn...

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Statistical Inference ECON 399 Neil Hepburn Contents 1 Introduction 1 2 Overview and Purpose 1 3 The Nature of Statistical Estimators 2 4 Sampling Distributions 2 5 Hypothesis Tests 7 6 Hypothesis Tests with Confidence Intervals 18 1 Introduction Introduction We now turn to the issue of inference and hypothesis testing This is where get to actually use OLS results to do something useful with 2 Overview and Purpose Overview OLS estimators on their own aren’t much good We need to be able to verify that they mean something If our reason for estimating a model is to test a theory, we need to be able to make use of those coefficient estimates If our purpose is forecasting, we need to be able to identify feasible ranges for the forecasted variable 1

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Overview Statistical inference and hypothesis testing allows us to do all that and more Later in the course when we look at diagnostic techniques for the validity of our regression models, we will make use of hypothesis testing methods What we develop here is absolutely fundamental 3 The Nature of Statistical Estimators The Nature of Statistical Estimators All too often students approach hypothesis testing by trying to memorize which test to use in which setting This is a very difficult way to go and what you “learn” seldom stays with you after you finish the course A better way (although seemingly harder at first) is to develop an intuitive understanding of the process Statistical Estimators are Random Variables The thing that you have to remember is that all of the statistical estimators that you have learned to compute (mean, standard deviation, median, OLS coefficients, correlation coefficients) are all random variables They depend on the particular values of the observations that you have sampled Take a different sample and you will get different estimates 4 Sampling Distributions Sampling Distributions Given that our statistical estimators are random variables, they fit into one of several sampling distributions. The most frequently encountered for our purposes are the standard nor- mal, the χ 2 ( chi-squared ) distribution, the F -distribution, and the t - distribution Before we begin our look at hypothesis testing we need to develop a solid understanding of sampling distributions. 2
The Standard Normal Distribution The normal distribution is the distribution that people think of when we talk about a “bell-shaped” curve The standard normal is a special case where the mean of the sampling distribution is zero and the variance (and standard deviation) are equal to one This tells us something about how likely it is to find a particular value of a random variable that is distributed as standard normal The Standard Normal Distribution The area under the standard normal curve is equal to 1 This is true for any sampling distribution This means that there is a 100% chance that the a randomly selected

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