Lecture 3 - Introduction to Econometrics Lecture 3...

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Introduction to Econometrics 3-1 Technical Notes Introduction to Econometrics Lecture 3: Hypothesis tests: bivariate probability distributions and simple regression Hypothesis tests Type I error - size Suppose we conduct a series of replications of a statistical experiment (for example, the estimation of the male wage premium considered in Lecture 2) and construct the test statistic t = coefficient/standard deviation for each replication. Then t is a random variable and has a sampling distribution: our procedure gives 10000 observations on this distribution 1 . We now look at the outcome of a t-test for a mean of 1.2777 (the ‘population’ value) in the wage experiment, and specifically at the number of times this hypothesis is rejected. This is equivalent to recording an absolute value of t $ (= {coefficient – 1.2777}/standard deviation) which is greater than the critical value. If we chose a 5% significance level, then for the samples of 132 observations this gives a critical value of 1.9784. Now we count the number of experiments for which abs( t $ ) > 1.9784: we find that this is true for 456 out of 10000 experiments, or 4.56% of the time. This gives an empirical measure of the probability of what is called ‘Type I error’, ie of rejecting the null (a mean of 1.2777) when it is true (in this case, as a result of the design of the experiment). This probability, which is a function of the investigator’s choice of a 5% significance level, is called the actual size of the test: the significance level (5% in this case) is also called the nominal size. The agreement between actual and theoretical values is fairly close for our experiment, but not perfect because of sampling error. Type II error – power We make a ‘Type II error’ when we accept the null, even though the alternative is true. For example, we might accept that the hypothesis that the mean wage premium for males was zero, even though the true value in the ‘population’ is 1.2777. The probability of a Type II error depends on the alternative considered, so it cannot be controlled directly by the investigator. An obvious procedure is to chose a test which minimises the probability of Type II error. This is more usually expressed by saying that we wish to maximise the power of the test, where power is defined by Power = 1 – Prob(Type II error ) The relationship between the different types of error, and the size and power of a test, can be set out in a simple table. The bold entries describe the type of error being made: the italic entries give the statistical terms for the associated probabilities. Accept H 0 Reject H 0 H 0 is true Correct decision Type I error Probability = size H 0 is false Type II error Correct decision Probability = power This table makes it clear that both size and power are related to the probability that a test will reject H 0 . We want this probability to be as small as possible if H 0 is true, but as large as possible if H 0 is false.
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