An Introduction to Econometrics by A. A. Walters (auth.)

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It is out-in-the-cold but not beyond-the-pale. Unfortunately the language of hypothesis testing gives us no way to describe how 'cold' or 'hot' a hypothesis is relative to its rivals. We obviously need a language of this kind - and, as we shall see, the concept of 'likelihood' provides a suitable way of attaching 'odds' of one hypothesis against another. The likelihood approach is a development of the method of estimating an unknown population parameter. It shows that, as a general proposition, it is much more efficient to estimate, using the language of likelihood, rather than to test hypotheses.

E. 21) STATISTICAL INFERENCE 49 estimate of the variance where we divide the sum of squares within classes by (n - k). We now form the ratio F = estimate of variance between classes estimate of variance within classes or in algebra: F~o. 22) Now suppose we have a population where class does not matter; and let us take one sample after another from this population. For each sample we can calculate the value F, and so one can imagine the sampling frequency distribution ofF being formed. One can discover the chances of getting a value of F in excess of any particular value; thus one can measure the probability that, from a population where class does not matter, we can get a value ofF as high or higher than a given value.

S) The estimate of the population variance is obtained by finding the sum of the squares of deviations from the sample mean and then dividing by (n - I) instead of the usual n. This has the effect of bringing the estimated variance, on the average, nearer to the true variance. The qualification 'on the average' is important; in any particular sample it may well be that the variance of that sample (obtained by dividing by n) exceeds the true variance. The vagaries of sampling makes this occurrence not a particularly rare event.

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