Are there situations in economics where a biased estimator is used for the estimation of some model?
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PROMPT:
Look up W.S. Gossett and read the story of his developing Student’s distribution. (Have a Guinness while reading about him if you’d like).
Are there situations in economics where a biased estimator is used for the estimation of some model? Please describe.
Lesson
This module focuses on how a statistic calculated from a sample is distributed. For example, if we know we have an iid sample from a Gaussian distribution, then the distribution of the sample mean is relatively straightforward because it is a linear combination of Gaussian distributed random variables (that is, each observation) and so the distribution of the mean would also be Gaussian with its mean that of the population from which the sample came. But what is the distribution of the sample variance? The distribution of the square of a statistic isn’t merely its distribution squared. It turns out that the distribution of the sample variance is chi-squared with degrees of freedom equal to the number of observations in the sample.
A special property of the Gaussian distribution is the statistical independence of the sample mean and sample variance. That property is unique to the Gaussian distribution. That’s a property that is often unknowingly exploited by researchers.
We don’t usually know the population mean and variance when we’re doing our data analysis. When drawing a sample from a Gaussian population with unknown mean and variance, the standardized, or Z-score, versions of the observations, that is, each of the observations has had the sample mean subtracted from it and that result is then divided by the sample variance, follow a Student’s t distribution. This distribution has an illustrious past. It was discovered by W.S. Gossett, who was employed for quality control by Guinness, the famous Irish brewery. He wrote a paper deriving it which was published in Biometrika, a British statistics journal. Because his employer didn’t want his name to be known, nor that they were using Gossett’s test for quality control, he used a pen name instead and signed the paper Student. (https://en.wikipedia.org/wiki/William_Sealy_Gosset (Links to an external site.))
We then discuss the very thorny issue of confidence intervals. This is one of the least understood and often frequently abused statistical concepts. We will spend some time trying to make sure that you understand the use of confidence intervals well enough not to get into trouble once you’ve left this class. (No guarantees, though).
Finally, we will discuss the concept of an unbiased estimator. Unbiasedness in estimation is a positive property for estimators, but there are situations where the mean-squared error of a biased estimator is smaller than that of an unbiased estimator. Also, we’ll see that in some situations that for some biased estimators, as the size of your sample grows, the bias gets smaller and the biased estimator converges to the unbiased estimator. An example of this is the sample variance calculated using the sample size in the denominator versus using one less than the sample size in the denominator in calculating the sample variance. The first sample variance is biased by the additive factor 1/n, while the second sample variance is unbiased, but the biased sample variance converges to the unbiased variance as the sample size increases.
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