1 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION We have seen that the variance of a random variable X is given by the expression above. Variance

2 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION Given a sample of n observations, the usual estimator of the variance is the sum of the squared deviations around the sample mean divided by n – 1, typically denoted s 2 X. Variance Estimator

3 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION Since the variance is the expected value of the squared deviation of X about its mean, it makes intuitive sense to use the average of the sample squared deviations as an estimator. But why divide by n – 1 rather than by n? Variance Estimator

4 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION The reason is that the sample mean is by definition in the middle of the sample, while the unknown population mean is not, except by coincidence. Variance Estimator

5 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION As a consequence, the sum of the squared deviations from the sample mean tends to be slightly smaller than the sum of the squared deviations from the population mean. Variance Estimator

6 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION Hence a simple average of the squared sample deviations is a downwards biased estimator of the variance. However, the bias can be shown to be a factor of (n – 1)/n. Thus one can allow for the bias by dividing the sum of the squared deviations by n – 1 instead of n. Variance Estimator

Variance 7 A similar adjustment has to be made when estimating a covariance. For two random variables X and Y an unbiased estimator of the covariance  XY is given by the sum of the products of the deviations around the sample means divided by n – 1. ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION Estimator Covariance

8 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION The population correlation coefficient  XY for two variables X and Y is defined to be their covariance divided by the square root of the product of their variances. Correlation

9 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION The sample correlation coefficient, r XY, is obtained from this by replacing the covariance and variances by their estimators. Correlation Estimator

10 The 1/(n – 1) terms in the numerator and the denominator cancel and one is left with a straightforward expression. ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION Correlation Estimator

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.7 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics