a) Population Regression Function vs Sample Regression Function. The Population Regression Function (PRF) is a description of the model that is thought to be generating the actual data and it represents the true relationship between the variables. The PRF embodies the true values of i?? 1 and i?? 2, and is expressed as: where Yi is the actual value obtained by adding the error term ui; or as where E(Y) may be regarded as the average or expected value of Y for a given value of X.
The PRF tells us how the mean of Y varies for different values of X. The population is the total collection of all objects to be studied. The population may be either finite or infinite, while a sample is a selection of just some items from the population. In general either all of the observations for the entire population will not be available, or they may be so many in number that it is infeasible to work with them, in which case a sample of data is taken for analysis.
(Brooks, 2002, pg. 112) The Sample Regression Function (SRF) It allows us to calculate the estimated value of Y for a given value of X. b) Error terms vs residuals. The error term – is the disturbance term of the PRF. It represents factors other than X that affect Y and is calculated as the difference between the actual data point and the estimated value of Y from the PRF. The residual is the difference between the actual value of Y and its fitted value:
In most cases we cannot calculate the parameters for PRF, therefore we cannot calculate the error term, which is why the residual is used more often in empirical studies. c) Regression coefficients vs estimators. An estimator is also known as a statistic, is simply a rule or formula or method that tells how to estimate the population parameter from the information provided by the sample at hand for example is an estimator of , is an estimator of(Gujarati, 2003, pg. 49) ,,andare regression coefficients; also know as intercept and slope coefficients. Estimators are like proxies of the real values, as opposed to coefficients which are the parameters of an equation.
BROOKS, Chris, 2002. Introductory econometrics for finance. New York: Cambridge University Press. GUJARATI, Damodar N. , 2003. Basic Econometrics. 4th ed. New York: McGraw-Hill. KOOP, Gary, . Analysis of Economic Data. Chichester, England: John Wiley & Sons. WOOLDRIDGE, Jeffrey M. , 2006. Introductory Econometrics. 3rd ed. Manson, USA: Thomson South-Western.