INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 101 Figure 1.1 is a depiction of homoscedasticity. Fora simple illustration, the sample in Figure 1.1 contains only five observations. Let us start with the first observation, where X has the value X 1 . If there were no disturbance term in the model, the observation would be represented by the circle vertically above X 1 on the line
The effect of the disturbance term is to shift the observation upwards or downwards vertically. The potentialdistribution of the disturbance term, before the observation has been generated, is shown by the normal distribution centered on the circle. The actual value of the disturbance term for this observation turned out to be negative, the observation being represented by the darkened indicator. The potential distribution of the disturbance term, and the actual outcome, are shown in a similar way for the other four observations. Although homoscedasticity is often taken for granted in regression analysis, in some contexts it maybe more reasonable to suppose that the potential distribution of the disturbance term is different for different observations in the sample. This is illustrated in Figure 1.2 where the variance of the potential distribution of the disturbance term is increasing as X increases. This does not mean that thedisturbance term will necessarilyhave a particularly large (positive or negative) value in anobservation where X is large, but it does mean that the a priori probabilityof having an erratic value will be relatively high. This is an example of heteroscedasticity, which means "differing dispersion. Mathematically, homoscedasticity and heteroscedasticity maybe defined Homoscedasticity:
