92 The Basics of financial economeTrics
estimated. Instead, nonlinear regression estimation techniques require the
use of optimization techniques to identify the parameters that best fit the
model. Researchers in disciplines such as biology and physics often have to
deal with nonlinear regressions.
Assumed Statistical Properties about the Error Term
The third assumption about the general linear model concerns the three
assumptions about the error term that we listed in Chapter 3 and repeat
below:
Assumption 1. The regression errors are normally distributed with zero
mean.
Assumption 2. The variance of the regression errors ()σε^2 is constant.
Assumption 3. The error terms from different points in time are inde-
pendent such that εt are independent variables for all t.Assumption 1 states that the probability distribution for the error term
is that it is normally distributed. Assumption 2 says that the variance of the
probability distribution for the error term does not depend on the level of
any of the independent variables. That is, the variance of the error term is
constant regardless of the level of the independent variable. If this assump-
tion holds, the error terms are said to be homoscedastic (also spelled homo-
skedastic). If this assumption is violated, the variance of the error term is said
to be heteroscedastic (also spelled heteroskedastic). Assumption 3 says that
there should not be any statistically significant correlation between adjacent
residuals. The correlation between error terms is referred to as autocorrela-
tion. Recall that we also assume that the residuals are uncorrelated with the
independent variables.
Tests for the Residuals Being Normally Distributed
An assumption of the general linear regression model is that the residuals
are normally distributed. The implications of the violation of this assump-
tion are:
■ (^) The regression model is misspecified.
■ (^) The estimates of the regression coefficients are also not normally dis-
tributed.
■ (^) The estimates of the regression coefficients, although still the best linear
unbiased estimators, are no longer efficient estimators.