Building and Testing a Multiple Linear Regression Model 101
of the independent variables used in the regression model (e.g., the prob-
lem of multicollinearity), (2) the linearity of the model, and (3) whether
the assumptions about the statistical properties of the error term are
warranted.■ (^) Visual inspection of a scatter diagram of each independent variable and
the regression residuals is a common approach for checking for linearity.
The presence of some systematic behavior in the residuals that depends
on the values of the independent variables might suggest that the rela-
tionship between the independent variable investigated and dependent
variable is not linear.
■ (^) The problem of a nonlinear functional form can be dealt with by trans-
forming the independent variables or making some other adjustment to
the variables.
■ (^) Testing for the assumptions about the error terms involves testing if
(1) they are normally distribution with zero mean, (2) the variance is
constant, and (3) they are independent.
■ (^) The implications of the violation of the normality assumption of the
error terms are threefold: (1) the regression model is misspecified, (2) the
estimates of the regression coefficients are not normally distributed, and
(3) although still best linear unbiased estimators, the estimates of the
regression coefficients are no longer efficient estimators.
■ (^) Three methodologies used to test for normality of the error terms are
the chi-square statistic, the Jarque-Bera test statistic, and analysis of
standardized residuals.
■ (^) In a linear regression model the variance of all squared error terms is
assumed to be constant, an assumption referred to as homoscedasticity.
■ (^) When the homoscedasticity assumption is violated, the variance of the
error terms is said to exhibit heteroscedasticity. Many time series data
exhibit heteroscedasticity, where the error terms may be expected to be
larger for some observations or periods of the data than for others.
■ (^) There are several tests that have been used to detect for the presence of
heteroscedasticity and there are several methodologies for constructing
models to accommodate this feature. Two of the most common meth-
odologies are the weighted least squares estimation technique and the
autoregressive conditional heteroscedasticity model (ARCH).
■ (^) The multiple linear regression model assumes that there is no statistically
significant correlation between adjacent residuals. This means that there
is no statistically significant autocorrelation between residuals.
■ (^) A time series is said to be autocorrelated if each term is correlated with
its predecessor so that the variance of each term is partially explained by
regressing each term on its predecessor. In time series analysis, this means
no significant autocorrelation between two consecutive time periods.