1. The independent variables do not form a linearly dependent set--i.e. the explanatory variables are not perfectly correlated. 2. Homoscedasticity--the probability distributions of the error term have a constant variance for all values of the independent variables (X i 's). Perfect multicollinearity is a violation of assumption (1).Heteroscedasticity is a violation of assumption (2)

Multicollinearity is a problem with time series regression Suppose we wanted to estimate the following specification using quarterly time series data: Auto Sales t =  0 +  1 Income t +  2 Prices t where Income t is (nominal income in quarter t and Prices t is an index of auto prices in quarter t. The data reveal there is a strong (positive) correlation between nominal income and car prices

0 (Nominal) income Car prices Approximate linear relationship between explanatory variables

Why is multicollinearity a problem? In the case of perfectly collinear explanatory variables, OLS does not work. In the case where there is an approximate linear relationship among the explanatory variables (X i ’s), the estimates of the coefficients are still unbiased, but you run into the following problems:  The estimates of the coefficients have high standard errors, weakening the capacity of the equation to produce accurate forecasts.  The aforementioned problem means small t-ratios and greater likelihood that the null hypotheses will not be rejected.  Muticollinearity means that the effect of the independent variables is mingled together--this makes it difficult for the researcher to disentangle the separate effects of the explanatory variables on the dependent variable.  Estimates of the coefficients (  ’s) are “unstable,” meaning that a comparatively small change in the data set can produce a big change in the estimate of the coefficient.

How do you know you have a problem with multicollinearity? +Do the estimates have high standard errors? Are the t-ratios microscopic? +Does the correlation matrix reveal a high correlation between explanatory variables--say, 0.70 or higher?

What can be done about multicollinearity? Increase sample size Delete one or more explanatory variables form your specification

3Heteroscedasticity sometimes shows up when we do regression analysis using cross-sectional data. 3Consider the following model: is the deterministic part of the equation and e i is the error term. Recall that we assume that E(  ) = 0

JAR #1JAR #2  = 0 Two distributions with the same mean and different variances

X1X1 X2X2 X2X2 X Y 0 P(  ) The disturbance distributions of heteroscedasticity

Household Income Spending for electronics Scatter diagram of ascending heteroscedasticity

Why is heteroscedasticity a problem?  Heteroscedasticity does not give us biased estimates of the coefficients--however, it does make the standard errors of the estimates unreliable. That is, we will understate the standard errors.  Due to the aforementioned problem, t-tests cannot be trusted. We run the risk of rejecting a null hypothesis that should not be rejected.