Assumptions of Regression Analysis

1
Assumptions of Regression Analysis

• The independent variables do not form a linearly
  dependent set--i.e. the explanatory variables are
  not perfectly correlated.
• Homoscedasticity--the probability distributions
  of the error term have a constant variance for
  all values of the independent variables (Xi's).

2
Perfect multicollinearity is a violation of
assumption (1).Heteroscedasticity is a violation
of assumption (2)
3
Multicollinearity is a problem with time series
regression
Suppose we wanted to estimate the following
specification using quarterly time series
data Auto Salest ?0 ?1Incomet
?2Pricest where Incomet is (nominal) income in
quarter t and Pricest is an index of auto prices
in quarter t.
The data reveal there is a strong(positive)
correlation betweennominal income and car prices
4
Approximate linear relationship between
explanatory variables
Car prices
0
(Nominal) income
5
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
  (Xis), the estimates of the coefficients are
  still unbiased, but you run into the following
  problems
• High standard errors of the estimates of the
  coefficientsthus low t-ratios
• Co-mingling of the effects of explanatory
  variables.
• Estimates of the coefficients tends to be
  unstable.

6
What do about multicollinearity

• Increase sample size
• Delete one or more explanatory variables

7
Understanding heteroscedasticity
This problem pops up when using cross sectional
data
8
Consider the following model
Yi is the determined part of the equation and
ei is the error term. Remember we assume in
regression that E(ei) 0
9
JAR 1
JAR 2
4
400
-4
-400
0
0
-2
200
-200
2
? 0
? 0
Two distributions with the same mean and
different variances
10
The disturbance distributions of
heteroscedasticity
f(x)
Y
X1
X2
X2
0
X
11
Scatter diagram of ascending heteroscedasticity
Spending for electronics
Household Income
12
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.