SCHEDULE OF WEEK 10
• Project 2 is online, due by Monday, Dec 5 at 03:00 am
• 2. Discuss the DW test and how the statistic attains
less/greater that 2 values in cases of positive/negative autocorrelation and then go over HW18-Q5.
• 3. Discuss multi-collinearity. Mention the disagreement
between F and individual t-tests similar to slides. Then the
|r|>0.8 for detecting the independent variables pairs we might
find problematic. Also how to decide which variable we drop
to cure the problem. Then do HW19-Q3.
ASSUMPTIONS IN SIMPLE LINEAR
REGRESSION
Slide 43, Page 77 in course packet.
1. Error term 𝜀 is Normal, with a mean 0,
2. Error term 𝜀 is Normal, with constant variance.
3. Error term is independent. (Durbin-Watson Test)
4. No outliers
5. No serious multicollinearity
DURBIN-WATSON TEST
• When you suspect the independence of error is violated,
specifically, you think it is AR(1)
AR(1): 𝑒𝑡 = 𝜃𝑒𝑡−1 + 𝜀𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝜀𝑡 ~𝑁(0, 𝜎)
• H0: No A/C
• Ha: 1st order A/C exists
• Statistics: d =
𝑛
2
𝑖=2(𝑒𝑖 −𝑒𝑖−1 )
𝑛 𝑒 2
𝑖=1 𝑖
. This value is always btw 0
and 4
• Two-tail test for first order A/C
DURBIN-WATSON TEST
• When conducting the test with significance level a, you
will be given dL,α and dU,α
• If 0<d < dL,α, there is statistical evidence that the error
terms are positively autocorrelated.
• If 4-dL,α<d <4, there is statistical evidence that the error
terms are negative autocorrelated
• If dU,α<d <4- dU,α, there is no statistical evidence for first
autocorrelation
• If dL,α < d < dU,α or 4-dU,α < d < 4-dL,α , the test is
inconclusive.
DURBIN-WATSON TEST
• Hwk 18 Q3
• Suppose that you want to investigate inflation in the US for the past
five years and you run a regression of annual inflation on annual
GDP. You decide to check the results for autocorrelation. The
residuals from the regression are printed below.
•
•
•
•
•
•
Year
2000
2001
2002
2003
2004
Residual
2
3
1
-2
-4
DURBIN-WATSON TEST
𝑛
2
𝑖=2(𝑒𝑖 −𝑒𝑖−1 )
𝑛
2
𝑖=1 𝑒𝑖
• Compute d =
•
•
𝑛
2 (𝑒𝑖
2
− 𝑒𝑖−1 ) =(3-2)^2+(1-3)^2+(-2-1)^2+(-4-(2))^2=18
𝑛 2
1 𝑒𝑖
=2^2+3^2+1^2+(-2)^2+(-4)^2=34
• So, d=18/34=0.529412
DURBIN-WATSON TEST
• Hwk 18 Q5
• Look at the example starting on page 87 (slide 18)
of the course packet. You want to know how the
weather affects ticket sales at a ski resort. You have
data on tickets sold and total snowfall and average
temperature over Christmas week for 20
consecutive years.
• Run a multiple regression of ticket sales on snowfall
and temperature and answer the following
questions.
MULTI-COLLINEARITY
• Course packet slide 26, page 89
• Multi-collinearity exists when independent variables, included
in the same regression, are linearly correlated to one or
another.
• This problem is considered serious when absolute value of
corr. coef. between any pair of independent variables exceeds
0.8. i.e. |r|>0.8.
MULTI-COLLINEARITY
• HWK 19 Q1
• Multicollinearity can best be described as the condition in which
the: independent variables in a regression have a high degree of
correlation with one another.
• Multicollinearity is likely to be causing problems with your
regression results if:
• the Overall Model F-test indicates a valid model but the
individual t-tests indicate none of the independent variables are
linearly related to the dependent variable
• the independent variables have one or more correlation
coefficients above .80
MULTI-COLLINEARITY
• HWK 19 Q1
• Multicollinearity can result in:
• an inability to interpret the estimates of the slope
coefficients
• an increased standard error of the individual slope
coefficients
• an increased p-value for the individual t-tests
• a decrease in the absolute value of the individual t-test test
statistics
MULTI-COLLINEARITY
• HWK 19 Q3
• This question is related to the example on pages 98-100 (slides
63-69) of your course packet. A real estate agent is interested
in estimating the value of a piece of lake front property. He
believes that price is a function of Lot Size (1000s of square
feet), Number of Mature Trees on the Lot, and Distance to the
Lake (in yards). He has collected data on the basis of recent
sales, which is provided here.