Welcome to Econ 420 Applied Regression Analysis

Study Guide

Week Twelve

Answer Key toAssignment 9 (30 points)

1. 11, Page 131No. The correlation coefficient r is not a slope from a line,like B is. It shows how STUDY and LIBRARY movetogether on a numerical scale, from –1 to 1. B is not onsuch a scale. B shows the movement in Y associated witha one-unit movement in X. If there is more than one independent variable, B is measured keeping the other independent variables constant. When r is calculated, none of the other independent variables present in the model are held constant.

Answer Key toAssignment 9 (30 points)

2. 13, page 132a.

– The correlation coefficient between INCOME and WEALTH is 0.82, which is high enough to indicate that there could be a multicollinearity problem, but it is not overwhelming evidence.

– Running a regression where INCOME is the dependent variable and WEALTH is the independent variable or vice versa gives an F-statistic (and a t-statistic) that is statistically significant at a 1% error level.

– The R2 is 0.67. This provides only mild support that there is a multicollinearity problem.

– The variance inflation factor is = 3. This indicates that multicollinearity is only a small problem, if it is a problem at all.

b. There is some evidence of multicollinearity, but it does not seem to be a big problem.

Answer Key toAssignment 9 (30 points)

2. 13, page 132 c. INCOME and WEALTH should be correlated to some extent, since

most people who have higher income will eventually have more wealth.

d. It might seem that the answers are contradictory, but they are not. It might seem that if you have INCOME and WEALTH in the same model, there should be multicollinearity, but in this particular data set, there is enough variation between INCOME and WEALTH that multicollinearity is not a big problem. There must be some people in the data set who have high income but have not accumulated as much wealth as you might expect. Perhaps there are others in the data set who have lower income but have more wealth than you would expect, because they are especially thrifty or they inherited wealth. As stated in the chapter, multicollinearity is a characteristic displayed by the data. This means that for any model, one sample could give results that exhibit multicollinearity, but a different sample might not.

Autocorrelation(Chapter 7- Up to Page 145)

• Suppose we are using time series data to estimate consumption (C) as a function of income (Y) and other factors

Ct = B1 + B2 Yt +…..+ et– Where t = (1, 2, 3, ….T)

– This means that • C1 = B1 + B2 Y1 +…. + e1, and• C2 = B1 + B2 Y2 +…. + e2

• …..• ……• CT = B1 + B2 YT+…. + eT ……

• One of the classical assumptions regarding the error terms is– No correlation among the error terms

• If this assumption is violated then autocorrelation becomes a problem.

First Order Autocorrelation

e2 = ρ e1 + u2

– That is, the error term in period 2 depends on the error term in period 1

– Where, u2 is a normally distributed error with mean of zero and constant variance

Second Order Autocorrelation

e3 = ρ1 e1 + ρ2 e2 + u3

– That is, the error term in period 3 depends on the error term in period 1 and the error term in period 2.

– Where, u3 is a normally distributed error with mean of zero and constant variance

Higher Order Autocorrelation

et = ρ1 et-1 + ρ2 et-2 + ρ3 et-3 + ….. + ut

– That is, the error term in period t depends on the error term in period t-1, the error term in period t-2, and the error term in period t-3,…etc.

– Where, ut is a normally distributed error with mean of zero and constant variance

Types of Autocorrelation

1. Positive• Errors form a pattern• A positive error is usually followed by another

positive error• A negative error is usually followed by another

negative error• More common

Example of positive autocorrelation

Types of Autocorrelation

2. Negative• A positive error is usually followed by a

negative error or visa-versa• Rare

Example of negative autocorrelation

Causes of Autocorrelation

• Wrong functional form

• Omitted variables

• Data error

• Lingering shock over time

Consequences of Autocorrelation

• Unbiased estimates but wrong standard errors– In case of positive autocorrelation standard

error of the estimated coefficients drops– Consequences?

Should we suspect Autocorrelation?

• If you are using time series data definitely• Easy to check

1. Run the regression

2. Plot residuals

3. If it looks like they are forming a pattern suspect autocorrelation

A Formal Test For First Order Autocorrelation

• Durbin-Watson test• Durbin Watson Stat. (d)

• It can be shown that d is approximately equal to 2 (1- ρ)• What is d under perfect positive correlation?

ρ = 1 d = 0• What is d under perfect negative correlation?

ρ = -1 d = 4• What is d under no autocorrelation?

ρ = 0 d = 2• What is the range of values for d?

0 to 4

EViews calculates d statistics • If d >2, you will to test for negative autocorrelation.• Null and alternative hypotheses

– H0: ρ≥0– HA: ρ<0

• Choose the level of significance (1% or 5%)• Critical dstat (page 320-323)• Decision rule

– If d>4-dL reject H0 there is significant negative first order autocorrelation

– If d< 4-dU don’t reject H0 there is no evidence of a significant autocorrelation

– if d is between 4 – dL and 4 – du the test is inconclusive.

• If d <2, test for positive autocorrelation.• Null and alternative hypotheses

– H0: ρ≤0– HA: ρ>0

• Choose the level of significance (1% or 5%)• Critical dstat (page 320-323)• Decision rule

– If d< dL reject H0 there is significant positive first order autocorrelation

– If d> dU don’t reject H0 there is no evidence of a significant autocorrelation

– if d is between dL and du the test is inconclusive.

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Assignment 10 (30 points)Due: Before 10 PM, Friday, November 16

• #5 and #6 and 9 page 156