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Marietta College

Spring 2011

Econ 420: Applied Regression Analysis

Dr. Jacqueline Khorassani

Week 13

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Tuesday, April 5

Exam 3: Monday, April 25, 12- 2:30PM

Leadership Q&A

David LeonhardtEconomics JournalistWashington BureauThe New York Times

TONIGHT 7:30pm

McDonough Gallery

Cosponsored by McDonough Center for Leadership & Business and the Economic Roundtable of the Ohio Valley

Is anyone interested in going to breakfast with him tomorrow 8

am, Lafayette Hotel

This is the last bonus opportunity of this semester

• 2 points for attending• 2-5 points per question• 2-10 points per summary

• Summaries are due before 5 pm on Friday, April 8 via an email attachment to me

• Total bonus points will be divided by 3 and added to your exams.

Return and discuss Asst 18# 12 Page 240a) • The estimated coefficients all are in the expected

direction• R2 bar seems fairly low.• Always check the significance at 10 percent or better• Coefficients of A , A2 and S are significant. • You can only interpret the magnitude of coefficients if

they pass the t-test of significance• Significance has to do with t-test• Importance has to do with the absolute value of

coefficient. 5

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b)

• It implies that wages rise at a declining rate with respect to age and eventually fall.

• Does not imply perfect collinearity (non-linear correlation).c)• Semilog (Ln W) is a possibility• The slope coefficient represents the percentage change in wage

caused by a one-unit increase in the independent variable (holding constant all the other independent variables).

• Since pay raises are often discussed in percentage terms, this makes sense.

• Phil & Yuan say, but what about the meaning of coefficient of A2 in a semi log function? (great point)

• Linda says, it depends on the purpose of the study (great point)

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d)• It’s a good habit to ignore (except to make sure that

one exists) even if it looks too large or too small. • Intercept picks up the mean of the error term & and

that is affected by omitted variables. e)• The poor fit and the insignificant estimated

coefficient of union membership are all reasons for being extremely cautious about using this regression to draw any conclusions about union membership.

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• Collect Asst 19# 5 Page 234– Including Part e (data is available online under

STOCK in Chapter 7)

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Imperfect Multicollinearity Problem

• What is it?• Let’s say you estimate an regression equation,

what makes you suspicious about possible mulit problem?

• What are the two formal tests we talked about before?

• Example• Income = f (wage rate, tax rate, hours of

work, ….)• Wage rate, tax rate and hours of work may

be all highly correlated with each other• Problem: simple correlation coefficient

may not capture this.

Sometimes 3 or more independent variables are correlated

• Regress each independent variable (say X1) on the other independent variables (X2, X3, X4)

• Then calculate VIF• VIF = 1 / (1- R2)• If VIF > 5 then X1 is highly correlated with the other

independent variables • Do the same for all of the other independent

variables

Test of Multicollinearity among 3 or more independent variables

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Asst 20• Data set: DRUGS (Chapter 5, P 157)• Estimate Equation 5.101. Before you run any formal tests, do you suspect an

imperfect mulitcollinearity problem? Why or why not?2. Examine the absolute values of the correlation coefficients

between the independent variables included in Equation 5.10. Do you find any evidence of muliticollinerity problem? Discuss.

3. Examine the VIF of the two most suspicious independent variables in Equation 5.1 based on what your found in Section 2 above. Do you find any evidence of muliticollinerity problem? Discuss

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Thursday, April 7• Exam 3: Monday, April 25, 12- 2:30PM• If you asked David Leonhardt questions on

Tuesday night, write it down and give it to me today.

• Summaries are due before 5 pm tomorrow via an email attachment.

• Bring laptops to class on Tuesday

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• Return and discuss Asst 19# 5 Page 234– Including Part e (data is available online under

STOCK in Chapter 7)

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(a)• You are correct but note that the null and

alternative hypotheses are not about beta hats, they are about betas.

(b)• It’s unusual to have a lagged variable in a cross-

sectional model.• BETA is lagged.

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Part C• Should we include EARN in the set of our

independent variables 1. Does the theory call for its inclusion? Yes, but a version

of it is in dependent variable exclude EARN2. Is the estimated coefficient of EARN significant in the

right direction? No exclude EARN3. As you include EARN, does the adjusted R squared goes

up? Yes include EARN4. As you include EARN, do the other variables’ coefficients

change significantly? Change somewhat! 5. As you include EARN, do AIC and SC go down? No

exclude EARN

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(d)• The functional form is a semilog left, which is

appropriate both on a theoretical basis and also because two of the independent variables are expressed as percentages.

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(e)• EARN, DIV , and Beta all can be negative, can’t

take their log.

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Return and discuss Asst 20• Data set: DRUGS (Chapter 5, P 157)• Estimate Equation 5.101. Before you run any formal tests, do you suspect an

imperfect mulitcollinearity problem? Why or why not?2. Examine the absolute values of the correlation coefficients

between the independent variables included in Equation 5.10. Do you find any evidence of muliticollinerity problem? Discuss.

3. Examine the VIF of the two most suspicious independent variables in Equation 5.1 based on what your found in Section 2 above. Do you find any evidence of muliticollinerity problem? Discuss

Dependent Variable: PMethod: Least SquaresSample: 1 32Included observations: 32

Variable Coefficient Std. Error t-Statistic Prob.

C 38.22131 6.387304 5.983951 0.0000GDPN 1.433680 0.214395 6.687108 0.0000CVN -0.594732 0.223947 -2.655679 0.0133PP 7.311330 6.123084 1.194060 0.2432DPC -15.62864 6.932635 -2.254359 0.0328IPC -11.38456 7.159258 -1.590187 0.1239

R-squared 0.811223 Adjusted R-squared 0.774920

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Do we suspect multicollinearity problem?What should we look for?R bar squared is high but we have two insignificant variables

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Correlation Matrix

GDPN CVN PP DPC IPCGDPN 1 0.86 0.21 0.17 -0.05CVN 1 0.12 0.31 0.06PP 1 -0.13 -0.21DPC 1 0.38IPC 1

Why are the diagonal values all 1?Why did I eliminate the values in bottom half of the table? Is multicollinearity a problem?

Dependent Variable: GDPNMethod: Least SquaresSample: 1 32Included observations: 32

Variable Coefficient Std. Error t-Statistic Prob. C 12.68648 5.187712 2.445486 0.0213CVN 0.901037 0.101695 8.860167 0.0000PP4.343201 5.432426 0.799496 0.4310DPC -4.112512 6.172508 -0.666263 0.5109IPC -3.883886 6.382854 -0.608487 0.5479

R-squared 0.766913

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VIF = 1/ (1-0.77)VIF = 4.34VIF<5 no serious multicollinearity problem

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Dependent Variable: CVNMethod: Least SquaresSample: 1 32Included observations: 32

Variable Coefficient Std. Error t-Statistic Prob. C -4.639597 5.415845 -0.856671 0.3992DPC 8.402889 5.733910 1.465473 0.1543GDPN 0.825806 0.093204 8.860167 0.0000IPC 2.662434 6.130964 0.434260 0.6676PP -1.333922 5.255631 -0.253808 0.8016R-squared 0.775957

VIF = 1/ (1-0.78)VIF = 4.54VIF<5 no serious multicollinearity problem

1. If your main goal is to use the equation for forecasting and you don’t want to do specific t- test on each estimated coefficient then do nothing.

◦ This is because multicollinearity does not affect the predictive power of your equation.

2. If it seems that you have a redundant variable, drop it.

◦ Examples◦ You don’t need both real and nominal interest rates in your

model◦ You don’t need both nominal and real GDP in your model

Remedies for Multicollinearity

3. If all variables need to stay in the equation, transform the multicollinear variables Example: Number of domestic cars sold = B0 + B1 average

price of domestic cars + B2 average price of foreign cars +…..+ є

Problems: Prices of domestic and foreign cars are highly correlated

Solution: Number of domestic cars sold = B0 + B1 the ratio of

average price of domestic cars to the average price of foreign cars +…..+ є

4. Increase the sample size or choose a different random sample

Remedies for Multicollinearity

Asst 21: Due Tuesday in classUse the data set FISH in Chapter 8 (P 274) torun the following regression equation:F = f (PF, PB, Yd, P, N)1) Conduct all 3 tests of imperfect

multicollinearity problem and report your results.

2) If you find an evidence for imperfect multicollinearity problem, suggest and implement a reasonable solution.

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Chapter 9 (Autocorrelation or Serial Correlation)

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

Ct = β1 + β2 Yt +…..+ єt– Where t = (1, 2, 3, ….T)

– This means that • C1 = β1 + β2 Y1 +…. + є1, and• C2 = β1 + β2 Y2 +…. + є2• …..• ……• CT = β1 + β2 YT+…. + єT ……

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

equation• If this assumption is violated then there is a problem of

pure serial correlation (autocorrelation).

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First Order Pure Autocorrelation

є2 = ρ є1 + 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 the mean of zero and constant variance

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Second Order Pure Autocorrelation

є3 = ρ1 є1 + ρ2 є2 + 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 the mean of zero and constant variance

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Higher Order Pure Autocorrelation

єt = ρ1 єt-1 + ρ2 єt-2 + ρ3 єt-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 the mean of zero and constant variance

• When the true (theoretical) regression line does not have an autocorrelation problem but our estimated equation does.

• Why?1. Specification error2. Wrong functional form3. Data error

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What is the Impure Serial Correlation?

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Types of Serial Correlation1. 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

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Example of positive autocorrelation

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Types of Serial Correlation

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

error or visa-versa• Less common

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Example of negative autocorrelation

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EViews allows you to see the residuals’ graph

• After you estimate the regression equation• Click on View on your regression output• Click on Actual, Fitted, Residual Table

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• Actual Fitted Residual Residual Plot• 23.5200 27.0183 -3.49829 |* . | . |• 25.9500 28.4120 -2.46201 | *. | . |• 25.9400 28.6804 -2.74042 | * . | . |• 27.2200 28.8564 -1.63643 | .* | . |• 27.8200 29.5686 -1.74857 | .* | . |• 29.7700 30.3005 -0.53051 | . * | . |• 32.0800 30.4455 1.63447 | . | *. |• 32.6200 31.9419 0.67814 | . | * . |• 32.8800 32.3974 0.48264 | . | * . |• 34.9000 32.2590 2.64105 | . | . * |• 36.8800 33.3502 3.52980 | . | . *|• 36.7400 35.0438 1.69615 | . | *. |• 38.4900 35.3550 3.13501 | . | . * |• 37.0100 33.6742 3.33580 | . | . * |• 36.9300 37.2539 -0.32386 | . *| . |• 36.7000 38.0231 -1.32312 | . * | . |• 39.8400 37.4913 2.34866 | . | .* |• 40.7100 39.4460 1.26398 | . | * . |• 43.1000 42.2249 0.87505 | . | * . |• 46.6400 44.7560 1.88398 | . | *. |• 46.9100 48.2692 -1.35924 | . * | . |• 48.4500 50.1672 -1.71724 | .* | . |• ………

What type of serial correlation may we have?

Negative residuals seem to be followed by other negative residuals suspect positive autocorrelation