Lecture 18 1

Econ 140Econ 140

Multiple Regression Applications III

Lecture 18

Lecture 18 2

Econ 140Econ 140Dummy variables

• Include qualitative indicators into the regression: e.g. gender, race, regime shifts.

• So far, have only seen the change in the intercept for the regression line.

• Suppose now we wish to investigate if the slope changes as well as the intercept.

• This can be written as a general equation:

Wi = a + b1Agei + b2Marriedi + b3Di + b4(Di*Agei) + b5(Di*Marriedi) + ei

• Suppose first we wish to test for the difference between males and females.

Lecture 18 3

Econ 140Econ 140Interactive terms

• For females and males separately, the model would be:

Wi = a + b1Agei + b2Marriedi + e

– in so doing we argue that would be different for males and females– we want to think about two sub-sample groups: males and females– we can test the hypothesis that the intercept and partial slope coefficients will be different for these 2 groups

abb ˆ and ˆ,ˆ21

Lecture 18 4

Econ 140Econ 140Interactive terms (2)

• To test our hypothesis we’ll estimate the regression equation above (Wi = a + b1Agei + b2Marriedi + e) for the whole sample and then for the two sub-sample groups

• We test to see if our estimated coefficients are the same between males and females

• Our null hypothesis is:

H0 : aM, b1M, b2M = aF, b1F, b2F

Lecture 18 5

Econ 140Econ 140Interactive terms (3)

• We have an unrestricted form and a restricted form– unrestricted: used when we estimate for the sub-sample groups separately– restricted: used when we estimate for the whole sample

• What type of statistic will we use to carry out this test?– F-statistic:

knknSSR

qSSRSSRF

U

UR

21

q = k, the number of parameters in the model

n = n1 + n2 where n is complete sample size

Lecture 18 6

Econ 140Econ 140Interactive terms (4)

• The sum of squared residuals for the unrestricted form will be:

SSRU = SSRM + SSRF

• L17_2.xls

– the data is sorted according to the dummy variable “female”

– there is a second dummy variable for marital status

– there are 3 estimated regression equations, one each for the total sample, male sub-sample, and female sub-sample

Lecture 18 7

Econ 140Econ 140Interactive terms (5)

• The output allows us to gather the necessary sum of squared residuals and sample sizes to construct the test statistic:

626.2466.0

224.1

633093.5495.7

3093.5495.7261.1621

knknSSR

qSSRSSRF

U

UR

– Since F0.05,3, 27 = 2.96 > F* we cannot reject the null hypothesis that the partial slope coefficients are the same for males and females

Lecture 18 8

Econ 140Econ 140Interactive terms (6)

• What if F* > F0.05,3, 27 ? How to read the results?

– There’s a difference between the two sub-samples and therefore we should estimate the wage equations separately

– Or we could interact the dummy variables with the other variables

• To interact the dummy variables with the age and marital status variables, we multiply the dummy variable by the age and marital status variables to get:

Wt = a + b1Agei + b2Marriedi + b3Di + b4(Di*Agei) + b5(Di*Marriedi) + ei

Irene O. Wong:Irene O. Wong:Irene O. Wong:Irene O. Wong:

Lecture 18 9

Econ 140Econ 140Interactive terms (7)

• Using L17_2.xls you can construct the interactive terms by multiplying the FEMALE column by the AGE and MARRIED columns

– one way to see if the two sub-samples are different, look at the t-ratios on the interactive terms

– in this example, neither of the t-ratios are statistically significant so we can’t reject the null hypothesis

Lecture 18 10

Econ 140Econ 140Interactive terms (8)

• If we want to estimate the equation for the first sub-sample (males) we take the expectation of the wage equation where the dummy variable for female takes the value of zero:

E(Wt|Di = 0) = a + b1Agei + b2Marriedi

• We can do the same for the second sub-sample (Females)

E(Wt|Di = 1) = (a + b3) + (b1 + b4)Agei + (b2 + b3) Marriedi

• We can see that by using only one regression equation, we have allowed the intercept and partial slope coefficients to vary by sub-sample

Lecture 18 11

Econ 140Econ 140Phillips Curve example

• Phillips curve as an example of a regime shift.

• Data points from 1950 - 1970: There is a downward sloping, reciprocal relationship between wage inflation and unemployment

W

UN

Lecture 18 12

Econ 140Econ 140Phillips Curve example (2)

• But if we look at data points from 1971 - 1996:

• From the data we can detect an upward sloping relationship

• ALWAYS graph the data between the 2 main variables of interest

W

UN

Lecture 18 13

Econ 140Econ 140Phillips Curve example (3)

• There seems to be a regime shift between the two periods

– note: this is an arbitrary choice of regime shift - it was not dictated by a specific change

• We will use the Chow Test (F-test) to test for this regime shift

– the test will use a restricted form:

– it will also use an unrestricted form:

– D is the dummy variable for the regime shift, equal to 0 for 1950-1970 and 1 for 1971-1996

Nt U

baW1

NNt U

DbU

bDbaW11

321

Lecture 18 14

Econ 140Econ 140Phillips Curve example (4)

• L17_3.xls estimates the restricted regression equations and calculates the F-statistic for the Chow Test:

• The null hypothesis will be:

H0 : b1 = b3 = 0

– we are testing to see if the dummy variable for the regime shift alters the intercept or the slope coefficient

• The F-statistic is (* indicates restricted)

Where q=2 kne

qeeF

2

2*2

ˆ

ˆˆ

Lecture 18 15

Econ 140Econ 140Phillips Curve example (5)

• The expectation of wage inflation for the first time period:

• The expectation of wage inflation for the second time period:

• You can use the spreadsheet data to carry out these calculations

NUbaDWE

1)0|(

NU

bbbaDWE1

)1|( 321

Lecture 18 16

Econ 140Econ 140

Relaxing Assumptions

Lecture 18

Lecture 18 17

Econ 140Econ 140Today’s Plan

• A review of what we have learned in regression so far and a look forward to what we will happen when we relax assumptions around the regression line

• Introduction to new concepts:

– Heteroskedasticity

– Serial correlation (also known as autocorrelation)

– Non-independence of independent variables

Lecture 18 18

Econ 140Econ 140CLRM Revision

• Calculating the linear regression model (using OLS)

• Use of the sum of square residuals: calculate the variance for the regression line and the mean squared deviation

• Hypothesis tests: t-tests, F-tests, 2 test.

• Coefficient of determination (R2) and the adjustment.

• Modeling: use of log-linear, logs, reciprocal.

• Relationship between F and R2

• Imposing linear restrictions: e.g. H0: b2 = b3 = 0 (q = 2); H0: + = 1.

• Dummy variables and interactions; Chow test.

Lecture 18 19

Econ 140Econ 140Relaxing assumptions

• What are the assumptions we have used throughout?• Two assumptions about the population for the bi-variate case:

1. E(Y|X) = a + bX (the conditional expectation function is linear); 2. V(Y|X) = (conditional variances are constant)

• Assumptions concerning the sampling procedure (i= 1..n) 1. Values of Xi (not all equal) are prespecified; 2. Yi is drawn from the subpopulation having X = Xi; 3. Yi ‘s are independent.

• Consequences are: 1. E(Yi) = a + bXi; 2. V(Yi) = 2; 3. C(Yh, Yi) = 0

– How can we test to see if these assumptions don’t hold?– What can we do if the assumptions don’t hold?

Lecture 18 20

Econ 140Econ 140Homoskedasticity

• We would like our estimates to be BLUE• We need to look out for three potential violations of the CLRM assumptions: heteroskedasticity, autocorrelation, and non-independence of X (or simultaneity

bias).• Heteroskedasticity: usually found in cross-section data (and longitudinal)• In earlier lectures, we saw that the variance of is

b̂

2

2

)ˆ(x

bV

– This is an example of homoskedasticity, where the variance is constant

Lecture 18 21

Econ 140Econ 140Homoskedasticity (2)

• Homoskedasticity can be illustrated like this:

constantvariance aroundthe regression line

Y

XX1 X2 X3

Lecture 18 22

Econ 140Econ 140Heteroskedasticity

• But, we don’t always have constant variance 2

– We may have a variance that varies with each observation, or

• When there is heteroskedasticty, the variance around the regression line varies with the values of X2i

Lecture 18 23

Econ 140Econ 140Heteroskedasticity (2)

• The non-constant variance around the regression line can be drawn like this:

XX1 X2 X3

Y

Lecture 18 24

Econ 140Econ 140Serial (auto) correlation

• Serial correlation can be found in time series data (and longitudinal data)

• Under serial correlation, we have covariance terms

– where Yi and Yh are correlated or each Yi is not independently drawn

– This results in nonzero covariance terms

h i

hiihi cccbV 2)(

Lecture 18 25

Econ 140Econ 140Serial (auto) correlation (2)

• Example: We can think of this using time series data such that unemployment at time t is related to unemployment in the previous time period t-1

• If we have a model with unemployment as the dependent variable Yt then

– Yt and Yt-1 are related

– et and et-1 are also related

Lecture 18 26

Econ 140Econ 140Non-independence

• The non-independence of independent variables is the third violation of the ordinary least squares assumptions

• Remember from the OLS derivation that we minimized the sum of the squared residuals

– we needed independence between the X variable and the error term

– if not, the values of X are not pre-specified

– without independence, the estimates are biased

0, given that 2 eXbge

Lecture 18 27

Econ 140Econ 140Summary

• Heteroskedasticity and serial correlation

– make the estimates inefficient

– therefore makes the estimated standard errors incorrect

• Non-independence of independent variables

– makes estimates biased

– instrumental variables and simultaneous equations are used to deal with this third type of violation

• Starting next lecture we’ll take a more in-depth look at the three violations of the CLRM assumptions