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### Multiple Regression Applications III

### Relaxing Assumptions

Lecture 18

Dummy 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.

Interactive terms

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

Wi = a + b1Agei + b2Marriedi + e

- in so doing we argue thatwould 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

Interactive 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

Interactive 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:

q = k, the number of parameters in the model

n = n1 + n2 where n is complete sample size

Interactive 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

Interactive terms (5)

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

- 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

Interactive terms (6)

Irene O. Wong:

- 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

Interactive 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

Interactive 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

Phillips 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

Phillips 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

Phillips 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

Phillips 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

Phillips 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

Lecture 18

Today’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

CLRM 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, c2 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: a + b = 1.
- Dummy variables and interactions; Chow test.

Relaxing 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) = s2; 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?

Homoskedasticity

- 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

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

X1

X2

X3

Homoskedasticity (2)- Homoskedasticity can be illustrated like this:

Y

constant

variance around

the regression line

Heteroskedasticity

- But, we don’t always have constant variance s2
- 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 X

Heteroskedasticity (2)

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

Y

X

X1

X2

X3

Serial (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

Serial (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

Non-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

Summary

- 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

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