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Lecture 15 1
Econ 140Econ 140
Multiple Regression Applications
Lecture 15
Lecture 15 2
Econ 140Econ 140Today’s Plan
• Two topics and how they relate to multiple regression
– Multicollinearity
– Dummy variables
Lecture 15 3
Econ 140Econ 140Multicollinearity
• Suppose we have the following regression equation:
Y = a + b1X1 + b2X2 + e
• Multicollinearity occurs when some or all of the independent X variables are linearly related
• Different forms of multicollinearity:
– Perfect: OLS estimation will not work
– Non-perfect: comes out of applied work - presents problems for inference and interpretation of the results.
– No test for detection - only possible to compare alternative specified forms of the model.
Lecture 15 4
Econ 140Econ 140Multicollinearity Example
• Again we’ll use returns to education where:
– the dependent variable Y is (log) wages
– the independent variables (X’s) are age, experience, and years of schooling
• Experience is defined as years in the labor force, or the difference between age and years of schooling
– this can be written: Experience = Age - Years of school
– What’s the problem with this?
Lecture 15 5
Econ 140Econ 140Multicollinearity Example (2)
• Note that we’ve expressed experience as the difference of two of our other independent variables
– by constructing experience in this manner we create a collinear dependence between age and experience
– the relationship between age and experience is a linear relationship such that: as age increases, for given years of schooling, experience also increases
• We can write our regression equation for this example:
Ln(Wages) = a + b1Experience + b2Age + e
Lecture 15 6
Econ 140Econ 140Multicollinearity Example (3)
• Recall that our estimate for b1 is
22122
21
221221
1
xxxx
yxxxxyxb
Where x1 = experience and x2 = age
• The problem is that x1 and x2 are linearly related
– as we get closer to perfect collinearity, the denominator will go to zero. – OLS won’t work!
Lecture 15 7
Econ 140Econ 140Multicollinearity Example (4)
• Recall that the estimated variance for is:
– So as x1 and x2 approach perfect collinearity, the denominator will go to zero and the expression for the the estimated variance of will increase
• Implications:– with multicollinearity, you will get large standard errors on partial coefficients– your t-ratios, given the null hypothesis that the value of the coefficient is zero, will be small
221
22
21
2222 ˆˆ 1
xxxx
xYXb
1b
1b
Lecture 15 8
Econ 140Econ 140More Multicollinearity Examples
• In L15_1.xls we have individual data on age, years of education, weekly earnings, school age, and experience– we can perform a regression to calculate returns given age and experience– we can also estimate bivariate models including only age, only experience, and only years of schooling– we expect that the problem is that experience is related to age (to test this, we can regress age on experience)
• if the slope coefficient on experience is 1, there is perfect multicollinearity
Lecture 15 9
Econ 140Econ 140More Multicollinearity Examples (2)
• On L15_2.xls there is a made-up example of perfect multicollinearity– OLS is unable to calculate the slope coefficients– calculating the products and cross-products, we find that the denominator for the slope coefficients is zero as predicted– If we have is an applied problem with these properties:
1) OLS is still unbiased
2) Large variance, standard errors, and difficult hypothesis testing
3) Few significant coefficients but a high R2
Lecture 15 10
Econ 140Econ 140More Multicollinearity Examples (3)
• What to do with L15_1.xls?– There’s simply not enough variation– We can collect more data or rethink the model– We can test for partial correlations between the X variables.– Always try specification checks.– Alternatively, try to re-scale variables so that the correlation is not the same.
Lecture 15 11
Econ 140Econ 140Dummy variables
• Dummy variables allow you to include qualitative variables (or variables that otherwise cannot be quantified) in your regression– examples include: gender, race, marital status, and religion– also becomes important when looking at “regime shifts” which may be new policy initiatives, economic change, or seasonality
• We will look at some examples:– using female as a qualitative variable– using marital status as a qualitative variable– using the Phillips curve to demonstrate a regime shift
Lecture 15 12
Econ 140Econ 140Qualitative example: female
• We’ll construct a dummy variable:
Di = 0 if not female i = 1, …n
Di = 1 if female
– We can do this with any qualitative variable– Note: assigning the values for the dummy variable is an arbitrary choice
• On L15_3.xls there is a sample from the current CPS– to create the dummy variable “female” we assign the value one and zero to the CPS’ value of two and one for sex, respectively– we can include the dummy variable in the regression equation like we would any other variable
Lecture 15 13
Econ 140Econ 140Qualitative example: female (2)
• We estimate the following equation:
ii DY 485.0975.5ˆ • Now we can ask: what are the expected earnings given that a person is male?
• Similarly, what are the expected earnings given that a person is female?
E(Yi | Di = 1) = a + b(1) = a + b
= 5.975 - 0.485 = 5.490
975.50|
)0(0|
ii
ii
DYE
abaDYE
Lecture 15 14
Econ 140Econ 140Qualitative example: female (4)
• We can use other variables to extend our analysis• for example we can include age to get the equation:
Y = a + b1Di + b2Xi + e
– where Xi can be any or all relevant variables
– Di and the related coefficient b1 will indicate how much, on average, females earn less than males
– for males the intercept will be – for females the intercept will be
a
1ˆ ba
Lecture 15 15
Econ 140Econ 140Qualitative example: female (5)
• The estimated regression found on the spreadsheet is
• The expected weekly earnings for men are:
• The expected weekly earnings for women are:
iii XDY 023.0656.0085.5ˆ
iii XbaDYE 2)0|(
iii XbbaDYE 21)()1|(
Lecture 15 16
Econ 140Econ 140Qualitative example: female (6)
• An important note:• We can not include dummy variables for both male and female in the same regression equation
– suppose we have Y = a + b1D1i + b2D2i + e
– where: D1i = 0 if male D1i = 1 if female
D2i = 0 if female D2i = 1 if male
– OLS won’t be able to estimate the regression coefficients because D 1i and D2i show perfect multicollinearity with intercept a
• So if you have m qualitative variables, you should include (m-1) dummy variables in the regression equation
Lecture 15 17
Econ 140Econ 140Example: marital status
• The spreadsheet (L15_3.xls) also estimates the following regression equation using two distinct dummy variables:
– where: D1i = 0 if male D1i = 1 if female
D2i = 0 if other D2i = 1 if married
• Using the regression equation we can create four categories: married males, unmarried males, married females, and unmarried females
eXbDbDbaY iii 32211
Lecture 15 18
Econ 140Econ 140Example: marital status (2)
• Expected earnings for unmarried males:
iiii XbaDDYE 321 )0,0|(
• Expected earnings for unmarried females:
iiii XbbaDDYE 3121 )()0,1|(
• Expected earnings for married males:
iiii XbbaDDYE 3221 )()1,0|(
• Expected earnings for unmarried females:
iiii XbbbaDDYE 32121 )()1,1|(
Lecture 15 19
Econ 140Econ 140Interactive terms
• So far we’ve only used dummy variables to change the intercept• We can also use dummy variables to alter the partial slope coefficients• Let’s think about this model:
ln(Wi )= a + b1Agei + b2Marriedi + e
– we could 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 partial slope coefficients will be different for these 2 groups
abb ˆ and ˆ,ˆ21
Lecture 15 20
Econ 140Econ 140Interactive terms (2)
• To test our hypothesis we’ll estimate the regression equation 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 15 21
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 15 22
Econ 140Econ 140Interactive terms (4)
• The sum of squared residuals for the unrestricted form will be:
SSRU = SSRM + SSRF
• L15.4.xls
– the data are 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 15 23
Econ 140Econ 140Interactive terms (5)
• The output allows us to gather the necessary sum of squared residuals and sample sizes to construct the estimate:
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 15 24
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
Lecture 15 25
Econ 140Econ 140Interactive terms (7)
• Using L15.4.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 not reject the null hypothesis
• We now know how to use dummy variables to indicate the importance of sub-sample groups within the data– dummy variables are also useful for testing for structural
breaks or regime shifts
Lecture 15 26
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(Wi|Di = 0) = a + b1Agei + b2Marriedi
• We can do the same for the second sub-sample (Females)
E(Wi|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