Econ 444, class 11. Robert de Jong 1. Monday November 6. Ohio State University. Econ 444, Wednesday November 1, class Department of Economics

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1 Econ 444, class 11 Robert de Jong 1 1 Department of Economics Ohio State University Monday November 6

2 Monday November 6 1 Exercise for today 2 New material: 1 dummy variables 2 multicollinearity

3 Exercise for today: Hypothesis testing Note: reject H 0 " means: we have statistical evidence against the hypothesis 1 Three possibilities: 1 Coefficient is approximately 3 standard errors away from 0 Reject H 0 2 t-value = > 1.96 Reject H 0 3 p-value not present: it would have been smaller than Same principle; only use instead of Coefficient is approximately 3 standard errors away from 0 Reject H 0 2 t-value = > Reject H 0 3 p-value not present: it would have been smaller than 0.01

4 3 Question is: is less than 1.96 standard errors away from 0.3? Answer: NO; so, reject H 0 4 Tests the hypothesis H 0 : β 0 = 0. That is, it tests whether if return on equity equals zero, the log of CEO salary equals 0. This means salary equals 1; this makes no sense. 5 Less than 1%. 6 t-value = coefficient standard error, so standard error = coefficient t-value = =

5 Why does the 1.96 standard error rule work? P(ˆβ se(ˆβ 1 ) β 1 ˆβ se(ˆβ 1 )) = P( ˆβ 1 β 1 se(ˆβ 1 ) 1.96) P( N(0, 1) 1.96) = Q: How do we replace the 95% by 99%? A: replace 1.96 (the 2.5% point of the standardnormal distribution) by (the 0.005% point)

6 Is the 1.96 ALWAYS correct? In small samples, the 1.96 may need to be replaced by a value drawn from the t distribution rather than the standardnormal You would need the 2.5% point from the t distribution This explains the name: t-value Small : say, sample size under 100

7 New material for today Dummy variables: variables that take on the value 0 and 1 Very often used; dummy or 0/1 variables capture qualitative information Examples: 1 male/female dummy: female equals 0 2 regional dummy: north equals 1 if individual lives in the north and zero otherwise 3 ethnicity: nonwhite equals 1 if individual is nonwhite 4 time period dummy: wartime equals 1 if the country is at war in the time period the observation is from

8 What does a dummy variable D i do? If D i = 1, this becomes Y i = β 0 + β 1 X i + β 2 D i + ǫ i while if D i = 0, we have Y i = (β 0 + β 2 ) + β 1 X i + ǫ i Y i = β 0 + β 1 X i + ǫ i If Y i is hourly wage in dollars, and D i is a female dummy variable, then β 2 is the estimated bonus (can be negative) to wage of being a female Conclusion: often dummy variables have coefficients that have a nice interpretation!

9 Dummy variable pitfalls Explain GPA of NYU students from mode of transportation to university; we have a variable that equals 1 if student walks or bikes, 2 if student takes subway, 3 if student drives What do we do with this information? A: we make should two dummy variables

10 Other dummy variable pitfall: Never use a dummy variable as your Y i variable Example: explain whether a female with small children participates in the labor force from spousal income, education level, etc. We need a technique different from linear regression in this case (we need a probit model )

11 Multicollinearity: failure of model assumption 6: No explanatory variable is a perfect linear combination of any other explanatory variable(s) (no perfect multicollinearity)." where Y i = β 0 + β 1 X i1 + β 2 X 2i + ǫ i Y i is expenditure on housing of individual #i X i1 is annual income in dollars of individual #i X 2i is annual income in thousands of dollars so, 1000 X 1i = X 2i always! Perfect multicollinearity: this is not allowed; Eviews gives error message near singular matrix

12 More problematic in practice: 1000 X 1i X 2i (someone rounded off) Eviews produces output; however we should realize there is a problem here near multicollinearity Other example of multicollinearity: including a dummy F i that equals 1 for all females in sample AND a dummy M i that equals 1 for all males in the sample Then, F i + M i = 1 always

13 No explanatory variable is a perfect linear combination of any other explanatory variable(s) (no perfect multicollinearity)." Why is this violated when M i and F i are included and M i + F i = 1? Y i = β β 1 X 1i + + β K X Ki + ǫ i A variable that is always 1 is also an explanatory variable Better therefore: No explanatory variable is a perfect linear combination of any other explanatory variable(s), including the constant (no perfect multicollinearity)."

14 Other example Regression of expenditure on housing on: 1 wage income; 2 investment income; 3 other income; 4 total income. What is the potential problem?

15 Dummy variable trap Suppose we have north", south", west and east dummy variables Q: can we include all four? A: NO; these four variables add up to 1! Solution: leave one out Analogy: in the female/male dummy variable case, we leave out either the male or the female dummy variable

16 So we can run a regression WAGE i = β 0 + β 1 X i1 + β 2 NORTH i +β 3 SOUTH i + β 4 WEST i + ǫ i In this case, all three dummy variables are 0 for an individual from the east; east is our reference category β 2 therefore is how much more someone in the north is expected to make, compared to an individual from the east