Chapter 4 - Logs, Polynomials, & Interaction Terms Flashcards
1) ln((A)(B)) =
2) ln(A/B) =
3) lne =
4) ln1 =
1) lnA + lnB
2) lnA - lnB
3) 1
4) 0
What to log and what not to log?
1) Currency values are often logged
2) Variables with larger integer values, population are often logged
3) Years are not logged
4) Percentages typically not logged
5) If the variable can take on a zero or a negative, not logged
When interpreting the coefficient of a estimated equation from a regression you conducted, what do you do?
Always remember to take the partial derivative
You run a regression and get the following equation: log(price) = 9.23 - 0.718log(pollution) + 0.306rooms
Interpret each of the coefficients, and use the correction formula on the log-lin interpretation.
For β1: On average, a 1% increase in the level of pollution leads to a 0.718% decrease in the price of the house, ceteris paribus.
For β2: On average, for every 1 additional room in a house, results in a 30.6% (100*β2%) increase in the price of the house, ceteris paribus.
Correction Formula: 100[(e^0.306) -1] = 35.8%
When interpreting log-lin models there is an approximation error that occurs as changes in log(y) become bigger and bigger. There is a correction formula for this, what is that correction formula?
%Δy = 100[(e^β) – 1]
Wage = e^(β0 + β1Ed + β2Exp)
Interpret the coefficients for Ed and Exp, where Ed = Years of education, and Exp = Years of work experience
1) ln both sides to get rid of the e.
2) ln(Wage) = β0 + β1Ed + β2Exp
3) Interpretation: On average, for every 1 additional year in education, wage increases by 100*β1%, ceteris paribus.
Same for experience.
You run a regression and get the following: prate = 97.32 + 5.02mrate + .314age - 2.66log(totemp) prate = participation rate in 401k mrate = 401k plan match rate age = age of 401k plan totemp = total number of firm employees Interpret the coefficients
For β1: On average, for every additional 1% matched rate in the 401k plan results in a 5.02% participation rate increase in the 401k plan, ceteris paribus.
For β2: On average, for every 1 additional year in the age of the 401k plan, results in a .314 participation rate increase in the 401k plan, ceteris paribus.
For β3: On average, a 1% increase in total firm employees results in a 0.0266 (β2/100) participation rate decrease in the 401k plan, ceteris paribus.
When dealing with polynomials in a regression what is the first step taken in order to understand the interpretation?
1) Take the derivative with respect to the variable that is being raised, x^2.
2) For interpretation how the x variable correlates with the y variable we must do it at a particular point, or a particular value of x. Typically the value of x is mean, but it can be any value of interest.
What is the general interpretation for the equation below, (interpret for x2)?
y = β0 + β1x1 + β2x2 + β3x2^2 + u
1) Take derivative with respect to x2: β2 +2β3x2
2) On average, a 1 unit change in x2 results in a β2 + 2β3x2 units change in y, ceteris paribus.
3) Never interpret β2 or β3 by themselves because they are both related to x2
Take the first and second derivative of the equation with respect to x2:
y = β0 + β1x1 + β2x2 + β3x2^2 + u.
What are the rules for knowing if the relationship is concave, convex, or linear?
1) First derivative: β2 + 2β3x2
2) Second derivative: 2β3
3) If β30, relationship between x2 and y is convex.
If β3=0 relationship between x2 and y is linear.
y = β0 + β1x1 + β2x2 + β3x2^2 + u
If the relationship between x and y is linear do we need to have a squared term? Give the hypothesis test to determine this.
1) There is no need to have the squared term, because the relationship is just that, linear.
2) H0: β3=0 and H1: β3≠0
If we reject the null hypothesis we should keep the squared term, and if we fail to reject the null hypothesis then we can drop it.
Conduct the t-stat test in the normal way.
y = β0 + β1x1 + β2x2 + β3x2^2 + u
What is the global significance test? (give the definition and the hypothesis test)
1) Test used to see if there is some relationship between x2 and y at any point int he data, or if x2 is ever correlated with y.
2) H0: β2=β3=0 and H1: At least one is not equal to zero
Must calculate the partial F-test with this test, and use a full and reduced model.
If we reject the null hypothesis then there is correlation between x2 and y
What is the local significance test? (give the definition and the hypothesis test)
First derivative: β2 + 2β3x2
1) Test used to see if there is some relationship between x2 and y at a specific point in the data.
2) H0: β2 + 2β3x2 = 0 and H1: β2 + 2β3x2 ≠ 0
If we reject the null hypothesis there is correlation between x2 and y at a specific value of x2
wage = 3.73 + 0.298expr - 0.0061expr^2 wage = wage rate expr = years of experience Interpret the coefficients. If experience is 5 years what is the interpretation?
1) First take the derivative: (∂wage)/(∂expr) = 0.298 - 2(0.0061)expr
2) On average, a 1 year increase in experience results in a 0.298-2(.0061)expr change in the wage rate, ceteris paribus.
3) On average, a 1 year increase in experience results in a $0.237 increase in the wage rate, ceteris paribus.
y = β0+β1x1+β2x2+β3x1x2+u
Take the first derivative with respect to x1, and then x2. Interpret the interaction term.
(∂y)/(∂x1) = β1 + β3x2
(∂y)/(∂x2) = β2 + β3x1
On average, a 1 unit change in x1 results in a β1 + β3x2 units change in y, ceteris paribus.
On average, a 1 unit change in x2 results in a β2 + β3x1 units change in y, ceteris paribus.
y = β0+β1x1+β2x2+β3x1x2+u
What is the hypothesis test to determine if we should keep the interaction term?
H0: β3=0 and H1: β3≠0
If we reject the null hypothesis then we should keep the interaction term in the regression. If we fail to reject, we drop the interaction term out.
y = β0+β1x1+β2x2+β3x1x2+u
What is the global significance test? (give the definition and the hypothesis test)
1) Test used to see if there is some relationship between x1 and y or x2 and y at any point in the data, or is x1 and x2 ever correlated with y.
2) H0: β1=β3=0 and H1: At least one is not equal to zero
2) H0: β2=β3=0 and H1: At least one is not equal to zero
Must calculate partial F-test and use a full and reduced model in doing hypothesis test. If we reject null hypothesis there is correlation between x1 and y, or x2 and y.
y = β0+β1x1+β2x2+β3x1x2+u
What is the local significance test? (give the definition and the hypothesis test)
1) Test used to see if there is some relationship between x1 or x2 and y at a specific point in the data. Use a specific value of x1 and x2.
2) H0: β1 + β3x2 = 0 and H1: β1 + β3x2 ≠ 0
2) H0: β2 + β3x1 = 0 and H1: β2 + β3x1 ≠ 0
If we reject null hypothesis there is correlation between x1 and y at a specific value of x2. Same applies for x2.
You run a regression and get the following equation: stndfnl = 2.05-0.0067atndrte-1.63priGPA-0.128ACT+0.296priGPA^2+0.0045ACT^2+0.0056priGPA*atndrte stndfnl=standardized final exam score atndrte=percentage of class attended priGPA=prior college GPA Interpret GPA coefficients and atndrte coefficients.
1) first take derivative with respect to GPA and then atndrte
2) (∂stndfnl)/(∂priGPA)=-1.36+2(0.296)priGPA+(0.0056)atndrte
2) (∂stndfnl)/(∂atndrte)=-0.0067+(0.0056)priGPA
3) On average a 1 point increase in prior GPA results in a -1.36+2(0.296)priGPA+(0.0056)atndrte change in the standard deviations from the mean final exam score, ceteris paribus.
3) On average, a 1 percentage point increase in attendance rate results in a -0.0067+(0.0056)priGPA change in the standard deviations from the mean final exam score, ceteris paribus.
You run a regression and get the following equation: stndfnl = 2.05-0.0067atndrte-1.63priGPA-0.128ACT+0.296priGPA^2+0.0045ACT^2+0.0056priGPA*atndrte stndfnl=standardized final exam score atndrte=percentage of class attended priGPA=prior college GPA 1) If priGPA =2.5 & atndrte = 90 whats the interpretation? 2) If priGPA =3.0 what is the interpretation?
1) first take derivative with respect to GPA and then atndrte
(∂stndfnl)/(∂priGPA)=-1.36+2(0.296)priGPA+(0.0056)atndrte
(∂stndfnl)/(∂atndrte)=-0.0067+(0.0056)priGPA
1) On average a 1 point increase in GPA results in a 0.624 increase in the standard deviations from the mean final exam score, ceteris paribus.
2) On average, a 1 percentage point increase in attendance rate results in 0.0101 increase in the standard deviations from the mean final exam score, ceteris paribus.
