STA 235H - Multiple Regression: Polynomials
Fall 2023
McCombs School of Business, UT Austin
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Some Announcements

Homework answer key will be posted on Tuesday/Wednesday.
- Make sure you check it out!
- Exercises: Multiple regression (e.g. Bechdel Test example), differences in associations between groups (e.g. luxury vs non-luxury cars depreciation).
Check personalized feedback for JITT 3, if included.
- Additional videos on material (and some R code) in Resources > Videos

2 / 18

Some Announcements

Homework answer key will be posted on Tuesday/Wednesday.
- Make sure you check it out!
- Exercises: Multiple regression (e.g. Bechdel Test example), differences in associations between groups (e.g. luxury vs non-luxury cars depreciation).
Check personalized feedback for JITT 3, if included.
- Additional videos on material (and some R code) in Resources > Videos

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Today

Roadmap of where we've been and where we're going.
Nonlinear models:
- Polynomial terms
Introduction to Causal Inference
- Potential Outcomes Framework

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Roadmap so farStarted the class with a review on simple linear regressions:Association between a variable XX and outcome YY
e.g. Revenue=β0+β1Bechdel+εRevenue=β0+β1Bechdel+ε

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Roadmap so farStarted the class with a review on simple linear regressions:Association between a variable XX and outcome YY
e.g. Revenue=β0+β1Bechdel+εRevenue=β0+β1Bechdel+ε

Followed by multiple regression:Partial association between XX and YY, when holding other variables constant.
e.g. Revenue=β0+β1Bechdel+β2Revenue+β3IMDB+εRevenue=β0+β1Bechdel+β2Revenue+β3IMDB+ε

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Roadmap so farStarted the class with a review on simple linear regressions:Association between a variable XX and outcome YY
e.g. Revenue=β0+β1Bechdel+εRevenue=β0+β1Bechdel+ε

Followed by multiple regression:Partial association between XX and YY, when holding other variables constant.
e.g. Revenue=β0+β1Bechdel+β2Revenue+β3IMDB+εRevenue=β0+β1Bechdel+β2Revenue+β3IMDB+ε

What if we want to compare differences in associations between groups?:Compare the association between XX and YY for group D=1D=1 and D=0D=0.
e.g. Price=β0+β1Year+β2Luxury+β3Year×Luxury+εPrice=β0+β1Year+β2Luxury+β3Year×Luxury+ε

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Roadmap so farWhat if our outcome YY is weird (e.g. not normally distributed)?If YY is skewed to the right (log-normal): Transform to log(Y)log(Y) to improve linearity assumption!
e.g. log(Price)=β0+β1Year+β2Luxury+β3Mileage+εlog(Price)=β0+β1Year+β2Luxury+β3Mileage+ε
Interpret coefficients as percent change (%)

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Roadmap so farWhat if our outcome YY is weird (e.g. not normally distributed)?If YY is skewed to the right (log-normal): Transform to log(Y)log(Y) to improve linearity assumption!
e.g. log(Price)=β0+β1Year+β2Luxury+β3Mileage+εlog(Price)=β0+β1Year+β2Luxury+β3Mileage+ε
Interpret coefficients as percent change (%)



What if our outcome YY is weird (e.g. binary)?e.g. Employed=β0+β1Age+β2Afam+β3NKids+εEmployed=β0+β1Age+β2Afam+β3NKids+ε
Interpret coefficients as change in probability (e.g. percentage points)

5 / 18

Roadmap so farWhat if our outcome YY is weird (e.g. not normally distributed)?If YY is skewed to the right (log-normal): Transform to log(Y)log(Y) to improve linearity assumption!
e.g. log(Price)=β0+β1Year+β2Luxury+β3Mileage+εlog(Price)=β0+β1Year+β2Luxury+β3Mileage+ε
Interpret coefficients as percent change (%)

What if our outcome YY is weird (e.g. binary)?e.g. Employed=β0+β1Age+β2Afam+β3NKids+εEmployed=β0+β1Age+β2Afam+β3NKids+ε
Interpret coefficients as change in probability (e.g. percentage points)

What if there isn't a linear relation between XX and YY?Include polynomial terms for XX

5 / 18

Roadmap so farWhat if our outcome YY is weird (e.g. not normally distributed)?If YY is skewed to the right (log-normal): Transform to log(Y)log(Y) to improve linearity assumption!
e.g. log(Price)=β0+β1Year+β2Luxury+β3Mileage+εlog(Price)=β0+β1Year+β2Luxury+β3Mileage+ε
Interpret coefficients as percent change (%)

What if our outcome YY is weird (e.g. binary)?e.g. Employed=β0+β1Age+β2Afam+β3NKids+εEmployed=β0+β1Age+β2Afam+β3NKids+ε
Interpret coefficients as change in probability (e.g. percentage points)

What if there isn't a linear relation between XX and YY?Include polynomial terms for XX

What if I want to know what is the effect of X on Y?Causal Inference!

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Adding polynomial terms

Another way to capture nonlinear associations between the outcome (Y) and covariates (X) is to include polynomial terms:
- e.g. $Y = β_{0} + β_{1} X + β_{2} X^{2} + ε$

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Adding polynomial terms

Another way to capture nonlinear associations between the outcome (Y) and covariates (X) is to include polynomial terms:
- e.g. $Y = β_{0} + β_{1} X + β_{2} X^{2} + ε$
Let's look at an example!

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Determinants of wages: CPS 1985

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Determinants of wages: CPS 1985

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Experience vs wages: CPS 1985

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Experience vs wages: CPS 1985

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + ε$

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Experience vs wages: CPS 1985

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$

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Mincer equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$

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Mincer equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$

Interpret the coefficient for education

$\log (W a g e) = 0.52 + 0.09 \cdot E d u c + 0.034 \cdot E x p - 0.0005 \cdot E x p^{2}$

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Mincer equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$

Interpret the coefficient for education

$\log (W a g e) = 0.52 + 0.09 \cdot E d u c + 0.034 \cdot E x p - 0.0005 \cdot E x p^{2}$

What is the association between experience and wages?

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Interpreting coefficients in quadratic equation

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Interpreting coefficients in quadratic equation

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Interpreting coefficients in quadratic equation

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

Pick a value for $E x p_{0}$ (e.g. mean, median, one value of interest)

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

Pick a value for $E x p_{0}$ (e.g. mean, median, one value of interest)

Increasing work experience from $E x p_{0}$ to $E x p_{0} + 1$ years is associated, on average, to a $({\hat{β}}_{2} + 2 {\hat{β}}_{3} \times E x p_{0}) 100$ % increase on hourly wages, holding education constant

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

Pick a value for $E x p_{0}$ (e.g. mean, median, one value of interest)

Let's put some numbers into it:

$\log (W a g e) = 0.52 + 0.09 \cdot E d u c + 0.034 \cdot E x p - 0.0005 \cdot E x p^{2}$

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

Pick a value for $E x p_{0}$ (e.g. mean, median, one value of interest)

Let's put some numbers into it:

$\log (W a g e) = 0.52 + 0.09 \cdot E d u c + 0.034 \cdot E x p - 0.0005 \cdot E x p^{2}$

Increasing work experience from 20 to 21 years is associated, on average, to a $(0.034 - 2 \times 0.0005 \times 20) \times 100 = 1.4$ % increase on hourly wages, holding education constant

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Interpreting coefficients in quadratic equation

$\log (W a g e) = β_{0} + β_{1} E d u c + β_{2} E x p + β_{3} E x p^{2} + ε$ What is the association between experience and wages?

Pick a value for $E x p_{0}$ (e.g. mean, median, one value of interest)

Let's put some numbers into it:

$\log (W a g e) = 0.52 + 0.09 \cdot E d u c + 0.034 \cdot E x p - 0.0005 \cdot E x p^{2}$

Increasing work experience from 20 to 21 years is associated, on average, to a $(0.034 - 2 \times 0.0005 \times 20) \times 100 = 1.4$ % increase on hourly wages, holding education constant

Note that in this case we are interpreting the association between Experience and Wages as a percent change, because Wages is in a logarithm!

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Let's go to R

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ReferencesIsmay, C. & A. Kim. (2021). “Statistical Inference via Data Science”. Chapter 6 & 10.
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STA 235H - Multiple Regression: Polynomials

Fall 2023

McCombs School of Business, UT Austin

Some Announcements

Some Announcements

Today

Roadmap so far

Roadmap so far

Roadmap so far

Roadmap so far

Roadmap so far

Roadmap so far

Roadmap so far

Adding polynomial terms

Adding polynomial terms

Determinants of wages: CPS 1985

Determinants of wages: CPS 1985

Experience vs wages: CPS 1985

Experience vs wages: CPS 1985

Experience vs wages: CPS 1985

Mincer equation

Mincer equation

Mincer equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

Interpreting coefficients in quadratic equation

References

Some Announcements

Help