04slides.pdf - More Details on the Simple Linear Regression...

This preview shows page 1. Sign up to view the full content.

You've reached the end of this preview.

Unformatted text preview: More Details on the Simple Linear Regression Model Tyler Ransom Univ of Oklahoma Jan 24, 2019 1 / 17 Today’s plan 1. Review reading topics 1.1 Units of Measurement 1.2 Functional Form 1.3 Conditions for Unbiasedness/Computation of standard errors 2. In-class activity: More practice running regressions and interpreting estimates 2 / 17 Units of measurement 3 / 17 Background - The three challenges of statistical inference are:1 1. Generalizing from sample to population (statistical inference) 2. Generalizing from control to treatment group (causal inference) 3. Generalizing from observed measurements to underlying constructs of interest (measurement) 1 Taken from Andrew Gelman’s blog. 4 / 17 Units of measurement - very important to know how y and x are measured in order to interpret regression functions - example: CEO salary and the company’s return on equity (roe). \ =963.191 + 18.501 roe salary N =209, R2 = .0132 - If salary is in thousands and roe is in percentage points, what is interpretation of βˆ 1 = 18.501? - What is the interpretation of βˆ 0 = 963.191? 5 / 17 Changing the units of measurement - What if now we decide to measure roe as a decimal instead of a percent? 6 / 17 Changing the units of measurement - What if now we decide to measure roe as a decimal instead of a percent? \ =963.191 + 1, 850.1 roedec salary N =209, R2 = .0132 where roedec = roe 100 6 / 17 Changing the units of measurement - What if now we decide to measure roe as a decimal instead of a percent? \ =963.191 + 1, 850.1 roedec salary N =209, R2 = .0132 where roedec = roe 100 - And what if salary is in dollars instead of thousands of dollars? 6 / 17 Changing the units of measurement - What if now we decide to measure roe as a decimal instead of a percent? \ =963.191 + 1, 850.1 roedec salary N =209, R2 = .0132 where roedec = roe 100 - And what if salary is in dollars instead of thousands of dollars? \ =963, 191 + 18, 501 roe salary N =209, R2 = .0132 6 / 17 Units, interpretation, and model performance - Notice how the R2 didn’t change at all when we changed the units! - Changing the units only changes the interpretation, not the performance of the model - Typically should choose units that correspond to plausible changes - e.g. typical ∆roe = 1 pct. point (pp), not 100 pp 7 / 17 Functional Form 8 / 17 Functional Form - Sometimes a linear function isn’t very realistic - e.g. a simple wage-education equation [ = − 5.12 + 1.43 educ wage N =759, R2 = .133 where wage is the hourly wage earned, and educ is years of education - What’s weird about this? 9 / 17 Functional Form - Sometimes a linear function isn’t very realistic - e.g. a simple wage-education equation [ = − 5.12 + 1.43 educ wage N =759, R2 = .133 where wage is the hourly wage earned, and educ is years of education - What’s weird about this? ? 1. educ = 0 ⇒ wage = −5.12 9 / 17 Functional Form - Sometimes a linear function isn’t very realistic - e.g. a simple wage-education equation [ = − 5.12 + 1.43 educ wage N =759, R2 = .133 where wage is the hourly wage earned, and educ is years of education - What’s weird about this? ? 1. educ = 0 ⇒ wage = −5.12 2. Constant return to education. Should be increasing! 9 / 17 The log transformation - Instead, consider using log(wage): log\ (wage) =1.142 + 0.099 educ N =759, R2 = .165 where log(·) is the natural logarithm X Now we don’t have negative wage when educ = 0 X Model allows for increasing returns to educ (but constant percentage effect) - Interpretation: 10 / 17 The log transformation - Instead, consider using log(wage): log\ (wage) =1.142 + 0.099 educ N =759, R2 = .165 where log(·) is the natural logarithm X Now we don’t have negative wage when educ = 0 X Model allows for increasing returns to educ (but constant percentage effect) - Interpretation: one-unit ↑ educ corresponds to ≈9.9% ↑ wage 10 / 17 Other uses of log - Can also put the log on the x variable (or both), See Table 2.3: Model Level-level Level-log Log-level Log-log Dep. Var. Indep. Var. Interpretation of β 1 y y log(y ) log(y ) x log(x) x log(x) ∆y ∆y %∆y %∆y = β 1 ∆x ≈ ( β 1 /100) [1%∆x] ≈ (100β 1 ) ∆x ≈ β 1 %∆x - Note: putting in a log changes the R2 completely - Use log to allow y and x to vary nonlinearly, but still be linear in parameters 11 / 17 Unbiasedness, standard errors 12 / 17 Gauss-Markov Assumptions 1. Linear in parameters 2. Random sampling 3. Var (x) > 0 4. E(u|x) = 0 5. Var (u|x) = σ2 (homoskedasticity) With (1)-(4) satisfied: OLS estimates are unbiased and With (5) satisfied: can easily compute standard errors 13 / 17 Are these crazy assumptions? On a scale of “not at all” to “absolutely”: Linear in parameters Not too crazy Random sampling Not crazy if cross-sectional data Var (x) > 0 Not at all crazy E(u|x) = 0 Absolutely crazy if observational data! Var (u|x) = σ2 Can be crazy, especially if time series / panel data 14 / 17 Why do we need to make these assumptions? You might wonder why we bother to make these assumptions - We do econometrics to learn something about a population of interest - We can’t learn much if we don’t make any assumptions! - Bothered by these assumptions? - Think: “tell how to conduct statistical inference on experimental data” 15 / 17 Variance of OLS estimators - Last time, we introduced the formulas for OLS estimators - Also interested in their variance - So we know how far away βˆ is expected to be from β - A big component of these estimators is σ2 = Var (u) SSR N−2 N 1 = uˆ 2i ∑ N − 2 i=1 σˆ 2 = 16 / 17 Variance of OLS estimators - Once we have σˆ 2 , we can obtain the SE of the β’s  Var βˆ 0 = 2 σ 2 ∑N i = 1 xi N ∑ i = 1 ( xi − x ) 2  σ2 σ2 = Var βˆ 1 = 2 SSTx ∑ i = 1 ( xi − x ) - Don’t worry about memorizing these formulas - Key takeaway: we can write them down in a fairly compact form - We can do that because of the assumptions we made 17 / 17 ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern