Lecture #8 & #9 Multiple regression

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1 Lecture #8 & #9 Multiple regression

2 Starting point: Y = f(x 1, X 2,, X k, u) Outcome variable of interest (movie ticket price) a function of several variables. Observables and unobservables. One or more hypotheses (needed)?

3 Vertical integration and ticket prices 1. Would other variables besides VI affect ticket prices? 2. Would the effect of VI on ticket prices depend on other characteristics of the theatre?

4 Multivariate/multiple regression Y = f X, u = β 0 + β 1 X + u E[Y X = x] = β 0 + β 1 x + u

5 Basic multiple regression Y = f X 1, X 2, u = β 0 + β 1 X 1 + β 2 X 2 + u E[Y X = x] = β 0 + β 1 x 1 + β 2 x 2

6 More structure - linear Y = β 0 + β 1 X 1 + β 2 X 2 This is the so-called population regression line (populaatio regressio). Y β 1 = dependent variable (vastemuuttuja) / endogenous variable. X i = independent variable i (selittävä muuttuja) / exogenous variable / regressor i. β 0, β 1, β 2 = parameters of the model.

7 Are all Xs born equal? Depends. Treatment variable = the one of primary interest. Control variable(s) = affects Y but we are not (so much) interested in this. Why include variables that we are not interested in?

8 What type of control variables matter, how & why? 1. cov(x 1, X 2 ) = 0 2. cov(x 1, X 2 ) 0 Key is whether the treatment variable and control variable are correlated or not.

9 What type of control variables matter, how & why? Why is this key? Recall መβ 1 = β 1 + ρ Xu σ u σ X Rewrite u = β 2 X 2 + v

10 What type of control variables matter, how & why? Rewrite cov X i, v = 0 Then መβ 1 = β 1 + β 2 ρ X1 X 2 σ X2 σ X1

11 What type of control variables matter, how & why? If the Xs are correlated, then the bias in β 1 depends on 1. What is the impact of X 2 on Y. 2. What is the correlation between the Xs. 3. How much variance X 2 has relative to X 1.

12 What type of control variables matter, how & why? So are we home if cov(x 1, X 2 ) = 0? Yes and no. If cov(x 1, X 2 ) = 0, then መβ 1 = β 1, but But adding X 2 decreases the standard error / increases the precision of መβ 1.

13 What type of control variables matter, how & why? A 2-variable model (App 6.2 in S&W). 2 explanatory variables and homosc. errors, ρ X1,X 2 = 0. Then = the variance of β 1. σ 2 β1 = 1 n Adding X 2 necessarily decreases σ u 2. σ u 2 2 σ X1

14 VI, TV, and ticket prices Let s look at the following model Highprice = β 0 + β VI VI + β TV TV + u VI TV mean sd N

15 Are VI and TV correlated? TV VI = 0 VI = 1 mean sd N

16 Are VI and TV correlated? Correlation coefficient (p-value 0.00).

17 VI cond on TV exposure 0 VI Years of TV data linear fit

18 Should we suspect that TV affects ticket price? TV and movies substitutes. increasing TV availability = lowering TV price. lower movie tickets.

19 Should we suspect that TV affects ticket price? TV increases awareness of movies. TV increases quality of movies. increasing TV availability = increasing attractiveness of movies. higher movie tickets.

20 Do VI and TV affect movie prices? Highprice vi = 0 vi = 1 mean sd N Highprice TV < 4 TV 4 mean sd N

21 Ticket prices cond on vi Movie Ticket Price VI data linear fit

22 Ticket prices cond on TV exposure Movie Ticket Price Years of TV data linear fit

23 First look univariate regressions Highprice = β 0VI + β VI VI + u Highprice = β 0TV + β TV TV + u

24 Highprice Highprice vi tv const N r F

25 Coeff of VI s.e. of VI p-value of VI Coeff of TV s.e. of TV p-value of TV nobs R2 Joint sign. (1) (2) vi tv const N r F City FE NO YES Size and significance (may) depend on control variables

26 (1) (2) vi tv const N r F City FE NO YES Highprice Highprice vi tv const N r F

27 Issues 1. How do the individual coefficients compare to univariate results? 2. What explains the difference(s)? 3. What about statistical significance of individual coefficients? 4. What about several / all coefficients? 5. What about R 2?

28 Issues 6. What is the interpretation of individual coefficients? 7. (under what assumptions) does OLS work? 8. How to choose which explanatory variables to include / exclude? 9. What if the world is more complicated than linear? 10. What all can go wrong, and how would I know / find out?

29 Issues 1. How do the individual coefficients compare to univariate results? 2. What explains the difference(s)?

30 Q1&Q2 multivariate vs. univariate Compare VI coefficient in the univariate to that in the multivariate regressions. What can you conclude? Recall መβ VI = β VI + β TV ρ XVI X TV σ XTV σ XVI

31 (1) (2) vi tv const N r F City FE NO YES VI TV mean sd N Highprice Highprice vi tv const N r F Corr(VI,TV) = -0.31

32 Q1&Q2 multivariate vs. univariate VI univariate bias Beta_TV corr(vi,tv) sd(tv) sd(vi) bias Beta_VI_univ Beta_VI_multiv Diff = Bias

33 Q1&Q2 multivariate vs. univariate Do the same for TV. What can you conclude?

34 Issues 3. What about statistical significance of individual coefficients? 4. What about the statistical significance of several / all coefficients?

35 Coeff of VI s.e. of VI p-value of VI Coeff of TV s.e. of TV p-value of TV nobs R2 Joint sign. Q3&Q4 statistical significance (1) (2) vi tv const N r F City FE NO YES Size and significance (may) depend on control variables

36 Q3&Q4 statistical significance Highprice = β 0 + β VI VI + β TV TV + u Can we test each coefficient individually as above? What about pairs (or more) of coefficients?

37 Q3&Q4 statistical significance Q#1: are both (β 0 and) β VI =0 and β TV = 0? F-test (and others). Cannot do this by looking at individual (t-) tests. Reason: 2 or more random variables need their joint distribution.

38 Q3&Q4 statistical significance F-test (under homosc.). For illustration only. F = (SSR restricted SSR unrestricted )/q SSR unrestricted /(n k unrestricted 1) F = 2 (R unrestricted 2 R restricted )/q 2 (1 R unrestricted )/(n k unrestricted 1) Modern software calculate the heterosc. robust F-test.

39 regr highprice vi tv, cluster(theaterid) Linear regression Number of obs = 2685 F( 2, 392) = Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 393 clusters in theaterid) Robust highprice Coef. Std. Err. t P> t [95% Conf. Interval] vi tv _cons

40 testparm vi tv ( 1) vi = 0 ( 2) tv = 0 F( 2, 392) = Prob > F =

41 Q3&Q4 statistical significance Q#2: what about β VI =β TV? Need either a direct test modern software allow this (easily). Or a trick (add and substract).

42 . test vi = tv ( 1) vi - tv = 0 F( 1, 392) = Prob > F =

43 Issues 5. What about R 2?

44 Q5 What about R 2? R 2 = ESS TSS = 1 SSR TSS R 2 increases (almost) surely as you add explanatory variables. Adjusted R 2 corrects for this. adjr 2 = 1 n 1 SSR n k 1 TSS = s u 2 s Y 2

45 Q5 What about R 2? n= #obs. k= #explanatory variables. Adjusted R 2 always < R 2.

46 Q5 What about R 2? High R 2 / increase in R 2 says nothing about causality. High R 2 does not mean your model does not have omitted variable bias. High R 2 does not mean you have the right set of explanatory variables. But, as you saw from the F-test formula, (changes in) R 2 are indicative.

47 Issues 6. What is the interpretation of individual coefficients?

48 Q6 interpretation of coefficients. Highprice= f VI, TV, u = β 0 + β VI VI + β TV TV + u E[Highprice VI = vi & TV = tv] = β 0 + β VI vi + β TV tv

49 Q6 interpretation of coefficients. E Highprice VI = vi & TV = tv = β 0 + β VI vi + β TV tv Take (partial derivative) of Highprice wrt. VI. Notice what is assumed of the (interaction) of the effect of VI and TV on Highprice. Effect of VI on Highprice, conditional on the value of TV.

50 Issues 7. (under what assumptions) does OLS work?

51 Q7 assumptions of OLS? A1: conditional distribution of u has mean zero given X. E u X = 0 Implies that u and X are uncorrelated (as cov u, X = 0).

52 Q7 assumptions of OLS? A2: X i, Y i i = 1,, n are i.i.d. The same concept as before, but now over a joint distribution of possibly a large # variables.

53 Q7 assumptions of OLS? A3: X i and Y i have nonzero finite fourth moments. Means that large outliers are (extremely) unlikely.

54 Q7 assumptions of OLS? A4: No perfect multicollinearity. What does this mean? Case #1: no two explanatory variables can be perfectly correlated. Case #2: no combination of explanatory variables can be perfectly correlated.

55 Q7 assumptions of OLS? Analogue: to solve a system of equations, you need as many equations as there are unknowns. Think of OLS: we get as many FOCs as there are parameters.

56 Q7 assumptions of OLS? Example of case #2: a constant and two dummy variables. Define X 1 = 1 if vi = 0 and 0 otherwise X 2 = 1 if vi = 1 and 0 otherwise This is the dummy-variable trap.

57 Q7 assumptions of OLS? Example of case #1: A 2-variable model (App 6.2 in S&W) 2 explanatory variables and homosc. errors. Then σ 2 β1 = 1 n ρ X1,X 2 σ u 2 2 σ X1 σ 2 β1 = the variance of β 1.

58 Q7 assumptions of OLS? σ 2 β1 = 1 n ρ X1,X 2 σ u 2 2 σ X1 Notice what happens when you increase ρ X1,X 2. A4 is also called the full rank condition.

59 Issues 8. How to choose which explanatory variables to include / exclude?

60 Q8 how to choose the expl. variables? Too few explanatory variables possible omitted variable bias. Too many variables multicollinearity and inflated se s. Note: too many requires correlation among explanatory variables. Can one test one s way out of this? No, but tests do help.

61 Q8 how to choose the expl. variables? There are tests of individual and of joint significance. Why cannot I run these on autopilot? Case #1: start from a small model, add variables according to some (statistical) criterion. Case#2: start from a large model, drop variables according to some (statistical) criterion.

62 Q8 how to choose the expl. variables? What goes wrong? 1. Statistical significance economic significance. 2. Statistical significance economic relevance. 3. You may end up with variables that are highly correlated with Y, but have no real connection to it. 4. Multiple testing leads to wrong (too good) test results.

63 Q8 The principled approach 1. Before touching your data, write down a protocol. 2. Base explanatory (control) variables on theory and existing knowledge. 3. Specify a testing protocol. 4. Execute

64 Q8 The practical approach 1. Try to be as close to the principled approach as possible. 2. Learning allowed and encouraged new/respecification. 3. Robustness testing.

65 Q8 Robustness testing 1. It is rarely the case that there is a right model that you can (re)cover. 2. Ask: are your results sensitive to small, well-justified changes to your model? 1. Adding (meaningful) variables. 2. Deleting variables. 3. Changing functional form. 4. Changing assumptions about the error term.

66 Issues 9. What if the world is more complicated than linear?

67 Q9 - What if the world is more complicated than linear? Well, make your model non-linear. Many ways to do this. Let s go through some of them. 1. Keeping variables the same, but making the function more complicated. 2. Transforming the variables.

68 Q9 Higher order polynomials p X = σ M i=0 α i X i = α 0 X 0 +α 1 X 1 +α M X M So thus far we have used a 1st order polynomial... Highprice= f VI, VI, u = β 0 + β VI VI + β TV TV + u

69 Q9 Higher order polynomials Would it make sense to add a 2nd order term for VI? What about TV? Highprice= f VI, TV, u = β 0 + β VI VI + β TV TV + β TV2 TV 2 + u

70 Q9 Higher order polynomials What is now the partial derivative wrt. TV? How to determine the degree of the polynomial? Start from a reasonable one (3 or 4). Test down. Notice: Here you have a prior plan and a clear sequence in mind.

71 Q9 Higher order polynomials Notice how there is a (big) difference in whether you take a higher order polynomial of a control variable vs. a treatment variable. Control variable: to make sure the treatment effect (=coefficient of the treatment variable) not biased because of wrong functional form on control variables. Treatment variable: to allow for different treatment effects ( heterogenous treatment effects ).

72 Q9 Example: Testing for U-shape A 2nd order polynomial gives you a U-shape. How to test? 1. Start from a 3rd order polynomial. 2. Test significance of 3rd order term (and joint sign. of 2nd and 3rd). 3. Drop 3rd order if insign., test 2nd order term.

73 Variable tv3 tv2 tv1 vi tv tv tv const N r F all F (TV2, TV3)

74 E[Highprice VI = 0] = expected ticket price when VI = 0 TV quart. quadr. linear

76 Q9 - What if the world is more complicated than linear? Can you do more than use polynomials? Yes... Y = f X + u Give f X any shape you like. We will skip this for now (semi- and nonparametric estimation).

77 Q9 Interactions What if there is reason to believe that the effect of X 1 depends on the value of X 2? Examples: 1. Returns to education different by gender. 2. Effect of R&D subsidies different by firm size.

78 Q9 Example: Effect of VI on ticket price depends on TV Highprice= f VI, TV, u = β 0 + β VI VI + β TV TV + u Highprice= f VI, TV, u = β 0 + β VI VI + β VI TV VI TV + β TV TV + u What is now the expected ticket price VI?

79 Variable (1) (2) vi vi x tv tv const N r F F(VI,VIxTV) 0.6 p-value 0.549

80 Q9 Example: Effect of VI on ticket price depends on TV How to calculate the effect of VI on ticket price? Now depends on the value of TV directly. Notice: 1. without the interaction VI TV, the effect of VI on ticket price independent of TV (= the same no matter what value TV takes). 2. not true any more with the interaction.

81 β VI VI + β VITV VI x tv

82 Q9 Transformations of variables What is a transformation of a variable? Use some g(x) instead of X. Most often use (natural) log of X. Sometimes 1/X. Always use a monotonic transformation.

83 Q9 Which variable to transform? Y, X, or both (all)? Using logs smooths the data, i.e., decreases the differences across different values that the variable takes. Taking logs allows negative values for a non-negative variable (if value < 1).

84 Y=Ln(X)

85 Log approximation to %-change ln Y + Y ln(y) Y Y

86 Q9 Which variable to transform? 1. Only Y Interpretation of β VI. Ln(Highprice) = β 0 + β VI VI + β TV TV + u Highprice = e β 0+β VI VI+β TV TV+u = e β 0 e β VIVI e β TVTV e u β VI = %-increase in Highprice due to a 1 unit increase in TV.

87 0.7 Accuracy of the %/ln - approximation % ln

88 Q9 Which variable to transform? 2. Only X Highprice = β 0 + β VI VI + β TV lntv + u Interpretation of β TV. β TV = change in Highprice due to a 1% increase in TV.

89 Q9 Which variable to transform? 3. Both Y and X ln(highprice) = β 0 + β VI VI + β TV lntv + u Interpretation of β TV. β TV = %-change in Highprice due to a 1% change in TV.

90 Variable linear loglinear linearlog loglog vi tv lntv const N r F

91 Q9 Interpretations of β TV Linear: a 1 unit increase in TV increases ticket prices by 0.064$ Loglinear: a 1 unit increase in TV increases ticket prices by 5.9% 0.06$. Linearlog: a 20% increase in TV increases ticket prices by = 0.04$ Loglog: a 1% increase in TV increases ticket prices by 0.188%. A 20% % 0.04$

92 Issues 10. What all can go wrong, and how would I know / find out?

93 Q10 What can go wrong? Internal validity. External validity.

94 Internal validity 1. Omitted variable bias. 2. Functional form misspecification (mistake). 3. Measurement error in variable(s). 4. Sample selection. 5. Simultaneous causality. 6. Non-homoskedastic errors.

95 Omitted variable bias Condition the one we have already discussed. Judicious choice of controls. New data. What else? we will get to these.

96 Functional form How can you be sure? 1. Tests between the functional forms you try. 2. Note: can easily test only those functional forms that are nested. Example #1: 1st and 2nd order polynomial - nested. Example #2: loglog and linear non-nested.

97 Functional form Try out different ones and check robustness of your results.

98 Measurement error in variables Case #1: Y measured with error, error random. Y obs = Y + error Let s have a look at our regression: Y = β 0 + β 1 X + u Y obs = Y + error = β 0 + β 1 X + u + error

99 Measurement error in variables Measurement error in Y not a big problem (as long as random). Leads to higher s.e., but no bias.

100 Measurement error in variables Case #2: X measured with error, error random. X obs = X + error Let s have a look at our regression: Y = β 0 + β 1 X obs + u Y = β 0 + β 1 (X + error) + u

101 Measurement error in variables One can show (see SW ch. 9.2): መβ 1 = σ X 2 σ 2 2 β 1 X + σ error σ X 2 σ 2 X +σ2 = signal to noise ratio. error The larger is the role of the error, the more is መβ 1 downward biased. = Attenuation bias.

102 Sample selection Your observations are not a random sample of the underlying population. Ex #1:Estimate the returns to entrepreneurship using 5 year old firms. The non-profitable exit. Ex #2: Estimate the returns to graduating quickly. Those who graduate quickly have unobservable skills that make them (un)attractive to employers.

103 Sample selection Ex #3: estimate effects of R&D subsidies. Firms that get subsidies not avg firms. Rule #1: think through and understand selection into your sample. We will discuss this later, but in general is an advanced topic.

104 Simultaneous causality Think of the determination of prices and quantities. Price affects how much is sold and produced. How much is bought and produced affects the price. simultaneous causality. We will come back to this.

105 Non-homoskedastic errors Heteroskedasticity. With sequential observations, maybe also correlation over time ( autocorrelation ). With e.g. geographical data, correlation across observation units ( clustering ). Affects statistical precision of individual coefficients, nothing else.

106 Variable homosk heterosk cluster: city cluster: theatre vi tv const N r F

107 External validity A branch of economics in which economic theory and statistical methods are fused in the analysis of numerical and institutional data Hood and Koopmans (1953, pp. xv.). Hood, W. C., and T. C. Koopmans (eds.), 1953, Studies in Econometric Method, Wiley.

108 External validity Any (material) change to any of the components of your study jeopardizes external validity. 1. Differences in (applicable) theory. 2. Differences in statistical method. 3. Differences in data (including in populations). 4. Differences in institutions.

109 External validity Is any study externally valid? Yes and no. Best to ensure internal validity, and conduct many studies.

Applied Statistics and Econometrics

Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple