ECON Introductory Econometrics. Lecture 7: OLS with Multiple Regressors Hypotheses tests

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1 ECON Introductory Econometrics Lecture 7: OLS with Multiple Regressors Hypotheses tests Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 7

2 Lecture outline 2 Hypothesis test for single coefficient in multiple regression analysis Confidence interval for single coefficient in multiple regression Testing hypotheses on 2 or more coefficients The F-statistic The overall regression F-statistic Testing single restrictions involving multiple coefficients Measures of fit in multiple regression model SER, R 2 and R 2 Relation between (homoskedasticity-only) F-statistic and the R 2 Interpreting measures of fit Interpreting stars in a table with regression output

3 Hypothesis test for single coefficient in multiple regression Thursday February 2 14:18: Page 1 analysis 1. regress test_score class_size el_pct, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(2, 417) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct _cons Does changing class size,while holding the percentage of English learners constant, have a statistically significant effect on test scores? (using a 5% significance level).

4 Hypothesis test for single coefficient in multiple regression analysis 4 Under the 4 Least Squares assumptions of the multiple regression model: Assumption 1: E (u i X 1i,..., X ki ) = 0 Assumption 2: (Y i, X 1i,..., X ki ) for i = 1,..., n are (i.i.d) Assumption 3: Large outliers are unlikely Assumption 4: No perfect multicollinearity The OLS estimators β j for j = 1,.., k are approximately normally distributed in large samples In addition t = β j β ( j0 ) N (0, 1) SE βj We can thus perform, hypothesis tests in same way as in regression model with 1 regressor.

5 Hypothesis test for single coefficient in multiple regression analysis 5 H 0 : β j = β j,0 H 1 : β j β j,0 Step 1: Estimate Y i = β 0 + β 1 X 1i β j X ji β 1 X ki + u i by OLS to obtain β j Step 2: Compute the standard error of β j (requires matrix algebra) Step 3: Compute the t-statistic Step 4: Reject the null hypothesis if t act = β j β j,0 ) SE ( βj t act > critical value or if p value < significance level

6 Hypothesis test for single coefficient in multiple regression analysis 6 Does changing class size, while holding the percentage of English learners constant, have a statistically significant effect on test scores? (using a 5% significance level) H 0 : β ClassSize = 0 H 1 : β ClassSize 0 Step 1: βclasssize = 1.10 Step 2: SE ( βclasssize ) = 0.43 Step 3: Compute the t-statistic t act = = 2.54 Step 4: Reject the null hypothesis at 5% significance level 2.54 > 1.96 and p value = < 0.05 Do we reject H 0 at a 1% significance level?

7 (R) 7 / / / / / / / / / / / / Statistics/Data Analysis Confidence interval for single coefficient in multiple regression 1. regress test_score class_size el_pct, robust Linear regression Number of obs = 420 F(2, 417) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct _cons % confidence interval for β ClassSize given in regression output 99% confidence interval for β ClassSize : ) β ClassSize ± 2.58 SE ( βclasssize 1.10 ± ( 2.21 ; 0.01)

8 Hypothesis tests on 2 or more coefficients 8 We add two more variables, both measuring what % of students come from families with low income meal pct i measures % of students in district eligible for a free lunch Tuesday February 7 11:33: Page 1 calw pct i measures % of students in district eligible for social assistance under a program called CALWORKS 1. regress test_score class_size el_pct meal_pct calw_pct, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons

9 Testing 2 hypotheses on 2 or more coefficients 9 H 0 : β meal pct = 0 H 1 : β meal pct 0 Step 1: βmeal pct = Step 2: SE ( βmeal pct ) = Step 3: t meal pct = = Step 4: Reject the null hypothesis at 5% significance level ( > 1.96 ) Step 1: βcalw pct = Step 2: SE H 0 : β calw pct = 0 H 1 : β calw pct 0 ( βcalw pct ) = Step 3: t calw pct = = 0.82 Step 4: Don t reject the null hypothesis at 5% significance level ( 0.82 < 1.96 )

10 Testing 1 hypothesis on 2 or more coefficients 10 Suppose we want to test hypothesis that both the coefficient on % eligible for a free lunch and the coefficient on % eligible for calworks are zero? H 0 : β meal pct = 0 & β calw pct = 0 H 1 : β meal pct 0 and/or β calw pct 0 What if we reject H 0 if either t meal pct or t calw pct exceeds 1.96 (5% sign. level)? If t meal pct and t calw pct are uncorrelated: Pr ( t meal pct > 1.96 and/or t calw pct > 1.96 ) = 1 Pr ( t meal pct 1.96 & t calw pct 1.96 ) = 1 Pr(t meal pct 1.96) Pr(t calw pct 1.96) = = > 0.05 If t meal pct and t calw pct are correlated situation is even more complicated!

11 Testing 1 hypothesis on 2 or more coefficients 11 If we want to test joint hypotheses that involves multiple coefficients we need to use an F-test based on the F-statistic F-statistic with q = 2 restrictions: when testing the following hypothesis H 0 : β 1 = 0 & β 2 = 0 H 1 : β 1 0 and/or β 2 0 the F-statistic combines the two t-statistics as follows ( ) F = 1 t1 2 + t2 2 2 ρ t1 t 2 t 1 t ρ 2 t 1 t 2 Equation illustrates that the F-statistic takes the potential correlation between the individual t-statistics into account In practice we use a software program to compute the F-statistic.

12 Tuesday February 7 14:44: Page 1 Testing 1 hypothesis on 2 or more coefficients (R) / / / / / / / / / / / / Statistics/Data Analysis 1. regress test_score class_size el_pct meal_pct calw_pct, robust Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons test meal_pct calw_pct ( 1) meal_pct = 0 ( 2) calw_pct = 0 F( 2, 415) =

13 Testing 1 hypothesis on 2 or more coefficients 13 We want to test hypothesis that both the coefficient on % eligible for a free lunch and the coefficient on % eligible for calworks are zero? H 0 : β meal pct = 0 & β calw pct = 0 H 1 : β meal pct 0 and/or β calw pct 0 The null hypothesis consists of two restrictions q = 2 It can be shown that the F-statistic with two restrictions has an approximate F 2, distribution in large samples Previous slide shows F = Table 4 (S&W page 807) shows that the critical value at a 5% significance level equals 3.00 (Pr H0 (F 2, > 3.00) = 0.05) This implies that we reject H 0 at a 5% significance level because > 3.00.

14 Printing is for personal, private use only. No part of this book may be reproduced or transmitted 14 wi Violators will be prosecuted. Testing 1 hypothesis on 2 or more coefficients

15 15 General procedure for testing joint hypothesis with q restrictions H 0 : β j = β j,0,..., β m = β m0 for a total of q restrictions H 1 : at least one of q restrictions under H 0 does not hold Step 1: Estimate Y i = β 0 + β 1 X 1i β j X ji β 1 X ki + u i by OLS Step 2: Compute the F-statistic Step 3: Reject the null hypothesis if F-statistic> Fq, critical with Pr ( ) F q, > Fq, critical = significance level

16 Tuesday February 7 16:02: Page 1 Testing a joint hypothesis with q=3 restrictions (R) / / / / / / / / / / / / Statistics/Data Analysis 1. regress test_score class_size el_pct meal_pct calw_pct, robust Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons test el_pct meal_pct calw_pct ( 1) el_pct = 0 ( 2) meal_pct = 0 ( 3) calw_pct = 0 F( 3, 415) =

17 Testing a joint hypothesis with q=3 restrictions 17 H 0 : β el pct = 0 & β meal pct = 0 & β calw pct = 0 H 1 : β el pct 0 and/or β meal pct 0 and/or β calw pct 0 Step 1: see previous slide Step 2: F-statistic= Step 3: We reject the null hypothesis at a 5% significance level because F-statistic> F critical 3, = 2.6

18 Testing a joint hypothesis with q=1 restriction 18 H 0 : β 1 = 0 H 1 : β 1 0 We have tested hypotheses that involve 1 restriction on 1 coefficient using the t-statistic We can also test these type of hypotheses using the F-statistic On slide 11 we saw that with q = 2 restrictions: ( ) F = 1 t1 2 + t2 2 2 ρ t1 t 2 t 1 t ρ 2 t 1 t 2 With q = 1 restriction we have F = t 2

19 Wednesday February 8 09:34: Page 1 Testing a joint hypothesis with q=1 restriction (R) / / / / / / / / / / / / Statistics/Data Analysis 1. regress test_score class_size el_pct meal_pct calw_pct, robust Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons test class_size ( 1) class_size = 0 F( 1, 415) = Prob > F =

20 The overall regression F-statistic 20 The overall F-statistic test the joint hypothesis that all the k slope coefficients are zero H 0 : β 1 = 0,..., β k = 0 for a total of q = k restrictions Tuesday February 7 11:33: Page 1 H 1 : at least one of q = k restrictions under H 0 does not hold 1. regress test_score class_size el_pct meal_pct calw_pct, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons Overall regression F-statistic equals

21 Testing single restrictions involving multiple coefficients 21 The % of students eligible for a free lunch and the % of students eligible for CALWORKS both measure whether students come from families with very low income Suppose we want to test whether coefficient on meal pct i equals the coefficient on calw pct i H 0 : β meal pct = β calw pct vs H 1 : β meal pct β calw pct There are two ways to perform this hypothesis test: 1 Transform regression model such that above H 0 is transformed to a null hypothesis that involves only 1 coefficient. 2 Test above restriction directly in software program (possible in Stata, not necessarily in all programs)

22 Testing single restrictions involving multiple coefficients 1.Transform regression model 22. H 0 : β 1 = β 2 vs H 1 : β 1 β 2 Suppose we have regression model with only two regressors: Y i = β 0 + β 1 X 1i + β 2 X 2i + u i add & subtract β 2 X 1i = β 0 + β 1 X 1i β 2 X 1i + β 2 X 1i + β 2 X 2i + u i rewrite = β 0 + (β 1 β 2 ) X 1i + β 2 (X 1i + X 2i ) + u i = β 0 + γx 1i + β 2 W i + u i Transformed hypothesis test: H 0 : γ = 0 vs H 1 : γ 0

23 Wednesday February 8 09:38: Page 1 Testing single restrictions involving multiple coefficients 1.Transform regression model 1. gen W = meal_pct + calw_pct regress test_score class_size el_pct meal_pct W, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct W _cons test meal_pct ( 1) meal_pct = 0 F( 1, 415) =

24 Testing Wednesday single February restrictions 8 10:00: involving Page 1 multiple coefficients 2. Test restriction directly in software program 1. regress test_score class_size el_pct meal_pct calw_pct, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons test meal_pct=calw_pct ( 1) meal_pct - calw_pct = 0 F( 1, 415) =

25 Measures of fit in multiple regression model 25 There are 3 measures that measure how well the OLS estimate of the multiple regression line describes or fits the data The Standard Error of the Regression (SER) SER = 1 n n k 1 i=1 û i The R 2 R 2 = ESS TSS = 1 SSR TSS The adjusted R 2 R 2 = 1 n 1 SSR n k 1 TSS

26 The SER in the multiple regression model 26 SER = 1 n û i n k 1 i=1 The SER estimates the standard deviation of the error term u i Tuesday February 7 11:33: Page 1 It measures spread of distribution of Y around regression line. n k 1 is degrees of freedom correction 1. regress test_score class_size el_pct meal_pct calw_pct, robust (R) / / / / / / / / / / / / Statistics/Data Analysis Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons The SER = Root MSE =

27 The R 2 27 The R 2 is the fraction of the sample variance of Y i explained/predicted by X 1i,..., X ki The R 2 is the ratio of the sample variance of Ŷi and the sample variance of Y i ) 2 n R 2 = ESS TSS = i=1 (Ŷi Y ( ) 2 = 1 SSR n TSS i=1 Y i Y In the multiple regression model the R 2 always increases when a regressor is added Also when this additional regressor explains very little of Y! Reason: whenever estimated coefficient on additional regressor is not exactly zero the SSR = n i=1 ûi is reduced.

28 The R 2 (R) / / / / / / / / / / / / Statistics/Data Analysis regress test_score class_size el_pct meal_pct, robust Linear regression Number of obs = 420 F(3, 416) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct _cons regress test_score class_size el_pct meal_pct calw_pct, robust Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons

29 The R 2 and the homoskedasticity-only F-statistic 29 There is a link between the F-statistic and the R 2 This link can be easily displayed when we assume homoskedastic errors: F homoskedasticity = (SSR restricted SSR unrestricted ) /q SSR unrestricted / (n k unrestricted 1) with SSR restricted the sum of squared residuals assuming the q restriction under H o hold and SSR unrestricted the sum of squared residuals assuming the q restriction under H o do not hold Since R 2 = 1 SSR we can also write TSS ( ) R 2 F homoskedasticity unrestricted Rrestricted 2 /q = ( ) 1 R 2 unrestricted / (n kunrestricted 1)

30 The R 2 and the homoskedasticity-only F-statistic 1. regress test_score class_size (R) 30 / / / / / / / / / / / / Statistics/Data Analysis Source SS df MS Number of obs = 420 F(1, 418) = Model Residual R-squared = Adj R-squared = Total Root MSE = test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size _cons regress test_score class_size el_pct meal_pct calw_pct Source SS df MS Number of obs = 420 F(4, 415) = Model Residual R-squared = Adj R-squared = Total Root MSE = test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons

31 The R 2 and the homoskedasticity-only F-statistic 31 H 0 : β el pct = 0 & β meal pct = 0 & β calw pct = 0 H 1 : β el pct 0 and/or β meal pct 0 and/or β calw pct 0 From previous slide: R 2 restricted = R 2 unrestricted = n = 420, q = 3 and k unrestricted = 4. F homoskedasticity = ( ) /3 ( ) / ( ) =

32 Wednesday February 8 10:58: Page 1 The R 2 and the homoskedasticity-only F-statistic (R) / / / / / / / / / / / / Statistics/Data Analysis 1. regress test_score class_size el_pct meal_pct calw_pct Source SS df MS Number of obs = 420 F(4, 415) = Model Residual R-squared = Adj R-squared = Total Root MSE = test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons test el_pct meal_pct calw_pct ( 1) el_pct = 0 ( 2) meal_pct = 0 ( 3) calw_pct = 0 F( 3, 415) =

33 The adjusted R 2 33 The R 2 increases whenever a regressor is added. also when regressor explains very little of Y and does not improve the fit of the model We can correct for this by deflating the R 2 which gives the adjusted R 2 R 2 = 1 n 1 SSR n k 1 TSS When we add a regressor: SSR = n i=1 û2 i decreases (whenever coefficient estimate is not exactly zero) k increases, which increases n 1 n k 1

34 Wednesday February 8 11:28: Page 1 The adjusted R 2 (R) / / / / / / / / / / / / Statistics/Data Analysis regress test_score class_size el_pct meal_pct, robust Linear regression Number of obs = 420 F(3, 416) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct _cons display e(r2_a) regress test_score class_size el_pct meal_pct calw_pct, robust Linear regression Number of obs = 420 F(4, 415) = R-squared = Root MSE = Robust test_score Coef. Std. Err. t P> t [95% Conf. Interval] class_size el_pct meal_pct calw_pct _cons display e(r2_a)

35 Interpreting measures of fit 35 The R 2 or adjusted R 2 tell you whether the regressors are good at predicting or explaining the values of the dependent variable in the sample of data. An R 2 (R 2 ) near 1 implies that X 1i,.., X ki produce good predictions of Y i An R 2 (R 2 ) near 0 implies that X 1i,.., X ki don t produce good predictions of Y i It is important not rely too much on the R 2 (R 2 ) Maximizing the R 2 (R 2 ) rarely answers an economically meaningfull question!

36 Interpreting measures of fit 36 A high R 2 (R 2 ) does NOT tell you whether: An included variable is statistically significant. you need to perform an hypothesis test using a t- or F-statistic The coefficients measure the true causal effect of the regressors The OLS estimators are only unbiased & consistent estimators if all Least Squares assumptions hold! There is omitted variable bias Omitted variable bias implies violation of 1st Least Squares assumption, this cannot be assessed by R 2 (R 2 ) You have chosen the most appropriate set of regressors Appropriate depends on question of interest When estimating a causal effect, OLS assumptions should hold

37 Interpreting *stars* in a table with regression output 37 Dependent variable: district average test scores Class size *** ** *** *** (0.519) (0.433) (0.270) (0.269) Percentage of English learners *** *** *** (0.031) (0.033) (0.036) Percentage with free lunch *** *** (0.024) (0.038) Percentage with CALWORKS (0.059) N Note: * significant at 10% level, ** significant at 5% level, *** significant at 1% level