1 / 22

Some Topics In Multivariate Regression

Some Topics In Multivariate Regression. Some Topics. We need to address some small topics that are often come up in multivariate regression. I will illustrate them using the Housing example. . Some Topics. Confidence intervals Scale of data Functional Form

devon
Download Presentation

Some Topics In Multivariate Regression

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Some Topics In Multivariate Regression

  2. Some Topics • We need to address some small topics that are often come up in multivariate regression. • I will illustrate them using the Housing example.

  3. Some Topics • Confidence intervals • Scale of data • Functional Form • Tests of multi-coefficient hypotheses

  4. Woldridgerefs to date • Chapter 1 • Chapter 2.1, 2.2,2.5 • Chapter 3.1,3.2,3.3 • Chapter 4.1, 4.2, 4.3, 4.4

  5. Confidence Intervals (4.3) • We can construct an interval within which the true value of the parameter lies • We have seen that • P(-1.96 ≤ t ≤ 1.96)=0.95 for large N-K • More generally:

  6. Interval b± tc *se(b) will contain b with (1-a)% confidence. • Where tc is “critical value” and is determined by the significance level (a) and the degrees of freedom (df=N-K) • For the case where N-K is large (>100) and a is 5% then tc = 1.96 • Same as the set of values of beta, which could not be rejected if they were null hypotheses • The range of possible values consistent with the data • A way of avoiding some of the ambiguity in the formulation of hypothesis tests • Formally: A procedure which will generate an interval containing the true value (1-a)% times in repeated samples

  7. Level Option • Stata command: regress … , level(95) • Note: in assignments I want you to do it manually regress price inc_pchstock_pc if year<=1997 Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 88.31 Model | 1.1008e+10 2 5.5042e+09 Prob > F = 0.0000 Residual | 1.5581e+09 25 62324995.9 R-squared = 0.8760 -------------+------------------------------ Adj R-squared = 0.8661 Total | 1.2566e+10 27 465423464 Root MSE = 7894.6 ------------------------------------------------------------------------------ price | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- inc_pc | 10.39438 1.288239 8.07 0.000 7.741204 13.04756 hstock_pc | -637054.1 174578.5 -3.65 0.001 -996605.3 -277503 _cons | 135276.6 35433.83 3.82 0.001 62299.24 208253.9 ------------------------------------------------------------------------------

  8. Scale (2.4 & 6.1) • The scale of the data may matter • i.e. whether we measure house prices in € or €bn or even £ or $ • Exercise: try this with housing or consumption examples • Basic model: yi = b1 + b2 xi + ui

  9. Change scale of xi : xi* = xi/c • Estimate: yi = b1* + b2* xi*+ ui • b2*= c.b2 • se(b2) = c.se(b2) • Slope coefficient and se change, all other statistics (t-stats, R2, F, etc.) unchanged.

  10. Change scale of yi : yi* = yi/c • Estimate y*i = b1* + b2* xi + ui • b2*= b2 /c • b1*= b1 /c • se(b2) = se(b2)/c • se(b1) = se(b1)/c • t-stats, R2, F unchanged • Both X and Y rescaled yi* = yi/c, xi* = xi/c • Estimate y*i = b1* + b2* x* + ui • If rescaled by same amount: • b1*= b1 /cse(b1) = se(b1)/c • b2 and se(b2) unchanged • t-stats, R2, F unchanged

  11. Functional Form (6.2) • Four common functional forms • Linear: qt = a +  pt+ ut • Log-Log: lnqt = a + lnpt + ut • Semilog: qt = a + lnpt + ut or lnqt = a + pt + ut • How to choose? • Which fits the data best (cannot compare R2 unless y is same) • Which is most convenient (do we want elasticity, rate of return?) • How trade-off two goals

  12. Elasticity and Marginal Effects

  13. Two housing models • The level variables: marginal effects regress price inc_pchstock_pc if year<=1997 Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 88.31 Model | 1.1008e+10 2 5.5042e+09 Prob > F = 0.0000 Residual | 1.5581e+09 25 62324995.9 R-squared = 0.8760 -------------+------------------------------ Adj R-squared = 0.8661 Total | 1.2566e+10 27 465423464 Root MSE = 7894.6 ------------------------------------------------------------------------------ price | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- inc_pc | 10.39438 1.288239 8.07 0.000 7.741204 13.04756 hstock_pc | -637054.1 174578.5 -3.65 0.001 -996605.3 -277503 _cons | 135276.6 35433.83 3.82 0.001 62299.24 208253.9 ------------------------------------------------------------------------------

  14. Log on log formulation regress lpricelinclh if year<=1997 Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 2, 25) = 86.21 Model | .791044208 2 .395522104 Prob > F = 0.0000 Residual | .11469849 25 .00458794 R-squared = 0.8734 -------------+------------------------------ Adj R-squared = 0.8632 Total | .905742698 27 .033546026 Root MSE = .06773 ------------------------------------------------------------------------------ lprice | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- linc | 1.67764 .2168253 7.74 0.000 1.23108 2.1242 lh | -2.011761 .5228058 -3.85 0.001 -3.0885 -.9350227 _cons | -7.039114 2.687196 -2.62 0.015 -12.5735 -1.504731 ------------------------------------------------------------------------------

  15. F-tests • Often we will want to test joint hypotheses • i.e. hypotheses that involve more than one coefficient • Linear restrictions • Three examples (using the log model) • H0: bH= 0 & bI= 0 H1: bH≠ 0 or bI≠0 • H0: bH= 0 & bI= 1 H1: bH≠ 0 or bI≠1 • H0: bH+bI= 1 H1: bH+bI ≠ 1

  16. 1. Test of Joint Significance • Example 1 is given the special name of “test of joint significance” • Could do K-1 t-tests, one on each of the K-1 variables • This would not be a joint hypothesis but a series of K-1 individual hypotheses • The two are not equivalent

  17. Why Joint Hypotheses matter • Recall the sampling makes the estimators random variables • Estimators of different coefficients are correlated random variables • All the coeff are estimated from same sample in any one regression • Making statements about one coefficient implies a statement about another • Formally P(b2=0).P(b3 =0)  P(b2=b3 =0)

  18. So the set of regressions in which both are zero is smaller than the set in which either one are zero • This intuition holds for more general hypotheses.

  19. Testing Joint Significance • As we look at all the variables it is natural to focus on the ESS • We form a test statistic • If the null hypothesis is true the ESS will be zero and RSS will be large

  20. So we can reject the null hypothesis if the test statistic is greater than zero • How much greater? • Greater than a critical value got from the F-distribution tables with three parameters • Significance level • Df1=K-1 • Df2=N-K • The test is so useful it is reported by stata

  21. Formal Procedure • State the Hypothesis we want to test H0: bH= 0 & bI= 0 H1: bH≠ 0 or bI≠0 • Calculate the test statistic assuming that null is true. 86.21 • Critical Value: • F(2,25)= 3.39 at 5% significance level • Stata: di invFtail(2,25,0.05) • As F>critical value we can reject the null hypothesis at the 5% signifacance level

More Related