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Date Created: 10/03/15
Wooldridge Introductory Econometrics 2d ed Chapter 4 Multiple regression analysis Inference We have discussed the conditions under which OLS estimators are unbiased and derived the variances of these estimators under the GaussMarkov assumptions The Gauss Markov theorem establishes that OLS estimators have the smallest variance of any linear unbiased estimators of the population parameters We must now more fully characterize the sampling distribution of the OLS estimators beyond its mean and variance so that we may test hypotheses on the population parameters To make the sampling distribution tractable we add an assumption on the distribution of the errors Proposition 1 MLR6 Normality The population error u is in dependent of the explanatory variables 31 ask and is normally distributed with zero mean and constant variance u N N u 02 This is a much stronger assumption than we have previously made on the distribution of the errors The assumption of normality as we have stated it subsumes both the assumption of the error process being independent of the explanatory variables and that of homoskedasticity For crosssectional regression analysis these six assumptions de ne the classical linear model The rationale for normally distributed errors is often phrased in terms of the many factors in uencing y being additive appealing to the Central Limit Theorem to suggest that the sum of a large number of random factors will be normally distributed Although we might have reason in a particular context to doubt I this rationale we usually use it as a working hypothesis Various transforrnations such as taking the logarithm of the dependent variable are often motivated in terms of their inducing normality in the resulting errors What is the importance of assuming normality for the error process Under the assumptions of the classical linear model normally distributed errors give rise to normally distributed OLS estimators bj N 6 Var lt12 1 which will then imply that b M N lt0 1 2 8b This follows since each of the bj can be written as a linear combination of the errors in the sample Since we assume that the errors are independent identically distributed normal random variates any linear combination of those errors is also normally distributed We may also show that any linear combination of the bj is also normally distributed and a subset of these estimators has a joint normal distribution These properties will come in handy in formulating tests on the coef cient vector We may also show that the OLS estimators will be approximately normally distributed at least in large samples even if the underlying errors are not normally distributed Testing an hypothesis on a single j To test hypotheses about a single population parameter we start with the model containing k regressors 3230311 223kku 3 2 Under the classical linear model assumptions a test statistic formed om the OLS estimates may be expressed as by Sb j N tn k l Why does this test statistichiffer from 2 above In that expression we considered the variance of bj as an expression including 0 the unknown standard deviation of the error term that is 02 In this operational test statistic 4 we have replaced 0 with a consistent estimate 3 That additional source of sampling variation requires the switch from the standard normal distribution to the t distribution with n k 1 degrees of freedom Where n is not all that large relative to k the resulting t distribution will have considerably fatter tails than the standard normal Where n k 1 is a large number greater than 100 for instance the t distribution will essentially be the standard normal The net effect is to make the critical values larger for a nite sample and raise the threshold at which we will conclude that there is adequate evidence to reject a particular hypothesis The test statistic 4 allows us to test hypotheses regarding the population parameter j in particular to test the null hypothesis H0 j 0 5 for any of the regression parameters The tstatistic used for this test is merely that printed on the output when you run a regression in Stata or any other program the ratio of 3 the estimated coef cient to its estimated standard error If the null hypothesis is to be rejected the tstat must be larger in absolute value than the critical point on the tdistribution The tstat will have the same sign as the estimated coef cient since the standard error is always positive Even if j is actually zero in the population a sample estimate of this parameter bj will never equal exactly zero But when should we conclude that it could be zero When its value cannot be distinguished om zero There will be cause to reject this null hypothesis if the value scaled by its standard error exceeds the threshold For a twotailed test there will be reason to reject the null if the tstat takes on a large negative value or a large positive value thus we reject in favor of the alternative hypothesis of j 7 0 in either case This is a twosided alternative giving rise to a twotailed test If the hypothesis is to be tested at eg the 95 level of con dence we use critical values from the tdistribution which isolate 25 in each tail for a total of 5 of the mass of the distribution When using a computer program to calculate regression estimates we usually are given the pvalue of the estimate that is the tail probability corresponding to the coef cient s tvalue The pvalue may usefully be considered as the probability of observing a tstatistic as extreme as that shown z flhe null hypothesis is true If the tvalue was equal to e g the 95 critical value the pvalue would be exactly 005 If the tvalue was higher the pvalue would be closer to zero and vice versa Thus we are looking for small pvalues as indicative of rejection A pvalue of 092 for instance corresponds to an 4 hypothesis that can be rejected at the 8 level of con dence thus quite irrelevant since we would expect to nd a value that large 92 of the time under the null hypothesis On the other hand a pvalue of 008 will reject at the 90 level but not at the 95 level only 8 of the time would we expect to nd a tstatistic of that magnitude if H 0 was true What if we have a onesided alternative For instance we may phrase the hypothesis of interest as H0 j gt 0 6 H A I j S 0 Here we must use the appropriate critical point on the tdistribution to perform this test at the same level of con dence If the point estimate bj is positive then we do not have cause to reject the null If it is negative we may have cause to reject the null if it is a suf ciently large negative value The critical point should be that which isolates 5 of the mass of the distribution in that tail for a 95 level of con dence This critical value will be smaller in absolute value than that corresponding to a twotailed test which isolates only 25 of the mass in that tail The computer program always provides you with a pvalue for a twotailed test if the pvalue is 008 for instance it corresponds to a onetailed pvalue of 004 that being the mass in that tail Testing other hypotheses about j Every regression output includes the information needed to test the twotailed or onetailed hypotheses that a population parameter equals zero What if we want to test a different hypothesis about the value of that parameter For instance we 5 would not consider it sensible for the mp0 for a consumer to be zero but we might have an hypothesized value of say 08 implied by a particular theory of consumption How might we test this hypothesis If the null is stated as H 0 I j Z CLj where aj is the hypothesized value then the appropriate test statistic becomes M m4 8 Sb and we may simply calculate that quantity and compare it to the appropriate point on the tdistribution Most computer programs provide you with assistance in this effort for instance if we believed that aj the coef cient on bdrms should be equal to 20000 in a regression of house prices on square footage and bdrms eg using HPRICEl we would use Stata s test command regress price bdrms sqrft test bdrms20000 where we use the name of the variable as a shorthand for the name of the coef cient on that variable Stata in that instance presents us with l bdrms 200000 F l 85 026 Prob gt F 06139 making use of an Fstatistic rather than a tstatistic to 6 perform this test In this particular case of an hypothesis involving a single regression coef cient we may show that this Fstatistic is merely the square of the associated tstatistic The pvalue would be the same in either case The estimated coef cient is 15198 19 with an estimated standard error of 9483517 Plugging in these values to 8 yields a tstatistic di b bdrms 2 O O O 0 se bdrms 5 O 6 3 3 2 O 8 which squared is the Fstatistic shown by the test command Just as with tests against a null hypothesis of zero the results of the test command may be used for onetailed tests as well as twotailed tests then the magnitude of the coef cient matters ie the fact that the estimated coef cient is about 15000 means we would never reject a null that it is less than 20000 and the pvalue must be adjusted for one tail Any number of test commands may be given after a regress command in Stata testing different hypotheses about the coef cients Con dence intervals As we discussed in going over Appendix C we may use the point estimate and its estimated standard error to calculate an hypothesis test on the underlying population parameter or we may form a con dence interval for that parameter Stata makes that easy in a regression context by providing the 95 con dence interval for every estimated coef cient If you want to use some other level of signi cance you may either use 7 the level option on regress eg regress price bdrms sqrft level 90 or you may change the default level for this run With set level All 1rther regressions will report con dence intervals With that level of con dence To connect this concept to that of the hypothesis test consider that in the above example the 95 con dence interval for bdrms extended om 3657 581 to 3405396 thus an hypothesis test With the null that bdrms takes on any value in this interval including zero Will not lead to a rejection Testing hypotheses about a single linear combination of the parameters Economic theory Will often suggest that a particular linear combination of parameters should take on a certain value for instance in a CobbDouglas production 1nction that the slope coef cients should sum to one in the case of constant returns to scale CRTS Q AL 1K 2E s 9 logQ logA I llogL glogK 3logE 11 Where K L E are the factors capital labor and energy respectively We have added an error term to the doublelog transforrned version of this model to represent it as an empirical relationship The hypothesis of CRTS may be stated as H0331 2 3Z1 10 The test statistic for this hypothesis is quite straightforward W N MM 11 8mmm and its numerator may be easily calculated The denominator 8 however is not so simple it represents the standard error of the linear combination of estimated coef cients You may recall that the variance of a sum of random variables is not merely the sum of their variances but an expression also including their covariances unless they are independent The random variables 91 b2 b3 are not independent of one another since the underlying regressors are not independent of one another Each of the underlying regressors is assumed to be independent of the error term u but not of the other regressors We would expect for instance that rms with a larger capital stock also have a larger labor force and use more energy in the production process The variance and standard error that we need may be readily calculated by Stata however from the variancecovariance matrix of the estimated parameters via the test command test caplaborenergyl will provide the appropriate test statistic again as an F statistic with a pvalue You may interpret this value directly If you would like the point and interval estimate of the hypothesized combination you can compute that after a regression with the l incom linear combination command lincom caplaborenergy will show those values but will only test that their sum is zero We may also use this technique to test other hypotheses than addingup conditions on the parameters For instance consider a twofactor CobbDouglas function in which you have only labor and capital and you want to test the hypothesis that labor s share 9 is 23 This implies that the labor coef cient should be twice the capital coef cient or H0 BL22BK07 12 H L2 0r 0 m H 0 3 3 L 23K 2 0 Note that this does not allow us to test a nonlinear hypothesis on the parameters but considering that a ratio of two parameters is a constant is not a nonlinear restriction In the latter form we may specify it to Stata s test command as test labor 2cap O In fact Stata will gure out that form if you specify the hypothesis as test labor2cap rewriting it in the above form but it is not quite smart enough to handle the ratio form It is easy to rewrite the ratio form into one of the other forms Either form will produce an Fstatistic and associated pvalue related to this single linear hypothesis on the parameters which may be used to make a judgment about the hypothesis of interest Testing multiple linear restrictions When we use the test command an Fstatistic is reported even when the test involves only one coef cient because in general hypothesis tests may involve more than one restriction on the population parameters The hypotheses discussed above even that of CRT S involving several coef cients still only represent one restriction on the parameters For instance if CRT S 10 is imposed the elasticities of the factors of production must sum to one but they may individually take on any value But in most applications of multiple linear regression we concern ourselves with joint tests of restrictions on the parameters The simplest joint test is that which every regression reports the socalled ANOVA F test which has the null hypothesis that each of the slopes is equal to zero Note that in a multiple regression specifying that each slope individually equals zero is not the same thing as specifying that their sum equals zero This ANOVA ANalysis Of VAriance Ftest is of interest since it essentially tests whether the entire regression has any explanatory power The null hypothesis in this case is that the model is y o u that is none of the explanatory variables assist in explaining the variation in y We cannot test any hypothesis on the R2 of a regression but we will see that there is an intimate relationship between the R2 and the ANOVA F S S E 2 1 Eig 13 F SSEk SSR n k 1 F RZk 1 R2ltn ltk1 where the ANOVA F the ratio of mean square explained variation to mean square unexplained variation is distributed as F TIL k 1 under the null hypothesis For a simple regression this statistic is F54 which is identical to 151an that is the square of the t statistic for the slope coef cient with precisely the 11 same 19 value as that t statistic In a multiple regression context we do not often nd an insigni cant F statistic since the null hypothesis is a very strong statement that none of the explanatory variables taken singly or together explain any signi cant fraction of the variation of y about its mean That can happen but it is often somewhat unlikely The ANOVA F tests k exclusion restrictions that all k slope coef cients are jointly zero We may use an Fstatistic to test that a number of slope coef cients are jointly equal to zero For instance consider a regression of 353 major league baseball players salaries from MLBl If we regress lsalary log of player s salary on years number of years in majors game syr number of games played per year and several variables indicating the position played f rstbase scndbase shrtstop thrdbase catcher we get an R2 of 06105 and an ANOVA F with 7 and 345 df of 7724 with a p value of zero The overall regression is clearly signi cant and the coef cients on years and game syr both have the expected positive and signi cant coef cients Only one of the ve coef cients on the positions played however are signi cantly different om zero at the 5 level scndbase with a negative value 0034 and a p value of 0015 The frstbase and shrtstop coef cients are also negative but insigni cant while the thrdbase and catcher coef cients are positive and insigni cant Should we just remove all of these variables except for scndbase The Ftest for these ve exclusion restrictions will provide an answer to that question 12 test frstbase scndbase shrtstop thrdbase catcher frstbase scndbase shrtstop thrdbase catcher 00 F 5 345 237 Prob gt F 00390 At the 95 level of signi cance these coef cients are not each zero That result of course could be largely driven by the scndbaseco cmnt test frstbase shrtstop thrdbase catcher l frstbase 00 2 shrtstop 00 3 thrdbase 00 4 catcher 00 F 4 345 156 Prob gt F 01858 So perhaps it would be sensible to remove these four Which even When taken together do not explain a meaningful fraction of the variation in lsalary But this illustrates the point of the joint hypothesis test the result of simultaneously testing several hypotheses that for instance individual coef cients are equal to zero cannot be inferred om the results of the individual tests If each coef cient is signi cant then a joint test Will surely reject the joint exclusion restriction but the converse is assuredly false AAAA 0000 GOOD U39IIPUJNH l3 Notice that a joint test of exclusion restrictions may be easily conduced by Stata s test command by merely listing the variables whose coef cients are presumed to be zero under the null hypothesis The resulting test statistic is an F with as many numerator degrees of freedom as there are coef cients or variables in the list It can be written in terms of the residual sums of squares SSRS of the unrestricted and restricted models F SSRT SSRW q SSRW n k 1 Since adding variables to a model will never decrease SSR nor decrease R2 the restricted model in which certain coef cients are not freely estimated om the data but constrained must have SSR at least as large as the unrestricted model in which all coef cients are datadetermined at their optimal values Thus the difference in the numerator is non negative If it is a large value then the restrictions severely diminish the explanatory power of the model The amount by which it is diminished is scaled by the number of restrictions q and then divided by the unrestricted model s 32 If this ratio is a large number then the average cost per restriction is large relative to the explanatory power of the unrestricted model and we have evidence against the null hypothesis that is the F statistic will be larger than the critical point on an F table with q and n k 1 degrees of freedom If the ratio is smaller than the critical value we do not reject the null hypothesis and conclude that the restrictions are consistent with the data In 14 14 this circumstance we might then reforrnulate the model with the restrictions in place since they do not con ict with the data In the baseball player salary example we might drop the four insigni cant variables and reestimate the more parsimonious model Testing general linear restrictions The apparatus described above is far more powerful than it might appear We have considered individual tests involving a linear combination of the parameters e g CRTS and joint tests involving exclusion restrictions as in the baseball players salary example But the subset F test de ned in 14 is capable of being applied to any set of linear restrictions on the parameter vector for instance that l 032 1 33 1 34 1 and g 1 What would this set of restrictions imply about a regression of g on X1 X2 X3 X4 X5 That regression in its unrestricted form would have k 5 with 5 estimated slope coef cients and an intercept The joint hypotheses expressed above would state that a restricted form of this equation would have three fewer parameters since l would be constrained to zero B5 to l and one of the coef cients 82 3 m expressed in terms of the other two In the terminology of 14 q 3 How would we test the hypothesis We can readily calculate SSRW but what about 88R One approach would be to algebraically substitute the restrictions in the model estimate that restricted model and record its SSRT value This can be done with any computer program that estimates a multiple regression but it requires that you do the algebra and transform the variables accordingly For 15 instance constraining B5 to 1 implies that you should form a new dependent variable y 1 X5 Alternatively if you are using a computer program that can test linear restrictions you may use its features Stata Will test general linear restrictions of this sort With the test command regress y x1 X2 X3 x4 x5 test x1 test x2x3x4l accum test X5l accum The nal test command in this sequence Will print an Fstatistic for the set of three linear restrictions on the regression for instance 1 years 00 2 frstbase scndbase shrtstop 10 3 sbases l0 F 3 347 3854 Prob gt F 00000 The accum option on the test command indicates that these tests are not to be performed separately but rather jointly The nal Ftest Will have three numerator degrees of freedom because you have speci ed three linear hypotheses to be jointly applied to the coef cient vector This syntax of test may be used to construct any set of linear restrictions on the coef cient vector and perform the joint test for the validity of those restrictions The test statistic Will reject the null hypothesis that the restrictions are consistent With the data if its value is large relative to the underlying Fdistribution l6
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