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Applied Econometrics

by: Jalyn Schaefer II

Applied Econometrics A,RESEC 213

Jalyn Schaefer II

GPA 3.84


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This 9 page Class Notes was uploaded by Jalyn Schaefer II on Thursday October 22, 2015. The Class Notes belongs to A,RESEC 213 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 40 views. For similar materials see /class/226561/a-resec-213-university-of-california-berkeley in Agricultural & Resource Econ at University of California - Berkeley.

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Date Created: 10/22/15
Irnbens Lecture Notes 1 ARE213 Spring 06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics ORDINARY LEAST SQUARES I ESTIMATION lNFERENCE AND PREDICTING OUTCOMES W 421 4 Let us review the basics of the linear model We have N units individuals rms or other economic agents drawn randomly from a large population On each unit we observe on outcome Y for unit 239 and a K dimensional column vector of explanatory variables X X1X2 XlK where typically the rst covariate is a constant X 1 for all 239 1 N We are interested in explaining the distribution of Y in terms of the explanatory variables X using a linear model Y 3X 8 1 In this equation is a K dimensional column vector In matrix notation Y X e where Y is an N dimensional column vector and X is an N gtlt K dimensional matrix with 2th row equal to X Avoiding vector and matrix notation completely K YI l Xil i n i K XiK i gi Z k39Xik8i k1 We assume that the residuals e are independent of the covariates or regressors and normally distributed with mean zero and variance 02 Assumption 1 eilXi N N0 02 We can weaken this considerably First we could relax normality and only assume indepen dence Irnbens Lecture Notes 1 ARE213 Spring 06 2 Assumption 2 e l X combined with the normalization that 0 We can even weaken this assumption further by requiring only mean independence Assumption 3 EelX 0 or even further requiring only zero correlation Assumption 4 Elsi X 0 We will also assume that the observations are drawn randomly from some population We can also do most of the analysis by assuming that the covariates are xed but this complicates matters for some results and it does not help very much See the discussion on xed versus random covariates in Wooldridge page 9 Assumption 5 The pairs XY are independent draws fram same distributian with the rst twa maments 0f X nite The ordinary least squares estimator for solves N l Yi 7 Xi 239 111111 12 6 This leads to B X X 1X Y The exact distribution of the ols estimator under the normality assumption in Assumption Imbens Lecture Notes 1 ARE213 Spring 06 3 Without the normality of the 8 it is dif cult to derive the exact distribution of However under the independence Assumption 2 and a second moment condition on e variance nite and equal to 02 we can establish asymptotic normality mm 7 6 N 0 02 EXX 1 Typically we do not know 02 We can consistently estimate it as Dividing by N 7 K 7 1 rather than N corrects for the fact that K 1 parameters are estimated before calculating the residuals Y 7 B Xi This correction does not matter in large samples and in fact the maximum likelihood estimator equal to A2 1 N A 2 am NEGlimo i1 is a perfectly reasonable alternative So in practice whether we have asymptotic normality or not we will use the following distribution for B 6 w M6 V estimated as V ElXX lY1 02 ElX XlYly 2 U2 N with corresponding estimator N 71 V z Z xXgt lt3 Imbens Lecture Notes 1 ARE213 Spring 06 4 Often we are interested in one particular coef cient Suppose for example we are inter ested in 6k In that case we have Bk Nlt 17ka7 where Vi is the 2394 element of the matrix V We can use this for constructing con dence intervals for a particular coef cient For example7 a 95 con dence interval for l would be Bk 7196 xVk Bk 7196 17 We can also use this to test whether a particular coef cient is equal to some preset number For example7 if we want to test whether 6k is equal to 017 we construct the t statistic t 3 i017 M and compare it to a standard normal distribution a normal distribution with mean zero and variance equal to one If we want to do a two sided test at the 10 level7 we would compare the absolute value of this t statistic to 16457 the 095 quantile of the standard normal distribution meaning that if Z has a standard normal N01 distribution7 the probability that Z is less than 1645 is equal to 095 Let us look at some real data The following regressions are estimated on data from the National Longitudinal Survey of Youth NLSY The data set used here consists of 935 observations on usual weekly earnings7 years of education7 and experience calculated as age minus education minus six Table 1 presents some summary statistics for these 935 observations The particular data set consists of men between 28 and 38 years of age at the time the wages were measured We will use these data to look at the returns to education Mincer developed a model that leads to the following relation between log earnings7 education and experience for individual 239 logearningsl l g educl g experl 64 exper 8139 Imbens Lecture Notes 1 ARE213 Spring 06 5 Table 1 SUMMARY STATISTICS NLS DATA Variable Mean Median Min Max Standard Dev Weekly Wage in 421 388 58 2001 199 Log Weekly Wage 594 596 406 760 044 Years of Education 135 12 9 18 22 Age 331 33 28 38 31 Years of Experience 136 13 5 23 38 Estimating this on the NLSY data leads to logvarr1gsi 4016 0092educl 0079experl 7 0002exper 0222 0008 0025 0001 The estimated standard deviation of the residuals is 039 041 In brackets are the standard errors for the parameters estimates7 based on the square roots of the diagonal elements of the variance estimate Using the estimates and the standard errors we can construct the con dence intervals For example7 a 95 con dence interval for the returns to education7 measured by the parameter 32713 01095 00923 7196 00087 00923 196 0008 007757 01071 The t statistic for testing 02 01 is 00923 7 0113901727 0008 so at the 90 level we do not reject the hypothesis that 02 is equal to 01 Now suppose we wish to use these estimates for predicting a more complex change For example7 suppose we want to see what the estimated effect is on the log of weekly earnings Imbens Lecture Notes 1 ARE213 Spring 06 6 of increasing a persons education by one year Because changing an individualls education also changes their experience in this case it automatically reduces it by one year this effect depends not just on 61 To make this speci c let us focus on an individual with twelve years of education high school and ten years of experience so that experZ is equal to 100 The expected value of this personls log earnings is logm1gsi 4016 0092 12 0079 10 7 0002 100 57191 Now change this personls education to 13 Their experience will go down to 9 and exper2 will go down to 81 Hence the expected log earnings is logvar 1gsi 4016 0092 13 0079 9 7 0002 81 57696 The difference is 02 7 03 7 19 04 00505 Hence the expected gain of an additional year of education taking into account the effect on experience and experience squared is the difference between these two predictions which is equal to 0051 Now the question is what the standard error for this prediction is The general way to answer this question is through the method The vector of estimated coef cients 0 is approximately normal with mean 6 and variance V We are interested in a linear combination of the 67s In this case the speci c linear combination is y 62 703 71964 X0 where A 0 1 71 719 Therefore because a linear combination of normally distributed random variables has a normal distribution V X3 N NW JVA where V is the variance in equation In the above example we have the following values for the covariance matrix V 00494 700011 700047 00001 00001 0000 00000 00006 00000 00000 Imbens Lecture Notes 1 ARE213 Spring 06 7 Hence the standard error of X0 is 000967 and the 95 con dence interval for X0 is 0031100693 The second method for getting an estimate and standard error for X0 is very easy in the linear case We are interested in an estimator for X0 To analyze this we reparametrize from 01 01 02 to 702031939 4 03 03 04 04 The inverse of the transformation is 01 02 Y031939 4 03 04 Hence we can write the regression function as logearningsl 01 y 03 19 04 educl 03 experi 04 experl2 81 31 Y educl 02 experi educi 04 experl2 19educi 8139 Hence to get an estimate for y we can regress log earnings on a constant7 education7 expe rience minus education and experience squared minus 19 times education This leads to the estimated regression function logearr1gsi 4016 0051 educl 0079 experl educi 7 0002 exper 19 educi 0222 0010 0025 0001 Now we obtain both the estimate and standard error directly from the regression output They are the same as the estimate and standard error from the delta method Imbens Lecture Notes 1 ARE213 Spring 06 8 Let us also look at a nonlinear version of this Suppose we are using a regression of log earnings on education The estimated regression function is logearr1gsi 50455 00667educ 00849 00062 The estimate for 02 is 62 01744 Now suppose we are interested in the average effect of increasing education by one year for an individual with currently eight years of education not on the log of earnings but on the m of earnings We could have regressed the level of earnings on years of education but that would not have delivered as good a statistical t At x years of education the expected level of earnings is Elearningsleduc x explt 1 32 39 0237 using the fact that if Z N NW 02 then ElexpZ expu 022 It is crucial here that we assume that elX N002 not just zero correlation with the covariates otherwise we could not calculate this expectation Hence the parameter of interest is 6 exp l g 9 022 7 exp l 62 8 022 Getting a point estimate for 6 is easy Just plug in the estimates for 6 and 02 to get expBl 02 9 022 7 expBl 02 8 322 199484 However we also may want a standard error for this estimate Let us write this more generally as 6 97 where y 02 We have an approximate distribution for y 6977 N0Q


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