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# Econometrics II EconS 512

WSU

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This 20 page Class Notes was uploaded by Maurine Kuhic on Thursday September 17, 2015. The Class Notes belongs to EconS 512 at Washington State University taught by Jonathan Yoder in Fall. Since its upload, it has received 81 views. For similar materials see /class/205979/econs-512-washington-state-university in Economic Sciences at Washington State University.

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Date Created: 09/17/15

Part 7Limited Dependent Variables November 187 2008 Part VII Discrete amp Limited Dependent Variables 1 Intro Categorical and Limited Dependent vari ables reading G207217 DM11 o discrete y can take on a set of discrete values 0172 o Censored true y are reported as a single value in a certain range 0 Truncation y missing for part of its range 0 Sample Selection y excluded or included in a systematic way 0 Duration y represents duration of an event or time until an event Maximum likelihood estimation has been the workhorse of limited dependent variable estimation7 so we Will cover the basics now 2 Discrete Dependent variables 0 Labor force participation l articipate17 not in labor force0 o Referendum results Yes17 no0 0 Transportation choice car07 bus17 bicycle27 etc Categories aren7t rank able Page 116 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 7Limited Dependent Variables November 18 2008 0 Answers to questions Strongly disagree neutral agree Strongly agree Likert scale Categories have ranking or order General approach model the probability of an event falling in a given category as a function of X and Proby j Fx 21 Models of Binary Choice People make a single decision between two choices and y takes the value of zero or one characterized as a conditional Bernoulli trial Pr0by 1lxy FOL3 Proby le6 17 Fx6 also Emmi 1 yiProbyilxi 0 X Probyi 0pm 1 X Probyi um Elyilxl39l Pr0byi1lX1 FX 6 Two theoretical constructs are useful conceptual tools 0 Index functions 0 Random utility models Index functions Although the data on outcomes are incomplete in some way eg y equals zero or one it is indicative of an unobserved latent continuous Page 117 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 187 2008 variable7 xi 7 51a Eigi yi 1 if gt 0 yi 0 if S 0 then Probyi gt 0 Prob5i lt Xi F Xi x is called an index function in this case The Random Utility Model Suppose we as researchers are able to measure utility imperfectly7 such that for two choices a and 127 we model utility as Ua 32X 8a and U12 bX 812 The suppose Y 1 if a is chosen then ProbY 1 ProbUa gt Ub Prob6a x 7 5a gt bX 7 51 Prob6a 7 gmx gt 5a 7 55 Prob5 lt x F6 xi 211 Linear Probability Model Recall that for a binary Bernoulli riVi7 Elyilxil PrObW 1lxi FOE ll Now suppose yi x 51 where 5139 N iiiid With 0 for all i and Page 118 7 WSU Econometrics 11 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 187 2008 yi 6 01 Then Elyilxl RIGX El x 6i X So the regression line can be interpreted as the conditional expectation of y given X7 or Proby llx a y o 2313 ooomx 2 N we RSI 02545 ndJRSu m n mm nusz 04034 03 306 04 02 no u y 0 o m 20 30 4o 50 so 70 an so we Pm I u ydxx u u u pngmx 212 Problems with the Linear Prob model Heteroskedasticity y is 0 or 17 and yi 7 x71 Therefore7 WM 1 91quot x 17 G x igiiyi 0i 11quot ElX XA You can show that given Varei 6Prob6i l 7 Smallest When 3 0 57 and larger as diverges from 05 Page 119 7 WSU Econometrics H 2007 Jonathan Yoda All gm xesexved Part 72Limited Dependent Variables November 187 2008 Predicted values The predicted values x can take on values greater than one or less than zero This is not consistent with the de nition of a probability 213 CDFbased Probability Models The discussion above about nonsensical predicted values suggests some character istics that a reasonable functional form should hold have pxli rnrooProby 1 l and Wig Proby 1 0 Any cumulative density function suffices As before7 for a binary response7 Elyilxl l0 X Fyi OlXl l1 X Fyi 1lxl F 6 x where F x is now de ned speci cally as a CDF 214 The Probit and Logit models Two commonly used distributions and their associated models are l the Normal distribution7 which leads to the Probit model 2 The Logistic distribution7 which leads to the Logit model Probit model f6 x pdf and F6 x CDF are 15 7 1 456 a 7 HQ a2 eXp 202 7 Q lt1gt 7 man 5 and a are not identi ed in a practical sense Prob 51 lt xi Prob lt so we normalize to a 1 without any loss of information unless heteroskedasticity Page 120 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 18 2008 exists eg a f 012 Therefore 0350 1 456 exp T i The marginal effect of ac on is The Logit model The Logistic CDF F6 x is e W X 22 Estimation of Binary Response models In practice estimation most often is based on Maximum Likelihood Assuming independent disturbances the the Likelihood function for a binary dependent variable is LPr0bY1 917Y2 y2mYn yn H FwdXi H 1 incx0 F6 xiyl 1 7 F xi1 yl and InL Z m Ian xn lt1 7 mm 7 Foe xn and since yi is binary 1nF x 2 mu 7 F6 xii Page 121 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 187 2008 Likelihood equations FOCs 91an 39fi 1 fi 7 9g 7yZE1iyz1iFi XZ70 n i 1 H H H Aixi 0 K X1i M are equivalent to or analogous to residuals in a least squares setting As in OLS7 these errors must be orthogonal to Xi Logit Ai 7 Ai Probit Ai ql ql X1 7 Where gt4i xi 1 if yi 1 4i 2 i 1 71 if yi 0 Where Fi and are the CDF and PDF evaluated at xii No analytical solution for Probit7 Logit numerical optimization required 23 Inference in Binary Response Models The covariance of the ML estimator is covi mz Iltr gtr1 iElHlVl H 821nL EL AM Auxix Logit a a 3 3 7 21 All x xix Probit Hessian of the LF is negative semide nite for both models unique global maximum is guaranteed Page 122 7 WSU Econometrics H 200 Jonathan Yoda All gm xesexved Part 7Limited Dependent Variables November 187 2008 Exercises 1 Prove that the likelihood function for the Logit model is globally concave 2 Show that the sum of the predicted probabilities equals the empirical sum of ones in the sample A number of estimators for Cov exist One is to use the empirical hessian H de ned above in place of Eg7 for the Logistic7 A A n A A 71 7 7 71 I 7 I I 39 Cov 7 H E H Al 1 A le Another commonlyused ML covariance estimator is the BHHH estimator vZRT WWI ghtx cacar Where 6 gly uq n n X K and aggL and gi is de ned above as M for both the Probit and Logit Greene interchanges gi and M Recall the FCC for max L 3 2lgixi 0 Observed gixi for each observation is a mean zero rv With a covariance Cov BlnL 33 231 Inference predicted probabilities Predicted probabilities are F F x nonlinear in 5 You can nd the asymptotic variance of a nonlinear function of parameters using the Delta method Page 123 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 187 2008 This value note it is a scalar dependent on a speci c set of observations on X can be used to generate con dence intervals for 232 Inference marginal effects The marginal effect of a change in M on is 8Eylx 7 8E B x i Bxk B x Bxk 5 KW Where is the PDF associated With Fl Marginal effects are data dependenti Two common methods of reporting marginal effects 0 Based on the sample means of X y f X or o The average of the marginal effectszfy These are asymptotically equivalent7 but Will give you different small sample results Marginal effects are nonlinear functions of so we use the Delta method to calculate Cov V le i note that 3 WWKW an an lt fgtltwzxgtflta gt B x 86 86 3 7 8f 7 BigX x fL K X K Page 124 7 WSU Econometrics H 200 Jonathan Yoda All gm xesexved Part 72Limited Dependent Variables November 187 2008 For the Probit and logistic7 the derivatives are Probit 62 7 x x 6A 63 Logistic a X 17 2A lt gt 17 2AA17 A X Plugging these into the general variance formula gives Probit 7 a l 7 x x l vXRT Q 1 7 x x l Logistic m A17 A2I 17 2A6x I 17 2Ax i This variance can then be used in Wald statisticsi Exercise Prove to yourself that marginal effects for the Logit are 8A x 7 Alta xgtlt1 7 A x x 111 i 903 f x 233 example con dence intervals for predicted probabilities and marginal effects A 1 0 A A 1 0 6 7 X so X1 Covm con dence interval from the Probit model x 7 V127 exp gm 012421 Page 125 7 WSU Econometrics 11 2007 Jonathan Yoda All gm xesexved Part 72Limited Dependent Variables November 187 2008 A A 1 0 VarF f2x vX 0059 0 2 0234 0 1 Now7 the predicted value for x is F x F1 fol 084 An asymptotic 95 con dence interval for 084 i 2050234 234 Hypothesis tests and goodness of t The Likelihood ratio test is often particularly convenient in binary choice mod A gt LR2i1nlt5 i1 i els Where and 1317 are the predicted probabilities from the unrestricted and restricted models respectively goodness of t Lots of measures McFadden s PseudoR2 is ln LR LRI171DLU 24 Speci cation of binary choice models We Will discuss two topics in particular The problem of heteroskedasticity7 and the choice between Probit and Logit 241 Heteroskedasticity In OLS7 o Excluded relevant variables leads to bias and inconsistency unless omitted X and included X are orthogonal 0 Failure to account for heteroskedasticity leads only to inef ciency7 not bias or inconsistency Page 126 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 18 2008 In Probit and Logit models excluded variables introduce bias regardless of or quot and t 39 39 leads to 39 39 t A binary response model with heteroskedasticity may be difficult to estimate so we rely on a Lagrange mul tiplier test for heteroskedasticityi LM review 0 the LM test statistic is based on the gradient slope ofthe ln L evaluated at the restricted estimates 0 We need to know the functional form of the unrestricted gradient but we only need to estimate the restricted parameter estimates not the unrestricted estimates 0 Particularly useful when the unrestricted model is tough to estimate 2 Construction of the LM statistic HypothesiZe Vare exp39y z Harvey s form where 2 does not include a constant Then the CDF representing predicted W 4in l ai exp 391 2 The Log likelihood looks just as before in terms of The rstorder conditions probabilities is are I alnL 9g 7 29139 exp739y Zixi mm 7 n 87 7 1291 expP39y zlzl7 X1 7 filtyi where gZ 7 With a homoskedasticity restriction y 0 exp iv2139 1 drops out of both terms so the sums of the gradients would no longer equal zero at the unrestricted max of Page 127 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 18 2008 lnLi let 2 gm gR n1 KKzgtlt1 211 gizd xi which to reiterate are the gradients of the likelihood given the constraint 1 0 Then LM gggvilgg N X2J where V Varl l 7ltXG GXgt and X x 2 242 Choice between Logit and Probit You might think about a Likelihood ratio test between the two but this is not valid because one form is not nested within another 7 there is no restriction involvedi Silva 2001 suggests the following method 1 choose one form of as a null hypothesis We7ll stick with the Probiti The alternative form of is the Logistic GDP 2 Compute the parameters of both models and calculate 7 p 2i A77 fiP where i is an observation index P and L denote the Probit and Logit respec tively pg are predicted probabilities from each respective model and 1313 are the PDF values for the null distribution the Standard Normal evaluated at Xz 3 Include 2 as a regressor in a second Probit model along with the original explanatory variables 4 If the parameter estimate associated with z is distributed asymptotically stan Page 128 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 187 2008 dard normal If signi cantly different from zero7 reject the null Probit in favor of the alternative The logistic 25 Proportions Data We can aggregated binary response data so each observation is the proportion of individuals in a group With a given response EG referendum data 52 pi 052 of voters in county 239 vote yes The dataset amounts to one observation for each of n counties 0 Let m F x be the true population proportion of yes votes7 o 1 7 m be the true proportion that votes no 0 Let 17f F x5i Tri5i Where 07 Where ni is the number of positive responses from Ni Bernoulli draws from a population Then ni is a binomial random variable With Nm and Varni N Ti1 7 m Given this setting7 Var5i 139 Sample size variation across groups leads to heteroskedasticity Consider two estimation approaches 1 transform our model into a linear function for GLS estimation 2 Maximum Likelihood estimation 251 Transformation to Log odds Based on a Taylorls series approximation7 F 1Pi m F 17ri ui 5i E Page 129 7 WSU Econometrics H 200 Jonathan Yoda All gm memes X W ui Part 72Limited Dependent Variables November 18 2008 F 1Pi is the inverse of F a CDF and is the PDF of F We can show that VARul F ng so the error term is still heteroskedastici Assuming a Logistic CDF Pi 7 lnlt1ipigt i xul This transformation is called the log odds For estimation transform the de pendent variable and run GLSi For GLS via weighted least squares the weights to multiply y and X by are wi niAZl 7 Aii If P equals zero or one lnPil 7 is not de ned Replace the zero one With 001 999 With lots of Us or ls you have a censoring problem 7 take another approach 252 Proportions data and Maximum Likelihood If you Consider that P 7 Ej yij Where yij are the underlying votes for each P the log likelihood function for proportions data is L H F xzv lt1 7 F x z w i1 H F xnsz HO F xnz1 Pz y1 y0 lnL ZnP1nFex1 7 131n17 F6 x Where n is the number of observations of proportions data and mi is the number of observations from Which each proportion Pl Was calculatedi Page 130 7 WSU Econometrics H 200 Jonathan Yoda All gm xesexved Part 72Limited Dependent Variables November 187 2008 26 Multiple categorical choices The Binary response model can be extended to a choice between 3 or more alter nativesi Choice between these alternatives can be related to the characteristics of the decision makers andor the characteristics of the choices 261 Multinomial Logit Consider 3 choices7 a7 b7 and c7 where these choices are associated with personal characteristics not choice characteristicsi Utility maximization Consider a choice between two alternatives among many 0 Suppose Uij xiJrEZj j for alternatives7 i for individuals7 and alternative k will be chosen over j if Uik gt Uiji 0 Suppose F f x exp7 exp7 1 x Type 1 extreme value Gumbel dis tribution It follows that Xi PM gt U2 Pla 7 gt w 7 am k k 1 k 1 k 1 zinexpngn o the three probabilities must sum to 1 0 One parameter vector in this case e is normalized to zero e x Proba Pa W7 e x Probb Pb W7 Probc PC 1 emx e gx Page 131 7 WSU Econometrics 11 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 18 2008 the marginal effects are J 813139 ijj e 2PM m e m 161 am The marginal effect of I on Pj could have the opposite sign of the coef cient betaij on I so coef cients themselves are dif cult to interpret The likelihood function is n L H P551 Pg P0031 i1 where Pai PM and PC are the probabilities as de ned above and da db and clC are mutually exclusive dummy variables that take the value 1 when each respective option is chosen MNL implies Independence of Irrelevant Alternatives HA The implied log odds for this model are Pa P P 1n aux 1n bx 1n rib 7 Lu x In Each case the relative size of the odds are not affected by the presence or absence of additional alternativesi If we were to add another choice very similar to choice a ln if gx is required by this model to remain the same Example In reality we might expect if to fall as people substitute toward the new choice We can use a Hausman test for HA 1 Estimate the MNL parameters with all alternatives then exclude a choice and estimate again Page 132 7 WSU Econometrics H 200 Jonathan Yoda All gm memes Part 72Limited Dependent Variables November 18 2008 2 H0 parameters don7t change Ha parameters do change 3 H r UV7 Vrl 1 r 7 3 N X206 where H all elements of H are within the parameter space of the restricted model 7 denotes the parameters and covariance estimates from the restricted model u denotes the unrestricted model and k is the number of regressors 4 Why K degrees of freedom If we reject HA then the multinomial logit is inappropriate Alternatives include the multivariate probit or the nested Logit Greene 1974 which we will not cover 262 shares with proportions data Suppose we took logs of both sides of the odds and replace 7r for each choice with the observed proportion or share P Pa Pa ln xa 51 ln xb 51 OLS if same regressors and no crossequation restrictions otherwise SUR Same heteroskedasticity issues may arise as in single alternative setting Aside Share equation estimation Demand system are often estimated ex penditure share form w pizi pizi X Despite being limited to the unit interval shares are often estimated as linear functions which represent rst order approximations to the true share functions 263 Using data on Characteristics of alternatives Where data are available on the characteristics of alternatives rather than or in addition to characteristics of the individuals making the choices the Conditional Page 133 7 WSU Econometrics H 200 Jonathan Yoda All gm xesexved Part 72Limited Dependent Variables November 187 2008 Logit model applies The model and its interpretation are very similar to the MNLi Covered in Greene Section 1712 27 Multiple ordinal categories 0 Likert scales 0 Income categories 0 Employment categories Assume the latent regression y x 5 F x x and there are J categoriesi Proby 0 Proby S 0 i x Proby 1 Prob0 lt y S m 7 ltIgtm 7 x 7 gt7ax Proby 2 Probwl lt yquot S 2 Mg 7 x 7 M1 7 x Pr0by J Probltw71 lt 9 1i POLng 7 x Where0ltM1ltM2ltltMJ1 The marginal effects for three choices7 With M 1 are BProby 7 0 T 7 7mm x W 7 l lt7 xgt 7 W 7 M W W 7 rm 0 Similar to the Multinomial Probit Logit7 except it imposes an ordinal rela tionship between choicesi Page 134 7 WSU Econometrics H 200 Jonathan Yoda All gm memes

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