question archive Recall the latent-variable probit model from class 9* = $3 +u
Recall the latent-variable probit model from class 9* = $3 +u
Subject:EconomicsPrice: Bought3
Recall the latent-variable probit model from class 9* = $3 +u. ulI ~ NIH. 1), y = 1(y* > D). where y'" is a latent variable lacking any meaningful units [e.g. utility values). (a) Note that we have assumed the variance of n is equal to one. Show that it does not make sense to assume u|:r: ~ N[U,02} and then tr},r to estimate or [or 02). (Hint: Figure nut Pr(y = l|r) and the associated log-likelihood function, and argue that E and (I can not be wparatelv estimated.) (b) Assume now that '9' does have meaningful units. Speci?cally, let's say that y'" is annual earnings (in dollars). We still do not observe y'" but rather we observe the ordered outcome 3; as follows: 1 if y' «c: sauna y: 2 if20,???£y'<4?:?m 3 if y' 33 40, 000 i. Would it make sense to run an OLS regression of 3; on 2:? Explain. ii. Now using the assumption that o|1= ~ NW, 02) [rather than u|:r: w NH), 1}): work out the probabilities Pr[y = 1|:r], PrIEy = 2|r], and Pr[y = 3'17). 1 iii. Provide the log-likelihood function 3(3), .5) that would be used to estimate ,8 and or, where b is the guess at {3 and s is the guess at o. iv. What is the meaning of a slope ?;— (assume its associated :3} is continuous and enters the model in just the ?j?lj term}? (Hint: Think about the latent- variable equation, not the probabilities for y outcomes.)
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