slides_3_distributions

# Indicator functions we have f x x 1 x 1 x log x 1 x

This preview shows pages 84–89. Sign up to view the full content.

indicator functions we have f X x 1 x 0 1 x  log x / 1 x 0 84

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A different choice that is common in more complicated econometric models is to replace the lognormal distribution with the truncated normal distribution, which is defined for x 0. We take the normal density and divide by the area to the right of zero. So, for x 0, the density is  x / 1 /  x / / When we combine this with P X 0 we get f X x 1 x 0 1 /  x / 1 x 0 85
An attractive feature of this density is that it contains the censored normal as a special case, namely, if 1 / . The censored and truncated normal distributions play important roles in the econometrics of missing data, where only part of a population is sampled. 86

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Similar devices are sometimes used for for discrete RVs, too. For example, one allows P X 0 to be different from what is implied for a standard discrete distribution, which is used for P X x for x 1,2, .... EXAMPLE : Suppose X is a count variable with P X 0 and f X x a truncated Poisson distribution, appropriately scaled, for x 0. We can write this as f X x 1 x 0 1 1 exp  exp x x ! 1 x 0 , x 0,1,2, .... 87
The division by 1 exp  ensures the density sums to unity because x 0 exp x x ! exp x 1 exp x x ! 1 or x 1 exp x x ! 1 exp . 88

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
If exp then we get back to the Poisson distribution, but allowing a free allows more flexibility. We can also parameterize the density as, say, f X x exp 1 x 0 1 exp  1 exp  exp x x ! 1 x 0 , x 0,1,2, .... for , 0 and then we get the Poisson density when . Such parameterizations can be useful for statistical calculations (later). 89
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern