For four of these five attributes, the relation assumed in classical regression analysis does not hold for Yʹ if it holds for Y.

 

Jacob: Are the five attributes explicitly listed?

 

Rachel: The five attributes are implicit in Fox’s discussion: symmetric, unimodal, normal distribution, constant variance, and linear relation.

I'm having a hard time with this HW. Any thoughts on how to go about reasoning this? How long an answer are we needing to give?

I also found this problem difficult.

[NEAS: Jacob: What are we supposed to explain for this homework assignment?

Rachel: If Y is normally distributed, Yʹ is lognormally distributed. If Y is symmetric, Yʹ is skewed, and the assumption of classical regression analysis does not hold. This problem is true for much actuarial work. In later modules, Fox explains how to solve this problem by generalized linear models. This pattern holds for all five items.]

 

I skipped this homework and came back to it after getting to the module on transformations. By considering the discussion of the log transformation in Module 6 (Chapter 4), I believe I've been able to figure out which characteristic is preserved by exponentiation. For example, if we use a log transformation to change the skewness and create symmetry, we would assume that the original function is skewed.

Nevertheless, I'm still at a loss as to how much of a reason is wanted. It all gets down to the nature of the function f(x)=e^x.

[NEAS: Correct. We are not looking for much explanation. These items become clear in the chapters on generalized linear models.]

Is a one sentence answer sufficient? I.E. to list which attributes it does and does not hold for?

[NEAS: A concise answer is sufficient.]

I am having a hard time answering this homework. There are no examples given even in the reference book. In my previous Microeconomic VEE, examples were given first in the modules and this will give us idea in answering our homework.

Adding examples will help us a lot. I hope NEAS will do this.

[NEAS: Fox discusses five attributes: symmetric, unimodal, normal distribution, constant variance, and linear relation. Explain (in a phrase) how each attribute relates to lognormal distributions. For example, the lognormal distribution is skewed, not symmetric.]

NEAS:  Could someone help me with the equal vs unequal spread?

Here are my thoughts:

If Y is normal with equal spread, then the conditional variance of Y,  sigma, is constant for all the Xs.

I realize that this does not necessarily mean the conditional variance of Y' will be constant for all the Xs, but wouldn't it be constant in the case where the mean of Y is constant for all the Xs?

Is their someway I can insert formulas on this discussion board to make my question clearer?

[NEAS: If the mean of Y is constant at all X points, the estimated beta is zero: Y = alpha + 0 * X + epsilon. Classical regression analysis assumes the means of Y differ by X (so beta is not zero), but the variance of the distribution of Y does not differ by X.]

 

If you take a look at the log-normal pdf (http://en.wikipedia.org/wiki/Log-normal_distribution) here, you can get an idea of how the lognormal distribution might be described in terms of skewness, heavy tails, etc...

 

 

I'm a little confused by one of the responses in a past post, that said we should comment on the features (e.g. skewness vs. symmetry) of a lognormal variable. The third feature is normal / heavy tailed so from that I thought that we would not assume that the variable Y' is lognormal except in that one part of the question. Should we assume Y is symmetric, has a single mode, is normal, has equal spread AND is linear? Or should we comment on each separately (e.g. Y given X has a single mode but is not necessarily normal)?

[NEAS: The normal distribution has the five attributes you mention. If the distribution is lognormal, which of the five attributes change?]

I believe I've been able to figure out which characteristic is preserved by exponentiation. For example, if we use a log transformation to change the skewness and create symmetry, we would assume that the original function is skewed.

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