You seem to be asking two questions:

1) Prediction vs. explanation:  MV regression, like any regression, can be used for both. But if you are solely interested in prediction, there may be other methods (more "black box"-y) that are better.

2) Sample size: The more complex the questions you are asking, the more data you need. There is no way around this for either prediction or explanation. However, while there are sometimes rules of thumb for how much data you need, it is not possible to state, for every problem, just how badly you will go wrong if you have too little.  There are some Monte Carlo studies, but each case will be at least a little different.

There is one quote I really like in data analysis: "if you feed a model crap, it is still crap that will come out" in other words, yes sample size and data quality- quality will affect whether your results are biased or not, sample size will give precision- with multivariate models, you do need a certain number of observations, especially if you incorporate several different explanatory variables. there are ways to calculate how many observations you need o answer specific questions, but this is beyond the scope of this answer

Thank you.

...

-  Please allow me to be more suscinct.  My question is as follows:

 "Is multivariate regression more useful for analysis than for prediction?"

From reference 2 it says:

"...if you ran a separate regression for each outcome variable, you would get exactly the same coefficients, standard errors, ...."

But from reference 1, I would think that to minimize that trace, one would not necessarily minimize each element on the diagonal. Is that true? I've never used it. 

If reference 2 is correct, then it would seem that the only reason for multivariate regression is analytical. -  Comments?   -  Thank you. 

1)  Oops!  I was thinking about "tradeoffs" on the elements in that trace, but if each element is already minimized, there is nothing to tradeoff, and the trace is then minimized ... being made up of all the minimum values.  SORRY.  (Nevermind, regarding that part.)

------------------------------------- 

2)  It seems that each response (y) variable should be studied for heteroscedasticity, rather than assume OLS.  How often do you actually see equal regression weights? 

3)  After looking into this, I am fairly convinced now that for Official Statistics, where totals are estimated from samples of finite populations, rather than performing an analytical study, that multivariate regression is not really useful for such a purpose.  I had been told by a biostatistician how disappointing he thought it was that he never saw multivariate regression used for such publications of Official Statistics, or at least that is what I thought he meant, and I wanted to check it out, to see if he had a point. 

Comments?  

Dear James

These discussion is interesting from two viewpoints:

1- Analysis or Prediction: the important thing before any

thought about the parameterizing is the data it self. There

are two types of data: Observational and experimental.

Most of the times we use observational data only to analyse

and that needs huge amount of data. But the main Idea of

experimental analysis is prediction. This is an important

thing we all neglect. So, If you have observational data you

can't rely on the analysis result for prediction. On the

other hand observational data are mostly nested data which

means the parameters that you gain in a system may not

apply to other one.

2- Multivariate analysis: I put off multipe part. the

important point in multivariate analysis is the relationships

between the "Y"s. SAS analyse them and says the result is

same as several univariate analysis because it assumes the

correlation between dependent variables are zero.

Multivariate analysis is a multidimentional analysis. If we

want the true multivariate changes we first need to deploy

a relationship between the Ys and determine the angle between the dimensions (which by default is 90 degrees). The simplest one is linear

correlation which leads to inflation factor. If there is

significant correlation between the Ys then we

overestimate the parameters by simple multivariate

analysis.

I hope the explanations help

Ehsan

Dear James

For your comment I should say I totally agree with that biostatistitian. Most researchers only extract 30 percent of the information that a dataset have because of using inappropriate statistical methods. In most JCR research papers I always see some researchers use a simple factorial design and only a comparison method like Duncan even for quantitative factors like different levels of pesticide. Nevertheless, most research papers have no financial evaluation of their results. Are the purposed method cost effective?????

For example, for a multivariate multiple analysis which have several dependent and several independent variables we'd better use canonical analysis methods or apply structural equation models rather than a simple regression.

OK. Thank you. That's fine when you are doing a study. But when publishing finite population estimated totals from weekly and monthly samples and annual censuses for official energy statistics, analysis is not the purpose. Periodic production of many tables of estimated totals and prices (ratios of estimated totals) is the goal. 

PS -  An example of a monthly publication of some of such official statistics is found through the following link. After some other information, tables of data are produced, and finally, relative standard errors are found in later tables.  There should be some Technical Notes toward the end. (There are substantial practical issues involved in production of these many reports, and many nonstatistician users and managers with whom to deal.) 

 https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.eia.gov/electricity/monthly/current_year/march2011.pdf&ved=0CB4QFjABahUKEwif0Z64voDHAhVE04AKHZvODPk&usg=AFQjCNHYaT6TCI1Ro9Xx1rgI6OhzvWHbIw&sig2=3Lp8fIcBBlkv4s7MyUBS-Q

If that link does not work, or takes too long, try the attached file or see the Energy Information Administration website for other publications.  Thank you. 

Dear Jemas, sorry for late reply. I agree with you. Based on my experience, I also think multivariate regression is more useful for analysis than for prediction. In forestry practice, the models for prediction generally do not have many independent variables.  Multivariate regression seems better for analyzing relationships among many variables. But sometimes, multivariate regression is also used for prediction, for example, the biomass models. In this case, the response variables are total biomass and component biomass, such as stem biomass, branch biomass, and foliage biomass, and prediction variables are usually the diameter, height and crown size (width and length). Because biomass data are generally heteroscedastic, so WLS  should be considered as you said.

"You could analyze these data using separate OLS regression analyses for each outcome variable. The individual coefficients, as well as their standard errors, will be the same as those produced by the multivariate regression." I don't think so. At least, only under some special  assumption, the statement is correct.

Well, I guess that if you are not looking for the relationships between the individual dependent variables, you don't need multivariate regression.  Please see the link below.  

Since I was wrong about that trace, as I noted above, these regressions apparently can be done separately and obtain the same predictions for each regression equation.  So references 1 and 2 above are consistent with each other.  

Thank you all. 

http://stats.stackexchange.com/questions/4517/regression-with-multiple-dependent-variables

http://stats.stackexchange.com/questions/4517/regression-with-multiple-dependent-variables