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"Under what condition (or conditions if you think it necessary) would one observe no change in the regression coefficient (e.g., b-hat Y on X1) for some variable when another variable is added to the regression equation?"

I think the answer is when the exogenous variables are perfectly uncorrelated - is that correct?

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Basically yes. This follows from the omitted variable bias problem. As you can see the bias depends on crossproduct of the variables in regression (in this case the intercept and your variable of interest) and the omitted variable. If the sample correlation of the variables is zero and the sample mean of the omitted variable is zero, then crossproduct is zero, hence there will be no change in the coefficient value. If true correlation and expectation of the omitted variable is zero then the expectation of this crossproduct is zero and bias is zero.

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  • $\begingroup$ It would be helpful to distinguish variables from the actual data. Even if the expected bias is zero, the actual bias is likely to be nonzero. In other words, the problem (as stated) seems to be about a dataset and not about an underlying model for the data by means of random variables, so the mention of expectation is a bit of a surprise. $\endgroup$
    – whuber ♦
    Commented Mar 1, 2011 at 16:54
  • $\begingroup$ @whuber, I had the formula in wikipedia page in mind. There the difference between estimate of the regression coefficient and its true value is expressed as sum of two terms. The bias is expectation of this sum. The expectation of one term is always zero, since it involves the regression error. The expectation of another term is zero if there is no correlation between the omitted variables and the variables in the regression. I modified the answer to reflect your comment. $\endgroup$
    – mpiktas
    Commented Mar 1, 2011 at 17:08
  • $\begingroup$ @whuber, you raised an interesting point. I've updated my question, but I'll need to check it. Unfortunately I am posting now via my phone, which rather limits my editing capabilities, so if I'll have something substantial to add it will be later. $\endgroup$
    – mpiktas
    Commented Mar 1, 2011 at 17:18
  • $\begingroup$ Mighty nice work from a telephone interface! $\endgroup$
    – whuber ♦
    Commented Mar 1, 2011 at 17:43

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