Does it make sense to purposefully run multiple regression with missing values for certain dummy variables? Let me explain.  You have a multiple regression using time series data within an econometrics model.  And, some of your variables are either dummy variables or interaction variables.  So, they take a value of 0 or a continuous value.  As an example, let's say one of your variable is the quarterly change in 5 year Treasury rate.  And, you disaggregate that variable into two.  One would cover the pre-Greenspan era from 1951 to 1986.  And, the other one would cover the Greenspan era to the present.  The first variable will show the quarterly change in such Treasury rates until mid 1986.  Thereafter, it will have a value of 0 or 0%.  Someone suggested that instead such a variable in the Greenspan era to the present should not have a zero value, but instead be blank.  In essence, creating purposefully missing values.  The argument is that this variable does not have a true value of 0% in the specified time frame.  Instead, it is truly absent.  And, this could affect the magnitude of its regression coefficient or impact on Y.  I question this rational on two counts.  First, I am not sure that any software could calculate such a regression with variables with different numbers of observations.  And, I think there is a very specific reason for that as outlined in my second argument.  Second, multiple regressions are resolved through matrix algebra including the use of covariance matrices and the inverting and transposing of such matrices.  And, those calculations are impossible with missing values on some of the variables.  In my mind, I believe I have already answered this question.  Can you tell me if this answer is correct.  Or am I missing something.       
 A: It's not that matrix computations are especially finicky: the plain old arithmetical operations they concisely represent are defined on numbers. You can't carry out a regression when an observation has a missing value on a predictor; you need to either substitute a number, or omit the predictor, or omit the observation.
It may assuage someone's doubts to work out the consequences of your coding scheme. For your predictor $x$, let
$$ x_1 =\left\{\begin{array}{ll}
x & \mathrm{pre-Greenspan}\\
0 & \mathrm{post-Greenspan}\\
\end{array}\right.$$
$$ x_2 =\left\{\begin{array}{ll}
0 & \mathrm{pre-Greenspan}\\
x & \mathrm{post-Greenspan}\\
\end{array}\right.$$
The expected value of the response $Y$ is given by the model
$$
\operatorname{E}Y  = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots
$$
where the $\beta$s are the coefficients you'll estimate. Now pre-Greenspan
$$\operatorname{E}Y  = \beta_0 + \beta_1 x + \ldots$$
while post-Greenspan
$$
\operatorname{E}Y  = \beta_0 + \beta_2 x + \ldots
$$
—the intercepts are equal & the slopes of $\operatorname{E}Y$ vs $x$ differ in the two time periods, which is, I gather, what you wanted. 
As you say, your method is equivalent to @Wayne's method without the indicator variable appearing as an isolated term, but only multiplied by $x$. Note the constraint that when $x$ is zero, $\operatorname{E} Y$ is the same in both time periods (given equal values of other predictors)—most people would want this violation of the marginality principle to have theoretical support as well as not being discrepant with the data.
A: Why not just calculate a regression for the data pre-Greenspan (using the data 1951-1986) and calculate another regression using the data post-Greenspan (using the data 1986 onwards)? If you have some assumptions on e.g. the intercept, you can still include that. Setting values to zero will naturally distort your regression.
And it is perfectly possible to calculate statistics and perform fitting with missing/incomplete data - just not with the usual matrix formalism (and as said, filling the missing values just to use the "inappropriate" tools is problematic) - in the case of unbalanced observations or missing data the estimators are then calculated as sums (of which matrix multiplication of course is a special case).
