Untangling lump samp panel data?

Edit: Context: I am estimating persistence analog Heckman (1981) based on firm level data. The endogenous variable is the distribution amount of profits directed to the owners. To estimate persistence, I plan to regress the endogenous variable against its past realization. To circumvent the initial conditions problem, I plan on using the Wooldridge method ("Simple solution to the initial conditions problem", 2005). So far, I dummied the endogenous variable and I thus apply a dynamic random effects discrete choice model.

I am using Stata 13 to estimate a simple probit model. I use panel data of secondary nature. The data is conducted via an annual survey. Unfortunately, the endogenous variable of interest is conducted for the past three years respectively. That is, each year the survey taker is asked to indicate the sum of the endogenous variable for the past two years and the current year.

Let's have an example. The picture below shows the data structure. The first row depicts the periods, the second the endogenous variable and its annual value. The following rows beneath show the past individual annual values and the three year sum - that is the observed value. So we have 12 observed periods in this example. Period t0 contains the sum of periods t-1; t-2; t-3 - in this case 3 + 6 + 7 = 16. For period t1 the survey taker discloses 16 yet again - though this time as the sum of periods t-2; t-3; t-0 and thus 6 + 7 + 3 = 16 . Each observed value thus consists of the sum of the two previous years and the current year. The picture below tries to show that via the "red" and "blue" shadows.

The problem being, I only see the three year sum - that is the "diagonal, bold values". I do not have the annual data. Given the outset of my research question, having the annual data would however be highly beneficial. Two questions arise:

(1) Is there any smart way how I can untangle the lump sum data? (2) If not, what can I do?

• Would you possibly have some starting data, such as the data for period $-2$ and the sum for periods $-1$ and $-2$? Two such pieces of information are needed to obtain a unique deconvolution of these windowed (or moving-average) sums. – whuber Jun 9 '15 at 17:28
• @whuber, unfortunately I don't. I personally do not see a arithmetic way to arrive at the unique and thus deconvolunized (cool word, thanks!) value. – Rachel Jun 9 '15 at 17:30
• OK, then we need to seek ways to avoid using that missing information. It is possible that you could obtain needed estimates without it, but that would depend on details of your model and what you're trying to do with it. There are many ways to approach this, but any edits you can make to the question to provide more context would help guide your readers and would-be respondents. – whuber Jun 9 '15 at 17:33
• Sure, no problem: I am estimating persistence analog Heckman (1981) based on firm level data. The endogenous variable is the distribution amount of profits directed to the owners. To estimate persistence, I plan to regress the one year lag of the endogenous variable on the endogenous variable. To circumvent the initial conditions problem, I plan on using the Wooldridge method ("Simple solution to the initial conditions problem", 2005). So far, I dummied the endogenous variable and I thus apply a dynamic random effects probit model – Rachel Jun 9 '15 at 18:06