Analysis approach: time-series or multiple models? I have data which (I think) does not warrant a time series analysis, but I am note sure whether my alternative analysis approach is acceptable:
The data consists of a numerical and a categorical predictor at time-point 0 and a predictand at time-point 0, 1, and 2. I want to test whether there is a relation (especially an interaction between the predictors) for the development of the predictand (predicted decline over time).
In a time series model this would generally look like this
$$y_t=X_t β + u_t$$
but in my case like this
$$y_t=X_0 β + u_t$$
My alternative approach would be to have three separate models, one for each time-point and somehow compare them. However, this does not feel right to me.
Any comments or suggestions on how to approach this analysis?
Edit:
Data example:
mydata <- read.table(header=TRUE, text="
  yt0   yt1   yt2   cond    xt0
  10    9     8     b       7
  12    12    11    a       3
  8     5     2     c       12
")

cond is factorial, x and y are numerical. 1 Case per condition just as example, in reality, I have 3 conditions with ~70-80 cases each.
P.S. It would be nice to use something that would be able to handle missing data in $y_t$
 A: I'm not sure if understand it correctly,
If you have data points for all your data in form of a categorical measurement:
time point 0, 1, 2 Then it is possible to compare all three groups and make three separated models on a variable.
Can you post your data? It is possible to do time series with dummy time series points which results in time series cross sectional data or time series panel data:
See the second post on research gate:
researchgate
From what you are describing these approaches seem possible
----------------------------UPDATE------------------------------
I'm sorry I needed a bit of time to think about it. And I have to revise my assumptions.
So you have three dependent variables y0, y1, y2 that are the same variable but at different time points. You have only one factor variable and a metric exogenous variable x0. If you would made three models for every y that would result in the same x0 for every y variable you have. That is the same as copying your variable x0 three times for the length of y1 and y2.
The fundamental question is: what is x0? is it ok, that there are no values for x1 and x2 that could be observed? or can x0 be assumed as equal as in the period of y1 and y2. Two examples:
Imagine y0 is the year 2000 and you have several points for y0 which reflect the days. If x0 is the temprature on a certain day during a hot season then it could be ok to assume the same value of x0 to x1 and x2 at these days at time period 0.
    row      amount of bees flying at time0           temperature
day1-2000                12                                19
day2-2000                12                                20
day3-2000                13                                21
day1-2001                11                                NA (but in case of three models we expect 19)

Then it could be possible to assume a similar x1 and x2 thus resulting in three models.
But how do you argue the decline? if you think x0 is something that should have a declining or bad outcome on y which was only active in y0 and should be still has an effect on y, then you have to consider transforming your variable with a reasonable carryover effect.
e.g. media example (with end of year 2000)
        row      sales due to media impact               media budget
    day29-2000                12                                40
    day30-2000                12                                50
    day31-2000                13                                21
    day01-2001                11                                 0 + (21*50% carryover which results in 10.5

In summary to give you a final advise: I would not model anything, I would look for a curvefitting approach in R. I think it would be possible to format your data to one y and transform your x with a certain amount of carryover and add the factor as another parameter in an assumed model function, which can be searched by the fitting function, that describes a declining or expiring effect. E.g. an exponential or linear decay https://www.thoughtco.com/exponential-decay-definition-2312215 and to find that effect, you could assume such a negative decay, thus the model function. The final curve fitting in R should give you a fittied function and thus R², residuals, all you want. The fundamental difference is, that you assume a model decay function in advance and let R find the fit and the decay. to your declining. And you wont have problems with transforming your missing variables in x0, as they will be transformed with an carryover effect.
What you should do is assume a decay function and let R fit it with your data by putting all ys into one column. The decay effect of x in your model will be estimated by the R curve fitting.
The required packages to try curvefitting in R are:
nlsr with the nlxb function:https://www.rdocumentation.org/packages/nlsr/versions/2019.9.7/topics/nlxb
and nls2 with the nls2 function:
https://www.rdocumentation.org/packages/nls2/versions/0.2/topics/nls2
At the end of the curve fitting you get the desired decay parameters to describe your decline to the function you chosed. However you have to calculate R² manual
If you wish more information on that topic, just ask me. But I think we are on the right way. I can even give you a code example of a nlxb curve fitting.
