# Linear dependence problem with time covariate and time dummies (in R)

This is related to a question I asked a couple weeks ago, but I've got a new question related to the same data. You can find the data and its accompanying explanation in the link provided.

I felt that a regression including year as a covariate along with year dummies would lead to a linear dependence problem, but I was told to try it anyway as

"the year dummies as independent variables [may] pick up year-specific random effects not accounted for by a time trend, e.g. for example the trend over all years could be down by say 2 percent per year which could apply to most years, but a negative macro shock in one particular year could make that year lie way off the regression line--a simple example of why the year dummies are not co-linear with a time trend."

This makes sense, I suppose, so I ran a regression that simply included year and year dummies for each year as the independent variables (including AR(1) corrections). This looked like the following:

> ## Generate YearFactor and AgeGroupFactor using factor()
>
> YearFactor <- factor(YearVar)
> AgeGroupFactor <- factor(AgeGroup)
>
> ## Check to see that YearFactor and AgeGroupFactor are indeed factor variables
>
> is.factor(YearFactor)
[1] TRUE
> is.factor(AgeGroupFactor)
[1] TRUE
>
> ## Run regressions with both time trend and year dummies to determine if a linear dependence problem exists.
>
> TrendDummies <- gls(PPHPY ~ YearVar + YearFactor, correlation=corARMA(p=1))
Error in glsEstimate(object, control = control) :
computed "gls" fit is singular, rank 13
> summary(TrendDummies)
Error in summary(TrendDummies) : object 'TrendDummies' not found
>


I interpret the error message "Error in glsEstimate(object, control = control) : computed "gls" fit is singular, rank 13" to mean that there indeed is a linear dependence problem in this case. Am I properly interpreting this?

Also, given the advice in quotes above, would my regression as constructed (if there were no linear dependence problems) capture the effects mentioned therein?

And finally, if I run the same regression as OLS with no AR(1) correlation structure, I do indeed get some results (instead of an error message). Any thoughts on that?