I am looking for guidance on whether I am approaching my problem correctly.
I have an annual time series { $x_{1}$, $x_{2}$, ..., $x_{t-1}$, $x_{t}$ }, where each observation is the estimated median rent price for one-bedroom apartments in an area. These data were derived from pooled cross-sectional samples collected each year, i.e., the median rent in 2015 is the median of a random sample of one-bedroom apartment rents collected in 2015, the median rent in 2016 is the median rent of a random sample of one-bedroom apartment rents collected in 2016, etc. I also have time series for the sample variances and sample sizes from each year.
I would like to fit a weighted least squares regression model to the median rent series and use it to predict the median rent of one-bedroom apartments next year (period $t+1$), using a measure of reliability in each sample estimate as the observation-level weights. Because median rent is increasing over time, the mean and variance of each year's sample is also increasing over time (non-stationary). To account for this, I assign each observation a weight equal to the sample median divided by the sample variance:
$$ w_{t} = \frac{{x_t}}{s_t^2} $$
This results in less reliable estimates being weighted less than more reliable estimates. Because median rent is non-stationary, for forecasting purposes I also first need to difference the series, then fit my model on this differenced series. In doing so, however, I also need to modify the weights so they measure the reliability of the change in median rent rather than the median rent itself. This is because a less reliable estimate of median rent in time $t-1$ may be followed by a more reliable estimate of median rent in time $t$ or vice versa, in which case the weight for $D1.x_{t}$ (e.g., $w_{x_{t}-x_{t-1}}$) should reflect that.
If I'm on the right track, then my question is what's the standard approach to calculate the weight for $x_{t}-x_{t-1}$? Should I just average the weights of $x_{t}$ and $x_{t-1}$? Should I use some kind of pooled variance like what you would use for a two-sample t-test? Something else?
