How can I get annual rates of change for combined trend estimates?

Question

I would like to combine trend indices (gained with different methods referring to the same subject, assuming they do not differ significantly) of two different time series and to derive the combined annual rate of change and standard error. To combine annual indices ($Y$) and standard errors ($S$) in the overlapping period I use the following formulas: $$ \begin{aligned} Y^*&=(Y_1/S_1^2+Y_2/S_2^2)\cdot S_1^2\cdot S_2^2/(S_1^2+S_2^2) \\ S^*&=\sqrt{S_1^2+S_2^2/S_1^2\cdot S_2^2} \end{aligned} $$ Here is an example dataset with trend estimates gained from generalized estimation equations (I do not have the raw data from which the estimates are calculated):

year      Y1        S1        Y2        S2
1990      91.3      12.0      NA        NA 
1991      81.8      8.7       NA        NA 
1992      65.9      6.9       NA        NA 
1993      80.8      8.0       NA        NA 
1994      81.7      8.1       NA        NA 
1995      90.0      8.5       NA        NA 
1996      113.7     10.0      NA        NA 
1997      117.5     10.2      NA        NA 
1998      139.9     11.6      NA        NA 
1999      120.0     9.9       NA        NA 
2000      130.1     10.6      NA        NA 
2001      115.1     9.4       NA        NA 
2002      105.2     9.2       NA        NA 
2003      83.9      7.5       NA        NA 
2004      105.8     8.9       NA        NA 
2005      111.0     9.5       85.7      7.2    
2006      100.0     0         100.0     0      
2007      105.3     9.4       104.0     6.1    
2008      93.5      8.6       102.3     5.5    
2009      117.0     10.8      111.5     6.2    
2010      107.0     10.4      108.5     6.1    

Now, I would like to get an estimate of the combined annual rate of change and its standard error for the time period of 1998 to 2008. Does anyone have any idea to share?

Answer 1

The $Y^*$ you propose is a weighted average of $Y_1$ and $Y_2$, with weights inversely proportional to their variances, which seems to me right. The problem then is the existence of missing data.

I think you might define your index as $Y^*$ when both $Y_1$ and $Y_2$ are observed, and just $Y_1$ or $Y_2$ if you only observe one. The problem might be that the resulting index would be non-homogeneous in variance, as in some periods it makes use of more information than in others. How large is this effect depends on the correlation among $Y_1$ and $Y_2$.

A more sophisticated method to derive an index from several overlapping an possibly incomplete time series might follow the method in

CASTROVIEJO, P.M. y TUSELL, F. (2007) Using redundant and incomplete time series for the estimation of cost of living indices, Review of Income and Wealth, vol. 53, p. 673-691.