# Compare slopes of multiple time-series but only one dependent variable

I am looking to determine if there are regional differences in single common measurement across time, using R.

The data is of annual frequency for 51 years (1961 to 2011) for seven regions.

The variable measured is per capita emissions.

Plotting the data visually suggests that not all regions exhibit the same trend (some appear constant, others increase/decrease while yet others look like their trends may be increasing - getting steeper).

What statistical test(s) should be used to determine if the slopes of these regional time-series trends are indeed different from each other? The measurements are likely not independent of each other within Region (behaviour is sticky), but we might presume that they are between Region...

If it helps, my dataset (in wide mts R format) is below.

Thanks!

Year    Europe Indus.Asia    Lat.Am NAfr.WCAsia NAm.Oceania       SSA  SSE.Asia
1961    1 0.3330287 0.08342768 0.2988371   0.1970832   0.6036110 0.2771904 0.1491002
1962    2 0.3399356 0.08739431 0.3050448   0.1957801   0.6072381 0.2774883 0.1519685
1963    3 0.3497737 0.09234678 0.3064860   0.1967561   0.6188448 0.2808335 0.1513993
1964    4 0.3506815 0.09567696 0.2994094   0.1964786   0.6400042 0.2846973 0.1527038
1965    5 0.3517937 0.10081783 0.2971080   0.1959702   0.6371043 0.2882228 0.1495615
1966    6 0.3655056 0.10409945 0.3022665   0.1959077   0.6513748 0.2845789 0.1506697
1967    7 0.3751051 0.10359561 0.3085563   0.1962348   0.6527076 0.2934402 0.1529256
1968    8 0.3847365 0.10244660 0.3181694   0.2015809   0.6682721 0.2886078 0.1558793
1969    9 0.3838484 0.10135446 0.3214369   0.2004127   0.6670239 0.2939225 0.1563187
1970   10 0.3936929 0.10550931 0.3190227   0.2011312   0.6736572 0.3029824 0.1582456
1971   11 0.3911788 0.10782480 0.3005018   0.2061686   0.6810935 0.3015153 0.1555705
1972   12 0.3888467 0.10796591 0.2995851   0.2042940   0.6862982 0.2964260 0.1543937
1973   13 0.4042535 0.11137749 0.2966723   0.1998279   0.6666261 0.2920530 0.1576311
1974   14 0.4114716 0.11116291 0.3081587   0.2041539   0.6770527 0.2975302 0.1573771
1975   15 0.4088594 0.11213601 0.3195318   0.2093902   0.6990652 0.2992955 0.1605997
1976   16 0.4124370 0.11137034 0.3329371   0.2110148   0.7338768 0.2997060 0.1594188
1977   17 0.4125538 0.11234512 0.3337914   0.2126674   0.7226528 0.3039447 0.1620297
1978   18 0.4198469 0.11851035 0.3397066   0.2194563   0.7101107 0.3064783 0.1637319
1979   19 0.4232864 0.12057965 0.3380299   0.2209554   0.6810473 0.3040716 0.1602157
1980   20 0.4218926 0.12148329 0.3505814   0.2280130   0.6721993 0.3029758 0.1621418
1981   21 0.4172976 0.12375568 0.3494269   0.2311256   0.6807635 0.3051217 0.1660875
1982   22 0.4223521 0.13089340 0.3373198   0.2317372   0.6819238 0.3058660 0.1661771
1983   23 0.4261567 0.13512950 0.3316797   0.2311859   0.6781721 0.2947787 0.1713330
1984   24 0.4327464 0.13905190 0.3364816   0.2352414   0.6853792 0.2891287 0.1726075
1985   25 0.4327769 0.14014033 0.3448070   0.2415244   0.6955721 0.2943122 0.1725749
1986   26 0.4384473 0.14360094 0.3596229   0.2449454   0.6966196 0.2941324 0.1753695
1987   27 0.4409924 0.14773305 0.3481004   0.2412722   0.6998172 0.2942876 0.1750773
1988   28 0.4418265 0.14960314 0.3485329   0.2437859   0.6900106 0.2968633 0.1778406
1989   29 0.4447421 0.15081434 0.3546949   0.2384495   0.6866791 0.3024815 0.1810750
1990   30 0.4304627 0.15639769 0.3570101   0.2413544   0.6841630 0.3049107 0.1816244
1991   31 0.4142964 0.15557267 0.3595744   0.2363494   0.6824669 0.3054693 0.1803897
1992   32 0.4099102 0.16369167 0.3620947   0.2492692   0.6920663 0.3002969 0.1849306
1993   33 0.3970080 0.17536967 0.3649345   0.2463124   0.6811545 0.3011804 0.1872747
1994   34 0.3827137 0.18347220 0.3754718   0.2482677   0.7008975 0.2940523 0.1884567
1995   35 0.3823503 0.19545531 0.3861121   0.2491750   0.6987574 0.2967518 0.1917361
1996   36 0.3844043 0.20075046 0.3952523   0.2507883   0.6990538 0.3023953 0.1937828
1997   37 0.3812992 0.20906418 0.3934072   0.2484025   0.6972838 0.3041432 0.1930203
1998   38 0.3798254 0.21622077 0.3870048   0.2545955   0.7007982 0.3047103 0.1946998
1999   39 0.3773650 0.22288219 0.4024141   0.2563572   0.7167966 0.3103474 0.1975481
2000   40 0.3792981 0.23270136 0.4036371   0.2594711   0.7132213 0.3080760 0.1984123
2001   41 0.3740450 0.23275714 0.4023274   0.2531616   0.7031493 0.3120869 0.1994249
2002   42 0.3778445 0.23755244 0.4036734   0.2609872   0.7121247 0.3163197 0.1958088
2003   43 0.3830538 0.24061360 0.4048166   0.2666359   0.7032099 0.3208936 0.2019877
2004   44 0.3879519 0.24239370 0.4106120   0.2687964   0.7123248 0.3294516 0.2035080
2005   45 0.3927197 0.24813350 0.4215930   0.2742297   0.7061623 0.3348791 0.2080581
2006   46 0.3947798 0.25229643 0.4313497   0.2810117   0.7046364 0.3409603 0.2142793
2007   47 0.3980420 0.25871431 0.4414887   0.2843196   0.7053338 0.3409665 0.2222694
2008   48 0.3975002 0.26601418 0.4477263   0.2842824   0.6871963 0.3479022 0.2245092
2009   49 0.4016810 0.27165581 0.4399089   0.2869924   0.6803058 0.3448820 0.2265546
2010   50 0.3940681 0.27902452 0.4439119   0.2939435   0.6725924 0.3563978 0.2337000
2011   51 0.3887767 0.28328307 0.4481589   0.3020620   0.6633836 0.3586240 0.2358033
• What do you think about my answer? Feb 10, 2017 at 11:11

One simple idea (perhaps too simple) would be to assume the time trends are linear in all the cases and then compare the slope coefficients.

Suppose you only had two series and three years (rather than seven series and 51 years), hence, $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$. Then you could build the following variables:

$z=(x,y)=(x_1,x_2,x_3,y_1,y_2,y_3)$,
$\text{intercept}_1=(1,1,1,0,0,0)$,
$\text{intercept}_2=(0,0,0,1,1,1)$,
$\text{time}_1=(1,2,3,0,0,0)$,
$\text{time}_2=(0,0,0,1,2,3)$

and estimate the following model

$$z = \beta_{0,1} \text{intercept}_1 + \beta_{0,2} \text{intercept}_2 + \beta_{1,1} \text{time}_1 + \beta_{1,2} \text{time}_2 + \varepsilon$$

and test for $\beta_{1,1}=\beta_{1,2}$.