Percent change: adding difference or calculating by date?

Question

I have data going over time showing funding provided by different states to a certain government programs. I want to compare them with each other and show over time who's giving more and who's giving less than they were in 2008 (result will be a line chart).

$$\begin{array}{l|c_1|c_2} \text{Year} & \text{Alabama} & \text{Alaska}\\ \hline 2008 & 196,342 &298,000\\ 2009 & 158,946 &318,500\\ 2010 & 143,242 & 333,600\\ 2011 & 142,050 & 348,598\\ \end{array}$$

Would I do this by getting the cumulative percent change? Or by checking each year by the earliest year in the dataset?

Ex. If I wanted to see how much Alabama has changed since 2008, is it better to do $\frac{2011-2008}{2008}$, or do that for each year (2009, 2010, 2011) and sum that change?

Answer 1

I would consider either cumulative percent change or its annualized version, to give you average yearly change, which may or may not be more meaningful depending on what you want to analyze and to whom you are presenting.

E.g. 3-year change in Alabama would be $$ \frac{X_{2011} - X_{2008}}{X_{2008}} = \frac{X_{2011}}{X_{2008}} - 1 $$ but annualized it is $$ \left(\frac{X_{2011}}{X_{2008}}\right)^{1/(2011-2008)} - 1 $$

Answer 2

Both methods lead to the same result. Let´s say you have 3 data for the years (end of the years) $2014, 2015,2016 : (x_1,x_2,x_3)=(100,110,121)$

Using all three data to calculate the cummulative growth rate from the beginning of $2015$ to the end of $2016$:

$$g=\frac{x_2}{x_1}\cdot \frac{x_3}{x_2}-1$$

$g=\frac{110}{100}\cdot \frac{121}{110}-1=1.1\cdot 1.1-1=1.21-1=0.21$

From the formula we see that $x_2$ can be cancelled. Thus we can use $x_1,x_3$ only:

$g=\frac{x_3}{x_1}=\frac{121}{100}-1=0.21$