Basics of Decomposition Analysis

I posted about "Basic decomposition" before. But at that time I was not so clear about it. Here is what I understand about it for sure.

Structural decomposition analysis(SDA) is an analytical method to find sources of dependent variable's change between two different time period. Sources of change are sought from basic relationship between dependent variable, independent variables and coefficients. Note: SDA always considers the possibility that coefficients change along with other variables. Change of coefficients is considered as a result from technological change. It is critical difference from growth accounting analysis (GAA), which presents technological change as a variable.

For example, let's say dependent variable Y can be expressed multiplication of independent variable X and coefficient a: Y= a*X. Now, we want to compare two different time period, t and t-1. There may be some difference in dependent variable (Yt - Yt-1), and we want to find sources of change.

Yt-Y(t-1) = [at*Xt] - [a(t-1)*X(t-1)] ---(1)

From this equation, we can consider three sources of change:

1. Change from independent variable, when coefficient is fixed to original time (t-1): Level change
   
2. Change from coefficient, when independent variable is fixed to original time (t-1): Technological change
3. Change from interaction of independent variable and coefficient change: Joint change

In equation form,

Difference of dependent variable
= [initial period (t-1)'s coefficient: a(t-1)] * [change of independent variables: Xt -X(t-1)] <- 1.
+ [change of coefficient: at - a(t-1)] * [initial period (t-1)'s independent variable: X(t-1)] <- 2.
+ [change of coefficient: at - a(t-1)] * [change of independent variables: Xt -X(t-1)] <- 3.

In order to present these sources, we need add and subtract [a(t-1)*Xt] + [at*X(t-1)] + [a(t-1)*X(t-1)] to the equation (1).

Yt-Y(t-1) = [at*Xt] - [a(t-1)*X(t-1)]
+ {[a(t-1)*Xt] + [at*X(t-1)] + [a(t-1)*X(t-1)]}
- {[a(t-1)*Xt] + [at*X(t-1)] + [a(t-1)*X(t-1)]}

= [a(t-1)*Xt] - [a(t-1)*X(t-1)]
+ [at*X(t-1)] - [a(t-1)*X(t-1)]
+ [at*Xt] + [a(t-1)*X(t-1)] - [a(t-1)*Xt] - [at*X(t-1)]

= a(t-1) * [Xt - X(t-1)] : 1. level change
+ X(t-1) * [at - a(t-1)] : 2. technological change
+ [at - a(t-1)] * [Xt - X(t-1)] : 3. Joint change

It looks complicated. But gist of analysis is straight forward; it considers three possible cases when we assume independent variable and coefficient change between two periods: (1) only independent variable changes with fixed coefficient, (2) only coefficient changes with fixed independent variable, (3) both change. About economic meaning of each change, I will explain later.

Understanding this basics, I took a look at the final demand decomposition again. But still it looks vague. One burning question is whether sigma Yi is equal to Y. It is. Than second term of the decomposition analysis is always zero. In order to answer this question, I need to practice with real IO tables.