15
Apr 17

## Error correction model

Error correction model: not a big deal but it is in the syllabus. This post derives from Matlab help, which has the clearest explanation.

Stochastic processes are divided into two broad categories: stationary and nonstationary ones. Stationary processes are to all processes what zero is to all numbers: a nonstationary process plus a stationary process is nonstationary. A difference of two nonzero numbers may give zero. Similarly, a difference of two nonstationary processes may give a stationary one. More generally, processes $\{X_{1,t}\},...,\{X_{n,t}\}$ are called cointegrated if there exist numbers $a_1,...,a_n$, not all of which are zero, such that

(1) $a_1X_{1,t}+...+a_nX_{n,t}=u_t$

is a stationary process. The linear combination on the left of (1) is called a cointegrating relationship.

Intuition. The source of nonstationarity is usually a trend (deterministic or stochastic). Variables can be cointegrated because they move around common trends. The cointegrating relationship reflects a long-term comovement tendency of stochastic processes (because (1) is true for all moments).

### Error correction model

Consider two cointegrated processes $\{X_{1,t}\},\{X_{2,t}\}$:

(2) $a_1X_{1,t}+a_2X_{n,t}=u_t.$

A short-term dynamics may be expressed by a model in differences such as

(3) $\Delta{X_{1,t}}=\beta\Delta{X_{2,t}}+v_t.$

The difference is defined by $\Delta{{X_t}}=X_t-X_{t-1}$. The differences are used in (3) to "clean" the variables of possible trends. An error correction model is obtained by adding to (3) a term proportional to (2):

(4) $\Delta{X_{1,t}}=c(a_1X_{1,t}+a_2X_{n,t})+\beta\Delta{X_{2,t}}+v_t=cu_t+\beta\Delta{X_{2,t}}+v_t.$

It balances a short-term dynamics of (3) with long-term tendency expressed by (2). Remember that before using (4) you need to run regression to prove (2).

A simple exercise from Dougherty is to prove that (4) is an autoregressive distributed lag model of order (1,1), that is, it has one lag of each of the dependent and independent variables.