Question

provide an overview of the theory and application of vector error correction model in financial econometrics. ensure you use some examples to elaborate your views as well as the key diagostics

Answer 1

A vector error correction model is a restricted vector autoregressive model (VAR) used for cointegrated non-stationary series. If the variables are stationary use the ordinary least square method directly. For a non-stationary series, we modify the VAR model to ensure consistent estimation. If x and y are stationary in their first differences and are cointegrated, then adjustments are made for the cointegration. The modified model is the vector error correction model (VEC).

Explanation:

A vector error correction model is a restricted vector autoregressive model (VAR) used for cointegrated non-stationary series.

The VAR model is a model used to express the interrelationships among the stationary variables. The VAR model requires the data to be stationary. If the data is found not to be stationary, we check again by taking the first differences of the time series.

For a non-stationary series, we modify the VAR model to ensure consistent estimation.

The vector error correction model is used when the variables are stationary in their first differences I (1)

For example:

Let us consider the time series variables yt and xt where each included variable is a determined by its own lag value and the lag value of the other included variable.

yt​=β10​+β11​yt−1​+β12​xt−1​+vty​xt​=β10​+β11​xt−1​+β12​yt−1​+vtx​​

This equations system is known as vector autoregression. Ordinary least square method can be used if the variables x and y are stationary. For non-stationary x and y, the estimation needs to be done on the differences I (1).

Δyt​=β10​+β11​Δyt−1​+β12​Δxt−1​+vtΔy​Δxt​=β10​+β11​Δxt−1​+β12​Δyt−1​+vtΔx​​

If x and y are stationary in their first differences and are cointegrated, then adjustments are made for the cointegration. The modified model is the vector error correction model (VEC).

The adjustments made for the cointegration restricts the variables to converge to the relationships, defined by cointegration, in the long run while allow it for short-run adjustments. The error term is the term showing cointegration. Correction of the deviations from the equilibrium (long-run) is done overtime through a chain of partial short run adjustments.

1.   If the variables are stationary use the ordinary least square method directly.

2.   The variables should be time-series.

3.   For non-stationary variables, check the first differences for stationarity.

4.   Check for the cointegration between the variables. Cointegration tests should be done.

5.   If x and y are the first differences and are cointegrated, then adjustments are made for the cointegration, the VEC model is applicable.