# Chapter 2 VAR Models

## 2.1 Reduced-Form VAR-Models

### 2.1.1 Notation

A VAR model consists of a set of \(K\) endogeneous variables \(Y_t=(y_{1t},...,y_{Kt})\) and we denote the order of lags by \(p\). Then the VAR(p) model is defined as

\[\begin{equation} Y_t=A_0+A_1Y_{t-1}+...+A_pY_{t-p}+u_t \end{equation}\] where \(A_i\), \(i=1,...,p\) are \(K\times K\) coefficient matrices and \(u_t\) is a K-dimensional innovation with mean \(0\) and variance \(\Sigma\).

We consider a simple example without any deterministic regressors and, 2 variables \((K=2)\) and one lag \((P=1)\): \[\begin{equation} Y_t=A_1Y_{t-1}+u_t \end{equation}\] where \(Y_t\), \(A_1\) and \(u_t\) take the following form \[\begin{equation} Y_t=\left[\begin{array}{c}Y_{1t}\\Y_{2t}\end{array}\right],\quad A_1=\left[\begin{array}{cc}A_{11}&A_{A12}\\A_{21}&A_{22}\end{array}\right] \end{equation}\]

### 2.1.2 The Companion Form

### 2.1.3 Stability

## 2.2 Structural VAR-Models

Structural VAR-Models (or, in short SVAR) have been introduced by () to replace large-scale macroeconometric models used at the time. Since then they have gained widespread applications in macroeconomic research and are used to study a number of important questions, such as: What is the effect of a monetary policy shock? What is the effect of a government spending shock?

This section explores the basic relationship between reduced-form VARs and structural VARs. The key difference between reduced-form VARs and structural VARs is that the latter allows for contemporaneous variables as explanatory variables. But this leads to the problem that structural VARs cannot be directly estimated but the contemporaneous relationship have to be identified using other means. This *identification* problem has led to a fruitful area of research and will be explored in a later chapter.