Excerpt

## Contents

1 Introduction

2 Building a recursive VAR

2.1 From the primitive system to the reduced form VAR .

2.2 Identification and the recursiveness aproach

2.3 Lag-lenght selection criteria

3 Application of the recursive VAR toolkit

3.1 Data description

3.2 The transmission mechanism of monetary policy

3.3 The price puzzle

3.4 Estimation of the model, determination of monetary shocks and impulse response functions

4 Conclusion

5 Appendix

## 1 Introduction

In the mid 70’s people started to doubt the validity of macroeconomic models as they were not able to forecast the worldwide recession due to the oil-price shock. These models needed an a priori seperation into endogenous or exogenous variables. This need for seperation was criticized by Sims (1980), who proposed as solution for this problem a Vector Autoregressive model (VAR). ”A VAR is an n-equation, n-variable linear model in which each variable is in turn explained by its own lagged values, plus current and past values of the remaining n-1 variables.”^{1} This offers the possibility that these variables influence each other mutually, which makes each of them endogenous.^{2}

Let us put some economic background to these definitions. As we focus on monetary policy, we might be interested in the mutual relation and behaviour of the interest rate (r) and inflation (π). For simplicity we just take these two variables with one lag into account. According to our setup we yield the following equation system:

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The equation system of (1) and (2) is called the primitive system, where (ϵrt) and (ϵπt) are uncorrelated white noise disturbances.

## 2 Building a recursive VAR

In this section we will build a recursive VAR step by step. In the first section the primitive system will be transformed into the reduced form VAR. Building on that, we will focus on the identification of the primitive system, which finally will yield us the recursive VAR. Finally we abandon the exogenous assumption about the lag-length which we set for simplicity to one. This will let us end up with the methodology, which will enable us to apply the VAR toolkit to a real world situation.

### 2.1 From the primitive system to the reduced form VAR

The problem with our primitive system is that if α12 and α22 are not equal to zero it is not possible to estimate its parameters. This is due to the indirect contemporaneous effect of ϵrt on πt and ϵπt on rt.^{3} using matrix algebra:

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Premultiplication by [Abbildung in dieser Leseprobe nicht enthalten] yields us to:

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where

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For notational purpose one can define ai0 as element i of vector A0, aij as the element in row i and column j of matrix A1, and eit as the element i of the vector et. We then can rewrite equation (3) and yield the following:

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The equation system, consisting of the equations (4) and (5), is called the VAR in standard form, or the reduced form VAR. As only lagged values occur in our reduced form VAR, it is now possible to estimate the coefficients consistently using OLS.^{4} Before we turn to the identification of the primitive system, it is worth to take a closer look on the residuals in the reduced form. The residuals e1t and e2t are given by:

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The variance and covariance of the residuals is given by:

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It becomes obvious that the error terms are composites of ϵrt and ϵπt. Furthermore we have to note that the error terms will be correlated, unless we have the (unlikely) case where α12=α21=0, which would be the case if the interest rate had no influence on inflation and vice versa.

### 2.2 Identification and the recursiveness aproach

Suppose we estimated the coefficients of the standard form VAR via OLS. We would by now know the six coefficients a10, a20, a11, a12, a21 and a22, as well as the variance of e1t and e2t and their covariance σe1t;e2t . The dilemma is that our primitive system, which we want to identify, has ten coefficients, which is one more than we obtain throughout the estimation of our reduced form VAR. According to this we are unable to recover all of the information which is present in the primitive system.^{5} Therefore, we have to restrict the primitive system. A restriction works in the way that we simply set one coefficient of the primitive system equal to zero. Let us restrict our primitive system by setting α12=0. The economic interpretation of this would be that inflation had no contemporaneous effect on the interest rate. On the opposite, the interest rate effects inflation contemporaneously.

This implies that the interest rate reacts with a lag to inflation.^{6} This kind of restriction, to identify the primitive system, is called recursive approach and was initially proposed by Sims (1980). Fact is, that we end up with a nine variable primitive system which we now could identify. The determination of the disturbance terms is now possible and yields in our case to:

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This points once more out that the shocks ϵrt and ϵπt effect inflation contemporaneously, while the interest rate is only effected by its own shock term. This triangular^{7} decomposition of the residuals is called Choleski decomposition and will be again of higher interest when we apply the recursive VAR to real world issues. Anyhow, it is important to notice that the restrictions which we impose to our system should be based on economic theory and should not be set arbitrarily, as we did it in our example.

### 2.3 Lag-lenght selection criteria

After having focused the methodology for solving a VAR under the recursiveness assump- tion, it is now time to drop our predefined lag length assumption. For simplicity we included only one lag in our model. The determination of the ”right” lag length, that yields to optimal results is somehow difficult. It has become common practice to include four lags in a VAR model. Four lags seem to deliver quite reasonable results and simul- taneously do not make the model too confusing. Beside this rule of thumb, two popular quantitative methods for lag selection will shortly be introduced in the following, namely the Akaike Information Criterion (AIC) and the Schwarz Bayesian Information Criterion (SIC). These criteria propose a method, which ”balances model fit and model complex- ity”.^{8} According to these criteria, the number of lags (p) to include in the model are the ones which minimize the value of the particular criterion. There is no final agreement in literature about the superiority of one of these criteria over the other. Anyhow, qualitative agreement was achieved, which points to an overfit of the data by the AIC while the SIC is likely to underfit data.^{9} Luetkepohl (1993) points out that forecasts based on the SIC are superior (though there is an underestimation of the lag length) to those of the AIC for a small sample size. This advantage vanishes when the sample size increases. Luetkepohl therefore suggests to ”compare the order estimates obtained by different criteria and possibly perform an analysis with different VAR orders”.^{10}

## 3 Application of the recursive VAR toolkit

In this section we will focus on the application of a recursive VAR on economic data. By that we will identify monetary shocks in the past and show the effect of an interest rate hike on output and inflation. The work in this section bases on the one of Boivin and Giannoni (2005)^{11}. In particular, we will extent their model estimation horizon, as we have by now more data available. This section is organized as follows: First, we will take a look at the data which we will use for our further analysis. Afterwards, we consider the alignment of the the variables in the VAR. This will be related to the description of the transmission mechanism of monetary policy. Before turning to the estimation of the model, we will take a look at the so called price puzzle, which will justify the inclusion of commodity prices in our analysis. The next step will then be the estimation of our recursive VAR. Having the estimated model we will finally be able to determine past monetary shocks and compute impulse response functions, which plot the response of the included macroeconomic variables to a monetary shock.

**[...]**

^{1} See Stock and Watson (2001) p. 1.

^{2} See Rinne and Specht (2002) p. 522.

^{3} Compare Gujarati (2003) p. 724.

^{4} See Enders (2004) p. 270.

^{5} Compare Enders (2004) p. 271.

^{6} Compare Christiano, Eichenbaum, Evans (1998) p. 5.

^{7} Triangular, as only zeros occur in the upper triangular of B.

^{8} See Koehler and Murphree (1998) p. 187.

^{9} Compare Koehler and Murphree (1998) p. 194.

^{10} See Luetkepohl (1993) p. 138.

^{11} The work of Boivin Giannoni (2005) will in the following be shortened by BG.

- Quote paper
- Felix Miebs (Author), 2006, Which are the Effects of Monetary Policy? Identifying Policy Shocks in recursive VARs, Munich, GRIN Verlag, https://www.grin.com/document/71446

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