Impulse response function

As discussed in the critique of VAR models, it is usually difficult to directly interpret the coefficients of an estimated VAR model. The impulse response function (IRF) is therefore often computed in order to study the interrelationships within the variables of a system (Griffiths and Lutkepohl, 1990). An impulse response function is taken as an essential tool in empirical causal analysis and policy effectiveness analysis, since it measures the time profile of the effect of a shock on the behaviour of a series and can be applied on a VAR model, when estimated from stationary data, as well as VEC model. However, the impulse response function modelled from the structural VAR approach for analyzing the monetary transmission mechanism is often criticized due to its assumption of a certain level of random behaviour from central banks (Bernanke and Mihov, 1998). Thus, the assumption of random behaviour in policy makers’ decisions must be taken with caution. Despite this assumption, SVAR models can still be used for tracing monetary shocks, since SVAR models trace the dynamics of the model thus, the shocks do not have to be large or persistent. Whether a VAR model, SVAR model or VEC model is used, the economic interpretation of monetary policy shocks is not straightforward or clear. Since monetary policy shocks can be generated from imperfect information that the central banks have about the current state of economy and the importance of output and inflation are relatively different in terms of moderating fluctuations, a certain level of random behaviour can be traced (Bernanke and Mihov, 1998). The random process and its fluctuations enable investigating the effect of monetary policy shocks on economic variables.

Since the impulse response function measures the time profile of the effect of a shock on the behaviour of a series, it can also be used for testing the reaction of consumer price inflation to a simulated shock to commodity prices in order to understand the relationship between variables. The analysis is carried out with regard to shocks of time-series rather than the series themselves. As noted, an impulse response function can be used in VAR, SVAR and VEC models with a similar technique but different results (Bisgaard and Kulahci, 2011).

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While IRFs from a stationary VAR expire over time, IRFs from a cointegrating VEC model do not always expire. Since each variable in a stationary VAR has a time invariant mean, and limits it causes, so that the effect of a shock to any one of the explanatory variables must expire in time so that the dependant variable can revert to its mean. However, if IRF is applied to time-series I(1), therefore the series is non-stationary but stationary in the first difference, in a cointegrating VEC model the effects of shocks will not expire over time. This is due to long-term equilibrium defined in the VEC model (Kennedy, 2003). In respect to the nature of series, there is an option for either a traditional impulse response or a generalized impulse response function. The generalized IRF is used, according to Koop et al. (1996), if treatment of the future is dealt with by using the expectation operator conditioned on only the history and/or shock. The impulse response constructed this way is therefore an average of what might happen given what happened in the present and past. However, a generalized impulse response function is strictly applicable only to stochastic time-series where shocks have a well defined meaning. Due to the volatility of commodity prices as explanatory variables, series can be assumed to be deterministic; therefore a traditional impulse response would be more relevant. The traditional IRS answers the question of what is the effect of a shock of size δ at time t on the series at time t+n under the condition of ceteris paribus.

The traditional impulse function can be expressed as the difference between the two different realizations of , where one realizations assumes time-series being hit by only one shock between t and t+n while the other realizations assumes the time-series is not hit by any shock between t and t+n therefore (Brooks, 2008):

-

+ =0, (4.29)

The impulse response function, whether applied to the VAR or VEC model, will therefore help to answer the question of what happens to consumer price inflation if there is a one unit shock to world food price, crude oil price and other series.

4.8.4.1 Variance decomposition

Another econometric tool in the VAR analysis for assessing the driving forces of cyclical fluctuations is variance decomposition. As explained by Seymen (2008), it gives the proportion of movements in the dependent variables that are due to their own shocks, versus shocks to the other variables. Indeed a shock to one of the variables will directly affect that

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variable; however, it will also affect other variables in the system through the dynamic structure of the VAR. Therefore, variance decompositions determine how much of the forecast error variance of a given variable is explained by innovations to each explanatory variable. Usually, it can be observed that own series shocks explain most of the forecast error variance of the series in a VAR (Asteriou and Hall, 2011). Therefore, variance decomposition function can be also presented as a demonstration of the forecast error variance and it also refers to the breakdown of the forecast error variance for a specific time horizon and is able to indicate which variables have short-term and long-term impacts on another variable of interest. In other words, it provides valuable information on what percentage of the fluctuation in a time-series is attributable to other variables at select time horizons.