Impulse Response Functions or IRFs are used to study the effects of shocks or impulses in a VAR or VECM system. It traces out one unit or one standard deviation shock to an endogenous variable and its effects on all the endogenous variables in a VAR or VECM, keeping all other variables and shocks constant.

For instance, consider a VAR system with income, consumption and investment as endogenous variables. Using impulse response functions, we can deliver one standard deviation or one unit shock to income. The effects of that shock can be seen in all the endogenous variables and equations of that VAR model. Hence, we can analyse the effects of income shock on consumption and investment behaviour over different time periods.

Impulse Response Functions are the most important application of VAR and VECM models. They are useful in understanding the dynamic behaviour of variables in a system. IRFs are important tools in predicting the effects of impulses/shocks and policy analysis.

## Vector Autoregression (VAR) specification: Reduced-Form VAR

Suppose, we have a VAR model of order 2 or VAR(2):

This model contains 3 endogenous variables. Each equation has 2 lags of income, consumption and investment as independent variables. The choice of lags or lag length can be made using various Information Criteria.

This model is known as a Reduced-Form VAR because the endogenous variables are determined only by their past values and the past values of other variables. This implies that no endogenous variable affects another in the same time period, i.e. there are no contemporaneous effects.

The error terms in a VAR system are referred to as impulses or innovations. The shocks or impulses to an endogenous variable are introduced through these error terms. Suppose, we wish to study the effects of income shock on the above VAR system. The shock is introduced in the 1st equation of income through its error term *u _{1t}*.

The shock or impulse and its effects can be traced using IRFs as follows:

## Impulse Response Functions (IRF)

IRFs can be easily estimated using statistical software packages. Let us assume one standard deviation shock to income. The shock will affect income (1st equation) in the same time period. However, the effects of that shock will be visible on other endogenous variables in the next time period. This happens because our VAR(2) system contains only the lags of variables, i.e. endogenous variables are explained by the previous 2-period lags. There are no contemporaneous effects or variables are not explained by the current period values.

Therefore, as the shock or impulse occurs, its effect on income will be seen in the same time period because it is an income shock. However, its effects on consumption and investment can be observed from the next time period onwards.

### Effect of Income shock on Investment

The graph shows that investment does not respond to income shock in the same time period. However, there is a positive relationship between income and investment. One standard deviation positive shock to income leads to a 0.4 standard deviation increase in investment in the first time period after the shock. In time period 2, the shock further causes a 0.5 standard deviation increase in investment. The effects of the shock start to reduce in the third time period and the impulse response function converges to zero. The effects of the shock die out in time period 5, where the lower bound confidence interval is zero.

The upper and lower bound confidence intervals show a range within which the impulse response may vary. The effects of the shock are significant in 5 time periods as confidence intervals are well above zero. Hence, we can say that any shock to income will have a positive impact on investment, other things remain constant. The effects of the shock will be significant and persist for 5 time periods after the shock occurs.

### Effect of Income shock on Consumption

Similar to investment, we can observe a positive relationship between income and consumption. One standard deviation increase in income due to shock leads to a 0.6 standard deviation increase in consumption. The effects of the shock are less in the second time period. The impulse response function converges to zero in the third time period as the effects of the shock die out.

As compared to consumption, the effects of income shock last longer on investment. Whereas, the effects fall to zero quickly in case of consumption.

## Forecast Error Variance Decomposition (FEVD)

In addition to IRFs, we also have the option to estimate Forecast Error Variance Decompositions (FEVD). FEVD shows the proportion of variations in an endogenous variable that is explained by a shock or impulse.

In the above example, we focussed on income shock and its effects on investment and consumption. Along with IRFs, we can obtain FEVD. For instance, FEVD for income shock on investment will show the proportion of variation in investment that is explained by income shock.

In this FEVD, we can observe that approximately 20% of the variations in investment are explained by income shock in the first time period. As we move further to the next time periods, the proportion of variation in investment explained by income shock goes on increasing. By time period 3, around 40% of the variations are explained by income shock. This proportion stabilises at approximately 40% beyond time period 3. The remaining variations in investment (unexplained by income shock) are explained by other shocks- consumption and investment shocks.

Similarly, we can construct FEVDs for investment in response to consumption shocks or investment shocks. These shocks explain the remaining proportion of variations in investment. For consumption shocks, we will see the proportion of variation in investment explained by consumption shock. Such analysis helps us decompose the effects of various shocks and trace those effects over time.

## Orthogonalized Impulse Response Functions (OIRFs) and SVAR

Orthogonalized-IRFs (OIRF) allow us to include contemporaneous effects of shocks or impulses. This means that we can use OIRFs to include the effects of shocks on endogenous variables in the same time period.

In the IRFs above, the income shocks affected investment and consumption in the next time period. This happened because we used a Reduced-Form VAR, where we included only past values of variables as independent variables in each equation. In reality, however, income should affect consumption and investment in the current time period as well. That is, any changes or shocks to income in the first equation will have an effect on investment and consumption in the same time period. Such a VAR model which includes contemporaneous effects can be stated as follows:

### Structural-VAR or SVAR

This model including contemporaneous effects is known as a Structural VAR or SVAR. It allows us to include more information in the model as compared to a Reduced-Form VAR because we have included current period variables in the model.

The first equation of income is similar to that of the Reduced-Form VAR as income is explained only by past or lagged values. The second equation for investment, however, contains contemporaneous effects of income on investment along with lagged or last values. The current period income affects investment instantly, i.e. in the same time period. Hence, the effects of any shocks to income will be observed on investment in the same time period.

Similarly, consumption is affected by income and investment in the same time period. Any shock to income or investment affects consumption in the same time period.

* NOTE:* we cannot include contemporaneous effects in all the equations because of the problem of Identification in simultaneous equation models. Hence, our model includes current period income to explain investment (2nd equation) and current period income and investment to explain consumption (3rd equation). The first equation of income does not include any current period variables. This type of structure is referred to as “Cholesky Decomposition”. This makes our SVAR model Exactly Identified.

### OIRFs (orthogonalized impulse response functions)

OIRFs are estimated after incorporating contemporaneous effects in the model. They give more information as compared to simple IRFs because they include current period effects as well.

** Important:** the ordering of variables has an impact on OIRFs. In our example, we have ordered our variables as income, investment and consumption. This implies that income will not be affected by other endogenous variables in the current time period. Consumption, on the other hand, is affected by both income and investment contemporaneously but does not have any effect on other variables in the current time period. Therefore, the ordering of variables determines which variables will affect others in the current time period. This decision must be based on some economic theory or relevant economic considerations.

#### Interpreting OIRFs (orthogonalized impulse response functions)

The interpretation of OIRFs is similar to IRFs. Let us take a look at some examples:

In the SVAR model and our ordering of variables, income affects investment contemporaneously. As a result, one standard deviation positive shock to income increases investment by approximately 0.5 standard deviations in the current time period or instantly. The effects of the shock increase in the first and second time period after the shock and start declining after. The effects of the shock converge to zero in the fifth time period. We can conclude that income has a positive and significant impact on investment. Moreover, the effects of any shock to income will persist on investment for 5 time periods in the future.

Based on the ordering of our variables, investment affects consumption in the same time period. It is evident from the above OIRF as a positive shock to investment reduces consumption in the current time period. The OIRF shows that one standard deviation positive shock to investment will decrease current consumption by 0.2 standard deviations. This negative effect is reduced in the first time period. In the second time period, we can observe that investment shock has a delayed positive impact on consumption. This means that positive investment shocks will initially reduce consumption in the current period, whereas, they will have a delayed positive impact on consumption after two time periods.

## Reliability of Impulse Response Functions

### VAR

Before estimating IRFs, OIRFs or FEVDs from VAR or SVAR model, we must ensure that:

- the endogenous variables in the VAR/SVAR model are stationary. If the variables are not stationary, the results of the model will not be reliable and we should not implement IRFs. Hence, we must determine the order of integration of variables before estimating VAR/SVAR. If the variables are level stationary or I(0), we can apply VAR to the original variables. If variables are integrated of order 1 or I(1), we must apply VAR after transforming the variables in their first deference form.
- VAR/SVAR model is stable. A stability check is necessary to avoid exploding forecasts or exploding IRFs. If the VAR model is stable, the IRFs will converge to zero after some time periods. If the VAR model is unstable, the IRFs will explode and will not be reliable. This happens when the variables are non-stationary and are not transformed appropriately.
- the model does not suffer from autocorrelation. If we detect autocorrelation, we must include more lagged variables in the model. The choice of lag length must ensure that autocorrelation is eliminated from the model.

The tests of stationarity, stability of VAR and autocorrelation test after VAR can be easily performed using statistical software packages.

** Note:** In VAR, the IRFs and OIRFs must converge to zero after some time periods. If they do not converge back to zero, the VAR model is not stable and reliable.

### VECM

Similar to VAR, we must check for autocorrelation and the stability of the VECM model. The VECM is applied in the presence of cointegration. As a result, we obtain both short-run and long-run coefficients from the model.

In VAR, the effects of shocks or impulses always converge to zero after some time periods. This is not the case for VECM models. The IRFs or effects of the shocks may or may not converge to zero in a VECM model. Hence, we have two different types of shocks or impulses in a VECM model:

when the IRfs or OIRFs converge to zero after some time periods, they are known as transitory shocks. The effects of these shocks die out after some time. For example, the effect of an income shock on consumption may become zero after a few time periods and it will not permanently increase consumption.*Transitory shocks:*the IRFs or OIRFs do not converge to zero in the case of permanent shocks. Instead they stabilize around a positive or negative value. Such shocks have a permanent effect on the response variable. For instance, an income shock may permanently shift investment upwards.*Permanent shocks:*