**A Mark Sadowski post**

What I am going to do next is to construct three bivariate Vector Auto-Regression (VAR) models to generate Impulse Response Functions (IRFs) in order to show what a shock to the monetary base leads to in terms of the price level and output. As mentioned in Part 1, the order of integration for the log levels of all four of our series (SBASENS, CPI, PCEPI and INDPRO) is one. With two unit roots per model, we are faced with a procedure that could lead to a VAR model in differences (a VARD), a VAR model in levels (a VARL), or a Vector Error Correction Model (a VECM).

Using the Augmented Dickey-Fuller (ADF) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests I find that the order of integration is zero for all four series after they are differenced (as expected). In order to render the IRFs easier to interpret, for the rest of this analysis I have multiplied the natural log of each series by 100.

Since there is no evidence of cointegration between the monetary base and CPI or PCEPI, we are really only faced with a choice between a VARD and a VARL in these two cases. This means we are confronted with a tradeoff between statistical efficiency and the potential loss of information that takes place when time series are differenced. In the interests of thoroughness, I am going to do it both ways.

First, let’s estimate the VARDs. Most information criteria suggest a maximum lag length of one in both VARDs. The LM test suggests that there is a problem with serial correlation at this lag length, but this problem disappears when the lag length is increased to three in both of these VARDs. The AR roots tables suggest that both VARDs are dynamically stable at this lag length.

Motivated by the dominant practice in the empirical literature on the transmission of monetary policy shocks, I am going to use a recursive identification strategy (Choleskey decomposition). Such a strategy means that the order of the variables affects the results. I will follow the traditional practice of ordering output and the price level before the monetary policy instrument in each vector. The response standard errors I will show are analytic, as Monte Carlo standard errors change each time an IRF is generated.

Here are the responses to a shock to the monetary base in the VARD including CPI as a variable. I’ll restrict the time period to 10 months as it isn’t of interest after that point.

And here are responses to a shock to the monetary base in the VARD including PCEPI as a variable.

In each case a positive shock to the rate of change in the monetary base generates a statistically significant positive response to the rate of change in the price level (i.e. the inflation rate) in the third month. Changing the impulse definition to Residual or Generalized Impulse doesn’t change the results much, if at all.

Now, let’s estimate the VARLs. Most information criteria suggest a maximum lag length of two in both VARLs. The LM test suggests that there is a problem with serial correlation at this lag length, but this problem disappears when the lag length is increased to three in both of these VARLs. The AR roots tables suggest that both VARLs are dynamically stable at this lag length.

Here are the responses to a shock to the monetary base in the VARL including CPI as a variable. In this case I’ll extent the time period to 48 months.

And here are responses to a shock to the monetary base in the VARL including PCEPI as a variable.

A positive shock to the monetary base generates a statistically significant positive response to the CPI in months three and four. A positive shock to the monetary base generates a statistically significant positive response to the PCEPI in months two through five. Changing the impulse definition to Generalized Impulse renders the responses statistically insignificant (there is little difference using Residual impulses), but we’ve already established that the monetary base has a statistically significant positive effect on CPI and PCEPI with the VARDs. And, as I said earlier, there is loss in statistical efficiency when estimating a VARL with unit roots.

The IRFs show that a 2.5% and 2.6% shock to the monetary base in month one leads to a peak increase in the CPI level of 0.12% and in the PCEPI level of 0.085% in month four respectively. In short, the empirical evidence does not seem to be very supportive of the Neo-Fisherite hypothesis.

Now let us turn our attention to industrial production.

In Part 1 we showed that there is evidence of cointegration between the monetary base and industrial production. This means, in addition to the option of estimating a VARD or a VARL, we may also estimate a VECM. But before discussing the pros and cons of doing this, let us estimate a VARD and VARL first. I should also mention at this point that, in a book edited by David Glasner, “Business Cycles and Depressions: An Encyclopedia”, Neil Ericsson (in an article entitled “Distributed Lags”) argues that estimating a model in first differences alone when cointegration exists discards the information contained in the levels and leads to model misspecification. Consequently a VARL is probably preferred to a VARD in this case, but it may still be useful to estimate both.

First let’s estimate the VARD. Most information criteria suggest a maximum lag length of three in the VARD. The LM test suggests that there is no problem with serial correlation at this lag length. The AR roots table suggests that the VARD is dynamically stable at this lag length. Here is the IRF for the VARD.

A positive shock to the rate of change in the monetary base generates a statistically significant negative response to the rate of change in industrial production in the third month, followed by a statistically significant positive response in the fourth month. Changing the impulse definition to Residual or Generalized Impulse doesn’t change the results much, if at all.

Now, let’s estimate the VARL. Most information criteria suggest a maximum lag length of four in the VARL. The LM test suggests that there is no problem with serial correlation at this lag length. The AR roots tables suggests that the VARL is dynamically stable at this lag length. Here is the IRF for the VARL.

A positive shock to the monetary base generates a statistically significant negative response to level of industrial production in the third month, followed by a statistically significant positive response in months eight through 17. Changing the impulse definition to Residual doesn’t change the results much, if at all. Changing the impulse definition to Generalized Impulse ** eliminates** the statistically negative response in the third month.

The IRF shows that a 2.5% shock to the monetary base in month one leads to a peak increase in industrial production of 0.43% in month 13.

Now, let’s talk about the pros and cons of estimating a VECM. The advantage of a VECM is that it can generate statistically efficient estimates without losing long-run relationships among the variables. Thus, if cointegration exists, and the true cointegrating relationship is known and can be given a theoretical interpretation, it’s generally acknowledged that a VECM should be estimated in the manner suggested by Johansen (1995).

On the other hand, if the true integrating relationship is unknown, imposing cointegration may not be appropriate. Imposing incorrect cointegrating relationships can lead to biased estimates and hence bias the IRFs derived from the VARL. In cases where there is no theory which can suggest the true cointegrating relationship or how it should be interpreted, it is probably better not to estimate a VECM. Moreover, Sims et al. (1990) show that when a cointegrating relationship exists, the systems dynamics can be estimated consistently with a VARL, and James Hamilton (1994) appears to agree (pp. 651-653). As a final note on the pros and cons of VARDS, VARLS and VECMs when modeling and forecasting with cointegrated variables, I highly recommend this “hands-on” paper by Tim Duy and Mark Thoma.

Nevertheless, I think it will still be interesting to estimate a VECM for industrial production. The following IRF is generated assuming a linear trend in the data and an intercept but no trend in the cointegrating vector. (Unfortunately VECM standard errors are not available in EViews.).

A positive shock to the monetary base generates a negative response to level of industrial production in the second and third month, followed by a positive response thereafter. A 2.4% shock to the monetary base in month one leads to a peak increase in industrial production of 0.51% in month 15. In this model the response of industrial production is remarkably persistent with the level of industrial production still up by 0.43% nearly four years later. Changing the impulse definition (to either Residual or Generalized Impulse) slightly ** increases **the response. In any case, as I’ve already implied, I don’t put much stock in this VECM.

Here are a couple of preliminary observations. The price level responses seem less persistent than what multivariable VARs estimated in “normal” times with short term interest rates as the instrument of monetary policy show. On the other hand, the output response seems somewhat more persistent.

In Part 3 I shall finally put output, the price level and the monetary base together in our baseline trivariate VAR model.