**A Mark Sadowski post**

In Part 1 we established that there is bidirectional Granger causality between the monetary base and CPI, PCEPI and industrial production. In Part 2, using bivariate Vector Auto-Regression (VAR) models to generate Impulse Response Functions (IRFs), we showed that a shock to the monetary base leads to an increase in the price and output level. In Part 3 we are going to estimate a trivariate VAR model to verify that the relationships we observed in Part 2 are robust to the inclusion of all three types of variables.

The first thing we have to do is to put all three types of variables into VARs in levels (VARLs) and test for cointegration.

Most information criteria suggest a maximum lag length of four for both the VARL that includes CPI as the measure of the price level, and the VARL that includes PCEPI as the measure of the price level. An LM test suggests that there is no problem with serial correlation in either VARL at this lag length. The AR roots tables suggest that the VARLs are dynamically stable, and the Johansen’s Trace Test and Maximum Eigenvalue Test both indicate that there exists one cointegrating equation in each VARL at this lag length. This probably shouldn’t be too surprising since we already have evidence that the monetary base and industrial production are cointegrated.

So, to reiterate what I argued in Part 2, although the order of integration for the log levels of all four of our series (SBASENS, CPI, PCEPI and INDPRO) is one, the fact that there is a cointegrating relationship between the three types of variables means that only estimating a VAR in first differences (a VARD) would discard the information contained in the levels and lead to model misspecification. Furthermore, since there is cointegration, we should probably estimate a Vector Error Correction Model (a VECM), since it can generate statistically efficient estimates without losing long-run relationships among the variables, as a VARL might. However, 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.

However, in the interests of thoroughness, I am going to estimate a VARD, a VARL and a VECM. And since we have two measures of the price level that means we will be estimating six models. But after we have done that, and have noted the results, the choice, at least for the moment, will come down to one between the two VARLs.

First, let’s estimate the VARDs. Most information criteria suggest a maximum lag length of three in both VARDs. The LM tests suggest that there is no problem with serial correlation at this lag length. And the AR roots tables suggest that both VARDs are dynamically stable at this lag length.

As I did in Part 2, I am going to use a recursive identification strategy (Cholesky decomposition), which is the dominant practice in the empirical literature on the transmission of monetary policy shocks. Such a strategy means that the order of the variables affects the results. I will follow the traditional practice of ordering output level first, the price level second, and the monetary policy instrument third in each vector. Changing the ordering of the variables would change the results (in this case there are six permutations), but since order doesn’t matter in the other impulse definitions I am mentioning (Residual and Generalized Impulses), we should be able to detect if this would have a significant effect. As before, the response standard errors I will show are analytic, since Monte Carlo standard errors change each time an IRF is generated.

Here are the responses to a shock to the monetary base in the trivariate VARD using CPI as the measure of the price level. 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 trivariate VARD using PCEPI as the measure of the price level.

In each case 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 as well as 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 look at the IRFs for the VARLs. As I already noted, most information criteria suggest a maximum lag length of four in both VARLs, the LM tests suggest that there is no problem with serial correlation at this lag length in either of these VARLs and 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 trivariate VARL using CPI as the measure of the price level. In this case I’ll extend the time period to 48 months.

And here are responses to a shock to the monetary base in the trivariate VARL using PCEPI as the measure of the price level.

In both cases 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. With PCEPI as the measure of the price level this is followed by a statistically significant positive response in the tenth month. In both cases a positive shock to the monetary base generates a statistically significant positive response to the price level in months three and four.

Changing the impulse definition to Residual doesn’t change the results much, if at all. However changing the impulse definition to Generalized Impulse results in a statistically significant positive response in only month four in the VARL with CPI as the measure of the price level, and results in the loss of the statistically significant positive response to industrial production in the VARL with PCEPI as the measure of the price level. Note that we’ve already established that the monetary base has a statistically significant positive effect on industrial production with the VARDs. As I said in Part 2, there is loss in statistical efficiency when estimating a VARL with unit roots.

The IRFs show that a positive 2.3% shock to the monetary base in month one leads to a peak increase in industrial production of 0.26% in month 11 in the VARL that uses CPI as the measure of the price level, and that a positive 2.4% shock to the monetary base in month one leads to a peak increase in industrial production of 0.27% in month 11 in the VARL that uses PCEPI as the measure of the price level. The IRFs also show that a 2.3% and 2.4% shock to the monetary base in month one leads to a peak increase in the CPI level of 0.10% and in the PCEPI level of 0.077% in month four respectively.

Now let’s estimate the VECMs. The following IRFs are generated assuming a linear trend in the data and an intercept but no trend in the cointegrating vector. Here are responses to a shock to the monetary base in the trivariate VECM using CPI as the measure of the price level. (As I noted in Part 2, unfortunately VECM standard errors are not available in EViews.).

Here are responses to a shock to the monetary base in the trivariate VECM using PCEPI as the measure of the price level.

In both cases a positive shock to the monetary base generates a negative response to level of industrial production in the second through fourth month, followed by a positive response thereafter. In both cases a 2.3% positive shock to the monetary base in month one leads to a peak increase in industrial production of 0.36% in month 14. In the first VECM, CPI has a peak increase of 0.092% in month four and then falls to a maximum decrease of 0.042% in month 16. In the second VECM, PCEPI has a peak increase of 0.069% in month four and then falls to a maximum decrease of 0.014% in month 17. In both cases changing the impulse definition to either Residual or Generalized Impulse doesn’t change the results much, if at all. But as I’ve already said, there is good reason to be skeptical of these VECMs.

Now the time has come to pick between the two VARLs.

First, the VARL that uses PCEPI as the measure of the price level has a statistically significant positive increase in the level of industrial production unlike the VARL that uses CPI as the measure of the price level. This makes the PCEPI VARL more consistent with the evidence from the more statistically efficient VARDs.

Second, since PCEPI is the deflator of the Personal Consumption Expenditure component of GDP it is arguably a better measure of the price level than the much more ad hoc CPI.

Third, and perhaps most importantly, the Federal Open Market Committee (FOMC) has used the PCEPI to frame its inflation forecasts since February 2000 (Page 4, Footnote 1) and since January 2012 the FOMC has explicitly targeted the inflation rate of PCEPI.

Thus PCEPI very likely represents a better measure of the preferences of the monetary authority as it sets the policy variable (i.e. the monetary base).

So the PCEPI VARL is the** Simple Baseline VAR for Studying the US Monetary Base and the Channels of Monetary Transmission in the Age of ZIRP**.

In upcoming posts I shall add other variables to the VAR that the monetary base Granger causes in the age of ZIRP, such as domestic stock indices, nominal Treasury yields, inflation expectations, the value of the dollar, and commercial bank credit and deposits.

“Here are responses to a shock to the monetary base in the trivariate VECM using CPI as the measure of the price level.”

But the curves below this sentence include one for PCEPI instead of CPI. Is the above sentence in error?

Obviously and trivially yes. It´s corrected