VFCI discrepancy with macro_dynamics

[matdehaven]

Copied from email chain:

Hi Fernando,

I figured out why we got two different VFCI-s when using the ‘vfciBusinessCycles’ and ‘macro_dynamics’ codebases.

There are two steps in the code for calculating h-period forward growth rates:

1) Calculate h period growth rates 2) Forward those growth rates by h periods

The ‘macro_dynamics’ code only ever forwards growth rates by 1 period. This means that the when looking at a variable like ‘fgr4.output’, it is actually the growth rate from t-3 to t+1. This is fine when looking at the baseline vfci, because then ‘fgr1.output’ is the growth rate from t to t+1.

I’ve put below a few figures that show when I change the code to instead forward 4 periods, you get the same figure as my code from ‘vfciBusinessCycles’.

I don’t think The Market Price of Risk paper makes use of any forward looking VFCI indexes, so it doesn’t change anything there. It does mean we still have to figure out why the VFCI_t+10 doesn’t look like a financial index for our paper (or just switch to using VFCI_t+1). I’ll see what I can figure out for that.

Best, Matt

‘macro_dynamics’ — Forward: 1, Growth rate 4

‘macro_dynamics’ — Forward: 4, Growth rate 4 (fixed)

‘vfciBusinessCycles’ — Forward: 4, Growth rate 4

‘vfciBusinessCycles’ — Forward: 10, Growth rate 10 — the one we have been using but doesn’t look like a financial index

[matdehaven]

Response from Fernando:

Hi Matt,

Thanks for figuring this out!

What the ‘macro_dynamics’ code is doing sounds wrong. I will double check to see if there was any reason to only forward one period even when we had growth rates over more periods, but I am pretty sure it is just an error.

And yes, as you say, we still have to figure out why the VFCI_t+10 doesn’t look like a financial index for our paper, or just switch to using VFCI_t+1.

Let me add one more option: maybe we can modify the VFCI_t+10 so that it looks like a financial index. We have been trying adding lags of consumption/output growth to the VFCI regression. My recollection is that adding lags didn't change the VFCI much. But maybe there are still a couple of things left to try. For example, if the dependent variable is output growth over 10 quarters, maybe we should add as independent variables lagged output growth over 10 quarters (rather than lags of one-quarter growth rates). There was also a bit of looking into adding lags in the mean vs vol regressions that I think we had left to do, like adding lags in one regression but not the other.

Best, Fernando

[matdehaven]

Need to check the alignment of the forecast error dates for the internal VFCI calculations.

I think that the internal VFCI calculation may need to lead the forecast errors before running the heteroskedastic regressions.

[matdehaven]

This was the case and when fixed the internal and external VFCIs align much better than before.

Need to go through and make certain the exact alignment is correct. It is still possible it needs to be lead/lagged by 1 period.

[matdehaven]

fit_het_reg_from_var is now correctly aligning the estimated VFCIs.

• [x] Need to check that the code still works when including additional lags and variables in the hetreg estimation.

[matdehaven]

We now have remarkable similarity between the external and internal VFCIs.

Comparison from Before (incorrect lead relationship)

Comparison after correction