## Meta-Residuals in Dynamic Factor Models

"...coherent assumptions on what is still invisible may increase our understanding of the visible."

Jean Baptiste Perrin in his Nobel Prize lecture

Welcome!

In this paper, meta-residuals are discovered that can be used for tracing the impulse transmission in a dynamic factor model (DFM). They lead to considerable improvements of forecasts of some series and, more important, to a complete new formula for impulse responses. This formula uses future residuals, which were until now supposed to be zero. The endogenous dynamic and inner structure of a macroeconomic system is revealed. Furthermore, the impulse responses developed here solve the price puzzle - the counterintuitive increase in inflation after a hike in the federal funds rate.

Below is a slideshow with the evolution of (the criteria for the) number of static factors in a DFM. The month in the title shows the last data point used for the corresponding computation.

Jean Baptiste Perrin in his Nobel Prize lecture

Welcome!

In this paper, meta-residuals are discovered that can be used for tracing the impulse transmission in a dynamic factor model (DFM). They lead to considerable improvements of forecasts of some series and, more important, to a complete new formula for impulse responses. This formula uses future residuals, which were until now supposed to be zero. The endogenous dynamic and inner structure of a macroeconomic system is revealed. Furthermore, the impulse responses developed here solve the price puzzle - the counterintuitive increase in inflation after a hike in the federal funds rate.

Below is a slideshow with the evolution of (the criteria for the) number of static factors in a DFM. The month in the title shows the last data point used for the corresponding computation.

Missing detail, some code, personal note

Preparing the presentation of meta-residuals for the R-meetup in Berlin in January 2018 showed me a missing detail that needs completion.

The definition of the meta-residuals in the individual part is missing. They are obtained in the regular way while extracting factors.

First, here is the code for the common component meta-residuals.

For the first forecasted period t+1 it is:

as pseudo code: last period in future fhat extracted – first period of forecasted fhat

R code: fe_resid_meta[current_ind,] <- last(na.omit(fhat_f_extr)) - first_forecasted_fhat

starting in t+2 :

as pseudo code: last period in future fhat extracted – fhat extracted, then forecasted one period

R code: fe_resid_meta[current_ind,] <- last(fhat_f_extr) - fhat_f_extr_f

The meta-residuals of the idiosyncratic part are given by the OLS residuals in the restricted factor extraction, the part of the code can be found here: snippet__id_meta-residuals.R

I wish you lots of fun and success with meta-residuals! The pseudo out-of-sample comparison of meta-residuals for slow moving variables might be spectacular!

Here are some thoughts for your way:

The code for the regular DFM forecast is here: 23_forecast_DFM_S&W.R , consider that beta_Paul (see code and article appendix) might computationally simplify things a bit.

Page 111 in the book “New Introduction to Multiple Time Series Analysis “(H.L., 2005; check online erratum of the book for this page) showed me how to program the inversion of the multivariate AR filter (I needed 2 weeks to program the whole function for asymptotic distributions of impulse responses for my master thesis; you don't need it in case you know how to inverse the filter, which I guess can be found also in other places like Hamilton).

The idea how to extract factors with constant coefficients came from the GAUSS function pc_factor(x,k) in the code of Stock and Watson 2005 , "Implications of dynamic factor models for VAR analysis", See line 209 and 210 in file Fcst.prc: “Calculation note:

Calculations are speeded by using the smaller of x*x' or x'x* “

and see also this. My comments (green, in German) state that Lamda is extracted and F computed vs. F extracted and lam computed.

I struggled a lot (in vain) to fix the sign of the meta-residuals. You might think about flipping the sign of the loadings - they are now "obtained" (but hold constant) trough PCA.

-1 * -1 = 1 remained the toughest thing to solve!

The publication is registered with the United States Copyright Office, registration number TXu002067929. There are some German similar counterparts.

A last thing: the Buddhist meditation concept of one-pointedness of the mind (Ekaggata, https://en.wikipedia.org/wiki/Ekaggata ) might be interesting to be seen with Figure 2 of the article, and probably interesting for psychologists working with dynamic factors models.

Good luck!