NAIRU V Estimation

by Marco Fioramanti and Robert Waldmann

This is the second to last post on the European Commissions DG -Ec-Fin estimates of cyclical unemployment for the purposes of calculating output gaps. This estimate is called unemployment minus the NAWRU (non accelerating wage inflation rate of unemployment). We will call unemployment - NAWRU "cyclical unemployment" even though it is agreed that the NAWRU is partly cyclical.

For several countries (including Italy) it is calculated with a time series model based on an accelerationist Phillips curve in which the change in wage inflation depends on cyclical unemployment. The model is fairly complicated with 11 paramters (estimated for Italy with 50 annual data points and 3 years of atheoretic forecasts). It is briefly described here based on this working paper.

The model attempts to fit 2 time series, unemployment and the change in the rate of increase of wages, and includes 4 disturbance terms. To be very brief, the expected acceleration of wage inflation is a linear function of cyclical unemployment and two lags of cyclical unemployment (the equation includes one of the disturbance terms). Cyclical unemployment is assumed to be an AR(2) (with the second of the disttubance terms) The NAWRU is assumed to be an I(2) second order random walk -- the drift of the NAWRU is itself assumed to be a random walk (so the disturbance to the drift and the disturbance to the level are the 3rd and 4th disturbance terms). The assumption that the drift is a random walk is crazy -- it always implies long term forecasts of unemployment less than zero or over 100%. The EC staff agree that this model can't be taken literally. They ignore it when making long term forecasts. However, the resulting estimates of cyclical unemployment are used to calculated output gaps.

Robert wrote "As one might guess, identification is a bit problematic. However, it is possible to convince a computer to estimate all the parameters." As we (mostly Marco) have attempted to do this for slightly modified models, we have discovered that it is very difficult to convince a computer to estimate all the parameters (DG -Ec Fin uses their own software). This (in addition to the usual procrastination) has caused a long delay between NAWRU IV and NAWRU V (this post).

The problem (at least for STATA addicts) is that STATA ends up at a corner attempting to set the variance of the disturbance to the drift of the NAWRU to zero. This means that estimated of the model as officially described using STATA's standard sspace command provides no empirical support for the theoretically unjustified assumption which has impossible long term implications. STATA (v. 11 to 14) refuses to report estimates after getting stuck in a corner (this is a feature not a bug).

Based on Robert's efforts to code a pseudo-annealing Kalman filter maximum likelihood estimator (which are not publishable even in a blog) we think the key issue is the imposition of an arbitrary maximum on the variance of the disturbance to cyclical unemployment. This will be the topic of NAWRU VI -- the final episode if and when the conclusion is based on the use of standard software.

But in this post, we want to discuss estimation of the model with the variance of that disturbance term set to zero -- that is -- estimation of a model in which the NAWRU is assumed to be a random walk with drift.

This model has less appalling implications for the long term. The NAWRU is not restricted to the range from 0% to 100% but it would be easy to impose this restriction (the standard approach would be to assume that the NAWRU is a martingale and the variance becomes small when the NAWRU is near the limits -- it is possible to assume that this state dependent variance is constant over the range experienced during the sample period so the model as written is valid). The fluctuations in the NAWRU remain exogenous and unexplained, but there is at least a literature on why the natural rate of unemployment might fluctuate.

This model has implications strikingly different from those of the EC DG- EcF Fin model. The The fitted NAWRU no longer tracks the business cycle. The variance of cyclical unemployment is much greater. The resulting fiscal dictates would have been very different if the EC had used our simpler model .

here un = NAWRU

ug = unemployment - NAWRU = "Cyclical Unemployment"

ddw = the acceleration of wage inflation

The estimate command is

(the constraints are identities such as unemployment = (unemployment-NAWRU) + NAWRU and the assumption that the NAWRU is not mean reverting)

matrix rjw=(1, 1, 1, 1, 1, 1.407003, -0.49923037, 0.1555339, 1, 1, -0.031958257, 0.048677339, -0.019232409, 0.032105009, 0.28977462, 0.000507234)

sspace (un L.un L.mu, state noconstant) ///

(mu L.mu, state noerror noconstant) ///

(ug1 L.ug, state noerror noconstant) ///

(ug2 L.ug1, state noerror noconstant) ///

(ug L.ug L.ug1, state) ///

(u un ug, noerror noconstant) ///

(ddw ug ug1 ug2, noconstant) if year >=1965 & year<=2017, ///

iterate(100) from(rjw) constraints(1 2 3 4 5 6 7) ///

covstate(di) covobserved(di) difficult

Here are the estimates

For what it's worth, the likelihood is larger than that reported by the EC. We don't think too much attention should be paid to those two numbers -- the reported likelihood depends on technical assumptions used to initialized the Kalman filter when there are nonstationary variables. The null that the disturbance to the NAWRU has mean zero (so the NAWRU is a trend) is not rejected. There is little evidence that ug is related to the acceleration of wage inflation.