NAIRU

Definition

The Non-Accelerating Inflation Rate of Unemployment (NAIRU) is the rate of unemployment consistent with stable inflation in the long run — neither persistently rising nor falling. In structural vector autoregression (VAR) terms (Zhao, undated; Blanchard-Quah 1989), the NAIRU is the component of actual unemployment whose associated structural disturbance has zero cumulated effect on inflation: it can permanently shift the unemployment rate without permanently affecting the price level, consistent with a vertical long-run Phillips Curve.

Key Ideas

The NAIRU embodies the long-run vertical Phillips Curve: monetary policy cannot permanently trade higher unemployment for lower inflation; any such attempt merely raises the price level.
It is not directly observable and must be inferred from the joint dynamics of unemployment and inflation.
Two major identification approaches: (1) reduced-form filtering via the Gordon "triangle" model or Kalman-filter methods (Staiger-Stock-Watson 1997); (2) structural VAR with a Blanchard-Quah long-run restriction (Zhao undated).
The NAIRU is time-varying: Laubach (2001) and Staiger-Stock-Watson (1997) document substantial variation across decades; the U.S. NAIRU fell sharply in the late 1990s.

How It Works

Structural VAR Identification (Blanchard-Quah 1989; Zhao undated)

The system is a bivariate VAR in $(\Delta\pi_t, u_t)$ : inflation is first-differenced because Augmented Dickey-Fuller (ADF) tests support I(1); unemployment is stationary. Lag length 4 (Likelihood Ratio (LR) test from max 5). Exogenous controls — constant, linear and quadratic trend, $\Delta$ import prices, $\Delta$ unit labour costs — partial out observable supply shocks before structural identification.

The reduced-form residuals are related to two orthogonal unit-variance structural disturbances:

$\begin{pmatrix} e_{\pi,t} \\ e_{u,t} \end{pmatrix} = C(0) \begin{pmatrix} \varepsilon_{1,t} \\ \varepsilon_{2,t} \end{pmatrix}$

The Vector Moving Average (VMA) $X_t = C(L)\varepsilon_t$ is recovered from the estimated VAR. The Blanchard-Quah long-run restriction imposes:

$\sum_{j=0}^{\infty} C_j^{11} = 0$

meaning the NAIRU disturbance $\varepsilon_1$ has zero cumulated effect on inflation (the long-run level of prices). Combined with the three moment conditions from $\Sigma = C(0)C(0)'$ (two unit variances and one zero covariance), this exactly identifies $C(0)$ .

Two structural components:

NAIRU disturbance $\varepsilon_1$ : aggregate-supply-like; permanently shifts unemployment; no long-run inflation effect.
Gap disturbance $\varepsilon_2$ : aggregate-demand-like; closes the unemployment gap; has a permanent effect on inflation.

NAIRU series: the counterfactual path of $u_t$ when $\varepsilon_2 \equiv 0$ throughout.
Core inflation: the counterfactual path of $\pi_t$ when $\varepsilon_1 \equiv 0$ — inflation driven entirely by demand; closely parallels the Quah-Vahey (1995) core inflation decomposition.

Empirical Results (U.S. Annual 1960–2000; Zhao undated)

NAIRU stable at 6–8% from 1960 through 1993; falls sharply to ~3% by 2000.
Long-run Phillips Curve: 6.8% pre-1997 → 4% post-1997.
NAIRU disturbance explains $<10\%$ of inflation forecast-error variance; gap disturbance dominates.
Impulse response: NAIRU shock initially lowers inflation (−0.4%), peaks at year 3, reverts to baseline — oscillatory but long-run neutral. Gap shock raises inflation permanently (+0.8% peak at year 1); unemployment takes 4+ years to recover.
95% bootstrap bands ~1.6–2% wide (1,000 replications) — tighter than univariate Phillips Curve (2.6%) or plain bivariate (2.3%).

Gordon's Triangle Model TV-NAIRU (Gordon 2005)

Gordon estimates a time-varying NAIRU $U^N_t$ within the triangle model inflation equation:

$p_t = a(L)p_{t-1} + b(L)(U_t - U^N_t) + c(L)z_t + e_{pt}, \qquad \sum a_i = 1$

The NAIRU evolves as a random walk: $U^N_t = U^N_{t-1} + \eta_t, \quad E[\eta_t] = 0, \quad \text{Var}(\eta_t) = \tau^2$

with $\tau^2$ estimated by maximum likelihood via the Kalman filter. Supply shocks $z_t$ (five variables: import prices, food-energy, medical care, Hodrick-Prescott (HP)-filtered productivity trend, Nixon controls) are crucial: omitting supply shocks forces $U^N_t$ above 8% in the 1970s to absorb unmodelled inflation variance.

Estimated U.S. NAIRU path: Stable at 5.6–6.3% during 1962–88; falls to 4.5% minimum in 1998; rises to 4.85% by 2004Q4. The late-1990s decline — consistent with the Staiger-Stock-Watson (1997, 2001) estimates — reflected favourable supply shocks (technology acceleration, falling import prices) rather than purely a structural shift in labour markets.

Why It Matters

The NAIRU is a key input to monetary policy: central banks target interest rates partly to keep unemployment near the NAIRU to avoid accelerating inflation. The structural VAR approach provides a model-free definition tied directly to the long-run Phillips Curve, without requiring a structural model of price-setting. Its time variation in the 1990s had significant policy implications — a falling NAIRU allowed the Fed to accommodate low unemployment without triggering inflation concerns.

Open Questions

Real-time identification: NAIRU estimates are subject to substantial real-time revision; the late-1990s decline was only clearly visible ex post.
Stability of the long-run restriction: the Blanchard-Quah restriction (demand shocks permanently affect inflation, supply shocks do not) may be violated if supply shocks affect inflation through expectation-anchoring failures.
Post-2000 behaviour: the U.S. NAIRU appeared to partially revert after 2000, questioning the permanence of the 1990s decline.
Natural rate vs. NAIRU: the structural NAIRU here is a statistical object; its connection to the equilibrium unemployment rate from search-and-matching theory requires a richer model.