Sticky information and inflation persistence: evidence from the U.S. data

Mankiw and Reis (2002) presented the SIPC as an alternative (theory of inflation) to the New Keynesian Phillips Curve (NKPC), which was criticized because of its lack of intrinsic persistence. Such criticisms pointed out that (i) the actual inflation responds gradually to monetary policy shocks, whereas the NKPC implies an immediate adjustment; (ii) output losses typically accompany a reduction in inflation, but this is not true with the NKPC; (iii) the NKPC implies that announced disinflation causes a boom, but in the actual economy the opposite is true. In general, if we assume that firms maximize profits and have rational expectations, then they would react to any exogenous shock by adjusting their prices as soon as they acquire information on the incoming shock and in each period, they would acquire all of the information available to identify new shocks. Hence, the effect of a shock on prices lasts for few periods and as a result, the only source of persistence in inflation dynamics is the one of exogenous shocks (e.g., cost push shocks, monetary policy shocks, demand shocks).

Note that for \(\lambda =1\), the SIPC encompasses the rational expectations model with monopolistic competition and flexible prices, which has repeatedly been shown in the literature to generate little endogenous persistence in inflation dynamics.

The whole estimation procedure is based on Molinari (2007).

Following MR, the SIPC is simulated by assuming that the growth rate of nominal output, i.e., \(\Delta m_{t}=\Delta y_{t}+\pi _{t}\), is exogenous and follows an autoregressive process of order 1. As a result, inflation from the SIPC can be easily simulated using its moving average representation as function of the exogenous errors in the autoregressive process. The calibration of the autoregressive coefficient and the variance of the exogenous shocks are taken from Reis (2006).

The average duration of information is \(D=\left( 1-(1-\lambda )\right) ^{-1}\). Thus, in the quarterly model used by MR, \(\lambda =0.25\) implies \(D=4\), i.e., four quarters.

Equation (3) follows after multiplying (2) by \(\left( \delta \varepsilon _{t-i}\right) ^{\prime }\) and taking the expectation conditional on the information at time \(t\). This calculation uses the fact that the errors are uncorrelated, i.e., \(E\left[ \varepsilon _{t}\varepsilon _{t-i}\right] =0\) for \(i\ge 1\). In Appendix B, the orthogonality conditions (3) are referred to as \(g_{1,t}\).

Given that \(Z_{t}\) is ergodic (see Proposition 1), then \(E\left[ \delta Z_{t}\cdot (\delta \varepsilon _{t-i})^{\prime }\right] =E\left[ \delta Z_{t-i}\cdot (\delta \varepsilon _{t})^{\prime }\right] \).

If \(\left\{ \widehat{\varepsilon }_{t},\widehat{A}_{i},\Sigma _{T}\right\} \) are consistent estimates of \(\left\{ \varepsilon _{t},A_{i},\Sigma \right\} \), it can be shown that the sample analog:converges almost surely to the population moment (3).

This follows immediately from the fact that the quadratic terms in (4) are all positive definite. Note that there is only one additional task required to compute (4) beyond the GMM estimation of (3), i.e., one needs to derive \(\frac{\partial g_{1,t}}{\partial \beta ^{\prime }}\) and to evaluate it at \(\{\beta _{T}^\mathrm{VAR},\theta _{T}^{2s}\}\). All of the other terms that appear in (4) are already computed in the two-stage estimation. \(\frac{\partial g_{1,t}}{\partial \lambda }\) and \(\Sigma _{g_{1}}^{-1}\) are delivered by the GMM algorithm that computes \(\lambda _{T}^{2s}\), and \(\left( E\frac{\partial g_{2,t}^{\prime }}{\partial \beta }\Sigma _{g_{2}}^{-1} E\frac{\partial g_{2,t}}{\partial \beta ^{\prime }}\right) \), the VCV matrix of VAR coefficients, is delivered by the algorithm that estimates the VAR(p) model.

See Sect. 4.2.

All data used in this paper come from the FRED-II database issued by the Federal Reserve Bank of St. Louis.

The output gap is the real GDP detrended with the HP filter.

The GMM model has a total of seven orthogonality conditions, including the contemporaneous moment (\(i=0\)). Results using a higher number of o.c. are quite similar to those reported and are available upon request.

In his baseline estimation, Coibion (2010) estimated \(\alpha \) and \(\lambda \) jointly, finding that they both are not significant. That result, however, is obtained using either subjective or out-of-sample VAR forecasts to generate the expectation terms present in the SIPC. When he uses in-sample forecasts (from a single VAR estimated over the entire period as the one estimated in this paper) instead, he finds estimates of \(\lambda \) significant and in the range [0.37, 0.42].

Kiley (2006) and Korenok (2008) express the SIPC in terms of marginal cost; therefore, \(\alpha \) does not appear in their estimated specifications.

See Reis (2006) and Woodford (2003), ch. 2.

For a more precise comparison, I estimated \(\lambda \) using the same information as used in previous papers, which turns out to be the information contained in the covariances between inflation and the first \(L\) lags of \(\delta Z_{t-i}\). In this case, using Eq. (2) and lagged \(\delta Z_{t-i}\) as instruments, I derive and estimate the following orthogonality conditions:where the infinite summation on the RHS of (6) is simulated at each step of the GMM algorithm, replacing \(\infty \) with \(J_{max}=120\). In the estimation of (6), \(\lambda _{T}^{2s}\) ranges between \(\left[ 0.30,0.41\right] \), a result in line with that obtained using lagged \(\varepsilon _{t}\) as instruments and very close to the estimates of Khan and Zhu (2006), Kiley (2006) and Korenok (2008).

The min RMSE VAR uses five extra variables to predict \(\pi _{t}\) and \(\Delta y_{t}\) with respect to the baseline VAR. As a result, the residuals \(\widehat{\varepsilon }_{t}\) from the min RMSE VAR are smaller than the ones of the baseline VAR.

Following similar GMM estimates in the literature, e.g., Galí and Gertler (1999), I use 19 instruments: a constant, 4 lags of inflation, 4 lags of output gap, and 2 lags of unemployment rate, interest rate, marginal cost, money growth, and term spread.

In this case, the GMM point estimates of \(\lambda _{T}^{2s}\) coincides with the estimate of the non-linear IV estimator, but with smaller variance.

I also estimated the model with \(\alpha = 0.15\) to control for a possible non-monotonic effect of \(\alpha \) on \(\lambda _{T}^{2s}\) within the \(\left[ 0.1,0.2\right] \) interval. These results are in line with those for \(\alpha =0.1\) presented in Table 8, and are available upon request.

The AR(2) model is extensively employed in the literature as a univariate benchmark model to predict inflation, e.g., in Khan and Zhu (2006) or Batchelor (1982), because it is the simplest univariate model with sufficiently rich dynamics to represent the actual inflation series.

See Angeloni et al. (2006).

This follows immediately from (3). If inflation persistence increases because of adaptive agents but structural \(\lambda \) remains constant, then in the misspecified o.c. (3) the RHS increases, and the estimated \(\lambda ^{2s}_{T}\) must decrease for the equality to hold.

Dupor et al. (2005) developed a similar model. They proposed a model of “dual stickiness” in which producers change prices sporadically and absorb the relevant information for price setting in random periods, as in the SIPC. Basically, Dupor–Kitamura–Tsuruga nested together Calvo’s sticky price framework with the sticky information of MR. As a result, inflation in period \(t\) is a function of all past periods’ expectations of future variables indexed from \(t+1\) onwards. The econometric strategy presented in Sect. 2.2 is computationally burdensome when applied to such a model, and therefore, in this paper I do not estimate Dupor–Kitamura–Tsuruga model but rather derive and estimate an alternative model with sticky information and adaptive agents.

The reduced form VAR(p) model used in Sect. 3.1 encompasses both the SIPC model and the hybrid versions derived in this section.

Andrews’s test looks for structural breaks with unknown timing by cutting the tails of the sample and computing recursively for each observation in the remaining middle part which is the most likely period where a break might have occurred.

The oil shock occurred in the middle of the sample and lasted sufficiently long to be included in each of the two sub-sample statistics calculated by Andrews’s test. Thus, if the effect of the oil shock were sufficiently large, any change in the inflation dynamics of one sub-sample, due to variations in the degree of sticky information, would not be detected by the test.

The VAR(p) model (5) is assumed to have errors \(\varepsilon _{t}\sim i.i.d. N\left( 0,\Sigma \right) \) and is estimated least square equation by equation.