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

Abstract

This paper analyzes the relationship between sticky information and inflation persistence by implementing a novel approach to estimate the Sticky Information Phillips Curve (SIPC). The degree of sticky information is estimated using a GMM estimator that matches the covariance between inflation and the shocks that affect firms’ pricing decisions. Although the SIPC contains an infinite number of terms, the theoretical covariances derived from the model have finite dimensions, thus allowing the estimation of the structural parameters without any truncation of the original model. This work shows that sticky information is significantly different if the model is estimated by matching inflation persistence or inflation variance. Previous empirical literature found that the SIPC model does not provide an accurate representation of the US postwar inflation. This paper qualifies such a finding by demonstrating that the SIPC is able to match the inflation persistence only at the cost of mismatching the inflation variance.

Appendices

A Proof of Proposition 1

Denoting the $j$-periods-ahead forecast error as $\displaystyle \varepsilon _{t\mid t-j}^{F}=Z_{t}-E\left[ Z_{t}\mid \Omega _{t-j}\right]$, the SIPC can be written as:

$$\begin{aligned} \pi _{t}=\frac{\alpha \lambda }{1-\lambda }y_{t}+\frac{\lambda }{1-(1-\lambda )} \delta Z_{t} -\lambda \sum _{j=0}^{\infty }\left( 1-\lambda \right) ^{j}\delta \varepsilon _{t\mid t-j-1}^{F} \end{aligned}$$

(14)

where $Z_{t}$ is any covariance stationary vector of variables that includes inflation and output gap, and $\delta$ is a $\left( 1\times n\right)$ row vector of zeros and constants that picks $\left( \pi _{t}+\alpha \Delta y_{t}\right)$ within $Z_{t}$. From the definition of $\delta Z_{t}$, we immediately see that Eq. (14) implies

$$\begin{aligned} \frac{\alpha \lambda }{1-\lambda }y_{t}+\alpha \Delta y_{t}=\lambda \sum _{j=0}^{\infty }\left( 1-\lambda \right) ^{j}\delta \varepsilon _{t\mid t-j-1}^{F} \end{aligned}$$

(15)

Using the Wold decomposition of $Z_{t}$, each forecast error can be written as:

$$\begin{aligned} \varepsilon _{t\mid t-j-1}^{F}=\sum _{i=0}^{j}A_{i}\varepsilon _{t-i} \end{aligned}$$

(16)

Next, using (16) to substitute for $\varepsilon _{t\mid t-j-1}^{F}$ in (15), we obtain:

$$\begin{aligned} \frac{\alpha \lambda }{1-\lambda }y_{t}+\alpha \Delta y_{t}=\lambda \sum _{j=0}^{\infty }\left( 1-\lambda \right) ^{j}\delta \sum _{i=0}^{j}A_{i}\varepsilon _{t-i} \end{aligned}$$

(17)

Moreover, notice that:

$$\begin{aligned}&\sum _{j=0}^{\infty }\left( 1-\lambda \right) ^{j}\delta \sum _{i=0}^{j}A_{i}\varepsilon _{t-i}\nonumber \\&\quad =\left( \delta \varepsilon _{t}+\left( 1-\lambda \right) \delta \varepsilon _{t} + \left( 1-\lambda \right) ^{2}\delta \varepsilon _{t}+\ldots \right) \nonumber \\&\qquad \quad + \left( \left( 1-\lambda \right) \delta A_{1}\varepsilon _{t-1}+\left( 1-\lambda \right) ^{2} \delta A_{1}\varepsilon _{t-1}+\ldots \right) + \ldots \nonumber \\&\quad =\frac{1}{1-\left( 1-\lambda \right) }\sum _{i=0}^{\infty }\left( 1-\lambda \right) ^{i}\delta A_{i}\varepsilon _{t-i} \end{aligned}$$

(18)

Hence, by plugging (18) into (17), we obtain

$$\begin{aligned} \frac{\alpha \lambda }{1-\lambda }y_{t}+\alpha \Delta y_{t}=\sum _{i=0}^{\infty }\left( 1-\lambda \right) ^{i}\delta A_{i}\varepsilon _{t-i} \end{aligned}$$

which proves the proposition.

B Proof of Proposition 2

Let $g_{1,t}$ be the set of orthogonality conditions (3), i.e.,

$$\begin{aligned} E[g_{1,t}]\equiv E\left[ \underset{m\times 1}{g_{1} \left( \lambda ,\beta _{T}^\mathrm{VAR},Y_{t} \right) }\right] = 0 \end{aligned}$$

(19)

where $m=L+1$ is the number of orthogonality conditions, $\beta \!=\!\mathrm{vec}\!\left( \left[ B_{1}^{\prime },\ldots ,B_{p}^{\prime }\right] ^{\prime }\right) , B_{j}$ are the matrices of parameters defined in (5), vec$\left( \cdot \right)$ is the column stacking operator, and $Y_{t}=\{y_{t},y_{t-1}\}$. Using the same notation, let $g_{2,t}$ be the orthogonality conditions used to estimate the VAR(p) model (5), i.e.,

$$\begin{aligned} E\left[ \underset{kn\times 1}{g_{2}\left( \beta ,X_{t}\right) }\right] =0 \end{aligned}$$

(20)

where $X_{t}$ are the $n$ endogenous variables of the VAR(p), and $k=np+1$.²⁹

The correct variance of $\lambda _{T}^{2s}$ should take into account the variance of the stochastic regressors $\beta _{T}^\mathrm{VAR}$, whereas the two-stage estimator mistreats $\beta _{T}^\mathrm{VAR}$ as fixed variables, therefore failing to account for the volatility of $X_{t}$ when computing the variance of $\lambda _{T}^{2s}$. To overcome this problem, I first construct a GMM model that jointly estimates $\{\lambda ,\,\beta \}$ pooling together the orthogonality conditions $\{g_{1,t}, g_{2,t}\}$. Next, I show that the point estimates of this estimator coincide with that of the two-stage estimator, even though the variances are different because the variance of $\lambda _{T}$ correctly incorporates the volatility of $X_{t}$ through the effect of $\beta$ on $g_{1,t}$, whereas the variance of $\lambda _{T}$ does not. Thus, I calculate the variance of $\theta _{T}$ in the pooled model and use it as the adjusted variance of $\theta _{T}^{2s}$.

The pooled GMM model

The vector of parameters $\left\{ \lambda , \beta \right\}$ can be jointly estimated by GMM using the pooled vector of o.c.

$$\begin{aligned} E\left[ \begin{array}{l} \underset{m\times 1}{g_{1,t}} \\ \underset{k n\times 1}{g_{2,t}} \end{array}\right] =0 \end{aligned}$$

(21)

where (19) and (20) are stacked column-wise, with a weighting matrix $W$ equal to:

$$\begin{aligned} W=\left( \begin{array}{ll} \Sigma _{g_{1}}^{-1} &{} O \\ O^{\prime } &{} I_{k n} \end{array}\right) \end{aligned}$$

(22)

The equivalence result

Denote the one-stage estimator of $\lambda$ from the pooled model (21) as $\lambda _{T}$, and note that $\displaystyle \frac{\partial E\left[ g_{2}\left( \beta ,X_{t}\right) \right] }{\partial \lambda }=0$ by construction and that the weighting matrix (22) is block diagonal. In this case two conditions hold. First, the objective function of the GMM estimation of (21), $\displaystyle J(\lambda _{T})=E(g_{1,t})\cdot \Sigma _{g_{1}}^{-1}\cdot E(g_{1,t}^{\prime })$, exactly coincides with the objective function of the two-stage GMM estimator $\lambda _{T}^{2s}$ that uses the inverse of its VCV matrix, $\Sigma _{g_{1}}^{-1}$, as the (optimal) weighting matrix. Second, the estimates $\beta _{T}$ coincide with the VAR model estimation, i.e., the equation by equation estimation of (20), which are the regressors used to estimate $\lambda _{T}^{2s}$ in the two-stage estimation.

Calculation of the variance of $\lambda _{T}$

Under the assumption that the covariance between $g_{1,t}$ and $g_{2,t}$ is zero, $W$ is the optimal weighting matrix for the model (21), and accordingly, the VCV matrix of $\theta _{T}$ is:

$$\begin{aligned} V\left( \begin{array}{l} \theta _{T} \\ \beta _{T} \end{array} \right) \!=\!\left[ T\cdot \underset{\equiv G}{\underbrace{E\left( \frac{\partial \left( \begin{array}{l} g_{1,t} \\ g_{2,t} \end{array} \right) }{\partial \left( \begin{array}{cc} \lambda&\beta ^{\prime } \end{array} \right) }\right) }}^{\prime }\underset{\equiv \Omega }{\underbrace{\left( E\left( \begin{array}{l} g_{1,t} \\ g_{2,t} \end{array}\right) \left( \begin{array}{ll} g_{1,t}^{\prime }&g_{2,t}^{\prime } \end{array} \right) \right) }}^{-1}E\left( \frac{\partial \left( \begin{array}{l} g_{1,t} \\ g_{2,t} \end{array} \right) }{\partial \left( \begin{array}{cc} \lambda&\beta ^{\prime } \end{array} \right) }\right) \right] ^{-1} \nonumber \\ \end{aligned}$$

(23)

where $G$ is a $\left( k n+m\right) \times \left( k n+1\right)$ matrix. The variance (23) can be written as:

$$\begin{aligned} V\left( \begin{array}{l} \lambda _{T} \\ \beta _{T} \end{array} \right) =\left[ E\left( \begin{array}{ll} \frac{\partial g_{1}^{\prime }}{\partial \lambda } &{} O^{\prime } \\ \frac{\partial g_{1}^{\prime }}{\partial \beta } &{} \frac{\partial g_{2}^{\prime }}{\partial \beta } \end{array} \right) \left( \begin{array}{ll} \Sigma _{g_{1}}^{-1} &{} O^{\prime } \\ O &{} \Sigma _{g_{2}}^{-1} \end{array} \right) E\left( \begin{array}{ll} \frac{\partial g_{1}}{\partial \lambda } &{} \frac{\partial g_{1}}{\partial \beta ^{\prime }} \\ 0 &{} \frac{\partial g_{2}}{\partial \beta ^{\prime }} \end{array} \right) \right] ^{-1}\Bigg /T \end{aligned}$$

where $\displaystyle \Sigma _{g_{h}}=E\left( g_{h,t}\cdot g_{h,t}^{\prime }\right)$ for $h=\{1,2\}$ and $O$ denotes a matrix of zeros of needed dimensions. After some algebra manipulation, this variance can be written as:

$$\begin{aligned} V\left( \begin{array}{l} \lambda _{T} \\ \beta _{T} \end{array} \right) =\left( \begin{array}{cc} E\frac{\partial g_{1,t}^{\prime }}{\partial \lambda }\Sigma _{g_{1}}^{-1}E \frac{\partial g_{1,t}}{\partial \lambda } &{} E\frac{\partial g_{1,t}^{\prime }}{\partial \lambda }\Sigma _{g_{1}}^{-1}E\frac{\partial g_{1,t}}{\partial \beta ^{\prime }} \\ E\frac{\partial g_{1,t}^{\prime }}{\partial \beta }\Sigma _{g_{1}}^{-1}E \frac{\partial g_{1,t}}{\partial \lambda } &{} E\frac{\partial g_{1,t}^{\prime }}{\partial \beta }\Sigma _{g_{1}}^{-1}E\frac{\partial g_{1,t}}{\partial \beta ^{\prime }}+E\frac{\partial g_{2,t}^{\prime }}{\partial \beta }\Sigma _{g_{2}}^{-1}E\frac{\partial g_{2,t}}{\partial \beta ^{\prime }} \end{array} \right) ^{-1}\Bigg /T\nonumber \\ \end{aligned}$$

(24)

In particular, using the simplifying formula for the inverse of partitioned matrices, the variance of $\lambda _{T}$ from Eq. (24) can be written as:

$$\begin{aligned} V(\lambda _{T})&= \left( \left( TV_\mathrm{na}\Bigg (\lambda _{T}^{2s})\right) ^{-1}-E\frac{ \partial g_{1,t}^{\prime }}{\partial \lambda }\Sigma _{g_{1}}^{-1}E\frac{ \partial g_{1,t}}{\partial \beta ^{\prime }}\left( E\frac{\partial g_{1,t}^{\prime }}{\partial \beta }\Sigma _{g_{1}}^{-1}E\frac{\partial g_{1,t}}{\partial \beta ^{\prime }}\right. \right. \nonumber \\&\left. \left. +E\frac{\partial g_{2,t}^{\prime }}{ \partial \beta }\Sigma _{g_{2}}^{-1}E\frac{\partial g_{2,t}}{\partial \beta ^{\prime }}\right) ^{-1}\cdot E\frac{\partial g_{1,t}^{\prime }}{\partial \beta }\Sigma _{g_{1}}^{-1}E\frac{\partial g_{1,t}}{\partial \lambda }\right) ^{-1}\Bigg /T \end{aligned}$$

(25)

where $\displaystyle V_\mathrm{na}(\lambda _{T}^{2s})=\left( T\cdot E\frac{\partial g_{1,t}^{\prime }}{\partial \lambda }\Sigma _{g_{1}}^{-1}E\frac{\partial g_{1,t}}{\partial \lambda }\right) ^{-1}$ is the non-adjusted variance of $\lambda _{T}^{2s}$ computed by the algorithm of the two-stage GMM estimation. Equation (25) proves the proposition.

Robustness analysis: tables

See Tables 8, 9, 10, 11, 12 and 13.

Table 8 GMM estimates of $\lambda$: fixed $\alpha = 0.1$

Table 9 GMM estimates of $\lambda$: QD filter

Table 10 GMM estimates of SIPC parameters: subjective expectations

Table 11 GMM estimates of the SIPC parameters: Real-time data

Table 12 GMM estimates of the hybrid SIPC model

Table 13 One-stage GMM estimator: the pooled model