Estimating a high-frequency New-Keynesian Phillips curve

Abstract

This paper estimates a high-frequency New-Keynesian Phillips curve via the generalized method of moments. Allowing for higher-than-usual frequencies strongly mitigates the problems of small-sample bias and structural breaks. Applying a daily frequency allows us to obtain estimates for the Calvo parameter of nominal rigidity over a very short period—for instance for the recent financial and economic crisis—which can then be easily transformed into their low-frequency equivalences. With Argentine data from the end of 2007 to the beginning of 2011 we estimate the daily Calvo parameter and find that on average prices remain fixed for approximately two to three months which is in line with recent microeconomic evidence.

Appendices

Appendices

1.1 Appendix A: Household’s optimization problem

Following Woodford (2003) and Galí (2008) we consider a standard intertemporal utility function of a representative household, which is additive separable in consumption $C_t$, in real money holding $M_t/P_t$ (where $P_t$ is the price level) and in leisure (labor supply $N_t$, respectively), subject to the underlying frequency in daily magnitudes:

$$\begin{aligned} U_t=E_t \sum ^{\infty }_{k=0} \beta (h_d)^k \left[ \frac{1}{1-\sigma }C^{1-\sigma }_{t+k}+\frac{1}{1-\psi }\left( \frac{M_{t+k}}{P_{t+k}}\right) ^{1-\psi }-\frac{1}{1+\eta }N^{1+\eta }_{t+ k}\right] , \end{aligned}$$

(17)

where $\sigma > 0$ represents the inverse intertemporal elasticity of substitution between present and future consumption of domestic goods, $\eta > 0$ is the inverse of the substitution elasticity of labor, $\psi >0$ stands for the inverse of the elasticity of money demand, $\beta (h_d) >0$ is the (frequency-dependent) discount factor and $E_t$ stands for the expectations operator.

The period-by-period budget constraint is characterized by money and bond holdings of the representative household and is given in real terms by

$$\begin{aligned} C_{t+k}&= -\frac{B_{t+k}}{P_{t+k}}-\frac{\Delta M_{t+k}}{P_{t+k}}+\frac{W_{t+k}}{P_{t+k}}N_{t+k} \nonumber \\ \qquad&+(1+i_{t+k-1})\frac{B_{t+k-1}}{P_{t+k}}+\Pi ^r_{t+k} + \frac{T^n_{t}}{P_t}, \end{aligned}$$

(18)

where $M_t (B_t)$ is the household’s nominal holding of money (one-period bonds). Bonds pay a nominal interest rate $i_t$ and $1+i_t$ represents the gross nominal interest rate. $W_t$ denotes the nominal wage. Real profits received from firms are equal to $\Pi _t^r$ and $T_t^n$ are nominal lump-sum taxes or transfers. Note that $\Delta M_{t+k}=M_{t+k}-M_{t+k-1}$ holds.

The household seeks to maximize (17) subject to (18) and the following cash-in-advance constraint:

$$\begin{aligned} P_{t+k} C_{t+k} = M_{t+k}, \end{aligned}$$

(19)

i.e., nominal consumption expenditures are not allowed to exceed the nominal money holdings of the household. This expression is given in real terms and under consideration of $y_t=c_t$ in Sect. 2. In can be easily shown that the optimality condition for the demand of real money holdings (in daily magnitudes) is given by:

$$\begin{aligned} m^r_t=\frac{M_t}{P_t}=\frac{1}{\psi } (\sigma y_t-\beta (h_d) i_t). \end{aligned}$$

(20)

1.2 Appendix B: Time series for the Argentine consumer price index (CPI)

See Fig. 1.

Fig. 1

The upper panel depicts the development in the Argentine CPI in quarterly magnitudes from 1995Q1 until 2010Q4. The lower panel depicts the corresponding Argentine CPI in daily magnitudes (scraped data index) from 2007Q4 until 2010Q4. Quarterly data is taken from Datastream$^\mathrm{\textregistered }$. Daily data is provided by www.inflacionverdadera.com