New Keynesian Phillips Curve with time-varying parameters

According to Mankiw and Reis (2002), in the purely forward-looking NKPC inflation is not sticky despite sticky price. Hence the model cannot explain inflation inertia where inflation responds to shocks with a delay and moves gradually. However, Cogley and Sbordone (2008) state that as long as one considers the time-varying inflation trend in a purely forward-looking NKPC, it can explain inflation inertia without introducing “backward-lookingness” or dynamic inflation indexation in the sticky-price model of Calvo (1983).

With the lagged inflation term in the model, it differs from purely forward-looking NKPC where inflation cannot react immediately in response to a shock. Instead, inflation exhibits inertia or adjusts gradually in response to a shock. As a result, including a lagged inflation term into the model introduces an extra channel for “intrinsic” inflation inertia.

Galí and Gertler (1999) argue output gap is a poor proxy for real economic activity. They note that inflation should lead output gap according to the theory of NKPC, but the empirical evidence shows output gap leads inflation.

Galí et al. (2005) estimate the model with fewer instrumental variables, and they obtain results similar to those of Galí and Gertler (1999). Although Galí et al. (2005) argue their results are robust to the number of instrumental variables, the estimates of the cyclical relationship between inflation and real economic activity are statistically insignificant in some cases.

Besides model structure, measurement of inflation expectation affects estimates. Roberts (1995) states that when future inflation expectations \(\left( E_{t}\pi _{t+1}\right) \) are proxied by surveys, he obtains statistically significant estimates of the cyclical relationship between inflation and real economic activity in a purely forward-looking NKPC. In line with Roberts (1995), Brissimis and Magginas (2008) obtain statistically significant estimates of the cyclical relationship in a hybrid NKPC when future inflation expectations are proxied by surveys.

Fuhrer (1997) uses a VAR (vector autoregression) approach to both estimate and test NKPC expanded to include extra lagged inflation terms. He finds the weights on those terms are quite large, and the hypothesis test that inflation depends only on expected future inflation is rejected for various measures of the output gap and the time lag. Rudd and Whelan (2005) criticize the model used by Galí and Gertler (1999) as misspecified, and argue that the conclusion of Galí and Gertler (1999) can be drawn even the true model is backward-looking specification. Moreover, Rudd and Whelan (2005) use GMM to estimate the NKPC in a closed-form solution, in which inflation depends on past inflation, the cumulative sum of future expected inflation and forcing variables, and they find the coefficient on lagged inflation is quite large, and the results are robust to the choice of proxies for real economic activity and the price index.

Different from Calvo’s sticky-price model, Mankiw and Reis (2002) propose a sticky information Phillips curve model without imposing the assumption of price rigidity that can explain the inflation inertia quite well. Mankiw and Reis (2002) compare a sticky information model with purely forward-looking NKPC and backward-looking NKPC. They find that the sticky information model is dominant in explaining several features of US inflation dynamics: Inflation inertia, the acceleration phenomenon and the contractionary feature of announced and credible disinflation.

Galí and Gertler (1999) divide the full sample into three overlapped subsamples, and it is hard to distinguish those samples in a high inflation period from the one in a low inflation period. In particular, I find their estimates of the cyclical relationship between inflation and real economic activity in one subsample are 15 times larger than those in the other two subsamples when inflation is calculated by a non-farm business (NFB) deflator.

This could be due to menu cost or informational frictions. Menu cost means that it is costly to change a posted price, and the informational frictions mean that individual firms cannot perfectly observe the aggregate price level.

Christiano et al. (2005) allow for completely dynamic inflation indexation in the standard Calvo sticky-price model, and this introduces a lagged inflation term into the model. To be more precise, they modify the sticky-price model by assuming the last period’s inflation is used to update prices of firms that are not able to adjust prices freely.

This differs from Galí and Gertler (1999) in that I extend the length of the data set and use the HP filter to extract the cyclical component of output.

In particular, instrumental variables such as the consumer price index (CPIAUCSL), effective Federal Fund rate (FEDFUNDS), and long-short interest rate spread (GS10-TB3MS) are monthly data. I take an average of those monthly series to obtain the quarterly data series.

The output gap in Galí and Gertler (1999) is defined by the quadratically detrended log of real GDP per capita. Technically, in order to get the quadratically detrended log of real GDP per capita, they first get the fitted values of the real GDP per capita \(\left( y_{t}\right) \) in log form by running the following regression, \(\ln y_{t}=\alpha _{0}+\alpha _{1}t+\alpha _{2}t^{2}+\varepsilon _{t}\). Then they get the desired results by doing the following calculation, \(\ln \hat{y_{t}}-\ln y_{t}\).

The measure of real marginal costs in Galí and Gertler (1999) is computed as the percentage deviations from the sample mean of the labor share of income \(\left( l_{t}\right) \), \(\frac{\left( l_{t}-\bar{l}\right) }{\bar{l}}\).

Lindé (2005) finds by Monte Carlo simulation that the GMM estimates in Galí and Gertler (1999) are biased. In contrast to Galí and Gertler (1999), his DSGE-based estimation approach shows that the output gap can be a good proxy for characterizing the cyclical relationship, and backward-looking price-setting behavior is relatively important in explaining US inflation dynamics.

The results of estimating NKPC become more consistent when using a Bayesian approach in DSGE models. Both Rabanal and Rubio-Ramírez (2005) and Smets and Wouters (2007) find a relatively important role for the forward-looking behavior on US inflation dynamics, in which they estimate small-scale and medium-scale New Keynesian models, respectively.