On the Granger causality between median inflation and price dispersion

Abstract

The Granger-causal relationship between the size and dispersion of fluctuations in sub-components of the US Consumer Price Index (CPI) is examined using both in-sample and out-of-sample tests and data from January 1968 to December 2008. Strong in-sample evidence is found for feedback between median inflation and price dispersion; the evidence for Granger-causation from median inflation to price dispersion remains strong in out-of-sample testing, but is less strong for Granger-causation in the opposite direction. The implications of these results for the variety of price-level determination models in the literature are discussed.

Keywords:

• Granger causality
• median inflation
• price dispersion
• out-of-sample testing

JEL Classification::

• C53
• E31
• E37

Notes

¹ In fact, any nonlinear dynamic pricing rule can induce a feedback relationship between mean/median inflation and price dispersion. See Subramanian and Kawachi (2004, pp. 79–80) for a similar argument in a different context.

² See, for example, Vining and Elwertowski (1976), Parks (1978), Reinsdorf (1994), Grier and Perry (1996), Parsley (1996) and Debelle and Lamont (1997).

³ Ashley (1981) uses the mean and SD as measures of inflation and price dispersion, respectively, and studies the Granger causality between these two variables over the period between January 1953 and June 1975 using a single out-of-sample test. In this study we use the sample median and interquartile range as measures of inflation and price dispersion, respectively, and look at a different sample period (January 1968 to December 2008) with both in-sample and a variety of out-of-sample tests. Despite the above differences, our findings with regard to the Granger causality between inflation and price dispersion are consistent with those in Ashley (1981), but richer due to the longer sample period and to the substantially more sophisticated testing techniques now available.

⁴ These CPI components are obtained from the Bureau of Labor Statistics – see the Appendix for details. The selection of the components used here is based on the availability of data over the entire sample period. Here we use seasonally unadjusted data simply because the usual seasonal adjustment procedures involve two-sided filtering which mixes together the past and future of a time series.

⁵ Relative importance figures for the 31 components of the CPI index are obtained from the Bureau of Labor Statistics.

⁶ Given a series of N observations, v ₁, v ₂, … , v_N , ranked from the smallest to the largest, the weighted percentile for i-th observation is:

where w_i is the weight of the i-th observation, , and . To find the value v corresponding to a given percentile l, we first find the observation number j where and then calculate the value as

⁷ The nonhomogeneity in the variance of the median inflation variable evident in Fig. 2(a) motivated some of the robustness checks described in the Section ‘Robustness checks’ below but was, in the end, not problematic.

⁸ This may seem like quite a large out-of-sample period (with 180 observations total), but this choice reflects the importance we attach to out-of-sample versus in-sample testing. Calculations in Ashley (2003) support the proposition that out-of-sample prediction period lengths in excess of 100 observations are worthwhile.

⁹ Seasonally un-adjusted monthly unemployment rate data are obtained from the Bureau of Labor Statistics.

¹⁰ This test is, of course, only justified if the usual regression assumptions of homoscedastic and serially uncorrelated model errors are valid. Here sufficient lags are added to the model so that the correlogram of the fitting errors is consistent with serially uncorrelated model errors and the fitting errors are tested for heteroscedastictiy using both the Breusch–Pagan–Godfrey test and the White test. Because the homoscedasticity assumption is problematic, White–Eicker (robust) SE estimates are used throughout.

^11 ‘Feedback’ between x_t and y_t is the case where both x_t fluctuations Granger cause future y_t fluctuations and y_t fluctuations Granger cause future x_t fluctuations. Because the Fourier transformation is a two-sided filter, it mixes together both past and future values of a time series. Consequently, the application of frequency domain methods to data exhibiting feedback is fraught – see Ashley and Verbrugge (2007, Section 3.6) for a detailed exposition of this point.

¹² Windows software to easily perform this decomposition is available from the authors; see Ashley and Verbrugge (2007, 2009) for all of the analytical and calculational details. But it is worth mentioning here that the problematic effects of possible feedback are eliminated by only ever using the last filtered observation obtained from a 36-month window moving through the data set; thus, the decomposition is effectively a one-sided filter. There are only 18 possible nonzero frequencies possible with a 36-month long window (rather than 36 components) because the component related to the sine of each frequency can sensibly be aggregated with the corresponding cosine component. Thus, it is feasible to estimate all 19 possible component coefficients with a typical monthly data set. On other hand, this does ‘use up’ 18 additional ‘degrees of freedom’ in the regression and 36 sample observations are consumed by the initial window. Less importantly, fluctuations with periods in excess of 36 months cannot be distinguished from one another.

¹³ As noted later in this section, we also test whether the out-of-sample forecasts obtained from the unrestricted model encompass those from the restricted one. Following Rogoff and Stavrakeva's (2008) comments, however, we do not give results based on this test a causal interpretation.

¹⁴ This direct bootstrap version of the test explicitly allows for a substantial contemporaneous cross-correlation between the two forecast errors and also (through the VAR) for serial correlation, which might be present due to model misspecification. It does not allow for heteroscedasticity in the errors, however: nowadays a wild bootstrap would be used, as implemented below. It does, on the other hand, implement a double-bootstrap which roughly quantifies the uncertainty in the inference due to the bootstrap approximation itself, which is only justified for large samples. Thus, this direct bootstrap test is preferable for short out-of-sample periods, whereas the kind of bootstrapping implemented here − for the GN test and the four others − is preferable for longer out-of-sample periods. See also footnote 16.

¹⁵ See West (1996), Clark and McCracken (2001, 2005), West (2006) and Clark and West (2006).

¹⁶ This is for rolling forecasts; for forecasts calculated recursively, the limiting distribution of the CW statistic is a bit more complex; see Clark and West (2007) for details. Effectively, per Paye (2010), these tests are focusing on testing the underlying causal structure rather than simply testing whether the forecasts from one model are more accurate than those of an another, as in the direct bootstrap test of Ashley (1998) discussed in footnote 14.

¹⁷ The autoregression for the aggregate inflation equation includes a linear trend, its first, second, third and 12th lags. The relative price dispersion equation is modelled as an Autoregressive (AR(3)) process and the seasonal difference of the change in the unemployment rate is modelled as an AR(4) process. These lag structures were chosen so as to minimize the BIC.

¹⁸ For simplicity, we fix the values of initial observations at their actual sample values.

¹⁹ The data on M2 _t , on IP _t and on TB3 _t are both seasonally unadjusted and obtained from Board of Governors of the Federal Reserve System H.6, G.17 and H.15, respectively. The appropriate lag lengths with which these variables enter the models is determined by minimizing the BIC. The first lag of M2 growth is included in the forecasting models for median inflation while its 4th and 5th lags are included in the forecasting models for price dispersion. The 1st and 12th lags of IP growth are included in the forecasting models for median inflation, and its 6th lag is included in the forecasting models for price dispersion. The first lag of TB3 is included in the forecasting models for median inflation and its second lag is included in the forecasting models for price dispersion. These estimation results are available upon request.

²⁰ See, for example, Fischer (1981) and Taylor (1981), for discussions of the likely effect of energy price shocks on the relationship between inflation and price dispersion. The three CPI components excluded from the calculation of median inflation and price dispersion are ‘fuel oil and other fuels’, ‘gas and electricity’ and ‘motor fuel’.

²¹ The break dates identified by Andrews’ supWald tests for the unrestricted aggregate inflation and relative price dispersion regressions are 1981M05 and 1973M01, respectively; more details on these results are available upon request.

²² The p-value plot for the ENC-NEW encompassing test is omitted because its interpretation in terms of Granger causality is murky. Qualitatively, for the test of price dispersion Granger-causing inflation, the ENC-NEW test p-value hovers in the range 0.05 to 0.10 until starting months in mid-2000 and then became even larger. For the ENC-NEW test of inflation Granger causing price dispersion, the p-value plot remains well below 0.05 until the middle of 2004 and then increases.