Fast micro and slow macro: Can aggregation explain the persistence of in‡ation? Filippo Altissimo (ECB and CEPR), Benoît Mojon (ECB and Université de la Méditerranée) Paolo Za¤aroni (Banca d’Italia) November 2004
Abstract We estimate a dynamic factor model for 404 disaggregate in‡ation series of the euro area CPI between 1985 and 2003. We then aggregate back the 404 estimated micro level in‡ation equations and decompose the dynamics of this aggregate into the e¤ects of the common and the idiosyncratic shocks. First, we …nd that one common factor accounts for 30 % of the overall variance of the 404 in‡ation series. Second, the propagation mechanism of this common shock is highly heterogenous across sub-indices. Third, the aggregation exercise, which mimics remarkably well the persistence observed in the aggregate in‡ation, demonstrates how the fast average micro adjustments translate into This paper would not have been possible without the kind help of a few individuals and the goodwill of their institution. We are extremely grateful to Laurent Baudry, Laurent Bilke, Herve Le Bihan and Sylvie Tarrieu (Banque de France), Johanes Ho¤man (Bundesbank) and Roberto Sabbatini and Giovanni Veronese (Banca d’Italia). We also would like to thank Anna-Maria Agresti and Martin Eiglsperger (ECB) for their outstanding assistance in the construction of historical time series of the German CPI sub-indices. Finally, we are grateful to Ignazio Angeloni, Steven Cecchetti, Michael Ehrmann, Frank Smets and participants to the Eurosystem In‡ation Persistence Network for helpful comments on previous presentations of this research. The opinions expressed here are those of the authors and do not necessarily re‡ect views of the European Central Bank or of the Banca d’Italia. Any remaining errors are of course the sole responsability of the authors.
1
the slow adjustment of euro area aggregate in‡ation. Fourth, we propose a new measure of core in‡ation for the euro area.
1
Introduction
Understanding the source and degree of persistence of prices is key both to improve our ability to forecast in‡ation and selecting structural models of the business cycle. Successful out-of-sample forecasting of in‡ation requires in …rst instance to decide the appropriate degree of persistence and, eventually, of non-stationarity of the data generating process. This a¤ects not only the point forecasts but, perhaps more importantly, impulse responses and predictive density forecasts. Turning to models of the business cycle, price stickiness is seen by many as the key ingredient that allow micro founded DSGE models to deliver the in‡ation and output persistence that we see in macroeconomic data. 1 Critics of sticky price models have stressed that these periodicities are far too large to make economic sense (see Chari, Kehoe and Mac Grattan (2001)) and inconsistent with the much faster average frequency of adjustments that can be observed in the micro data (Bils and Klenow (2004), Dhyne et al. (2004)). However, the theory of cross-sectional aggregation of dynamic processes (Robinson (1978), Granger (1980), Forni and Lippi (1997) and Za¤aroni (2004)) has showed that slow macroeconomic adjustments may very well be consistent with much faster average speed of adjustments at the micro level.2 1
This research agenda has actually been very successful because it obtained plausible estimates of structural /policy invariant parameters from the data. Gali and Gerlter (1999) write: "we estimate all the structural parameters of the model using conventional econometric methods. The coe¢ cients in our structural in‡ation equation are ’mongrel’ functions of (a) [. . . ] model primitive (s): the average duration that an individual price is …xed (i.e., the degree of price ’stickiness’) ... See also Sbordone (2003), Galì and Gertler (1999), Chritiano et al.(2003), Smets and Wouters (2003) among many others. 2 The long adjustment of the aggregate appears to be well approximated by a long memory stationary process, whose autocorrelation function decays hyperbolically toward zero as the lag increases. Such a slow rate of decay has, in the case of in‡ation, led many empirical studies to infer non-stationarity.
2
This paper uses this theoretical apparatus to analyze the euro area in‡ation, its persistence and how it relates to micro level persistence.3 We estimate a dynamic factor model for N = 404 sub-indices of the euro area CPI between 1985 and 2003. In the model, each in‡ation series depends on a macroeconomic shock, “a common factor”and a sub-index speci…c “idiosyncratic factor”. We then aggregate back the 404 models of micro level in‡ation and decompose the dynamic of this aggregate into the e¤ects of the common and the idiosyncratic shocks. We …nd …rst that one common factor accounts for 30 % of the overall variance of the 404 series. This share is twice as large if one focuses on low frequencies, i.e. on the persistent components of the series. Second, the propagation mechanism of shock at micro level across sectors is highly heterogenous. This heterogeneity is the prerequisite for the aggregation mechanism to be maximally e¤ective. Third, the implied persistence from the aggregation exercise mimics remarkably well the persistence observed in the aggregate in‡ation. In particular, it turns out that the cross-sectional distribution of the micro parameters implies an autocorrelation function of the aggregate CPI in‡ation which decays hyperbolically toward zero. Altogether, the high volatility and low persistence, observed on average at the level of in‡ation sub-indices, is consistent with the aggregate smoothness and persistence. Finally, a joint output of the analysis is that the aggregate of the e¤ects of the common shock provides a new measure of core in‡ation that is derived from a formal model of common and idiosyncratic shocks that a¤ect in‡ation. The paper is organized as follows. The following section presents the elements on cross-section aggregation. Section 3 presents the data used in the empirical analysis and Section 4 addresses the presence of common factors across the price sub-indices. Section 5 introduces the micro models and 3
Aggregation is the main mechanism underlying our analysis and its key role is evident when one looks at the way in which the CPI is linked to the individual goods and services indexes. For instance, in the US the Bureau of Labor Statistics collects prices on around 80000 goods and services each month, which are divided into 350 categories called entry level items; those data is aggregated up to the overall CPI. The link between the monthly micro price quotes for each entry level item, whose relative frequencies of changes has been analyzed by Bils and Klenow (2002) and others, and the aggregate CPI implies at least two layers of aggregation: The …rst one goes from the individual price records to the price index of the relevant subcategory. The second one relates the subcategories to the aggregate CPI. This paper focuses on this second layer.
3
the estimation results, while Section 6 links the micro parameters to the aggregate persistence. Section 7 concludes.
2
Aggregation of heterogeneous AR(1) models
In this section we examine, by means of a series of numerical examples, the most important results on contemporaneous aggregation of heterogeneous ARMA models, when the number of units gets arbitrarily large (see Robinson (1978) and Granger (1980)). For sake of simplicity, let the ith agent or ith subindex be described by an AR(1) model: yit =
i yit 1
+
(1)
it ;
where both the coe¢ cients and the random shocks vary across individuals. When considering an arbitrarily large number of units, a convenient way to allow for heterogeneity is to assume that the coe¢ cients i are i:i:d: random drawn from some underlying, …nite dimensional, distribution F ( ). Stationarity of yit then requires j i j< 1 a:s: or, alternatively, that F ( ) has support ( 1; 1). Finally, we assume that the random shock is the sum of a common and of an idiosyncratic component it
= ut +
(2)
it ;
with the ut being an i:i:d: sequence (0; 2u ) and it being an i:i:d: sequence (0; 2 ). The i;t are also assumed independent across individuals. In view of (2) and linearity of the model one gets the following decomposition for the aggregate (L denotes the lag operator) 1 X ut 1X = + n i=1 1 n i=1 1 iL n
Yn;t
n
it iL
= Un;t + En;t ;
meaning that the aggregate could be separated in a common and idiosyncratic component. Although the statistical properties of each unit are well-de…ned, conditioning on i , knowledge of the entire history of each xit , or even of a …nite number n of them, is uninformative for F ( ). However, when looking at 4
an arbitrarily large number of units, F ( ) will then entirely determine the properties of the limit aggregate, which we de…ne as the limit of the Yn;t for n ! 1. It is well know that summing a …nite number of ARMA processes yields again an ARMA process. For example, the sum of n distinct AR(1) models, with di¤erent auto-regressive parameters, yields an ARMA(n; n 1). However, when n goes to in…nity, it turns out that for absolutely continuous F ( ), the limit of Xn;t will not belong to the class of ARMA processes, in contrast to the individual xit . To illustrate the main results, let us consider a possible way of parameterizing F 0 ( ) = f ( ) as a Beta distribution, with parameters p; q > 0, ( B 1 (p; q) p 1 (1 )q 1 ; 0 < 1; f( ) = 0; otherwise: where B(p; q) denotes the Beta function of order p; q. For sake of comparison, we consider only Beta distributions satisfying E
(3)
= ;
i
q. The q parameter measures for some 0 < 1 which implies p = 1 how dense is the distribution of the micro parameters i around the unit root. This is going to be a key element of the results. First, let us focus on the common component. This can be written as Un;t = ut + ^ 1 ut setting for every k
+ ^ 2 ut
1
1X ^k = n i=1
2
+ ::: + :::;
(4)
n
k i:
When n gets large, by the strong law of large numbers, each ^ k will converge a:s: to the population moments of the i : ^k !
k
= E(
k i)
a:s: for n ! 1:
It turns out that (under suitable regularity conditions) the Un;t will converge in mean square to the limit aggregate Ut = ut +
1 ut 1
5
+
2 ut 2
+ :::+
(5)
as n goes to in…nity.4 From (5), the dynamic pattern of the k represents the impulse response of the common shocks ut on the aggregate. We now make a numerical evaluation of the impulse response function of the limit aggregate. For the Beta density tedious calculations yield to: k
=
(1
q + k)
(1 q )
q)
( 1 q + k)
(1
;
(6)
where (x) indicates the Gamma function. Table 1 reports the dynamic pattern of k for various values of q. We compare the results with the case of homogeneous AR(1), setting the autoregressive coe¢ cient equal to . In this case, the impulse response will be given by k . The …rst half of the table reports the results for = 0:8. The second half of the table considers = 0:95. In this way, we can compare the e¤ect of aggregation with an homogeneous, yet very persistent, case. The e¤ect of aggregation is dramatic: the impulse response function of the aggregate process (common component) decays towards zero very slowly, compared with the homogeneous coe¢ cient case. This is true even for unrealistically large values of , such as = 0:95. Note that the smaller is q, the larger is the mass of the distribution around the unit root and the slower will the effect of random shocks fade away. Finally, choosing a di¤erent has a greater e¤ect on the constant parameter cases rather than on the heterogeneous case. In other words, the average impulse response is markedly di¤erent from the impulse response of the average. Further insights can be obtained by looking at the corresponding analytic results. When 0 < (2 1) < 1 for k of (6) one gets5 k
ck
q
as k ! 1:
Again, an hyperbolic behavior arises whose intensity is now directly determined by the Beta parameter q, in agreement with the …ndings of Table 1. In particular, the smaller is q, so the more dense is the distribution f ( ) around unit root and so more micro units are very persistent and the slower will the k converge towards zero (recall that q > 0 always). 4
This representation holds for the Beta distribution case when q > 1=2; see Za¤aroni (2004). p 5 This result follows by using Stirling’s formula, (x) 2 e x+1 (x 1)x 1=2 as x ! 1 in (6).
6
Table 1 Impulse response functions of limit aggregate Ut k
k
q=
0:2
0:3
f( ) 0:7
1
3
0:8 0:70 0:57 0:47 0:29 0:26
0:8 0:67 0:48 0:33 0:12 0:09
0:8 0:66 0:44 0:28 0:07 0:05
0:8 0:65 0:37 0:17 0:01 0:01
0:95 0:91 0:82 0:73 0:49 0:45
0:95 0:91 0:79 0:67 0:33 0:27
0:95 0:90 0:79 0:65 0:27 0:22
0:95 0:90 0:78 0:62 0:15 0:11
= 0:8 1 2 5 10 50 200
0:8 0:64 0:33 0:11 1:4 10 4:1 10
5 20
0:8 0:72 0:61 0:54 0:39 0:36 = 0:95
1 2 5 10 50 200
0:95 0:90 0:77 0:60 0:08 3:5 10
5
0:95 0:91 0:83 0:76 0:58 0:54
7
Table 2 Asymptotic behaviour of varn (En;t ).
n 10 100 1; 000 5; 000 10; 000 20; 000
q=
0:2
1:4 3:6 2:8
f( ) 0:7
0:3
1:6 106 104 108 1 1
0:4 5:4 8:3 103 1:8 103 1:2 104 4:6 103
3:1 1:4 4:3 3:7 1:6
0:3 10 10 10 10 10
1
2 2 3 3 3
5:2 3:7 8:4 4:2 2:2
0:2 10 10 10 10 10
3
2 3 4 4 4
1:6 1:5 3:1 1:6 7:9
0:2 10 10 10 10 10
3 3 4 4 5
The symbol 1 indicates a large value (computer out‡ow) for varn (En;t ).
Furthermore in view of the linearity of the set-up, this characterization of the impulse response has a neat representation in terms of the auto-covariance function (acf) and spectral density of the limit aggregate Ut (see the above references for further details). Second let us focus on the idiosyncratic component En;t . Table 2 reports the asymptotic behavior of varn (En;t ) =
n 1 X 1 n2 i=1 1
2 i
;
for various values of n. Hereafter, varn (:) indicates the variance operator, for given parameter values ( 1 ; :::; n ). The most, somewhat striking and counter-intuitive result that appears from Table 2 is that this component, made by averaging perfectly independent and stationary (a:s:) units, does not necessarily vanish. In fact, varn (En;t ) gets smaller as n increases for the Beta distribution case for q = 0:7; 1; 3. In contrast, varn (En;t ) gets arbitrarily large for q = 0:2; 0:3. It turns out that this is precisely what should happen. One can then ask which component is the dominant one with respect to aggregate dynamics. In Table 3, we look at the behavior of the ratio Rn =
varn (En;t ) varn (Un;t )
as n gets large. It turns out that Rn gets smaller as n increases for the Beta distribution case for q = 0:7; 1; 3. For the cases where the idiosyncratic variance tends to explode (q = 0:2; 0:3), instead, Rn is stable, in the sense 8
Table 3 Asymptotic behaviour of Rn .
n
q=
10 100 1; 000 5; 000 10; 000 20; 000 The symbol
0:2
0:3
0:9 0:51 0:49 0:48 0:17
0:9
f( ) 1
0:7
4:9 1:2 3:6 1:5 9:9
0:3 10 10 10 10 10
2 2 3 3 3
1:7 3:7 9:4 3:6 1:6
0:1 10 10 10 10 10
3
2 3 4 4 4
1:1 1:1 2:3 1:1 5:7
0:1 10 10 10 10 10
2 3 4 4 5
indicates an undetermined value (computer out‡ow) for Rn .
that it does not diverge nor converge towards zero. This suggests that also the variance of Un;t tends to explode and, moreover, at the same rate. This is to say that in the non-stationary case the common and the idiosyncratic component have precisely the same importance in determining aggregate ‡uctuations. This has been formally established in the theory (see Appendix B). Model (1) represents an extremely simpli…ed set-up. In fact, we consider below, a more complicated and realistic model than (1). However, it turns out that, as far as the analysis of aggregation is concerned, the results that apply to an AR(1) set-up equally apply to higher-dimensional models, which must include an auto-regressive component, with no qualitative di¤erences. Summarizing, we have found that allowing for heterogeneity across the auto-regressive micro parameter, has a dramatic e¤ect on the behavior of the aggregate impulse response function. In fact, the e¤ect of random shocks to the aggregate decays much slowly compared with any individual AR(1). Second, the e¤ect of the idiosyncratic shocks might not be negligible on aggregate ‡uctuations. The key ingredient appears to be the relative importance of the very persistent micro unit or, in other words, the shape of the cross-sectional distribution f ( ) around 1, which in the Beta distribution is dictated by the parameter q. When q > 1 the density f will have little mass around unit root (i.e. f ( ) goes to zero for approaching 1), the aggregate process is stationary but the autocovariance has an hyperbolic decay with short memory. When 1=2 < q < 1, large mass of the f ( ) will be around
9
unit root (i.e. f ( ) " 1 for approaching 1) the process is still stationary but displays long memory. Both cases can be formally represented as var(Ut ) < 1; cov(Ut ; Ut+k )
c k1
2q
as u ! 1;
(7)
and one can easily see that for 1=2 < q < 1 the cov(Ut ; Ut+k ) are not summable, which is the classical symptom of long memory. In other words, the smaller is q, the larger is the frequency of units whose i is close to unity, and the more persistent is the limit aggregate. When 0 < q < 1=2 the aggregate process will be not stationary. For instance this is evident since the autocovariance function does not even go to zero as k increases. This characterization remains true not only for the Beta distribution but for any distribution whose behavior around unit root can be parametrized by (1 )q 1 :
3
The data
The data consist of 404 seasonally adjusted quarter over quarter (q-o-q) in‡ation rates of CPI sub-indices from France, Germany and Italy.6 In this section, we …rst discuss our choice of data and sample period. We then show that our sample of sectoral prices is representative of the full set of prices that compose the euro area in‡ation. Finally, we present descriptive statistics on sectoral and aggregate in‡ation and their persistence.
3.1
Choice of data and sample period
We use CPI data rather than HICP because the latter are available only since 1995. However, because earlier data are not readily available in all the countries of the euro area, we limited our data to France, Germany and Italy. These countries together account for roughly 70 % of the euro area population and consumption. We focus on the post 1985 data for two reasons. First, the German data are not available beforehand. Second, many studies have showed that the mid-eighties marked a signi…cant break in the mean in‡ation in most OECD countries (Benati, 2004, and Corvoisier and Mojon, 2004). In particular, the 6
We present the source and the construction of the series in the data appendix.
10
latter paper shows that the in‡ation of the Italian CPI, the French CPI and the Euro area CPI all admit a break in their mean in the mid-eighties. More relevant for our study of CPI sub-indices, Bilke (2004) shows that almost all the 148 French sectoral in‡ation rates (exactly our data as far as France in concerned) admit a break in their mean around 1984 and very infrequently at any other time of the 1972-2003 sample period. This is sharp evidence in favour of the view that the breaks in the mean of French in‡ation of 1984 is indeed due to the major shift in the French monetary policy regime.7 This break a¤ects the in‡ation rates of all the goods and services of the consumption basket about a year after the government took a series of measures ( including price and wage controls and aggressive defense of the peg to the D-Mark) which marked the beginning of the F ranc f ort policy.8 In Italy, the mid-eighties also corresponds to a tighter commitment to deliver lower in‡ation. Gressani, Guiso and Visco (1988) write: “the [mid-1980s in‡ation reduction] was the result of a complex process aimed at bringing in‡ation under control, which involved [. . . ] the entrance into the EMS, a steady and restrictive monetary policy, the self-restrained wage earners, [. . . ] and the management of publicly controlled prices”. As regards Germany, where the monetary policy regime was more stable, the breaks in the mean of in‡ation of two largest trade partners is an major event in itself. For all these reasons, we deem the 1985-2003 sample as appropriate to 7
Three other papers have applied break tests to sectoral data may obtain the rejection of the hypothesis that breaks have a common "macroeconomic" factor that would a¤ect all sectors at the same time. Clark (2003) also estimate the e¤ects of allowing for an early 1990’s break in the mean of US sectoral in‡ation. He …nds such a break is signi…cant for about half of his sub-indices, the break is much less widespread across sectors than in the case of the 1984 case estimated by Bilke (2004). Lunnemann and Mathä (2004), who focus on the e¤ect of the launch of the euro, in January 1999, on the dynamics of sectoral in‡ation, cannot reject a break in the dynamics of in‡ation at this date for 56 % of the sectors. Cecchetti and Debelle (2004) who use larger sub-aggregates than the other studies, cover most OECD countries. 8 We can also very well identify the political process that led to this major shift in the French monetary policy. Attali (1993) describes how, in the spring of 1983, Jacques Delors, Pierre Mauroy and a few others convinced François Mitterand to opt for the F ranc f ort policy and stick to it.
11
study the persistence of in‡ation in the euro area. This choice of sample leaves us up to 78 observations of q-o-q in‡ation rates for each of the subindices.
3.2
Coverage of the euro area in‡ation
We now show that our three-countries coverage is representative of the euro area in‡ation. We …rst compare the time series of the aggregates that we can reconstruct from the 404 well-behaved in‡ation series9 and the o¢ cial euro area in‡ation. A preliminary step in this comparison is to show that limiting the investigation to Italian, French and German sub-indices gives a reasonable approximation of the euro area aggregate. This is shown in Figure 1, which reports the CPI in‡ation of each country as well as their aggregate. The latter is de…ned as a weighted average of the three countries in‡ation (1995 PPP-GDP weights of 0.422, 0.296 and 0.282 for Germany, France and Italy, respectively). In Figure 2, we plot the reconstructed in‡ation aggregates (of the well behaved series) and the o¢ cial CPI in‡ation rates and in Table 4, we compare some of their key features. The table reports the mean and the standard deviations of the in‡ation time series as well as four measures of the persistence of the series: the lag one autocorrelation, the largest root of a …tted AR(4), the largest root of the best …tting ARMA10 , and …nally, the sum of the autoregressive coe¢ cients (SARC thereafter) of the AR(4) 11 . Our estimates of the SARC, which has become the most popular measure of persistence, are actually very to the estimates reported in previous research on in‡ation persistence for sample periods similar to ours (see in particular, Benati (2004), Bilke (2004), Gazinski and Orlandi (2004) and Levin and Piger (2004)). 9
The three CPI original databases together comprise 470 sub-indices. 66 of these were not suitable for estimation of an ARMA model either because they have too few observations (e.g. some sub-indices are available only since 2000), or because they correspond to items which prices are set at discrete intervals (e.g. Tobacco or Postal services). We are left with 404 "well-behaved" series that are proper to be modelled as ARMA processes. 10 Best ARMA(p; q) as selected by an AIC criteria with 0 p; q 4. 11 All four measures are estimated on the qoq in‡ation time series for the 1985q22003q3 sample. All estimates were compiled in Matlab. In particular, the AR(4) and the ARMA(4) roots were estimated with the ARMAX procedure.
12
Both Figure 2 and Table 4 convey that the o¢ cial and the reconstructed aggregates are very similar. We notice however di¤erences between the persistence of the o¢ cial in‡ation and the aggregates that we reconstructed. This gap in persistence is largest between the French and the Italian o¢ cial and reconstructed indices for the lag one autocorrelation and the SARC. These di¤erences may be due to our method of reconstruction of the aggregate which di¤er form the construction of the o¢ cial indices in mainly two respects. First, we focus only on the well behaved sub-indices, and second, we use …xed weights while the weights of o¢ cial indices are adjusted every year to re‡ect changes in the composition of the consumption basket. Unfortunately, none of these points can be dealt within our model of the aggregate persistence as a function of individual series’persistence. However, given that the di¤erences in persistence between the euro area aggregate of 404 sectoral in‡ation rates and "o¢ cial" EA3 CPI in‡ation is very small, we consider that our reconstructed aggregate of 404 time series is indeed very good approximation of the euro area aggregate in‡ation.
13
3.3
Descriptive statistics on the 404 sectoral in‡ation series
We now turn to the properties of the sub-indices in‡ation. Figure 3 reports the cross-section averages of selected descriptive statistics that has been initially computed for each of the 404 annualized q-o-q in‡ation time series. The mean of individual sectors in‡ation is concentrated around the mean in‡ation of their weighted aggregate of 2:3, the un-weighted mean of the means being 2:5. Average in‡ation between 1985 and 2003 is negative only in a few sectors. Turning to standard deviations, sectoral in‡ation is noticeably more volatile than aggregate in‡ation. The average standard deviation is equal to 3 %, i.e. three time as large the one of the aggregate. This much higher volatility is a common feature of the in‡ation rates of sectoral prices as shown in Bilke (2004), Clark (2003) and Lünnemann and Mathä (2004a, 2004b) for CPI sub-indices as well as for sectoral PPI (Ernst and Mojon, 2005). The four charts in the second and the third rows of Figure 3 report the distribution and the cross-secion mean of the four measures of persistence that were introduced in the previous section. All four distributions are left skewed with a mode between .5 and 1. They are however not completely similar. The largest root of the ARMA, which corresponds to the most general model, is more concentrated close to one from below. We also note that the two measures of the largest root are on average larger than either the autocorrelation coe¢ cient and the SARC. This is to be expected given that the second and the third roots of the in‡ation processes are negative for a large proportion of the 404 series. In Table 5, we compare these distributions to the persistence of the aggregate. All measures indicate a large gap between the persistence of the aggregate and the typical/average persistence of sectoral in‡ation.12 The former is even larger than the 75th percentile of the sector’s persistence for the SARC, the ARMA. The di¤erence between the average sectoral in‡ation persistence and the aggregate in‡ation persistence is relevant. To illustrate this point, it is useful to recall that two AR(1) processes with a root of 0.93 and 0.65 respectively would respond very di¤erently to shocks: four quar12
The di¤erent measures of persistence are positively correlated with one another. In particular, the cross-sectional correlations of SARC with the largest root of an AR(4) model and the lag one autocorrelation are close to 1.
14
ters after a temporary 1.0 percent shock the …rst process would still be 0.7 percent above baseline against 0.12 for the second. We stress two additional observations. First, the weighted mean of the sub-indices persistence is smaller than the (un-weighted) mean. This indicates that sub-indices with a larger weight in the CPI are less persistent than the average. Second, we observe sharper di¤erences in persistence across the main groupings of the CPI (processed food, unprocessed food, energy, nonenergy industrial goods-NEIG- and services) than across countries. The gap between the ARMA largest root of the in‡ation process of energy prices (0.44) and one of the NEIG (0.78) is much wider than between the root associated to the ARMA processes …tted on the in‡ation of German, French and Italian prices. Comparing the main groupings of CPI sub-indices across countries, we also …nd that the "sectoral" hierarchy as the one of the euro area applies within each country.13 To conclude this …rst description of the data, we …nd clear evidence that the in‡ation rates of the individual sub-indices are way more volatile and much less persistent than the in‡ation rate of the aggregate CPI index.
4
The common component in the cross section
This section assesses the two elements that play a crucial role in the crosssection aggregation of time series: the presence of common shocks and the heterogeneity in the propagation mechanism of those shocks. This implies that, using the notation of Section 2, there is a common shock ut and that the i are di¤erent across i. We start by investigating the presence of common shocks in the cross-section of the in‡ation sub-indices before we demonstrate heterogeneity in the propagation of this common shock across sectors. 13
The averages of each grouping for each country is reported in the annex table A1, in the data appendix. These results are not directly comparable to earlier studies on sectoral in‡ation because these report the estimated persistence of CPI main groups aggregates and not average of indi…dual items persistence for groupings of the CPI sub-indices. One exception is Lünnemann and Mathä (2004) who however focus on a much smaller sample than we do.
15
16
Following the recent developments in the literature of large cross-sections (see Stock and Watson 1998, and Forni et al. 2002) and Clark (2003) application of this approach to US disaggregate consumption de‡ators, we estimate the principal components of the autocovariance structure of the data to assess the presence and relative importance of common driving factors (shocks). Figure 4 presents the spectrum of …rst ten dynamic principal components of the 404 in‡ation time series.14 while Figure 5 shows the relative contribution of those ten dynamic principal components in explaining, at each frequency (periodicity), the overall variance of the data 15 . Furthermore, Table 6 below reports the …rst ten normalized eigenvalues associated with the contemporaneous variance-covariance matrix of the data. Table 6: Static principal components 1 2 3 4 5 6 7 8 9 10 0.29 0.07 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.02 Source: Authors’calculations. This analysis indicates the presence of some strong common feature in the data. In particular, the …rst dynamic factor accounts around 60 per cent of the variability of the data at long-term periodicity, while its relative importance decays for short-term ‡uctuations (Figure 5). Furthermore, it is clear from Figure 4 that this common factor is the main driver of the in‡ation persistence observed in the data. Indeed, the …rst principal component is the only one which shows high persistence as indicated by the size of its spectral peak at zero frequency.16 These result is very consistent with Clark(2003), who stress that the common "factor" of the disaggregate in‡ation rates is more persistent that the idiosyncratic components. The other principal components (second to tenth) together account for a much smaller share of the 14
The autocovariance function up to eight lags has been used in the construction of the multivariate spectral matrix. The data has been standardized to have unit variance before estimating the multivariate spectra. 15 Denoted as F (w) the co-spectrum of the data, the i (w) spectrum of the ith dynamic principal component is the ith largest eigenvalue of the F (w) at each frequency w: The P relative contribution of the ith dynamic principal component is given by i (w)= i i (w): 16 The height of the spectrum at frequency zero is a non-paramteric measure of the persistence of a time series.
17
variance than the …rst principal component. These principal components are equally relevant at all frequencies, as indicated by their ‡at patterns in the Figures17 . On the basis of these results, we opt for a factor model of the sectoral in‡ation series that admits a single common shock. Retaining the assumption of linearity, we model the subindex in‡ation as the sum of two components, one common and one idiosyncratic as: yit =
0i
+
it
+
it
with i = 1; :::; N
(8)
where it is the common component, which equals i (L)ut when one assumes only the possibility of one common shock ut , and it is a stationary idiosyncratic component, orthogonal to the common one. Therefore i is a lag polynomial term which represent the way in which the common shock a¤ects the yit process. Having provided supporting evidence relative to the presence of a common shock, we are in the position to assess whether the propagation mechanism of the common shocks in the cross-section of price sub-indices is homogenous across items.18 In other words, we check whether we can reject that i (L) = (L) 8i. Still resorting to spectral analysis, we study the covariance at di¤erent frequencies between the …rst dynamic principal component and the 404 series, as measured by the cross-spectrum. We focus on three frequencies: zeros, =6 (three years periodicity) and =2 (yearly periodicity) respectively for the sake of space. Figure 6 reports the distribution of the cross-spectrum values at the three selected frequencies. We interpret the di¤erences in the mode and the shape of the histograms across the three frequencies as a strong indication that there is large heterogeneity in the way common shocks get transmitted across sub-sectors in‡ations. The following section models this microeconomics level heterogeneity. 17
Even if the static factor analysis does not directly provide indication on the number of common shocks present in the cross section, see (Forni et al., 2000). 18
Here we go beyond the analysis of Clark (2003), which does not investigate the role of heterogeneity at the micro level for the persistence of aggregate in‡ation.
18
5
The model and its estimation
5.1
The model
The quarterly rate of change of each sectoral price sub-indices is assumed to behave accordingly to the following parameterization of the dynamic factor structure in (8): yit =
0i
+
=
0i
+
with ut i:i:d: (0; 1) and lag operator satisfy i (z)
6= 0;
i (L)ut i (L)
Ai (L)
ut +
i:i:d: (0;
it
i (z)
+
i2 ).
(9)
it i (L)
it ;
Ai (L)
The above polynomials in the
6= 0; Ai (z) 6= 0 for j z j
1;
where i (L)
= (
0i
+
Ai (L) = (1 i (L)
= (1 +
1i L
+
1i L 1i L
2i L 2i L
+
2i L
2
+
2
2
3i L 3i L
+
3i L
3
+
3
3
4i L 4i L
+
4i L
4
4
4
);
);
):
The only restriction that this implies is that each of the above polynomials is at most of the fourth order, while lower order may also be obtained since some coe¢ cients can be estimated to be zero. Therefore, each yit behaves as a stationary ARM A(pi ; qi ) with a possibly non zero mean, where 1 pi ; qi 4. Note that we are imposing that the common part, involving the ut , and the idiosyncratic part, involving the i;t , have an identical autoregressive structure. Moreover, for sake of simplicity, we are also assuming that both the common and the idiosyncratic component have an MA component of identical order qi , implying iqi +1 = iqi +1 = 0 and so on. These assumptions simplify greatly the estimation procedure and, at the same time, do not appear overly restricting in terms of the implied dynamics. On the other hand, we allow full heterogeneity of all coe¢ cients so that, for example, 0i ; can vary for each i = 1; :::; N .
5.2
Estimation strategy 19
Given a sample (y1 ; y2 ; ::::; yN ), with yi = (yi1 ; ::; yit ; ::; yiT )0 , we estimate the parameters by means of a multi-stage procedure as follows: (i) …rst, for each unit i, we estimate an ARMA(pi ; qi ) with non-zero mean but without distinguishing between the common and the idiosyncratic component. In fact the sum of two moving average components, i (L)ut
+
i (L) it
Bi (L)zit ;
also turns out to be an M A(qi ) for a qi -th order polynomial Bi (L) and an innovation sequence zi;t . However, we do not need to specify the coe¢ cients of Bi (L) as a function of the coe¢ cients i (L); i (L), in order to obtain consistent estimate of 0i and of the coe¢ cients of Ai (L); ^i (L)^ (ii) second, we average across i the estimated MA component B zi;t . (N ) This yields an estimate x bt of ! N 1 X (N ) xt i (L) ut ; N i=1 where the approximation improves as N grows since, as N ! 1, the idio(N ) syncratic component vanishes. Then we …t an M A(qi ) to the estimated x bt and obtain an estimate of the common innovation u^t ; (iii) third, using the u^t as an (arti…cial) regressor, we …t a M AX(qi ; qi ) process (an MA process with exogenous regressors) to each Bi (L)zit , in order to obtain consistent estimate of the coe¢ cients of i (L) and i (L). Such procedure, although clearly ine¢ cient, does not require any distributional assumption. Implicitly it requires that both N; T to diverge to in…nity with N=T decreasing toward zero, or at most with N; T diverging at the same rate. Estimation of each of the ARMA and MA processes is carried out using the Kalman …lter. The order qi ; pi of the models are estimated in stage (i) based on the Akaike criteria. The codes, available from the authors upon request, are written in MatLab.
5.3
Estimation Results
Given that the full model comprise more than six thousand parameters (15 for each of the 404 time series), this section comments on a selection of the most relevant ones. 20
Figure 7 shows the estimated time series of the common shock ut . The estimated shock appears su¢ ciently white, with a non-signi…cant low order autocorrelation as well as with non-signi…cant ARCH e¤ects, corroborating the i:i:d: hypothesis. We then report in Figure 8 the cross-sectional distribution of a commonly used measure of persistence, namely the signed modulus of the maximal autoregressive root. We did sign such modulus so that we could distinguish between the e¤ect of a negative root from a positive one and also consider the e¤ect of complex roots. The results indicate that the distribution is very dense near unity but it has also a long tail to the left, indicating some heterogeneity in the propagation mechanism. This has sound implications in term of the e¤ect of aggregation. In particular, it implies that the degree of memory of the CPI in‡ation (the aggregate of the yit ) appears as a sort of long memory. We further elaborate on this point in the next section. Figures 9 and 10 compare the distribution of the …rst loading of the common shock, which is a measure of the size of the e¤ect of the common shock in the individual time series, with the cross-sectional distribution of the standard deviation of the idiosyncratic components. The results clearly indicate that the idiosyncratic volatility i is substantially larger than the common shock volatility, in fact ten times larger so that is markedly dominates at micro level. The median of the distribution of the estimated …rst loading is 0:04 whereas we obtain 0:48 for the standard deviation of the idiosyncratic component. This con…rms strikingly that most of the variance of sectoral prices is indeed due to sector speci…c shocks.
5.4
Which CPI sub-indices have the most persistent in‡ation rates?
This section identi…es the sectors of which the response to the common shock is the most persistent. As indicated in section 2, this identi…cation is important because the persistence of the aggregate is mainly driven by the (potentially few) sub-indices which in‡ation process admits the largest roots. Given that our data comprises 404 series, we cannot, for the sake of space, report a full list of all the sub-indices of which the price are the most persistent. Moreover, the de…nition of the sub-indices entering the national CPIs are not 21
strictly comparable. We therefore simply provide the country and Main CPI groupings origin of the most persistent in‡ation processes in Table 6. The table describe the proportion of in‡ation series of each of the group for which the largest root the ARMA that loads the common shock is larger than 0.9 and 0.95 respectively. From this …rst description of the series, the most striking conclusion is that there is neither a country nor a group of CPI items that appear to concentrate a sharply higher proportion of the most persistent in‡ation series. Looking at the in‡ation processes with a largest root superior to 0.9, Italy and the Non-energy industrial goods appear as the most frequently persistent groups. However, this picture changes for the 111 in‡ation series with a largest root superior to 0.95. Only the unprocessed food appear as consistently less frequently among the most persistent sectors. These results should however be taken with caution as …ner groupings of the CPI items may reveal that some parts of the Services or of the NEIG appear more persistent consistently across countries. We leave this further investigation for future research though.
22
6
Persistence of aggregate in‡ation
Given the estimates of the micro parameters, we are now in the position to replicate the exercise performed in Section 2 on the case of the simple AR(1) model and have a better description of the e¤ect of aggregation in terms of the implied degree of persistence of the aggregate series. To this end, we will proceed in two di¤erent ways: (i) …rst, reconstruct the aggregate as an exact sum of the individual micro time series and in this way we exactly recover the contribution of the common shocks to the aggregate time series and to aggregate persistence; (ii) second, we exploit the theoretical link between the distribution of the largest autoregressive root and the autocovariance structure of the aggregate to infer the dynamic properties of the latter, similarly to the exercise of Section 2. We start by constructing the aggregate data as a weighted average of the individual subindex and using our estimates of the model in (9) one gets: Yn;t =
n X
wi
yit
wi
b0i + ut
i=1
=
n X i=1
b0 + b (L)
n X
wi
i=1
ut + bt
b i (L)
bi (L) A
+
n X i=1
wi
bi (L)
bi (L) A
it
where the wi are the …xed CPI weights. So the aggregate in‡ation is decomposed into two components, one associated with the common shocks, ut ; and its propagation mechanism, b (L); and a second associated with the micro idiosyncratic process, t : Figure 11 shows the reconstructed aggregate, Yn;t ; versus its common component, b (L) ut : There is an high correlation between the two components, around 0.76, but the former is clearly more volatile pointing to the fact that the idiosyncratic component t is still relevant in the aggregate. We however stress that given that the idiosyncratic component has little persistence, b (L) ut constitute a new measure of core in‡ation that can be of direct relevance for monitoring and forecasting in‡ation. 19 19
Again, for the sake of space, we leave the compararison of this new measure of core in‡ation with the ones already available for future research. See Camba-Mendez (2003) for a recent review of the literature on core in‡ation and the appeal of such measures of underlying in‡ation for central banks.
23
The aggregate propagation mechanism, b (L); is a weighted average of the propagation at micro level, b i (L) and its estimates can be used to recover the autocovariance structure of the component of the aggregate in‡ation that is driven by the common shock, i.e.: b (L) ut : Figure 12 shows the autocovariances of Yn;t and the b (L) ut , while Figure 13 reports the average autocovariance of micro process.20 The autocovariance of the reconstructed common components tracks remarkably well the covariance structure of the aggregate data, in particular in term of its decay. The fact that the variance of the aggregate in‡ation is larger than the one of the common component, indicates that the component associated with micro idiosyncratic noise is still relevant in the aggregate data, even if it seems to have little or no dynamic structure. More remarkable is the contrast of the results of Figure 12 with the one of Figure 13. Average contemporaneous variance of the micro data is an order of magnitude larger than the one of the aggregate data or of the common component in the aggregate data but it decays to zero very quickly. Finally, Figure 14 compares the autocorrelation of the aggregate, the one of the common component and the average autocorrelation of the data. The same conclusion emerges, the common component is the main driver of the dependence of the aggregate data. The above discussion has been based on the analysis of the exact autocovariance structure of the common component, given that in the exercise at hand we have the knowledge of all the relevant micro process. However similar conclusion can be inferred by analyzing the cross-sectional distribution of the maximal root of the autoregressive part Ai in the same manner as discussed in Section 2. Figure 15 shows the nonparametric estimate of such distribution. The following features emerge. First, the support of the distribution is [ 1; 1]. Only ten units exhibit an estimated value greater than 1. Second, the distribution appears uni-modal, with the great majority of individuals displaying a positive auto-regressive coe¢ cient and a mode of 0.85. Third, 186 time series (47 %) have the maximal root larger than 0.90 and 111 time series (27 %) larger than 0.95. Finally, the behavior near unity of such 20
Two graphs were needed given the di¤erent scale of the results.
24
distribution can be approximated as f^( )
c(1
)
0:12
as
!1 ;
implying that our estimate of the q as in Section 2 is equal to 0:88: Following (7) in Section 2 and the results in Za¤aroni (2004), it is possible to show that the autocorrelation function (ACF) of the common component of the aggregate, b (L) ut ; as a consequence has a decay that satis…es: cov( b (L)
ut ; b (L)
ut+k )
ck
0:76
as k ! 1
(in the long memory jargon, this would imply a parameter of long memory of d = 0:12): Therefore, the common component of the aggregate in‡ation appears to be a stationary but long memory process. Therefore, the ACF decays toward zero as a power law (very slowly), and thus markedly di¤erent from the behavior of the individual income processes. Moreover, we estimate the similar memory parameter using now the aggregate reconstructed data. The estimate of the memory parameter appear very close to the one recover by the micro structure, with cov(Yn;t ; Yn;t+k )
ck
:92
as k ! 1:
with standard deviation of the estimate of 0:24 (in the long memory jargon, this would imply a parameter of long memory of d = 0:04):21 The above results are framed in term of autocovariance function and they show the tight link between the micro dynamic and macro one. However to better highlight the e¤ect of aggregation on persistence of shocks we consider the following exercise. We construct a mean ARMA process, as the ARMA process having roots equal to the mean roots of the 404 estimated ARMAs and we then compare the autocovariance and impulse response function with the one of the common component, b (L) ut : The idea of the exercise is to see the aggregate response to shock in case the propagation mechanism is equal across agents, i.e., to quantify the e¤ect of heterogeneity and aggregation. Figures 16 and 17 show the autocovariance function of this mean ARMA with 21
Hubrich and Hassler (2003) estimate a long memory parameter d of 0.19 from monthly EU12 HICP in‡ation over the same sample period.
25
the one of the common component, b (L) ut ; and the impulse response for the mean ARMA versus the one implied by b (L): The exercise is quite instructive. In the case of a homogeneity of the micro propagation mechanism, after 20 periods a shock ut would be completely absorbed, while in reality, due to the presence of heterogeneity and of some very persistent micro units, after 20 around 30% of the original shock has not been absorbed. To conclude, we can claim that the analysis of the micro determinants of the aggregate in‡ation support the view that aggregate in‡ation in our sample period can be regarded as a stationary but long memory process. We have shown that starting from very simple ARMA process at micro level we have been able to properly reconstruct the dynamic property of the aggregate. In doing this, we have shown that the micro volatility and low persistence can be squared with the aggregate smoothness and persistence.
7
Conclusion
In this paper, we build on the heterogeneity in the in‡ation dynamics across CPI sub-indices and investigate the role played by cross-sectional aggregation in explaining some of the di¤erence observed between micro and macro evidence regarding in‡ation dynamics. We focus in particular on the link between CPI sub-indices and the aggregate CPI. We estimated time series models for 404 sub-indices of “items/sectors” of euro area CPI between 1985 and 2003. We …tted ARMA processes at micro level distinguishing the propagation mechanism of the common and idiosyncratic shocks. Our …rst result is that the propagation mechanism of shock at micro level across sectors is very heterogenous. This heterogeneity implies a non trivial link between sectoral and aggregate persistence. We perform an aggregation exercise and compute the aggregate persistence as a function of the 404 sectors persistence. The persistence that we obtain through this aggregation exercise mimics remarkably well the persistence observed in the aggregate in‡ation. Our model therefore reconciles the high volatility and low persistence, observed on average at the level of sectoral in‡ation with the smoothness and persistence of the aggregate. Finally, the model also provides a new measure of core in‡ation which is de…ned as the 26
e¤ects of the common shock on the disaggregate in‡ation rates. Altogether, this paper demonstrated the importance of heterogeneity and aggregation for understanding the persistence of in‡ation at the macroeconomic level. We leave the design of stylized models of the business cycle that can be consistent with both heterogeneity at the micro level and the implied persistence at the macro level for future research.
27
Data appendix The French CPI sub-indices The French data consist of 161 monthly sub-indices, available since 1972. We thanks to Baudry and Tarrieu (2003) who back dated the INSEE CPI (1998 base year). The later is publicly available since 1990 for 148 sub-indices while the prices of 13 items enter the CPI basket only after this date. Baudry and Tarrieu used the 1980 base year CPI to build 141 Laspeyres chained indices all the way to 1972 while 7 prices are available only since 1987. A full list of the sectors is available in Baudry and Tarrieu (2003) in French and in Bilke (2004) in English. The German CPI sub-indices The German data consist of 100 prices of the 3 Digits classi…cation of HICP sub-indices. These prices are available monthly from early 90s to 2004. To back date the HICP, we used about 150 3-digits sub-indices of the 1990 base year CPI data, which are available between 1985 and 1995. The procedure is the following. First, Agresti and Eiglsperger (2004) identi…ed 83 matches between one HICP sub-index and one or more CPI-base year 1990 sub-indices. Second, we regress the former on the latter for the 53 overlapping observations between January 1991 and May 1995. The regressions are based on y-o-y in‡ation rates. The …t is very high (adjusted Rsquare superior to .95) for a large majority of cases, where apparently the same items are surveyed in the two database. In ten cases we do not proceed to the back dating because the …t of the regression is poor (for adjusted R-Squares inferior to 0.65). Finally, the back dated HICP between 1985 and 1990 are then obtained by applying the estimated coe¢ cients of the above mentioned regressions to the corresponding CPI-base year 1990 sub-indices. This left us with 73 time series since 1985. We added other 27 series which start only 1991. The Italian CPI sub-indices The Italian data consist of 167 monthly indices underlying the Italian CPI constructed by ISTAT. The data, kindly provided by the Bank of Italy Research Department, start in 1980 and were rebased to 1995 equal to 100. The full list, and corresponding description, of the sectors is available upon request from the authors.
28
Seasonal adjustment The data were seasonally adjusted with TRAMO-SEATS. The main advantage of this routine is that it is used regularly for the o¢ cial HICP statistics, and that it allows for integrated seasonal components. This latter aspect is particularly important for our data because EUROSTAT has required that, from the mid-1990’s on, the biannual sales campaigns are re‡ected in the HICP sub-indices of interest.22 In the seasonal adjustment procedure, we utilized the longest available monthly series, i.e. 1972-2004 for France, 1981-2003 for Italy and 1985-2004 for Germany. The monthly price level has been transformed into quarterly average and the analysis has been performed on quarter on quarter in‡ation rates. We use this frequency both to limit the noise of the monthly series and to compare directly our results with the business cycle literature. Also, the most prominent studies of sectoral in‡ation in the euro area (e.g. Lünnemann and Mathä, 2004) and elsewhere Cecchetti and Debelle (2004).
Data cleaning We further clean our data according to the following steps. First, we eliminate all the series that start only in the mid 1990’s. This is because these series do not have enough degrees of freedom to carry out the estimation of ARMA models. We exclude from our sample 8 French and 9 German subindices that are available only after 1995, 1998 or 2000. Second, we eliminate all the series which are adjusted at rare discrete steps. The typical such series include the price of Tobacco or mailing services. We exclude 6 German, 4 French and 12 Italian sub-indices of this type. Altogether we keep 404 sub-indices out of the 444 available in the initial dataset, 377 of which are available from 1985 Q1 to 2003Q4. A last step before, the estimation is to corrections of outliers in the in‡ation series. As a matter of facts, many sectoral sub-indices are subject to one, two or three level shifts in the period. These translate into unusually large observations of in‡ation. Some of these level shift may indicate re-basing the basket of the sub22
This may a¤ect in particular our French and our German data. As a matter of facts, the Banca d’Italia keeps track of the prices both with the new method and consistently with the historical data. And we used the historically consistent series.
29
index, other are true changes in the price level. While we cannot distinguish between them, we …lter out the outliers by replacing observations that are more than 3 standard deviations away from the time series mean of each in‡ation series by a local median observation. This outlier correction mainly eliminates discrete shifts in the levels of the price indices.
30
31
Figures
Figure 1: Aggregate y-o-y in‡ation rates
32
Figure 2: Aggregate q-o-q "o¢ cial" and reconstructed in‡ation rates
33
Figure 3: Characteristics of 404 sectoral in‡ation rates
34
Figure 4: Spectra of the …rst ten dynamic principal components.
Figure 5: Normalized spectra of the …rst ten dynamic principal components
35
Figure 6: Histogram of the cospectra with the …rst principal component frequencies 0, pi/3 and pi/2.
36
Figure 7: Estimated common shock ut :
Figure 8: Histogram of largest autoregressive root
37
Figure 9: Histogram of …rst loading of the common shock,
Figure 10: Histogram of the idiosyncratic standard deviation
38
0i
i:
Figure 11: Reconstructed Aggregate and its common component
Figure 12: Autocovariance of aggregate in‡ation and of its common component
39
Figure 13: Average autocovariance of the subindexes in‡ation process
Figure 14: Autocorrelation of the aggregate, of the common compronent and the average autocorrelation of the micro data.
40
Figure 15: Nonparametric estimate of the distribution of the largest autoregressive root
41
Figure 16: Autocovariance of common component and of mean ARMA
Figure 17: Impulse response of common component and of mean ARMA to a common shock 42
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45
