Estimating Monetary Inflation

Transcription

1 Estimating Monetary Inflation Fredrik NG Andersson Lund University Preliminary and Incomplete: Please do not quote or cite without the author s permission. Abstract Inflation targeting has become a popular monetary policy regime during the last 15 years. The central bank does however not determine the inflation rate alone. Nonmonetary events such as shocks, fiscal policy decisions etc can create short term fluctuations in the inflation rate for which the bank is not responsible. The real time problem for the bank is therefore to estimate a monetary inflation rate for which it is responsible. This paper introduces five wavelet based signal extraction algorithms to separate nonmonetary events from the underlying monetary inflation pressure. Compared to commonly used estimates no items are excluded from the price index and relative price changes, which often are ignored, are hence accounted for. Applied to consumer price data from the United States, the United Kingdom and the Eurozone an evaluated against Excluding Food and Energy and a trimmed mean estimate using three criteria, transparency, similarity of mean and forecasting ability the proposed wavelet based estimates have nice properties. Keywords: Monetary inflation, Signal to noise, Wavelets JEL-codes: E31, C82 Fredrik NG Andersson gratefully acknowledge funding from the Jan Wallander and Tom Hedelius foundation; Project number P :1 Department of Economics, Lund University, Box 7082, Lund, Sweden ngf.andersson@nek.lu.se Telephone: +46 (0) Fax: +46 (0)

2 Introduction Maintaining the purchasing power of money has historically been none of the main objectives for the central bank. With the introduction of inflation targets this objective has been made explicit in many countries such as the United Kingdom, the Eurozone, Australia and Sweden. Although inflation is commonly measured as the percentage increase in the consumer price index (CPI), headline inflation (CPI-inflation) and monetary inflation is not necessarily the same, at least not in the short run. Various forms of shocks, seasonal effects, fiscal policy changes etc. causes short term fluctuations in the headline inflation rate. Since the bank is not responsible for these nonmonetary events and cannot affect them, they are viewed as noise by the central banker. The real time problem for the bank is the estimate an underlying monetary inflation pressure for which it is responsible using the noisy price data available. Various estimation approaches have been proposed. These, however, often ignore medium to long run relative price changes which can cause a bias in the monetary inflation estimate. This paper introduces five wavelet based signal extraction algorithms to extract the monetary inflation signal from the noisy data. De-noising each price series individually and then weighting the items together using the original budget weights solves the potential problem medium to long run relative price changes can cause. Applied to consumer price data from the United States, the United Kingdom and the Eurozone and evaluated against two commonly used estimates Excluding Food and Energy and a trimmed mean estimate these new wavelet based perform well. From a microeconomic perspective headline inflation can be viewed as measuring the cost of living, how much it costs to maintain a certain utility level. From a monetary policy perspective inflation could be tied to the purchasing power of money rather than the cost of living. Fisher (1922) argues that since also other prices than consumer prices are affected by 1

3 the monetary policies, an inflation index where everything purchased or purchasable is included yields a better estimate of the monetary inflation pressure. Designing such a broad price index, although desirable, is difficult. Many inflation targeting central banks such as Bank of England, European Central Bank and the Swedish Riksbank therefore uses a limited set of consumer price index instead. In the long run, however, if inflation is exclusively caused by excess money growth the difference between a broad all including price index and the consumer price index should be quite small. Although it is reasonable to assume that the cost of living and the purchasing power of money is closely linked in the long run, short term fluctuations can cause headline inflation and monetary inflation to diverge in the short run. Nonmonetary events such as shocks, seasonal effects and fiscal policy changes etc. causes variation in headline inflation for which the bank is not responsible. Since the central bank can only affect monetary inflation it must disregard those short run fluctuations and base its policy decisions on an estimate of monetary inflation rather than headline inflation. Estimating monetary inflation can be viewed as a signal extraction problem. The available price data includes both information about the monetary inflation signal and the noise caused by the various shocks. The problem is to extract the inflation signal from the noise. For this purpose various estimators have been proposed. Most of them are based on the assumption that all prices can be divided into two components. One related to monetary policy and the other an idiosyncratic shock component with expected value zero. A commonly used estimate is the Excluding Food and Energy (XFE) estimate. It is assumed that different items of the price index contains different signal to noise ratios. A volatile item is assumed to contain a lower signal to noise ration than a less volatile item and by excluding the most volatile items, in this case food and energy prices, the noise level is reduced and a better estimate of monetary inflation can be obtained. 2

4 Diewert (1995) takes that argument one step further and discard the original budget weights all together and creates new weights based on the relative volatility in each item. The new inflation index, the neo-edgworthian index, gives items with a low volatility a high weight and volatile items a low weight. An advantage with the neo-edgworthian index compared to XFE, is that no items have been removed so no information has been lost. These estimates assume a symmetric distribution of the idiosyncratic component. The observed cross sectional distribution of the price changes is however often skewed. Bryan and Pike (1991) argue that the median price change should be a better estimate when the distribution of the prices is nonnormal. Bryan and Cecchetti (1994) develop a model where firms face menu costs and only adjust their prices if the shocks are greater than some threshold level. If not all firms adjust their prices at once this can cause a skewness in the cross sectional distribution of the prices. As a solution to the problem they propose a limited influence or a trimmed mean (TM) estimator. By excluding items with the largest and the smallest price changes during a period a normal distribution can be created approximately. The difference compared to XFE is that the TM estimate does not necessarily remove the same items from the price index every period. These estimates treat relative price changes as short term movements that have expected value zero and they are therefore included in the idiosyncratic component. However, it may be desirable to also allow for medium to long run relative price changes as well. These can be caused by long term changes in preferences or varying productivity growth between different sectors of the economy. Since these are not short run movements they do not have expected value zero at every point in time and must therefore be dealt with in the estimation process. Excluding certain items from the price index or removing the original budget weights can cause a bias in the monetary inflation estimate. Clements and Izan (1987) introduce the 3

5 possibility of medium to long term relative price changes. In their model, however, these are not time dependent and are assumed to continue for ever. This paper introduces a new estimator of monetary inflation. It is assumed that all prices consist of three parts. One related to monetary policy, one capturing medium to long term relative price changes (which can be time dependent) and an idiosyncratic shock component with expected value zero. Monetary inflation is estimated by first applying a wavelet denoising algorithm to the data. Under the assumption that the idiosyncratic shocks are normally distributed for each item of the price index, these algorithms remove the shocks. The relative price changes are then removed by weighting the price series together using the budget weights. Due to the household s budget restriction all medium to long term relative price changes cancel out in the weighting process. There are two advantages with this approach. First, the possibility of medium to long run relative prices can be introduced. Secondly, in the long run headline inflation and estimates of monetary inflation are equal to each other by construction. Since it is assumed that headline inflation in the long run is an unbiased estimator of money induced inflation this is an important feature. Using consumer price data from the United States, the United Kingdom and the Eurozone covering the period 2001:4-2006:7, these new wavelet based monetary inflation estimates are compared to two commonly used estimates XFE and TM. Following the discussion in Rick and Steindel (2005) and Wynne (1999) three criteria are used to evaluate them transparency, similarity of means and forecasting ability. On all three criteria the wavelet based estimates perform very well. Although novel in this context the wavelet based signal extraction algorithms presented in this paper are not new. Previously they have been applied in fields such as for example mathematics and image processing. See for example Donoho and Johnson (1994) and Jansen 4

6 and Bultheel (1999). For a discussion related to economics see for example Gencay and Selcuk (2001) or Crowley (2007). The rest of the paper is organized as follows. Section 2 discusses trend inflation and the evaluation criteria. Section 3 gives a brief overview of wavelets, Section 4 presents the denoising algorithms and they are applied to the empirical data in Section Trend inflation Let π it represent inflation in item i=1,,i of the price index at time t=1,..,t. Assume it can be decomposed into three components. A common inflation component ( π ) caused by the central bank s monetary policies, a medium to long run relative price change component 1 (r it ) and a short run idiosyncratic shock component (x it ) with expected value zero and varianceσ. The data generating process can be written as t 2 i π = π + r + x (1) it t it it The relative price changes can be both short, medium and long run processes. Long run changes in preferences or varying productivity growth between different sectors of the economy can cause relative price changes which last for more than one period. Thus r it does not necessarily have expected value zero at every point in time. However, due to the household s budget restriction it must hold that I j= 1 b 0 (2) it r it where b it is the budget weight for good i at time t. It should be noted however that I it j= 1 r 0. (3) 1 Since the consumer price index only updates the budget weights once a year short term relative price changes are assumed to be random shocks and are included in the idiosyncratic shock component. 5

7 Headline inflation, H π t, is the weighted inflation rate of all items I H π = b (π + r + x ). (4) t j= 1 it it it it The expected value of headline inflation is then given by I I H [ t ] E[ bitπ it ] = E bit ( π t + rit + xit ) I [ ] = E[ b ( π + r )] E[ π ] E π = it t it = i= 1 i= 1 i= 1 t (5) Since the relative price changes disappear when the items are weighted together using the budget weights expected headline inflation equals monetary inflation. In the long run, headline inflation is therefore an unbiased estimator of monetary inflation. However, if the distribution of the idiosyncratic components vary between the items of the index, the cross sectional distribution of the shocks can be skewed at a given point in time. Therefore in the short run headline inflation can be a biased estimator of monetary inflation. Depending on the nature of the shocks it can either be biased upwards or downwards, this bias can also be time varying. The problems caused by medium to long run relative price changes and a skewed cross sectional distribution of the idiosyncratic components can be solved by removing the shocks from the individual price series first and then weigh together all the prices using the budget weights. Once the shocks are removed the skewness problem disappears and weighting each item using its budget weight eliminates all relative price changes. Once the shocks have been removed Eq. (5) is thus an unbiased and efficient estimator of monetary inflation. 4.1 Evaluating monetary inflation estimates There is no commonly agreed upon method to extract a monetary inflation signal from noisy data and the problem of which method to use is treated as an empirical problem. Following, in 6

8 spirit, Rich and Steindel (2005) and Wynne (1999) this paper uses three criteria; transparency, similarity of mean and forecasting ability, to evaluate the different estimates. It is likely that the household is more familiar with cost of living inflation than monetary inflation from its every day activities. For the public to understand the bank s policies and for the bank to gain support for its policies it is important that the public understands how the monetary inflation estimate is created, what it measures and how it is different from headline inflation. In other words there has to be a large degree of transparency surrounding the estimate. If the estimate is not transparent it can become difficult for the policy maker to build support for its policy decision during periods when headline and monetary inflation deviates from each other. A further test for a monetary inflation estimate is whether it has the same mean as headline inflation. In the long run when the shocks are zero, headline inflation returns to monetary inflation. Hence, the short term variations in the inflation rate are merely fluctuations around the monetary inflation rate and once the effect of these shocks has died out headline inflation equals monetary inflation. Thus the mean in headline inflation should equal the mean in the estimated monetary inflation. Finally, including the trend estimate in a forecasting model of future headline inflation should reduce the forecasting error. This follows again from that the expected value of the shocks are zero which implies that headline inflation returns to monetary inflation once the effect of the shock has disappeared. Once future policy decisions have been accounted for, present monetary inflation and future headline inflation should be closely related. 3. Wavelet Transformation Time series are often analyzed in the time domain. This is convenient when analyzing its behaviour at specific points in time and how this relates to the behaviour of other time series. It may, however, also be interesting to study the series behaviour at different frequencies. This 7

9 facilitates, among other things, the isolation of seasonal effects and the distinction between short and long run fluctuations. It is easier to identify these frequency patterns and related them to frequency patterns detected in other series in the frequency domain than the time domain. Fourier transformations are commonly used to convert the time series from the time domain to the frequency domain. This transformation is achieved by representing the series using a set of sine and cosine functions at different wavelengths (frequencies) and amplitudes. Sine and cosine functions are, however, homogenous over time in the sense that the same peak-trough-peak pattern of amplitudes for a given frequency band keeps reproducing itself along the time axis. For a time series affected by non-recurring events such as a structural break or temporary shocks this homogeneity assumption may be too restrictive. A wavelet transformation is similar to a Fourier transformation in that it converts the time series from the time domain to the frequency domain. In a wavelet transformation, however, the homogeneity assumption is relaxed so that the amplitude for a given frequency band can change across the time axis. This is achieved by dividing the time axis into smaller segments which are analyzed individually. A discrete wavelet transformation (DWT) transforms the data not into individually frequencies but frequency bands. These frequency bands are often called scales in the wavelet literature. A low scale captures high frequency fluctuations and a high scale the low frequency fluctuations. There is in other words an inverse relationship between frequency and scale. Unlike the Fourier transformation the width of the frequency bands are not the same for all scales. Scale 1 (which captures the highest frequency component) covers the frequencies ¼ to 1 1 ½ and scale j the frequencies and j j. In the frequency domain the time series (x t ) is represented by the wavelet coefficients (v J, w jk ). The time series can be restored to the time domain using the wavelet functions (φ t, ψ jkt ) 8

10 x t J 2 1 = v φ + w ψ t=1,,t, J=ln(2)/ln(T) (6) J t j= 1 k= 0 j jk jkt φ t captures the time series lowest frequency component and is sometimes called the father wavelet. ψ jkt is referred to as the mother wavelet and it is used to capture the time series frequency information at scale j. A restriction imposed by the DWT is that the number of observations that can be included in the transformation must equal 2 J where J is an integer. If this restriction isn t fulfilled observations must either be removed or added until the restriction is met. Scale (frequency band) j is in the frequency domain represented by the wavelet coefficients w jk. Since all the frequency components are orthogonal to each other an individual scale can be transformed to the time domain independently of the other scales. When a scale is represented in the time domain it is commonly referred to detail (d j ) instead of a scale. This distinction between scale and detail is usually made to distinguish between which domain the frequency component is represented in. The detail is defined by j 2 1 j = w jk k = 0 d ψ (7) jkt The time series lowest frequency component which in the frequency domain is represented by v J can in the time domain be represented by s = φ (8) J v J t s J is commonly referred to the J-th level smooth. Eq. (6) can thus be written as x = s J + d j J j= 1 (9) Sometimes it is not necessary or interesting to decompose the time series into all possible frequency components. Assume the analysis only requires that Q<J scales are extracted from 9

11 the data. The remaining J-Q details and the J-th level smooth can then be combined into one component. It is referred to as the Q-th level smooth and is defined by s Q = s J + J d j j= Q+ 1 (10) The scales (or details) are orthogonal to each other which imply that the variance of the time series is the sum of the variance of the J-th level smooth and all the details. J var( x) = var( s J ) + var( d j ) (11) j= 1 There are many different wavelet specifications available such as the Haar and the Daubechie wavelets. The choice of wavelet function can influence the frequency decomposition. These differences are however often small but the robustness of the decomposition could be tested by applying more than one wavelet specification. A complication which some wavelet specifications encounter, such as the Daubechie but not the Haar wavelet, is that the end of the sample the transformation requires more observations than are available. This problem can be solved by either circularly connecting the boundary points so that observation 1 also represents the unobserved observation T+1, or mirroring the data so that observation T-1 also represents observation T+1. The wavelet coefficients affected by the boundary assumption they should be analyzed carefully. 4. Wavelet methods for removing noise Assuming the idiosyncratic shocks (x it ) are normally distributed these can be removed asymptotically by applying a wavelet based de-noising algorithm. Following the discussion in Section 2, an estimate of monetary inflation and headline inflation should share the same long run developments. The price series long run path is caused by their low frequency components and the noise is located in the high frequencies, although some of the inflation signal may also be present at these frequencies. The wavelet de-noising algorithms are thus 10

12 applied to the price series high frequency component only while the low frequency component is assumed to contain a part of the signal and no noise. Depending on the denoising problem at hand the exact definition of high versus low frequency has to be determined on an individual basis and the robustness of the choice tested by comparing different choices. There are two commonly used sets of de-noising algorithms. The first set which assumes the signal (D t,) to be deterministic and the noise (ε t ) a random variable with mean zero. The other set assumes both the signal (C t ) and the noise to be random variables. The deterministic models can be represented by (12) and the random signal model by (13) where X, D, C and ε are (T 1), T being the number of observations. X = D + ε (12) X = C + ε (13) Neither D, C or ε are readily observable and must be estimated using either one of the algorithms discussed below. 4.1 Deterministic Signal A DWT of (12) yields the wavelet coefficients, ω l, l=1,...t. ω = WD + Wε = d + e (14) where W is the (T T) wavelet transformation matrix, d the deterministic component represented in the wavelet domain and e the noise also represented in the wavelet domain. Since the DWT is a linear transformation, d and e have the same properties as D and ε. The de-noising algorithm presented in Percival and Walden (2006, ch10) is based on minimizing the loss function (15) 11

13 ^ 2 2 m = X D + (15) γ mδ where D^ = W' J and J is the vector of wavelet coefficients that have been altered according to the de-noising algorithms which will be discussed later and δ is a penalty for including too many coefficients. The loss function is based on preserving as much of the original timeseries features as possible with as few wavelet coefficients as possible. As shown by Donoho and Johnstone (1994), choosing δ correctly will imply that all noise that is independent and normally distributed can be removed, at least asymptotically, and only the signal remains. As already mentioned the de-noising algorithm is only applied to the high frequency component. Assume the time series is decomposed into Q high frequency components and the remaining J-Q frequency components are defined as the low frequencies 2. The following denoising algorithms (16)-(21) are then applied to the wavelet coefficients representing scales 1,,Q. It can be showed that minimizing (15) is equivalent to setting all small wavelet coefficients related to scales 1,,Q to zero. The remaining wavelet coefficients related to the high frequencies can either be kept as they are (which is called hard thresholding) or they can be altered in some way to make the transition less dramatic between being removed completely by the threshold or being unchanged. For the hard threshold the de-noising algorithm is given by 0, if ωl δ ω l = (16) ωl otherwise Soft thresholding removes the small wavelet coefficients and reduces the value of the others. This algorithm is represented by (17) 2 Scales 1,,Q are thereby defined as high frequency and the Q-th level smooth as the low frequency component. This relates back to the definition in Eq. (9). 12

14 { ωl }( ωl ) + ωl = sign δ (17) where + 1 if ωl > 0 sign { ω l } = 0 if ωl = 0 (18) 1 if ωl < 0 and ωl -δ if ωl δ 0 l = + 0 if ω δ < 0 (19) l ( ω δ ) Using soft thresholding, coefficients which pass the threshold level are reduced towards zero, which makes the difference between being removed or included less severe. Mid thresholding is a combination of the soft and the hard thresholding algorithms. This algorithm sets small coefficients to zero, reduces the size of some medium sized coefficients and allows large coefficients to remain as they are. Large is here defined as twice the threshold level. This algorithm can be written as { ωl }( ωl ) + + ωl = sign δ (20) where sign{ ω l } is defined as in (18) and where ω ( δ ) = ( ω δ ) + l if ω 2δ l ω l 2 if 0 < ωl < 2δ + l (21) 0 otherwise 2 If the random noise in equation (12) is assumed to be i.i.d. and distributed N (0, ), then all noise can be removed asymptotically by defining 3 σ ε δ 2σ 2 log( N) (22) = ε 3 See Donoho and Johnstone (1994) 13

15 The variance of the noise can be estimated with a median absolute deviation estimator (MAD). Assuming that the first scale only contains noise this variance can be estimated through ˆ σ mad = median ω 1,0, ω 2, , K, ω T 1, 1 2 (23) where ω 1, k, k=0,,t/2-1, are the wavelet coefficients related to the first scale. The constant rescales the estimate to be an estimator of normally distributed white noise (Percival and Walden, 2006 ch10). 4.2 Random Signals If the signal is also assumed to be a random variable, the data generating process is given by (13). In the wavelet domain this is represented by ω = R + e (24) R is assumed to follow a sparse signal model R [ I = 1] = p and [ I = 0] = p d j, n ( j, n G j,t P j, t 2 = 1 I ) N(0, σ ) P 1 (25) p represents the probability that the signal is zero. When p is close to zero the signal is assumed to be zero most of the time and when p is close to one different from zero. The noise is distributed e d 2 = N(0, σ ) (26) j, t e This de-noising algorithm does not remove coefficients, as was the case in the deterministic model but rescales them based on a shrinkage rule determined by the signal s noise rations. The shrinkage rule is given by 14

16 ω shr j, n b = ω j, n (27) 1+ c where σ p σ G + σ G e 2 2 b = and c = exp( ω, /(2 )) j nb σ e (28) σ + σ (1 p) σ G e e An estimate ofσ can be obtained, as before, from the MAD estimator (23) and 2 e ˆ σ 2 ˆ σ ˆ ω = 1 p 2 2 σ e G (29) where N j G 2 * ˆ σ ω = ω * j, n, N = N j N n= 0 j= 1 (30) The exact choice of p is subjective and different values should be applied to test for robustness. 5. Empirical Analysis Monetary inflation is estimated for the United States, United Kingdom and the Eurozone using non-seasonally adjusted monthly CPI-data. The data is collected from the Bureau of Labor Statistics for the United States, Eurostat for the Eurozone and National Statistics for the United Kingdom. Due to redefinitions of CPI and the restriction imposed by the DWT, the analysis is based on 64 monthly observations between 2001:4 and 2006:7. For the United States the CPI is divided into 160 categories, the Eurozone 87 and the United Kingdom 84. As a comparison to the wavelet based estimators two commonly used estimators are also constructed, XFE 4 and a TM estimate 5. Two definitions of inflation are applied, monthly 4 Estimates of XFE are collected from the Bureau of Labor Statistics for the United States, Eurostat for the Eurozone and National Statistics for the United Kingdom They are not constructed by the author. 5 Specifically the TM estimate removes items corresponding to 16% of the budget. 15

17 inflation (31) and yearly inflation (32) each normalized to yearly values. If the trend estimate captures medium to long term inflation the trend estimates should be more or less the same irrespective of whether they are based on monthly or yearly inflation. ( ln( ) ln( CPI )) monthly inf CPI 1 (31) it = it it ( ln( ) ln( CPI )) 100 yearly inf CPI 12 (32) it = it it i denotes component i in the CPI. Three different definitions of high versus low frequency are tested. With T=64 a maximum of 6 frequency components can be extracted. If Q=4 it assumed that all shocks have died out after 16 months. If Q is increased to 5 it can take them up to 32 months to disappear and when Q=6 64 months 6. For the random signal model three choices of p are presented 0.99, 0.995, These are chosen after experimenting with p=0.5, 0.75 and A value below 0.99 resulted however in almost no de-noising at all. Furthermore the sensitivity of the results with respect to the chosen wavelet function is studied by using two different wavelet specifications a Haar wavelet and a Daubechie wavelet 7. The estimates of yearly monetary inflation are plotted in Figures 1-3, and monthly monetary inflation in Figures 4-6. Statistics of means and standard deviations for the individual series are available in Tables 1-4. The differences between the two wavelet specifications are in general small. Therefore the figures only present the estimates based on the Daubechie wavelet. The tables however include statistics for both the Haar and the Daubechie wavelets. [TABLES 1-4 ABOUT HERE] [FIGURES 1-6 ABOUT HERE] 6 All three choices are tested however they all yield more or less the same results implying that the noise is located in primarily in the first four scales. Therefore only the results for Q=5 are presented. 7 More specifically the Daubechie 4 wavelet. 16

18 In general most of the fluctuations contained in the first three frequency components are either removed or rescaled. This can be interpreted as fluctuations which after eight months have disappeared are primarily noise. For monthly inflation scale four (8-16 months fluctuations) also contain some noise however it also contains part of the signal. For yearly inflation the de-noising algorithms leaves this scale almost intact. This implies that fluctuations which have disappeared after a year are just noise in the data. How much noise and how much signal the different scales contain vary between the different items of the price index. For some items scale 1-3 are just noise, for some scale three also contains a part of the signal while others also have some noise in scale four. Since the nature of the shocks is so different cross the items it is not possible to remove it by simply applying some smoothing process to the data. Monetary inflation estimates based on smoothing headline inflation either removes too little of the noise or also a part of the inflation signal. The monetary inflation estimates are less volatile than headline inflation. In the United States the volatility is reduced from 4.44 in monthly headline inflation to between 0.7 and 2.38 depending on algorithm and wavelet function. The decline is similar for the United Kingdom and the Eurozone, from 3.64 to between and from 3.31 to respectively. Naturally since both XFE and TM are based on removing volatile items from the CPI these also reduce the volatility in the inflation estimate. However the decrease is in general smaller than for the wavelet based estimates. Furthermore the mean in XFE and TM deviate more from the mean in headline inflation than any of the wavelet estimators do. The greatest difference is observed for the United States where the mean in XFE is 0.7 percentage points below the mean in headline inflation. Since the sample is rather short the mean in the wavelet estimates also deviates from the mean in headline inflation although the difference are much 17

19 smaller than for XFE or TM, within percentage points from the mean in headline inflation. XFE and TM can at times deviate from headline inflation for long periods of time. In the United States they are constantly below headline inflation from late 2002 until the end of In the Eurozone all monetary inflation estimates and headline inflation are closely related until early Headline inflation and the wavelet based estimators then slowly increases while XFE and TM decreases slightly. From late 2004 until the end of the sample XFE and TM are about 0.5 percentage points below any of the wavelet based estimators. In the United Kingdom the differences between TM and the wavelet estimators are not as large as in the United States or the Eurozone. The TM estimate is slightly lower at the end of the sample but are otherwise indicating the same monetary inflation pressure. XFE however, deviate substantially from all other estimates. It is above yearly headline inflation from early 2002 until early 2005, in January 2003 almost twice as high as the other estimates. At the end of the sample the wavelet estimators, XFE and TM are indicating different yearly monetary inflation pressures. In the United States the wavelet estimators point at a monetary inflation pressure between 3 and 3.5 percent while XFE and TM are about 1 percentage point below, 2%-2.5%. In the United Kingdom the wavelet estimators have been indicating an increased inflation level since late 2004 and are approaching 2.5% at the end of the sample. XFE and TM are indicating if not decreasing monetary inflation at least a stable inflation level. In the Eurozone headline inflation and all monetary inflation estimators are very stable at the end of the sample. However, according to the wavelet estimates inflation is above ECB s target of 2% while they are fulfilling it according to XFE and TM. Monthly headline inflation is affected by a lot of noise in all three currency areas and thereby much more volatile than yearly inflation. The wavelet based estimators are however able to remove much of this noise and the differences between the monetary inflation 18

20 estimates based on monthly inflation are small compared to those based on yearly inflation. Although less volatile than monthly headline inflation both XFE and TM are still substantially more volatile the wavelet estimators. There are also great differences between the estimates depending on whether they are applied to monthly or yearly inflation. These results are not surprising since neither XFE nor TM de-noises the items which are included in the monetary inflation estimate. The empirical analysis is based on non-seasonally adjusted data and before either XFE or TM are estimated the seasonal effects should be removed. This is however not necessary for the wavelet estimators which removes the seasonal effects while de-noising the items. There are some differences between the different wavelet estimators. Hard thresholding is more volatile than any of the other estimators and it deviates slightly from the other wavelet estimates particularly for monthly inflation. The mean are however the same. Otherwise the differences are small between the different algorithms. There are also only small differences between the deterministic models and the random variable models. For yearly monetary inflation the differences are almost non existent. For the random model increasing p makes the monetary inflation estimate smoother. For monthly inflation there can be some small differences between the estimates at times, however they are almost identical for yearly inflation. 5.1 Forecasting The forecasts of headline inflation are based on Eq. (33). π ( + π (33) forecast H t+ h = α h + β h π t π t ) H t where h=1,6,12 is the forecasting horizon. This is the same forecasting model that is applied in Rich and Steindel (2005), Clark (2001) and Hogan, Johnson and Lafléche (2001) among others. The model does not include any policy variable and the forecasting horizons have been 19

21 chosen to reflect it. It is unlikely that monetary policy can affect inflation within a year so excluding a policy variable from the model will have very little impact on the results. For a longer forecasting horizon a policy variable should be included. Only yearly inflation is forecasted since monthly inflation contains too much noise to be forecasted. Forecasting monthly inflation is more an exercise in forecasting the noise than the inflation signal. The forecasts are based on a moving window technique. For an estimate of headline inflation at period t+h the inflation model is estimated using the observations up to period t. For a forecast of headline inflation t+h+1, the inflation model is re-estimated using observations up to period t+1. With this technique all information that would have been available is used but no future information is used. As a comparison to the monetary inflation estimates inflation is also forecasted using a random walk. Root Mean Squared Errors (RMSE) are presented in tables 5-7. [TABLES 5-7 ABOUT HERE] The wavelet based estimates of monetary inflation have in general a smaller RMSE than XFE, TM or the random walk. XFE and TM have a smaller RMSE than any of the wavelet estimates in only two cases, US 1 month ahead and UK 12 months ahead. Otherwise the wavelet forecasts are better, especially for the 12 month ahead forecasting horizon. In the United States the RMSE is about 0.5 percentage points smaller for any of the wavelet estimates than XFE, TM or the random walk. RMSE is about twice as high in the Eurozone for the XFE and TM estimates than for the wavelet estimates. In the United Kingdom the differences are much smaller and RMSE is within the same region irrespective of estimator. For the shorter horizons the differences between the different estimates are much smaller than for the longer forecasting horizon. In the Eurozone all the wavelet estimates have a smaller RMSE. In the United States and the United Kingdom at least on of the wavelet 20

22 estimates have a smaller RMSE than XFE, TM or the random walk. But the differences are small. For the United States and the Eurozone the random variable (p=0.999) model has the smallest RMSE. But the differences between the different algorithms are small. Nor are there any large differences between estimates based on the Haar wavelet and those based on the Daubechie wavelet. Sometimes the RMSE is smaller when the Haar wavelet is applied and sometimes the Daubechie wavelet performs better but there is no over all pattern. 6. Conclusion This paper introduces five new estimates of monetary inflation. Unlike many commonly used estimates such as Excluding Food and Energy and the Trimmed Mean estimate these do not exclude the possibility of medium to long run relative price changes. Therefore they also perform better when evaluated using the criteria transparency, similarity of means and forecasting ability than XFE or TM do. They are more transparent because they only remove short term fluctuations, they do not ignore certain items of the price index. It is easier for the policy maker to motivate why he or she should ignore certain short term fluctuations if these are both positive and negative over time. If say food and energy prices increases faster than other prices for a couple of years, it is more difficult to motivate why these price changes should be ignored. If headline inflation in the long run is linked to monetary policy the mean in headline inflation should be the same as the mean in the monetary inflation estimates. The wavelet based estimates has by construction the same long run mean although some small difference may appear in the short run. As can be observed particularly in the United States both XFE and TM can be biased especially when the same items are excluded all the time. Although the TM estimate can exclude different items over time, the probability that food and energy prices are excluded are high since these have a higher volatility than many of the other items. Over 21

23 the sample period 2001:4-2006:7 food and energy prices are almost constantly excluded in the TM estimate. The wavelet based estimates are generally also better at forecasting future inflation especially at the long forecasting horizon. In the short run the differences are smaller although the wavelet estimates always perform well. In conclusion the wavelet estimates perform well. Empirically they are more transparent than XFE and TM, they have the same mean as headline inflation and they are if not better at least as good at forecasting future headline inflation. And theoretically they stand on a more solid foundation since relative price changes are not ignored.. 22

24 References Bryan M., Cecchetti S. and Wiggins R. (1997). Efficient inflation estimation. NBER working paper no Bryan M. and Pike C. (1991). Median Price Changes: an alternative approach to measuring current monetary inflation. Federal Reserve Bank of Cleveland Economic Commentary December 1. Clark T. (2001). Comparing Measures of Core Inflation. Reserve Bank of Kansas City Economic Review v.86. no2. Crowley P.M. (2007). A Guide To Wavelets for Economists. Journal of Economic Surveys Vol. 21. No. 2. Dolmas J. (2005). Trimmed Mean PCE Inflation, Federal Reserve Bank of Dallas Working Paper Donoho D. and Johnstone I.M. (1994). Ideal spatial adaptation by wavelet shrinkage, Biometrika v. 81, pp Fisher (1922). The Purchasing Power of Money: Its Determination and Relation to Credit, Interest, and Crises. The Macmillan Company. New York. Gencay R. and Selcuk F. (2001). An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press. Hogan S., Johnson M., Laflèche T. (2001). Core Inflation Bank of Canada Technical Report 89. Jansen M. and Bultheel A. (2001). Empirical Bayes Approach to Improve Wavelet Thresholding for Image Noise Reduction. Journal of the American Statistical Association Vol. 96. No 454, pp Percival D. and A. Walden (2006). Wavelet Methods for Time Series Analysis, Cambridge University Press, New York. Rich R. and Steindel C. (2005). A Review of Core Inflation and an Evaluation of Its Measures, Federal Reserve Bank of New York Staff Report no 236. Wynne M. (1999). Core Inflation: A Review of Some Conceptual Issues, European Central Bank Working Paper no 5. 23

25 Figure 1. United States Yearly Inflation

26 Figure 2. United Kingdom Yearly Inflation 25

27 Figure 3. Eurozone Yearly Inflation 26

28 Figure 4. United States Monthly Inflation 27

29 Figure 5. United Kingdom Monthly Inflation 28

30 Figure 6. Eurozone Monthly Inflation 29

31 TABLE 1 YEARLY INFLATION STATISTICS D(4) WAVELET United States United Kingdom Eurozone Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. CPI Inflation Random Variable p= Random Variable p= Random Variable p= Hard Thresholding Mid Thresholding Soft Thresholding Ex. Food and Energy Trimmed Mean 16% TABLE 2 YEARLY INFLATION STATISTICS HAAR WAVELET United States United Kingdom Eurozone Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. CPI Inflation Random Variable p= Random Variable p= Random Variable p= Hard Thresholding Mid Thresholding Soft Thresholding Ex. Food and Energy Trimmed Mean 16%

32 TABLE 3 MONTHLY INFLATION STATISTICS D(4) WAVELET United States United Kingdom Eurozone Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. CPI Inflation Random Variable p= Random Variable p= Random Variable p= Hard Thresholding Mid Thresholding Soft Thresholding Ex. Food and Energy Trimmed Mean 16% TABLE 4 MONTHLY INFLATION STATISTICS HAAR WAVELET United States United Kingdom Eurozone Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. CPI Inflation Random Variable p= Random Variable p= Random Variable p= Hard Thresholding Mid Thresholding Soft Thresholding Ex. Food and Energy Trimmed Mean 16%

33 Random Variable p=0.990 TABLE 5 ROOT MEAN SQUARED ERRORS UNITED STATES Random Variable p=0.995 Random Variable p=0.999 Hard Thresh. Mid Thresh. Soft Thresh. Trim. Mean 16% Ex. Food and Energy Random Walk h= h= h= h= h= h= Haar D(4)

34 Random Variable p=0.990 TABLE 6 ROOT MEAN SQUARED ERRORS UNITED KINGDOM Random Variable p=0.995 Random Variable p=0.999 Hard Threshold Mid Threshold Soft Threshold Trimmed Mean 16% Ex. Food and Energy Random Walk h= h= h= h= h= h= Haar D(4)

35 Random Variable p=0.990 TABLE 7 ROOT MEAN SQUARED ERRORS EUROZONE Random Variable p=0.995 Random Variable p=0.999 Hard Threshold Mid Threshold Soft Threshold Trimmed Mean 16% Ex. Food and Energy Random Walk h= h= h= h= h= h= Haar D(4)