Using Wavelets to Uncover the Fisher Effect

Transcription

1 Department of Economics Discussion Paper Using Wavelets to Uncover the Fisher Effect Frank J. Atkins and Zhen Sun University of Calgary September 003 Department of Economics University of Calgary Calgary, Alberta, Canada TN 1N4 This paper can be downloaded without charge from

2 USING WAVELETS TO UNCOVER THE FISHER EFFECT by Frank J. Atkins Zhen Sun Department of Economics The University of Calgary Calgary, Alberta Canada TN 1N4 This Draft September, 003 ABSTRACT In this paper we apply a new approach to eliminate spurious regression resulting from long-memory fractionally integrated processes. Instead of fractionally differencing the series, as is commonly done in practice, we apply the discrete wavelet transform (DWT) to the series and then estimate the standard Fisher equation regression in the wavelet domain. This is motivated by the approximate de-correlation property of the DWT for long memory processes. Wavelets have some characteristics that make them particularly suitable as a vehicle for analyzing economic and financial data. Because of the translation and scale properties, non-stationary in the data is not a problem when using wavelets. Further, because of the flexibility in choice of basis function, that is, choice of wavelet function, and because of the property of narrow compact support, wavelets are particularly well suited to handle complex signals that involve cusps, discontinuities, and rapid changes in modeling regime. In the empirical section of this paper, we use the CPI inflation rate and 5 interest rates varying in maturity for both Canada and the United Sates. In terms of the Fisher regression, the results indicate that the degree of fit of the regression of interest rates on inflation rate increases as we move to longer time scales, and the slope coefficient rises as the time scale of analysis increases. In the longest time scale, the slope coefficients in the United States are statistically close to 1. The regression results reveal that the Fisher effect cannot be identified at the short time scale. JEL Classification: C3, E43 *Corresponding Author Frank Atkins Department of Economics University of Calgary, 500 University Dr. N.W. Calgary, Alberta TN 1N4 atkins@ucalgary.ca

3 1. Introduction The empirical relationship between nominal interest rates and inflation which falls under the broad rubric of the Fisher effect has a long history in applied econometrics. Generally speaking, the econometric holy grail in this literature is to uncover evidence that changes in the (expected) inflation rate bring about one for one changes in the nominal interest rate. Early studies investigating the hypothesis generally concluded in favour of the proposed relationship (see Fama (1975), Nelson and Schwert (1977), Mishkin (1981, 1988) and Fama and Gibbons (198)). 1 However, several studies later argued that the relationship does not hold strongly for certain periods of time or when the test is performed on other country beside United States (see Barsky (1987), Summers (1983), and Huizinga and Mishkin (1984, 1986) and Kandel, Ofer and Sarig (1996)). It has been suggested by Diebold and Innoue (001) that time series econometrics went through a unit root boom in the 1980s, followed by a fractional integration boom in the 1990s. A great deal of the Fisher effect literature followed these developments. There was a good reason for this, as the ideas of integer integration and co-integration, as well as fractional integration and co-integration, fit well with the theory of the Fisher effect. In macroeconomics we would expect a long run relationship between nominal interest rates and inflation, but not necessarily a short run relationship. Therefore, the idea that two series share some long memory component is appealing. The concepts of integration and co-integration are clearly connected with the concept of spurious regression. Indeed, the alternative hypothesis in a typical Engle-Granger co-integration hypothesis is spurious regression. Recently, Tsay and Chung (000) have extended the theoretical analysis of spurious regression from unit root processes to the class of long-memory fractionally differenced processes. Tsay and Chung demonstrated that spurious regression can arise among a wide range of long-memory fractionally differenced processes, even in cases where both the dependent variable and regressor are stationary, as long as their orders of differencing sum up to a value greater than 0.5. Based on this, Tsay and Chung conclude that it is long memory or strong dependence, instead of nonstationarity or lack of ergodicity, that causes spurious effects and that the usual differencing procedure may not be able to completely eliminate spurious effects if the data series are not only nonstationary, but possess strong long memory. In this paper, we apply a new approach to eliminate spurious regression between long-memory fractionally differenced processes. Instead of differencing the series, as is commonly done in practice, we propose to apply the discrete wavelet transform (DWT) to the series and then estimate the regression 1 Here early is a relative term, as the empirical literature actually goes as far back as Fisher (1896).

4 in the wavelet domain. This is motivated by the approximate decorrelation property of the DWT for long memory processes. In addition, we exploit an even more important property of wavelets that involves the separation of time scales of variation into a sequence of scales that can be decomposed orthogonally. We maintain that the relationship between inflation and interest rates may be better expressed in terms of restrictions to given time scales. We investigate the Fisher effect in different time scales, obtained from reconstruction of the data by discrete wavelet transform (DWT). Wavelets have also some characteristics that make them particularly suitable as a vehicle for analyzing economic data. Because of the translation and scale properties, nonstationarity in the data is not a problem when using wavelets. Further, because of the flexibility in choice of basis function, that is, choice of wavelet function, and because of the property of narrow compact support, wavelets are particularly well suited to handle complex signals that involve cusps, discontinuities, and rapid changes in regime. The paper is organized as follows. Section briefly surveys the literature on the several applications of wavelets in economics, and introduces the reader to the rudiments of wavelet theory. Section 3 presents estimates of the fractional order of integration of the various interest rates and the inflation rate using Jensen s (1999) wavelet OLS estimator applied to the original data series. We then repeat this estimation using the data generated from the discrete wavelet transform of the original series. We then estimate a series of regression of the nominal interest rate in inflation using the filtered data. Section 4 contains a summary.. A Brief Overview of Wavelets The study of wavelets as a distinct discipline started in the late 1980s. Wavelet theory has since inspired the development of a powerful methodology, which includes a wide range of tools such as wavelet transforms, multi-resolution analysis, time-scale analysis, time-frequency representations with wavelet packets, signal processing, data compression, medical imaging, turbulence and numerical analysis are only a few examples from a long list of disciplines in which wavelets have been successfully employed. Wavelets are mathematical expansions that transform data from the time domain into different layers of frequency levels. Compared to standard Fourier analysis, they have the advantage of being localized both in time and in the frequency domain, and enable the researcher to observe and analyze data at different scales.

5 .1 Some Recent Economic Applications It has long been recognized that economic decision-making is dependent on the time scale involved, and economists emphasize the importance of discerning between long-run and short-run behaviour. Wavelets offer the possibility of going beyond this simplifying dichotomy by decomposing a time series into several layers of orthogonal sequences of scales using Mallat s (1989) multi-scale analysis. These scales can then be analyzed individually and compared across different series. An effort along these lines is illustrated in the paper by Davison et al. (1998), who investigated U.S. commodity prices. Even though the differences across scales were not pursued fully, the authors did consider the different properties of the wavelet coefficients across scales and calculated a measure of the relative importance of the coefficients between scales. In Ramsey and Lampart (1998a, 1998b) two relationships were examined, that between expenditure and income and that between money and income. With respect to the former, the claim that the relationship would vary and that the relevant variables would differ across scales was confirmed. The real interest rate was discovered to be a significant variable only for the longest time scales and only for durable goods. For both durable goods and for non-durable goods, the degree of fit and the strength of the relationship declined monotonically as the scale decreased. At certain scales the relationship between expenditure and income was seemingly more complex than a simple linear relationship. Recently, in Gencay et al (00), the empirical results on the money income relationship were confirmed by employing the same techniques, but using slightly different data definitions, for the U.S., United Kingdom, Japan, and Austria. For all countries (except Austria) the Gencay et al qualitatively matched the Ramsey and Lampart results. Austria was different in that the first three scales, which had money caused income, but at the longest scales the feedback mechanism prevailed. Tkacz (001) used Jensen s (1999) wavelet estimator to ascertain the fractional integration order of interest rates for the United States and Canada. Tkacz finds that most interest rates are mean reverting in the very long run, with the fractional order of integration increasing with term to maturity.

6 . Basic Wavelet Theory There are two types of wavelets defined on different normalization rules: father waveletsφ and mother waveletsψ. The father wavelet integrates to 1 and the mother wavelet integrates to 0: φ ( t) dt = 1 (1) ψ ( t) dt = 0 () Roughly speaking, the father wavelets are good at representing the smooth and low frequency parts of a signal, and the mother wavelets are useful in describing the detail and high-frequency components. Thus, they are used in pairs within a family of wavelet functions, with father wavelets used for the trend components and the mother wavelets for all the deviations from the trend. Any function f (x) in L ( R) to be represented by a wavelet analysis can be built up as a sequence of proections onto father and mother wavelets generated from φ and ψ through scaling and translation as follows: x k φ, k = φ( x k) = φ( ) (3) x k ψ, k = ψ ( x k) = ψ ( ) (4) The wavelet representation of the signal or function f (x) in L ( R) can then be given as: f AJ k J, k + d J, kψ J, k + d J 1, kψ J 1, k + + =, d1, kψ 1, k k k k k φ Λ (5) where J is the number of multi-resolution components, and k ranges from 1 to the number of coefficients in the specified component. The coefficients A J, k, d J, k,λ,d1, k, are the wavelet transform coefficients given by the proections AJ, k = φ J, k f dx (6)

7 d, k = ψ, k f dx, for = 1,, Λ, J (7) The multi-resolution decomposition of a signal can then be defined by using the product of the crystals and the corresponding wavelet atoms: A J = AJ, kφ J, k (8) k D = d, kψ J, k, for = 1,, Λ, J (9) k The functions are called the smooth signal and the detail signals, respectively, which constitute a decomposition of a signal into orthogonal components at different scales. Similarly to the wavelet representation of a signal in L ( R), a signal f (x) can now be expressed in terms of these signals: f x) = A + D + D + Λ D ( ) (10) ( J J J x As each term in equation (10) represent components of the signal f (x) at different resolutions, it is called a multi-resolution decomposition (MRD). The coarsest scale signal A J (x) represents a coarse scale smooth approximation to the signal. 1 Adding the detail signal D J (x) gives a scale J approximation to the signal, A J 1, which is a refinement of the coarsest approximation A J (x). Further refinement can sequentially be obtained as: A 1 = A + D 1 = AJ + DJ + DJ 1 + Λ + D (11) signal f (x). The collection{ A J, AJ 1, AJ, Λ A1 } provides a set of multi-resolution approximations of the

8 3. Empirical Results The empirical analysis makes use of monthly data on inflation and interest rates for the United States and Canada. The inflation rate in each country is measured as the month-to-month change in the consumer price index multiplied by 100. For the U.S. we use the Federal funds rate, the 90-Day Treasure bill rate and as well as the 1, 3, 5 and 10 years Government bond yields. For Canada, we use the Bank of Canada rate and the 91-Day Treasure Bill rate, as well as the 1-3, 3-5, 5-10 and 10+ years bond Yields. All data are taken from Federal Reserve Economic Data (FRED) and CANSIM II database, respectively. The data is monthly and runs from Sep 1959 to Apr Fractional Integration Order of the Original Series In this section, we estimate the fractional integration order of each time series using the wavelet OLS estimator suggested by Jensen (1999). Let x be the fractional integrated process, I(d), defined by t ( 1 L) = ε (1) d xt t d Where ε i. i. d(0, σ ), d is any real value and ( 1 L) is fractional integrating operator defined t by the binominal expansion (1 L ) d Γ( d + 1)( L) = 1+ = Γ( d + 1) Γ( ) d( d 1) d( d 1)( d ) 3 = 1 dl + L L +Λ Λ (13)! 3! Here L denotes the lag operator and Γ the gamma operator.

9 Building on the work of Tefwik and Kim (199) and McCoy and Walden (1996), Jensen (1999) demonstrates that, if X() denotes the variance of the wavelet coefficient at scale, then an estimate of d can be obtained from applying ordinary least squares to LnX ( ) = Lnσ dln (14) Tables 1 and present estimates of the fractional integration parameter of the data under consideration, using the HAAR wavelet and three different degrees of smoothing for the Daubechies wavelet.

10 Table 1: Wavelet Estimates d from equation (14) United States Variable Haar Daubechies-4 Daubechies-1 Daubechies-0 Inflation Rate (0.071) (0.0600) (0.0663) (0.078) Federal Funds Rate (0.1000) (0.059) (0.064) (0.0889) 90-Day T. Bill (0.097) 0.85 (0.0480) (0.0616) (0.087) 1 Year Bond Rate (0.0894) (0.0436) (0.0616) (0.0877) 3 Year Bond Rate (0.0671) (0.0458) (0.0671) (0.1068) 5 Year Bond Rate (0.0608) 10 Year Bond Rate (0.0566) (0.0500) (0.0557) (0.0700) (0.076) (0.104) (0.1659) Table : Wavelet Estimate of d from equation (14) Canada Variable Haar Daubechies-4 Daubechies-1 Daubechies-0 Inflation Rate (0.0831) (0.097) (0.0947) 0.40 (0.1000) Bank Rate (0.0693) (0.0557) (0.078) (0.196) 91-Day T Bill (0.0640) (0.0656) (0.0866) (0.1735) 1-3 Year Bond Yield (0.059) (0.0574) (0.085) (0.1700) 3-5 Year Bond Yield) (0.0510) (0.0566) (0.0831) (1706) 5-10 Year Bond Yield 10+ year Bond Yield (0.0510) (0.0539) (0.0574) (0.0600) (0.0837) (0.0906) (0.1789) (0.1836)

11 The results on Tables 1 and have some interesting features. First, the estimates of the fractional integration parameter for the various interest rate series are qualitatively similar to those attained by Tkacz (001). Generally speaking, the estimates of d increase with the maturity of the interest rate. It appears that the rates of short maturity, say one year or less, do not follow unit root processes, while the interest rates with longer maturities have estimates of d which are within two standard errors of unity. The second interesting feature of these estimates is that the inflation rate in each country appears to have a much smaller order of integration than any of the interest rates. 3. Applying the Discrete Wavelet Transform In this section, we apply the discrete wavelet transform described above to decompose the series into five different time scales. The sub series D and A in a multi-resolution analysis form an additive decomposition of the original time series: J J f = A J + D (15) = 1 Each sub-series is associated with changes at scale λ =, while A is associated with J weighted averages over scales of (see Percial and Mofeld (1997) for more details). The first scale of wavelet coefficients D 1 D is filtering out the high-frequency fluctuations by essentially looking at adacent differences in the data. The results are shown on Tables for the United States and on Tables for Canada. J

12 Table 3.1 Wavelet Estimate of the Fractional Integration Order U.S. Inflation Rate Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.087) D (0.0678) D (0.0794) D (0.141) D (0.035) A (0.958) (0.0480) (0.796) (0.1744) (0.737) (0.1718) (0.1490) (0.050) (0.394) (0.1581) (0.6107) (0.1847) (0.6030) (0.0458) (0.0819) (0.141) (0.1304) (0.439) (0.5416) Table 3. Wavelet Estimate of the Fractional Integration Order U.S. Federal Funds Rate Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0894) D (0.0656) D (0.1175) D (0.0316) D (0.3703) A (0.3795) (0.0700) (0.0458) (0.1175) (0.6710) (0.168) (0.1497) (0.1000) (0.0943) (0.1063) (0.3843) (0.398) (0.4345) (0.1360) (0.087) (0.0877) (0.179) (0.951) (0.3881) Table 3.3 Wavelet Estimate of the Fractional Integration Order U.S. 90 Day Treasury Bill Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0877) D (0.0539) D (0.0954) D (0.100) D (0.4343) A (0.3583) (0.1044) (0.063) (0.0781) (0.0700) (0.760) (0.1386) (0.1446) (0.0866) (0.0854) (0.1960) (0.4337) (0.4935) (0.0959) (0.0686) (0.0933) (0.1987) (0.997) (0.3619)

13 Table 3.4 Wavelet Estimate of the Fractional Integration Order U.S. 1 year Bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0843) D (0.0616) D (0.1131) D (0.0557) D (0.5685) A (0.3704) (0.087) (0.044) 0.56 (0.0794) (0.0700) (0.780) (0.1375) (0.1319) (0.0480) 0.06 (0.1058) (0.1703) (0.5586) 0.95 (0.5364) (0.1183) (0.0500) (0.165) (0.63) (0.63) (0.368) Table 3.5 Wavelet Estimate of the Fractional Integration Order U.S. 3 Year Bond Yield Variables Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.1670) D (0.0755) D (0.1179) D (0.148) D (0.9464) A (0.38) (0.0678) (0.083) (0.0678) (0.1039) (0.987) (0.170) (0.0990) (0.083) (0.0954) (0.1817) (0.7078) (0.8016) (0.1334) (0.033) (0.1466) (0.711) (0.983) (0.389) Table 3.6 Wavelet Estimate of the Fractional Integration Order U.S. 5 Year Bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.1568) D (0.0768) D (0.1077) D (0.194) D (1.0551) A (0.3153) (0.0600) (0.083) (0.0748) (0.1086) (0.3391) (0.674) (0.0794) (0.083) (0.0794) (0.168) (0.7776) (1.037) (0.19) (0.0300) (0.19) (0.87) (0.3103) (0.3167)

14 Table 3.7 Wavelet Estimate of the Fractional Integration Order U.S. 10 Year Bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.1709) D (0.0787) D (0.1039) D (0.3437) D (0.8663) A (0.3116) (0.0700) (0.041) (0.0768) (0.1643) (0.414) (0.3357) (0.0843) (0.044) (0.0557) (0.335) (0.8857) (1.017) (0.117) (0.0469) (0.1086) (0.366) (0.3400) (0.3061) Table 4.1 Wavelet Estimate of the Fractional Integration Order Canada Inflation Rate Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0975) D (0.0775) D (0.0883) D (0.1658) D (0.141) A (0.395) (0.0894) (0.1039) (0.087) (0.534) (0.3688) (0.83) (0.0883) (0.0980) (0.0954) (0.4551) (0.0819) (0.4194) (0.097) (0.1000) (0.141) (0.81) (0.4309) (0.5378) Table 4. Wavelet Estimate of the Fractional Integration Order Canada Bank of Canada Rate Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0656) D (0.100) D (0.0949) D (0.1533) D (0.4968) A (0.3461) (0.0469) (0.1319) (0.140) (0.1170) (0.713) (0.311) (0.0608) (0.30) (0.185) (0.173) (0.3481) (1.0099) (0.0616) (0.156) (0.0917) (0.1670) (0.6391) (0.376)

15 Table 4.3 Wavelet Estimate of the Fractional Integration Order Canada 91 Day Treasury Bill Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0436) D (0.105) D (0.1315) D (0.1616) D (0.5311) A (0.958) (0.0656) (0.063) (0.11) (0.1876) (0.919) (0.38) (0.0686) (0.0686) (0.687) (0.330) (0.3910) (0.975) (0.0490) (0.087) (0.1439) (0.191) (0.6496) (0.860) Table 4.4 Wavelet Estimate of the Fractional Integration Order Canada 1-3 Year bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.063) D (0.0608) D (0.100) D (0.374) D (0.6534) A (0.890) (0.108) (0.0490) (0.165) (0.409) (0.370) (0.3375) (0.11) (0.064) (0.474) (0.3755) (0.7531) (0.8843) (0.1487) (0.071) (0.1556) (0.3936) (0.404) (0.31) Table 4.5 Wavelet Estimate of the Fractional Integration Order Canada 3-5 Year Bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.133) D (0.0964) D (0.1643) D (0.3114) D (0.6704) A (0.857) (0.081) (0.0387) (0.1407) (0.764) (0.3783) (0.3350) (0.081) (0.0583) (0.037) (0.461) (0.714) (0.883) (0.087) (0.078) (0.96) (0.31) (0.396) (0.391)

16 Table 4.6 Wavelet Estimate of the Fractional Integration Order Canada 5-10 Year Bond Yield Crystal: Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.0943) D (0.1015) D (0.1459) D (0.3557) D (0.657) A (0.901) (0.0938) (0.0616) (0.1691) (0.506) (0.3971) (0.343) (0.0954) (0.076) (0.317) (0.4399) (0.718) (0.8776) (0.1091) (0.0849) (0.15) (0.3114) (0.366) (0.38) Table 4.7 Wavelet Estimate of the Fractional Integration Order Canada 10+ Yer Bond Yield Variables Observations Haar Daubechies-4 Daubechies-1 Daubechies-0 D (0.1780) D (0.1616) D (0.1685) D (0.3059) D (0.565) A (0.3050) (0.0656) (0.0707) (0.165) (0.408) (0.481) (0.3617) (0.0608) (0.0794) (0.1965) (0.3647) (0.8181) (0.880) (0.0800) (0.1153) (0.58) (0.3370) (0.3500) 1.00 (0.3980) The striking result from the above tables is that all of the estimates of the fractional integration parameter using the detail coefficients are smaller than those of the original time series using the Jensen estimator. Notice that all of the fractional integration orders of the detail coefficients are between 0.5 and 0.5, which make them as stationary time series with finite variance. The implication is that any shock to the DWT data will decay much faster than in the original time series However, caution must be exercised when discussing inference in these results. Notice that the standard error of the estimates increases from high-frequency to low-frequency. This is due to the smaller sample size associated with the low-frequency components. As we can see, at the coarsest detail and approximation coefficients, and A, there are only 16 observations in the sample compared with D observations in the finest detail coefficients D1.

17 3.3 Fisher Regressions in the Wavelet Domain A further interesting feature of the above results is that the sum of the integration orders (inflation rate plus one interest rate) is below 0.5 in absolute value. Therefore, according to the results in Tsay and Chung (000), we can regress a crystal of the nominal interest rate on a crystal of the inflation rate and thereby avoid any spurious regression that may result from any long memory properties possessed by these series. This is the methodology proposed by Fan and Whitcher (001) to eliminate spurious regression between long memory fractionally integrated processes. Given the above, the discrete wavelet transform is an ideal tool to eliminate the long memory property of time series data and to analyze relationships at different timescales. Instead of looking at the relationship between inflation and interest rate averaged over all timescales, we examine the relationship at each timescale separately. Let I A ), R( A ), I( D ) and R D ) represent the inflation ( J J rate (I) and the nominal interest rate (R) at each scale, as determined by the wavelet decomposition. We can then estimate a sequence of regressions: ( or R( A ) = α + β I( A ) + ε (16) J J J J J R( D ) = α + β I( D ) + ε, = 1,, Λ J (17) We run theses regressions for different maturities of the interest rates and inflation rate both for United States and Canada for the individual wavelet crystals. These results are shown on Tables and For both countries, for all interest rates, the slope coefficients are close to 0 for short time scales, (D1 and D) and increase and become statistically significant for the longer time scales. For the U.S., we cannot reect 1.0 for D5 crystals for all interest rates. Interestingly, for Canada, for the longer scale data only the Bank of Canada rate and the 91 Treasury Bill rate are close to one. The other slope coefficients for longer scale data, although statistically greater than 0, are still more than standard errors below unity. The regression results are consistent with the conclusion that the Fisher effect cannot be identified at the short time scale, but there is a positive relationship at the long time scale.

18 Table 5.1 U.S. Regressions of Federal Funds Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.038) D (0.0746) D (0.196) D (0.3785) D (0.6950) A (5.5696) Slope (0.0104) (0.0303) (0.0673) (0.0975) (0.088) (0.1953) R Table 5. U.S. Regressions of Treasury Bill Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0190) D (0.0591) D (0.15) D (0.736) D (0.5656) A (4.880) Slope (0.0083) (0.040) (0.05) (0.0704) (0.1699) (0.1693) R

19 Table 5.3 U.S. Regressions of 1 Year Bond Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0198) D (0.0570) D (0.1560) D (0.81) D (0.681) A (5.3108) Slope (0.0086) 0.07 (0.03) (0.0535) (0.074) (0.1887) (0.186) R Table 5.4 U.S. Regressions of 3 Year Bond Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0171) D (0.0487) D (0.191) D (0.607) D (0.644) A (5.569) Slope (0.0075) (0.0198) (0.0443) (0.0671) (0.1935) (0.1950) R

20 Table 5.5 U.S. Regressions of 5 Year Bond Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0156) D (0.0436) D (0.1165) D (0.456) D (0.651) A (5.617) Slope (0.0068) (0.0177) 0.80 (0.0340) (0.063) (0.1959) (0.1969) R Table 5.6 U.S. Regressions of 10 Year Bond Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0133) D (0.0367) D (0.0973) D (0.97) D (0.631) A (5.718) Slope (0.0058) (0.0149) (0.0334) (0.0591) (0.1896) (0.006) R

21 Table 6.1 Canada Regressions of Bank of Canada Rate on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.047) D (0.0696) D (0.1888) D (0.5363) D (1.1006) A (5.8695) Slope (0.0060) (0.0170) (0.0455) (0.1367) (0.305) (0.1951) R Table 6. Canada Regressions of 91 Day Treasury Bill Yield on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.01) D (0.0685) D (0.1798) D (0.5337) D (1.0937) A (6.359) Slope (0.0054) (0.0167) (0.0433) (0.1361) (0.91) (0.103) R

22 Table 6.3 Canada Regressions of 1-3 Year Bond Yield on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.07) D (0.0640) D (0.1685) D (0.3377) D (0.8889) A (5.597) Slope (0.0055) (0.0156) (0.0406) (0.0861) (0.186) (0.1859) R Table 6.4 Canada Regressions of 3-5 Year Bond Yield on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.003) D (0.0573) D (0.1494) D (0.870) D (0.7937) A (5.3159) Slope (0.0049) (0.0140) (0.0360) (0.073) (0.166) (0.1767) R

23 Table 6.5 Canada Regressions of 5-10 Year Bond Yield on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0178) D (0.0493) D (0.1361) D (0.533) D (0.7579) A (5.49) Slope (0.0043) (0.010) (0.038) (0.0646) (0.1587) (0.1743) R Table 6.6 Canada Regressions of 10+ Year Bond Yield on Inflation Rate Using Individual Crystals Crystal Observations Intercept D (0.0153) D (0.043) D (0.1170) D (0.13) D (0.7794) A (5.181) Slope (0.0037) (0.0103) (0.08) (0.0564) (0.163) (0.1735) R

24 4. Conclusions In this paper we have used multi resolution analysis to filter data via the discrete wavelet transform. Wavelets have some characteristics that make them particularly suitable as a vehicle for analyzing economic data. In particular, wavelets are localized in both time and scale, which gives them a distinct advantage over typical frequency domain applications. We maintain that the relationship between inflation and interest rates may be better expressed in terms of restrictions to given time scales, which we obtain from reconstruction of the data by discrete wavelet transform. The conclusion from the preceding empirical analysis is that the results are broadly consistent with the predictions of the Fisher effect. There appear to be a long run relationship between nominal interest rates and inflation in both Canada and the United States. In the short run, our results show almost no relationship between these variables. In the United States, the estimated coefficient from a Fisher regression is approximately unity over cycles of several years. In Canada, the coefficient appears to be somewhat less than unity.

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