# Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets

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## Abstract

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**JEL**C1, C3, C5, E3

## 1. Introduction

## 2. Financial Indicators and Economic Activity in the Standard Classical Approach

OLS-Dependent variable: $\mathbf{\Delta}{\mathit{y}}_{\mathit{t}\mathbf{+}\mathbf{4}}$; 1973:4–2009:4 | ||||
---|---|---|---|---|

Coefficients | HAC s.e. | t-Ratio | p-Value | |

const | 1.737 | 1.125 | 1.543 | 0.125 |

$\Delta {y}_{t}$ | 0.471 | 0.278 | 1.697 | 0.092 * |

$\Delta {y}_{t-1}$ | 0.138 | 0.190 | 0.729 | 0.467 |

$\Delta {y}_{t-2}$ | –0.070 | 0.215 | –0.326 | 0.745 |

$\Delta {y}_{t-3}$ | 0.124 | 0.254 | 0.488 | 0.626 |

$RF{F}_{t}$ | –0.087 | 0.140 | –0.625 | 0.533 |

$T{S}_{t}$ | –0.703 | 0.203 | –3.461 | 0.001 ** |

$C{S}_{t}$ | –0.141 | 0.516 | –0.274 | 0.785 |

${\overline{R}}^{2}$ = 0.235 - s.e. reg. = 1.986 - DW = 0.45 | ||||

LM test up to order 4 = 123.1 (p-value = 1.04e–43) | ||||

Breusch-Pagan test-LM = 20.01 (p-value = 0.005) | ||||

Test RESET-F(1, 136) = 3.52 (p-value = 0.063) |

## 3. Wavelet-Based Exploratory Data Analysis

#### 3.1. Continuous Wavelet Transform

**Figure 2.**CWT plot for real output, real interest rate, term spread and credit spread (from top to bottomt). The color code for power ranges from blue (low power) to red (high power) with significant regions associated with warmer colors (red, orange and bright green). A black contour line testing the wavelet power 5% significance level against a white noise null is displayed as is the cone of influence, represented by a shaded area corresponding to the region affected by edge effects.

- The power of the industrial production was very high at scales corresponding to business cycle frequencies, that is 2 to 10 years, until mid-1980s. After that, the power at all frequencies steadily decreased, with an exception in the last decade, when it was again quite high at the scale corresponding to a 6 years period (in the literature this evidence is referred to as the “Great Moderation” [40]);
- For the real interest rate there is evidence of high (orange) and very high (red) power at scales corresponding to periods greater than 4 years (and between 2 and 4 years at the very beginning of the sample, i.e., 2nd half of the seventies); there is also evidence of a significant region at short scales, between 1 and 2 years in early 80s in correspondence with the structural change in the monetary policy, when the Fed implemented a very restrictive monetary policy as a reaction to the inflationary pressures of the second oil shock (Volcker disinflation period [41]);
- For the term spread there is clear evidence of very high and high power at scales between 4 and about 10 years, a sort of “optimal” scale, and, like in the real interest rate case, also evidence of power (bright green) at short scales, that is 1 to 2 years, in early 80s;
- Finally, for the credit spread there is evidence of high power at scales corresponding to periods between 4 and 8 years, particularly at the beginning and end of the sample period.

#### 3.2. Discrete Wavelet Transform

- Time-Scale analysis using scalograms and time-frequency plots provides clear evidence of the changing nature of the signals;
- The continuous wavelet transform does a good job of revealing the scaling characteristic (major scales) present in the data;
- The time-frequency plots are useful in detecting sharp changes in the high- frequency component of the data, like structural breaks.

**Figure 3.**Time-Frequency plots: Time is measured along the horizontal axis and frequency along the vertical axis. Each wavelet coefficient is represented by a rectangular area with fine scale coefficients occupying tall thin boxes and coarse scale coefficients flat wide boxes. The modulus of the wavelet coefficient determines the gray level of each rectangle.

**Figure 4.**The auxiliary ${\tilde{D}}_{k}$ test for j = 1 (upper panel) and j = 2 (lower panel) for output (left), term spread (center) and real interest rate (right).

## 4. Detection and Location of Structural Breaks through the Test of Homogeneity of Variance

D-Statistic critical values comparison: | |||||
---|---|---|---|---|---|

WF | Scale | N | 10% | 5% | 1% |

LA(8) | 1 | 27 | 0.3146 | 0.3509 | 0.4208 |

2 | 61 | 0.2127 | 0.2361 | 0.2866 | |

D(4) | 1 | 30 | 0.2995 | 0.3336 | 0.3974 |

2 | 63 | 0.2102 | 0.2346 | 0.2820 | |

Haar | 1 | 32 | 0.2908 | 0.3250 | 0.3903 |

2 | 64 | 0.2086 | 0.2323 | 0.2796 |

**Table 3.**Results of testing for homogeneity of variance using the LA(8), D(4) and Haar wavelet filters.

D-Statistic: | |||||
---|---|---|---|---|---|

WF | Scale | Output | Real int. rate | Term spread | Credit spread |

LA(8) | 1 | 0.3980 * | 0.6584 | 0.5017 | 0.3445 ** |

2 | 0.4289 | 0.5713 | 0.4726 | 0.1576 | |

D(4) | 1 | 0.3233 ** | 0.6343 | 0.4401 | 0.3364 * |

2 | 0.5065 | 0.5866 | 0.2622 * | 0.3014 | |

Haar | 1 | 0.4233 | 0.6236 | 0.4143 | 0.3536 * |

2 | 0.4406 | 0.4409 | 0.3399 | 0.2572 * |

Scale 1 | |||
---|---|---|---|

WF | output | real int. | term |

rate | spread | ||

LA(8) | 83:I | 82:III | 82:III |

D(4) | 82:III | 82:III | 82:III |

Haar | 83:III | 82:II | 82:III |

Scale 2 | |||

WF | output | real int. | term |

rate | spread | ||

LA(8) | 84:I | 82:IV | 83:II |

D(4) | 84:I | 82:IV | 83:II |

Haar | 84:IV | 81:III | 82:IV |

## 5. Time Scale Relationships: Measuring Individual Effects at Different Horizons

- The current stance of monetary policy, as measured by the real interest rate, is significantly
**negatively**related to future output at the shortest scales, ${D}_{2}$ and ${D}_{3}$, (associated to 4 and 8 quarters, respectively), but**positively**related in the long-run, ${S}_{4}$ (i.e., periods greater than 16 quarters); - The shape of the yield curve is
**positively**related to future output in the shorter run, at scale ${D}_{3}$, associated to 8 quarters, but at longest scales, i.e., ${D}_{4}$ and above, there is evidence of a**negative**relation between the term structure and future output; - The credit spread is significantly
**negatively**related to future output at the longest scales only, i.e., ${D}_{4}$ and above.

Dependent variable: ${\mathit{\u03f5}}_{\mathbf{\Delta}\mathit{y}\mathbf{;}\mathbf{(}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}\mathbf{,}\mathit{t}}$; 1973:1–2009:4 (T = 144) | |||||
---|---|---|---|---|---|

${\u03f5}_{RIR;(TS,CS),t}$ | coeffs | HAC s.e. | t-ratio | p-value | Adj-${R}^{2}$ |

Aggregate | –0.0538 | 0.1515 | –0.3556 | 0.7226 | –0.0043 |

${S}_{4}$ | 0.2584 | 0.0733 | 3.527 | 0.0006 *** | 0.1393 |

${D}_{4}$ | –0.4196 | 0.2687 | –1.562 | 0.1206 | 0.0801 |

${D}_{3}$ | –1.1956 | 0.2732 | –4.377 | 2.32e–05 *** | 0.2006 |

${D}_{2}$ | –0.4962 | 0.1069 | –4.641 | 7.80e–06 *** | 0.1455 |

${D}_{1}$ | –0.0652 | 0.1376 | –0.4740 | 0.6362 | –0.0036 |

${\u03f5}_{TS;(RIR,CS),t}$ | coeffs | HAC s.e. | t-ratio | p-value | Adj-${R}^{2}$ |

Aggregate | –0.8951 | 0.1944 | –4.604 | 8.95e- *** | 0.1721 |

${S}_{4}$ | –0.8318 | 0.2100 | –3.961 | 0.0001 *** | 0.3766 |

${D}_{4}$ | –1.0630 | 0.3390 | –3.135 | 0.0021 *** | 0.2514 |

${D}_{3}$ | 1.0708 | 0.3278 | 3.267 | 0.0014 *** | 0.1196 |

${D}_{2}$ | 0.3648 | 0.3096 | 1.179 | 0.2406 | 0.0179 |

${D}_{1}$ | 0.1486 | 0.2085 | 0.7131 | 0.4770 | –0.0022 |

${\u03f5}_{CS;(RIR,TS),t}$ | coeffs | HAC s.e. | t-ratio | p-value | Adj-${R}^{2}$ |

Aggregate | –0.4913 | 0.5734 | –0.8568 | 0.3930 | 0.0143 |

${S}_{4}$ | –1.1937 | 0.4244 | –2.813 | 0.0056 *** | 0.1392 |

${D}_{4}$ | –1.9878 | 0.6594 | –3.015 | 0.0030 *** | 0.2302 |

${D}_{3}$ | –1.3714 | 0.8000 | –1.714 | 0.0887 * | 0.0378 |

${D}_{2}$ | –0.1250 | 0.4606 | –0.2714 | 0.7865 | –0.0055 |

${D}_{1}$ | 0.0824 | 0.2141 | 0.3849 | 0.7009 | –0.0048 |

## 6. Conclusions

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**MDPI and ACS Style**

Gallegati, M.; Ramsey, J.B.; Semmler, W.
Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets. *Axioms* **2013**, *2*, 182-207.
https://doi.org/10.3390/axioms2020182

**AMA Style**

Gallegati M, Ramsey JB, Semmler W.
Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets. *Axioms*. 2013; 2(2):182-207.
https://doi.org/10.3390/axioms2020182

**Chicago/Turabian Style**

Gallegati, Marco, James B. Ramsey, and Willi Semmler.
2013. "Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets" *Axioms* 2, no. 2: 182-207.
https://doi.org/10.3390/axioms2020182