Contents

1. Introduction
2. Spectral analysis
3. Statistical detrending methods
4. Deterministic trends
5. Stochastic filters
6. Beveridge-Nelson decompositions
7. Wavelet decompositions

Introduction to decompositions

• Most time series exhibit repetitive or regular behaviour over time
• Hence a great part of the study of this data is conducted in the time domain, with ARIMA or state-space models
• Another important phenomena of time series is that they may be decomposed into periodic variations of the underlying phenomenon
• Shumway & Stoffer (2011) suggest these frequency decompositions should be expressed as Fourier frequencies that are driven by sines and cosines
• This follows the tradition of Joseph Fourier

Identifying the business cycle

• These decompositions may be used to describe the stylised facts of the business cycle:
• Business cycle refers to regular periods of expansion and contraction in major economic aggregate variables (Burns and Mitchell, 1946)
  • i.e. persistence of economic fluctuations and correlations (or lack thereof) across economic aggregates
• We say that a turning point occurs when the business cycle reaches local maximum (peak) or local minimum (trough)
• May also be used to identify the output gap (difference between potential and actual output)
• It is important to note that when seeking to measure the business cycle, there are no unique periodicities that are of relevance
• In addition, most economic time series are both fluctuating and growing, which makes the decomposition quite difficult

Introduction to decompositions

• We may imagine a system responding to various driving frequencies by producing linear combinations of sine and cosine functions
• The frequency domain may be considered as a regression of a time series on periodic sines and cosines
• This lecture considers a few widely used methods that are used to decompose economic time series

Spectral Analysis

• A time series could be considered as a weighted sum of underlying series that have different cyclical patterns
• The total variation of an observed time series will be the sum of each underlying series, which may vary in different frequencies
• Spectral analysis is a tool that can be used to decompose the variation of a time series into different frequency components

Spectral Analysis

• Consider an example of three quarterly time series variables, $y_t$, $x_t$ and $\upsilon_t$
• Where the term $\upsilon_t \sim \mathsf{i.i.d.} \mathcal{N} (0,\sigma)$
• If $x_t = y_t^{\top} \beta + \upsilon_t$, then after regressing $y_t$ on $x_t$, we would expect to find that the coefficient would be large and significant, provided that $\sigma$ is not too large
• The rationale for this is that $x_t$ contains information about $y_t$, which is reflected by the coefficient value for $\beta$
• From an intuitive perspective, frequency domain analysis involves regressing a time series variable, $x_t$, on a number of different periodic frequency components
• This would allow us to identify which frequency component is contained in $x_t$

Spectral Analysis

• To define the rate at which a series oscillates, we define a cycle as one completed period of a sine or cosine functions $$\begin{eqnarray} y_t =A \cos (2 \pi \omega t+\phi) \end{eqnarray}$$
• for $t= 0,\pm1,\pm2, \ldots$, where $\omega$ is a frequency index, defined in cycles per unit of time, which is an expression of $T$
• $A$ determines the height or amplitude of the function and the starting point of the cosine function is termed the phase, $\phi$

Spectral Analysis

• When seeking to conduct some form of data analysis, it is usually easier to use a trigonometric identity of this expression which may be written as, $$\begin{eqnarray} y_t =U_1 \cos (2 \pi \omega t) +U_2 \sin (2 \pi \omega t) \end{eqnarray}$$
• where $U_1 =A \cos \phi$ and $U_2 =-A\sin \phi$ are often taken to be normally distributed random variables
• The above random process is also a function of its frequency, defined by the parameter $\omega$
• The frequency is measured in cycles per unit of time
• For $\omega= 1$, the series makes one cycle per time unit
• For $\omega=.50$, the series makes two cycles per time unit

Spectral Analysis

• To see how the spectral techniques can be used to interpret the regular frequencies in the series, consider the following four periodic time series $$\begin{eqnarray} x_{1,t} &=&2\cos (2\pi t6/100)+3\sin (2\pi t6/100) \\ x_{2,t} &=&4\cos (2\pi t30/100)+5\sin (2\pi t30/100) \\ x_{3,t} &=&6\cos (2\pi t40/100)+7\sin (2\pi t40/100) \\ y_{t} &=&x_{1,t}+x_{2,t}+x_{3,t} \end{eqnarray}$$

Spectral Analysis

• Sorting out of the essential frequency components in a time series, including their relative contributions, constitutes one of the main objectives of spectral analysis
• One way to accomplish this objective is to regress sinusoids that vary at the different fundamental frequencies on the data
• This is represented by the periodogram (or sample spectral density) and may be expressed as, $$\begin{eqnarray} P(j/n) = \frac{2}{n} \sum^{n}_{t=1} y_t \cos \big(2 \pi t \; j/n \big)^2 + \frac{2}{n} \sum^{n}_{t=1} y_t \sin \big(2 \pi t \; j/n \big)^2 \end{eqnarray}$$
• It may be regarded as a measure of the squared correlation of the data with sinusoids oscillating at a frequency of $\omega_j =j/n$, or $j$ cycles in $n$ time points

Spectral Analysis

• An interesting exercise would be to construct the $x_1$ series from $y_t$, which may be regarded as actual data
• To do so we need to filter out all components that lie outside the chosen frequency band of $x_1$
• Such a filter could operate with the aid of a regression model that contains the information that relates to a particular frequency (although there are more convenient ways of going about this)
• Hence, cycles with frequencies corresponding to $x_2$ and $x_3$ would be excluded, while cycles with frequency corresponding to $x_1$ will be maintained (i.e. can pass through the filter)

Methods for decomposing a time series

• Various detrending methods provide different estimates of the cycle
• Appropriate transformation should depend on the underlying dynamic properties
  • Usually a good idea to consider whether a series has a stochastic trend (i.e. unit root)
• Assume that an economic time series can be decomposed into trend, $g_{t}$, and cycle, $c_{t}$: $$\begin{eqnarray} y_{t}=g_{t}+c_{t} \end{eqnarray}$$
• where we abstract from a noise and seasonal component
• Estimates of $g_{t}$ and $c_{t}$ may be obtained from various univariate detrending methods

Deterministic trends & filters

• Early methods to decompose economic variables assumed that the (natural) growth path for the economy was largely deterministic

• Therefore, the trend cycle decomposition was described as, $$\begin{eqnarray} y_{t} &=&g_{t}+c_{t} \\ \widehat{g}_{t} &=&\widehat{\alpha }_{0}+\widehat{\alpha }_{1}t+\widehat{\alpha }_{2}t^{2}+ \ldots \\ \widehat{c}_{t} &=&y_{t} - \widehat{g}_{t} \end{eqnarray}$$

  • where the trend, $\widehat{g}_{t}$, is found by simple estimation techniques
  • cycle corresponds to the residual in the series
• When we assume a linear trend, $|\alpha_{1}| >0$ and $\alpha_{2}=0$
• For a quadratic trend, $|\alpha_{1}| >0$ and $|\alpha_{2}| >0$

Deterministic trends & filters

• Previous graph displays the logarithm of South Africa GDP with linear trend and cycle
• However, productivity growth has not been perfectly log-linear (i.e. constant growth rate) and far from smooth
• In addition, there are several structural breaks such as the oil price shock in 1973/1974 & recent GFC
• To allow for a possible structural break in the trend, we could estimate, $$\begin{eqnarray} \widehat{g}_{t}=\widehat{\alpha }_{0}+\widehat{\alpha }_{1}t+\widehat{\alpha}_{2}DS_{t}(j)+\widehat{\alpha }_{3}DL_{t}(k) + \ldots \end{eqnarray}$$
• where $DS_{t}(j)$ and $DL_{t}(k)$ are dummy variables that capture the change in the slope or the level of the trend in periods $j$ and $k$
• Hence, $DS_{t}(j)=t-j$ and $DL_{t}(k)=1$, if $t>j$ or $t>k$, while it would be zero otherwise

Deterministic trends & filters

• Identifying structural breaks could be problematic
• Detrending an integrated process with a deterministic trend may result in the introduction of a spurious cycle
• See Nelson-King (1981) for details

Stochastic trends & filters

• Let $g_{t}$ represent a moving average of observed $y_{t}$
• We can extract the trend component, $g_{t}$, by applying $$\begin{eqnarray} g_{t}=\overset{n}{\underset{j\text{ }=\text{ }-m}{\sum }}{\omega}_{j}y_{t-j} \end{eqnarray}$$
• where $m$ and $n$ are positive integers and $\omega_{j}$ are weights in the $G(L)$ polynomial $$\begin{eqnarray} G(L)=\overset{n}{\underset{j\text{ }=\text{ }-m}{\sum }}\omega_{j}L^{j} \end{eqnarray}$$
• where $L$ is defined so that $L^{j}y_{t}=y_{t-j}$ for positive and negative values for $j$

Stochastic trends & filters

• The cyclical component is the difference between $y_t$ and $g_t$ $$\begin{eqnarray} c_{t}=[1-G(L)]\text{ }y_{t}\equiv C(L)\text{ }y_{t} \end{eqnarray}$$
• where $C(L)$ and $G(L)$ are linear filters
• Weights are chosen to add up to one, $\sum_{j=-m}^{n}c_{j}=1$, so that the level of the series is maintained
• The moving-average filter with weight $1/5$, may be obtained by filtering over the {moving window} of five observations $$\begin{eqnarray} g_{t}=\frac{1}{5} \sum^{2}_{j=-2} y_{t}=\frac{1}{5} \big(y_{t-2}+y_{t-1}+y_{t}+y_{t+1}+y_{t+2}\big) \end{eqnarray}$$
• Will produce a smooth stochastic trend if the underlying data has such a trend

Hodrick-Prescott (HP) filter

• The HP filter has been a widely used approach to extract cycles in economic data
• Extracts a stochastic trend, $g_{t}$, for a given value of $\lambda$, which is the smoothing parameter
• Seeks to emphasize true business cycle frequencies
• The filter can be obtained as the solution to the following problem: $$\begin{eqnarray} \min_{g_{t}} \sum^{T}_{t=1} \; \left[\left(y_{t}-g_{t}\right)^{2} + \lambda \Big\{ \left( g_{t+1}-g_{t}\right) - \left(g_{t}-g_{t-1}\right) \Big\}^{2} \right] \end{eqnarray}$$
• This minimization problem has a unique solution, and the filtered series, $g_{t}$ has the same length as $y_{t}$
• Termed a low-pass filter as it only models low frequency data

Hodrick-Prescott (HP) filter

• The smoothness is determined by $\lambda$, which penalizes the acceleration in the growth component
• If $\lambda \rightarrow \infty$, the lowest minimum is achieved when variability in the trend is zero (as in the case of a linear trend)
• If $\lambda =0$ there is more variation in the trend, such that there will be no cycle
• Hodrick and Prescott argue that $\lambda =1600$ is a reasonable choice for U.S. quarterly data
• However, it is not necessarily the case that it can be universally applied, to other variables or output of other components
• Specifies a typical cycle of between eight to ten years when using traditional values for $\lambda$

Hodrick-Prescott (HP) filter

• Another concern is the end-of-sample problem:
  • trend is close to observed data at the beginning and end of the sample
  • problematic when we are at the peak of a cycle
  • some researchers use forecasts to generate additional data at the end of the series
• King & Rebelo (1993) note that the HP-filtered cyclical component contains both forward and backward differences
• As a result, the end of sample properties are poor when you do not have an observation for $t+1$ or $t-1$
• Method is also criticised on the basis that the smoothness of the stochastic trend component has to be determined a priori

Band pass filters

• Band pass filters introduced to economic data by Baxter & King (1999) and Christiano & Fitzgerald (2003)
• Identify all components that correspond to the chosen frequency band that has an upper and lower limit
• Need to determine the periodicity of the business cycles one wants to extract
• This is usually expressed within the frequency domain

Frequency-domain

• Consider a time series, $$\begin{eqnarray} y_{t}=A\cos (2\pi \omega t) \end{eqnarray}$$
• where $A$ is the amplitude (height) of the cycle
• $\omega$ is the frequency of oscillation (the number of occurrences of a repeating event per unit of time)
• $2 \pi$ measures the period of the cycles
• $t$ is the time
• Hence, if $y_{t}=A\cos (2\pi t)$, we will observe one cycle over the data sample
• By increasing $\omega$, we increase the number of cycles

Band Pass Filters

• An intuitive measure of frequency is the amount of time that elapses per cycle, $\lambda$ $$\begin{eqnarray} \lambda =2\pi /\omega \end{eqnarray}$$
• Where we have quarterly data, for $\omega$ corresponding to a cycle length of 1.5 years
• Set $\lambda=6$ quarters per cycle and solve for $6=2\pi /\omega_{h}$ $$\begin{eqnarray} \omega_{h}=2\pi /6=\pi /3 \end{eqnarray}$$
• Similarly, the frequency corresponding to a low frequent cycle length of 8 years is: $$\begin{eqnarray} \omega_{1}=2\pi /32=\pi /16 \end{eqnarray}$$

Baxter and King (1999)

• Baxter and King (1999) decompose a time series into three periodic components: trend, cycle, and irregular fluctuations
• Business cycles were defined as periodic components whose frequencies lie between 1.5 and 8 years per cycle
• Periodic components with lengths longer than 8 years were identified with the trend
• Periodic components of less than 1.5 years were identified with the irregular component $$\begin{eqnarray} B(\omega ) &=&1\text{ for }\omega \in \lbrack \pi /16,\pi /3]\text{ or }[-\pi /3,-\pi /16 ]\\ &=&0\text{ otherwise} \end{eqnarray}$$
• Hence, the interval $B(\omega ) = [\pi /16,\pi /3]$ can be interpreted as the business cycle frequency
• The interval $[0,\pi /16]$ corresponds to the trend and $[\pi /3,\pi ]$ defines irregular fluctuations

Band Pass Filters

• While Baxter and King favour a 3-part decomposition, other economists prefer a two-part classification
• This may be incorporated in this setup, where $$\begin{eqnarray} H(\omega ) &=&1\text{ for }\omega \in \lbrack \pi /16,\pi ]\text{ or }[\text{-}\pi ,-\pi /16] \\ &=&0\text{ otherwise} \end{eqnarray}$$
• The trend component is still defined in terms of fluctuations lasting more than 8 years
• Cyclical component now consists of all oscillation lasting 8 years or less
• This is known as a high pass filter, as only higher frequency components are captured in $H(\omega )$
• As with the HP filter one has to decide on the preferred frequencies for the cycles a priori
• There are potential end-of-sample problems, but usually eliminate the estimates at the start and end

Beveridge-Nelson Decomposition

• Beveridge & Nelson (1981) model the trend as a random walk with drift, and the cycle is treated as stationary process with zero mean
• To perform this decomposition, let $y_{t}$ be integrated of first order, so that its first difference, $\Delta y_{t}$ are stationary
• Assume it has the following moving average representation $$\begin{eqnarray} (1-L)y_{t}=\Delta y_{t}= \mu +B(L)\varepsilon_{t} \end{eqnarray}$$

Beveridge-Nelson Decomposition

• The BN decomposition explores the following
• First, define the polynomial, $$\begin{eqnarray} B^{\ast }(L)=(1-L)^{-1}[B(L)-B(1)] \end{eqnarray}$$
• where $B(1)=\overset{\infty }{\underset{s=0}{\sum }}B_{s}$
• Rewriting this polynomial in terms of $B(L)$, gives $$\begin{eqnarray} B(L)=[B(1)+(1-L)B^{\ast }(L)] \end{eqnarray}$$
• and substituting into the above yields $$\begin{eqnarray} \Delta y_{t}=\mu +B(L)\varepsilon_{t}= \mu +[B(1)+(1-L)B^{\ast}(L)]\varepsilon_{t} \end{eqnarray}$$

Beveridge-Nelson Decomposition

• For the decomposition, $y_{t}=g_{t}+c_{t}$, it follows that $\Delta y_{t}=\Delta g_{t}+\Delta c_{t}$
• Therefore a change in the trend component of $y_{t}$ equals $$\begin{eqnarray} \Delta g_{t}=\mu +B(1)\varepsilon_{t} \end{eqnarray}$$
• and the change in the cyclical component is, $$\begin{eqnarray} \Delta c_{t}=(1-L)B^{\ast }(L)\varepsilon_{t} \end{eqnarray}$$
• where we see that the trend follows a random walk with drift
• This expression can be solved to yield $$\begin{eqnarray} g_{t}=g_{0}+\mu t+B(1)\overset{t}{\underset{s=1}{\sum }}\varepsilon_{s} \end{eqnarray}$$

Beveridge-Nelson Decomposition

• As such, the trend consists of both a deterministic term $$\begin{eqnarray} g_{0}+\mu t \end{eqnarray}$$ and a stochastic term $$\begin{eqnarray} B(1)\overset{t}{\underset{s=1}{\sum }}\varepsilon_{s} \end{eqnarray}$$
• For $B(1)=0$, the trend reduces to a deterministic case
• where for $B(1)\neq 0$, the stochastic part indicates the long-run impact of a shock $\varepsilon_{t}$ on the level of $y_{t}$

Beveridge-Nelson Decomposition

• The cyclical component is stationary and is given by $$\begin{eqnarray} c_{t}=B^{\ast }(L)\varepsilon_{t}=(1-L)^{-1}[B(L)-B(1)]\varepsilon_{t} \end{eqnarray}$$
• Beveridge & Nelson (1981) showed that the stochastic trend could also be interpreted as the long-term forecast from RW plus drift model
• Cycle is the stationary process that reflects the deviations from the trend

Beveridge-Nelson Decomposition

• To estimate the BN decomposition in practice, assume an $AR(1)$ process for the growth rate of output $$\begin{eqnarray} \Delta y_{t}=\phi \Delta y_{t-1}+\varepsilon_{t}, \end{eqnarray}$$
• where we ignore the constant term
• Assuming $\phi <1,$ the $AR(1)$ process can be written in terms of the infinite order $MA(q)$ process where we find $B(L)$, $B(1)$ and $B^{\ast }(L)$ as $$\begin{eqnarray} B(L) &=&\frac{1}{1-\phi L} \\ B(1) &=&\frac{1}{1-\phi } \\ B^{\ast }(L) &=&(1-L)^{-1}[B(L)-B(1)]=\frac{\phi }{(1-\phi )(1-\phi L)} \end{eqnarray}$$

Beveridge-Nelson Decomposition

• Solving in terms of $y_{t}$ $$\begin{eqnarray} y_{t}=(1-L)^{-1}[B(1)+(1-L)B^{\ast }(L)]\varepsilon_{t} \end{eqnarray}$$
• which can be rewritten as $$\begin{eqnarray} y_{t}=B(1)(1-L)^{-1}\varepsilon_{t}+(1-L)^{-1}[B(L)-B(1)]\varepsilon_{t} \end{eqnarray}$$
• Substituting in for the $AR(1)$ solution derived above, we have $$\begin{eqnarray} y_{t} &=&g_{t}+c_{t} \\ &\Downarrow & \\ y_{t} &=&\frac{1}{1-\phi }(1-L)^{-1}\varepsilon_{t}+\frac{-\phi }{(1-\phi L)(1-\phi )}\varepsilon_{t} \end{eqnarray}$$

Beveridge-Nelson Decomposition

• The advantage of Beveridge-Nelson method is that it is appropriate when a series is difference-stationary
• It also allows the series to contain a unit root that can be highly volatile
• However, it has the disadvantage of being rather time-consuming to compute
• In addition, one has to choose between different $ARMA(p,q)$ models that may give quite different results
• Misrepresenting an $I(2)$ process as an $I(1)$ process may generate excess volatility in the trend

Summary

• Many economic and financial applications make use of decompositions for nonstationary time series, which are transformed into a permanent and a transitory component
• Could use a linear filter where trend is perturbed by transitory cyclical fluctuations
• The Hodrick-Prescott (HP) filter is the most popular way to extract business cycles
• The HP filter extracts a stochastic trend for a given value of the parameter $\lambda$
  • Trend moves smoothly over time and is uncorrelated with the cycle
  • Results are not robust to the value of the smoothness parameter
• Another popular method used to measure the business cycle is the band pass (BP) filter
• The filter removes (filters out) all the components in a series except those that correspond to the chosen frequency band

Summary

• In the Beveridge and Nelson (BN) decomposition, the permanent component is shown to be a random walk with drift
• The transitory component is stationary process with zero mean, which is perfectly correlated with the permanent component
• Different decompositions provide different results and should be interpreted with caution
• Usually a good idea to consider different options before drawing conclusions

Wavelet transformations

• Spectral decompositions define the rate at which the time series oscillates
• Results in the loss of all time-based information
• Assumes that the periodicity of all the components is consistent throughout the entire sample
• This may not be the case:
  • Gabor (1946) developed the Short-Time Fourier Transform (STFT) technique
  • Involves the application of a number of Fourier transforms to different subsamples
  • Precision of the analysis is affected by the size of the subsample
• Large subsample to identify changes in low frequency
• Small subsamples to identify changes in higher frequency

Wavelet transformations

• Wavelet transformations capture features of time-series data across different frequencies that arise at different points in time
• Wavelet functions are stretched and shifted to describe features that are localised in frequency and time
  • Could be expanded over a relatively long period of time when identifying low-frequency events
  • Could be relatively narrow when describing high frequency events
• Involves shifting various wavelet functions with different amplitudes over the sample of data
• One is then able to associate the components with specific time horizons that occur at different locations in time
• Wavelets use scales rather than frequency bands, where the highest scale refers to the lowest frequency

Wavelet transformations

• Early work with wavelet functions dates back to Haar (1910)
• See, Hubbard (1998) and Heil (2006) for a detailed account of the history of wavelet analysis
• For computation most studies currently employ the multiresolution decomposition of Mallat (1989) and Strang (1996)
• Early applications of wavelet methods in economics include the work of Ramsey (1997), which made use of a wavelet decomposition of exchange rate data to describe the distribution of this data at different frequencies

Wavelet transformations

• To describe this technique, consider a variable that is composed of a trend and a number of higher-frequency components
• The trend may be represented by a father wavelet, $\phi(t)$
• The mother wavelets, $\psi(t)$, are used to describe information at lower scales (i.e. higher frequencies)
• One could then describe variable $x_t$ as $$\begin{eqnarray} x_t = \sum_k s_{0,k} \phi_{0,k} (t) + \sum_{j=0}^{J} \sum_k d_{J,k} \psi_{J,k} (t) \end{eqnarray}$$
• where $J$ refers to the number of scales, and $k$ refers to the location of the wavelet in time
• The $s_{0,k}$ coefficients are termed smooth coefficients, since they represent the trend, and the $d_{J,k}$ coefficients are termed the detailed coefficients, since they represent finer details in the data

Wavelet transformations

• Mother wavelet functions, $\psi_{J,k} (t), \dots, \psi_{1,k} (t)$, are then generated by shifts in the location of the wavelet in time and scale $$\begin{eqnarray} \psi_{j,k} (t) = 2^{-j/2} \psi \left(\frac{t-2^{j}k}{2^j}\right), \;\; j=1,\dots,J \end{eqnarray}$$
• where the shift parameter is represented by $2^{j}k$ and the scale parameter is $2^{j}$
• As depicted in the daublet wavelet functions, smaller values of $j$ (which produce a smaller scale parameter $2^{j}$), would provide the relatively tall and narrow wavelet function on the left
• Larger values of $j$, the wavelet function is more spread out and of lower amplitude
• After shifting this function by one period, we produce the function that is depicted on the right

Wavelet transformations

• Wavelet functions may be:
  • Smooth - decompose data into trend, cycle, noise (or various cycles)
  • Peaked - identify peak and trough of cycle
  • Square - identify structural breaks
• Use smooth functions that include daublets, coiflets and symlets
• Multiresolution techniques are used for computation, which includes the maximum overlap discrete wavelet transform (MODWT)

Wavelet transformations – Summary

• Advantages:
  • Can be applied to data of any integration order
  • Has the benefits of spectral techniques without losing time support (very useful when identifying changes in the process at different frequencies)
  • Can include a number of bands, which are additive