Comparison of detrending methods

Transcription

1 Comparison of detrending methods Gabrielle Larsson and Tamás Vasi Department of statistics, Uppsala University Supervisor: Daniel Preve June 4, 2012 ABSTRACT This thesis examines the difference between the extracted cyclical components of some macroeconomic time series using four popular detrending methods HP, BK, CF and FOD. We use different approaches to compare their differences. A standard examination of the cyclical component is applied. We also take a frequency domain approach and examine the sample spectra for each cycle. Moreover, impulse responses and the correlation between the cyclical components extracted by each detrending method are studied. We conclude that for quarterly data HP, BK and CF produce similar cycles. However, when considering annual data the HP diverges from the BK and CF. The FOD extracts cycles that are not similar to those of the other three examined filters. Keywords: Business cycles, detrending methods.

2 Introduction To correctly estimate business cycles is essential for macroeconomic research. Canova (1998) gives two reasons for its importance. First, estimated business cycles help economists to calculate fluctuations caused by different types of economic activities. Second, the estimated business cycles give economists a benchmark to examine the validity of theoretical models. However, the measurement of business cycles involves the controversial practice of detrending time series, de Haan et al. (2008). Several methods of extracting the business cycle from a given time series have been developed, but none of them leads to inferior results to one another. According to Canova (1998), different detrending methods do not estimate the same cyclical components. The estimation of the business cycle is complicated because its movements are composed by low frequency trends and high frequency noise patterns. In order to deal with this issue different methods vary in the handling of trends and noise and thus the diverse results, Estrella (2007). Although there is a common agreement that business cycles ought to be identified as deviations from the trend of a time series there are fundamental disagreements about the connection between the trend and the cycle, Canova (1998). Early studies assumed that the trend and cyclical component could be studied separately. This was based on the intuition that short run and long run economic fluctuations had different impacts on the economy, Bjornland (2000). Nevertheless, this assumption was soon disputed by Nelson & Plosser s (1982) findings. Based on several historical time series for the US the authors could not reject the hypothesis of a unit root. The existence of unit roots in macroeconomic data thus implies that economic shocks will not only affect the cyclical component but will also have permanent changes on the trend component, Bjornland (2000). Considering this fact researchers are confronted with the complex task of separating the two components and therefore resulting in different filter methods. On the other hand, Burnside (1997) does not consider the different outcomes to be a major concern. He claims that the fundamental issue is that there is no common agreement among economists about the definition of a business cycle. As long as there are disagreements there will always be filters that are designed in various ways to extract the cyclical component. This thesis analyses and compares different detrending methods by applying them to macroeconomic data. We evaluate four frequently used filter methods the Baxter King (BK) band pass filter, the Christiano Fitzgerald (CF) band pass filter, the Hodrick Prescott (HP) filter and the first order difference (FOD). Our guidelines for an appropriate filter are mainly based on Baxter & King s (1999) requirements. They argue that the following properties should be achieved by a filter in order to consider it as a good filter. The filter should have the ability to eliminate unit roots and the capacity to remove the cyclical component. In other words the filter should isolate fluctuations of the data at a certain frequency, de Haan et al. (2008). Furthermore, we add that the filter ought to accurately extract turning points as an additional requirement for a good filter. In order to assess this necessity we use the NBER s (National Bureau of Economic Research) dating of recessions as a benchmark. Our aim is not to determine which of our examined filter is most appropriate. Instead we attempt to examine their differences. How much do these filtering methods differ? Are economists mislead by the 2

3 fact that they believe that all filters produce the same outcome? For this reason we also investigate the correlation between the extracted cyclical components. This thesis is partly based on four studies; Canova (1998), Bjornland (2000), Baxter & King (1999) and Estrella (2007). Canova (1998) investigates a set of US macroeconomic data using a variety of filtering methods. He concludes that different filtering methods produce different output. Bjornland (2000) confirm Canova s (1998) finding when she studies macroeconomic data for Norway using both a time domain and a frequency domain approach. Furthermore, Baxter & King (1999) compares the HP, FOD and the BK on US quarterly and annual macroeconomic data sets. They conclude that overall, FOD diverges considerably from the other two filters but, HP and BK extract similar cyclical components for quarterly data. On the other hand the similarities in the extracted components between the HP and the BK diminish when applied to annual data. Estrella (2007) argues that the similarities between the outputs for the different filters depend generally on the underlying data as the filters extract different frequencies from the data. The thesis is structured in the following way. In the first section we present the data, the different filters and our way to approach the research problem. First we follow Bjornland (2000) and divide our analysis into two parts: a time domain analysis and a frequency domain analysis. The time domain analysis illustrates the persistence of a business cycle for each filter method. Besides we make a standard examination of the cyclical components which include, for instance, the cyclical component s volatility. Our frequency domain analysis focuses, on the other hand, on which frequencies are emphasized by the different detrending methods. Moreover, we follow Canova s (1998) approach and include an Impulse Response Function (IRF) in order to determine how long a business cycle tends to last according to each filter. Besides, this method helps us to compare the time taken for each filter to react to a shock. We further evaluate by comparing recessions according to the examined filters with NBER as a benchmark. Furthermore, we attempt to answer the question if these filtering methods do produce similar results by examining how their cyclical components correlate with each other. We find that for quarterly data the HP, BK and the CF filters do extract similar cycles. However, for annual data the difference between the HP and the two band pass filters increases. For this reason we argue that conclusions drawn from the results obtained either by the HP, BK or the CF filter on quarterly data should not differ significantly. Nevertheless, caution ought to be taken when interpreting annual data. Furthermore, we conclude that the FOD produces cycles that clearly diverge from the other three examined filters. We therefore argue that drawing conclusions from cyclical components obtained by the FOD should be interpreted with caution. 3

4 2 Method 2.1 Description of the detrending methods In this thesis we aim to investigate the differences and the similarities of different detrending methods. We examine and compare the most widely used filters. According to van Aarle et al. (2008) and de Haan et al. (2008) frequently used filters are the BK, CF, HP filter and the FOD. Consequently our focus is on these filters The Baxter King filter The Baxter King (BK) filter is an approximation of an ideal band pass filter, Guay & St Amant (1997). As time series can be decomposed into various frequency components the ideal band pass filer extracts frequencies within predetermined ranges and eliminates all other frequencies. The ideal band pass filter is characterized to require an infinite set of data, Christiano & Fitzgerald (2003). The BK filter specifies the business cycle as fluctuations within specific ranges. The filter adopts Burns and Mitchell s (1946) definition of a business cycle for which a business cycle lasts no less than 6 quarters and no longer than 32 quarters. Furthermore, the filter assumes a symmetric finite moving average series where the cycles are passed through a predetermined lower and upper bond i.e. through the periods 6 and 32 quarters. The BK filter is a linear filter which eliminates trend components and high frequency irregular components while it aims to retain the intermediate cyclical components, Baxter & King (1999). The filter decomposes a time series irregular component., into three components: a trend, a cycle and an, (2.1) where is the trend component, the cyclical component and is the irregular component, Baxter & King (1999). A new time series is generated when a finite symmetric moving average is applied, (2.2) where are fixed constants or weights and is the maximum lag length. In order to extract the cyclical components from the time series with help of the above defined symmetric moving average the BK filter uses weights that add to zero,. This is the trendelimination property, Baxter & King (1999). When using weights that add up to zero the moving average has certain elimination properties and generates a stationary time series. This is an important feature as economic time series tend to be non-stationary, Baxter & King (1999). Baxter & King derive their filter through a frequency domain perspective. The starting point for the BK filter is the Cramér representation theorem which states that a covariance stationary process can be written as ( ), (2.3) 4

5 under suitable conditions. Here, the time series is expressed as the integral of random periodic components, ( ), where ω is radians per unit time. The random periodic components at different frequencies are uncorrelated. If Cramér theorem is applied to Equation (2.2) we get ( ) ( ), (2.4) where ( ) ( ) is the frequency response function of the filter. This could be interpreted as how much responds to at a given frequency with respect to the weight ( ), to the random periodic component ( ). It is important to note that, for the BK filter, ( ) has the value zero at frequency zero i.e. ( ) = 0. The frequency response function of the filter is also a band pass filter since it only passes frequencies that are within a pre-determined frequency band, Baxter & King (1999). The cyclical component is extracted in the following way where the weights function, Guay & St Amant (1997)., (2.5) are found by the inverse Fourier transform of the frequency response ( ) ( ) (2.6) Assuming Burns and Mitchell s (1946) definition has to be between and, the pre-specified duration for a business cycle. The BK filter is built up by two low pass filters as it has two pre-determined frequency bands, and. For this reason two frequency response functions have to be specified, ( ) for (zero otherwise), ( ) for. In order to obtain the weight,, for the cyclical component we extract ( ) from ( ) and thus we get a new frequency response function ( ) for or and zero otherwise. The can then be obtained by using Equation The Christiano Fitzgerald filter The Christiano and Fitzgerald filter (CF) is like the BK filter is also an approximation of an ideal band pass filter and has therefore basic similarities with it. BK assumes a symmetric moving average while in order to approximate an ideal band pass filter the CF on the other hand assumes that the time series, follows a random walk without drift, Christiano & Fitzgerald (2003). The cyclical component is estimated as follows for where, (2.7) ( ) ( ) 5

6 ,, and are the cut off lengths of the cycle, 6 and 32 respectively for quarterly data, Christiano & Fitzgerald (2003). That is, cycles that are longer than but shorter than are considered to be the cyclical component. Furthermore, we note that the CF filter puts different weights to each observation and hence the filter is not symmetric, Haug & Dewald (2004). A contrast to the BK is that the BK assumes the weights are fixed regardless of the number of observations. Besides, the CF is consistent compared to the BK as it converges to an ideal band pass filter as the sample size,, increases. The reason is that the approximation error of the weights diminishes as the sample size increases, Haug & Dewald (2004) The Hodrick Prescott filter The Hodrick Prescott (HP) filter is a smoothing method which aims to obtain a smooth component from the trend. Assume that we have: (2.8) where is the given time series which is decomposed into a trend component, and a cyclical component,. The trend and the cyclical components cannot be observed and for this reason the HP filter is used to estimate these components, Hodrick & Prescott (1997). Assuming that is stochastic over time and uncorrelated with, the trend component is estimated through the following minimization problem: { } { ( ) [( ) ( )] } (2.9) The first part of Equation (2.9) is a measurement of the goodness of fit, the squared sum of the difference between the time series variable and the trend ( ). In the second part, the variation in growth rate of the trend component is being measured, Marcet & Ravn (2004). The HP filter aims to minimize the distance between the given series and the trend while the growth rate of the trend component is minimized, Nilsson & Gyomai (2011). The multiplier,, is a positive number which penalizes the variability in the growth component series. A high value on gives a more linear (smoother) trend and allows for more variation in the cyclical component. For equal to zero the trend component is equivalent to the actual series,. The key determinant for the minimization problem (2.9) is the value of. For quarterly data Hodrick & Prescott (1997) suggest to be This value is widely used among researchers when dealing with quarterly data, Marcet & Ravn (2004). However, there are disagreements about the value of for annual data. There are a number of recommendations about the value 6

7 of. For instance, Baxter & King (1999) finds 10 whereas Ravn & Uhlig (2002) suggests 6.25 for annual data. Nevertheless, we intend to follow the standard practice for annual data for which is, according to Mohr (2005) and Backus & Kehoe (1992), 100. According to them there are two reasons for using =100: first, this value is the most frequently used in the literature and second, because the European Central Bank uses this value when approximating business cycles, Mohr (2005) The first order difference The first order difference (FOD) is a straightforward and commonly used method due to the fact that it removes a unit root component from the time series. To compute the FOD the following equation is used: (2.10) where is the observed time series. The trend is: (2.11) The cyclical component is the difference between the observed time series and the trend: (2.12) The underlying assumption is that the trend component is a random walk with no drift and that the cyclical component is stationary. The two components are uncorrelated, Canova (1998). 2.2 Data The analysis is based on four different macroeconomic data sets; GDP, consumption, investment and inflation for the United States. These data sets are frequently used in studies of business cycles, de Haan et al. (2008) and van Aarle et al. (2008). We investigate the cycles for both annual and quarterly data. We intend to answer the question if the filters produce different cycles when applied for different periods. For this reason we use two quarterly and two annual data series. GDP and consumption is measured quarterly and investment and inflation is measured annually. GDP is taken from the National Bureau of Economic Research (NBER), measured in billions of chained 2005 U.S dollar (adjusted for inflation over time). The time period is from 1947Q1 to 2011Q4. From the Bureau of Economic Analysis (BEA) we use quarterly consumption (1950Q1-2011Q4) measured in billions of U.S. Dollar, in 2005 prices. For investment and inflation the data is taken from the World Bank and is measured annually. Inflation is measured during the period 1961 to 2010 and investment consists of data from 1960 to We use the logarithm of the data sets GDP, consumption and investment. Before we proceed further we test if the series are stationary. The data sets are tested for a unit root and we conclude that we can not reject the null hypothesis that the data is non-stationary at 5% significance level 1. Following this conclusion one desirable requirement is that the 1 ADF unit root test, see table 2.1 in Appendix 2. 7

8 detrending methods succeed to remove the unit root from the time series. This is investigated in a later section. 2.3 Description of our analysis Standard examination of the cyclical components In this section we aim to investigate the basic properties of the extracted cyclical components. First, as previously mentioned, macroeconomic data tend to be non-stationary and for this reason a central requirement for a filter is to have the ability to remove unit roots. Therefore we perform unit root tests of each extracted cyclical components. Second, we follow Canova s (1998) approach and examine the first three moments of each extracted component. We ask ourselves which of the filtering methods produce cycles that are in line with economical intuitions. Lastly, we take a time domain approach proposed by Bjornland (2000) and analyze the correlation properties of the cyclical components. We also check the duration of the business cycles for each extracted series. Does the persistence of business cycles depend on which filter is used? Frequency domain analysis Several articles such as Canova (1998), Estrella (2007) and Bjornland (2000) examine the filters through a frequency domain approach. In this paper we intend to investigate the sample spectrum of our examined filters. Our purpose with spectral analysis is to identify the frequencies on which the cyclical components are based. The question is whether the extracted cyclical components consist of low- and intermediate frequencies. If so, then the detrending methods have been able to remove the cyclical behavior of the time series in a proper way. Spectral analysis assumes that the Fourier transform exists which is true for a stationary process with absolutely summable autocovariance function,, Wei (1994). For this reason it is necessary for the examined filters to render series without unit root components. The spectrum of the cyclical components is estimated in the following way: ( ) ( ) ( ), (2.13) where ( ) is the estimated spectrum and is the sample autocovariance at lag k. Even if ( ) is an asymptotically unbiased estimator of ( ) it is not a consistent estimator and should thus be used by caution when estimating ( ), Wei (1994). An increased number of observations will therefore not reduce its variance, Cryer & Chan (2008). And so in order to reduce the variance of the sample spectrum, we smooth the spectrum using the Bartlett s Window, see for example Bjornland (2000). The consequence of smoothing is that the variance is reduced but, the estimator of ( ) becomes biased, Wei (1994). For a more detailed discussion of the spectrum see the Appendix 1, Section

9 2.3.3 Impulse response function The impulse response function (IRF) is commonly used to study the effect of a shock to the economy by one unit from macro economic variables. In this thesis we have two reasons for considering the IRF. Firstly, to study the time it takes for the peak response to occur after a shock and also to measure the magnitudes of the peak response (compared to the size of the shock). We are not interested in interpreting the response from the shocks but to see how the responses differ (or not) between the detrending methods. Is there any difference in the response between the detrendning methods? Secondly, we want to find the length of a cycle as a response to the shock. There are several ways of measuring the cycle length and the result from the IRF is compared with the frequency domain analysis. An IRF traces out the effect on the response variable when being positively shocked by the impulse variable by one unit. The effect is measured on periods in the future and all other potential shocks are set to zero, van Aarle et al. (2008). When shocking the variables in the model there are several responses, the lagged dependent variable is affected as well as the other endogenous variables in the model. The IRF is usually obtained from a Variance Autoregressive model (VAR). In our model the original and cyclical time series are used as endogenous variables (and a constant is used as an exogenous variable). Consider the following VAR (1) systems with two equations:, (2.14), (2.15) where and appear as lagged variables in their own equations as well as lagged exogenous variable in the other equation. This shows that when there is a shock in, there will also be an effect on. Suppose that at time t there is a shock to by one standard deviation,, and is not shocked. Moreover, suppose that for period there is non shock in either or, Carter Hill et al. (2008). The response to the shock for future time periods will then be: Time (when the shock occurs): The effect on is and on there is no effect Time : The effect on is and the effect on is Time : The effect on is ( ) and the effect on is In future periods, the effect from the shock will die out for typical values of the 9

10 In our analysis the original times series is the impulse and the cyclical component of the series is the response. The decomposition method is measured in one standard deviation of the residual, analogues to Canova (1998). Consider a VAR (k) system with two equations:, (2.16), (2.17) where is a constant ( ), is the original time series and is the cyclical component of the time series. In the equations above, K is the number of lags and is a stochastic error term ( ). To decide the number of lags in the model we use the Akaike Information Criterion (AIC). The number of recommended lags varies among the different detrending methods and could be seen in Appendix 2, Table 2.2. It ranges between 2 and 4 for annual data and it is between 3 and 8 for quarterly data, common values for quarterly respectively annual data, Kilian (2001). 3 Empirical results 3.1 Standard examination of the cyclical components In a study by Nelson and Plosser (1982) the hypothesis of a unit root in several macroeconomic data could not be rejected. Their finding was fundamental in a way that they contradicted the previous thought that a time series will not drift away from its mean value. As a consequence each shock has a permanent effect on the series, Bjornland (2000). Because of the non-stationarity the behavior of the series depends on a specific time period and has thus little practical value when considering the business cycle. We therefore investigate the first requirement of a good filter, proposed by Baxter and King (1999), that a filter should have the ability to remove unit roots. For reason of space the plots and tables are only presented for GDP in this section. The results for consumption, investment and inflation are reported in the Appendix 2, Section 2.4. ADF ( ) PP ( ) BK-filter -6,258* -4,709* CF-filter - 2-5,077* HP-filter -7,634* -5,791* FOD -10,985* -8,001* Table 3.1. Unit root tests of the cyclical components obtained by the different detrending methods for GDP. * indicates the rejection of the null hypothesis at a 5 % significance level of a unit root. In order to examine whether the extracted cyclical components have unit roots we use the ADF test. According to the ADF unit root test the null hypothesis that the cyclical components have a unit root is rejected for GDP. We report similar results for consumption, inflation and investment, see Appendix 2, Table 2.3, 2.6 and The error message Near singular matrix occurred when the Eviews software was used. 10

11 However, Perron (1989) shows that the results from the ADF test are not reliable if structural changes are occurring in the observed time series. Considering the extracted cyclical components we spot several potential structural breaks. For this reason we follow Estrella (2007) and also apply another unit root testing, the Phillips-Perron (PP) test. This is a nonparametric alternative to the ADF test, Gujarati & Porter (2009). Nevertheless, even with the PP test we reject the null hypothesis that there is a unit root in the examined cyclical components. Conclusively, there is evidence that the different filtering methods have been able to remove unit roots from the time series. Therefore all of our detrending methods have satisfied our first requirement for a good filter. We pursue further to consider a visual examination of the extracted business cycles. Figure 3.1 displays the results obtained from the four filters when applied to GDP. BK-filter CF-filter q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 HP-filter q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 FOD-filter q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 Figure 3.1 Business-cycles obtained from the BK, CF, HP and FOD filters. There are visual similarities between the BK, CF and HP. It seems like the band pass filters and the smoothing linear filter resemble each other. However, it is observed that the two band pass filters extract smoother cycles compared to the HP. Examining specific recessions we note that the economic contraction in 1949 was substantial according to all three filters. Nevertheless, we note that the HP depicts this economical downturn to be relatively more severe than compared to the band pass filters. Moreover, our result supports the finding of Canova (1998) as the three filters illustrate the economic downturn in 1979 due to the oil crisis only as a minor slowdown in the economy. For the recent financial crisis, , the BK, CF and the HP show a significant decline in the economy. We conclude that visually these three filters produce similar cyclical components. 11

12 BK-filter CF-filter HP-filter* FOD* Standard deviation 0,0156 0,0163 0,0169 0,0010 Skewness 0,4441-0,1983-0,5608-0,1334 Kurtosis 3,1006 3,1068 3,4956 4,3762 Table 3.2 Summary statistics, * indicates a rejection at the 5% significance level of the null hypothesis that the cycles are normally distributed (Jarque-Bera test). The cyclical component extracted by the first order difference deviate from the outputs of the other three detrending methods. We observe a lower variability of the output obtained by the FOD filter. This finding is supported by Canova (1998) who notes that the FOD generates relatively lower cyclical components. The reason for this is clarified in our frequency domain approach in the next Section. When examining specific recessions it is observed that the FOD filter shows similar large contractions in 1949 and However, the filter responds to the oil price shock in 1979 in a more severe way than the other three filters which coincide with the result found by Canova (1998). Considering the behavior of the extracted components for consumption, inflation and investment we observe, not surprisingly, that the two band pass filters resemble each other. Furthermore, the cycles extracted from quarterly data by the HP filter have similar patterns as the band pass filters. However, for annual data the HP filter displays higher volatility than the BK and CF filter. Annual cycles obtained by the FOD seem to be even more volatile than for the HP filter but, for quarterly data the FOD extracts cycles with lower volatility. Nevertheless, for data on annual basis the cycles extracted by the FOD do show more similar fluctuations with the other filters. We also examine the third and fourth moments for GDP. Several works, such as Neftci (1984) and Falk (1986), have suggested that business cycles are asymmetric. The economic intuition behind this is that contractions tend to last for shorter time periods than expansions but are at the same time more severe and thus generating the asymmetric behavior of the time series, Neftci (1984). Studying Table 3.2 we note that for the HP, CF and for the FOD filters the skewness is slightly negative which implies that the distribution has a long left tail. The BK on the other hand has a positive skewness indicating a long right tail. Additionally, we could not reject the null hypothesis for the two band pass filters that the extracted business cycles are normally distributed. This suggests that the extracted cyclical components using these two filters are symmetric. In contrast, the cyclical components extracted by the HP and FOD filters appear to be asymmetric. Even though our results for the two band pass filters refute the economic theory of asymmetric business cycles there are studies, such as De Long & Summers (1986) that found no evidence for asymmetries in macroeconomic data. As a result of the ambiguity of the symmetrical behavior of the business cycle we conclude that the issue with asymmetry does not have a first order importance. 12

13 Lag BK-filter CF-filter HP-filter FOD 1 0,894 0,906 0,843 0, ,652 0,658 0,601 0, ,356 0,338 0,328 0, ,088 0,035 0,086-0, ,096-0,186-0,100-0,140 Table 3.3 Summary of autocorrelations up to lag 5 for each detrending method. Finally, we approach our comparison of the filters using a time domain analysis. We follow Bjornland (2000) and investigate the first five lags of the sample autocorrelation. When the autocorrelation in the extracted cycles is positive it tells something about the persistence of the business cycle, Bjornland (2000). Observing the cyclical components from GDP we conclude that the cycles extracted by the HP, BK and the CF filters behave in a similar way indicating slowly decaying autocorrelations. The duration of a business cycle for these filters is approximately 4 lags. However, the FOD produces lower autocorrelations and thus also lower persistence. Once again, the explanation for this phenomenon is discussed in the next section. Investigating the autocorrelations for annual data, inflation and investment, we conclude that the autocorrelations are changing between positive and negative values indicating a noisy pattern. In contrast, the cyclical components extracted by the HP, CF and BK filters for consumption based on quarterly data indicate slowly decaying positive autocorrelations. 3.2 Frequency domain analysis In this section we examine the cyclical components extracted by the different filters using a frequency domain approach. A good filter should preserve intermediate frequencies and eliminate high frequencies (noise) and low frequencies, Baxter & King (1999). The detrended GDP, inflation, investment and consumption are investigated and compared using spectral analysis. Moreover, as previously mentioned, spectral analysis should only be performed if the data is stationary. In Section 3.1 we showed that the cyclical components are stationary and therefore can be analyzed using spectral analysis. 13

14 Spectral Density of BK_cycle Spectral Density of HP_cycle BK filter Frequency from 0 to PI HP filter Frequency from 0 to PI CF filter Spectral Density of CF_cycle Frequency from 0 to PI FOD filter Spectral Density of FOD_cycle Frequency from 0 to PI Figure 3.2 Spectral density of detrended GDP. Observing the sample spectra for the detrended quarterly GDP it is noticed that the two band pass filters and the HP filter have extracted the cyclical component at similar frequencies. The spectrum of the HP filter has its power at relatively low frequencies while the high frequencies are negligible. Our result corresponds somewhat to the result found by Bjornland (2000). She also concludes that HP obtains frequencies at low and intermediate frequencies. Nevertheless, this comparison should be taken with caution as her study is investigating GDP for Norway. Additionally, the power at zero frequency seems to be removed by the HP filter. This suggests that the trend component is extracted by the filter, Wei (1994). The reason why zero frequencies are removed is because the HP filter places zero weights on these frequencies, Baxter & King (1999). Considering the two band pass filters we see that they have power at relatively low intermediate frequencies. While the BK filter has power at zero frequency the CF filter displays no power at this particular frequency. Hence the BK filter does not entirely eliminate the trend component. Therefore our result contradicts Baxter & King s (1999) claim that their designed BK does place zero weight at zero frequencies. However, we note that in this study we deal with sample spectra which might have been the cause of the different result. The HP, BK and CF filters display similar sample spectra as their peaks are at approximately the same frequencies for both quarterly GDP and quarterly consumption. The reason for this is that the HP filter is generally considered as a good approximation of a band pass filter for quarterly data. Defining the smoothing parameter for the HP as =1600 the cyclical component obtained by the filter should be similar to the output from the band pass filters, Baxter & King (1999). This is emphasized by the previous section where we found that the 14

15 extracted cycles by the HP, CF and BK follow similar patterns for quarterly data. Nevertheless, we also reported in the previous Section that the cycles extracted by the BK and the CF filters appears to be smoother than the HP filter. The reason is that the HP filter put weight, even if it is low, on high frequencies whereas the two band pass filters do not put any weight on these frequencies. On the other hand the sample spectrum for the FOD filter differs from the other filters. The peak seems to be at higher frequencies. Moreover, the filter appears to emphasize both high and low frequencies. Additionally, compared to the other filters the FOD filter put significantly more weight both to low and high frequencies. The fact that the FOD highlights high frequencies was observed in the previous section where we saw that the cycles extracted by FOD had spiky appearances with low volatility. We extend our frequency domain analysis and investigate the periodogram of the sample spectra for the cyclical components extracted from GDP, for results see the Appendix 2, Section 2.5. This helps us to identify an approximation of the length of a business cycle according to each filter. The HP, CF and BK filter once again indicate similar results. The data from the HP filter suggests that a business cycle lasts approximately quarters which is strikingly similar to the result obtained using the BK filter, quarters. By comparison, the CF filter shows a slightly different length of a business cycle. According to this filter a business cycle persists for 26 quarters. These reasonably similar business cycle lengths obtained by the filters yet again emphasizes that the HP filter is a good approximation of a band pass filter for quarterly data. In contrast, the FOD once again diverges from the rest of the examined filters. According to this filter a business cycle persists for roughly 8.93 quarters. This is a result that significantly differs from the other filters. Nevertheless, the investigated filter methods do have business cycles which are within the business cycle span defined by Burns & Mitchell (1946). We continue by investigating the sample spectra for annual data, see the Appendix 2, Section 2.5 for results. The two band pass filters exhibit similar spectral densities for annual data. However, the trend component still persists in the investment series filtered by the BK-filter. Moreover, our result seems to contradict Baxter & King s (1999) finding that the HP filter is a poor estimate of a band pass filter for annual data. The authors claim that the reason for this is that there are controversies of what value should have for annual data. However, our results do not indicate any significant differences between these filters as their peaks are roughly at the same frequency. We note that in their study they set =10 whereas we followed the commonly used value =100. Nevertheless, we observe that for inflation the HP filter put more emphasize on lower frequencies than the band pass filters indicating a higher volatility, see previous Section. Our result seems to be the opposite from those of Baxter & King (1999). According to Baxter & King (1999) inflation is generally more volatile than other macroeconomic data and for this reason the HP filter allows more high frequency components to pass while the band pass filters remove these components. 15

16 The FOD peaks at similar frequencies for annual data as the other three filters. This indicates that for annual data the FOD extracts the cyclical components at similar frequencies. We point this out in the previous section where we observed that for annual data the cyclical components extracted by the FOD resemble somewhat the other three filters output. On the other hand, as seen for quarterly GDP, the filter still emphasizes relatively more high and low frequencies than the HP, CF and the BK filters. Our frequency domain analysis is based on the smoothed sample spectrum which is a biased estimator of ( ) and hence we acknowledge the limitation of this part of our analysis. Therefore the comparison with other studies should be taken with caution. 3.3 Impulse response function In a study by Canova (1998) a comparison of the reaction from a shock to the cyclical component of the time series is performed for different filter methods using impulse response functions. From IRF Canova (1998) also measures the length of a cycle as a response to the shock. He concludes that the reaction is to some extent method dependent and also varying among different data sets. The same procedure is performed in this thesis and the result is compared with Canova s (1998) as well as with those from the frequency domain analysis in the previous section. The first part is the analysis of the length of the GDP cycle and the second part is a comparison of the time to peak response after a shock by one standard deviation. The length of the cycle is defined as the time needed to complete a cycle measured in quarters. In the second column of Table 3.4 the duration of a GDP cycle as response to the shock is presented. The length varies remarkably among the different methods, for BK the cycle is more than three years longer than for the other filters. All cycle lengths are within the range proposed by Burns and Mitchell (1946). Canova (1998) did a similar IRF study where he finds that a business cycle lasts for 20 and 6 quarters for HP and FOD respectively. Our result differs with one year for HP. The result for FOD show a more than doubled length, 14 quarters compared to 6 by Canova. In the frequency domain analysis the business cycle was found to last for approximately 22 quarters for BK and HP, compared to the results using the IRF the difference is 1.5 years. For the CF filter the persistence of the cycle is twice as long, 26 quarters compared to 13 by IRF. FOD has duration of 9 years in the periodogram while the length is 14 quarters in Table 3.4. The result in the periodogram for HP and FOD is more similar to that obtained by Canova (1998). However, Canova used a different data set and therefore the similarities should be taken with caution. We can thus agree with the conclusion drawn by Canova (1998): different detrending methods yield cycles of different average duration in the data. 16

17 Method Cycle length* Quarter and height at the first peak response GDP Consumption Investment Inflation BK , , , ,5140 CF , , , ,2649 HP , , , ,5571 FOD , , , Table 3.4 IRF results for all detrending methods. *Where cycle length is measured in quarters. It is also of interest to investigate the time it takes for the first peak to occur after the shock, this is reported in Table 3.4 (columns 3-10). The quarter where the first peak occurs after a shock varies among the different methods and data sets. FOD has the fastest peak response (4 quarters) for GDP and consumption while the slowest are the HP- and CF-filter for inflation. The responses are visualized in Figure 3.3 for GDP and in the Appendix 2, Section 2.6, for the other data sets. 20 BK-filter 20 CF-filter HP-filter FOD Figure 3.3 Impulse responses for GDP and the four detrending methods. The response is shown for 25 quarters from when the shock occurred. The BK, HP and FOD filters produce similar results with a first peak response between 4 to 9 quarters after the shock; the length of the peaks is about 3 quarters. After the peak response the effect diminish, the method with the shortest time to peak does also diminish the fastest. The HP filter responds negative to the shock at the first quarters. The graph for the CF filter is oscillating over time and is not affected by the shock until the third quarter. Firstly, it has a negative response and then it peaks at the 16th quarter. The graph for the CF filter has a different movement suggesting that this filter is not responding as much to the shock as the other methods. 3 The error message Near singular matrix occurred when the Eviews software was used. 17

18 For consumption the peak response occurs within three years for all methods. The responses are in general much smaller than the size of the shock and especially small for CF. Also for consumption HP is initially responding negative to the shock. The first response for inflation is negative for BK, CF and HP. The peak response is after 24 quarters for CF and HP compared to 16 for BK. After the peak responses all series oscillates before they die out. The responses to a shock in investment are the most equal for the different detrending methods. The peak response occurs somewhere between 12 to 16 quarters after the shock. For the BK and the CF the responses are similar, the first effect is negative and the series oscillate and dies out. The HP has a peak response and then dies out while FOD oscillates. As Canova (1998) concludes, the effect for the different methods is varying. The magnitude of the response for CF is smaller than for the other methods and significantly smaller than the size of the shock. In general the magnitude of the peak response is smaller than the shocks, the opposite was concluded by Canova (1998). However, it is worth mention that the idea of computing an IRF for these filters is not flawless. Burnside (1997) criticizes the IRF analysis made by Canova (1998) arguing that the procedure is impossible to draw any real conclusions from. 3.4 Evaluation using NBER A possible benchmark for evaluating the business cycle components extracted using the various filters is to compare how well they match the recessions according to the NBER, Estrella (2007). NBER take a different approach to measure recessions. Although their guideline for recession is identified as the general definition of two or more quarters of real GDP decline they also include a range of other indicators. For instance, NBER put a lot of emphasize on various monthly data and employment, NBER (2012). Furthermore, it is interesting to examine how the theoretically based approach used by NBER compares to our filters. In the following section we therefore compare how the business cycles extracted using the examined filters resemble the recessions according to NBER. The shaded areas in the figures below indicate recessions according to NBER s measurements. Visually BK s recessions differ from NBER. Our result coincides with Estrella (2007) that in some instances the series produced by the BK tends to peak before the shaded regions then fall to the lowest point at the end of each recession. Moreover, according to the BK the recessions were more severe during the early nineties and CF and HP filters indicate a similar pattern. Although the turning points correspond to NBER chronology we observe that there are more recessions according to the series extracted by both the CF the HP filters given NBER s recessions. Visually, even if the series recession extracted by the FOD seems to reasonably correspond to NBER this filter turns out to be the least accurate when detecting recessions. 18

19 .06 BK-filter.06 CF-filter q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 HP-filter q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 FOD q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q q1 1954q3 1962q1 1969q3 1977q1 1984q3 1992q1 1999q3 2007q1 Figure 3.4 Cyclical component of GDP, the shaded lines are NBERs measure of recession. Of course visual examination could be argued to be a questionable procedure. For this reason we follow Estrella s (2007) approach and estimate a probit equation in which the recessions according to NBER are the binary dependent variables. Our aim is to examine the pseudo in order to obtain a measure of goodness of fit. BK-filter 0,059 0,818 CF-filter 0,070 0,808 HP-filter 0,098 0,827 FOD 0,419 0,897 Table 3.5 Measures for goodness of fit, McFadden and, in probit equations Table 3.5 provides us with the pseudo estimated by each probit equation. Examining and we observe that FOD has the highest value. Furthermore, we note that the HP-filter produces relatively higher pseudo values than the two band pass filters. Although Estrella (2007) applied his own modified pseudo see Estrella (1997), our result is in line with his study. Estrella (2007) includes filters such as the BK and the HP and recognizes that the HP produces a better relative fit than the BK-filter. On the other hand, we should be cautious when interpreting the obtained pseudo. What actually matters when using binary regression is the sings of the coefficients and their statistical significance. The goodness of fit is thus of secondary importance, Gujarati & Porter (2009). For this reason we ought to be careful when drawing the conclusion that FOD 19

20 produces business cycle components that best matches the recessions according to the NBER specially when considering the visual examination. 3.5 Correlation between cyclical components Until now the comparison of the different filters has mainly been based on how similar they perform and react using different statistical procedures. Another measurement of interest is to see how equal the cyclical component is between the methods. In Table 3.6 the sample correlation matrix for the cyclical components of GDP is reported. The correlations between the BK, HP and CF-filters are high (>0.88) while FOD has a low correlation with the other detrending methods (<0.22). Van Aarle et al. (2008) reports a similar result when studying GDP in the EURO area; high correlations for BK, HP and CF and low correlation for FOD with the other methods. The correlation matrix for consumption (Appendix 2, Section 2.7) is very similar to that for GDP, high correlations for BK, CF and HP-filter and low for FOD. BK CF HP FOD BK CF HP FOD * Table 3.6 Correlation matrix for the cyclical components of GDP extracted using the different detrending methods. *Non significant correlation at a 5% significance level. According to Baxter and King (1999) the correlation between BK and HP for GNP is estimated to be As argued in the previous sections, the HP filter approximates a band pass filter for quarterly data and because of this fact many similarities has been shown that can help to explain why the correlations between HP and the band pass filters are high. Besides, de Haan et al. (2008) conclude that HP and the band pass filters commonly yield similar results. In the sample correlation matrix, Appendix 2, Section 2.7, for investment and inflation the observed correlation between FOD and the other detrending methods is much higher (>0.38) compared to GDP and consumption. As well as for GDP and consumption the observed correlations for BK with HP and CF are high while the correlation between CF and HP is lower (<0.82). The results confirm and emphasize the findings of the previous sections. There are similarities between HP and band pass filters, this could be seen visually as well as when analyzing the correlations between the methods. FOD is the method that differ the most from the other filters. Compared to the other methods, the volatility of the FOD is much lower for quarterly data and this is also where the main difference is observed. For investment and inflation the data is measured annually and the number of observations is lower and therefore the results for the four detrending methods are more equal. 20