Empirical business cycle research typically commences with the extraction of a so-called deviation cycle using a time-series smoothing filter. This methodology is appealing for its pragmatism; it is easy to implement, and the output it produces is conveniently interpreted as percentage deviations from the natural level of output. However, recent literature offers staunch criticism of deviation cycle analysis, especially with regards to the assumption implicitly underlying it – that business cycle fluctuations are restricted to distinct intervals on the frequency domain.

Despite its lack of a basis in theory, the analysis of deviation cycles over particular frequency ranges may still yield useful stylised business cycle facts. This, however, hinges on whether the information that a frequency filter captures consistently aligns with relevant theory-based business cycle concepts. Whether this is the case is an empirical matter, and herein lies the rationale for our research.

We investigate the informational content of South Africa’s output deviation cycles.

We extract deviation cycles at standard high- and medium-frequency ranges (denoted as short- and medium-term deviation cycles respectively) and analyse their informational overlap with the components of an alternative theory-based estimate of the business cycle, decomposed into demand, supply, domestic and foreign sources of business cycle dynamics.

Our findings suggest that the contents of deviation cycles extracted over a high-frequency range do not neatly correspond to the transitory ‘demand-driven’ business cycle, while cycles extracted over a medium-frequency range correspond closely to the combined path of permanent output shocks.

One should thus be cautious of drawing strong conclusions about the nature of business cycles from filter-based deviation cycle estimates, particularly if the objective of the study relies on assuming that high-frequency deviation cycles correspond to transitory demand shocks.

Empirical business cycle research typically commences with the extraction of a so-called deviation cycle using a time-series smoothing filter. This methodology is appealing for its pragmatism; it is easy to implement, and the output it produces is conveniently interpreted as percentage deviations from the natural level of output. However, recent literature offers staunch criticism of deviation cycle analysis, especially with regard to the assumption implicitly underlying it: that business cycle fluctuations are restricted to distinct intervals on the frequency domain. If permanent shocks are a significant driver of the business cycle (as real business cycle theory suggests), business cycle dynamics may be inextricably linked to the low-frequency permanent component of output, presenting challenges for the core assumption underlying frequency-based business cycle analysis (Canova

Despite the lack of a neat alignment between method and theory, the analysis of deviation cycles over particular frequency ranges may still yield useful stylised business cycle facts. This however hinges on whether the information that a frequency filter captures, consistently aligns with relevant theory-based business cycle concepts. Whether this is the case is an empirical matter, and herein lies the rationale for our research. We investigate the informational content of South Africa’s output deviation cycles extracted at standard high- and medium-frequency ranges (denoted as short- and medium-term deviation cycles respectively) by comparing them with the components of an alternative theory-based estimate of the business cycle, decomposed into demand, supply, domestic and foreign sources of business cycle dynamics.

Our theory-consistent estimate of the business cycle consists of structural shocks to real output, which we estimate via an open-economy structural vector autoregressive (SVAR) model and identify by imposing long-run restrictions in the style of Blanchard and Quah (

Subsequent to obtaining our estimates of the business cycle, the bulk of our analysis centres on simple Pearson correlations between our statistically identified components of the business cycle and our short- and medium-term deviation cycles estimates. While rudimentary, we deem this approach appropriate and sufficiently robust, given that we are comparing information extracted from the same time series. However, we supplement our analysis of the informational content of deviation cycles extracted over medium-range frequencies by testing for cointegration with the decompositions of our benchmark SVAR business cycle estimate. We thereby take advantage of the apparent nonstationarity of our medium-term deviation cycles estimate to determine what information is sufficient to render this time series stationary. Given that cointegration in a univariate setting implies that the two series contain the same underlying stochastic trend (Engle & Granger

Harding and Pagan (^{1}

Deviation cycle analysis is based on the decomposition of a time series into a growth component and a cyclical component. When applying this decomposition to real output data, the cyclical component is regarded as a measurement of the business cycle, and the permanent component is often interpreted as a measure of Lucas’s (^{2}

The widespread usage of the deviation cycle method makes due consideration of its weaknesses a worthy concern. In this regard, Harding and Pagan (

Despite this identification deficit, deviation cycles may still yield useful stylised facts and insights into business cycle dynamics. Frequency filters are advantageously flexible, allowing researchers to check the robustness of their results by isolating and analysing cyclical variation in real output at various frequency ranges, and recent research based on deviation cycle analysis has used this flexibility to conduct business cycle research that investigates and accounts for the impact of frequency range choices on stylised business cycle facts. For example, Comin and Gertler (

It is on account of this problem that we provide this evaluation of the informational content of deviation cycles in South African real output. We extract deviation cycles over frequencies conventionally used to capture short- and medium-term business cycle movements and compare them with business cycle estimates obtained from a structural econometric model. As discussed below, our SVAR estimate of the business cycle can be decomposed into transitory and permanent shocks (loosely interpreted as demand and supply shocks) and into domestic and foreign shocks. Comparison with these decompositions of the business cycle allows us to observe the extent to which deviation cycles, extracted at different frequencies correspond to these sources of business cycle fluctuations in South African real output.

We use the Christiano and Fitzgerald (

As noted previously, filters are statistical instruments with no basis in economic theory. However, an alternative approach to estimating business cycles that grapples directly with the identification problem is the SVAR identification strategy developed by Blanchard and Quah (

Following this work, various authors investigate the sources of business cycles using VAR estimation and the Blanchard-Quah identification strategy, and some have subsequently extended the framework to a greater number of variables, thus increasing the number of distinct structural shocks that may be estimated. For instance, Karras (

The study by Du Plessis et al. (

Variable selection is the first step to achieving identification via the Blanchard-Quah methodology. Following Clarida and Gali (_{t}) along with two sources of demand shocks, namely government expenditure as a percentage of GDP (_{t}) and the real interest rate (_{t}). Tests for unit roots in these variables were performed and the results are excluded for brevity (results are available upon request). We find a unit root in all series except for the real interest rate.

The model estimated by Du Plessis et al. (

Variable section is important for obtaining a well-specified VAR, but it should be noted that our interest lies in the evolution of the structural shocks underlying these variables and the extent to which they determine the evolution of real output in particular. We are not interested in the causal parameters of these variables as determinants of real output. Additional variables that may constitute further sources of transitory and permanent shocks abound, but we have chosen to limit our specification to these five variables on account of data availability, the precedent set by Du Plessis et al. (

The next step in following Blanchard and Quah (_{t} is difference stationary, we hold the argument maintained by Du Plessis et al. (^{3}_{t} in levels destabilises the system.

Our system of equations can be represented as a vector moving-average process of the form _{t} = _{t}, where

_{0} + _{1}_{2}^{2} + _{3}^{3} … in the lag operator _{n} for _{t}_{−}_{n} on _{t}. The matrix _{t}.

Each of the five structural shocks contained in the vector _{t} are assumed to be independently, identically distributed and serially uncorrelated. We will refer to these, from left to right, as the oil price shock _{t} to impact _{t} to an arbitrary extent. Consequently, given the current state of the matrix _{t} = _{t} is an unidentified VAR. However, we can identify the structural shocks in _{t} by placing a sufficient number of restrictions on _{t} via imposing theory-based restrictions on the long-run impact of the shocks _{t} on the variables _{t}.

In addition to standard normalisation assumptions on _{t}, the remaining assumptions required to achieve identification are a set of restrictions on _{t} are independent and serially uncorrelated, we can identify _{t} by imposing restrictions on the matrix:

This we obtained by setting _{t} to a single pulse of all five elements of _{t}. In its current state, the matrix _{t}.

Following Balcilar and Tuna (_{12} = 0. Next, the assumption that South Africa is a small open-economy implies that domestic shocks to domestic variables have no long-run impact on foreign variables, and as such that _{13} = _{14} = _{15} = _{23} = _{24} = _{25} = 0. Lastly, we assume that monetary shocks _{34} = _{35} = _{45} = 0. Incorporating the above on the matrix

From these assumptions we have a sufficient number of restrictions to estimate the matrix _{t}. For the sake of brevity, we do not discuss the process of obtaining _{t} in detail here, as the process of moving from restrictions on _{t} is a matter of mere computation now that we have restricted _{t} = _{t}, and where _{t} is its vector of reduced-form disturbances. Once we have estimated _{t} = _{t}, we can write _{t} (Clarida and Gali

With _{t} identified we can assess the informational content of deviation cycles extracted at different frequency ranges. We compare the content of deviation cycles with a range of combinations of structural shocks, with particular emphasis on the path of transitory shocks, that is, the demand-based business cycle as defined in Du Plessis et al. (_{t} (

As a final note on our methodology, we acknowledge that the results of our analysis rest on whether or not the Blanchard-Quah identifying assumptions hold for our five variable VAR. With respect to the assumptions of spherical and serially uncorrelated error terms, we can (and do) test this, but we unfortunately cannot test whether our restrictions on the long-run impact matrix are valid. Our analysis also relies on the general efficacy of the Blanchard-Quah identification strategy. Lippi and Reichlin (

We estimate deviation cycles and SVAR-based structural shocks to real output on a sample period from 1961Q2 until 2015Q3, chosen on the basis of data availability. We define real output, our variable of interest, as real GDP measured at a quarterly frequency. Quarterly data for South African real output and government expenditure and monthly data for the repo rate were obtained from the South African Reserve Bank (SARB

Quarterly real output for the US, the United Kingdom (UK), Australia and Japan (all obtained from the International Monetary Fund’s International Financial Statistics Database,

Monthly and quarterly South African consumer price index (CPI) and the quarterly series of the West Texas Intermediate (WTI) spot oil price and the rand-dollar exchange rate (used to convert the spot oil price to rand) were obtained from Quantec’s EasyData database. We calculate the quarterly real interest rate from monthly data using the ‘within-quarter’ formula from Du Plessis, Smit and Sturzenegger (

In the final specification of our SVAR, the real rand-denominated oil price, our proxy for global output and domestic real output, are specified in log-differences; the ratio of government expenditure to GDP and the real interest rate are included in levels.

We apply the CF filter to the log of South African real GDP data. The results are reported in

Short- and medium-term deviation cycles.

Several features of

For our theoretical benchmark, we estimate the SVAR described above at six lags. We conducted standard specification tests for normality and autocorrelation on the unrestricted VAR (the results have been omitted for brevity and are available upon request). Tests for autocorrelation were deemed passable, but it should also be noted that the unrestricted VAR did not pass tests for the joint normality of the residuals. This result seemed to follow primarily from world output, whose residual series is platykurtic on account of the great recession. Adding additional lags did not correct this misspecification. We do not attempt to correct for this finding by including outlier dummy variables, as this would unduly reduce the information contained in the residual series (and hence in our estimates of the vector of structural shocks). Furthermore, we maintain that non-normality is not so problematic in this context. While the assumption that the residuals are uncorrelated is necessary for the structural decomposition of the estimated residuals into structural shocks (Clarida & Gali

Pearson correlation coefficients for shocks and deviation cycles.

Cumulative shocks | Deviation cycles | ||
---|---|---|---|

Short-term | Medium-term | Medium-term (first differences) | |

Real interest rate shocks |
−0.1942 |
−0.1091 | −0.0311 |

Government expenditure shocks |
0.0616 | 0.1848 |
0.1291 |

Domestic output shocks |
0.1109 | 0.6137 |
0.3166 |

Global output shocks |
0.0619 | 0.1591 |
0.0842 |

Oil price shocks |
0.214 |
0.3842 |
0.2751 |

Domestic shocks | 0.1167 |
0.6534 |
0.3349 |

External shocks | 0.222 |
0.4069 |
0.2845 |

Transitory shocks | 0.0196 | 0.166 |
0.1174 |

Permanent shocks | 0.1998 |
0.7244 |
0.4251 |

All five pure shocks | 0.2086 |
0.7672 |
0.4385 |

Note: This table contains estimates of Pearson correlation coefficients for the variables mentioned. All calculations were obtained using R.

, Statistical significance at 10%;

, Statistical significance at 5%;

, Statistical significance at 1%.

With reference to short-term deviation cycles, we find that there is substantial correlation with the negative demand shocks that correspond to the real interest rate

Structural shocks and short-term deviation cycles: (a) medium-term deviation cycles and transitory shocks; (b) medium-term deviation cycles and permanent shocks; (c) Medium-term deviation cycles and domestic shocks; (d) medium-term deviation cycles and external shocks.

In contrast with these results, it is interesting to note the correlation coefficient corresponding to

In sum, the two factors most strongly related to short-term deviation cycles are transitory shocks associated with the real interest rate and permanent shocks associated with real oil prices. Short-term deviation cycles, frequently interpreted as transitory demand shocks, do not appear to be strongly related with the combined path of (domestic) transitory shocks. We regard these findings as cursory evidence corroborating the critique of Harding and Pagan (

Turning now to observations regarding medium-term cycles, permanent shocks (with a correlation of 0.7244) account for a far greater proportion of deviation cycle variation at this frequency than do transitory shocks (0.166). The varying extents of overlap between medium-term cycles and transitory and permanent shocks is evident in

Structural shocks and medium-term deviation cycles: (a) ; (b) ; (c) ; (d)

It is particularly interesting to note that the combination of transitory shocks is more strongly correlated with medium-term cycles than with short-term cycles. In fact, barring the correlation reported for the real interest rate, it seems that the cycles and shocks are more strongly related in the medium term for all of the reported combinations of structural shocks – this observation is robust, holding for the correlations reported for both the levels and differences of these series. Tentatively, we regard the finding that the combined path of transitory shocks is more strongly correlated with medium-term cycles than with short-term as a corroboration of the notion that medium-term deviation cycles are less prone to discarding relevant business cycle information.

Considered along the domestic-external dichotomy, we find that the combined path of domestic shocks to output has a reportedly higher correlation with medium-term cycles than do external shocks – these series are depicted in

In order to further investigate the overlap between medium-term cycles and different combinations of structural shocks, we test for cointegration between these series. We test the stationarity of the medium-term cycles and find them to be difference stationary. Thus, if a regression of these cycles on a series of structural shocks yields stationary residuals, this might be regarded as an indication that the cycles capture the same stochastic trend as the shocks. For these tests, we capture the medium-term component of the business cycle by simply removing the long-term component – that is, we use the information captured in the frequency range 1–200 quarters. This leads to better behaved specification tests than for results obtained using medium-term cycles as defined earlier but does not change the conclusions we draw from our tests for cointegration.

Pretests and test for cointegration.

Cumulative shocks | ADF |
Joint normality tests (VAR in levels) | Cointegrating vectors ( |
ADF |
||||
---|---|---|---|---|---|---|---|---|

J-B test | Skewness | Kurtosis | Shock | Cycles | ||||

Real interest rate shocks () | 0.379 | - | - | - | - | - | ||

Government expenditure shocks () | 0.411 | - | - | - | - | |||

Domestic output shocks () | 0.981 | - | - | - | - | |||

Global output shocks () | 0.615 | - | - | - | - | - | ||

Oil price shocks () | 0.391 | - | - | - | - | |||

Domestic shocks | 0.95 | - | - | - | - | |||

External shocks | 0.363 | - | - | - | - | |||

Transitory shocks | 0.408 | - | - | - | - | |||

Permanent shocks | 0.898 | - | - | - | - | - | ||

All five pure shocks | 0.831 | - | - | - | - | - | ||

Real interest rate shocks () | - | - | - | - | 14.837 | 6.885 | 0.373 | 0.667 |

Government expenditure shocks () | - | - | - | - | 20.83 |
1.131 | 0.601 | 0.839 |

Domestic output shocks () | - | - | - | - | 21.831 |
1.611 | 0.81 | 0.385 |

Global output shocks () | - | - | - | - | 24.536 |
6.977 | 0.609 | 0.706 |

Oil price shocks () | - | - | - | - | 22.109 |
5.026 | 0.224 | 0.466 |

Domestic shocks | - | - | - | - | 18.861 |
1.918 | 0.678 | 0.424 |

External shocks | - | - | - | - | 18.027 |
4.536 | 0.165 | 0.457 |

Transitory shocks | - | - | - | - | 21.553 |
1.234 | 0.588 | 0.83 |

Permanent shocks | - | - | - | - | 45.01 |
3.36 | 0.114 | 0.035 |

All five pure shocks | - | - | - | - | 41.977 |
3.775 | 0.086 |
0.027 |

Note: In addition to

, Statistical significance at 10%;

, Statistical significance at 5%;

, Statistical significance at 1%.

ADF, Augmented Dickey-Fuller; VAR, vector autoregressive mode; OLS, ordinary least squares.

On account of the non-robustness of the Johansen procedure under non-normally distributed errors, we opt to test for cointegration via the Engle and Granger (

In sum, our tests for cointegration yield an interesting refinement to the findings reported in

Our comparison of deviation cycles with business cycles identified by an open-economy SVAR model produced several tentative, but interesting, conclusions. We did not find evidence that the high-frequency deviation cycle neatly corresponds to the purely transitory demand-based business cycle as measured by the open-economy SVAR. Rather, permanent shocks to output (often interpreted as supply shocks) seem to constitute an important source of variation in the high-frequency deviation cycle. Medium-term cycles were more strongly related with all shocks and combinations thereof, except shocks to the real interest rate. However, it seems that medium-term deviation cycles primarily capture business cycle information driven by permanent shocks to output, as suggested by our tests for cointegration.

Our findings suggest that deviation cycle analysis should be interpreted with caution. The medium-term deviation cycle seems to provide a good approximation of the South African business cycle, if one is interested in studying cycles derived from both transitory and permanent shocks, that is, demand- and supply-side variation in output. However, we did not find that short-term deviation cycles capture distinctly transitory, demand-driven information, as has been assumed in some applications. One should thus be cautious of drawing strong conclusions about the nature of business cycles from filter-based deviation cycle estimates, particularly if the objective of the study relies on assuming that high-frequency deviation cycles correspond to transitory demand shocks.

Willem Boshoff would like to acknowledge financial support from Economic Research Southern Africa (ERSA) for an earlier version of the paper, which appeared as a working paper: ERSA Working Paper 200. Willem Boshoff would like to thank Laurie Binge for research assistance on an earlier revision and would like to thank Adrian Pagan for enlightening discussions on an early version of this article, which led to the current version

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

W.H.B. was responsible for the concept and earlier econometrics. L.M. dealt with the econometrics.

Deviation cycles are also sometimes referred to as growth cycles; see Canova (

See for instance Agénor, McDermott and Prasad (2000) or Rand and Tarp (2002).

See Du Plessis et al. (