Trends and Cycles in Macro Series: The Case of US Real GDP

In this paper we propose a new modelling framework for the analysis of macro series that includes both stochastic trends and stochastic cycles in addition to deterministic terms such as linear and non-linear trends. We examine four US macro series, namely annual and quarterly real GDP and GDP per capita. The results indicate that the behaviour of US GDP can be captured accurately by a model incorporating both stochastic trends and stochastic cycles that allows for somedegree of persistence in the data. Both appear to be mean-reverting, although the stochastic trend is nonstationary whilst the cyclical component is stationary, with cycles repeating themselves every 6 – 10 years.


Introduction
In this paper we put forward a new modelling framework for macro series that allows for two singularities (or poles) in the spectral density function, one corresponding to the long-run or zero frequency (i.e. to the long-run evolution of the series), the other to a non-zero frequency (and related to a cyclical pattern repeated approximately every 6 -10 years). The proposed model is very general: it extends the classical framework based on (non-cyclical and cyclical) unit roots by allowing for fractional integration, and considers both deterministic and stochastic patterns, at the zero and cyclical frequencies, including both long-and short-memory components. The cyclical patterns are modelled using Gegenbauer processes.
The suggested methodology is then applied to analyse the stochastic behaviour of four US macro series, namely annual and quarterly GDP and GDP per capita. The results indicate that the behaviour of US GDP can be captured accurately by a model incorporating both stochastic trends and stochastic cycles that allows for some degree of persistence in the dynamics of the series. Both appear to be mean-reverting, although it is found that the stochastic trend is non-stationary whilst the cyclical component is stationary, with cycles repeating themselves every 6 -10 years. Deterministic (linear and non-linear) terms were also incorporated into the model but were not found to be statistically significant in any case.
The layout of the paper is as follows. Section 2 briefly reviews the main approaches to modelling GDP found in the literature. Section 3 presents the statistical model. Section 4 discusses the data and the empirical results. Section 5 offers some concluding remarks.

Literature Review
GDP, whether nominal, real or per capita, is typically a non-stationary variable in most developed countries. For many years, the standard modelling approach was to use deterministic functions of time, usually of a linear form, as in the following specification: where {y t , t = 1, 2, …, T} is the observed (GDP) series, α and β are the coefficients on an intercept and a linear time trend respectively, and x t is assumed to be covariance stationary, usually of the ARMA form, to capture short-run and cyclical patterns in the data. Therefore, the process followed by x t can be represented as where ) L ( φ and ) L ( θ stand for the AR and MA components of the series respectively. This modelling framework was dominant in the literature until the publication of a very influential paper by Nelson and Plosser (1982), who examined fourteen US macroeconomic series and by applying the tests developed by Fuller (1976) and Dickey and Fuller (1979) found evidence of unit roots and came to the conclusions that the behaviour of these variables except one could be better described in terms of stochastic trends, that is, as in the following model including an intercept: where x t is I(0) and can be represented as in (2). 1 This model has been widely employed in the macro literature and in the last twenty years many additional unit root tests have been developed (Phillips and Perron, 1988;Elliot et al., 1996;Ng and Perron, 2001;etc.). These two specifications, i.e. the deterministic trend model as in (1) and the stochastic trend model as in (3), can coexist within the same framework if x t in (1) contains a unit root, the main difference between the two models being the treatment of shocks, which have transitory effects in the case of (1) but permanent ones in the case of (3). However, a process may display nonstationary, persistent behaviour but still be mean-reverting as in the I(d) models with a differencing parameter d lying in the interval [0.5, 1). In such models, x t is specified as where d can be any real value and u t is I(0) (defined as in footnote 1). Variants of this model have been used to analyse the behaviour of GDP in various countries (see, e.g., Michelacci and Zaffaroni, 2000, Mayoral, 2006, Gil-Alana, 2010, Caporale and Gil-Alana, 2013, Caporale and Skare, 2014. Cyclicality is another important feature of GDP series. There exists a large literature using different methods such as time-varying transition probabilities (TVTP) Markov-switching regime models (see, e.g., Simpson et al., 2001), band pass filters (Christiano and Fitzgerald, 1999), etc. A similar approach to equation (4) can also be used to allow stochastic cyclical processes to be fractional as in the following model, where L stands for the lag operator (i.e., Lx t = x t-1 ),  Gray et al. (1989Gray et al. ( , 1994 showed that x t in (5) is (covariance) stationary if d < 0.5 for │μ = cosw r │< 1 and if d <0.25 for│μ│= 1. This process implies the existence of a pole or singularity at a non-zero frequency which corresponds to the cyclical pattern. Special cases of this model were analysed by Athola and Tiao (1987) and Bierens (2001) setting d = 1, and by Gil-Alana (2001), DePenya and Gil-Alana (2006) and others allowing d to take fractional values. In this paper we combine the two models given by equations (4) and (5) in a single framework that is presented in the following section.

A New Statistical Model for Trends and Cycles
We propose a very general specification that incorporates both deterministic and stochastic trends, not only at the zero frequency but also at the non-zero (cyclical) frequencies, allowing for both long-and short memory components. The model is the following: where f is a function that can also be non-linear, depending on time and the unknown parameter vector φ; L is the lag operator; d 1 and d 2 are the integration orders of the long-run and the cyclical frequency respectively, where w r = 2πr/T with r = T/j, j indicates the number of periods per cycle and r the frequency with a singularity or pole in the spectrum; u t is I(0) and displays weak dependence (as in equation (2) but replacing x t with u t ), and ε t is i.i.d. N(0, σ 2 ). Therefore, the vector of parameters to be Robinson (1994) had previously proposed a general testing framework that includes as a special case a very similar specification to ours. He considered the following model with a linear deterministic component: where z t is a (kx1) vector of deterministic terms and/or weakly exogenous variables and s represents the number of cyclical and/or seasonal patterns observed in the data. Within this set-up, he tested the null hypothesis: Assuming now that s = 2, the second equation in (7) becomes the same as the second one in (6), with the two coefficients d 1 and d 2 referring respectively to the long run and cyclical components of the series. In this context, a LM test of (8) in (6) can be defined as where T is the sample size, and where λ j = 2πj/T, and the summation in * in the above equations is over all frequencies which are bounded in the spectrum. I(λ j ) is the periodogram of t û defined as: where the last term of the above equation, for some special non-linear cases such as those presented in the following section, can be expressed in a linear way as , with T * as a suitable subset of the R q Euclidean space. Finally, g u is a known function coming from the spectral density of u t : Note that these tests are parametric and, therefore, they require specific modelling assumptions about the short-memory specification of u t . In particular, if u t is a white noise, g u ≡ 1, whilst if it is an AR process of the form φ(L)u t = ε t , g u = |φ(e iλ )| -2 , with σ 2 = V(ε t ), with the AR coefficients being a function of τ.
The point estimates were obtained by choosing over a grid the values of d 1 , d 2 and r that minimise Robinson's (1994) test statistic. They were found to be almost the same as those obtained by maximising the Whittle function in the frequency domain (Dahlhaus, 1989). The confidence intervals were calculated by choosing the values of the differencing parameters for which the null hypothesis could not be rejected at the 5% level.
Under very general regularity conditions, Robinson (1994) showed that for this particular version of his tests: where "→ d " stands for convergence in distribution. Therefore, unlike in the case of other procedures, this is a classical large-sample testing situation. If this test is carried out in the context of (6) the null H o will be rejected against the alternative H a : As mentioned before, despite the potentially nonlinear structure of the first equation in (6), its interaction with the second equation will make it linear for some specific nonlinear structures, such as the Chebyshev polynomials in time presented in the following section. 2

Empirical Analysis
We examine the following four series:   non-stationary behaviour. This is confirmed by their correlograms (Figure 2) and the periodograms (Figure 3), the former decaying slowly and the latter exhibiting their highest values at the smallest frequencies. Figure 4 displays the same four series in first differences, with the corresponding correlograms and periodograms (displayed in Figures 5 and 6 respectively) providing evidence of cyclical patterns.

[Insert Figures 4 -6 about here]
We start by considering a linear model with a time trend allowing for unit roots and fractional degrees of integration, specifically: where the errors are assumed to follow in turn a white noise and an autocorrelated process. However, instead of imposing a parametric ARMA structure on u t , we employ a non-parametric method due to Bloomfield (1973) such that the error term is specified exclusively in terms of its spectral density function, which is given by where σ 2 is the variance of the error term and m indicates the number of short-run dynamic terms, usually low (e.g., 1 or 2), which approximates highly parameterised ARMA models with very few parameters, and producing autocorrelations decaying exponentially as in the AR case. Moreover, it is stationary for all range of parameters unlike in the AR case.
Tables 1 and 2 display the estimates of d, along with their corresponding 95% confidence bands, for the three cases of i) no deterministic terms, ii) a constant, and iii) a constant and a linear time trend, assuming in turn that u t is a white noise (Table 1) and autocorrelated as in the model of Bloomfield (Table 2).

[Insert Tables 1 and 2 about here]
In the white noise case the time trend is significant in all cases except annual GDP per capita, and the estimates of d are significantly above 1, ranging from 1.31 (quarterly GDP) to 1.45 (annual GDP per capita). When allowing for (weak) autocorrelation as specified by Bloomfield (1973), the time trend is significant in all four cases, and the estimated values of d are still significantly above 1 but smaller.
Given the significance of the time trend in most cases, next we investigate whether it might be non-linear by using an approach based on Chebyshev polynomials in time that has been shown to perform well in the context of the tests of Robinson (1994) for fractional integration (Cuestas and Gil-Alana, 2016 with m indicating the order of the Chebyshev polynomial P i,T (t) defined as: (see Hamming (1973) and Smyth (1998) for a detailed description of these polynomials). Bierens (1997) and Tomasevic et al. (2009)

[Insert Table 3 about here]
Table 3 displays the estimated coefficients of the Chebyshev polynomials in time along with the estimates of d for the case of uncorrelated errors (almost identical results were obtained with autocorrelated (Bloomfield) disturbances). To allow for some degree of nonlinearity, we set m equal to 3, therefore θ 2 and θ 3 are the coefficients corresponding to the nonlinear trends. Nonlinear behaviour is only found in the case of the quarterly real GDP series with the two nonlinear coefficients (θ 2 and θ 3 ) being statistically significant at the 5% level; θ 2 is also found to be significant in the case of the two real GDP per capita series (annual and quarterly) but not for annual real GDP.
Further, the estimated values of d are all significantly higher than 1, ranging from 1.25 (quarterly real GDP) to 1.37 (annual real GDP per capita). 3 Next we examine the possibility of a cyclical pattern in the data and for this purpose we consider a model specified as in (6): the coefficients of the first equation on the deterministic terms were not found to be statistically significant in any case, both with a linear time trend and when allowing for nonlinearities by using Chebyshev polynomials in time. Therefore, we estimate models for both the original and the demeaned series but without time trends, assuming in turn that u t in (6) is a white noise process, an AR(1) process, and finally follows the exponential spectral specification of Bloomfield (1973). The results are presented in Tables 4, 5 and 6 respectively.

[Insert Tables 4, 5 and 6 about here]
In the case of white noise errors (Table 4), the estimated value of d for the original series is 10 years at the annual frequency, and 7 and 13 quarters respectively for real GDP and real GDP per capita at the quarterly frequency. The estimated value of r for the demeaned series is 12 at the annual frequency, and 7 and 8 respectively for real GDP and real GDP per capita at the quarterly frequency. It is also noteworthy that d 1 is systematically higher than d 2 , which indicates that the long-run frequency is relatively more important than the cyclical one. Specifically, d 1 ranges between 0.55 (annual real GDP, original data) to 1.24 (quarterly real GDP per capita, demeaned data), while d 2 oscillates around 0, being significantly positive for the original series at the annual frequency as well as for both annual and quarterly real GDP per capita in the case of the demeaned series. Table 5 displays the results under the assumption of AR(1) errors. In this case the estimated value of r is 10 for the four annual series, whilst it is 7 and 10 respectively for quarterly real GDP and real GDP per capita. Moreover, the estimated value of d 1 is much lower than in the previous case, and is not statistically different from zero for some of the original series. This might be a consequence of the competition with the AR(1) parameter in describing the degree of persistence in the long run structure of the data. For the demeaned series the values of d 1 are significant but smaller than those reported in Table 4. Besides, d 2 is now statistically significant in all cases, which implies the presence of a cyclical pattern.
Finally, Table 6 displays the results under the assumption that the error term follows the non-parametric specification of Bloomfield (1973). We consider these the most reliable evidence since this model allows for a certain degree of weak autocorrelation without affecting the estimation of the remaining parameters. The estimated value of r is now 10 for the two annual series, regardless of whether raw or demeaned data are used. The two differencing parameters are significantly different from zero in one of the four cases examined, with the value of the long-run parameter d 1 being around 0.60 and that of d 2 about 0.2 for the original data and slightly higher (about 0.3) for the demeaned series; for quarterly real GDP, r = 7, d 1 is around 0.7 and d 2 is close to zero for the original data but equal to 0.14 (and statistically significant) for the demeaned data; for quarterly real GDP per capita, r = 13, d 1 is equal to 0.66 for the original data and 0.91 for the demeaned data, and d 2 is statistically insignificant in both cases.
To summarise the main findings: first, there is evidence that when modelling GDP one should allow for long memory and fractional integration instead of restricting the differencing parameters to be either zero or one; second, both the zero (or long-run) frequency and the other non-zero (cyclical) frequencies play a role, at least in some cases. Awareness of the latter point is important for the purpose of evaluating the effects of shocks, which can affect not only the long run but also the cyclical structure.
However, our modelling approach has the limitation of not being able to discriminate between the effects on the two components, since the error term is of a multiplicative nature and is based on both. Future work will aim to address this issue.

Conclusions
This paper proposes a new statistical model for macro series that captures both their long-run behaviour and their cyclical properties by including two poles in their spectrum, in addition to both linear and non-linear deterministic trends. The adopted framework is very general since it also allows for fractional degrees of integration and both long-and short-run memory components, and is suitable for modelling any macro series of interest. As an illustration, in this study it is applied to analyse four US real GDP series (annual and quarterly, per capita as well) and it is found to capture very well the stochastic properties of this series. In particular, both stochastic trends and stochastic cycles are found to be significant, both being mean-reverting, but the former being nonstationary and the latter stationary with cycles repeating themselves every 6 -10 years. There is also evidence of persistence. Future work will extend this modelling framework with the aim of distinguishing between different types of shocks affecting trends and cycles separately while still allowing for a flexible degree of persistence.