When are trend–cycle decompositions of GDP reliable?

Abstract

In this paper, we examine the results of GDP trend–cycle decompositions from the estimation of bivariate unobserved components models that allow for correlated trend and cycle innovations. Three competing variables are considered in the bivariate setup along with GDP: the unemployment rate, the inflation rate, and gross domestic income. We find that the unemployment rate is the preferred variable to accompany GDP in the bivariate setup to obtain accurate estimates of its trend–cycle correlation coefficient and the cycle. We show that the key feature of the unemployment rate that allows for reliable estimates of the cycle of GDP is that its nonstationary component is small relative to its cyclical component. Using quarterly GDP and unemployment rate data from 1948:Q1 to 2019:Q1, we obtain a trend–cycle decomposition of GDP that resembles the conventional CBO estimates; we find positively correlated trend and cycle components.

Appendices

Appendix A: Effect of the trend–cycle correlation on the contribution of the cycle to the variance of output growth

With the variance decomposition of the change in output growth given by

$$\begin{aligned}&\%\ \text {of}\ {{\,\mathrm{var}\,}}\left( \Delta y_t\right) \ \text {explained by} \ {{\,\mathrm{var}\,}}\left( \Delta c_t\right) \\&\quad =100\times \frac{{{\,\mathrm{var}\,}}\left( {\mathbb {E}} \left( \Delta y_t| \Delta c_t\right) \right) }{{{\,\mathrm{var}\,}}\left( \Delta y_t\right) } \\&\quad =100\times \frac{{{\,\mathrm{var}\,}}\left( \Delta c_t\right) \left( 1+\frac{\rho _{\eta ^y\varepsilon }\sigma _{\eta ^y}\sigma _{\varepsilon }}{{{\,\mathrm{var}\,}}\left( \Delta c_t\right) }\right) ^2}{\sigma _{\eta ^y}^2 + {{\,\mathrm{var}\,}}\left( \Delta c_t\right) + 2\rho _{\eta ^y\varepsilon }\sigma _{\eta ^y}\sigma _{\varepsilon }}, \end{aligned}$$

one can obtain the partial derivative with respect to $\rho _{\eta ^y \varepsilon }$, yielding the following expression:

$$\begin{aligned} \frac{2\sigma _{\eta ^y}^2\sigma _{\varepsilon }^2\left( \sigma _{\eta ^y} +\sigma _{\varepsilon }\rho _{\eta ^y\varepsilon }\right) \left( \sigma _{\varepsilon }\tilde{\phi } +\sigma _{\eta ^y}\rho _{\eta ^y\varepsilon }\right) }{\sigma _{\varepsilon }^2\tilde{\phi } \left( \sigma _{\eta ^y}^2+\sigma _{\varepsilon }^2\tilde{\phi }+2\sigma _{\eta ^y} \sigma _{\varepsilon }\rho _{\eta ^y\varepsilon }\right) ^2}, \end{aligned}$$

(A.1)

where

$$\begin{aligned} \tilde{\phi } = \frac{1-\phi _2}{(1+\phi _2)(1-\phi _1-\phi _2)(1+\phi _1-\phi _2)}. \end{aligned}$$

We note that $\tilde{\phi }>1$ for $\phi _2\in (-1,0)$ and $\phi _1 \in (0,1-\phi _2)$. Therefore, a sufficient condition for the derivative (A.1) to be positive, i.e., that the contribution of the cycle to the variance decomposition of output is increasing with the value of the trend–cycle correlation, is

$$\begin{aligned} \sigma _{\varepsilon }< \sigma _{\eta ^y} < \tilde{\phi }\sigma _{\varepsilon }. \end{aligned}$$

Appendix B: Monte Carlo results from the bivariate model for the other correlation coefficients

In this section, we assume the benchmark parameterization in Table 1 and vary the sample size (Fig. 8) and the variance of the trend of the accompanying variable (Fig. 9), as we did in Sect. 4.1.1. We report the distributions of the posterior mean estimates of the different correlation coefficients of the bivariate model.

Fig. 8

Frequency distribution of correlation coefficients varying sample size

Fig. 9

Frequency distribution of correlation coefficients varying variance of accompanying variable

Appendix C: Mapping between the coefficients of the two- and one-cycle models when the latter is the true data generating process

Under the DGP, we have that $c_t^y = c_t$ and $c_t^x = \theta c_t$, where $c_t = \phi _1 c_{t-1} + \phi _2 c_{t-2} + \varepsilon _t$, as in Eq. (18). Hence, the autocovariance function of $c_t^x$ is as follows:

$$\begin{aligned} \gamma _j^x = \theta ^2 \gamma _j, \end{aligned}$$

for all j, where $\gamma _j = {{\,\mathrm{cov}\,}}(c_t,c_{t-j})$, and the autocorrelation function is as shown below:

$$\begin{aligned} r_j^x = r_j, \end{aligned}$$

for all j, where $r_j = {{\,\mathrm{corr}\,}}(c_t,c_{t-j})$. As a consequence, we have that $\phi ^x_1 = \phi _1$ and $\phi _2^x = \phi _2$. Moreover, because $\gamma _0^x = \theta ^2 \gamma _0$, we obtain that

$$\begin{aligned} \sigma _{\varepsilon ^x}^2 = \theta ^2\sigma _\varepsilon ^2. \end{aligned}$$

(C.1)

In addition, we can prove that the correlation coefficient between the cycle innovations in the estimated two-cycle model should be (plus or minus) one if the DGP has only one cycle, as follows. First, because $c_t^x = \theta c_t = \theta c_t^y$ in the estimated model, we know that

$$\begin{aligned} {{\,\mathrm{cov}\,}}(c_t^y,c_t^x) = \theta \gamma _0. \end{aligned}$$

(C.2)

Second, using the Wold decomposition and the fact that $\phi ^x_1 = \phi _1=\phi ^y_1$ and $\phi _2^x = \phi _2 = \phi _2^y$, we also know that

$$\begin{aligned} {{\,\mathrm{cov}\,}}\left( c_t^y,c_t^x\right)&= {{\,\mathrm{cov}\,}}\left( \sum _{j=0}^\infty \psi _j \varepsilon _{t-j}^y,\sum _{j=0}^\infty \psi _j\varepsilon _{t-j}^x\right) , \nonumber \\&= \sigma _{\varepsilon ^y\varepsilon ^x}\sum _{j=0}^\infty \psi _j^2, \end{aligned}$$

(C.3)

where $\psi _j$, for $j=0,1,\ldots ,\infty$, are the coefficients of the Wold decomposition of an AR(2) process with coefficients $\phi _1$ and $\phi _2$. Combining (C.2) and (C.3), we obtain that

$$\begin{aligned} \sigma _{\varepsilon ^y\varepsilon ^x} = \theta \sigma _{\varepsilon ^y}^2. \end{aligned}$$

(C.4)

Hence, from (C.1) and (C.4), the correlation between the cycle innovations in the estimated two-cycle model would be given by the following:

$$\begin{aligned} \rho _{\varepsilon ^y\varepsilon ^x}&= \frac{\sigma _{\varepsilon ^y,\varepsilon ^x}}{\sigma _{\varepsilon ^y}\sigma _{\varepsilon ^x}}, \\&= \frac{\theta \sigma _{\varepsilon ^y}^2}{sign(\theta )\theta \sigma _{\varepsilon ^y}^2}, \\&= sign(\theta )\times 1. \end{aligned}$$

Finally, all the other correlation coefficients between innovations in the estimated two-cycle model should be zero if the same holds in the DGP.

Appendix D: Mapping between the coefficients of the one- and two-cycle models when the latter is the true data generating process

Under the one-cycle estimated model, using (16), we have that

$$\begin{aligned} \Delta x_t = \theta \Delta c_t + \eta ^x_t, \end{aligned}$$

and

$$\begin{aligned} \theta = \frac{{{\,\mathrm{cov}\,}}(\Delta x_t,\Delta c_t)}{{{\,\mathrm{var}\,}}(\Delta c_t)}. \end{aligned}$$

Under the DGP,

$$\begin{aligned} \Delta x_t&= \Delta c_t^x + \eta _t^x, \\ \Delta c_t&= \Delta c_t^y. \end{aligned}$$

where

$$\begin{aligned} \Delta c_t^x&= \tilde{\phi }_1^xc_{t-1}^x + \phi _2^x c_{t-2}^x + \varepsilon _t^x, \end{aligned}$$

(D.1)

$$\begin{aligned} \Delta c_t^y&= \tilde{\phi }_1^yc_{t-1}^y + \phi _2^y c_{t-2}^y + \varepsilon _t^y, \end{aligned}$$

(D.2)

with $\tilde{\phi }_1^j = \phi _1^j-1$ for $j=x,y$. Hence,

$$\begin{aligned} \theta&= \frac{{{\,\mathrm{cov}\,}}(\Delta c_t^x + \eta _t^x,\Delta c_t^y)}{{{\,\mathrm{var}\,}}(\Delta c_t^y)}, \\&= \frac{{{\,\mathrm{cov}\,}}(\Delta c_t^x,\Delta c_t^y)}{{{\,\mathrm{var}\,}}(\Delta c_t^y)} +\frac{{{\,\mathrm{cov}\,}}(\eta _t^x,\Delta c_t^y)}{{{\,\mathrm{var}\,}}(\Delta c_t^y)}, \\&=\frac{\sigma _{\varepsilon ^x\varepsilon ^y}}{\sigma ^2_{\varepsilon ^y}} \frac{\sum _{i=0}^{\infty }\tilde{\psi }_i^x\tilde{\psi }_i^y}{\sum _{i=0}^{\infty } \tilde{\psi }_i^{y^2}} + \frac{\sigma _{\eta ^x\varepsilon ^y}}{\sigma ^2_{\varepsilon ^y}} \frac{1}{\sum _{i=0}^{\infty }\tilde{\psi }_i^{y^2}}, \\&=\rho _{\varepsilon ^y\varepsilon ^x}\frac{\sigma _{\varepsilon ^x}}{\sigma _{\varepsilon ^y}} \frac{\sum _{i=0}^{\infty }\tilde{\psi }_i^x\tilde{\psi }_i^y}{\sum _{i=0}^{\infty } \tilde{\psi }_i^{y^2}} + \rho _{\varepsilon ^y\eta ^x}\frac{\sigma _{\eta ^x}}{\sigma _{\varepsilon ^y}} \frac{1}{\sum _{i=0}^{\infty }\tilde{\psi }_i^{y^2}}, \end{aligned}$$

where $\tilde{\psi }_i^j$ for $i=0,1,2,\ldots ,\infty$ and $j=x,y$ are the coefficients of the Wold decomposition of the AR(2) processes (D.1) and (D.2), respectively.

Appendix E: Estimation results from alternative specifications

This section presents the results from all the bivariate specifications considered in the paper and adds the specification proposed by Sinclair (2009) in which there are specific trends and cycles in real GDP and the unemployment rate. Figure 10 depicts the estimated GDP cycles and their 68% credibility intervals, Fig. 11 shows the estimated mean trend output growth rate and its 68% credibility interval, and Table 7 contains the posterior mean estimates of the parameters of the different models.

Fig. 10

Smoothed GDP cycle

Fig. 11

Mean trend output growth rate

Table 7 Posterior mean estimates

Fig. 12

Estimation results under heteroskedasticity

Appendix F: GDP–unemployment rate model estimation results under heteroskedasticity

As a sensitivity exercise, we estimate the GDP–unemployment rate model of Sect. 5 allowing for heteroskedasticity. In particular, we assume that the variance–covariance matrix of the innovations of the model suffers a structural break starting in 1986, a period denominated as the Great Moderation. This specification implies that both variances and correlation coefficients are allowed to differ between the two parts of the sample. Results appear in Fig. 12.

As can be seen, the output gap estimate is very similar to that obtained under the homoskedasticity assumption shown in Fig. 7, except perhaps for a reduction in the uncertainty around the estimate starting around the second half of the 1980s, as a consequence of the lower variance estimates whose estimates appear in Table 8. In general, all the innovations of the model experience a reduction in their variance during the Great Moderation compared with the previous period.

Table 8 Real GDP–unemployment rate UC model estimates under Heteroskedasticity

Regarding the correlation coefficients, their posterior mean estimates keep their signs compared with the homoskedastic case. However, splitting the sample has consequences on the uncertainty around them. In particular, the trend–cycle correlation coefficient of output has credible sets that now include zero.