A Systemic Approach to Estimating the Output Gap for the Italian Economy

Abstract

Estimating potential output and the corresponding output gap plays a key role, not only for inflation forecasting and the assessment of the economic cycle, but also for the fiscal governance of the European Union (EU). Potential output is, however, an unobservable and extremely uncertain variable. Empirical measurements differ considerably depending on the econometric approach adopted, the specification of the data generating process and the dataset used. The method adopted at the EU level, which was agreed within the Output Gap Working Group, has been subject to considerable debate. The fiscal councils of the various Member States contribute to the discussion over the output gap modelling. This paper aims at estimating the potential output of the Italian economy, using a combination of five different models proposed by the relevant literature. More specifically, in addition to a statistical filter, we use unobserved components models based on the Phillips curve, the Okun law and the production function. The approach adopted allows to reconcile the parsimony of the econometric specification with the economic interpretation of the results. Estimates of the output gap obtained with the five selected models present important properties: low pro-cyclicity, stability with respect to the preliminary data and consistency with the economic theory. The use of multiple models also enables the construction of confidence bands for the output gap estimates, which are helpful for policy analysis. In the empirical application for Italy, estimates and forecasts of the output gap recently produced by relevant organisations tend to fall within the confidence interval calculated on the basis of the five selected models.

Appendix

The European Commission Model

In the CAM, the estimates of the potential level of four variables are aggregated within the framework of the production function (see Havik et al. 2014). Total Factor Productivity—TFP (A), unemployment rate (U), participation rate (PR) and hours worked per worker (H). For the last two series, a univariate decomposition using the Hodrick-Prescott filter is adopted. The greatest modelling effort has been devoted to the specification and estimation of the two bivariate models for the two remaining components. The TFP model is treated with Bayesian inference, while that for the NAWRU is based on a maximum likelihood approach. The output decomposition model is not log-additive. A mixed multiplicative-additive approach is adopted (as far as the NAWRU is concerned). The CAM obtains the potential value of TFP using a bivariate structural model that links the logarithm of the Solow residual to a cyclical composite indicator (the CUBS, obtained as a weighted average of measurements from qualitative surveys of capacity utilisation in the manufacturing sector and the climate of confidence in construction and services).

Potential labour input (in terms of total hours worked) is obtained first by factoring L into the product of four components:

$$\begin{aligned} L = H \times E\times PR \times POP, \end{aligned}$$

where H is the number of annual hours per employed person, E is the employment rate, PR is the participation rate (active population of working age) and POP is the population of working age (those aged 15–74).

The potential level for PR and H is obtained by applying the Hodrick-Prescott filter, while for $E=1-U$, where U is the unemployment rate, the following decomposition is used

$$\begin{aligned} E_P =(1-\text{ NAWRU}), \end{aligned}$$

where the subscript P indicates the potential level of the variable. In addition to information drawn from the characteristics of the series and extracted through decomposition of unobserved components, the natural unemployment rate (Non-Accelerating Wage Rate of Unemployment) is obtained using information from the Phillips curve that links the growth rate of wages to cyclical unemployment, represented as a second-order autoregressive process. We thus have

$$\begin{aligned} \begin{array}{lll } L_P &{} =&{} H_P\times (1-\text{ NAWRU})\times PR_P\times POP. \\ \end{array} \end{aligned}$$

The EC’s estimate of the NAWRU for Italy is based on the accelerationist version of the Phillips curve (see Havik et al. 2014), such that a reduction in the inflation rate in wages is associated with a positive unemployment gap (i.e. the NAWRU is below the observed rate). The estimate is then corrected by an ad hoc procedure for anchoring it to a long-term reference value obtained using a panel estimate that explains the unemployment rate as a function of structural variables for the labour market. The constraints adopted by the EC give rise to procyclical estimates of the NAWRU. Fioramanti (2015) has studied the role played by the constraints imposed on the variance of the model’s stochastic disturbances, highlighting how the choice of upper and lower limits crucially determines the evolution of NAWRU over time.

The Bivariate Model with Output and Inflation

The bivariate model presented in “The Bivariate Model with Output and Inflation” section can be represented as follows:

$$\begin{aligned} \begin{array}{llll} y_t &{}=&{} \mu _{t} + \psi _{t}, &{} t= 1,\ldots ,n, \\ &{} &{} &{} \\ \mu _{t} &{}=&{} \mu _{t-1} +\beta _{t-1} + \eta _{t}, &{} \eta _{t} \sim \text{ IID } N(0,\sigma ^2_{\eta }),\\ \beta _{t} &{}=&{}\beta _{t-1} + \zeta _{t}, &{} \zeta _{t} \sim \text{ IID } N(0,\sigma ^2_{\zeta }),\\ &{} &{} &{} \\ \psi _{t} &{}=&{}\phi _1 \psi _{t-1} + \phi _2 \psi _{t-2} + \kappa _{t}, &{} \kappa _{t} \sim \text{ IID } N(0,\sigma ^2_{\kappa }),\\ &{} &{} &{} \\ \pi _t&{} =&{} \gamma _e \pi _t^e + \pi _t^*+ \theta _0 \psi _{t} + \theta _1 \psi _{t-1} +\sum _{k=1}^K \beta _k x_{kt} + \epsilon _{\pi t}, &{}\epsilon _{\pi t} \sim \text{ IID } N(0,\sigma ^2_{\epsilon \pi }),\\ \pi _t^* &{}=&{} \pi _{t-1}^* + \eta _{\pi t} &{}\eta _{\pi t} \sim \text{ IID } N(0,\sigma ^2_{\eta \pi }),\\ \end{array} \end{aligned}$$

(7)

in which the GDP series, $y_t$, is decomposed into trend $\mu _{t}$ and cycle $\psi _{t}$ and the inflation series $\pi _t$, measured using the GDP deflator, follows a standard Phillips curve, with an inertial component, $\pi _t^*$, expectations $\pi _t^e$, the output gap and a number of exogenous variables $x_{kt}$, such as the terms of trade and the oil price. The extension with the shock in the cycle is obtained by substituting the specification of the cycle with the following: $\psi _{t} =\phi _1 \psi _{t-1} + \phi _2 \psi _{t-2} + \kappa _{t}+ \lambda I(t = \tau )$, with $\tau = 2009$.

Table 2 shows the maximum likelihood estimates of the parameters of the model based on the series from 1970 to 2018. The regression coefficient associated with inflation expectations is not significantly different from 1. The estimated period of the cycle is very high (the estimated cyclical frequency is not different from zero).

The introduction of an intervention variable in the cycle equation—in the specification with cyclical shocks—reduces the estimated variance of cyclical shocks. The coefficient associated with the variable (in the last row of Table 2) is negative and highly significant. Furthermore, it substantially improves the fit of the model, as shown in Table 3 which reports the value of the log-likelihood, the Wald test for the joint significance of the two coefficients of the Phillips curve, together with the test of the hypothesis of the long-term neutrality of the output gap, $H_0:\theta _0+\theta _1 = 0$, and a number of diagnostics concerning the GDP equation and calculated on the standardised residuals of the Kalman filter: the Ljung-Box autocorrelation test with 4 lags, the Jarque-Bera normality test, the Goldfeld and Quandt heteroscedasticity test and the first-order conditional heteroscedasticity test (ARCH (1)).

In particular, for the specification with a cyclical shock, the output gap has a significant impact (at the 5% level) on the level of inflation, but not on the acceleration of prices. In general, the introduction of the shock improves the statistics.

Table 2 Maximum likelihood estimation of the bivariate model (1).

Table 3 Diagnostics and goodness of fit

The Trivariate Model with Output, Inflation and Unemployment

The trivariate model is an extension of the bivariate and is obtained by adding the relationship linking the unemployment gap to the output gap to the specification presented in “The Bivariate Model with Output and Inflation” section:

$$\begin{aligned} \begin{array}{llll} U_t &{}=&{} \mu _{ut} + \psi _{ut}, &{} t= 1,\ldots ,n, \\ &{} &{} &{} \\ \mu _{ut} &{}=&{} \mu _{u,t-1} +\beta _{u,t-1} + \eta _{ut}, &{} \eta _{ut} \sim \text{ IID } N(0,\sigma ^2_{\eta u}),\\ \beta _{ut} &{}=&{}\beta _{u,t-1} + \zeta _{ut}, &{} \zeta _{ut} \sim \text{ IID } N(0,\sigma ^2_{\zeta u}),\\ &{} &{} &{} \\ \psi _{ut} &{}=&{}\phi _u \psi _{u,t-1} + \delta _0 \psi _{t} + \delta _1 \psi _{t-1} + \kappa _{ut}, &{} \kappa _{ut} \sim \text{ IID } N(0,\sigma ^2_{\kappa u}),\\ \end{array} \end{aligned}$$

(8)

where $U_t$ is the unemployment rate and $\psi _{u,t-1}$ is the unemployment gap. The unemployment gap model is sufficiently generic and contains a number of particularly interesting cases. Where $\delta _0 = \delta _1 = 0$, the unemployment gap model becomes purely idiosyncratic. Okun law can manifest itself in two ways: rewriting

$$\begin{aligned} \psi _{ut} =\frac{\delta _0 + \delta _1 L}{1-\phi _u L} \psi _{t} + \frac{\kappa _{ut}}{1-\phi _u L}, \end{aligned}$$

we see that under the restrictions $\phi _u=0$ and $\delta _1=0$, $\psi _{ut}= \delta _0 \psi _t +\kappa _{ut}$. If $\sigma ^2_{\kappa u} = 0$, the relationship between the gaps is strictly proportional, with the output gap generating the common cycle with unemployment. Otherwise, in case $\delta _1 = -\delta _0\phi _u$, $\psi _{ut}$ has one component proportional to the output gap and a specific component represented by an AR(1) process.

The likelihood associated with the unrestricted model is equal to − 188.73. The maximum likelihood estimates for the parameters are reported in Table 4. The model estimated for the output gap is more persistent than the bivariate model ($\phi _1$ and $\phi _2$ are 1.3687 and − 0.4699 respectively) and characterised by cyclical shocks with low variabilities. The relationship connecting inflation to the gap through the Phillips curve seems to have loosened, but the output gap is significantly linked to the unemployment gap, as shown by the Wald test of the hypothesis $H_0:\delta _0+\delta _1=0$ reported in Table 5. The coefficient $\phi _u$ is high and significant, and the restriction $\phi _u=0$ is clearly rejected.

Table 4 Maximum likelihood estimates of the trivariate model (1).

Table 5 Trivariate model; diagnostics and goodness of fit

The Integrated Multivariate Model Based on the Production Function

Starting with (6), the integrated multivariate model can be represented as follows. Let $\mathbf {y}_t = (f_t, a_t, h_t, e_t, c_t)'$, where the series $c_t$ represents the CUBS composite indicator, $\varvec{\mu }_t = (\mu _{ft}, \mu _{ht}, \mu _{at}, \mu _{et})',$$\varvec{\psi }_t = (\psi _{ft}, \psi _{ht}, \psi _{at}, \psi _{et})',$$\psi _t=\varvec{\gamma }' \varvec{\psi }_t,$$\varvec{\gamma }= (1, \alpha , \alpha , \alpha )'$.

$$\begin{aligned} \begin{array}{llll} \mathbf {y}_t &{} = &{} \varvec{\mu }_t+\varvec{\psi }_t &{} t = 1, \ldots , n,\\ &{} &{} &{} \\ \varvec{\mu }_t &{}=&{} \varvec{\mu }_{t-1} + \varvec{\beta }_{t-1} &{} \\ \varvec{\beta }_t &{}=&{} \varvec{\beta }_{t-1} + \varvec{\zeta }_t &{} \varvec{\zeta }_{t} \sim \text{ IID } N(\text{0 },\Sigma _{\zeta }),\\ &{} &{} &{} \\ \varvec{\psi }_t &{}=&{} \phi _1 \varvec{\psi }_{t-1} + \phi _2\varvec{\psi }_{t-2} + \varvec{\kappa }_t &{} \varvec{\kappa }_{t} \sim \text{ IID } N(\text{0 },\Sigma _{\kappa }),\\ &{} &{} &{} \\ c_t &{}=&{} \theta _c \psi _{ft} + \epsilon _{ct} &{}\epsilon _{c t} \sim \text{ IID } N(0,\sigma ^2_{\epsilon c}),\\ &{} &{} &{} \\ \pi _t&{} =&{} \gamma _e \pi _t^e + \pi _t^*+ \theta _0 \psi _{t} + \theta _1 \psi _{t-1} +\sum _{k=1}^K \beta _k x_{kt} + \epsilon _{\pi t}, &{}\epsilon _{\pi t} \sim \text{ IID } N(0,\sigma ^2_{\epsilon \pi }),\\ \pi _t^* &{}=&{} \pi _{t-1}^* + \eta _{\pi t} &{}\eta _{\pi t} \sim \text{ IID } N(0,\sigma ^2_{\eta \pi }),\\ \end{array} \end{aligned}$$

(9)

where it is assumed that $\Sigma _\zeta$ is a diagonal matrix, while $\Sigma _\kappa$ is a full matrix. The multivariate cycle has scalar coefficients and is such that the output gap also has an AR(2) representation with the same coefficients. The CUBS replicates that used in the CAM’s TFP model. The maximum likelihood estimation of the matrix $\Sigma _\kappa$ implies correlations between relatively small cyclical disturbances:

	f	a	h	e
f	1.00	−  0.05	−  0.30	0.32
a	−  0.05	1.00	0.18	−  0.26
h	−  0.30	0.18	1.00	0.10
e	0.32	−  0.26	0.10	1.00

The complexity of the model and the small number of observations, compared with the large number of parameters, makes the estimation especially difficult and the measurement of standard errors using the Delta Method imprecise. Future applications envisage the use of bootstrap simulation techniques to overcome this drawback.