Predicting recessions with a frontier measure of output gap: an application to Italian economy

Among the variety of different approaches attempting to model and forecast economic recessions, we will focus on those that employed the parametric binary choice approach and find that a good model for the prediction of the US recessions is a parsimonious model with only one of a few predictors, the most important of which is the interest rate spread and one discrete variable, the lagged dependent variable. The roots of this approach go back to at least the seminal work of Estrella and Mishkin (1995, 1998), who thoroughly investigated various parametric models with many variables and concluded that the best forecasts resulted from a parsimonious probit model involving only one explanatory variable, the lagged spread. Duecker (1997) confirmed this result, yet also found that including the lagged dependent variable among regressors substantially improved the predicting power of the Estrella and Mishkin (1995, 1998) approach, especially for the recessions of the 1970s and 1990s that were missed by various other forecasting methods. Overall, the analyses in Estrella and Mishkin (1995, 1998) and Duecker (1997) suggest that their parsimonious model outperforms many alternative models that included many variables to gain a high in-sample fit, yet happened to be poorly forecasting the future. Also see Kauppi and Saikkonen (2008) for further refinements and more references and discussions.

This paper contributes to the empirical literature on predicting recessions by adding two novelties: (i) we apply a nonparametric dynamic time series discrete response model suggested by Park et al. (2017) and (ii) we use a new measure of output gap as one of the recession predictors. In particular, we employ a robust nonparametric frontier panel data model proposed by Mastromarco and Simar (2015) to estimate the time-dependent conditional efficiency of countries and use this as a measure of output gap.¹ In a macroeconomics context, where countries are producers of output (i.e., GDP), given inputs (e.g., capital, labor), and technology, inefficiency can be identified as the distance of the individual production from the frontier. This frontier can be estimated by the maximum output of the reference country regarded as the empirical counterpart of an optimal boundary of the production set. Hence, at least on intuitive grounds, we might interpret the inefficiency as a measure of output gap with respect to the potential output of the technological frontier.

Output gap is traditionally obtained as a deviation from a statistical measure of trend. One of the earliest and currently widely used statistical methods for measuring the output gap is based on measuring the output trend calculated by fitting a polynomial in time to the output, the residual being the estimated cycle. This method imposes a strong prior on the smoothness of the trend. Another popular statistical approach uses a filter, Hodrick and Prescott (1997), to identify the trend and the cycle. The trend measure in this case is smooth but not deterministic. The Baxter and King (1999) filter defines the cycle as having spectral power in pre-specified frequencies. However, Murray (2003) stresses that this filter extracts an estimate of the cycle which includes some trend shock. Other statistical approaches need a model to identify the stochastic trend component. These statistical methods do not require smoothness but impose the restriction of no correlation between the cycle and the trend, which may lack theoretical support. Beveridge and Nelson (1981) suggest a measure of trend as a long run forecast of an ARMA model. The unobserved components model extracts an estimate of the trend and cycle using the Kalman filter (Harvey 1985; Watson 1986; Clark 1987).

Differently from the statistical methods, the economic approaches estimate the output gap in the framework of the production function (for example Galí and Gertler 1999). Recently, various studies (Kuttner 1994; Gerlach and Smets 1999; Apel and Jansson 1999; Roberts 2001; Basistha and Nelson 2007; Basu and Fernald 2009) tried to combine the statistical approach with the economic approach by estimating the unobserved components of the multivariate model. These approaches do not impose smoothness or restrictive correlation structure, but estimate the output gap based on the empirical implications of the forward-looking Phillips curve.

Often, potential output is referred to as the production capacity of the economy. In our framework of the frontier model, potential output refers to the maximum level of output that can be produced for a given level of inputs, using full employment and capital utilization. The gap between the potential and actual outputs is interpreted as a measure of inefficiency which in our paper also captures the varying factor utilization over the cycle. The approach is closely linked to the production theory based approach in measuring the output gap. We cast our empirical model in frontier form, treating the gap as an unobserved variable—efficiency scores—estimated using nonparametric frontier methods. In pursuing an economic based approach, we avoid imposing strong priors on the smoothness of the trend or cycle, and the restrictive correlation structure between the trend and the cycle shocks.

Furthermore, parametric modeling may suffer from misspecification problems when the data generating process is unknown, as is usual in the applied studies. We propose a unified nonparametric framework for accommodating simultaneously the problem of model specification uncertainty and time dependence in the panel data frontier model. Specifically, we estimate the panel data frontier model using a flexible nonparametric two step approach to take into account the time dependence. Following recent developments in nonparametric conditional frontier literature (Florens et al. 2014; Mastromarco and Simar 2015, 2018), we adapt the nonparametric location-scale frontier model, where we link production inputs and output to time. In the first step we clean the dependence of inputs and outputs on time factors. These time factors capture the correlation among units. By eliminating the effect of these factors on the production process, we mitigate the problem of dependence across our time units and we are able to estimate a nonparametric frontier model from the panel data. (In the application we illustrate this approach for the data on 16 OECD countries.) In the second step, we estimate the frontier and the efficiency scores using inputs and outputs whitened from the influence of time.

The main idea of this paper is to merge the interesting streams of literature described above: the novel nonparametric methods to estimate frontier efficiency of an economy as a new measure of output gap and the novel nonparametric method to estimate the probability of an economic recession. We do this by deploying a generalized nonparametric quasi-likelihood method in the context of dynamic discrete choice models for time series data (Park et al. 2017). To illustrate the new framework, we use data from 1995 to 2019 with quarterly frequency and find that our model using either nonparametric or the linear probit model, applied frequently in this context, is able to offer additional insights into the literature.

The paper is organized as followed. Section 2 presents the methodology. Specifically, Sect. 2.1. explains the nonparametric discrete choice models for time series to predict recessions. Section 2.2. introduces our proposed measure of output gap and explains time-dependent conditional efficiency scores and the nonparametric estimation. This section elucidates the location-scale models to eliminate the influence of common time factors and external variables. Section 3 illustrates an empirical application for the case of Italian economy. Section 4 gives concluding remarks.

In this section, we summarize the elements from Park et al. (2017) (hereafter PSZ) that are needed in our setup to forecast economic recessions. The model should provide the elements for analyzing the behavior of a discrete variable in a time series setup. The approach is nonparametric.

Suppose we observe \(({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t},Y^{t})\), \(t=1,\ldots ,T\), where \(\left\{ ({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t},Y^{t})\right\} _{t=-\infty }^{\infty }\) is a stationary random process. We assume as in PSZ that the process satisfies strong mixing conditions that typically allows time dependence which disappears at a geometrical rate when the time lags are too large.²

The response variable is binary taking the values 0 and 1; in our set-up, \(Y=1\) for a recession and \(Y=0\) is otherwise. The vector of covariates \({{\varvec{X}}}^{t}\) is of dimension r and of continuous type, whereas \({{\varvec{Z}}}^{t}\) is a discrete vector of dimension k. The components of \({{\varvec{Z}}}^{t}\) may be lagged values of the response Y, e.g., \(Y^{t-1},\ Y^{t-2}\). The idea is to estimate the mean functionSince Y is binary, we haveA key ingredient in these discrete choice models is the link function g, which is a strictly increasing function, defining the function f asIn parametric models, it is assumed that \(f({{\varvec{x}}},{{\varvec{z}}})\) takes a parametric form, and then, \(m({{\varvec{x}}},{{\varvec{z}}})=g^{-1}(f({{\varvec{x}}},{{\varvec{z}}}))\). Thus, a wrong choice may jeopardize the estimation of m. In nonparametric settings, \(f({{\varvec{x}}},{{\varvec{z}}})\) will be locally approximated by some local polynomial around \(({{\varvec{x}}},{{\varvec{z}}})\), so the choice of g is much less important. Approximating locally the functions \(g_{1}(m({{\varvec{x}}},{{\varvec{z}}}))\) or \(g_{2}(m({{\varvec{x}}},{{\varvec{z}}}))\) for two different link functions \(g_{1}\) and \(g_{2}\) does not make much difference. One may simply take the identity function, though since the range of the target m is [0, 1], we will choose a link that guarantees the correct range (like Probit or Logit). Now, given the link g and the sample \(\left\{ ({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t},Y^{t})\right\} _{t=1}^{T}\), we see from (2.2) that the log-likelihood of f is given by \(\sum _{t=1}^{T}\ell \left( g^{-1}\left( f({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t})\right) ,Y^{t}\right) \) where \(\ell (\mu ,y)=y\log \left( \frac{\mu }{1-\mu }\right) +\log (1-\mu )\).

Let \(({{\varvec{x}}},{{\varvec{z}}})\) be a fixed point of interest at which we want to estimate the value of the mean function m, or equivalently of its transformed function f. In a nonparametric approach, we will apply local smoothing techniques to the observations \(({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t})\), which are in the neighborhood of \(({{\varvec{x}}},{{\varvec{z}}})\). As explained in PSZ, this leads to weighting the observation \(({{\varvec{X}}}^{t},{{\varvec{Z}}}^{t})\) near \(({{\varvec{x}}},{{\varvec{z}}})\) by some kernel. For the continuous variables (X), usual continuous kernels (Gaussian, Epanechnikov, etc.) can be used, while for the discrete variables (Z), some appropriate discrete kernels have to be used. Here we use the product kernel \(w_{c}^{t}({{\varvec{x}}},{{\varvec{z}}})\times w_{d}^{t}({{\varvec{z}}})\) defined aswhere \(\mathbb {1}(A)\) denotes the indicator function such that \(\mathbb {1}(A)=1\) if A holds and zero otherwise and \(\gamma _{l}\in [0,1]\) is the bandwidth for the \(j{\mathrm{th}}\) discrete variable, while for the continuous kernels, we havefor a symmetric kernel function K and two bandwidth, \(h_{j}(1)>0\) and \(h_{j}(2)>0\), corresponding to the two groups denoted as z(1) and z(2), for each jth continuous variable. The discrete kernel is in the spirit of Aitchison and Aitken (1976), except that it is standardized to be between 0 and 1. The continuous kernel is a generalized kernel proposed by Li et.al. (2016), which allows different bandwidths for the continuous variables across various groups defined by the values of \({{\varvec{Z}}}\), thus allowing for more flexibility in terms of the fitted curvatures in the two groups. It is worth noting that when \(\gamma _{l}=0\), one performs a separate estimation for each group identified by the values of \(Z_{l}\). When \(\gamma _{l}=1\), one considers that \(Z_{l}\) is irrelevant and so all the groups are pooled together, although different bandwidths for continuous variables may still imply different curvatures in the two groups.

For approximating \(f(\cdot ,\cdot )\) locally near the point \(({{\varvec{x}}},{{\varvec{z}}})\), we will not make use of the link function, nor of the likelihood function. The local approximation is linear in the direction of the continuous variable and constant in the direction of the discrete variables. To be specific, we havewhere \(f_{j}({{\varvec{x}}},{{\varvec{z}}})=\partial f({{\varvec{x}}},{{\varvec{z}}})/\partial x_{j}\). So the local approximation can be viewed as a first order Taylor’s expansion of f in \({{\varvec{x}}}\), near \(({{\varvec{x}}},{{\varvec{z}}})\).

To estimate \(f({{\varvec{x}}},{{\varvec{z}}})\) and its partial derivatives \(f_{j}({{\varvec{x}}},{{\varvec{z}}})\), we thus maximizewith respect to \(\beta _{0}\) and \(\beta _{j}\), \(j=1,\ldots ,r\). The solutions \({\hat{\beta }}_{0}={\widehat{f}}({{\varvec{x}}},{{\varvec{z}}})\) and \({\hat{\beta }}_{j}={\widehat{f}}_{j}({{\varvec{x}}},{{\varvec{z}}})\) for \(j=1,\ldots ,r\). Then, an estimator of the mean function \(m({{\varvec{x}}},{{\varvec{z}}})\) is obtained by inverting the link function: \({\widehat{m}}({{\varvec{x}}},{{\varvec{z}}})=g^{-1}({\hat{\beta }}_{0})\).

The theory in PSZ shows that the asymptotic properties of the estimators do not much depend on the choice of the link function, as long it is smooth enough and strictly increasing, because the estimation is performed locally. We will choose below the probit link, i.e., \(g(s)=\Phi ^{-1}(s)\), where \(\Phi \) is the cumulative distribution function of the standard normal distribution. So we have to maximize in \((\beta _{0},\beta _{j})\), \(j=1,\ldots ,r\)The properties of the resulting estimators follow from PSZ. In summary, under certain regularity assumptions and with the optimal order of the bandwidths, \(h_{c,j}:=(h_{j}(1)+h_{j}(2))/2\propto T^{-1/(r+4)}\) and \(\gamma _{l}\propto T^{-2/(k+4)}\), Theorem 3.1 in PSZ establisheswhere \({\bar{h}}_{c}=\prod _{j=1}^{r}h_{c,j}\) and the variance V has a complicated expression which depends on the properties of the data generation process (DGP) (see PSZ for details). We see from (2.9) that the optimal bandwidths balance, as often the case, is between the square of the bias terms and the variance.

We propose as an output gap our measure of inefficiency. The output gap is an economic measure of the difference between the actual output of an economy and its potential output. Potential output is the maximum amount of goods and services an economy can turn out when it is most efficient—that is, at full capacity. Often, potential output is referred to as the production capacity of the economy. In the context of this paper, we assume that a country is the producer of an output (i.e., GDP), given inputs (e.g., capital, labor), and available technology. The inefficiency is defined as the distance between the actual production and its maximum or frontier potential, given the inputs and technology.³

As explained above, we would like to use the level of inefficiency of the country for a particular year by considering the so-called conditional inefficiency (Cazals et al. 2002; Daraio and Simar 2005; Mastromarco and Simar 2015). Inputs here are Capital (K) and Labor (L), and the output is the GDP (Q), and we have quarterly data \(t=1,\ldots ,T\) for 16 OECD countries. Evaluating the marginal efficiency measures by considering the so-called meta-frontier of the 3-dimensional cloud of T points \(\{(K_{t},L_{t},Q_{t})\}_{t=1}^{T}\) would not make too much sense since the technology certainly varies over the years. We will rather consider the conditional efficiency measure where we condition on the time period. This enables us to take into account that production factors adjust to fluctuations of aggregate demand and supply with time delays due to market regulations and price stickiness.⁴

As suggested in Mastromarco and Simar (2015), to introduce the time dimension we consider indeed, with some abuse of notation, time as a conditioning variable W and we define the attainable production set at time t as the support of the conditional probabilitywhich can be interpreted as the probability of observing, at time t, a production plan dominating a given point \((\xi ,\zeta ,\eta )\). So, the feasible technology \(\Psi ^{t}\) can be defined asFinally, this leads to consider for the output orientation the conditional efficiency scorewhich is known as the Farrell–Debreu output oriented efficiency measure (see, e.g., Kumar and Russell 2002, for its use in a related context but using a simpler estimator). Nonparametric estimators of these efficiency scores have been developed and their asymptotic properties are well known (see e.g. Jeong et al. 2010). Here, we will follow the approach suggested by Florens et al. (2014) which has some advantages described below.

In the first step, a flexible nonparametric model is used to whiten the inputs (K, L) and the output Q from the effect of time W. We have the following modelwhere we assume that \((\varepsilon _{K},\varepsilon _{L},\varepsilon _{Q})\) are ‘independent’ of time W, with \({{\mathbb {E}}}[\varepsilon _{\ell }]=0\) and \({{\mathbb {V}}}[\varepsilon _{\ell }]=1\) for \(\ell =K,L,Q\). The estimation of the mean and variance functions is done by local polynomial smoothing as explained in detail in Florens et al. (2014). They suggest also a bootstrap test for testing the assumption of independence, but in our application below we will evaluate various correlations (Spearman, Pearson, and Kendall) to check if this assumption is reasonable.

In our application, we first use the local-linear methods to estimate the mean functions \(\mu _{\ell }(t)\), \(\ell =K,L,Q\). From the squared residuals, we estimate the variance functions \(\sigma _{\ell }^{2}(t)\) by local constant methods (to avoid negative variances). Finally, Florens et al. (2014) define the estimated ‘pure’ inputs and the estimated ‘pure’ outputs aswhich are ‘pure’ in the sense of being filtered from time dependence. In this ‘pure units space,’ we can compute the output directional distance to the efficient frontier.⁵ Since the output here is univariate, the efficient frontier in pure units is the functionso that the directional distance of a point \((e_{K},e_{L},e_{Q})\) to the frontier is simply given bywhere the value zero indicates the point \((e_{K},e_{L},e_{Q})\) is on the efficient frontier. Under the location-scale assumptions, it can be proved that the conditional frontier in original units can be recovered as (see Florens et al. 2014, for details)so that the gap in the output to reach the frontier level is given byThe nonparametric estimators of these various elements are obtained by plugging the estimators of the mean and variance functions derived above. One of the main advantages of this location-scale approach is that for estimating the functions \((\mu _{\ell }(t),\sigma _{\ell }(t))\) we require only smoothing in the center of the data in a standard regression setup. As pointed out in Bădin et al. (2019), a direct estimation of \(\lambda (\xi ,\zeta ,\eta |t)\) requires delicate problems of optimal bandwidths selection for estimating the support of the conditional \(H_{K,L,Q|W}(\xi ,\zeta ,\eta \;|\;W=t)\).

So at the end of this step of efficiency estimations, we end up in practice with estimated efficiency scores in the pure units \(\delta (e_{K_{t}},e_{L_{t}},e_{Q_{t}})\) and, if wanted, the measures of the gaps in original units of the DGP, i.e., \(G_{Q}(K_{t},L_{t},Q_{t}|t)\) at each observation \(t=1,\ldots ,T\). These values (eventually lagged) will be used to improve the prediction of a recession in our application below.

Real data samples contain in general some anomalous data, and the estimated frontier obtained by these nonparametric techniques can be fully determined by these outliers or extreme data points, jeopardizing the measurement of inefficiencies, potentially leading to unrealistic results. Cazals et al. (2002), Daouia and Simar (2007), in the frontier literature, propose an approach which aims to keep all the observations in the sample but replace the frontier of the empirical distribution by (conditional) quantiles or by the expectation of the minimum (or maximum) of a sub-sample of the data. This latter method defines the order-m frontier that we will use here.

In brief, the partial output frontier of order-m is defined for any integer m and for input values \(e_{K_{t}},e_{L_{t}}\), as the expected value of the maximum of the output of m units drawn at random from the populations of units such that \(\varepsilon _{K}\le e_{K},\varepsilon _{L}\le e_{L}\). Formally,where the \(\varepsilon _{Q,it}\) are drawn from the empirical conditional survival function \({\widehat{S}}_{\varepsilon _{Q}|\varepsilon _{x}}(e_{Q}|{{\varepsilon }}_{x,it}\le e_{x})\). This can be computed by Monte Carlo approximation or by solving a univariate numerical integral (for practical details see Simar and Vanhems 2012).

If m increases and converges to \(\infty \) and \(n\rightarrow \infty \), it has been shown (see Cazals et al. 2002) that the order-m frontier and its estimator converge to the full frontier, but for a finite m, the frontier will not envelop all the data points and so is much more robust than the Free Disposal Hull (FDH) to outliers and extreme data points (see, e.g., Daouia and Gijbels 2011, for the analysis of these estimators from a theory of robustness perspective). Another advantage of these estimators is that besides the fact that their limiting distribution is normal, they achieve the parametric rate of convergence (\(\sqrt{n}\)).

There are different ways to measure the spread to be used in the models that we consider here. For the US economy, it is often (albeit not always) measured as the difference between the 10-year US Treasury bond rate and the 3-month US Treasury bill rate, though there are other variants (e.g., see Park et al. (2020) and references there in). For other countries, including those in the EU, there appears to be no ‘one-fit-all’ rule on how to best measure the spread, as it may depend largely on the country of interest or even the time period considered. Here we choose to measure it as the difference between the 10-year Italy Treasury bond rate and the 10-year Germany Treasury bond rate, in per cent per annum. The logic behind using this measure of spread is grounded in the belief that the 10-year yield on German bonds is typically considered as the benchmark for the Euro area since they are viewed by investors as a risk-free market asset, at least in relative terms.⁶ The data for this variable were sourced from OECD.stat Monthly Monetary and Financial Statistics (MEI).⁷ However, again, we acknowledge that other measures of the spread can be tried, and some of them potentially may work better for some countries, yet not others, or differ across different periods for the same country. In fact, finding such a measure of spread that would serve as the best predictor for a given country may be a research question in itself and we leave it for future research endeavors.

The variables on recessions are constructed as following. We use the Composite Leading Indicators from OECD Reference Turning Points and Component Series data, which is analogous to the information from the Business Cycle Dating Committee of NBER typically used for timing the recessions in the USA.⁸ In particular, note that the OECD identifies months of the so-called turning points (peaks and troughs) of the business cycle. The periods between a peak and a trough that follows it are then deemed as the recessionary periods (\(Y_{t}=1\)), while the periods between a trough and a peak that follows it are deemed as the expansionary periods (\(Y_{t}=0\)). To be more precise, since the turning points are announced for a particular month while we use quarterly data, to construct this time series we use the following rule: the recession begins on the quarter of the month of the peak and ends on the quarter of the month on the trough.⁹

To construct our measure of output gap, we need to go beyond the data on Italy and consider a few other countries that may be deemed as relevant peers for Italy, to estimate a relevant technological frontier. For this illustrative exercise, we choose the following OECD countries: Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Israel, Italy, South Korea, Netherlands, New Zealand, Norway, Spain, Sweden and UK.¹⁰

The data for these countries were sourced from OECD.stat (OECD Quarterly National Accounts) and include 99 quarterly observations from (1995 : Q1) till (2019 : Q2), on capital, labor and output.¹¹ To be precise, the output Q is proxied by gross domestic product (GDP), and is measured in millions of US dollars, at 2015 constant price level. For the labor input L, we use the number of employed persons (in thousands) seasonally adjusted. Meanwhile, the capital K is also measured in millions of US dollars at 2015 constant price and constructed applying the perpetual inventory method (PIM) by using the real investment series (gross fixed capital formation).¹² As often suggested with macroeconomic data, all these variables are transformed in logarithms before the frontier estimation.

The Italian economy is one of the oldest in the World, with roots going thousands years back to at least the Roman Empire. Through its long evolution on its way to the modern days, it has witnessed a myriad of ‘ups and downs’ of its economy—what is now usually referred to as Business Cycles. In a broad sense, even a book cannot give a full picture of this interesting country and its economy, yet a brief snapshot on the recent years might be useful here.¹³

Despite the long and great history, fairly developed institutions, and relatively high level of physical and human capital, the Italian economy has been fairly stagnant during the last three decades, the period we focus on in this study. For example, in Fig. 1 we depict the growth rate of Italian GDP during 1995–2019.¹⁴ Note that for the late 1900s, the figure exhibits negative growth in Q2 (2nd quarter) of 1996 and Q1 (1st quarter) of 1998. In the Q1 (1st quarter) of 2009, as the figure reveals, GDP growth registers the largest negative value, and by the Q3 (3rd quarter) of 2009 the economy began to re-grow slightly. In the Q3 (3rd quarter) of the year 2011, Italy’s growth was negative till Q1 (1st quarter) of 2013; then, Italy’s economy recovered with positive economic growth rates, but in Q1 of 2019 it starts to contract again. Here it is worth noting that, similarly as with the NBER data on recessions in the US, the OECD data on recessions in Italy (highlighted with gray shadow in Fig. 1) are not the same as the casual definition of a recession being two consecutive quarters of negative growth, but are based on the identification of the turning points of the business cycle, as described above.

Various reasons have been advocated in the literature as explanations for such poor economic performance of Italy. One of them is the lagging productivity growth relative to its peer countries. In particular, it was argued that insufficient productivity growth may be pivotal to Italy’s competitiveness problem, witnessed by the continual erosion of world export market shares and the limited ability to attract foreign direct investment (Faini et al. 2004). These problems appear to be particularly relevant in Italian manufacturing industries where productivity has been low and international competitiveness has worsened over the recent decades (Bassanetti et al. 2004; Aiello et al. 2011; Pellegrino and Zingales 2017). For example, Pellegrino and Zingales (2017) credit the inability of Italian firms to take full advantage of the information and communication technology revolution as one of the key reasons for the poor productivity or what they dubbed as ‘Italy’s productivity disease.’ In turn, and as for many other failures or successes of a country, the existence and persistence of this ‘disease’ appear to be due to specific institutional aspects; or, as Pellegrino and Zingales (2017) put it:

“While many institutional features can account for this failure, a prominent one is the lack of meritocracy in the selection and rewarding of managers. ...the prevalence of loyalty-based management in Italy is not simply the result of a failure to adjust, but an optimal response to the Italian institutional environment. Italy’s case suggests that familism and cronyism can be serious impediments to economic development even for a highly industrialized nation.”

Clearly, disentangling the true reasons for the recessions in Italy is well beyond the scope of this paper, if at all possible. What seems more feasible, however, is to compare or benchmark Italy to some of its peers—as we do via the proposed output gap measure explained above—in the hope that it may potentially help in providing some useful information for predicting upcoming recessions via the dynamic choice models.

Turning the attention to the spread dynamics, one can also note that Fig. 2 reveals the spread between the 10-year Italy Treasury bond rate and 10-year Germany Treasury bond rate increases during the period of low economic growth. This indicates a lack of confidence of investors in the Italian economy due to the deterioration of potential determinants of the spread, namely the current or expected macroeconomic fundamentals, such as fiscal policy, international risk, liquidity conditions, sovereign credit ratings, to mention a few. Again, note that while in some periods the dynamics of the spread to some extent matches the upcoming changes in the recession indicators (highlighted with gray shadow), the relationship appears to be not very strong, e.g., relative to what we found in the literature for the recessions in the USA (see Park et al. 2020 and references therein).

Here, we first have to run three location-scale models for K, L, Q, respectively, to clean the effect of time W.¹⁵ This provides the ‘pure’ inputs and ‘pure’ output, \(\{({\widehat{\varepsilon }}_{K_{t}},{\widehat{\varepsilon }}_{L_{t}},{\widehat{\varepsilon }}_{Q_{t}})\}_{t=1}^{T}\) as explained above. The correlations of these ‘pure’ inputs/output with time are given in Table 1 (where \(X_{1}=K,X_{2}=L,Y=Q\) and \(Z=W\)). Clearly these correlations are very small so we can infer that the assumption of independence between \((\varepsilon _{K},\varepsilon _{L},\varepsilon _{Q})\) and W, which is part of our location-scale model, seems reasonable.

Robust measures of efficiency scores, providing the gaps in ‘pure’ units were computed with \(m=1500\). This choice was done for letting less than 25% of points above the order-m frontier, as shown in Fig. 3. Note that from the values of \(m=1500\) and m \(\rightarrow \) \(\infty \) (the full FDH frontier), all the results are quite similar.

The resulting efficiency scores \({\widehat{\delta }}_{m,t}\) are shown in Fig. 4, which illustrates that most of the time, the time effect has indeed been cleaned from the production process. We see also that most of the values of \({\widehat{\delta }}_{m,t}\) are positive and some take very small (near zero) negative values. Figure 5 exhibits the time path of output gaps in original units (in logs and re-scaled by their mean). Figure 6 reports the values of the gap in original units for each country in our sample at the first period of observation (1995: Q1) and last period (2019: Q2).

We give in “Appendix C” the full table of results for all the time periods. The table also indicates the gaps \(G_{t}\) in original units of the DGP, as defined above (in log scale and re-scaled by their mean). Figure 9 in “Appendix C” reports the values of output gap in original units for all countries in our analysis for the first and last year of the observation period.

Our next step is to fit the prediction model described above by estimating the parametric linear probit model and the nonparametric model of PSZ to the data described above. In particular, we fit the following model: where \(X_{1,t}={Sp}_{t-r_{1}}\) is the spread lagged by \(r_{1}\) periods, \(X_{2,t}=\Delta _{G,t-r_{2}}\) is the first difference of the estimates of output gaps (production efficiency) lagged by \(r_{2}\) periods. Finally, \(Z_{t}=Y_{t-r_{3}}\), where we recall that \(Y_{t}\) is the dichotomous dependent variable, defined as \(Y_{t}=1\) if “Italian economy is in recession” in the quarter t and 0 otherwise and \(r_{3}\) is its chosen lag. Finally, for smoothing \(Z_{t}\) in the nonparametric approach we use the complete smoothing technique suggested by Li et.al. (2016), allowing different bandwidths for the continuous variables in the two groups determined by the values of Z, as described in Sect. 2.1.

Even though there are only three potential predictors in the general specification (3.1), many variations in it are possible that are based on different subsets of predictors and different choices of lags for each predictor. In the following sub-sections, using data on Italy we briefly show and discuss how a model selection can be done in such situations.¹⁶

We now proceed with the out-of-sample forecasts, to see if we can have a reasonably good prediction of the recession periods (one-period and two-periods ahead), using the data from the beginning till 2016:Q1 from either the parametric or nonparametric approaches.²²

The forecasts of the recessions are displayed in Fig. 8. In most cases (and on average), we can observe a slightly better forecast value for the parametric approach, both for the case of the one-period ahead and for the two-periods ahead forecasts. In particular, note that, with the one-period ahead forecasts, both approaches correctly and somewhat similarly warn about the recession in Q1-2018–Q2-2019, with the parametric approach slightly outperforming. Both approaches with the one-period ahead forecasts correctly alert us to the non-recession (or expansions) in Q1-2016 through Q3-2017, though both miss on warning about the start of the recession in Q4-2017, while they manage to warn correctly about the subsequent quarters being in the recession.