1 Introduction

In recent years, due to the availability of data on a vast number of correlated macroeconomic and financial variables collected regularly by statistical agencies, there has been an increasing interest in modeling large systems of economic time series. Therefore, econometricians have to deal with datasets consisting of hundreds of series, thus making the use of large dimensional dynamic factor models (DFMs) more attractive than the usual vector autoregressive (VAR) models, which usually limit the number of variables; see Boivin and Ng (2006). DFMs were introduced by Geweke (1977) and Sargent and Sims (1977) with the aim of representing the dynamics of large systems of time series through a small number of hidden common factors, and are mainly used for one of the following two objectives: first, forecasting macroeconomic variables and second, estimating the underlying factor in order to carry out policy-making (e.g., the business cycle; lagging, coincident, and leading indicators; or impulse-response functions, among other aspects). Another interesting application is to use the common factors as instrumental variables or exogenous regressors in panel data analysis. See Bai and Ng (2008), Stock and Watson (2011), and Breitung and Choi (2013) for a review of existing literature and applications of DFMs.

Note that macroeconomic time series are generally non-stationary and frequently cointegrated; see, for example, Kunst and Neusser (1997). On the other hand, a cointegrated system can be expressed in terms of a factor representation; see Stock and Watson (1988), Vahid and Engle (1993), Gonzalo and Granger (1995). Furthermore, these authors show that the common factor representation implies that the variables of the system are cointegrated if the common factors are I(1) and the individual effects are I(0). However, in practice, the cointegration results are not frequently used to predict macroeconomic variables, given that we differentiate each element of the factor representation individually.

Empirically, it is interesting to determine the number of common factors of the economy, given that we can summarize a large number of macroeconomic variables in few common trends.Footnote 1 Furthermore, by estimating the common factors we can observe the orthogonal dynamics of the economy, which is very important for macroeconomic policy. These common factors, for example, may be related to specific groups of variables, and it is interesting to analyze mechanisms of transmission to other groups of variables. Additionally, we can estimate the confidence intervals for the loading weights, common factors, and common components, which allow us to carry out statistical inference in order to better understand economic phenomena.

The stochastic common trends represent the long-run behavior of the variables. Previous studies related to this paper are Duy and Thoma (1998), who evaluate the improvement of the cointegration relationship in VAR models; Eickmeier and Ziegler (2008), who examine the use of the factor-augmented vector autoregressive (FAVAR) in order to predict macroeconomic and financial variables, focusing on the US economy. They conclude that on average, factor forecasts are slightly better than other models. Banerjee et al. (2014) incorporate the error correction term in FAVAR models, showing that their approach generally offers higher forecasting accuracy relative to the FAVAR. Also, Eickmeier et al. (2014) prove different specifications of FAVAR models, using time-varying coefficients (tv) and stochastic volatility errors. The main conclusion is that the FAVAR models, regardless of the specification, outperform univariate models on in-sample forecasts. When they evaluate the out-sample performance, there is no improvement with respect to the benchmark model. More recently, Hindrayanto et al. (2016) study the forecasting performance of a four factor model approach for large datasets. They conclude that a collapsed DFM is the most accurate model. Finally, Wilms and Croux (2016) propose a sparse cointegration method for a large set of variables by setting some coefficients in the cointegration relationship at exactly zero. They conclude that their method leads to significant forecast accuracy with respect to the method proposed by Johansen (1988, 1991). Other samples of reference are Stock and Watson (2002a, b), Marcellino et al. (2003), Peña and Poncela (2004), Reijer (2005), Schumacher (2007), Giannone et al. (2008), Eickmeier et al. (2014), Lahiri et al. (2015), and Panopoulou and Vrontos (2015), who show that DFMs are useful in order to reduce the forecast errors with respect to the traditional models, as autoregressive approaches and linear regressions with macroeconomic diffusion indexes.Footnote 2 It is important to comment that, in a similar way as we have proposed in this study but in a stationary framework, Bräuning and Koopman (2014) propose to use collapsed dynamic factor analysis in order to predict target variables of the economy, exploiting the state-space representation between the target variables and the common factors. They conclude that the forecast accuracy is improved with respect to benchmark models.

However, most empirical applications have been carried out for the US or Euro economies. Therefore, we propose a FAVAR model to predict Mexican economic activity using concepts of cointegration such that we forecast I(1) time series where the estimated common factors are the common trends of the economy. We readily exploit the long-run relationship between the common factors and the specific variable that we wish to predict.Footnote 3

We contribute to the literature in two ways: (1) empirically in determining the number of common trends of the Mexican economy and (2) the use of the common trends in order to predict economic activity. This is an advantage with respect to other alternatives, given that the FAVAR with cointegrated variables is very straightforward to implement computationally. Furthermore, we also direct our attention to the predictive capacity of macroeconomic variables and their common trends. In this way, we use the traditional method of principal components (PC) and additionally, partial least squares (PLS).

The rest of this paper is structured as follows. In Sect. 2, we summarize the historical behavior of the Mexican economy. In Sect. 3, we briefly explain the DFM and the FAVAR representation. In Sect. 4, we describe the estimation methods used. In Sect. 5, we describe the data, present the cointegration results, their descriptive predictive features, the empirical strategy to obtain small forecast errors, and the results obtained. Finally, we conclude in Sect. 6.

2 The Mexican economy

Mexico is one of the world’s leading emerging countries. According to data from the International Monetary Fund (IMF), it is the 11th largest economy and the largest emerging country outside the BRICs, i.e., Brazil, Russia, India, and China (Rodriguez-Pose and Villarreal 2015). In 2015, Mexico’s population ascended to 119.9 million and its GDP per capita stood at 9005.0 US dollars (World Bank 2016b).

After a prolonged period of sustained economic growth, the country began decelerating in the late 1970s. Following a sovereign default, Mexico encountered a severe economic crisis from 1982 to 1985. Subsequently, the Mexican government put forward a series of market reforms that culminated with the implementation of the North American Free Trade Agreement in 1994. These reforms included the privatization of previously government-operated firms, fiscal reforms, and opening the country to trade and foreign direct investment (FDI), among other actions (Kehoe and Ruhl 2010). Mexico’s economic transformation was successful in reducing inflation, maintaining fiscal discipline, reducing its external debt burden, and increasing trade as a share of GDP (Hanson 2010). Nonetheless, this was not accompanied by high levels of economic growth, as between 1982 and 2015 the country’s GDP grew at an average annual rate of 2.3% (World Bank 2016a).

Moreover, in recent decades the Mexican economy has regressed on different fronts. In 1995, Mexico was responsible for 7.0% of trade among economies classified by the IMF as emerging and developing, behind China at 12.7%. By 2008, China’s share was still the largest and totaled 22.3%, while Mexico dropped to third at 5.4%, behind Russia. Additionally, in 1995, China was the largest recipient of FDI to emerging and developing economies, accounting for 33.4%, while Mexico was second, receiving 8.5%. By 2008, Mexico’s position had weakened considerably, falling to seventh at 3.2% (Kehoe and Ruhl 2010). Factors, such as rigidities in the labor market, an inefficient financial sector, and a lack of contract enforcement, have limited Mexico’s capacity to benefit from its reforms and therefore hindered the country’s economic growth (Kehoe and Ruhl 2010).

As an economy open to world markets, Mexico hosts many modern firms, notably in the automobile, aerospace, and foods and beverages sectors, which employ highly skilled, well-educated, and well-remunerated workers. Nonetheless, this represents only a small part of the economy and is concentrated in a few regions of the country. Another segment of the economy is characterized by high levels of informality, low-skilled work, low productivity levels, and out-of-date technologies. Unregistered firms in the informal sector employ close to half of Mexico’s labor force, with workers that frequently lack access to a good education, reliable health care, and affordable financial services, conditions that strongly hamper their accumulation of human capital. Lastly, a third segment of the economy is made up of companies that, for long periods, have been protected from competition, particularly dominant firms in the energy and telecommunications sectors (Dougherty 2015).

During the global economic crisis of 2008–2009, Mexico’s GDP suffered a significant contraction, which was accentuated by a reduction in the amount of remittances the country receives from migrant workers based in the US and the outbreak of influenza A(H1N1) throughout the country. Nevertheless, due to an upgraded macroeconomic policy framework and careful regulation of the financial system, Mexico did not suffer the type of financial and fiscal crises experienced in other countries (Schwellnus 2011).

In the years following the global economic crisis, the Mexican economy was characterized by a persistent trend of increasing debt-to-GDP, which grew from 29.0% in 2007 to 50.5% in 2016, and a significant decline in the volume of oil production (World Bank 2016a). Output volatility in Mexico remained high, which can have high costs for individuals and for long-term growth. Furthermore, in Mexico, temporary disturbances in output are usually accompanied by temporary reductions in consumption, since a substantial share of the population is credit-constrained and the social safety net is generally weak. This is costly for individuals who prefer a smooth path of consumption and are averse to periods of unemployment or poverty (Schwellnus 2011).

Finally, during the last decade, Mexico has implemented a more ambitious and wide-ranging innovation policy aimed at getting closer to the technological frontier and generating higher levels of economic growth (Rodriguez-Pose and Villarreal 2015). By 2016, the expansion of economic activity was highly dependent on private consumption as low levels of investment and export demand were scarcely contributing to growth. In the medium term, economic and financial stability and an increase in external competitiveness, derived from the depreciation of the country’s currency, are expected to boost private investment, exports, and hence economic growth (World Bank 2016b).

In this context, it is very important to formulate an econometric approach in order to successfully predict Mexican economic activity, which takes into account a large number of macroeconomic and financial variables associated with the behavior of the economy. Hence, we use a large non-stationary DFM in which the variables are summarized in the common trends of the Mexican economy. These common trends are used in a FAVAR representation, evaluating the out-sample forecast error one and two steps ahead during a recent phase of the economy.

3 The models

In this section we introduce notation and describe the non-stationary DFM and the FAVAR model in order to use the common trends to predict a specific macroeconomic variable.

3.1 Non-stationary dynamic factor model

Suppose that N large economic time series \(Y_t = (y_{1t},\ldots , y_{Nt})^{\prime }\), observed from \(t=1,\ldots ,T\), are I(1) and any two series \(y_{it}\) and \(y_{jt}\) are cointegrated, such that the idiosyncratic component \(\varepsilon_t = (\varepsilon_{1t},\ldots , \varepsilon_{Nt})^{\prime }\) is stationary. In this case, and following Stock and Watson (1988), there is a common factor representation as follows:

$$Y_t = PF_t + \varepsilon_t,$$
(1)

where \(P=(p_{1}^{{\prime }},\ldots ,p_{N}^{{\prime }})^{{\prime }}\) is the \(N\times r\) (\(r < N\)) matrix of factor loadings, where \(p_{i}=(p_{i1},\ldots ,p_{ir})\) and \(F_{t}=(F_{1t},\ldots ,F_{rt})^{{\prime }}\) is the \(r \times 1\) vector of common factors, or in this case, the common trends.

Given that any two series of \(Y_t\) are cointegrated in the sense of Engle and Granger (1987), the dynamic of the common factors and the idiosyncratic components are the following:

$$F_t = F_{t-1} + \eta_t$$
(2)
$$\varepsilon_t = \varGamma (L)\varepsilon_{t-1} + a_t$$
(3)

where the factor disturbances, \(\eta_{t}=(\eta_{1t},\ldots ,\eta_{rt})^{{\prime }}\), are \(r\times 1\) vectors, distributed independently from the idiosyncratic noises for all leads and lags. Furthermore, \(\eta_{t}\) and \(a_{t}\) are white noises with positive definite covariance matrices \(\varSigma_{\eta }\) and \(\varSigma_{a}\), respectively. Additionally, \(\varGamma (L)\) collected the \(N\times N\) matrices containing the autoregressive parameters of the idiosyncratic components where L is the lag operator such that \(L^{k}\varepsilon_t = \varepsilon_{t-k}\). These autoregressive matrices satisfy the usual stationarity assumptions.

The DFM in Eqs. (1)–(3) is not identified because for any \(r\times r\) non-singular matrix H, the system can be expressed in terms of a new loading matrix and a new set of common factors as follows:

$$Y_t = P^{*}F^{*}_t + \varepsilon_t$$
(4)
$$F^{*}_t = F_{t-1}^{*} + \eta_t^{*} ,$$
(5)

where \(P^{*}=PH\), \(F^{*}_t = H^{-1}F_t\), and \(\eta_t^{*} = H^{-1}\eta_t\). The DFM in Eqs. (4) and (5) is observationally equivalent to that in Eqs. (1) and (2). A normalization is necessary to solve this identification problem and uniquely define the factors. In the context of PC factor extraction, it is common to impose the restriction \(P^{\prime }P/N = I_r\) and \(F^{\prime}F\) being diagonal, where \(F = (F_1,\ldots ,F_T)^{\prime }\) is a \(T\times r\) matrix of common factors. Alternatively, we can set the restrictions \(F^{\prime}F/T^2 = I_r\) and \(P^{\prime }P\) being diagonal; see Bai (2004) for identification issues in the non-stationary case.

In Eqs. (1)–(3), if \(\varGamma = 0\) and \(\varSigma_a\) is diagonal, then the DFM is known as strict; see Bai and Ng (2008). On the other hand, when \(\varGamma \ne 0\) and \(\varSigma_a\) is diagonal, there is serial correlation in the idiosyncratic noises and in this case the DFM is called exact; see Stock and Watson (2011). Chamberlain and Rothschild (1983) introduce the term approximate model when the idiosyncratic term does not need to have a diagonal covariance matrix. Forni et al. (2000) and Stock and Watson (2002a) generalize the approximate model, allowing for weak serial and cross-correlation. Furthermore, Bai and Ng (2002) and Bai (2003) also allow for heteroskedasticity in both the time and cross-sectional dimensions.

3.2 The factor-augmented autoregressive model

Once the non-stationary DFM is specified, we can use the common trends to predict a specific macroeconomic variable, denoted as \(x_t\); see Stock and Watson (2005), Banerjee et al. (2013), among many others. It is important to comment that if forecasting is the objective, although (\(x_t, F_t)^{\prime }\) are integrated or cointegrated, the FAVAR specification is quite convenient. Note that the optimal conditional expectation does not require stationarity or stability in the system. The theoretical justification is given by Lütkepohl (2006). More recently, Barigozzi et al. (2016) discuss several situations where a VAR estimated in levels is equivalent to a cointegrated VAR. Basically, when estimating a VAR in levels we can consistently estimate the parameters of a cointegrated VAR. Therefore, the FAVAR is provided as follows:

$$\left( \begin{array}{c} x_t \\ F_t \end{array} \right) = \varPhi (L) \left( \begin{array}{c} x_{t-1} \\ F_{t-1} \end{array} \right) + v_t,$$
(6)

where

$$\varPhi (L) = \left( \begin{array}{ccccc} \varPhi_{11}^{(k)} &{} \varPhi_{12}^{(k)}&{} \varPhi_{13}^{(k)} &{} \dots &{} \varPhi_{1r+1}^{(k)} \\ 0 &{} \varPhi_{22}^{(k)} &{} 0 &{} \dots &{} 0 \\ 0 &{} 0 &{} \varPhi_{33}^{(k)} &{} &{} 0 \\ \vdots &{} \vdots &{} &{} \ddots &{} \\ 0 &{} 0 &{} 0 &{} &{} \varPhi_{r+1r+1}^{(k)} \end{array} \right) L^{k+1},$$

for \(k = 1,\ldots , p\), where \(v_t \sim N(0, \varSigma_v)\). Intuitively, common economic trends that summarize its long-run behavior are used to predict a specific macroeconomic variable. For a similar representation but in the stationary context, see Bräuning and Koopman (2014). Note that expression (6) is the prediction equation; for example, assuming \(p = 1\), for any h step ahead, then

$$x_{t+h} = \varPhi_{11}x_{t-(h+1)} + \varPhi_{12}F_{1,t-(h+1)} + \cdots + \varPhi_{1r+1}F_{r+1,t-(h+1)} + v_{t+h}.$$
(7)

Additionally, we can introduce deterministic components and seasonal and exogenous variables in Eq. (6).

4 Estimation

The estimation of the parameters of Eq. (6) is obtained using restricted ordinary least squares (OLS). Furthermore, it is necessary to define the number of common factors and their factor estimates. In this section we describe some traditional methods to determine the number of non-stationary common factors. Additionally, we describe PC factor extraction and comment on an alternative method to estimate the common factors, known in literature as PLS.

4.1 Determining the number of factors

It was previously assumed that r is known, however in practice it is frequently necessary to estimate it. In this subsection we describe two procedures to determine r in the context of large DFMs, namely the Onatski (2010) procedure and the ratio of eigenvalues proposed by Ahn and Horenstein (2013). There are other alternatives to determine r, the Bai and Ng (2002) information criteria, the respective correction proposed by Alessi et al. (2010), and the procedure carried out by Kapetanios (2010), among many other approaches. Moreover, in the context of non-stationary DFMs, Ergemen and Rodriguez-Caballero (2016) use the procedure given by Hallin and Liska (2007) to determine the number of regional and global factors allowing fractional differencing in \(Y_t\). However, and following Corona et al. (2017a), they show that the finite sample performance of the proposed procedures in this paper exhibits a good performance when we use data in first differences or levels and the common factors are I(1).Footnote 4

4.1.1 Onatski (2010) procedure

The Onatski (2010) procedure is based on the behavior of two adjacent eigenvalues of \({\hat{\varSigma }}_Y = T^{-1}Y^{\prime }Y\), for \(j = 1, \ldots , r_{\max }\). Intuitively, it is reasonable that when N and T tend to infinity, the difference between \({\hat{\lambda }}_j - {\hat{\lambda }}_{j+1}\) tends to zero while \({\hat{\lambda }}_r - {\hat{\lambda }}_{r+1}\) diverges to infinity, where \({\hat{\lambda }}_i\) is the i-largest eigenvalue of \({\hat{\varSigma }}_Y\). Onatski (2010) points out the necessity of determining a “sharp” threshold that separates convergent and divergent eigenvalues, denoted as \(\delta\). The author gives the empirical procedure to determine the threshold and in practice, this approach is more robust in presence of non-stationarity and when the factors are weak or when the proportion of variance attributed to the idiosyncratic components is larger than the variance of the common component. We denote this estimator as \({\hat{r}}_{\mathrm{ED}} = \max \lbrace j \le r_{\max } : {\hat{\lambda }}_j - {\hat{\lambda }}_{j + 1} \ge \delta \rbrace.\).

4.1.2 The Ahn and Horenstein (2013) ratios of eigenvalues

Ahn and Horenstein (2013) develop the consistency of the estimation of r when they use the ratio of two adjacent eigenvalues of \({\hat{\varSigma }}_Y\) under the traditional assumptions of PC factor extraction. The two criteria provided by the authors are based on maximizing with respect to \(j = 1, \ldots , r_{\max }\) the following ratios:

$$\text{ER}(j)=\frac{{\hat{\lambda }}_{j}}{{\hat{\lambda }}_{j+1}},$$
(8)
$$\text{GR}(j)=\frac{ \ln (1+{\hat{\lambda }}_{j}^{*})}{\ln (1+{\hat{\lambda }}_{j+1}^{*})},$$
(9)

where \({\hat{\lambda }}_{0}=\frac{1}{m}\sum_{i=1}^{m}{\hat{\lambda }}_{j}/\ln (m)\) and \({\hat{\lambda }}_{j}^{*}={\hat{\lambda }}_{j}/\sum_{i=k+1}^{m}\hat{\lambda }_{i}\). The value of \({\hat{\lambda }}_{0}\) has been chosen following the definition of Ahn and Horenstein (2013), according to which \({\hat{\lambda }}_{0}\rightarrow 0\) and \(m{\hat{\lambda }}_{0}\rightarrow \infty\) as \(m\rightarrow \infty .\)

4.2 Principal components factor extraction

The most popular method to extract static common factors is based on PC given that it does not require assumptions of the error distribution, and the estimation of the common component is consistent, among many other properties; see Bai (2003). This procedure separates the common component from the idiosyncratic noises by considering a cross-sectional averaging of the variables within \(Y_t\) such that when N and T tend simultaneously to infinity, the weighted averages of the idiosyncratic noises converge to zero, with only the linear combinations of the factors remaining. Therefore, this method requires that the cumulative effects of the common component increase proportionally with N, while the eigenvalues associated with the idiosyncratic components remain bounded.

The PC estimator of \(F_t\) can be derived as the solution to the following least squares problem:

$$V_r(P,F)=\min_{F_1,\ldots ,F_T,P}(NT)^{-1}\sum_{t=1}^{T}(Y_t-PF_t)^{\prime }(Y_t-PF_t)$$
(10)

subject to the restrictions \(F^{\prime }F/T^2 = I_r\) and \(P^{\prime }P\) being diagonal, where \(F = (F_{1},\ldots , F_{T})^{\prime }\) is a \(r \times T\) matrix of common trends. The minimization problem in (10) is equivalent to maximizing tr\([F^{\prime }(YY^{\prime })F]\) where \(Y = (Y_1,\ldots ,Y_T)^{\prime }\) with the dimension \(T\times N\). This maximization problem is solved by setting \(\hat{F}\) equal to T times the eigenvectors corresponding to the r largest eigenvalues of the \(T\times T\) matrix \(YY^{\prime }\). The corresponding PC estimator of P is given byFootnote 5:

$$\hat{P} = \frac{Y^{\prime } \hat{F}}{T^2}.$$
(11)

Bai (2003) deduces that the rate of convergence and the limiting distributions of the estimated factors, factor loadings and common component is the stationary DFM when the cross-section and time dimensions tend towards infinity. Under the assumptions considered in this paper, Bai (2004) shows the consistency of \(\hat{F}_t\), \(\hat{P},\) and the common component. The finite sample performance of \(\hat{F}_t\) is analyzed by Bai (2004) and recently by Corona et al. (2017b). The first author concludes that \(\hat{F}_t\) is a close estimation of \(F_t\) when the idiosyncratic errors are stationary and even if the sample size is moderately small (i.e., \(N = 100\) and \(T = 40\)). The second authors consider structure dependence in the idiosyncratic errors and show that when the idiosyncratic errors are I(0) we can obtain close estimations of \(F_t\) even if N is small and the variance of \(\varepsilon_t\) is large. When the idiosyncratic errors are I(1), \(\hat{F}_t\) works poorly. In this case, it is convenient to use the estimator given by Bai and Ng (2004).Footnote 6 Other alternatives to estimate the common factors without differencing them are given by Barigozzi et al. (2016) and Corona et al. (2017b).

Note that the estimation of \(F_t\) disregards the dynamic of Eqs. (2) and (3). The reason for this is that we estimate the common factors from static representation in the factor model. Corona et al. (2017b) incorporate the dynamic of Eq. (2) in a second step to obtain smooth estimates of the common factors using the Kalman filter. However, in this study we focus on a large DFM from static representation. For a technical discussion of the analogies and differences between static and dynamic representation, see for example Bai and Ng (2007). Thus, our approach is related to Bai (2004) and Bai and Ng (2004), who study non-stationarity in DFMs under a static representation.

4.3 Partial least squares

In addition to the estimation of \(F_t\) by PC, we consider the PLS estimation, which takes into account the effect of a dependent variable. In a similar manner to Fuentes et al. (2015), we estimate the common factors in economic time series using PLS, denoted as \({\tilde{F}}_t\). Intuitively, the idea is to find the orthogonal latent variables that maximize the covariance between Y and \(x = (x_1, \dots , x_T)^{\prime }\). The estimation process is iterative, and the first step consists in the eigenvalue decomposition of the following \(T \times T\) matrix:

$$M = Yxx^{\prime }Y^{\prime }$$
(12)

Then, the first common factor, \({\tilde{F}}_{1t}\), is the first eigenvector associated to the first eigenvalue of M. The second common factor is estimated from the residual matrix of \(e = Y - B{\tilde{F}}_1^{\prime }\). To obtain the following common factors, the process is repeated \(r-2\) steps using the \({\tilde{F}}_{1t}, \dots , {\tilde{F}}_{r-1t}\) common factors in each step.

5 Empirical analysis

In this section we present the data from the Mexican economy used in this study, describe the methodology to evaluate the accuracy of the common trends in order to predict the Mexican economy, determine the number of common trends, and analyze their dynamic by evaluating the performance of the FAVAR model proposed in Sect. 2 with respect to benchmark models.

5.1 Data

Initially, we consider 511 macroeconomic and financial variables obtained from the Banco de Información Económica (BIE) of the Insituto Nacional de Geografía y Estadística (INEGI), Mexico’s national statistical agency. The analysis covers from March 2005 to April 2016, hence, \(T = 133\). The blocks of variables are considered according to the INEGI division, and additionally, we compare this division with respect to the National Institute’s Global Econometric Model, which considers nine blocks with a total of 67 variables.Footnote 7

According to our approach, it is necessary that all variables are integrated of order one, so that, if factors are found, these are the common trends of the observations. In this case, we consider only I(1) variables according to the Augmented Dickey–Fuller (ADF) test.Footnote 8 When needed, the time series have been deseasonalized and corrected by outliers using X-13ARIMA-SEATS developed by the US Census Bureau.Footnote 9 Following Stock and Watson (2005), outliers are substituted by the median of the five previous observations.

Finally, according to these non-stationary conditions, we work with the following database of \(N = 211\) variables (number between parentheses)Footnote 10:

  • Balance of trade (19)

  • Consumer confidence (18)

  • Consumption (9)

  • Economic activity (13)

  • Employment (5)

  • Financial (35)

  • Industrial (58)

  • International (17)

  • Investment (8)

  • Miscellaneous (18)

  • Prices (11).Footnote 11

Furthermore, we define \(x_t\) as the Global Index of Economic Activity (IGAE, Indicador Global de la Actividad Económica).Footnote 12

5.2 Estimating the common trends

We apply the criteria described in this article to detect the number of factors. We use an \(r_{\max } = 11\). The results indicate that \({\hat{r}}_{\mathrm{ER}} = {\hat{r}}_{\mathrm{GR}} = 2\) and \({\hat{r}}_{\mathrm{ED}} = 5\).Footnote 13 Given that Onatski (2010) is more robust in the presence of non-stationarity, see Corona et al. (2017a), we work with this number of factors.Footnote 14

Figure 1 plots the results of the ratios and differences of eigenvalues with the respective threshold. Note that, intuitively, it is congruent that the ratio of eigenvalues of Ahn and Horenstein (2013) determines one and two common factors, given that the first and second differences of eigenvalues are large, while the others are practically zero. However, the estimation of the “sharp” threshold is one of the main contributions from Onatski (2010), which as we have mentioned, consistently separates convergent and divergent eigenvalues. Furthermore, the five common factors explain 79.19\(\%\) of the total variability. Specifically, the first common factor explains 46.14\(\%\), the second 19.59\(\%\), the third 6.2\(\%\), the fourth 4.75\(\%,\) and the fifth 2.51\(\%\).

Fig. 1
figure 1

Determination of the number of factors through the Onatski (2010) procedure, \({\hat{\lambda }}_i - {\hat{\lambda }}_{i+1}\) (top panel) and ratio of eigenvalues (bottom panel), \({\hat{\lambda }}_{i+1}/{\hat{\lambda }}_{i}\), for \(i = 1,\ldots , 6\)

Full size image

Figure 2 plots the behaviors between \(\ln x_t\) (deseasonalized) and each common factor extracted by PC and PLS. We observe that the first common factor, for each procedure, is similar to \(\ln x_t\) with contemporaneous linear correlations of 0.95 for PC and 0.96 for PLS. The second common factors are slightly correlated with \(\ln x_t\), having contemporaneous correlations of 0.13 and 0.08 for PC and PLS, respectively. On the other hand, \({\tilde{F}}_{3t}\) and \({\tilde{F}}_{5t}\) have an inverse behavior, although the correlations with respect to \(\ln x_t\) are around \(-0.01.\) In the other cases, the estimated common factors are positively related to \(\ln x_t\) but the linear correlations are drastically small. These facts indicate that contemporaneously, only the first two common factors are associated with economic activity. It is interesting to mention that these two factors explain 65.73% of the total variability. Note that it is complicated to establish the predictive capacity between the common factors, and it is necessary to evaluate it in the FAVAR model. Furthermore, it is clear that the sample period includes the economic crisis of 2008–2009; however, Stock and Watson (2011) show that the PC estimator of the factors is consistent even with certain types of breaks or time variation in the factor loadings.

Fig. 2
figure 2

Top left panel \(\langle lnx_t, F_{1t} \rangle\), top right panel \(\langle \ln x_t, F_{2t} \rangle\), middle left panel \(\langle \ln x_t, F_{3t} \rangle\), the middle right panel \(\langle \ln x_t, F_{4t} \rangle\) , and bottom left panel \(\langle \ln x_t, F_{5t} \rangle\). The blue color refers to PC and the red color to PLS (color figure online)

Full size image

Figure 3 shows the weighted average contribution of each variable block in the common factors estimated by PC. Each color bar represents the contribution of the percentage of explanation of each variable group with respect to each common factor, denoted as follows:

$$\hat{\mu }_{jg} = \sum_{i_g = 1}^{N_g}|\hat{p}_{i_gjg}|/N_g \quad \text{for } j = 1, \ldots r, \text{ and } g = 1, \ldots , G,$$

where \(\hat{p}_{i_gjg}\) is the loading weight of each group of variables; G is the number of blocks of variables; and \(N_g\) is the number of variables in each group. Specifically, the order of the groups of variables is computed as \(r^{-1}\sum_{j=1}^{r}{\hat{\lambda }}_{j}\hat{\mu }_{jg}\) where \({\hat{\lambda }}_j\) is the variance contribution of each common factor. The block that most explains the common factors is the miscellaneous group, economic activity, and balance of trade blocks. On the other hand, prices, employment, and consumer confidence are the least relevant groups of variables. Note that this importance is in terms of the loading contribution and it is not interpreted as predictive power. Specifically, the first common factor is more correlated with the IGAE of tertiary activities (0.99), the second common factor with the economic situation with respect to last year (0.94), the third common factor with oil exports (0.78), the fourth common factor with edification (0.70), and the fifth common factor with the food industry (0.53).

Fig. 3
figure 3

Weighted mean \(\hat{\mu }_{jg} = \sum_{i_g = 1}^{G}|\hat{p}_{i_gjg}|/N_g \text{ for } j = 1,\ldots , r, \text{ and } g = 1, \ldots , G,\) where G is the number of blocks of variables and \(N_g\) is the number of variables in each group

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In order to evaluate if the common factors are the common trends of \(x_t\), we carry out the cointegration exercise. The possible cointegration relationship is given by:

$$\begin{aligned} {\hat{v}}_t= & {} \ln x_t - \underset{(0.0000)}{4.6369} - \underset{(0.0000)}{0.0719}\hat{F}_{1t} - \underset{(0.0000)}{0.0098}\hat{F}_{2t} \nonumber \\ &-\,\underset{(0.0000)}{0.0003}\hat{F}_{3t} + \underset{(0.0000)}{0.0008}\hat{F}_{4t} + \underset{(0.0000)}{0.0002}\hat{F}_{5t}. \end{aligned}$$
(13)

We estimate the ADF test with its respective p value (in parentheses), obtaining the following results:

$$\text{ADF test}{:} -3.4382\ (0.01)$$

Then, we can verify that the common trends of a large dataset of the economic variables are cointegrated with economic activity. Furthermore and following Bai and Ng (2004), first, we carry out a Panel Analysis of Non-stationarity in Idiosyncratic and Common Components (PANIC) on the idiosyncratic errors obtained using the “differencing and recumulating” method in order to disentangle the non-stationarity in this component. Second, we apply a PANIC to the idiosyncratic components estimated using data in levels as we have proposed in this study. In the first analysis, we obtain a p value of 0.1171 while in the second, we obtain a p value of 0.0000. Although in the first case \({\hat{\varepsilon }}_t\) is statistically non-stationary, the p value is around the uncertainty zone. Furthermore, we apply the variant of the ADF test proposed by Bai (2004) with the aim of detecting how many of the five common factors are non-stationary. As we expected, the tests show that the five common factors are non-stationary. Therefore, we conclude that the idiosyncratic terms are stationary and the common factors are non-stationary; hence, we can also argue that the elements of \(Y_t\) are cointegrated and the common factors are the common trends of \(Y_t\) and \(x_t\).

Note that Eq. (13) is the static version of Eq. (7). The goal of this exercise is to determine whether \(F_t\) are the common trends of \(x_t\) and \(Y_t\). We use this information to forecast the target variable with the FAVAR model presented in Sect. 2.

5.3 Evaluating the use of common trends to predict Mexican economic activity

It is important to have an empirical strategy to adequately predict the target variable. Consequently, with the aim of selecting the forecast model, we consider all possibilities of FAVAR models, \(\sum_{i=1}^r {}^rC_i\times 396\), where \(^rC_i\) is the binomial coefficient \(\left( {\begin{array}{c}r\\ i\end{array}}\right)\) and 396 is obtained as the product of 11 seasonal dummies (3–12 and none), 3 deterministic specifications in the FAVAR model (none, constant and trend), and 12 lags (1–12). Therefore, the lag order, the seasonal dummies, the deterministic component, and the factors are directly determined by minimum out-sample forecast error. The training sample covers from March 2015 to April 2016, such that we forecast 12 periods (1 year) for \(h = 2\). This forecast period is one of relative economic and political stability in Mexico. For example, during this time frame, the annual growth rate of Mexico’s GDP in any quarter was never lower than 2.3% and never higher than 2.8% (INEGI 2016). Moreover, the country’s economic performance tends to be more volatile in times close to presidential elections in Mexico, and to a lesser degree in the US, which do not coincide with our forecast period. Statistically, this period represents around 10% of the number of observations and discounting the degrees of freedom, we are able to represent 25% of \(T-K\) where K is the number of parameters in the FAVAR model.

For each model, we compute the forecast error. The forecasts are dynamic, so that we update \(T+1\) in each month. We selected the model that minimizes the Root Mean Square Error (RMSE).Footnote 15 Furthermore, we focus on the models that give a forecast error lower than a threshold. This threshold is determined directly using Eickmeier et al. (2014) as a reference. In their work they predict several macroeconomic variables using FAVAR models, FAVAR-tv, FAVAR-tv with stochastic volatility errors and univariate models. For one and two step ahead, forecasting the US GDP, the RMSE values in in-sample forecasts are 0.76 and 0.80 considering all periods for \(h = 1\) and \(h = 2,\) respectively. However, we select a threshold of 0.5 to obtain more accurate forecasts. Note that, if we predict \(\varDelta \ln x_t\) for the first step ahead, the forecast error is \(e_{T + 1} = (\ln x_{T + 1} - \ln x_{T}) - (\ln x_{T+1}^{f} - \ln x_T) = \ln x_{T + 1} - \ln x_{T+1}^{f}\), i.e., it is equivalent to forecasting the level of the first step ahead. Therefore, we focus on the levels of the IGAE.

5.4 Using common trends to predict Mexican economic activity

First, in order to descriptively evaluate the predictive capacity of each variable, we calculate the cross-correlation between \(y_{it-h}\) and \(x_t\) and consequently:

$$\rho ^{*}_h = \text{corr}(y_{it-h},x_t)|\lbrace \max (h \le 12:\text{Prob}(\text{corr}(y_{it-h}, x_t))<\alpha )\rbrace,$$

where \(\alpha = 0.05\). Figure 4 plots the results of the previous equation. Note that the top panel plots the \(\rho ^{*}_h\) with the confidence interval. The middle panel shows their corresponding maximum significant lag, \(\max (h \le 12:\text{Prob}(\text{corr}(y_{it-h}, x_t))<\alpha )\) and bottom panel presents the mean absolute correlations for each block of variables. Note that the block of variables most highly correlated with the future of \(x_t\) is the miscellaneous one. It is interesting to note that all significant variables of this block are positively correlated with the IGAE. Other interesting variable blocks are the economic activity and financial groups. On the other hand, the blocks least correlated with the future of \(x_t\) are consumer confidence, employment, and prices.

Fig. 4
figure 4

The top panel plots the \(\rho ^{*}_h\) with the confidence interval. The middle panel shows their corresponding maximum significant lag, \(\max ( h \le 12:\text{Prob}(\text{corr}(y_{it-h}, x_t)) < 0.05),\) and the bottom panel presents the mean absolute correlations for each block of variables

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It is important to state that the PANIC carried out on the idiosyncratic errors and common factors shows that the elements of \(Y_t\) are cointegrated; hence, it is reasonable to expect that the correlations between \(y_{it}\) and \(x_t\) are not spurious.

Once the models are estimated, we review the forecast errors lower than the threshold in the training sample. Figure 5 plots the historical behavior of the RMSE for the selected predictive models. The top panel plots the RMSE for \(h = 1\). Note that the PC gives slightly more accurate results than PLS. Furthermore, the dispersion of the RMSE for PLS is larger than PLS and neither approach presents outliers. The bottom panel shows the results for \(h = 2\). We can see that for both procedures, the forecast errors are slightly increased with respect to \(h = 1\). It is interesting to mention that for both h the tails from both procedures are intercepted. This graph is important because this behavior is expected for the following two predicted months.

Fig. 5
figure 5

Box plots for the RMSE of out-sample forecasts for the selected models. The top panel is \(h = 1\) and the bottom panel \(h = 2\). The blue color refers to PC and the red color to PLS (color figure online)

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Using the models from Fig. 5 we predict two steps ahead: May and June 2016. Note that we have n models, such that it is necessary to combine the forecasts. Thus we propose a weighted average, obtaining loadings similar to the PLS procedure by solving the following optimization problem:

$$\rho_1 = \underset{w,b}{\max \text{cor}}(X^{f}w, xb) \quad \text{subject to} \quad \text{Var}(X^{f}w) = \text{Var}(xb) = 1,$$

where \(w = (w_1,\ldots , w_n)\). In order to normalize the loading weights, we carry out the following scaling: \(w^{*} = n(\sum_{i = 1}^{n}w_i)^{-1}w\), such that the loading weights are between 0 and 1. Figure 6 shows the forecast density, the predictions, and the observed data. We plot the confidence interval to 95\(\%\). Note that for \(h = 1\), PLS is more accurate than PC, while for \(h = 2\), PLS is less accurate. The models are centered on the mean of the distribution. Moreover, the forecast density acquires the observed data. Furthermore, focusing on \(h = 2\), the distribution of PC has two modes and the predicted data tend towards the center of the distribution. On the other hand, for PLS, it tends towards the median of the distribution.

Fig. 6
figure 6

Forecast densities, forecast points (red color), and observed data (blue color) (color figure online)

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An interesting question is: which common trends are helpful to reduce forecast error? To this end, we compute the following coefficients through OLS:

$$\partial e_t | D_{ti} = 1< 0 \quad \text{and} \quad \text{Prob}(\partial e_t | D_{ti} = 1) < 0.10 \qquad \text{for } i = 1,\ldots , N_{\mathrm{m}},$$

where \(N_{\mathrm{m}}\) is the number of models for each procedure. In other words, we carry out a linear regression between the forecast errors according to each procedure and dummy variables that specify the combination of common trends \(F_{1t}\), \(F_{2t}\), \(F_{3t}\), \(F_{4t},\) and \(F_{5t}\) for both procedures. Figure 7 plots the result for each procedure in each h. We can see that, in PC, \(F_{1t}\) is a very important common factor to reduce the forecast error for \(h = 1\), whereas \(F_{1t}\), \(F_{2t}\), \(F_{4t},\) and \(F_{5t}\) are important for \(h = 2\). Furthermore, in the PLS approach \(F_{1t}\), \(F_{2t}\), \(F_{4t},\) and \(F_{5t}\) are relevant common factors for \(h = 1\) whereas \(F_{1t},\) and \(F_{2t}\) are for \(h = 2\). In fact, for both procedures the interaction of all common factors is important to reduce the forecast errors. This result completes the conclusions when we analyze Figs. 1 and 4, where all common factors are helpful to reduce the forecast errors.

Fig. 7
figure 7

Effects of variables on forecast errors: \(\partial e_t | D_{ti} = 1 < 0\) and \(\text{Prob}(\partial e_t | D_{ti} = 1) < 0.10\). The blue color refers to PC and the red color to PLS (color figure online)

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In order to evaluate the forecast accuracy in “real time,” we predict two steps ahead (May and June 2016). Figure 8 shows the forecast accuracy of the following models: (i) PC, (ii) PC using only factors that contribute to reducing the forecast error denoted as PC (2), (iii) PLS, (iv) PLS (2), (v) the average between the first and third models, and (vi) the average between the second and fourth models. We can see that for \(h = 1\) PLS, PLS (2) and the forecast average of PC and PLS are the most accurate model with 0 forecast error for May 2016. For \(h = 1\), PC gives a forecast error of 0.1 in June 2016. In conclusion, and with reference to Fig. 5, the results are as expected given that PC and PLS give forecast errors lower than the selected threshold.

Fig. 8
figure 8

Observed forecast error for May 2016 (\(h = 1\)) and June 2016 (\(h = 2\)). PC (blue color), PC 2 (deep blue color), PLS (red color), PC 2 (deep red color), average of forecasts (gray color), and average 2 of forecasts (black color). The number 2 indicates that the models only considered the variables that satisfy \(\partial e_t | D_{ti} = 1 < 0\) and \(\text{Prob}(\partial e_t | D_{ti} = 1) < 0.10\) (color figure online)

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It is interesting to mention the historical behavior of the models. Hence, we first analyze the RMSE of the benchmark models: the Autoregressive Integrated Moving Average (ARIMA) and the macroeconomic diffusion index (Stock and Watson 2002b). Figure 9 plots the results for the out-sample training. For \(h = 1\), we observe that the RMSE interval is between 0.82 and 1.07 for the ARIMA model, while for the macroeconomic diffusion index it is between 0.62 and 0.83. For \(h = 2\) the errors are slightly reduced in both cases; however, the macroeconomic diffusion index has better results. Hence, note that the inclusion of the factor in linear models reduces the forecast error. Then, we would expect the FAVAR models to show a small RMSE.Footnote 16

Fig. 9
figure 9

Forecast Errors for benchmark models: April 2015–2016. ARIMA (red color) and Macroeconomic diffusion index (blue color). The top left panel plots \(h = 1\) and the bottom right panel plots \(h = 2\) (color figure online)

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Figure 10 plots step-by-step the forecast errors of the FAVAR models considering the approach presented in this work. Note that the RMSE interval of PC for \(h = 1\) is between 0.27 and 0.67 and for \(h = 2\) between 0.29 and 0.59. The RMSE mean values are 0.47 and 0.44 for each h, respectively. Note that in both h, February 2016 is the outlier forecast error. On the other hand, for PLS and \(h = 1\), the forecast errors are between 0.34 and 0.60 and for \(h = 2\), the confidence interval is between 0.3 and 0.64. In this case, the mean of RMSE is 0.47 for each h, respectively. Note that the improvement with respect to the ARIMA model and macroeconomic diffusion index is relevant, above all when the factors are estimated using PC. Note that the forecast errors are very similar between PC and PLS.

Fig. 10
figure 10

Forecast Errors for FAVAR models: April 2015–2016. PC (red color) and PLS (blue color). The top left panel plots \(h = 1\) and the bottom right panel plots \(h = 2\) (color figure online)

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A question of interest is: are the models consistent through the training sample? We can obtain different models in each step ahead, in which case the consistency of the predictors may be questionable. Taking into account only the selected models, we can observe in Fig. 11 that all models are robust in \(h = 1\). In fact, 76\(\%\) of the models are within the threshold and for \(h = 2\), the behavior of the predictions is similar to May 2016. Note that the predictions for May and June 2016 are carried out with information up to April 2016. Hence, we present the forecast in \(h = 1\) for May 2016 and \(h = 2\) for June 2016. The model is not updated as in the previous forecasts. It is interesting to note that February 2016 was the most complicated month to predict; however, the robustness of the selected models is reasonable. The reason why February 2016 was a complicated month to predict can be explained by the fact that this month has 29 days, and the seasonal variables and the dynamics between the variables do not account for this effect.

Fig. 11
figure 11

Robustness of selected models. Red points indicate that the model satisfies the empirical threshold in the specific forecast month. A red cross indicates that there is no information to compute the statistic (color figure online)

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6 Conclusions and further research

In this paper we estimated the common trends of the Mexican economy using a large dataset of macroeconomic variables. Using cointegration concepts, we estimated the common trends using I(1) variables through PC, determining the number of common factors according to the Onatski (2010) and the Ahn and Horenstein (2013) procedures. Alternatively, we estimated the common factors using PLS.

We find that 211 macroeconomic and financial variables can be summarized in at most \({\hat{r}} = 5\) common trends, which are cointegrated with Mexican economic activity. Furthermore, we use the common trends in a FAVAR model with the aim of predicting Mexican economic activity. We statistically evaluate the predictive capacity of the common trends, where we can see that each common factor reduces the forecast error. Additionally, we observe that the forecast error is reduced with respect to the ARIMA and factor-augmented regressions, such that the common trends are useful to predict Mexican economic activity.

An important conclusion is that the macroeconomic common trends can be used in more sophisticated models in order to reduce the forecast errors. Additionally, note that in this study we used data in levels in order to estimate the common factors. This is empirical evidence on the use of DFMs when we do not transform the data to stationarity.

Economically, the more relevant groups of variables in the determination of the common trends are the miscellaneous group, the economic activity, balance of trade, and the industrial sector. The first group of variables basically comprises tourism information and variables related to the automotive industry. In this context, the behaviors of the external sector and internal demand are very important to predict the behavior of the global economy. It is reasonable to expect that external uncertainty and disincentives in the internal market can be dangerous for future economic growth. In other words, these groups of variables indicate the future movements of Mexican economic activity.

A future line of research is to use the common trends in factor error correction models (FECM), and FAVAR with time-varying coefficients, among many other models, combining the predictions with different combinations of forecasting methods. Furthermore, we consider a unit root test that takes into account the structural break that occurred between 2008 and 2009 in order to refine the procedure to select I(1) variables. Consequently, it is interesting to study the non-stationarity of the estimated common factors in the presence of breaks. Additionally, we will take into account the possibility of sparse cointegration for forecasting proposes.