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A dynamic factor model for the Mexican economy: are common trends useful when predicting economic activity?

Latin American Economic Review volume 26, Article number: 7 (2017) Cite this article

Abstract

In this paper we propose to use the common trends of the Mexican economy in order to predict economic activity one and two steps ahead. We exploit the cointegration properties of the macroeconomic time series, such that, when the series are I(1) and cointegrated, there is a factor representation, where the common factors are the common trends of the macroeconomic variables. Thus, we estimate a large non-stationary dynamic factor model using principal components (PC) as suggested by Bai (J Econom 122(1):137–183, 2004), where the estimated common factors are used in a factor-augmented vector autoregressive model to forecast the Global Index of Economic Activity. Additionally, we estimate the common trends through partial least squares. The results indicate that the common trends are useful to predict Mexican economic activity, and reduce the forecast error with respect to benchmark models, mainly when estimated using PC.

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.

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.

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.

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.

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).

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.

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

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 \({\hat{r}}_{\mathrm{ED}} = \max \lbrace j \le r_{\max } : {\hat{\lambda }}_j - {\hat{\lambda }}_{j + 1} \ge \delta \rbrace.\).

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 \(m\rightarrow \infty .\)

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.

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.

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.

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

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
figure1

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
figure2

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
figure3

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.

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.

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
figure4

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
figure5

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
figure6

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
figure7

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
figure8

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
figure9

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
figure10

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
figure11

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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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.

Notes

  1. 1.

    The common factors are equivalent to the common trends when we assume a static representation in the factor model. Barigozzi et al. (2016) develop the econometric theory for non-stationary DFMs with large datasets under different assumptions of the dynamic of the common factors.

  2. 2.

    Certainly, macroeconomic diffusion indexes are related to DFMs. However, in this study we make a distinction between multivariate predictive approaches and the classic diffusion index forecasts given by Stock and Watson (2002b). Furthermore, we focus on a DFM from a static point of view. Once the common factors are estimated and their non-stationarity disentangled, we use the factors in the proposed FAVAR representation.

  3. 3.

    A related study is Caruso (2015), who examines the flow of conjunctural data relevant to assess the state of Mexico’s economy. Specifically, the author exploits the information embedded in macroeconomic news from both Mexico and the US, in a model constructed to nowcast Mexican real Gross Domestic Product (GDP) based on DFMs and Kalman filters.

  4. 4.

    Alternatively, Bai (2004) proposes an information criteria for data in levels, modifying the \(\text{PC}(k)\) information criteria of Bai and Ng (2002) by multiplying the penalty function by \(\alpha_T = T/[4 \log \log (T)].\)

  5. 5.

    Alternatively, if we choose the restrictions \(P^{\prime }P/N = I_r\) and \(F^{\prime }F\) being diagonal, the estimator of the matrix of factor loadings, \(\hat{P}\), is \(\sqrt{N}\) times the eigenvectors corresponding to the r largest eigenvalues of the \(N\times N\) matrix \(Y^{\prime }Y\), with estimated factor matrix \(\hat{F} = Y\hat{P}/N\). The difference is only computational, these latest restrictions are less costly when \(T > N\), whereas \(F^{\prime }F/T^2 = I_r\) with \(P^{\prime }P\) being diagonal are less costly when \(T < N\).

  6. 6.

    According to Bai and Ng (2004), if \(\varepsilon_t\) is not stationary, it is necessary to differentiate \(Y_t\) to consistently estimate \(F_t\) using \(\hat{F}_t = \sum_{s=1}^t\hat{f}_s\) where \(\hat{f}_t\) is the PC factor estimated when using data in first differences.

  7. 7.

    https://nimodel.niesr.ac.uk.

  8. 8.

    See Barigozzi et al. (2016) for a similar approach.

  9. 9.

    https://cran.r-project.org/web/packages/seasonal.pdf.

  10. 10.

    See Annex with the description of each block of variables included.

  11. 11.

    It is well known that the prices can be I(2). In our sample period seven prices are I(2) and they are differenced.

  12. 12.

    Note that the block of economic activity includes the components of the aggregate IGAE. Hence, this specific variable is not included in the block.

  13. 13.

    Additionally, we estimate the Bai and Ng (2002) information criteria using data in levels and first-differenced data. In both cases, the three criteria tend to \({\hat{r}} = r_{\max }\).

  14. 14.

    For the US economy, Alessi et al. (2010) determine six common factors using the database from Stock and Watson (2005).

  15. 15.

    As we have mentioned in the introduction, the goal of this paper is to forecast Mexican economic activity. To this end, we have two measures: quarterly, GDP; and monthly, IGAE. In an early analysis, we prove that IGAE is perfectly correlated with GDP. In this case, given the monthly frequency data, it is more convenient to forecast the IGAE, with \(x_t\) as the target variable.

  16. 16.

    Alternatively, we can consider RMSE in terms of benchmark models like several authors. However, we decide to use the RMSE to easily verify whether the FAVAR models give RMSEs lower than the selected threshold of 0.5.

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Acknowledgements

Partial financial support from the CONACYT CB-2015-01-252996 is gratefully acknowledged.

Author information

Affiliations

  1. Departmento de Estadística, Universidad Carlos III de Madrid, Calle Madrid, 126, 28903, Getafe, Madrid, Spain

    Francisco Corona

  2. Centro de Investigación en Matemáticas, A.C., Jaliso s-n, Mineral de Valenciana, 36240, Guanajuato, Guanajuato, Mexico

    Graciela González-Farías

  3. Departamento de Estudios Económicos, El Colegio de la Frontera Norte, Carretera Escénica Tijuana-Ensenada Km 18.5, 22560, Tijuana, Baja California, Mexico

    Pedro Orraca

Authors
  1. Francisco Corona
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  2. Graciela González-Farías
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  3. Pedro Orraca
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Corresponding author

Correspondence to Francisco Corona.

Annex: Variables in the dynamic factor model

Annex: Variables in the dynamic factor model

\(\#\) Short Long Block Log SA T
1 EXP TOT Total absolute value exports Balance of trade Yes Yes 1
2 EXP PET TOT Oil exports Balance of trade Yes Yes 1
3 EXP PET CRU Crude oil exports Balance of trade Yes Yes 1
4 EXP PET OTR Other oil exports Balance of trade Yes Yes 1
5 EXP NO PET Non-oil exports Balance of trade Yes Yes 1
6 EXP NO PET AGR Farming exports Balance of trade Yes Yes 1
7 EXP NO PET EXTRAC Extractive exports Balance of trade Yes Yes 1
8 EXP MAN AUT Automotive exports Balance of trade Yes Yes 1
9 EXP NO MAN RES Manufacture exports Balance of trade Yes Yes 1
10 IMP TOT PET Total oil imports Balance of trade Yes Yes 1
11 IMP CON TOT Consumption goods imports Balance of trade Yes Yes 1
12 IMP CON PET Consumption goods oil imports Balance of trade Yes No 1
13 IMP CON NO PET Consumption goods non-oil imports Balance of trade Yes Yes 1
14 IMP INT PET Intermediate goods oil imports Balance of trade Yes Yes 1
15 IMP BK Capital goods imports Balance of trade Yes Yes 1
16 REM TOT Total remittances Balance of trade Yes Yes 1
17 REM MO Money orders remittances Balance of trade Yes Yes 1
18 REM TRANS ELECT Electronic transfers remittances Balance of trade Yes Yes 1
19 REM EFEC Cash and in-kind remittances Balance of trade Yes Yes 1
20 I CONF Consumer confidence index Consumer confidence Yes Yes 1
21 I CONF COMP HOG 12 Index: compared to this household’s economic situation 12 months ago, how do you think your situation is at the moment? Consumer confidence Yes Yes 1
22 I CONF FUT COMP PAI 12 Index: how do you foresee this household’s economic situation in 12 months’ time, compared to the current situation? Consumer confidence Yes Yes 2
23 I CONF COMP PAI 12 Index: how do you consider the country’s economic situation today compared to 12 months ago? Consumer confidence Yes Yes 1
24 I CONF FUT COMP PAI 12 Index: how do you foresee the country’s economic situation in 12 months’ time? Consumer confidence Yes Yes 1
25 I CONF POS FUT Index: comparing the current economic situation with that of a year ago, how do you consider at the present moment the possibilities that you or any of the members of this household Consumer confidence Yes Yes 1
26 I CONF B Consumer confidence index (balance) Consumer confidence Yes Yes 2
27 COMP 12 B Other complementary index: how would you describe your economic situation compared to 12 months ago? Consumer confidence Yes Yes 1
28 FUT 12 B Other complementary index: and how do you think your economic situation will be in 12 months, compared to the current situation? Consumer confidence Yes Yes 1
29 POSI 12 Other complementary index: at this moment, are you more able to buy clothes, shoes, food, etc., than a year ago? Consumer confidence Yes Yes 1
30 POSI 12 VAC Other complementary index: do you consider that during the next 12 months you or any of the members of this household will afford to go on vacation? Consumer confidence Yes Yes 1
31 POS AHO Other complementary index: are you currently able to save some of your income? Consumer confidence Yes Yes 1
32 COMP C 12 Other complementary index: how do you foresee your economic conditions to save in 12 months’ time compared to current conditions? Consumer confidence Yes Yes 1
33 COMP PRE Other complementary index: compared with the previous 12 months, how do you think prices will behave in the country in the next 12 months? Consumer confidence Yes Yes 1
34 COMP 12 EMP Other complementary index: do you think that employment in the country in the next 12 months is going to increase, remain the same, or decrease? Consumer confidence Yes Yes 1
35 AUT 12 Other supplemental index: are any members of this household or you planning to buy a new or used car in the next 2 years? Consumer confidence Yes Yes 1
36 PLA AUT 2 Other complementary index: are any members of this household or you planning to buy, build or remodel a home in the next 2 years? Consumer confidence Yes Yes 1
37 PLA CONST 2 Other complementary index: (balance) compared with the previous 12 months, how do you think prices will behave in the country in the next 12 months? Consumer confidence Yes Yes 1
38 IVF Physical volume index total Consumption Yes Yes 2
39 IVFBST IVF goods and services of national origin total Consumption Yes Yes 2
40 IVFB IVF goods and services of national origin goods Consumption Yes Yes 1
41 IVFS IVF goods and services of national origin services Consumption Yes Yes 2
42 IVFBI IVF imported goods Consumption Yes Yes 1
43 IVFAT IVF total accumulative Consumption Yes Yes 1
44 IVFNB IVFAT goods and services of national origin goods Consumption Yes Yes 1
45 IVFNS IVFAT goods and services of national origin services Consumption Yes Yes 2
46 IVFBIMP IVFAT imported goods Consumption Yes Yes 1
47 IGAE A 1 IGAE primary activities Economic activity Yes Yes 1
48 IGAE A 2 IGAE secondary activities total Economic activity Yes Yes 1
49 IGAE A 21 IGAE secondary activities 21 mining Economic activity Yes Yes 1
50 IGAE A 22 IGAE secondary activities 22 Economic activity Yes Yes 1
51 IGAE A 31–33 IGAE secondary activities 31–33 Economic activity Yes Yes 1
52 IGAE A 3 IGAE tertiary activities total Economic activity Yes Yes 1
53 IGAE A 43–46 IGAE tertiary activities 43–46 trade Economic activity Yes Yes 1
54 IGAE A 48 49 51 IGAE tertiary activities 48–49–51. Transport, mail, and storage; mass media information Economic activity Yes Yes 2
55 IGAE A 52–53 IGAE tertiary activities 52–53 financial and insurance services; real estate and rental services of movable and intangible goods Economic activity Yes Yes 1
56 IGAE A 54–56 IGAE tertiary activities 54–55–56 professional, scientific and technical services; corporate; business support services and waste management and remediation services Economic activity Yes Yes 2
57 IGAE A 61–62 IGAE tertiary activities 61–62 educational services; health and social work services Economic activity Yes Yes 1
58 IGAE A 71–81 IGAE tertiary activities 71–81 cultural and sporting recreation services, and other recreational services; other services except government activities Economic activity Yes Yes 2
59 IGAE 72 IGAE tertiary activities 72 temporary accommodation and food and beverage preparation services Economic activity Yes Yes 1
60 PEO Economically active population: occupied population Employment Yes Yes 1
61 PED Economically active population: unemployed population Employment Yes Yes 1
62 PO S AGR Occupied population by agriculture economic activity sector Employment Yes Yes 1
63 PO CONS Occupied population by construction economic activity sector Employment Yes Yes 1
64 PO MAN Occupied population by manufacturing industry economic activity sector Employment Yes Yes 1
65 TIIE Interbank interest rates, bank collection costs, and CETES performance Interbank interest rate (TIIE) Financial Yes Yes 1
66 CETES Interbank interest rates, bank funding costs, and CETES performance cost of time deposits of liabilities denominated in US dollars (CCP-dollars) Financial Yes Yes 1
67 TC Exchange rate of peso against dollar and UDIS value Interbank (sale) Financial Yes Yes 1
68 I TC R Mexican peso real exchange rate index Financial Yes Yes 1
69 I TC R VAR Mexican peso real exchange rate index annual change Financial No No 1
70 IPCBMV Index of prices and quotes of the Mexican stock exchange maximum Financial Yes No 1
71 TIIL International interest rates: Libor rate Financial Yes No 1
72 M1 M1 money and coins held by the public Financial Yes Yes 2
73 M1 CB M1 foreign currency checking accounts at resident banks Financial Yes Yes 2
74 M1 DAP M1 demand deposits of savings and loan companies Financial Yes Yes 1
75 M2 CIRT Internal funding of resident banks total Financial Yes Yes 1
76 M2 CIBR Domestic funding from resident banks Financial Yes No 1
77 M2 CIBR ME Internal collection of resident banks in foreign currency Financial Yes Yes 1
78 M2 VP GF Securities issued by the Federal Government total Financial Yes Yes 1
79 IPAB IPAB total Financial Yes Yes 1
80 IPAB S IPAB securities held by the siefores Financial Yes Yes 1
81 IPAB EPP IPAB securities held by public and private companies Financial Yes Yes 1
82 M2 SIEFORES Other public securities held by the siefores Financial Yes Yes 1
83 M2 EPP Other public securities owned by public and private companies Financial Yes Yes 1
84 M2 V P T Private securities total Financial Yes Yes 1
85 M2 V P SIEFORES Private securities held by the siefores Financial Yes No 1
86 M2 V P EPP Private securities owned by public and private companies Financial Yes Yes 2
87 FON F SIEFORES Savings funds for withdrawal from SIEFORES total Financial Yes Yes 1
88 FON F SIEFORES BM Retirement funds at the Banco de Mexico total Financial Yes Yes 1
89 SIEFORES Retirement funds at the Banco de Mexico total Financial Yes Yes 1
90 M3 Domestic financial assets held by non-residents total Financial Yes No 2
91 M3 BRT Collection of resident banks total Financial Yes Yes 1
92 M3 BRMN Collection of resident banks in national currency Financial Yes Yes 1
93 M3 BRME Collection of resident banks in foreign currency Financial Yes Yes 1
94 M3 VP T Government securities held by non-residents total Financial Yes No 2
95 M3 GF Securities issued by the Federal Government total Financial Yes Yes 2
96 M3 IPAB Public securities held by non-residents securities issued by the IPAB Financial Yes Yes 1
97 M4 Acquisition of branches and agencies of Mexican banks abroad total Financial Yes Yes 1
98 M4 DR Deposits of residents total Financial Yes Yes 1
99 M4 DNR Deposits of non-residents total Financial Yes No 1
100 IVFI 212 Mining of metallic and non-metallic minerals, except oil and gas Industrial Yes Yes 1
101 IVFI 213 Mining-related services Industrial Yes Yes 1
102 IVFI 23 Construction total Industrial Yes Yes 2
103 IVFI 237 Construction of civil engineering works Industrial Yes Yes 2
104 IVFI 238 Specialized construction works Industrial Yes Yes 1
105 IVFI 311 Manufacturing 311 industry Industrial Yes Yes 2
106 IVFI 312 Manufacturing 312 beverages and tobacco industry Industrial Yes Yes 1
107 IVFI 313 Manufacturing 313 manufacture of textiles and textile finishing Industrial Yes Yes 1
108 IVFI 314 Manufacturing 314 manufacture of textiles, except apparel Industrial Yes Yes 1
109 IVFI 315 Manufacturing 315 manufacture of clothing Industrial Yes Yes 1
110 IVFI 316 Manufacturing 316 tanning and finishing of leather, manufacture of leather and leather substitutes materials Industrial Yes Yes 1
111 IVFI 321 Manufacturing 321 wood industry Industrial Yes Yes 1
112 IVFI 322 Manufacturing 322 paper industry Industrial Yes Yes 1
113 IVFI 323 Manufacturing 323 printing and related industries Industrial Yes Yes 1
114 IVFI 324 Manufacturing 324 manufacture of petroleum products and coal Industrial Yes Yes 1
115 IVFI 325 Manufacturing industries 325 chemical industry Industrial Yes Yes 1
116 IVFI 326 Manufacturing 326 Plastic and rubber industry Industrial Yes Yes 1
117 IVFI 327 Manufacturing 327 manufacture of non-metallic mineral products Industrial Yes Yes 1
118 IVFI 331 Manufacturing industries 331 basic metal industries Industrial Yes Yes 1
119 IVFI 332 Manufacturing 332 manufacture of metal products Industrial Yes Yes 1
120 IVFI 333 Manufacturing 333 manufacture of machinery and equipment Industrial Yes Yes 1
121 IVFI 334 Manufacturing 334 manufacture of computer equipment, communication, measurement and other electronic equipment, components and accessories Industrial Yes Yes 1
122 IVFI 335 Manufacturing 335 manufacture of electrical fittings, electrical apparatus, and electric power generation equipment Industrial Yes Yes 1
123 IVFI 336 Manufacturing 336 manufacture of transport equipment Industrial Yes Yes 1
124 IVFI 337 Manufacturing 337 manufacture of furniture, mattresses, and blinds Industrial Yes Yes 1
125 IVFI 339 Manufacturing 339 other industrial manufacturing Industrial Yes Yes 1
126 IVFIA Accumulated total of industrial activity Industrial Yes Yes 1
127 IVFIA 21 Accumulated mining total mining Industrial Yes Yes 1
128 IVFIA 212 Accumulated mining of metallic and non-metallic minerals, other than oil and gas Industrial Yes Yes 1
129 IVFIA 213 Accumulated mining-related services Industrial Yes Yes 1
130 IVFIA 22 Accumulated total generation, transmission and distribution of electricity, water and gas supply by pipelines to the final consumer Industrial Yes Yes 1
131 IVFIA 222 Accumulated Water supply and piped gas supply to the final consumer Industrial Yes Yes 1
132 IVFIA 23 Accumulated construction total Industrial Yes Yes 1
133 IVFIA 236 Accumulated building Industrial Yes Yes 1
134 IVFIA 237 Accumulated construction of civil engineering works Industrial Yes Yes 1
135 IVFIA 238 Accumulated specialized construction work Industrial Yes Yes 1
136 IVFIA 31–33 Accumulated manufacturing industry total Industrial Yes Yes 1
137 IVFIA 311 Accumulated 311 food industry Industrial Yes Yes 1
138 IVFIA 312 Accumulated 312 beverages and tobacco industry Industrial Yes Yes 1
139 IVFIA 313 Accumulated manufacturing industry 313 Industrial Yes Yes 1
140 IVFIA 314 Accumulated manufacturing industry 314 Industrial Yes Yes 1
141 IVFIA 315 Accumulated manufacturing industry 315 Industrial Yes Yes 1
142 IVFIA 316 Accumulated manufacturing industry 316 tanning and finishing of leather Industrial Yes Yes 1
143 IVFIA 321 Accumulated manufacturing industry 321 Industrial Yes Yes 1
144 IVFIA 322 Accumulated manufacturing 322 Paper industry Industrial Yes Yes 1
145 IVFIA 323 Accumulated manufacturing 323 printing and related industries Industrial Yes Yes 1
146 IVFIA 324 Accumulated manufacturing 324 manufacture of petroleum products and coal Industrial Yes Yes 1
147 IVFIA 325 Accumulated manufacturing industries 325 chemical industry Industrial Yes Yes 1
148 IVFIA 326 Accumulated manufacturing 326 plastic and rubber industry Industrial Yes Yes 1
149 IVFIA 327 Accumulated manufacturing 327 manufacture of non-metallic mineral products Industrial Yes Yes 1
150 IVFIA 331 Accumulated manufacturing 331 basic metal industries Industrial Yes Yes 1
151 IVFIA 332 Accumulated manufacturing 332 manufacture of metal products Industrial Yes Yes 1
152 IVFIA 333 Accumulated manufacturing 333 manufacture of machinery and equipment Industrial Yes Yes 1
153 IVFIA 334 Accumulated manufacturing 334 manufacture of computer, communication, measurement and other electronic equipment, components, and accessories Industrial Yes Yes 1
154 IVFIA 335 Accumulated manufacturing 335 manufacture of accessories, electrical apparatus, and electric power generation equipment Industrial Yes Yes 1
155 IVFIA 336 Accumulated manufacturing 336 manufacture of transport equipment Industrial Yes Yes 1
156 IVFIA 337 Accumulated manufacturing 337 manufacture of furniture, mattresses, and blinds Industrial Yes Yes 1
157 IVFIA 339 Accumulated manufacturing 339 other manufacturing Industrial Yes Yes 1
158 IPI EUA Indices of United States industrial production International Yes Yes 1
159 TD EUA United States unemployment rates International Yes No 2
160 OM EUA United States monetary offer International Yes No 2
161 TI 3M Interest rate USA 3 months International Yes Yes 1
162 TI 6M Interest rate USA 6 months International Yes Yes 1
163 TI 1M Interest rate USA 1 year International No Yes 2
164 TI 2A Interest rate USA 2 years International Yes No 1
165 TI 3A Interest rate USA 3 years International Yes No 1
166 TI 5A Interest rate USA 5 years International Yes No 1
167 TI 10A Interest rate USA 10 years International Yes No 1
168 TI 20 Interest rate USA 20 years International Yes No 1
169 TI FF Interest rate USA federal funds International Yes No 1
170 I REMUN EUA Manufacturing remunerations USA International Yes Yes 2
171 RESER EUA Total reserves (not including gold) USA International Yes No 1
172 EXP EUA Total exports USA International Yes No 1
173 IMP EUA Total imports USA International Yes No 1
174 BC EUA Balance of trade USA International No No 1
175 INV T Investment total Investment Yes Yes 1
176 INV CON Investment residential Investment Yes Yes 2
177 INV CON NR Investment non-residential Investment Yes Yes 2
178 INV MET Investment total machines and equipment Investment Yes Yes 1
179 INV ME Investment machines and equipment Investment Yes Yes 1
180 INV MEO Investment machines, equipment, and others Investment Yes Yes 1
181 INV MET Investment transporting equipment (imported) Investment Yes Yes 1
182 INV MAEO Investment machines, equipment and others (imported) Investment Yes Yes 1
183 TUR VS Total balance Miscellaneous Yes Yes 1
184 TUR IT Total income Miscellaneous Yes Yes 1
185 TUR TI International tourists Miscellaneous Yes Yes 1
186 TUR ET Total expenditure Miscellaneous Yes Yes 1
187 TUR TIT Total expenditure (international tourists) Miscellaneous Yes Yes 1
188 TUR VIT Total income (volume) Miscellaneous Yes Yes 1
189 TUR ITIT Total income (international tourists) Miscellaneous Yes Yes 1
190 TUR ET V Total expenditure (volume) Miscellaneous Yes Yes 1
191 TUR ET TUT Total expenditure (international tourists) Miscellaneous Yes Yes 1
192 TUR GM IT Expenditure average (income total) Miscellaneous Yes Yes 1
193 TUR GM I TUT Expenditure average international tourists (income total) Miscellaneous Yes Yes 1
194 TUR GM ET Expenditure average (expenditure total) Miscellaneous Yes Yes 1
195 TUR GM E TIT Expenditure average international tourists (expenditure total) Miscellaneous Yes Yes 1
196 AUT Production total automotive Miscellaneous Yes Yes 1
197 AUT CAM T Sales to the public. Total automotive Miscellaneous Yes Yes 1
198 AUT CAM TT Sales to the public. Total buses Miscellaneous Yes Yes 1
199 AUT CAM TN Sales to the public. Total buses (national) Miscellaneous Yes Yes 1
200 AUT CAM TI Sales to the public. Total buses (imported) Miscellaneous Yes Yes 2
201 INPC National index of consumer prices Prices Yes Yes 1
202 INPC 4 INPC clothing, footwear, and accessories Prices Yes Yes 2
203 INPC 5 INPC furniture, household appliances, and accessories Prices Yes Yes 2
204 INPC 5 INPC health and personal care Prices Yes Yes 2
205 INPC 6 INPC transportation Prices Yes Yes 2
206 INPC S INPC underlying Prices Yes Yes 2
207 INPC SM INPC underlying goods Prices Yes Yes 2
208 INPP 21 National index of producer prices (INPP) mining without oil Prices Yes No 1
209 INPP 22 INPP generation, transmission and distribution of electricity, water and gas supply by pipelines to the final consumer Prices Yes Yes 1
210 INPP 23 INPP construction Prices Yes No 1
211 INPP 48-49 INPP transportation, post and storage Prices Yes Yes 2
  1. Log denotes logarithmic transformation, SA means seasonal adjustment, and T if 1 denotes that \(y_t \sim I(1)\) and 2 that \(\varDelta y_t \sim I(1)\)

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Corona, F., González-Farías, G. & Orraca, P. A dynamic factor model for the Mexican economy: are common trends useful when predicting economic activity?. Lat Am Econ Rev 26, 7 (2017). https://doi.org/10.1007/s40503-017-0044-7

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Keywords

  • Dynamic factor models
  • Common trends
  • Factor-augmented vector autoregressive model
  • Partial least squares
  • Forecast error

JEL Classification

  • C38
  • C53
  • E00