A dynamic factor model for the Mexican economy: are common trends useful when predicting economic activity?
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 nonstationary 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 factoraugmented 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.
Keywords
Dynamic factor models Common trends Factoraugmented vector autoregressive model Partial least squares Forecast errorJEL Classification
C38 C53 E001 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 policymaking (e.g., the business cycle; lagging, coincident, and leading indicators; or impulseresponse 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 nonstationary 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.^{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 longrun 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 factoraugmented 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 timevarying coefficients (tv) and stochastic volatility errors. The main conclusion is that the FAVAR models, regardless of the specification, outperform univariate models on insample forecasts. When they evaluate the outsample 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.^{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 statespace 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 longrun relationship between the common factors and the specific variable that we wish to predict.^{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 (RodriguezPose 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 governmentoperated 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, welleducated, and wellremunerated 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, lowskilled work, low productivity levels, and outofdate 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 debttoGDP, 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 longterm growth. Furthermore, in Mexico, temporary disturbances in output are usually accompanied by temporary reductions in consumption, since a substantial share of the population is creditconstrained 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 wideranging innovation policy aimed at getting closer to the technological frontier and generating higher levels of economic growth (RodriguezPose 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 nonstationary 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 outsample 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 nonstationary DFM and the FAVAR model in order to use the common trends to predict a specific macroeconomic variable.
3.1 Nonstationary dynamic factor model
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 crosscorrelation. Furthermore, Bai and Ng (2002) and Bai (2003) also allow for heteroskedasticity in both the time and crosssectional dimensions.
3.2 The factoraugmented autoregressive model
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 nonstationary 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 nonstationary DFMs, Ergemen and RodriguezCaballero (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).^{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 ilargest 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 nonstationarity 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
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 crosssectional 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.
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 nonstationarity in DFMs under a static representation.
4.3 Partial least squares
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.^{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.^{8} When needed, the time series have been deseasonalized and corrected by outliers using X13ARIMASEATS developed by the US Census Bureau.^{9} Following Stock and Watson (2005), outliers are substituted by the median of the five previous observations.

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).^{11}
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 \({\hat{r}}_{\mathrm{ED}} = 5\).^{13} Given that Onatski (2010) is more robust in the presence of nonstationarity, see Corona et al. (2017a), we work with this number of factors.^{14}
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 outsample 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 \(TK\) 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).^{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, FAVARtv, FAVARtv with stochastic volatility errors and univariate models. For one and two step ahead, forecasting the US GDP, the RMSE values in insample 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
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.
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 factoraugmented 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 timevarying 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 nonstationarity of the estimated common factors in the presence of breaks. Additionally, we will take into account the possibility of sparse cointegration for forecasting proposes.
Footnotes
 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 nonstationary DFMs with large datasets under different assumptions of the dynamic of the common factors.
 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 nonstationarity disentangled, we use the factors in the proposed FAVAR representation.
 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.
 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.
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.
 8.
See Barigozzi et al. (2016) for a similar approach.
 9.
 10.
See Annex with the description of each block of variables included.
 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.
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.
Additionally, we estimate the Bai and Ng (2002) information criteria using data in levels and firstdifferenced data. In both cases, the three criteria tend to \({\hat{r}} = r_{\max }\).
 14.
 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.
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.
Notes
Acknowledgements
Partial financial support from the CONACYT CB201501252996 is gratefully acknowledged.
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