Are daily financial data useful for forecasting GDP? Evidence from Mexico 1

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1 Are daily financial data useful for forecasting GDP? Evidence from Mexico 1 Luis M. Gomez-Zamudio Raul Ibarra * Banco de México Banco de México Abstract This article evaluates the role of using financial data sampled at high frequencies to improve short-term forecasts of quarterly GDP for Mexico. The model uses both quarterly and daily sampling frequencies while remaining parsimonious. In particular, the mixed data sampling (MIDAS) regression model is employed to deal with the multifrequency problem. To preserve parsimony, factor analysis and forecast combination techniques are used to summarize the information contained in a dataset containing 392 daily financial series. Our findings suggest that the MIDAS model that incorporates daily financial data leads to improvements in quarterly forecasts of GDP growth over traditional models that either rely only on quarterly macroeconomic data or average daily frequency data. The evidence suggests that this methodology improves the forecasts for the Mexican GDP notwithstanding its higher volatility relative to that of developed countries. Furthermore, we explore the ability of the MIDAS model to provide forecast updates for GDP growth (nowcasting). JEL Classification: C22, C53, E37. Keywords: GDP Forecasting, Mixed Frequency Data, Daily Financial Data, Nowcasting. 1 We thank Bernardo Guimaraes (the editor), Juan R. Hernandez, Jorge Herrera, Gonzalo Rangel, two anonymous referees and seminar participants at Banco de México and El Colegio de México for valuable comments. Jose A. Jurado and Andrea Miranda provided excellent research assistance. Support provided by CONACYT is gratefully acknowledged. The views on this paper correspond to the authors and do not necessarily reflect those of Banco de México. * Corresponding author. Banco de Mexico, Direccion General de Investigacion Economica, Av. 5 de Mayo 18, Centro, Mexico City 06059, Mexico. address: ribarra@banxico.org.mx. 1

2 1. Introduction Forecasting influences the economy as a whole, as individuals and policy makers rely upon predictions of macroeconomic variables to make decisions. Thus, it is fundamental that the predictions are a good approximation of the realizations of the variable of interest. In turn, the accuracy of these forecasts depends on the information set and the forecasting model. Financial data, such as stock prices and interest rates, contain potentially useful information for making predictions due to its forward looking nature. There are, however, some challenges that must be addressed to exploit this type of data. The first is the fact that financial information is sampled at a much higher frequency than macroeconomic variables (e.g., GDP). These macro variables are typically available on a quarterly basis, whereas many financial variables are sampled on a daily basis. The standard approach to use this information to make forecasts is to average the high frequency financial data in the quarter, i.e., a flat aggregation weighting scheme, to be able to estimate a regression with quarterly data. This method, however, might not be optimal, for instance, if more recent data are more informative, in which case it should receive a higher weight than earlier data. A simple linear regression would require estimating a large number of parameters, thus leading to high estimation uncertainty. One possible way to overcome this difficulty is to use the mixed data sampling (MIDAS) approach proposed by Ghysels et al. (2004, 2007). The MIDAS approach consists of regressions that allow the forecasted variable and the regressors to be sampled at different frequencies, using distributed lag polynomials to achieve parsimony. This family of models has been used in recent literature, such as in Clements and Galvão (2008) and Marcellino and Schumacher (2010), to improve the accuracy of predictions of quarterly GDP with monthly indicators. More recently, the specific usage of financial data paired with the MIDAS model to forecast GDP growth has been 2

3 explored in Andreou et al. (2013). In short, these articles have concluded that the use of mixed frequency data improves forecast accuracy. A second challenge is how to incorporate all the available information in such a way that the model remains parsimonious. In this regard, some methods are potentially useful to deal with large datasets of financial variables such as factor models and forecast combinations, as well as a wide variety of model parameterization options that considerably reduce the number of estimated coefficients. Factor models are useful to summarize the information content of large data sets with a few common factors (Stock and Watson, 2002). Forecast combinations have been found in empirical studies to improve accuracy over individual forecasts by exploiting information from a set of models rather than relying on a single model (Timmermann, 2006). In this paper, we employ factor models and forecast combinations as complementary approaches. That is, we use forecast combinations of models estimated with different factors extracted from the group of financial variables. To the extent of our knowledge, the MIDAS approach has not been previously applied to developing economies to forecast GDP. It is relevant to do so, because the volatility of economic variables in these countries tends to be higher, which affects forecast accuracy. In this paper, we investigate whether the proposed methodologies lead to improvements in short-term forecasting of the Mexican GDP growth rate. For this purpose, a large set of 392 financial variables was obtained from Bloomberg. These variables can be grouped in the following categories: commodities, equities, corporate risk, foreign exchange and fixed income. This dataset will be used as the main information source. Because of the large number of variables, factor analysis is used to summarize all the information. Using these factors, the MIDAS model is estimated and forecasts are obtained for different specifications at horizons of one and four quarters ahead. The performance of the 3

4 MIDAS models is then compared to traditional benchmark models that only use quarterly macroeconomic data. 2 In addition, forecast combinations are carried out to further improve accuracy. We also present the GDP forecasts from a MIDAS regression model using a monthly dataset of macroeconomic variables as in Marcellino and Schumacher (2010). Thus, we are able to assess the role of daily financial variables compared to the approach of using only monthly variables. The most important result is that the inclusion of daily financial data and the use of the MIDAS regression model to forecast quarterly GDP growth does improve accuracy over traditional models for Mexico. Furthermore, in line with existing literature, we find that forecast combinations are effective at improving the predictive ability of a set of models. We conclude that the methodologies described herein are successful at incorporating additional information while preserving parsimony. Our article also provides statistical comparisons of the forecasting ability of the MIDAS model. First, we investigate whether the MIDAS model that incorporates daily financial data leads to improvements for quarterly forecasts of GDP growth over traditional models that rely only on quarterly macroeconomic data. Second, we would like to find out how the MIDAS model compares against a flat aggregation weighting scheme. Third, we explore the ability of the MIDAS model to provide forecast updates of GDP growth using recent information (nowcasting). The results show that the model with financial data and quarterly macroeconomic data outperforms a model that only employs quarterly macroeconomic variables. 2 There are alternative methods of using high frequency data to predict quarterly GPD growth, such as bridge models (Baffigi et al. 2004), state space models (Mariano and Murasawa, 2003) and factor models (Giannone et al., 2008). While bridge models and state space models rely on small sets of variables, factor models allow exploiting large datasets by summarizing the information into a few common factors. Our paper is focused exclusively on MIDAS models, although comparisons of forecasts from MIDAS models with some of these methods would clearly be of interest for future research. 4

5 Furthermore, we find that the MIDAS model outperforms the flat aggregation scheme in terms of accuracy. The MIDAS model is useful to provide updates of GDP growth, although the forecasts with leads seem to have a similar predictive accuracy compared to the short-run forecasts without leads. The rest of the article is organized in the following way. Section 2 introduces the MIDAS regression model, factor analysis and forecast combination. An overview of the dataset is shown in Section 3. Section 4 presents the results. Section 5 concludes the article. Lastly, the Appendix provides a detailed description of the dataset and supplemental results. 2. Methodology 2.1. The MIDAS Model To illustrate the MIDAS model, consider two of the variables used in this article: Mexican GDP growth as the dependent variable and the Goldman Sachs Commodity Index (GSCI) of silver as the independent variable. GDP growth is sampled quarterly, while the GSCI index is sampled daily. Now, define Y Q t = GDP t and X D m,t = GSCI t, where Q stands for quarterly, D for daily and m is the number of trading days in a quarter. Using this notation, a prediction of the GDP growth h periods into the future with the model proposed by Ghysels et al. (2004, 2007) has the following form: Q,h = μ h + Y t+h Q p ρ h j+1 Y Q t j + β h q θ D w h Y 1 X 1 m 1 D h j=0 i=0 i+j m X m i,t j + u j=0 t+h. This model has a constant, the traditional AR terms with p Y Q quarterly lags of the dependent variable denoted as ρ h D j+1, and a term that incorporates q X times m daily θ lags for the independent variable. The term multiplying the daily variable w h i+j m 5

6 deserves special attention. This term is the weighting scheme that will reduce the number of parameters to estimate and lead to a more parsimonious model. Five of the weighting schemes available will be presented in this subsection as explained in Ghysels et al. (2007). As mentioned above, these schemes significantly reduce the number of parameters to estimate. a) The U-MIDAS is an unrestricted version in the sense that every high frequency lag has its own coefficient to estimate. It can be useful when m is small. A desirable characteristic of these weights is that they can be estimated using traditional ordinary least squares. b) The Normalized Beta probability function has the following form consisting of three parameters, w i (θ 1, θ 2, θ 3 ) = a θ1 1 i (1 ai ) θ 2 1 N θ1 1 a (1 ai i ) θ 2 1 i=1 + θ 3, where a i = (i 1), with i = 1,2,, m. This scheme can be made more (m 1) parsimonious by restricting the first parameter to be unitary and/or the third parameter to be zero. 3 If all of these parameters are unrestricted, this weighting scheme is called Beta Non Zero. As defined before, m denotes the number of high frequency lags used in the regression. c) The normalized exponential Almon lag polynomial consists of two parameters represented as, 3 The beta function described above follows from Ghysels (2015) and approximates the beta function described in Galvão et al. (2013) as Beta(θ 1, θ 2 ) = a i θ1 1 (1 ai ) θ 2 1 Γ(θ 1 +θ 2 ) Γ(θ 1 )Γ(θ 2 ), where Γ is the gamma function. 6

7 w i (θ 1, θ 2 ) = exp (θ 1i+θ 2 i 2 ) m i=1 exp (θ 1 i+θ 2 i 2 ) As with the previous weighting scheme, the second parameter can be restricted to be zero. d) The Almon lag polynomial is unable to identify the parameter β, therefore, βw i (θ 0,, θ P ) = P p=0 θ p i p. The order of the polynomial p is chosen by the researcher. e) The step functions are also unable to identify β: βw i (θ 0,, θ P ) = θ 1 I i [a0,a 1 ] + P p=2 θ p I i [ap 1,a p ],. with a 0 = 1 < a 1 < < a p = m. I is an indicator function with a value of one whenever i lies within the specified interval, and is zero otherwise. Excluding the U-MIDAS and the Almon lag polynomial, the above schemes are estimated by nonlinear least squares. As described in Ghysels et al. (2007), the exponential Almon lag and the Beta probability functions are flexible enough to accommodate various shapes, such as slow-declining, fast-declining or humpshaped patterns. In contrast, the unrestricted MIDAS and the step-function schemes impose less structure on the function. Thus, these schemes can result in nonmonotonic-shapes, possibly associated with mean reverting effects of high frequency variables on the dependent variable. The advantage of those schemes is that they can be estimated through OLS, but require a larger number of parameters to estimate. As will be shown in our results, we find that the beta function seems to perform better in terms of forecasting accuracy. Figure 1 shows various shapes of the Beta weighting function for several values of the parameters. As can be seen, the rate of decay is governed by the values of the parameters. 7

8 0.2 Figure 1: Beta polynomial weighting function 0.15 =1, =5 =1, =10 =1.5, =5 Weight Lags (days) Note: The figure plots the weights on the first 63 lags of the beta polynomial function for different values of the parameters. As a comparison, the more traditional way of using high frequency data is to make an average, which is called a flat aggregation scheme. In our case, that would mean averaging the GSCI daily index for each quarter, i.e., assigning the same weight to all the lags in a quarter. Although this scheme has been widely used in the literature, it may not be optimal for time series that exhibit memory decay. Thus, the MIDAS regression allows us to choose the optimal shape of the weights Factor Models Following Stock and Watson (2002), we use factor models to condense the information of a large number of variables into a few factors. Stock and Watson (2002) have found that factor models are useful to improve the forecasts of key macroeconomic variables, such as output and inflation. The goal is to obtain a small set of factors that explains an important part of the variation in the entire set of variables. Formally, suppose there is a large set of variables X that will be used for 8

9 forecasting. This set contains N variables with T observations each. It is possible that N>T. The goal is to find a set of factors F and a set of parameters Λ that best explain X. The factor model can be written as: X t = ΛF t + e t, where e t are idiosyncratic disturbances with limited cross-sectional and temporal dependence. Another way to look at a factor is to think of it as an unobservable variable that explains an important part of the variation of the observed variables. To estimate the factors, Stock and Watson (2002) propose the use of the method of principal components which consists of minimizing the following expression: V(F, Λ ) = (NT) 1 (X it λ F i t t ) 2 i, where λ i is the ith row of Λ. Most of the literature has focused on extracting factors at low frequencies, such as quarterly or monthly data. Following this approach, we will extract factors from a large set of daily financial variables. Once the factors are estimated, they are incorporated into the MIDAS regression as a high frequency variable. For instance, if we use the factor that explains the largest variation of the entire set of financial variables, denoted as F 1, as the high frequency regressor, our MIDAS regression model can be written as: Q,h = μ h + Y t+h Q p ρ h j+1 Y Q t j + β h q θ D w h Y 1 X 1 m 1 1 h j=0 i=0 i+j m F m i,t j + u j=0 t+h. In our case, the first factor accounts for 23% of the variability of the 392 daily time series used. The first 5 factors explain 42.7% of underlying variation. Section 4 presents more details about the dataset. Following Marcellino et al. (2003), the series are standardized before the factors are obtained, by subtracting their means and dividing by their standard deviations. 9

10 This is necessary as a wide variety of series are employed and they differ in their units of measurement. There are two approaches that can be used to estimate the factors, namely the static and the dynamic methods. In this article, we employ the static method. Following Stock and Watson (2002, 2008), the static method is at the same time parsimonious and robust to having temporal instability in the model, as long as the instability is relatively small and idiosyncratic Forecast Combinations To employ the information contained in several of the estimated factors without increasing the number of parameters in the model, we use forecast combination methods. In this way we can include the information contained in an important number of explicative variables. By preserving parsimony, we achieve lower parameter uncertainty, thus improving forecasting accuracy. Furthermore, an important reason to use forecast combinations is to construct forecasts from a relatively large number of possible parameterizations of the MIDAS model. Thus, forecast combinations deal with the problem of model uncertainty by using information from alternative models instead of focusing on a single model. A survey on forecast combination methods can be found in Timmermann (2006). As a general result in the literature, forecast combinations improve forecast accuracy (Timmermann, 2006). Following Andreou et al. (2013), we present a few combinations that improve the Root Mean Squared Forecast Error (RMSFE) of the individual predictions. Formally, Q,h Y CM,t+h n h Q,h i=1 Y i,t+h. = w i,t Thus, a forecast combination Y CM,t+h can be interpreted as a weighted average of the n forecasts Y i,t+h for the horizon h of n models. Again, an important decision is 10

11 to select the weighting scheme. For this purpose, we need to think in terms of a loss function. Formally, a combination of n forecasts is preferred to a single forecast if, E[L(Y i,t+h Q,h, Y t+h )] > min for i = {1,2,, n}. Q,h C( ) E[L(C(Y 1,t+h, Y 2,t+h Q,h,, Y n,t+h Q,h ), Y t+h )], In the inequality above: L is a loss function that relates the forecasted and the observed values. Intuitively, the loss function is expected to grow as the forecasted value drifts further from the actual value. C on the other hand, is the combination function that relates the individual forecasts. Thus, we would like to select a function C that minimizes the expected loss, and the forecast combination would be preferred if the expected value of the loss function for that combination is smaller than each of the expected losses for each of the individual forecasts. Given the previous assumptions, the solution is a linear combination of individual forecasts. To finish this derivation let us denote Y t+h Q,h a vector containing all h individual forecasts and w t+h a vector of parameters. Then, the combination function can be rewritten as C(Y t+h Q,h h ; w t+h ). The last step requires to define a loss function. Following Andreou et al. (2013), the Mean Squared Forecast Error (MSFE) is used as it has been found to provide the highest improvement in forecasts. Thus, the MSFE weights are selected by analyzing the historical forecasting performance of the model and assigning to each of them a weight inversely proportional to their MSFE Nowcasting The MIDAS models have the ability of incorporating recent information to improve the forecasts. To understand this, suppose that current quarter GDP growth needs to be predicted. If we are one month into the current quarter, that is, at the end of 11

12 January, April, July or October, we will have at least 21 trading days (1 month) of daily data to forecast quarterly economic growth. Using the information up to date to forecast the next value of a variable of interest is called nowcasting. Formally, the MIDAS model is augmented with leads in the following way: Q,h = μ h + ρ h j+1 Y t+h p Y Q 1 j=0 Q Y t j m 1 + β h [ θ w D i m X m i,t+1 h + u t+h i=(3 Jx) m 3 q D X 1 m 1 θ + w h i+j m j=0 i=0 D X m i,t j The new term has two noticeable aspects. First, the subindex t+1 for the financial variable X D implies that the forecasting equation includes high frequency information generated during the present quarter. The other important thing to notice is the values of i and Jx. Let s suppose m=63, that means there are 63 trading days in a quarter. If the first month of the quarter has just finished, there are 21 days of data available, thus, Jx=1 needs to be selected to obtain the appropriate limits of the sum. As opposed to traditional nowcasting that involves state-space models potentially implying a large number of parameters and measurement equations, the MIDAS approach provides a parsimonious framework to deal with a large number of high frequency predictors Forecast Evaluation To compare the forecasting ability of alternative models, we use the Diebold and Mariano (1995) test. That is, we test for the null hypothesis that two different models have the same forecasting ability. To that end, we define a quadratic ] 12

13 forecast loss function for model i as g(u i,t ) = u 2 i,t. Under the null hypothesis, both models have equal forecasting ability, that is: H o : g(u 1,t ) = g(u 2,t ) Diebold and Mariano (1995) first define the difference between the loss functions for two alternative models as d t = g(u 1,t ) g(u 2,t ). Then, they propose the following test statistic: d DM = var(d ) Where d is the sample mean of d t and var(d ) is defined as var(d ) = γ 0 +2γ γ q H 1. H is the number of forecasted periods and γ j = cov(d t, d t j ). The statistic has a t-student distribution with H-1 degrees of freedom. The p-values shown later in the paper are derived from a regression with robust errors of d t on a constant and testing whether the constant is statistically significant Alternative Models To analyze the relative performance of the MIDAS model, we estimate the following alternative models: an autoregressive (AR), a random walk (RW), a vector autorregresive (VAR) and a Bayesian vector autoregressive (BVAR) model. We also compare our results to the Survey of Professional Forecasters. The aforementioned models and survey have been widely used by both central banks and the empirical literature as benchmarks for GDP forecasting (Chauvet and Potter, 2013). Both the AR and the RW models contain seasonal dummy variables. The order of the AR model was chosen using the AIC/BIC criteria, resulting in one autoregressive lag. 13

14 VAR models represent a systematic way to capture the dynamics and comovements of a set of time series without restricting for a specific functional form and have been particularly useful for forecasting purposes since the influential paper by Sims (1980). The VAR model can be written as: p Y t = A 0 + A i Y t i + ε t i=1 Where Y t is the vector of variables being forecasted, A i are the matrices of coefficients to estimate and ε t is a vector of residuals. The variables included in the VAR model are the growth rate of GDP, quarterly inflation rate, interest rate and US GDP growth rate. 4 To determine the number of lags p we use the AIC and set the maximum number of lags to four. The model can also contain seasonal dummy variables that are not included in the equation above for simplicity. A limitation of VAR models is that they often imply a large number of parameters to estimate, resulting in a loss of degrees of freedom, thus leading to inefficient estimates and lower forecasting performance. To deal with this limitation, we estimate a Bayesian VAR (BVAR) model (Litterman, 1986; Doan et al., 1988). The idea is to use an informative prior to shrinking the unrestricted VAR model towards a parsimonious naïve benchmark, thus reducing parameter uncertainty and improving forecasting accuracy. Previous studies have found that BVAR models have a good forecasting performance compared to conventional macroeconomic models for different countries and periods, including Litterman (1986), McNees (1986), Artis and Zhang (1990), Bańbura et al. (2010), among others. 4 Herrera-Hernandez (2004) and Capistran and Lopez-Moctezuma (2010) find that US GDP is useful to improve Mexican GDP forecasts in a VAR framework. 14

15 A BVAR model requires specifying the mean and standard deviation of the prior distribution of the parameters. In particular, we follow the Minnesota prior, in which each variable follows a random walk around a deterministic component (Litterman, 1986). If the model is specified in first differences, this prior specification shrinks all of the elements of A i toward zero. This implies that each variable depends mainly on its own first lag. In addition, the Minnesota prior incorporates the belief that more recent lags should provide more reliable information than more distant ones and that own lags explain more of the variation of a given variable than lags of other variables in the equation. The prior beliefs are imposed by setting the following moments for the prior distribution of the parameters:, j = i k2τ E[(A k ) ij ] = 0, V[(A k ) ij ] = λ 2 γ 2 2 σ i { k 2τ σ 2, otherwise j Thus, the Minnesota prior can be described by three hyperparameters, the overall tightness parameter λ, the relative cross-lags parameter γ and the decay parameter τ. Changes in these parameters imply changes in the variance of the prior distribution. The overall tightness parameter λ indicates the tightness of the random walk restriction, or the relative weight of the prior distribution with respect to the information contained in the data. For λ = 0, the data does not influence the estimates. As λ, the posterior estimates converge to the OLS estimates. The parameter γ < 1 indicates the extent to which the lags of other variables are less informative than own lags. The parameter τ 0 captures the extent to which more recent lags contain more information than more distant ones. Thus, the factor 1/k 2τ represents the rate at which prior variance decreases with increasing lag length. σ 2 i /σ 2 j accounts for the different scale and variability of the series. σ i and σ j are λ 2 15

16 estimated as the standard errors of an univariate AR regression for each variable. Finally, we use a non-informative (diffuse) prior for the deterministic variables. The BVAR model is estimated using Theil s mixed estimation method (Theil and Goldberger, 1961). The hyperparameters are chosen based on forecasting performance. In particular, we estimate the BVAR model for the combinations resulting from setting the following parameters: λ={0.1,0.2}, γ={0.3,0.5}, τ=1, and the number of lags p={1,2,3,4}. 5 From these 16 combinations of hyperparameters, we select the combination that minimizes the RMSFE in a pseudo out-of-sample forecasting exercise. To provide further evidence of the forecasting accuracy of the MIDAS model, our forecasts are also compared with those of the Survey of Professional Forecasters, which is maintained by Banco de Mexico. Capistran and Lopez-Moctezuma (2010) find that the forecasts from this survey outperform forecasts from traditional univariate and multivariate time series models. There are about 30 survey participants, including financial, consulting and academic institutions. Capistran and Lopez-Moctezuma (2010) provide an in depth description of this survey. We use the consensus forecast for the GDP growth rate, defined as the mean across forecasters. For the forecasting period used in this paper, the data are only available at the one quarter ahead horizon. We also generate the GDP forecasts using a DSGE model considered in Rubaszek and Skrzypczyński (2008). DSGE models allow to represent an economy based on microfounded decisions and have been widely used by central banks as a tool for policy analysis and forecasting. Some studies that provide evidence about the 5 Those values for the hyperparameters have been used in previous literature (e.g., Dua and Ray, 1995; LeSage, 1999; Canova, 2007). 16

17 forecasting accuracy of DSGE models are Smets and Wouters (2004), Adolfson et al. (2007), Liu et al. (2009), Rubaszek and Skrzypczyński (2008), Edge et al. (2010) and Del Negro and Schorfheide (2013). The model used in this paper can be regarded as a benchmark DSGE that has proven useful to forecast GDP in the literature (Rubaszek and Skrzypczyński, 2008; Del Negro and Schorfheide, 2013). The economy consists of households that maximize lifetime utility, firms that maximize profits and a monetary authority that cares for price and output stability. The model is subject to demand, productivity and monetary shocks. The model includes three core equations: a dynamic IS curve, a forward looking Phillips curve and a monetary policy rule, which determine the path for output, prices and interest rates. The utility function of the representative household is an increasing function of consumption and a decreasing function of labor. The households receive wages from labor, dividends from owned firms and interest from bonds. There are perfectly competitive final goods producers that use a continuum of differentiated intermediate goods as inputs, taking their price as given. The intermediate goods producers operate under monopolistic competitive conditions and hire labor as the only input. Nominal rigidities are introduced following the sticky price framework of Calvo (1983). Finally, the central bank follows a Taylor rule in which the interest rate responds to changes in inflation and the output gap. The model is characterized by the following system of equations: ĥ t = 1 σ [R t Π t+1 + ε t+1 D ε t D ] + ĥ t+1 Π t = δ 1 + βδ Π t 1 + β 1 + βδ Π t+1 + (1 βξ)(1 ξ) mc (1 + βδ)ξ t 17

18 R t = γr t 1 + (1 γ) (γ π Π t + γ y (y t y t 1 )) + η t M ε t i = ρ i i ε t 1 + η t i, i = {D, S} The first equation is the dynamic IS curve. The second equation represents the dynamic Phillips curve, while the third equation presents the interest rate rule followed by the monetary authority. The laws of motion for the demand and supply shocks are represented in the last equation. A hat above a variable indicates a deviation from its steady state value. y t is detrended output, h t is habits, R t is the nominal interest rate, Π t is the inflation rate, mc t is the marginal cost of ouput, σ is the inverse of the intertemporal elasticity of substitution, λ is the habit formation parameter, ξ measures the degree of price stickyness, δ is the degree of price adjustment, β is the discount factor, γ and ρ i are persistence parameters, and η t i are IID white noise disturbances, for i = {D, S, M}. Habit formation is given by (1 λ)ĥ t = y t λy t 1. Marginal costs can be written as mc t = σĥ t + φl t ε t S, where labor supply is given by L t = y t ε t S and φ is the elasticity of labor supply. The system is transformed into a state-space representation, where the observable variables are expressed in terms of the model variables. The model is estimated using a Bayesian approach. The assumptions about the prior distributions of the parameters follow those of Rubaszek and Skrzypczyński (2008), which in turn are consistent with those of Ireland (2004), Smets and Wouters (2004) and Del Negro et al. (2007). The priors are combined with the conditional density of the observables to obtain the posterior distribution. The posterior densities are estimated with the Markov chain Monte Carlo algorithm for each quarter of the 18

19 evaluation sample, using draws. Table B.1 in Appendix B shows the prior and posterior distribution of the parameters Data We use three databases in our analysis at different sampling frequencies: daily, monthly and quarterly. The daily database is divided into 5 different categories of financial information: commodities (166 series), equities (94 series), foreign exchange (27 series), corporate risk (53 series) and fixed income (52 series). As previously stated, the dependent variable is the Mexican GDP. Most of these daily financial series have been found in the literature to be good predictors of output growth (Andreou et al., 2013). The period of estimation is 1999Q1-2009Q4 and the period of forecasting is 2010Q1-2013Q4, i.e., 16 quarters. Although the sample is relatively small for nonlinear least squares estimation, Bai et al. (2013) provide evidence based on Monte Carlo simulations showing that the forecasting performance of MIDAS regression models may not be affected. The time series of the Mexican GDP, though not as long as that of developed countries, is available since Nevertheless, the estimation period is effectively shorter because an important number of financial variables is available from 1999 onwards. Although it might be a short period for forecasting purposes, it allows for the inclusion of useful daily information. Moreover, we use a sample period during which Mexico has followed exclusively a floating exchange regime and exclude the 1995 economic crisis from the estimation period, which could affect our estimations. The database constructed is primarily a subset of the time series suggested by Andreou et al. (2013), which has been shown to provide good predictive content 6 We have checked the sensitivity of the forecasting results on the prior distributions of the parameters. The results are essentially unchanged. 19

20 for US GDP. Nonetheless, there are a few notable remarks regarding the Mexican data. First, the CETES 28 day rate is included in the fixed income group. 7 It was especially important to include information on the interest rate for Mexico since it is the primary monetary policy instrument. While data for other CETES maturities is not available for the study period, US bonds and bills should partially compensate this lack of information. Second, the foreign exchange rates are expressed in terms of Mexican pesos. Furthermore, in terms of equity, two indicators are of relevance, MEXBOL, an index of the Mexican Stock Exchange, and VIMEX, a measure of expected volatility in the Mexican Stock Market. Finally, some of the financial variables specific for Mexico that could be relevant to forecast GDP, such as bonds and commodities, are unavailable for the entire study period and thus were not included in our study. For detailed information concerning the series used, please refer to the Appendix. All the financial information was retrieved from Bloomberg. Following Marcellino et al. (2003), some of the series employed here were transformed because they were nonstationary. We tested the hypothesis of unit roots by means of an augmented Dickey Fuller (ADF) test with 12 lags. Nonstationary series were transformed to first log-differences. Then, to ensure stationarity the transformed series were tested for unit roots using the ADF test. In general, we transform commodity prices, stock prices and exchange rates into daily returns (i.e., first log differences). Interest rates for US corporate bonds are transformed to first differences. Domestic interest rates are found to be stationary in levels. The forecasting variable, i.e., the GDP growth rate, is not seasonally adjusted. Therefore, regressions are estimated using seasonal dummy variables. 7 CETES are Mexican Treasury Bills, that is, debt issued by the federal government through the Ministry of Finance and Banco de México (the Mexican Central Bank). 20

21 Another important set of information included in our regressions is the quarterly macro data. This set comprises 20 macro variables whose high explanatory and predictive power for GDP has been previously documented (Andreou et al., 2013). In particular, this set contains information such as price indexes, international trade variables, inflation rate and economic activity indexes for Mexico and the US. Part of this set of variables is available on a monthly basis. To transform these variables into quarterly data, monthly data are averaged for every quarter. In addition to the daily set of financial variables, a dataset of monthly macro data is used as the high frequency data for the MIDAS regression. This set consists of 18 variables, such as price indexes, economic activity indexes for Mexico and US CPI. The same procedure is followed to preserve parsimony, i.e., a set of factors is estimated and different forecasts using each factor are combined to obtain the final forecast. The Mexican data were obtained from the National Institute of Statistics and Geography (INEGI, by its acronym in Spanish) and Banco de México (the Mexican Central Bank). US data were obtained from the Federal Reserve Bank of St. Louis. 4. Results 4.1. Forecasting Exercise and Model Selection Before presenting results for the forecasting exercise, a few points that require further clarification will be discussed. First, a recursive window is used for all the model specifications and horizons, unless otherwise stated. For instance, consider the forecast i, with i = 0, 1,.., n 1, where n is the number of one-step ahead forecasts. Then, the start date of estimation is fixed at 1999Q1, whereas the end date changes with each forecasted value, which is 2009Q4+i. Thus, the model is estimated each time the window changes and the forecasts are computed one-step ahead. This window grows with each forecasted point as it includes the next 21

22 observed value. The recursive window is expected to improve the forecasts over a fixed estimation window, as each new estimation includes more recent information. The second important aspect is to specify whether the exercise is in real time or not. That is, as GDP is subject to revisions (as well as other macroeconomic variables used as regressors), the data actually available at a particular quarter may differ from the final values that will be released by statistical offices. Although it would be of interest to perform a real time forecasting exercise by using the vintages of data that were actually available to the forecasters, real time data for Mexico are unavailable. Thus, we use revised data in our estimations. However, our models are still comparable in the forecasting evaluation exercise as all of them use the same information. As discussed in the introduction section, an interesting capability of the MIDAS model is nowcasting, which allows to forecast using current date information. Suppose that every month has exactly 21 trading days and that all the daily information used is available at the end of the day, as it occurs with most of the financial variables. Then, it would be theoretically feasible to obtain 63 progressive forecasts for next quarter s GDP. The problem is that quarterly GDP figures are published with a lag of two months on average. This underscores the importance of nowcasting as a technique able to incorporate new information to the forecast, up to the publishing of an official value for the variable of interest. By the time the first quarter GDP has been published, two months of information will be available into the next quarter. The model allows the user to use this extra information to forecast the second quarter and it does more, because it is also capable of using data up to the date the second quarter GDP is officially published. If we assume the publishing lag of two months also applies to next quarter, then up 22

23 to 5 months of useful information could be used to estimate and forecast the second quarter GDP. Since there is a wide variety of specifications available, we use the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to select the number of lags for both the autoregressive terms and the high-frequency terms. In our preferred forecasting framework, we use the information from five factors. In particular, we follow a similar approach as in Andreou et al. (2013) and use a forecast combination from the five models estimated with each of the five factors extracted from the entire set of financial variables. That is, we use both factor models and forecast combinations to deal with the large dataset of financial variables. We use the beta function as it presented in most cases the lowest RMSFE. In addition, the variance of the RMSFE of this weighting scheme is smaller. The tests to identify the best models were implemented using a maximum of 5 lags of the dependent variable and 1 to 6 lags of the independent factor (q D X ). As the number of trading days in a quarter is m=63, the maximum number of daily lags is 63 6=378. The selection of the models was done following the Bayesian information criteria (BIC). As explained before, regardless of the high frequency lags specified, the model estimates only 2 parameters for the Beta weighting scheme Forecasting Results Table 1 presents forecasts for different specifications estimated for two different forecasting horizons: 1 quarter ahead (h = 1) and 1 year ahead (h = 4). Out of the alternative benchmark models, the BVAR and the SPF have the best forecasting performance at the one quarter ahead horizon. The DSGE and the BVAR models have the best forecasting performance at the four quarters ahead horizon. As expected, the RW model shows the highest RMSFE. The forecasting accuracy of 23

24 the BVAR model is consistent with previous studies for different countries, including Artis and Zhang (1990) and Bańbura et al. (2010). Similarly, our result about the predictive accuracy of the DSGE model at the one year horizon is consistent with previous literature (Del Negro and Schorfheide, 2013). The Table also presents the relative RMSFE of the MIDAS model with respect to the benchmark AR model. As can be seen, the RMSFE of the MIDAS model that employs the first factor is outperformed by the benchmark models. A possible explanation is that the benchmark models contain macroeconomic variables that have a good predictive content to forecast GDP which are not contained in the MIDAS model. In the last part of this subsection, we will present an exercise that incorporates macroeconomic variables into the MIDAS model to provide evidence of the forecasting ability of this methodology and the use of high frequency data. Factor estimation is also applied to each group of financial variables. From this decomposition, 5 factors are extracted, one for each of the 5 groups of financial variables. Table 1 shows the forecasting results with the first factor of each group. We use the Beta weighting scheme and select the number of lags using Akaike and Bayesian Information Criteria. We only include the first factor in each regression as the variables in each group are highly related among them. Even though this is a parsimonious weighting specification, the predictive power for all variable groups, except for exchange rates, do not seem to improve over the benchmark models. In other words, the uncertainty associated with parameter estimation for these specifications outweighs the additional predictive power incorporated through the individual sets of financial series. The role of the exchange rates to forecast GDP could be explained in part by the status of Mexico as a small open economy. On the other hand, the other sets of variables might be less related to Mexican GDP dynamics as they are for US GDP. 24

25 Table 1: RMSFE comparison for models with no leads h=1 h=4 Model RMSFE RMSFE as RMSFE as RMSFE % of AR % of AR Alternative models AR RW VAR BVAR SPF DSGE Factor 1 Beta (p=2, q=6) Commodities F1 Beta (p=1, q=1) Equities F1 Beta (p=3, q=5) Corporate F1 Beta (p=1, q=2) FX F1 Beta (p=1, q=1) Fixed Income F1 Beta (p=1, q=5) Forecast Combinations Factors 1 to 5 Beta Best AIC/BIC Note: The table shows the root mean square forecast error (RMSFE) for h = 1 and h = 4 step ahead horizons of the GDP for the sample 1999Q1-2013Q4. Estimation period: 1999Q1-2009Q4. Forecasting period: 2010Q1-2013Q4. The RMSFE are also presented as a percentage of the AR. First, the forecasts are estimated for each of the alternative models described in the paper. Second, the table shows the results for the MIDAS model using the first daily factor of the 392 financial variables shown in Appendix A. Then, the forecasts are also estimated using the first factor of each group of financial variables. Finally, a forecast combination based on the first five factors is presented. A recursive window is used for all estimations. While it is readily apparent from the dataset that corporate risk and fixed income are two groups that focus mainly on the US economy and even though there are some variables such as interest rates for the Mexico, these do not seem to provide sufficient information to predict Mexican GDP growth by themselves. Equities might also present a similar problem. Although forecasting accuracy does not seem to improve when the individual groups of financial variables are included in the model, when all variables are included together and the factors contain mixed information it is clear that they are successful at improving the forecasting accuracy of the model. The last section of 25

26 the table presents a forecast combination based on the MSFE using five MIDAS specifications, one for each of the first five factors. Each of these models is optimal in the AIC-BIC sense but for different factors. As expected, the combination yields a lower RMSFE. This improvement can be explained by the fact that it considers the information contained in each factor. The results suggest that forecast combinations improve the accuracy of different information sets. The goal of the final part of this section is to investigate whether introducing daily financial data into a MIDAS regression framework is useful for forecasting GDP beyond macroeconomic data. We also compare the forecasting accuracy of the MIDAS model with the traditional models that take a simple average of daily financial data, i.e., a flat aggregation scheme. Table 2 contains a summary of the RMSFE for several models. The model denoted as FAR (factor autorregresive) under quarterly macro data incorporates the quarterly macro data to the AR model using a factor model as in Stock and Watson (2002). In particular, we extract 3 quarterly macroeconomic factors from the database of 20 quarterly macroeconomic series described in the data section. As a result, the first 3 factors explain nearly 76% of the overall variation of the 20 quarterly macroeconomic series. 8 These estimated factors augment the benchmark AR model to obtain the FAR models. A second family of MIDAS models is presented as monthly macro + quarterly macro data. This family consists of models where the high frequency variables used for the estimation of the model are the same set of monthly macroeconomic variables that were averaged using a flat aggregating scheme. Flat is used to denote the family of models that use a flat 8 Ibarra (2012) finds that, for the case of Mexico, the estimated factors from a broad set of macroeconomic variables for the period are highly related to relevant subsets of key macroeconomic variables such as output and inflation. That is, the estimated factors seem to be informative and interpretable from an economic point of view. Our results are consistent with those findings. 26

27 aggregation scheme for high-frequency data as well as quarterly macro data. In other words, the values for all trading days of the daily financial assets within the quarter were averaged to obtain a single value per quarter. Combined MIDAS is used to refer to a combination using the MSFE of 5 MIDAS specifications: one for each of the first 5 factors. Finally, the models denoted as financial data incorporate the information contained in the 392 daily financial series. As before, the RMSFE for different specifications is presented in Table 2. The results show that adding quarterly data to the AR model improves forecasting accuracy in terms of the RMSFE at both horizons. In particular, the factor model that includes quarterly macroeconomic data outperforms the AR, VAR, BVAR and SPF forecasts. That is, quarterly macroeconomic data such as consumption, investment, trade, inflation and foreign macroeconomic variables seem to provide important information to predict future GDP. Similarly, the monthly macro data improve forecast accuracy at the one quarter ahead horizon. However, the gains in terms of accuracy seem to be lower compared to those obtained from adding quarterly macroeconomic data. The results of adding financial data are of particular interest. As the RMSFE for these specifications show, including these variables within a combined MIDAS model improves forecasting accuracy. Notably, the results also suggest that gains in terms of RMSFE derived from the inclusion of financial data are larger under a MIDAS regression scheme than under a flat aggregation scheme. 27

28 Table 2: RMSFE comparisons of alternative models not seasonally adjusted GDP h=1 h=4 RMSFE as RMSFE as RMSFE RMSFE Model % of AR % of AR Traditional Models AR RW VAR BVAR SPF DSGE Quarterly macro data FAR Monthly macro + quarterly macro data Beta (p=4, q=3) Combined MIDAS Financial data Flat Beta (p=2, q=6) Combined MIDAS Financial data + quarterly macro data Flat Beta (p=2, q=1) Combined MIDAS Note: The table shows the root mean square forecast error (RMSFE) for h = 1 and h = 4 step ahead horizons of the GDP for the sample 1999Q1-2013Q4. Estimation period: 1999Q1-2009Q4. Forecasting period: 2010Q1-2013Q4. The RMSFEs are also presented as a percentage of the AR. The 5 MIDAS forecasts estimated from each of the daily factors are combined to obtain the Combined MIDAS. A recursive window is used for all forecasts. The last part of Table 2 shows the results of adding the financial data to the specifications that include quarterly macro data. The results illustrate that adding daily financial data with a MIDAS regression scheme improves forecasting accuracy over a traditional model that contains only quarterly macro data at both forecast horizons. Another important result is that forecast combinations of MIDAS regression models based on different groups of financial variables improve forecasting accuracy. 28

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