Maximum likelihood estimation of large factor model on datasets with arbitrary pattern of missing data.

Transcription

1 Maximum likelihood estimation of large factor model on datasets with arbitrary pattern of missing data. Marta Bańbura 1 and Michele Modugno 2 Abstract In this paper we show how to estimate a large dynamic factor model on datasets with an arbitrary pattern of missing data. The framework allows to handle efficiently and in an automatic manner sets of indicators characterized by different publication delays, frequencies and sample lengths. This can be relevant e.g. for young economies for which many indicators are compiled only since recently. We also show how the unexpected part of new data releases is related to the forecast revision. This can be used e.g. to trace the sources of the latter back to the individual series in the case of simultaneous releases. The methodology is applied to the short-term forecasting and backdating of the euro area GDP based on a large panel of monthly and quarterly data. Keywords: Factor Models, Forecasting, Large Cross-Sections, Missing data, EM algorithm. JEL classification: C53, E37. The authors would like to thank Domenico Giannone, Siem Jan Koopman, Michele Lenza and Lucrezia Reichlin and the seminar participants at Banca d Italia, the ECB, ISF 2008, CEF 2008, the conference on Factor Structures in Multivariate Time Series and Panel Data in Maastricht and 5th Eurostat Colloquium on Modern Tools for Business Cycle Analysis. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the European Central Bank. 1 European Central Bank and ECARES, Université Libre de Bruxelles; marta.banbura@ecb.int; contact author; 2 European Central Bank and ECARES, Université Libre de Bruxelles; michele.modugno@ecb.int.

2 1 Introduction In this paper we show how to estimate a dynamic factor model on large cross-sections with an arbitrary pattern of missing data. More precisely, we adopt a maximum likelihood approach and show how to modify the steps of the Expectation-Maximisation algorithm to the case with missing data. The framework can be used for extracting information from large datasets including e.g. mixed frequency or short history indicators or for real-time applications in which data arrive with delays and in a nonsynchronous manner. We also show how the unexpected part of a data release is related to the forecast revision and illustrate how this link can be used to understand the sources of the latter in the case of simultaneous releases. Using a large number of indicators in order to asses the current and the future state of the economy is becoming a standard practice. In this context, factor models have emerged as an effective tool for extracting information from many indicators in a parsimonious way. 1 fundamental assumption underlying those models is that most of the co-movement of the variables in a given dataset can be summarized by only few factors. This assumption seems to be justified in the case of macroeconomic and financial data. Forecasting macroeconomic variables is one of the domains where factor models show a great potential. 2 This paper builds on the large dynamic factor model of Giannone, Reichlin, and Small 2008). It has been implemented in several countries and proved to perform well in short-term forecasting, see e.g. Bańbura and Rünstler 2007) for the euro area GDP, Aastveit and Trovik 2007) for Norwegian GDP, Matheson 2007) for New Zealand GDP and inflation. It allows for explicit modelling of the dynamics of the factors 3 and for missing data at the end of the sample. This feature can be exploited e.g. to extract information from timely indicators in real-time forecasting. 4 In Giannone, Reichlin, and Small 2008) the estimation is based on principal components and performed on a balanced panel. The Therefore it is not well suited for datasets including e.g. mixed frequency variables or series of different lengths. In order to handle efficiently and in an automatic manner large datasets with an arbitrary pattern of data availability we adopt the maximum likelihood approach. Our framework can be particularly relevant for the euro area or other young economies for which many series have been compiled only since recently e.g. euro area Purchasing Managers Surveys). Moreover as the series measured at a lower frequency can be interpreted as high frequency indicators with missing data, mixed frequency 1 See Forni, Hallin, Lippi, and Reichlin 2000) and Stock and Watson 2002a) for the theoretical foundations. 2 See e.g. Bernanke and Boivin 2003); Boivin and Ng 2005); D Agostino and Giannone 2006); Forni, Hallin, Lippi, and Reichlin 2005, 2003); Giannone, Reichlin, and Sala 2004); Marcellino, Stock, and Watson 2003); Stock and Watson 2002a,b). 3 shown to be important in the simulation study of Doz, Giannone, and Reichlin 2005); 4 in contrast to models based on balanced datasets as in e.g. Forni, Hallin, Lippi, and Reichlin 2000); Stock and Watson 2002a); 2

3 datasets can be easily handled. This can be important for two reasons: first, the information in the indicators sampled at a lower frequency e.g. consumption, employment) can be used to extract the factors; second, the forecasts of the former can be easily obtained. Apart from data availability considerations the maximum likelihood estimation is more appealing than the two-step method because it is more efficient and, more importantly, it provides framework for imposing restrictions on the parameters. For example, Reis and Watson 2007) and Modugno and Nikolaou 2008) impose restrictions on the parameters of the dynamic factor model; the former in order to identify the pure inflation, the latter in order to forecast the yield curve using the Nelson-Siegel exponential components framework. Maximum likelihood approach has been used in the classical factor analysis for small datasets see e.g. Geweke, 1977; Sargent and Sims, 1977; Watson and Engle, 1983). For large datasets estimation based on principal components has been applied, see e.g. Stock and Watson 2002a,b); Forni, Hallin, Lippi, and Reichlin 2000); Bai 2003). Recently, Doz, Giannone, and Reichlin 2006) have shown that maximum likelihood is consistent, robust and computationally feasible also in the case of large cross-sections. To find the maximum likelihood estimates for the high-dimensional parameter space we use the Expectation-Maximisation EM) algorithm. It was proposed by Dempster, Laird, and Rubin 1977) as an iterative solution for maximum likelihood estimation in problems with missing or latent data. It was first applied for a dynamic factor model by Watson and Engle 1983) on a small cross-section without missing data. Doz, Giannone, and Reichlin 2006) have shown that the EM is a valid approach to maximum likelihood estimation for large cross-sections as it provides reliable results and each iteration is easy to implement and computationally inexpensive. We show how the EM steps derived in Watson and Engle 1983) should be modified in the case with missing data. 5 We also show how the dynamics of the idiosyncratic component can be modelled. The simulation results suggest that the methodology provides reliable results even if the fraction of missing data is substantial. We apply the methodology to short-term forecasting of the euro area GDP on the basis of large panel of monthly and quarterly indicators. We use a monthly dataset similar to the one in Bańbura and Rünstler 2007) or ECB 2008) and augment it by short history and quarterly series. The results are comparable to those in Bańbura and Rünstler 2007) based on model of Giannone, Reichlin, and Small 2008). This means that, while the additional series do not provide further information, our methodology allows us to analyse datasets including the mixed frequency and short history indicators and obtain their forecast in a unified framework. In the following exercise we illustrate how the news in the consecutive releases of different groups of variables revise the GDP forecast. In particular, this enables us to produce statements like 5 Shumway and Stoffer 1982) show how to implement the EM algorithm for a state space form with missing data, however only in the case in which the matrix linking the states and the observables is known. 3

4 e.g. the GDP forecast was revised up because of higher than expected release of.... Finally, we show that the framework can be used for backdating. In particular, the back estimates of GDP are fairly close to the true values. Related literature includes Mariano and Murasawa 2003) who use a quasi-newton method to estimate a small dynamic factor model by maximum likelihood on monthly and quarterly series. This framework has been recently used by Camacho and Perez-Quiros 2008) to obtain the real-time estimates of the GDP from monthly indicators. Schumacher and Breitung 2008) forecast GDP from large number of monthly indicators using the EM approach as proposed by Stock and Watson 2002b), which does not exploit the dynamics of the factors. Proietti 2008) estimates a factor model 6 from a balanced panel by EM for interpolation of expenditure and output components of the GDP and shows how to incorporate relevant accounting and temporary constraints. As for the interpolation and backcasting, Angelini, Henry, and Marcellino 2006) propose methodology based on large cross-sections. In contrast to their procedure our method exploits the dynamics of the data and is based on maximum likelihood which allows for imposing restrictions and is more efficient for smaller cross-sections. The paper is organized as follows. Section 2 presents the model, discusses the estimation and explains how the news content can be extracted. Section 3 describes the empirical application. Section 4 concludes. The technical details, data description and convergence checks are provided in the Appendix. 2 The model Let y t = y 1,t, y 2,t,..., y n,t ), t = 1,..., T denote a stationary n-dimensional vector process standardised to mean 0 and unit variance. We assume that y t follows an approximate factor model given by: y t = Λf t + ɛ t, ɛ t N0, Ψ), 1) where f t is a r 1, r n vector of common factors and ξ t is the idiosyncratic error, uncorrelated with f t at all leads and lags. Contrary to the case of the exact dynamic factor model, the lack of serial and cross-correlation of the idiosyncratic error is not necessarily required, see below. Moreover it is assumed that the common factors f t follow a stationary VAR process of order p: f t = A 1 f t 1 + A 2 f t A p f t p + u t, u t N0, Q). 2) 6 a la Watson and Engle 1983) and a la Stock and Watson 2002b); 4

5 2.1 Estimation In this section we derive the Expectation-Maximisation EM) steps under the assumption of the exact factor model: y t = Λf t + ξ t, ξ t N0, R), 3) where R is diagonal and ξ t is serially uncorrelated. As argued in Doz, Giannone, and Reichlin 2006) these assumptions could be too restrictive, in particular in the case of large cross-sections. They study the approximate factor model, allowing weak serial and cross-correlation in the idiosyncratic component and show that as n, T the factors can be consistently estimated by quasi maximum likelihood, where the miss-specified model is the exact factor model see Doz, Giannone, and Reichlin, 2006, for the technical details). Consequently, the estimators considered below are asymptotically valid also in the case of the approximate factor model. In the main text we set for simplicity p = 1, the case of p > 1 is discussed in the Appendix. Accordingly, the system given by 3) and 2) can be written in a state space form with the latent factors as states: y t = Λf t + ξ t, ξ t N0, R), f t = A 1 f t 1 + u t, u t N0, Q). 4) As f t are unobserved the explicit maximum likelihood estimators of the parameters θ = Λ, A 1, R, Q) are in general not available. At the same time, when n is large, a direct numerical maximisation of the likelihood is computationally difficult because of the large number of parameters. 7 Doz, Giannone, and Reichlin 2006) propose to circumvent the computational complexity by means of the EM algorithm. It offers a solution to problems for which incomplete or latent data yield the likelihood intractable. The essential idea is to write the likelihood as if the data were complete and to fabricate the missing data in the expectation step. 8 Let us denote the joint log-likelihood of y t and f t, t = 1,..., T by ly, F ; θ), where Y = {y 1,..., y T } and F = {f 1,..., f T }. Given the available data Ω T, 9 the EM algorithm converges towards the maximum likelihood estimates in a sequence of two alternating steps: 1. E-step - the expectation of the log-likelihood conditional on the data is calculated using the estimates from the previous iteration θr): Lθ, θr)) = E θr) ly, F ; θ) ΩT ; 7 Recently, Jungbacker and Koopman 2008) have shown how to reduce the computational complexity related to estimation and smoothing if the number of observables is much larger than the number of factors. 8 The EM algorithm was proposed by Dempster, Laird, and Rubin 1977). For overview see e.g. McLachlan and Krishnan 1996). 9 Ω T Y because some observations in y t can be missing. 5

6 2. M-step - the parameters are re-estimated through the maximisation of the expected loglikelihood with respect to θ: θr + 1) = arg max Lθ, θr)). 5) θ For the model given by 4) the maximisation of 5) results in the following expressions for Λr + 1) and A 1 r + 1): Λr + 1) = A 1 r + 1) = T E θr) yt f t Ω ) T T E θr) ft f t Ω ) 1 T, 6) T E θr) ft f t 1 Ω ) T T E θr) ft 1 f t 1 Ω ) 1 T. 7) Notice that these expressions resemble the OLS solution to the maximum likelihood estimation for auto-) regressions with complete data with the difference that the sufficient statistics are replaced by their expectations. The r + 1)-iteration covariance matrices are computed as the expectations of sums of squared residuals conditional on the updated estimates of Λ and A 1 : 10 1 T yt ) ) ) ΩT Rr + 1) = diag E T θr) Λr + 1)f t yt Λr + 1)f t 8) 1 T = diag E T θr) yt y t Ω T T Λr + 1) E θr) ft y t Ω )) T and Qr + 1) = 1 T T E θr) ft f t Ω T T A1 r + 1) E θr) ft 1 f t Ω ) T. 9) When y t does not contain missing data we have that E θr) yt y t Ω T = yt y t and E θr) yt f t Ω T = yt E θr) f t Ω T. 10) Finally, the conditional moments of the latent factors, E θr) f t Ω T, E θr) f t f t Ω T, E θr) ft 1 f t 1 Ω T and E θr) ft f t 1 Ω T, can be obtained through the Kalman smoother for the state space representation 4) with the parameters θr)), see Watson and Engle 1983). However, when y t contains missing values we can no longer use 10) when developing the expressions 6) and 8). Shumway and Stoffer 1982) derive the modifications for the missing data case but only with known Λ. We provide the EM steps for the general case with missing data. 10 Note that Lθ, θr)) does not have to be maximised simultaneously with respect to all the parameters. The procedure remains valid if M-step is performed sequentially, i.e. Lθ, θr)) is maximised over a subvector of θ with other parameters held fixed at their current values, see e.g. McLachlan and Krishnan 1996), Ch. 5. 6

7 Let W t be a diagonal matrix of size n with i th diagonal element equal to 0 if y i,t is missing and equal to 1 otherwise. As shown in the Appendix, Λr + 1) can be obtained as vec Λr + 1) ) T ) 1 = E θr) ft f t Ω T ) T Wt vec W t y t E θr) f t Ω T. 11) Intuitively, W t works as a selection matrix, so that only the available data are used in the calculations. Analogously, the expression 8) becomes 1 T Rr + 1) = diag W t y t y T tw t W t y t E θr) f t Ω T Λr + 1) W t W t Λr + 1)E θr) ft Ω T y t W t + W t Λr + 1)E θr) ft f t Ω T Λr + 1) W t + I W t )Rr)I W t )) ). 12) Again, only the available data update the estimate. I W t in the last term selects the missing observations. For example, when for some t all the observations in y t are missing, the period t contribution to Rr + 1) would be Rr)/T. It is easy to see that with W t I, 11) and 12) coincide with the complete data expressions obtained by plugging 10) into 6) and 8). Note that the static factor model is a special case of the representation considered above with A 1 = = A p = 0 in 2). In this case the same expressions for Λr + 1) and Rr + 1) apply and the conditional expectations can be derived explicitly, see the Appendix. Note that this approach is different from the EM based method proposed by Stock and Watson 2002b) to compute the principal components from unbalanced datasets. In the latter case, the objective function is proportional to the expected log-likelihood under the assumption of fixed factors and homoscedastic idiosyncratic component Modelling the serial correlation in the idiosyncratic component In the previous section the EM steps were derived under the assumption of the exact factor model. However in certain applications, like e.g. forecasting, it can be advantageous to take the dynamics of the idiosyncratic component into account, cf. e.g. Stock and Watson 2002b). There are different approaches to modelling of the idiosyncratic serial correlation. For example, Reis and Watson 2007) include lags of the observables into the measurement equation and alternate between two steps - they estimate the coefficients on the lags conditional on the remaining parameters and vice versa. Jungbacker and Koopman 2008) propose to use the Kalman smoother to estimate the auto-) regression parameters as additional states in an augmented state space form. Those approaches are however not appropriate in the case of arbitrary missing data pattern. Instead, we propose to represent the idiosyncratic component by an AR1) process and add it to the state vector. 7

8 More precisely, we assume that ɛ i,t, i = 1,..., n in 1) can be decomposed as: ɛ i,t = ɛ i,t + ξ i,t, ξ i,t N0, κ), ɛ i,t = α i ɛ i,t 1 + e i,t, e i,t N0, σi 2 ), 13) where κ is a very small number. 11 Combining 1), 2) and 13) results in the new state space form: y t = Λ f t + ξ t, ξ t N0, R), f t = Ã1 f t 1 + ũ t, ũ t N0, Q), 14) where f t = ft ɛ t ut, ũ t = e t, Λ = Λ I A1 0, Ã 1 = 0 diagα 1,, α n ), Q = Q 0 0 diagσ 2 1,, σ2 n) and R is a fixed matrix with κ on the diagonal. It follows, that the expressions for A 1 r + 1) and Qr + 1) remain as above while the one for Λr + 1) needs to be modified as follows: vec Λr + 1) ) T ) 1 = E θr) ft f t Ω T T Wt vec W t y t E θr) f t Ω T + Wt E θr) ɛt f t Ω ) T, with θ = Λ, Ã, Q), see the Appendix for the derivations. Furthermore, the r + 1)-iteration estimates of the autoregressive parameters of the idiosyncratic component are given by: T α i r + 1) = E θr) ɛi,t ɛ ) T i,t 1 Ω T E θr) ɛi,t 1 ɛ ) 1 i,t 1 Ω T, σ 2 i r + 1) = 1 T T E θr) ɛi,t ɛ i,t Ω T αi r + 1) T E θr) ɛi,t 1 ɛ ) i,t Ω T. The conditional moments involving ɛ t can be obtained from the Kalman smoother on the augmented state space given by 14). Notice that augmenting the state vector by the idiosyncratic component increases the dimension of the former. This can slow down the Kalman filter but it does not cause any computational problems in our applications. Jungbacker, Koopman, and van der Wel 2009) show how to speed up the Kalman filter recursions by alternating between the representation of Reis and Watson 2007) and the one given by 14) depending on the availability of the data in y t. Depending on the fraction of missing data, this can lead to substantial computational gains, however comes at the cost of more complex, time-varying state space representation. 11 This allows us to write the likelihood analogously to the exact factor model case, see the Appendix. 8

9 2.2 Forecasting, backdating and interpolation Given the estimates of the parameters ˆθ = ˆΛ, Â1, ˆR, ˆQ) and the data set Ω T the best linear predictions: Py T +h Ω T ), h 0 can be obtained from the Kalman filter. In case some of the observations in y t are missing, the corresponding rows in y t and ˆΛ and the corresponding rows and columns in ˆR) are skipped when applying the Kalman filter cf. Durbin and Koopman, 2001). The interpolates and backdata h < 0) can be also simply obtained from the Kalman smoother. To deal with mixed frequencies, low-frequency series are treated as high-frequency variables with missing data. 2.3 News and forecast revisions In this section we show how the unexpected content, i.e. the news, in a data release is linked to the resulting forecast revision. Let Ω v 1 and Ω v be two consecutive vintages of data, consequently Ω v 1 Ω v. 12 Let I v denote the news in Ω v with respect to Ω v 1. For example, let us assume that the difference between Ω v 1 and Ω v is the release of the European Commission EC) Surveys for the period t i. The news is I v = y EC t i P yt EC ) i Ω v 1, where y EC t i is the vector of the new observations. Assume that we are interested in how this news revises the GDP forecast for the period t j. As I v Ω v 1 we can write and in particular ) P yt GDP j Ω v }{{} new forecast P Ω v ) = P Ω v 1 ) + P I v ) = P ) yt GDP j Ω v 1 }{{} old forecast +P y GDP t j I v }{{} news In other words, the updated forecast can be decomposed into the sum of the old forecast and of the contribution from the news in the latest release. To compute the latter we use the fact that ) P yt GDP j I v = E Furthermore, given the model 4) we can write y GDP t j y GDP t j = λ GDP f tj + ξ GDP t j, I v = y EC t i I v ) E I v I v) 1 Iv. ). y EC t i Ω v 1 = λ EC fti f ti Ω v 1 ) + ξ EC t i, 12 In what follows, we do not take into account the revisions and changes in the parameter estimates. The influence of those factors needs to be analysed separately. 9

10 where λ GDP and λ EC are the rows of Λ corresponding to GDP and EC Surveys, respectively. 13 It can be shown see the Appendix) that: ) E = λ GDP Ef tj f tj Ω v 1 )f ti f ti Ω v 1 ) λ EC and y GDP t j I v E I v I v) = λec Ef ti f ti Ω v 1 )f t i f ti Ω v 1 ) λ EC + R EC, where R EC is a diagonal matrix with elements of R corresponding to the EC Surveys. The expectations Ef tj f tj Ω v 1 )f ti f ti Ω v 1 ) and Ef ti f ti Ω v 1 )f ti f ti Ω v 1 ) can be obtained from the Kalman filter, see the Appendix. Consequently, we can find a vector B such that the following holds: ) yt GDP j Ω v }{{} new forecast = yt GDP j Ω v 1 +B }{{} old forecast yt EC i yt EC i Ω }{{ v 1 } news. 15) This enables us to trace the sources of forecast revisions. 14 More precisely, in the case of a simultaneous release of several groups of) variables it is possible to decompose the resulting forecast revision into contributions from the news in individual groups of) series, see the illustration in Section In addition, we can produce statements like e.g. after the release of the EC Surveys, the forecast of GDP went up because the indicators turned out to be on average) higher than expected Empirical application In the empirical part of the paper we apply the methodology described above in the context of backdating and short-term forecasting the euro area GDP. In particular we show that, with our methodology, it is possible to include quarterly variables and short-history monthly indicators in our panel and to analyse their informative content through the news concept. Moreover we analyse the effect of modelling the idiosyncratic as AR1). 3.1 Data The dataset contains in total 117 indicators and was downloaded in March The detailed description including the list of the series, their availability and applied transformations is 13 For the case with the idiosyncratic component following an AR process, f t and λ should be simply replaced by f t and λ, respectively. 14 Note, that the contribution from the news is equivalent to the change in the overall contribution of the series to the forecast the measure proposed in Bańbura and Rünstler, 2007) when the correlations between the predictors are not exploited in the model. Otherwise, those measures are different, see the Appendix for the details. In particular, there can be a change in the overall contribution of a variable even if no new information on this variable was released. Therefore news is a better suited tool for analysing the sources of forecasts revisions. 15 If the release concerns only one group or one series, the contribution of its news is simply equal to the change in the forecast. 16 This holds of course for the indicators with positive entries in B. 10

11 provided in the Appendix. For the purpose of the empirical exercise the dataset is divided into three categories. The first group consists of 74 indicators similar to the one described in Bańbura and Rünstler 2007) or used at the ECB for the short term forecast, cf. ECB 2008). It contains information on world prices, trade, industrial production, European Commission surveys, retail trade, labour, financial, US and interest rates. All the series in this group date back at least to January 1993, but most of them have a longer history. We refer to this group as the benchmark dataset. Notice that as the benchmark dataset does not contain any short or low frequency series, a balanced panel can be easily obtained by simply discarding the last few rows with missing observations caused by the publication delays. Therefore principal components based estimation as in Giannone, Reichlin, and Small 2008) and Bańbura and Rünstler 2007) can be also applied, see below. The second group contains 20 monthly short-history indicators, namely the Purchasing Managers surveys. They are available from August 1997 or July Finally, the quarterly group includes 23 quarterly variables with different time spans) on real GDP and its expenditure components, real value added, employment, unit labour cost and European Commission survey on capacity utilisation. Following Giannone, Reichlin, and Small 2008) the monthly indicators are transformed so as to be in line with the nature of the quarterly variables which are differences of 3 month totals). The empirical applications are performed on two different datasets. The first is composed of the series included in the benchmark dataset plus the GDP. The second dataset includes all the available time series i.e. from the benchmark, short-monthly and quarterly groups). Moreover on both datasets we run the model with and without the autoregression in the idiosyncratic components. This implies that we will present results for four different specifications labelled as: Ben: the results are obtained using the benchmark dataset and the representation with an i.i.d. idiosyncratic component given by 4); Ben-I : the results are obtained using the benchmark dataset and the specification with an AR1) representation for the idiosyncratic component given by 14); All: the results are obtained using all the available data and the representation with an i.i.d. idiosyncratic component; All-I : when the results are obtained using all the available data and the specification with an AR1) representation for the idiosyncratic component. 11

12 To deal with mixed frequency data it is assumed that y t are observed at a monthly frequency and the quarterly indicators are to be understood as monthly series with missing values in the first and second month of each quarter. Consequently, the GDP figure for a given quarter is assigned to the last month of this quarter. 3.2 Forecast evaluation Given the trade-off between the timeliness and the information content, in the evaluation we want to account for the differences in the publication delays. In this, we aim at replicating as closely as possible the real-time forecasting exercise. As the real-time vintages are not available for all the variables of interest and whole evaluation period, we perform the so called pseudo-real time exercise. It means that we use the final figures as of March 2009, however at each point of the evaluation sample we follow the data availability pattern specific to this point in time. More precisely, in each month we follow the availability pattern specific to the middle of the month after the data on industrial production are released). For example, in mid-february the last available figure on industrial production would be for December of the previous year, while for survey data, which are much more timely, there would be already numbers for January. Accordingly, in the middle of each month the publication lag for industrial production and surveys is two and one month, respectively. Consequently when we evaluate the model in e.g. January, the data for industrial production end in November while for surveys they are available up to December. The same mechanism is applied to all the variables. The procedure for quarterly variables follows a similar logic modified to take into account the quarterly frequency of the releases, see e.g. Bańbura and Rünstler 2007) for more formal explanation. We evaluate the accuracy of 1-, 0- and -1-quarter ahead projections. 0-quarter-ahead projection, so called nowcast, means that we project the current quarter. Analogously, -1-quarterahead projection, so called backcast, refers to the previous quarter. These are relevant as the euro area GDP is released only around 8 weeks after the end of the respective quarter and so projections are meaningful up till the second month of the following quarter. Note that, the available information changes depending on the respective month within a given quarter. Therefore we evaluate separately forecasts made in the first, second or third month of a quarter. For example, 1-quarter-ahead forecast made in the first month means that we project e.g. the second quarter relying on the information available in January i.e. first month of the first quarter), for the third quarter using the information available in April, etc As GDP is assumed to be observed in the third month of the corresponding quarter, 1-quarter-ahead forecast made in the first month will correspond to a 5-month forecast horizon; 1-quarter-ahead forecast made in the second month to a 4-month horizon, etc.; with the backcast made in the second month corresponding to a -2-month forecast horizon, cf. Angelini, Camba-Méndez, Giannone, Rünstler, and Reichlin 2008); 12

13 For the measure of prediction accuracy we choose the root mean squared forecast error RMSFE). The estimation sample starts in January We choose a recursive estimation which means that the sample length increases each time that more information becomes available. The evaluation sample is 2000 Q1 to 2008 Q3 18 and it is divided into two subperiods. The training sample 2000 Q Q4 is used to choose the parameterisation for each of the four specifications. For the number of factors we assume 1 r 5, while we keep the number of lags fixed, p = Given r and p, the specifications are evaluated in the period 2004 Q Q3. As these subsamples are relatively short, we also present the results for the best parameterisation ex-post, i.e. with the lowest RMSFE on the whole evaluation sample 2000 Q Q3. Table 1 presents the results for the backcast, the nowcast and 1-quarter-ahead forecasts based on the four specifications, along with the parameterisations chosen. For the benchmark dataset, we also report the results obtained using the two-step 2S) approach based on principal components proposed by Giannone, Reichlin, and Small 2008) and applied by Bańbura and Rünstler 2007) for the euro area. Figure 1 shows the true GDP growth and the nowcasts obtained using different specifications. We can see that the accuracy of the projections based on different specifications is comparable. There seem to be some small improvements from modelling the idiosyncratic components as an AR1). As can be seen on Figure 1 the models accounting for a serially correlated idiosyncratic component seem to better capture the downturn of 2002/03. Further, the forecast based on the 2-step method seems to be a bit more volatile. As for the composition of the data there are no gains in forecast accuracy for GDP from augmenting the dataset by the additional indicators. On the other hand, the accuracy of forecasts based on the extended dataset deteriorates only slightly. Consequently, we can include the additional series in the dataset and obtain their forecasts or back estimates). This also allows us to analyse the news they provide, see the next section. 3.3 News and forecast revisions for 2008 Q4 Let us recall from Section 2.3 that the news is understood as the unexpected part of a data release. In the following exercise, on the example of the fourth quarter of 2008, we illustrate how the projection evolves with new data releases and what are the contributions of the news in Q4 was excluded from the evaluation sample because of the extreme value of the GDP in this period - more than 4 standard deviations from its mean. Although there is not much evidence of this happening in our experiments, it could bias the results towards the models that were accurate in this particular quarter; 19 Increasing p to 2 has not led to improvements in forecast accuracy; 13

14 Figure 1: Nowcast of the GDP GDP 2S Ben All Ben-I All-I Mar-00 Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07 Mar-08 14

15 Table 1: RMSFEs for different estimation approaches and datasets With training Best ex-post 2S Ben All Ben-I All-I 2S Ben All Ben-I All-I r, p 3,1 2,1 2,1 4,1 4,1 2,1 2,1 2,1 5,1 4,1 1-quarter ahead Month Month Month Nowcast Month Month Month Backcast Month Month Average Notes: RMSFEs of forecasts, nowcasts and backcasts produced with a factor model estimated using the two-step methodology on the benchmark dataset 2S), using the maximum likelihood on the benchmark dataset Ben), on the benchmark dataset with idiosyncratic components modelled as AR1) Ben-I ), on the entire dataset All), on the entire dataset with idiosyncratic components modelled as AR1) All-I ). r and p refer to the number of factors and their lags, respectively. For the 2S method it is also possible to impose restrictions on the rank of the covariance matrix of the factors. In both cases it was chosen to be 1. With training means that r was chosen on the training sample 2000 Q Q4 and the model was evaluated on 2004 Q Q3. Best ex-post refers the results for the best parameterisation on the entire evaluation sample 2000 Q Q3. different groups of series to the forecast revisions. We compare the news from the specification Ben and All. The first forecast is performed in July 2008 corresponding to the 1-quarter-ahead forecast in the first month) and we revise the projection each month as new data arrive. The last estimate is obtained in February 2009 and the actual GDP for the fourth quarter was released in March. At each step we break down the forecast revision into the contributions of the news from different data groups using the formula 15). 20 In other words, the difference between two consecutive forecasts is the sum of the contributions of the news from all the variables plus the effects of model re-estimation. Figure 2 shows the evolution of the forecast as the new information arrives, the actual value of the GDP for the fourth quarter and the decomposition of the revisions. For the sake of the news analysis the series are grouped into following groups: Industrial Production indicators IP), European Commission EC) surveys, financial series Fin), US data, Purchasing Managers surveys PMI, only included in All) and Other The contribution of a group of series is the sum of the contributions of the series within this group. 21 See the Appendix for the list of series in each group. Here Fin contains also commodity prices. Other contains all remaining series plus the effect of re-estimation. 15

16 We can see that the news from both models are pretty consistent. On average, the biggest contributions to the revisions come from the European Commission surveys. In contrast, those from the industrial production have a sizeable effect on the projection only in the case of the backcast. This is in line with the results of Bańbura and Rünstler 2007) on the important role of soft data for the GDP projections when the hard data for the relevant periods are not yet available. Another observation is that the European Commission surveys and Purchasing Managers surveys do not always carry the same information. In particular the PMI news in December and January were positive meaning that the actual outcome of the surveys was less negative than predicted by the model). 3.4 Backdating the GDP A useful feature of our framework is that the estimates of the missing observations in the panel can be obtained from the Kalman smoother in an automatic manner. This enables us to use it e.g. in order to backdate a short history series or to interpolate quarterly variables. In this section we evaluate the model at backdating the GDP. For this purpose we modify the datasets described above by discarding all the observations on GDP prior to March Further, we estimate the parameters of the models and obtain the estimates of the missing values of GDP from the Kalman smoother. Figure 3 plots the back estimates of the GDP based on four considered specifications and the actual quarterly growth rate of the GDP. Table 2 reports the root mean squared errors RMSE) of the backdated GDP obtained with the different specifications. Table 2: RMSEs of back estimates of the GDP Ben All Ben-I All-I Notes: RMSEs of the back estimate produced with a factor model estimated on the benchmark dataset Ben), on the benchmark dataset with the AR1) for the idiosyncratic component Ben-I ), on the entire dataset All) and on the entire dataset with the AR1) for the idiosyncratic component All-I ). All the results are obtained with the GDP growth figures prior to March 2000 discarded. As we can see from Table 2 and Figure 3, independently of the dataset or the model used, the back estimates seem to capture the movements of the GDP, giving reasonable estimates of the past values of the series and the different specifications yield comparable results. Contrary to the results for the forecast the additional variables seem to improve the accuracy and AR1) idiosyncratic to deteriorate it. 16

17 Figure 2: Contribution of news to forecast revisions for 2008 Q4 Dataset Ben Other US News Fin News EC News IP News TRUE Fcst Jul-08 Aug-08 Sep-08 Oct-08 Nov-08 Dec-08 Jan-09 Feb-09 Mar09 Dataset All Other PMI News US News Fin News EC News IP News TRUE Fcst Jul-08 Aug-08 Sep-08 Oct-08 Nov-08 Dec-08 Jan-09 Feb-09 Mar09 17

18 Figure 3: Back estimates of the GDP Ben All Ben-I All-I GDP Mar-93 Mar-94 Mar-95 Mar-96 Mar-97 Mar-98 Mar-99 Mar-00 Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07 18

19 4 Summary This paper proposes a methodology for the estimation of a large dynamic factor model for datasets with arbitrary pattern of missing values. For that we adopt the maximum likelihood approach, which is shown by Doz, Giannone, and Reichlin 2005) to be consistent as well as computationally feasible for large cross-sections once the Expectation-Maximisation EM) algorithm is applied. We show how the steps of the EM algorithm should be modified in the case of missing data. We also propose how to account for the dynamics of the idiosyncratic component. In addition, we derive the link between the unexpected component of data releases and the resulting forecast revision. We apply this methodology for the short-term forecasting of the euro area GDP on the basis of large monthly dataset. Thanks to the flexibility of the framework in dealing with missing data, short history and quarterly variables can be incorporated e.g. Purchasing Managers surveys, GDP components or labour statistics). The effect of including these indicators in the large monthly dataset similar to the one used in e.g. Bańbura and Rünstler 2007) or ECB 2008) is evaluated in an out-of-sample forecast exercise. The results indicate that the additional indicators do not improve the precision of the projections. On the other hand, they can be analysed and forecast in a single model. We also compare the models with and without the explicit modelling of the serial autocorrelation of the idiosyncratic component. Both approaches yield similar results, with small improvements of the former. Those improvements could be more sizable in the case of monthly variables, which we do not forecast here, see e.g. Stock and Watson 2002b). It is a possible extension of the current model left for future research. Finally, we show that the framework can be easily used for back estimation. In particular, the back estimates of the GDP are fairly close to the true values. References Aastveit, K., and T. Trovik 2007): Nowcasting Norwegian GDP: The role of asset prices in a small open economy, Norges Bank Working Paper 2007/09. Angelini, E., G. Camba-Méndez, D. Giannone, G. Rünstler, and L. Reichlin 2008): Short-term forecasts of euro area GDP growth, Working Paper Series 949, European Central Bank. Angelini, E., J. Henry, and M. Marcellino 2006): Interpolation and backdating with a large information set, Journal of Economic Dynamics and Control, 3012),

20 Bai, J. 2003): Inferential Theory for Factor Models of Large Dimensions, Econometrica, 711), Bańbura, M., and G. Rünstler 2007): A look into the factor model black box. Publication lags and the role of hard and soft data in forecasting GDP., Working Paper Series 751, European Central Bank. Bernanke, B. S., and J. Boivin 2003): Monetary policy in a data-rich environment, Journal of Monetary Economics, 503), Boivin, J., and S. Ng 2005): Understanding and Comparing Factor-Based Forecasts, International Journal of Central Banking, 3, Camacho, M., and G. Perez-Quiros 2008): Introducing the EURO-STING: Short Term INdicator of Euro Area Growth, Banco de España Working Papers 0807, Banco de España. D Agostino, A., and D. Giannone 2006): Comparing alternative predictors based on large-panel factor models, Working Paper Series 680, European Central Bank. Dempster, A., N. Laird, and D. Rubin 1977): Maximum Likelihood Estimation From Incomplete Data, Journal of the Royal Statistical Society, 14, Doz, C., D. Giannone, and L. Reichlin 2005): A two-step estimator for large approximate dynamic factor models based on Kalman filtering, Manuscript, Université Libre de Bruxelles. 2006): A Quasi Maximum Likelihood Approach for Large Approximate Dynamic Factor Models, Working Paper Series 674, European Central Bank. Durbin, J., and S. J. Koopman 2001): Time Series Analysis by State Space Methods. Oxford University Press. ECB 2008): Short-term forecasts of economic activity in the euro area, in Monthly Bulletin, April, pp European Central Bank. Forni, M., M. Hallin, M. Lippi, and L. Reichlin 2000): The Generalized Dynamic Factor Model: identification and estimation, Review of Economics and Statistics, 82, ): Do Financial Variables Help Forecasting Inflation and Real Activity in the Euro Area?, Journal of Monetary Economics, 50, ): The Generalized Dynamic Factor Model: one-sided estimation and forecasting, Journal of the American Statistical Association, 100,

21 Geweke, J. F. 1977): The Dynamic Factor Analysis of Economic Time Series Models, in Latent Variables in Socioeconomic Models, ed. by D. Aigner, and A. Goldberger, pp North-Holland. Giannone, D., L. Reichlin, and L. Sala 2004): Monetary Policy in Real Time, in NBER Macroeconomics Annual, ed. by M. Gertler, and K. Rogoff, pp MIT Press. Giannone, D., L. Reichlin, and D. Small 2008): Nowcasting: The real-time informational content of macroeconomic data, Journal of Monetary Economics, 554), Jungbacker, B., S. Koopman, and M. van der Wel 2009): Dynamic Factor Analysis in The Presence of Missing Data, Tinbergen Institute Discussion Papers /4, Tinbergen Institute. Jungbacker, B., and S. J. Koopman 2008): Likelihood-based Analysis for Dynamic Factor Models, Tinbergen Institute Discussion Papers /4, Tinbergen Institute. Koopman, S. J., and J. Durbin 2000): Fast filtering and smoothing for multivariate state space models, Journal of Time Series Analysis, 213), Marcellino, M., J. H. Stock, and M. W. Watson 2003): Macroeconomic forecasting in the Euro area: Country specific versus area-wide information, European Economic Review, 471), Mariano, R., and Y. Murasawa 2003): A new coincident index of business cycles based on monthly and quarterly series, Journal of Applied Econometrics, 18, Matheson, T. 2007): An analysis of the informational content of New Zealand data releases: the importance of business opinion surveys, Reserve Bank of New Zealand Discussion Paper 2007/13. McLachlan, G. J., and T. Krishnan 1996): The EM Algorithm and Extensions. John Wiley and Sons. Modugno, M., and K. Nikolaou 2008): The forecasting power of international yield curve linkages, ECB Mimeo. Proietti, T. 2008): Estimation of Common Factors under Cross-Sectional and Temporal Aggregation Constraints: Nowcasting Monthly GDP and its Main Components, Manuscript. Reis, R., and M. W. Watson 2007): Relative Goods Prices and Pure Inflation, NBER Working Paper

22 Rubin, D. B., and D. T. Thayer 1982): EM Algorithms for ML Factor Analysis, Psychometrica, 47, Rünstler, G., and F. Sédillot 2003): Short-term estimates of euro area real GDP by means of monthly data, Working Paper Series 276, European Central Bank. Sargent, T. J., and C. Sims 1977): Business Cycle Modelling without Pretending to have to much a-priori Economic Theory, in New Methods in Business Cycle Research, ed. by C. Sims. Federal Reserve Bank of Minneapolis. Schumacher, C., and J. Breitung 2008): Real-time forecasting of German GDP based on a large factor model with monthly and quarterly Data, International Journal of Forecasting, 24, Shumway, R., and D. Stoffer 1982): An approach to time series smoothing and forecasting using the EM algorithm, Journal of Time Series Analysis, 3, Stock, J. H., and M. W. Watson 2002a): Forecasting Using Principal Components from a Large Number of Predictors, Journal of the American Statistical Association, 97, b): Macroeconomic Forecasting Using Diffusion Indexes., Journal of Business and Economics Statistics, 20, Watson, M. W., and R. F. Engle 1983): Alternative algorithms for the estimation of dynamic factor, mimic and varying coefficient regression models, Journal of Econometrics, 23,

23 A Data Description Number Block Name Transf. Availab. Benchmark dataset 1 World market prices of raw materials in Euro. Index Total, excluding Energy 2 Jan-80 2 Commodity WORLD-MKT PRICES, ENERGY RAW MAT., CRUDE OIL 2 Jan-80 3 Prices GOLD PRICE, US DOLLARS/FINE OUNCE,LONDON FIXING 2 Jan-80 4 Brent Crude-1 Month Fwd,fob US$/BBL converted in euro 2 May-85 5 Trade with World all entities), Export Value SA, not w.d. adj. 2 Jan-80 6 Trade Trade with World all entities), Export Value, SA, not w.d. adj. 2 Jan-80 7 Trade with World all entities), Export Value, SA, not w.d. adj. 2 Jan-80 8 Trade with World all entities), Export Value, SA, not w.d. adj. 2 Jan-80 9 Retail trade, except of motor vehicles and motorcycles - W.d. and SA 2 Jan Total industry - w.d. and SA 2 Jan Total Industry excluding construction) - W.d. and SA 2 Jan Manufacturing - W.d. and SA 2 Jan Construction - W.d. and SA 2 Jan Industrial Total Industry excluding Construction and MIG Energy - W.d. and SA 2 Jan Production Energy excluding NACE Rev.1 Section E - W.d. and SA 2 Jan MIG Capital Goods Industry - W.d. and SA 2 Jan MIG Durable Consumer Goods Industry - W.d. and SA 2 Jan MIG Energy - Working day and SA 2 Jan MIG Intermediate Goods Industry - Working day and SA 2 Jan MIG Non-durable Consumer Goods Industry - W.d. and SA 2 Jan Manufacture of basic metals - W.d. and SA 2 Jan Manufacture of chemicals and chemical products - W.d. and SA 2 Jan Manufacture of electrical machinery and apparatus n.e.c. - W.d. and SA 2 Jan Manufacture of machinery and equipment n.e.c. - W.d. and SA 2 Jan Manufacture of pulp, paper and paper products - W.d. and SA 2 Jan Manufacture of rubber and plastic products - W.d. and SA 2 Jan Industry Survey: Industrial Confidence Indicator - SA 1 Jan Industry Survey: Production trend observed in recent months - SA 1 Jan Industry Survey: Assessment of order-book levels - SA 1 Jan Industry Survey: Assessment of export order-book levels - SA 1 Jan Industry Survey: Assessment of stocks of finished products - SA 1 Jan European Industry Survey: Production expectations for the months ahead - SA 1 Jan Commission Industry Survey: Employment expectations for the months ahead - SA 1 Jan Surveys Industry Survey: Selling price expectations for the months ahead 1 Jan Consumer Survey: Consumer Confidence Indicator - SA 1 Jan Consumer Survey: General economic situation over last 12 months - SA 1 Jan Consumer Survey: General economic situation over next 12 months - SA 1 Jan Consumer Survey: Unemployment expectations over next 12 months - SA 1 Jan Consumer Survey: Price trends over last 12 months - SA 1 Jan Consumer Survey: Price trends over next 12 months - SA 1 Jan Construction Survey: Construction Confidence Indicator - SA 1 Jan Construction Survey: Trend of activity compared with preceding months - SA 1 Jan Construction Survey: Assessment of order books - SA 1 Jan Construction Survey: Employment expectations for the months ahead - SA 1 Jan Construction Survey: Price expectations for the months ahead - SA 1 Jan Retail Trade Survey: Retail Confidence Indicator - SA 1 Jan Retail Trade Survey: Present business situation - SA 1 Jan Retail Trade Survey: Assessment of stocks - SA 1 Jan Retail Trade Survey: Expected business situation - SA 1 Jan Retail Trade Survey: Employment expectations - SA 1 Apr New passenger car - W.d. and SA 2 Jan Unemployment rate, Total all ages), Total male & female), SA, not w.d. adj 1 Jan Index of Employment, Construction; SA, not w.d. adj 2 Jan Labour Index of Employment, Manufacturing; SA, not w.d. adj 2 Oct Index of Employment, Total Industry; SA, not w.d. adj 2 Jan Index of Employment, Total Industry excluding construction); 2 Oct US, INT.RATE, MONEY-MKT, TREAS.BILLS, 3-MONTH, MKT YIELD 1 Jan US, YIELD, SECOND.MKT, US TREASURY NOTES & BONDS, 10 YEARS 1 Jan US, UNEMPLOYMENT RATE, SA 1 Jan US US, INDUST.PROD., TOTAL EXCL. CONSTRUCTION, SA 2 Jan US, EMPLOYMENT, CIVILIAN, SA 2 Jan US, RETAIL TRADE, VALUE NAICS DEF.), SA 2 Jan US, PRODUCTION EXPECTATIONS IN MANUF., NSA 1 Jan US, CONSUMER EXPECTATIONS INDEX, NSA 1 Jan US, M2, SA 2 Jan Index of notional stocks M1 - SA 2 Jan Money Index of notional stocks M2 - SA 2 Jan Index of notional stocks M3 - SA 2 Jan Index of loans - SA 2 Jan ECB Nominal effective exch. rate, core group of currencies against Euro 2 Jan Financial EUROSTOXX 50 RHS) 2 Dec S&P 500 COMPOSITE - PRICE INDEX 2 Jan year govt. bonds 1 Jan Reuters.Money market.euro.euribor 1 Jan-80 Short Monthly dataset 75 Composite - employment 1 Jul Composite - new orders 1 Jul Composite - output 1 Jul Composite - input prices 1 Jul Manufacturing - employment 1 Aug Purchasing Manufacturing - new orders 1 Aug Managers Manufacturing - new export orders 1 Aug Surveys Manufacturing - output 1 Aug Manufacturing - purchasing manager index 1 Aug Manufacturing - productivity 1 Jan Manufacturing - quantity of purchases 1 Aug Manufacturing - supplier delivery times 1 Aug Manufacturing - stocks of finished goods 1 Aug Manufacturing - stocks of purchases 1 Aug-97 23