Nowcasting Euro-Area GDP growth using multivariate models

Transcription

1 Nowcasting Euro-Area GDP growth using multivariate models Silvia Lui *, Gian Luigi Mazzi ** and James Mitchell * * National Institute of Economic and Social Research ** European Commission, Eurostat December 2013 Abstract This paper presents a unified multivariate model to produce nowcasts of an aggregate variable which nests aggregate and disaggregate modelling approaches. We consider various approximations to this, in general, infeasible multivariate model given that, in practice, estimation of the multivariate model becomes infeasible as the number of countries and number of indicator variables increases. These approximations to the infeasible but efficient multivariate model include the mixed approach of Hendry & Hubrich (2011) on which we focus; but we also discuss alternatives such as the use of global VAR models, factor models, combination methods and Bayesian large-n (shrinkage) VAR models. We then undertake outof-sample simulations using real-time vintage data to establish the utility of the mixed approach relative to the disaggregate and aggregate approaches which are commonly employed. We find that (i) aggregate nowcasts, which are conditional on aggregate indicators, fail to outperform disaggregate nowcasts which aggregate (take a weighted average) of country-level nowcasts. This implies aggregate indicators do not convey as much information as the disaggregate indicators; (ii) nowcasts from the mixed approach prove more accurate than both aggregate and disaggregate nowcasts. This suggests that disaggregate components convey useful information about the aggregate; (iii) a comparison between the EA nowcasts produced by the mixed approach (using the German smoothed estimates as an indicator) at t+30 days and comparable US estimates, reveals that the US estimates have a RMSE 0.34 and the EA ones of This seems quite encouraging remembering that, unlike the US estimates, our EA nowcasts at t + 30 days are entirely model-based, rather than survey-based; (iv) although disaggregate nowcasts perform worse than 1

2 nowcasts from the mixed approach at t + 15 and t + 30 days, their performance is actually not much (and not statistically significantly) worse. And they do produce more accurate nowcasts at t 30, t 15 and t + 0 days; and (v) irrespective of the modelling approach used, the nowcasts failed to predict the severity of the recession. This suggests that further modification of the nowcasting model is required to accommodate abrupt changes in economic conditions. Of course, it is well known that forecasting at times of great (structural) change is challenging and the failure of our nowcasts to keep up with these highly atypical events is not, per se, an indictment of our recommendation that the multivariate approach be employed when nowcasting aggregates. 2

3 1 Introduction Flash estimates (nowcasts) and/or forecasts can be computed using many different modelling approaches, in particular when interest rests with an aggregate variable, such as Euro-area GDP growth. Nowcasts can be computed at the aggregate level, with an aggregate model adopted and an aggregate nowcast produced directly by conditioning on aggregate indicators and/or lags of the aggregate itself. This is the so-called direct approach. In contrast, one could produce nowcasts or forecasts at a disaggregate level (the so-called indirect approach), say the sectoral or country level, using a disaggregate model which conditions on disaggregate information and then aggregates up to produce a nowcast or forecast for the aggregate. Alternatively, recently a mixed approach has been proposed and adopted by Hendry & Hubrich (2011), in which an aggregate forecast is produced by conditioning on both the aggregate and disaggregate indicators. The existing literature has considered all three of these approaches in empirical applications. Empirical evidence as to which approach is best is mixed. In fact, Hendry & Hubrich (2011) discuss how performance varies across forecasting or nowcasting situation. In this paper, we first unify these three alternative approaches to producing nowcasts of an aggregate variable. We nest them within a general multivariate framework for forecasting and nowcasting Euro Area aggregates, and consider various approximations to this, in general, infeasible multivariate model. In practice, estimation of the multivariate model becomes infeasible as the number of countries and number of indicator variables increases. These approximations to the infeasible but efficient multivariate model include the use of global VAR models, the aforementioned mixed approach, factor models, combination methods and Bayesian large-n (shrinkage) VAR models. We focus on use of mixed models. Throughout the mixed frequency nature of real-time data, an essential feature when nowcasting quarterly aggregates like GDP, is also accommodated. The generic multivariate model which we present exploits fully the utility of disaggregate information when nowcasting the aggregate and, in particular, accounts for interdependence among the disaggregates. In theory it is therefore the optimal approach, in the sense that it must deliver a Root Mean Squared Error (RMSE) at least as low as either the aggregate, disaggregate or mixed approaches. Under certain conditions, which we discuss, the multivariate model reduces to the disaggregate or aggregate models. 3

4 2 The relationship between Aggregate, Disaggregate and Multivariate Models In this section, we discuss how the disaggregate and aggregate models are nested in a multivariate model. We also provide a description of different approximations to the multivariate model when the number of series being modelled is too large, and discuss how the different modelling approaches are related to each other. First of all, let x t denotes a N-dimensional vector, x t = (x 1t, x 2t,, x Nt ). Each x it is a k-dimensional vector that contains the domestic variables of country i (i = 1,, N). x it can contain a single element, such as domestic GDP growth, log y it. It can also contain multi-elements. For example, if x it contains the growth rate of GDP, IP and survey data of country i, it becomes a three-dimensional vector, x it = ( log y it, log ip it,m, log bal it,m ) (we let m denotes month. The notation distinguishes between GDP growth, and growth of IP and survey data due to the fact that the latter two are monthly variables). We consider the following generic multivariate linear model, where one lag is assumed for notational ease only, x t = Θx t 1 + u t (1) x 1t x 2t. = Θ 11 Θ 12 Θ 1N Θ 21 Θ 22 Θ 2N x 1t 1 x 2t 1. + u 1t u 2t. (2) x Nt Θ N1 Θ N2 Θ NN x Nt 1 The diagonal elements of Θ, Θ ii contain the coefficients on domestic variables in country i s system of k i equations. The off diagonal elements of Θ, Θ ij, contain the coefficients of country j s variables in country i s equation, i j. We further assume the errors, u t follow a multivariate normal distribution, so u t N (0, Σ), and u Nt Cov (u 1t, u 1t ) Cov (u 1t, u 2t ) Cov (u 1t, u Nt ) Cov (u Σ = 2t, u 1t ) Cov (u 2t, u 2t ) Cov (u 2t, u Nt ) Cov (u Nt, u 1t ) Cov (u Nt, u 2t ) Cov (u Nt, u Nt ) Cov (u it, u it ) is the covariance matrix of the errors in country i s system of equations, while Cov (u it, u jt ) is a matrix containing the covariances between the variables of coun- 4 (3)

5 try i and country j. The element in Cov (u it, u jt ), namely Cov (u ilt, u jst ) denote the covariance of the lth variable of country i and the s th variable of country j. Our aim is to nowcast or predict the aggregate N w i x it. To produce such a nowcast or forecast requires us to estimate the above system. It is known that estimation of (1) is only feasible if N is small. When N gets big, estimation of this system involves estimation of a large number of parameters and will become infeasible. In the case when N, the above generic model relates to the Infinite-Dimensional VAR (IVAR) of Chudik & Pesaran (2009). In order to make estimation feasible, certain restrictions on the model are necessary, in one form or another. In the IVAR analysis of Chudik & Pesaran (2009), and the GVAR analysis of Pesaran et al. (2004), Dees et al. (2007) and Pesaran et al. (2009), the importance of restrictions on Σ, the assumption of weak cross-sectional dependence on the errors, is emphasised. Chudik & Pesaran (2009) propose a way to estimate IVAR models by shrinking part of the parameter space in the limit as the number of endogenous variables tends to infinity. They give an example of a VAR model in which the parameters can be decomposed into a component with fixed elements (that do not vary with the number of endogenous variables) that captures the neighbour effects; and another nonneighbour component that captures the remaining interactions. Under the assumption of weak cross-sectional dependence of the errors, the non-neighbour component converges in quadratic mean to zero. Whereas in the GVAR analysis, estimation is carried out on a country-by-country basis, so weak dependence when N is sufficiently large and the idiosyncractic errors are weakly correlated is essential. Nowcasts of the aggregate using a multivariate framework allows us to handle the crosssectional dependence, though they are assumed weak, conditioning on such dependence is still essential. The cross-country covariances are assumed non zero, i.e. Cov (u it, u jt ) 0 Pesaran et al. (2004) discuss three separate but interrelated ways the GVAR model allows for cross-country and cross-variable interactions that is useful to understand how forecasts produced by the multivariate framework are different from those from the aggregate model and disaggregate models. First of all, it is through the dependence of domestic country variables x it on the foreign country variables x jt. Secondly, the country-specific variables depend on common global exogenous variables. Finally, via the contemporaneous dependence of the shocks of country i on the shocks of country j. And such dependence is measured by the cross-country covariances Σ ij = Cov (u it, u jt ), for i j. It can be seen that in the generic multivariate model, each partial system of x it is a VAR(1) of the domestic country i. However, the interrelations of these N partial systems i=1 5

6 come in two ways (1) Θ ij 0 and (2) Cov (u it, u jt ) 0. A forecast of the aggregate produced by this multivariate system can be defined as N w i E (x it Ω t 1 ), where Ω t 1 = {x it 1 } N i=1 (4) i=1 where w i denotes the weight assigned to country i. In the case when Θ ij = 0 and Cov (u it, u jt ) = 0. (1) is simply a stack of countryspecific VAR(1) models which are uncorrelated to each other. Estimation of all N partial systems is no different from estimating each disaggregate model separately/independently. A forecast of the aggregate is produced, in the disaggregate approach, by summing up the set of disaggregate nowcasts given by N w i E (x it x it 1 ) (5) i=1 Both disaggregate forecasts from disaggregate models and forecasts from multivariate models can be conditional on disaggregate/country-specific components. Now consider the case when aggregate forecasts are produced directly from an aggregate model. Using the notation above, we define the aggregate model as N N w i x it = φ w i x it 1 + u A t (6) i=1 i=1 where u A t denotes the error of the aggregate model. The aggregate nowcast is given by ( N ) N E w i x it w i x it 1 (7) i=1 i=1 We can see that the information sets, on which these three forecasts, in (4), (5) and (7) are based, are different. So the pertinent question is, when will these three forecasts (the aggregate, disaggregate and multivariate model forecasts) be equivalent to each other? 6

7 2.1 Equivalence between the aggregate, disaggregate and multivariate approaches Taylor (1978) in his comment on Geweke (1978) uses a simple model to explain temporal and sectoral aggregation of seasonally adjusted data. Although our attention is confined to the three forecasts above, his example can be useful to illustrate the conditions when they are equivalent. Suppose there are two disaggregate components of the aggregate, x 1 and x 2 (for simplicity, we ignore the time subscript). The illustration of Taylor (1978) assumes both series consist of a seasonal and nonseasonal part. We modify this example by simply assuming each series consists of an observed (with subscript A) and unobserved component (with subscript B). x i = x A i + x B i, i = 1, 2 (8) Further assume the pair of observed and unobserved components, { } x A 2 i and { } x B 2 i=1 i i=1 are uncorrelated, both have zero means and covariance Σ A and Σ B respectively. Σ A and Σ B represent the variance-covariance matrix between the observed and unobserved components across the two disaggregate series. Suppose our interest is to obtain the aggregate of the observed component x =x A 1 +x A 2. Let x denotes a prediction of the aggregate. The optimal prediction of the aggregate should be the one that minimises the prediction error E [(x x) x 1, x 2 ] So the optimal prediction should be given by the mean of the aggregate conditional on the disaggregate components x 1 and x 2. That is x (M) = E [x x 1, x 2 ] = E [ x A 1 + x A 2 x 1, x 2 ] = E [ x A 1 x 1, x 2 ] + E [ x A 2 x 1, x 2 ] (9) of However the prediction error will not be minimised if the prediction takes the values x (D) = E [ x A 1 x 1 ] + E [ x A 2 x 2 ] (10) or x (A) = E [ x A 1 + x A 2 x 1 + x 2 ] (11) We can see that (9), (10) and (11) are analogous to (4), (5) and (7), respectively. That 7

8 is, x (M) = multivariate (or mixed) nowcast (12) x (D) = disaggregate nowcast (13) x (A) = aggregate nowcast (14) Taylor (1978) points out that the reason (9) is optimal, as compared to (10) is that the former is conditional on all available disaggregate components while the latter is not. Even though the prediction (or nowcast) in (5) and (10) are the aggregate of the predictions of the disaggregate components, the prediction of the disaggregate component is formed by conditioning on a limited set of information. As we can see from (5), nowcasts of the disaggregate of country i are conditional on its own domestic components, similarly as in (10). Whereas the nowcast produced for each country, before aggregation, from the multivariate approach (4) is conditional on the disaggregate components of all N countries, similarly in (9). In other words, x (M) allows for interdependence among countries and variables, so that the disaggregate information can be fully utilised. The relative gain in efficiency by conditioning the aggregate forecast on all disaggregate components (in other words, via a multivariate framework) is large when there is a high degree of heterogeneity in the disaggregate series being aggregated. In the case when the disaggregate series are highly correlated, the relative gain in efficiency of such forecast is low. Taylor (1978) further explains the situation when the three predictions are equivalent. This can be appreciated by supposing that { } x A 2 i and { } x B 2 i=1 i follow a joint normal i=1 distribution. Rewrite (9), (10) and (11) as x (M) = a 1 x 1 + a 2 x 2 (15) x (D) = b 1 x 1 + b 2 x 2 (16) x (A) = c (x 1 + x 2 ) (17) When Σ A = Σ B then a 1 = a 2 = b 1 = b 2 = c. In this case the three predictions are equivalent. Recall Σ A and Σ B represent the two sources of variations in the disaggregate series. Σ A = Σ B implies the disaggregate series exhibit common variation; there is a 8

9 common factor which explains all of their variation. If the covariance between x A 1 and x A 2, and the covariance between x B 1 and x B 2 is zero, and there is no homogeneity between the two disaggregates, then the forecast from the multivariate model (in other words, the efficient forecast) is equivalent to the forecast from the disaggregate model and x (M) = x (D). Applying this to our models above, this implies Cov (u it, u jt ) = 0 in (3). The degree of commonality between the disaggregates relates to the cross-sectional dependence literature, see Pesaran (2006), and also the literature on factor approximations; see Forni et al. (2000) and Stock & Watson (2002b). So when the disaggregate series exhibit a certain degree of heterogeneity in their movement, the multivariate model will be preferred as the different interactions among the disaggregate series can be accounted for. In such a case, we should expect the RMSE of the multivariate model to be less than that of the other approaches. This explains the conditions how and when the aggregate and the disaggregate models are special cases of the multivariate model. 2.2 Approximations to the multivariate model Multivariate models therefore benefit from minimum RMSE, as opposed to both aggregate and disaggregate models when the disaggregate components are heterogenous. However, estimation of the multivariate model when N is large is always a challenge, in practice. Many existing studies have essentially proposed various methodologies to tackle the issue of dimensionality in multivariate models. Those methods include factor models (see Forni et al. (2000), Stock & Watson (2002a) and Stock & Watson (2002b), and also a review by Eklund & Kapetanios (2008)); the Global VAR (GVAR) model of Pesaran et al. (2004), Dees et al. (2007), and Pesaran et al. (2009), in which estimation is done at the disaggregate level before the set of disaggregate VARs are stacked up to construct the GVAR model; the Infinite-Dimensional VAR (IVAR) of Chudik & Pesaran (2009) in which dimensionality is resolved by shrinking part of the parameter space in the limit as the number of endogenous variables tends to infinity; the Large Bayesian VAR of Banbura et al. (2010) that uses Bayesian shrinkage by imposing restrictions on the prior distribution of the model parameters. Moreover, the so-called mixed approach by Hendry & Hubrich (2011) which places the disaggregate indicators in the aggregate model to produce the aggregate forecasts; and finally, the forecast combination approach in which many models are estimated relating the disaggregates to the aggregate and then combined to produce an optimal forecast or nowcast for the aggregate that averages out model uncertainty; see, 9

10 for example, Jore et al. (2010), and Ravazzolo & Vahey (2010). Although the primary purposes of the two latter two approaches was not to render estimation of the multivariate model feasible, they have nevertheless provided this advantage. The following sections consider these various approximations to the multivariate model and explain how they are related to each other. And it also discusses possible extensions of these methodologies that one might consider Factor models Factor models have been extensively used as an approximation to multivariate model to resolve the problem of dimensionality; see for example Forni et al. (2000), Stock & Watson (2002a) and Stock & Watson (2002b). Factor models assume the comovement among data series within a large dataset can be summarised by a few unobserved common factors. Consider again the N-dimensional vector x t = (x 1t, x 2t,, x Nt ), and each x it is a k-dimensional vector that contains the domestic variables of country i. As discussed before, when N gets large (or the number of disaggregate series included in each x it gets large), the size of the dataset x t gets large. Estimation of the multivariate model for x t, (1), then becomes infeasible. The existing literature adopts two common approaches to resolve the problem of dimensionality. One option is to shrink the parameter space, the other way is to reduce the size of the dataset. The IVAR analysis of Chudik & Pesaran (2009) has developed a link between data and parameter shrinkage using the concept of cross-sectional dependence. Their work shows that by imposing restriction on the parameters, one can achieve data shrinkage. In fact, using the concept of cross-sectional dependence, they show how the dominant unit in the IVAR model can become a dynamic factor. Now suppose x t has a factor structure and can be represented by a factor model x t = ΛF t +υ t (18) where F t summarises the comovement among the variables/indicators in x t. The common factors can either be static or dynamic. For simplicity, we assume x it contains only one single series (for example only one domestic variable for each country i), and each of the disaggregate series consists of two components: x it = x C it+x I it 10

11 where x C it represents the components (or variation) in x it that are common for all i. x I it is the component in x it that is unique to unit i. So the collection of the commonality, { x C it } N i=1 gives us ΛF t,. the common components in x t. In the case when the disaggregate series are homogenous or when they exhibit high correlation, we should expect the magnitude of ΛF t to become big, in other words the set { xit} C N to be big. in other words, i=1 the approximation of x t is sufficient. Consider the variance of the x t in the factor model (18) is given by V ar (x t ) = ΛV ar (F t ) Λ +V ar (υ t ) In the case when x t = ΛF t, that is all the common factors have captured all variation in x t, the magnitude of the error υ t will be small. As a result, V ar (x t ) ΛV ar (F t ) Λ A prediction of x t should be given by E (x t F t ) (19) ( E x t { ) xit} C N i=1 Compare this with a prediction given by a multivariate model which conditions on all available disaggregate series E ( ) ( x t {x it } N i=1 = E x t { xit} C N, { ) x I N i=1 it} i=1 (20) (19) and (20) will give us the same value only if { xit} I N is negligible. In which case i=1 x it = x C it and x t = ΛF t. Then minimising the variance of the multivariate model is thus no different from minimising the variance of the factor model, although one may experience efficiency loss which arises from having to estimate the factor loadings Global VAR model The GVAR model developed by Pesaran et al. (2004), Dees et al. (2007), and Pesaran et al. (2009) is particularly useful when modelling disaggregate series and at the same time allowing for interdependence among countries and disaggregate variables. In the GVAR setting, estimation is carry out at the country-level via a VAR model with foreign weakly exogenous variables, i.e. the VARX* model. The resulting N country-specific VARX* models are then stacked up to form a GVAR model via a link matrix which specifies the 11

12 weights. Assuming there are N + 1 countries of interest, consider a VARX* model for country i (i = 0, N), ignoring the time trend that is included in the original GVAR model, x i,t = a i0 + Φ i0 x i,t 1 + Λ i0 x i,t + Λ i1 x i,t 1 + ν i,t (21) where the vector x i,t is a k i 1 vector containing country i s variables, with i = 1,, N. x i,t is a ki 1 vector containing some foreign country variables, which are assumed exogenous. For example, x i,t can contain country i s growth of GDP, IP and survey data, in which case x it = ( log y it, log ip it,m, log bal it,m ). Φ i0 is a k i k i matrix of coefficients on the lag of x it, and Λ i0 and Λ i1 are k i ki matrices of the coefficients on the foreign variables and their lags. ν i,t is assumed serially uncorrelated with mean zero and cross-sectionally weakly dependent. Let Cov (ν ilt, ν jst ) be the covariance of the l th variable in country i and s th variable in country j. N Cov(ν ilt, ν jst ) j=0 Weak dependence implies 0, as N, i, j, s. N The foreign variables for country i, x i,t are represented by the weighted averages of the disaggregate components of the rest of the N +1 countries. A fundamental element in the construction of GVAR using the N VARX* of individual countries is the link matrix, W i, that contains the weights used to link up the variables in each country-specific VARX* with the endogenous variables in all N +1 VARX*. As explained in Pesaran et al. (2004), Dees et al. (2007), and Pesaran et al. (2009), the GVAR is derived by first constructing a global vector x t that contains the endogenous variables in the N country-specific VARX*. That is, x t = (x 1t, x 2t,, x Nt ) with dimension k = N k i. The country-specific variables can be written in terms of a global vector, x t i=1 z it = W i x t ; i = 0, 1,, N (22) The subscript i of the weight matrix suggests that different weight matrix is used across countries, which reflects different relationships between the domestic variables of each country with the indicators in the global vector. The original GVAR defines the matrix W i by the trade weights. In practice, we can experiment with the use of different weights. Using the above definitions, equation (21) can be written as A i z it = a i0 +B i z it 1 +ν it (23) 12

13 where A = (I k, Λ i0 ), and B = (Φ i0, Λ i1 ). Using (22), (23) becomes A i w i x t = a i0 +B i W i x t 1 +ν it (24) The GVAR model is thus derived by stacking up all N + 1 country-specific VARX* models to yield Gx t = a 0 +Hx t 1 +ν it, and x t = G 1 a 0 +G 1 Hx t 1 +G 1 ν it (25) where a 0 = a 00 a 10., G = A 0 W 0 A 1 W 1., H = B 0 W 0 B 1 W 1., ν t,m = ν 0,t,m ν 1,t,m.. a N0 A N W N B N W N ν N,t,m So estimation of the parameter matrices in GVAR is done via a two step procedure. In the first step, we estimate the N + 1 VARX* model for each country to obtain the parameter matrices in the country-specific models. The parameter matrices in GVAR can then be computed simultaneously by using the link matrix that contains the predetermined weight for each country and the parameter matrices in the N + 1 VARX* models. Notice that the GVAR uses weights which are not estimated but pre-determined. As aforementioned, one can try out different weights on the foreign variables to observe how the significance of individual coefficients (as shown by the t-statistics), as well as the fit of the model changes when different weights are used. The result of this experiment can provide an insight into what weights are optimal for VARX* estimations. GVAR allows for cross-country, cross-variable dependence, that is Cov (ν ilt, ν jst ) is non-zero, though this cross-sectional dependence is assumed to be weak. In the case when Cov (ν ilt, ν jst ) = 0 and the parameter matrices of the foreign variables become zero, i.e. Λ i0 =Λ i1 = 0, the system reduces to N + 1 independent system of country-specific disaggregate model. The GVAR thus nests the disaggregate models. (26) GVAR as an approximation of a global factor model Although it seems that GVAR tackles the curse of dimensionality differently from factor approximation as a first 13

14 glance, this is not the case. As shown by Dees et al. (2007), the GVAR is related to a global factor model. In their paper, they present the GVAR model as an approximation of a global factor model in terms of observed global factors (F G t ) and unobserved global factors (F U t ) as follows x it = α i + Λ ig F G t + Λ iu F U t + η it (27) x it is again a k i 1 vector containing country i s variables. In our sort of exercise, it might contain GDP growth, IP and survey data. η it is a k i 1 vector containing other variables that are specific to country i, e.g. the lags of x it, and some other domestic variables. Λ ig and Λ iu are the factor loadings for the observed global factor and the unobserved global factor, respectively. Again we ignore the linear time trend present in their model. If the model does not contain the unobserved global factor, estimation would be simple as the model for country i is unrelated to the other N models. Of course linkage of all the country-specific model is still present in the GVAR setting through the observed global factor. But in the first step of the computational procedure, one can estimate all N + 1 country-specific model separately. Whereas, with the presence of the unobserved global factor, estimation is more complicated as the unobserved global factor itself needed to be estimated alongside the parameters matrices in (27). Computation becomes burdensome, in particular, if N is large. However, Dees et al. (2007) suggest proxying the unobserved common factor using the cross-sectional means of the country-specific variables and the observed global factors by following Pesaran (2006), as an alternative to factor estimation via the Kalman filter. In the factor model literature, unobserved common factors are usually defined as the unobserved comovement among series, in other words, they represent the cross-sectional dependence among the data series. The cross-section mean is a way of summarising those comovements. Thus including it in the model can mop up cross-sectional dependence in the data. In fact, Dees et al. (2007) have proved formally how cross-sectional means of the country-specific variables and the observed global factors can proxy the unobserved global factors. Aggregating the country-specific model in (27) using weight W i, so (27) becomes x t = α i + Λ ig F G t + Λ iu F U t + η t (28) 14

15 where x it = N W i η it. i=1 N W i x it, α i = i=1 N W i α i, Λ ig = i=1 N W i Λ ig, Λ iu = i=1 N W i Λ iu, and η it = Since they assume the first difference of η it, η it η it 1, is an absolutely summable (vector) moving average of some idiosyncratic shocks u it, and u it IID(0, I). So it follows that the first difference of η it is also an absolute summable (vector) moving average. That is η it η it 1 = η t = ( N ) w i Γ il u it l i=1 They apply Lemma A.1 in Pesaran (2006), assume the variance of η t is bounded by some fixed constant K, V ar( η t ) K <, and W i satisfies some conditions (see p.5 of Dees et al. (2007) for details), then η t converges to 0 in quadratic mean when N. Thus η it converges in quadratic mean to a time invariant variable η. Then they arrive at F U t q.m. l=0 ( ΛiU ΛiU ) 1 ΛiU ( x t α i Λ ig F Gt η t ) if the matrix of the factor loading Λ iu has full rank. The above proof supports their argument of using cross-sectional means of the country-specific variables and the observed global factors to proxy the unobserved global factors. i=1 (29) (30) Using disaggregate information in the aggregate model We have discussed above the factor approximation and the GVAR model both as approximations to the multivariate model. Both methods aim to reduce the dimension of the disaggregate information set (in our case country-level disaggregates) to make estimation feasible. Therefore, both methods can be seen to have nested disaggregate models, with the former summarising the disaggregate information by common factors, the latter directly estimating disaggregate models then using the output to construct the global model. In both methods, aggregate information is neither used as the dependent or explanatory variable. This is in contrast to the so-called mixed approach of Hendry & Hubrich (2011). Although the mixed approach is not primarily used to tackle the dimensionality problem, it can be seen as an approximation to the multivariate model, since it models the aggregate directly but conditioning on the disaggregate constituents of the aggregate. 15

16 Hendry & Hubrich (2011) use an aggregate model, developed from a series of disaggregate series, to explain to what extent including extra disaggregate components can improve the fit of the aggregate model. Following their analysis, we re-consider the generic model (1) x t = Θx t 1 + u t (31) where x t = {x it } N i=1. For simplicity, we assume the k dimensional vector x it has only one element, so k = 1. Let w t be a vector of weights. So the aggregate is given by the weighted sum of the disaggregate series in x t. Let X t denotes the aggregate, so X t = w t x t. Pre-multiplying (31) by the weight vector allow us to derive a model in which aggregate X t depends on its lags X t 1 and the lag of the disaggregate components x t 1 : w t x t = w t Θx t 1 + w t u t (32) X t = w t Θx t 1 + ρw t 1 x t 1 ρw t 1 x t 1 + w t u t (33) X t = ρw t 1 x t 1 + (w t Θ ρw t 1 ) x t 1 + w t u t (34) X t = ρx t 1 + (w t Θ ρw t 1 ) x t 1 + w t u t (35) Hendry & Hubrich (2011) point out that the aggregate model can only be improved systematically by including additional disaggregate components up to the extent that the parameters of the disaggregate components in (35) are constants, i.e. (w t Θ ρw t 1 ) = c, and the disaggregate components are significant in explaining the aggregate. Let γ t = w t Θ, then the disaggregate components are informative in explaining the aggregate in (35) so long as γ it ρw it 1 for each of the disaggregate component, i. The optimal prediction of the aggregate is given by its conditional mean, that is E (X t X t 1, x t 1 ) (36) or E ( ) w t x t w t x t 1, {x it } N i=1 (37) We can see that (37) is based on a richer information set than (7). Let γ t = w t Θ, disaggregate components are informative in explaining the aggregate in (35) so long as γ it ρw it 1 for each of the disaggregate component, i. In their paper, Hendry & Hubrich (2011) explain and show analytically how one can 16

17 achieve more accurate forecasts by including disaggregate components of the aggregate in the information set. Thus they enrich the information set on which the aggregate forecast or nowcast is conditioning. Their results show differences in the performance of different forecasting approaches can be explained by: (i) changes in the collinearity among the disaggregate constituents that in turn affects the bias-variance trade-off in forecast model selection; (ii) changes in the proportion of the variance of the disaggregate constituents that is captured by the common factor; (iii) changes in the unconditional moments of the aggregate being forecasted. When the aggregate and the disaggregate series exhibit a considerable amount of variability, use of disaggregate variables to forecast the aggregate can improve forecast accuracy. This is an argument used also to distinguish the condition when the multivariate model that comprises disaggregate information is preferred to the aggregate model that disregards any disaggregate information, as discussed in previous section. Hendry & Hubrich (2011) also undertake an empirical analysis of the relative forecast accuracy of aggregating disaggregate forecasts, aggregate forecasts produced by aggregate model, and aggregate forecasts produced by the aggregate model conditioning on both the aggregate and disaggregate variables. The findings of their empirical applications to forecast Euro Area inflation show weak evidence indicating that the disaggregate information helps improve forecast accuracy for the aggregate. But the findings of their application to forecast US inflation show contradicting evidence. These results lead them to conclude that whether conditioning on disaggregate information increases forecast accuracy depends on the forecasting situation Forecast combination Forecast combination offers a means of integrating out model uncertainty. In other words, it is an insurance against selecting an inappropriate model. It is natural to consider forecast combination as an approximation of the multivariate modelling approach in forecasting or nowcasting aggregate; especially if the aggregate forecast is produced by combining disaggregate forecast (e.g. as a weighted sum in point forecasting, or as a weighted combination of disaggregate densities in density forecasting). The paper by Ravazzolo & Vahey (2010) is one example. They propose an ensemble approach to construct density forecast for an aggregate as a linear combination of the disaggregate component forecast densities. Each component forecast is produced from an autoregressive linear time series model for a single disaggregate series using the Linear Opinion Pool (LOP). To explain their LOP method, suppose the aggregate of interest 17

18 has N components. For example, say the aggregate is Euro Area GDP growth, so the N components are the GDP growth rates of the countries in the Euro Area. Their method starts off by first estimating the disaggregate for each country i as an AR to obtain disaggregate forecast densities for each country. They then adopt a post-processing step to estimate recursively a (in-sample) regression of the aggregate on the expected value of the predicted density from each disaggregate AR model. Country i s forecast density for the aggregate is thus defined as the disaggregate forecast density corrected for the bias, with the bias given as the intercept obtained from the (in-sample) regression of the aggregate on the expected value of the predicted density. The forecast density of the aggregate is thus given by the disaggregate ensemble density, which is the weighted sum of all N country s forecast densities from the disaggregate (component model), with the weight being recursively updated, and thus varying across time. It can be seen that from the Ravazzolo & Vahey (2010) approach that it is related to the mixed approach proposed by Hendry & Hubrich (2011), in which forecast for the aggregate is conditioning on its disaggregate constituents. The forecast density of the aggregate is a combination of N forecast densities of the aggregate produced by N disaggregate component models. In contrast to their method, we will consider fully utilising the disaggregate information of individual countries in the Euro Area in our study via a double combination approach. Consider again the generic multivariate model as presented in (??). Suppose our objective is to produce density forecast of Euro Area GDP growth. We consider obtaining density forecasts for each country s y it, from not only a regression of y it on the variable specific to country i, i.e. x it, but also from the N 1 regressions of y it on x jt, for j = 2,, N, and i j. Then for each country i, will have N country-specific models. Once we have obtained the forecast densities y it for i = 1,, N of for all N models for each country, we combine the density forecasts from the N models to obtain a density forecast for country i. The density forecast for the aggregate is then produced by combining all the disaggregate density forecasts for all N countries. In fact, the parameters of x jt on the regression of y it can provide shares of each elements of the foreign variables in x jt on y it. This, in turn, provide insight into weights assigned to different countries as suggested by the data. Besides, we might also consider the weight combination strategy of Jore et al. (2010) to ascertain how much weight to put on different density forecasts or nowcast. This approach of combining information across the different disaggregate components can be compared with the alternative of selecting one individual model (perhaps based 18

19 on just one disaggregate or a group of preferred disaggregates) and using this to nowcast/forecast. Thereby one can relate findings, which consider the forecast density, to the literature on the merits of pooling versus selecting and specifying individual models (but in the context of point forecasts); see Marcellino & Schumacher (2008) Bayesian methods As discussed above, both GVAR models and factor models provide a means to solve the problem of dimensionality. An alternative approach to resolve the problem is Bayesian shrinkage. Banbura et al. (2010) develop Large Bayesian Vector Autoregressions (Large BVAR) which apply Bayesian shrinkage to handle VAR models with many endogenous variables. Prior distributions of the parameters in VAR with restrictions on the first and the second moment are imposed. Giannone & Reichlin (2009) discuss the relationship between BVAR and GVAR. They point out that both approaches require co-movement in the set of data series under study, which in turn, relates to the unobserved common factor literature. 3 Handling mixed frequency data We produce nowcasts from the aggregate, disaggregate and mixed approaches using a mixed-frequency VAR model which can accommodate mixed-frequency data with the ragged-edge problem. This is an issue which must be accounted for, in particular when our interest is to forecast or nowcast quarterly Euro Area GDP growth using IP and survey data - that are published monthly. Different methods have been proposed to handle mixed frequency data when real-time forecasting and nowcasting. A typical example being the bridge equation in which a two-equation system is used to forecast a quarterly variable e.g. GDP growth, with the second equation comprising the forecasting model for the monthly variable e.g. Industrial Production (see for example Salazar & Weale (1999) and Baffigi et al. (2004)). Another example which has received an increasing attention is the MIDAS regression which is a parsimonious way of linking multiple indicators available at a higher frequency to a PEEI of interest available at a lower frequency. See, for example, Clements & Galvo (2009), Kuzin et al. (2009a), and Kuzin et al. (2009b). In this section, we will present a Mixed Frequency VAR (MFVAR) model for forecasting and nowcasting EA aggregates. We will also present other extensions to this model. 19

20 3.1 Mixed Frequency VAR model: An overview In this section we describe a mixed frequency VAR model that relates quarterly GDP growth to monthly indicator variables, while imposing the quarterly aggregation constraint for GDP. Since our interest is to nowcast Euro area GDP growth, one of the indicator variables we have chosen is Euro area IP growth, since this is published monthly and known to be a good indicator for GDP. Another indicator we have chosen is the monthly survey data from the DG-ECFIN, published at the end of the month concerned. We consider the Economic Sentiment Indicator (ESI). We outline the monthly variable VAR model below. And we explain how the aggregation constraint can be incorporated into the model to develop a Mixed Frequency VAR model (MFVAR). Our real-time simulation exercises, below, will be based on different specifications and modifications of these MFVAR models, to accommodate the evolving nature of the real-time information set available for nowcasting Monthly VAR model As an example, let x t,m denote a 3-dimensional vector x t,m = ( log y t,m, log ip t,m, log bal t,m ), where y t,m and ip t,m denote Euro area GDP, Euro area Industrial Production, whereas bal t,m denotes a monthly survey data. So vector x t,m contains the monthly growth rate of these three variables. While t and m represent quarter and month respectively (t = 1, T ; m = 1, 2, 3). We further assume that we are at end of quarter t, so m = 3. Suppose the variables are related via a VAR model of order 1 in the following form: x t,3 = log y t,3 log ip t,3 log bal t,3 ; u t,3 = x t,3 = Ξ + Θx t,2 + u t,3 (38) c 1 θ 11 θ 12 θ 13 ; Ξ = ; Θ = u y t,3 u ip t,3 u bal t,3 So the three equation system of the trivariate VAR(1) model is c 2 c 3 θ 21 θ 22 θ 23 θ 31 θ 32 θ 33 log y t,3 = c 1 + θ 11 log y t,2 + θ 12 log ip t,2 + θ 13 log bal t,2 + u y t,3 (39) log ip t,3 = c 2 + θ 21 log y t,2 + θ 22 log ip t,2 + θ 23 log bal t,2 + u ip t,3 (40) log bal t,3 = c 3 + θ 31 log y t,2 + θ 32 log ip t,2 + θ 33 log bal t,2 + u bal t,3 (41) 20

21 The elements in u t,3 are assumed to have a standard normal marginal distribution with mean 0 and variance σ 2 j, j = y, ip, bal. They are also assumed to follow a trivariate normal distribution, with mean 0 and covariance matrix Σ: Σ = σ 2 y σ y,ip σ y,bal σ y,ip σ 2 ip σ ip,bal (42) σ y,bal σ ip,bal σ 2 bal Notice that the VAR(1) model, (38), is a monthly model, which means all variables involved are monthly indicators. In reality, industrial production (IP) and survey data (Bal) are released monthly. However, the GDP data are released quarterly. So monthly GDP growth log y t,3 is unobserved; what is observed is quarterly GDP growth, log y t. So when one relates the quarterly GDP growth to the monthly growth in IP and survey data, it brings about the problem of mixed frequency data. If one s interest is to nowcast quarterly GDP growth using monthly indicators, the VAR model above has to be modified to accommodate the mixed frequency nature of the variables. We impose the temporal aggregation constraint following Mariano & Murasawa (2003), and thereby extend the VAR model into a mixed frequency VAR model (MFVAR) which reflects the quarterly aggregation of monthly variables for both domestic and foreign GDP growth. We now explain the mechanics of imposing the aggregation constraint The aggregation constraint Suppose we estimate the unobserved monthly interpoland of GDP denoted as y t,m, where t indicates a particular quarter (t = 1,, T ) and m indicate a month within that quarter (m = 1, 2, 3). The interpolation is a monthly equation that relates the unobserved y t,m to the observed monthly indicator, i.e. the monthly IP and monthly survey data. To allow for non-linearity in the function of y t,m, we let h t,m = h(y t,m ). Although h t,m is unobserved but the quarterly aggregates of {y t,m } is observed. Let y t denotes the quarterly GDP, y t = 3 y t,m (43) m=1 Applying the mean value theorem, we have h (ỹ t,m ) = h(y t,m) h(y t ) y t,m y t (44) 21

22 = h(y t,m ) = h(y t ) + h (ỹ t,m )(y tm y t ) (45) where y t = yt 3 is the monthly average in quarter t and the value ỹ t,m lies between y t,m and y t. Aggregating h(y t,m ) across all months in quarter t, we have 3 3 h t,m = 3h(y t ) + h (yt,m)(y t,m y t ) (46) m=1 = 3h(y t ) The derivative h (ỹ t,m ) is approximately constant across the months in the tth quarter 3 if the error of approximation sum to zero, i.e. (y t,m y t ) = 0 ( for example, (y t,m y t ) m=1 m=1 is relatively small). Applying the logarithmic transformation to (46), and 3 log y t,m = 3 log y t 3 log 3 m=1 log y t = log y t,m + log 3 m=1 Taking the log difference of y t, the log3 term cancels out. This becomes analogous to 3 Mariano & Murasawa (2003) who assume log y t = 1 log y 3 t,m. The log difference of y t is given by log y t log y t 1 = 1 3 working through the algebra gives us m=1 3 log y t,m 1 3 m=1 3 log y t 1,m m=1 log y t = 1 3 log y t, log y t,2 + log y t, log y t 1, log y t 1,2 (47) where log y t,m is the monthly GDP growth of month m in quarter t. (47) shows the quarter GDP growth rate is the weighted average of the monthly growth rates. 22

23 3.1.3 State-space representation of the MFVAR model As one can see from (47), the right hand side variables, namely monthly GDP growth in different months of quarter t and t 1, are unobserved. A natural way to formulate and estimate the VAR(1) model in (38) subject to the aggregation constraint (47), given the unobserved monthly GDP growth components, is to adopt a state-space representation. The unobserved monthly GDP growths are summarised in the state vector. The transition equation is a monthly equation. While the measurement equation states the aggregation constraint that ensures the unobserved monthly GDP growth rates add up to their quarterly value. The mixed frequency nature of the data can then be fully accommodated. Once again assuming we are at the end of quarter t, i.e. when m = 3. Consider the state-space model of the following form Y t,m = Zα t,m + ɛ t,m (48) α t,m = Tα t,m 1 + η t,m where Y t,m = log y t log ip t,3 log bal t,3 α t,m = log y t,3 log y t,2 log y t,1 log y t 1,3 log y t 1,2 log ip t,3 (49) log bal t,3 The measurement equation relates the quarterly GDP growth to its monthly elements via the aggregation constraint, and therefore is an identity. The error term, ɛ t,m, is zero. The hyperparameter matrix in the measurement equation is 1/3 2/3 1 2/3 1/3 0 0 Z = (50) The transition equation represents the monthly conditional model (38). The transition 23

24 equation written in matrix form is thus log y t,3 log y t,2 log y t,1 log y t 1,3 log y t 1,2 log ip t,3 log bal t,3 = c c 2 c 3 θ θ 12 θ θ θ 22 θ 23 θ θ 32 θ 33 log y t,2 log y t,1 log y t 1,3 log y t 1,2 log y t 1,1 log ip t,2 log bal t,2 + u y t, u ip t,3 u bal t,3 (51) The state-space representation of the MFVAR is estimated by Maximum Likelihood via the variance decomposition using the Kalman filter. 24

25 3.2 MFVAR model used when nowcasting Euro Area GDP growth In this section we describe the different specifications of the mixed frequency VAR model that we use for our real-time simulation exercise, that relates quarterly GDP growth to monthly indicator variables, while allowing the quarterly aggregation constraint for GDP growth to be satisfied simultaneously. We outline the steps we have taken in producing (i) nowcasts from aggregate models in which the nowcast is only conditioned on aggregate indicator variables; (ii) disaggregate nowcasts in which country-level nowcasts are aggregated up to produce nowcast for the EA aggregate; and (iii) nowcasts from the mixed approach in which disaggregate information is incorporated directly into the aggregate model when producing nowcasts. When nowcasts are produced using either the aggregate or the disaggregate approaches, our model contains only one quarterly variable, which is GDP growth (as well as the monthly indicators). However, when nowcasts are produced using the mixed approach in which EA aggregate GDP growth is related to not only the EA monthly indicators but also to the country-level quarterly GDP growth data (either from a single individual EA12 country, or a weighted average of all the countries), the mixed frequency nature of the data becomes more complicated, since the quarterly aggregation constraint needs to be satisfied for both the EA aggregate GDP growth and disaggregate GDP growth. In this case we adopt a two-step estimation approach, which will be explained later in this section Aggregate nowcasts We use a three-variable MFVAR model to produce nowcasts for the EA aggregate. x t,3 in the monthly aggregate model, (38), which contains only the EA aggregates, such that in the previous notation x t,3 = ( ) log yt,3 EA, log ip EA t,3, log balt,3 EA, where the superscript EA means the variables are the EA aggregates; monthly EA GDP growth is treated as an unobserved variable. Both the growth rate of EA IP and EA survey data are monthly variables, while the observed EA GDP growth is a quarterly variable. The aggregation constraint (47) needs to be satisfied for aggregate GDP growth. We estimate the MFVAR model in the state-space form, that is, equation (48), described in the previous section, with the measurement equation satisfying the aggre- 1 For more details about the Economic Sentiment Indicator, see The Joint Harmonised EU Programme of Business and Consumer Surveys: User Guide (updated 4 July 2007), European Commission Directorate General For Economic and Financial Affairs. 25