A Monthly Indicator of the Euro Area GDP

Transcription

1 A Monthly Indicator of the Euro Area GDP Tommaso Proietti 1 Università di Roma Tor Vergata and GRETA, Venice Cecilia Frale 2 Università di Roma Tor Vergata Final Report 1 Dipartimento S.E.F. e ME.Q., Via Columbia 2, Roma, Italy. tommaso.proietti@uniroma2.it. 2 Dipartimento S.E.F. e ME.Q., Via Columbia 2, Roma, Italy and Ministry of the Economy and Finance, via XX Settembre 97, Roma, Italy. cecilia.frale@uniroma2.it.

2 Abstract This technical report illustrates the methodology adopted for the estimation of a monthly indicator of the euro area real gross domestic product. We carry out the disaggregation of the chain-linked quarterly value added from the output side for six branches of the NACE Rev. 1 - Level A6 classification and from the expenditure side for the main GDP components. With the sole exception of the agricultural branch, for which we use univariate disaggregation methods, due to the lack of related monthly indicators, we adopt a parametric dynamic factor model formulated at the monthly level, taking the temporal aggregation constraint into account. The multivariate models are cast in the state space form and computational efficiency is achieved by implementing univariate filtering and smoothing procedures. The chained-linked total GDP at basic and market prices results from the contemporaneous aggregation of the disaggregated estimates for the six branches (GDP at basic prices), plus taxes less subsidies (at market prices), via a multistep procedure that exploits the additivity of the volume measures expressed at the prices of the previous year. From the expenditure side we also proceed to the disaggregation of final consumption expenditures, gross capital formation, imports and exports by means of separate dynamic factor models; again, a multistep procedure is used to aggregate the estimates into a GDP measure at market prices. The final estimate of the euro area GDP at market prices is then obtained by combining the two estimates with optimal weights, reflecting their relative precision. Keywords: Temporal Disaggregation. Multivariate State Space Models. Dynamic factor Models. Kalman filter and smoother. Chain-linking.

3 Contents 1 Introduction 2 2 The information set Output side Expenditure side Methodology Regression based univariate methods State space representation of the disaggregated time series models The Chow-Lin model The Litterman and Fernández models Temporal aggregation and State space form Statistical treatment of the univariate case Maximum Likelihood Estimation Diagnostic checking and disaggregated estimates Drawbacks of regression based methods Multivariate methods The Stock and Watson dynamic factor model The advantages of the dynamic factor approach State space representation Temporal aggregation Univariate treatment of filtering and smoothing for multivariate models Diagnostic checking and disaggregated estimates Chain-linking and temporal disaggregation 24 5 Empirical results: temporal disaggregation of GDP Some preliminary empirical evidence The output side Agriculture, hunting and fishing (AB) Industry (CDE) Construction (F) Trade, transport and communication services (G-H-I) Financial services and business activities (J-K)

4 5.2.6 Other services (L-P) Taxes less subsidies on products The expenditure side Final consumption expenditure Gross capital formation External Balance Monthly gross domestic product Conclusions 40 A Production schedule 85 B Description of computer programmes 85 List of Tables 1 Monthly indicators employed for the disaggregation of sectorial value added Monthly indicators employed for the disaggregation of GDP from expenditure side Summary of monthly indicators methodology Revision history for Industrial value added (years ) Index of coincident indicators for Industry (Branches C-D-E): parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Construction (Branch F): parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Trade, transport and communication services (Branches G-H-I): parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Financial services and business activities (Branches J-K): parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Other services (Branches L-P): parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Final consumption expenditure: parameter estimates and asymptotic standard errors, when relevant

5 11 Index of coincident indicators for Gross capital formation: parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Imports: parameter estimates and asymptotic standard errors, when relevant Index of coincident indicators for Exports: parameter estimates and asymptotic standard errors, when relevant Monthly gross domestic product from the output side: estimates, standard errors and growth rates Monthly gross domestic product from the expenditure side: estimates, standard errors and growth rates Monthly gross domestic product at market prices: combined estimates, combined standard errors and growth rates List of Figures 1 Temporal disaggregation of value added of branch A-B: Agriculture, hunting and fishing, Eurozone12, Temporal disaggregation of value added of branch C-D-E: Industry, Eurozone12, Fernàndez model using six monthly indicators Temporal disaggregation of value added of branch C-D-E: Industry, Eurozone12, SW dynamic factor model Four monthly indicators for the disaggregation of the value added of the Construction branch (code F of the Nace rev. 1) Temporal disaggregation of value added of branch F: Construction, Eurozone12, SW dynamic factor model Monthly indicators for the disaggregation of the value added of the Trade, transport and communication services branches (codes G H I of the Nace rev. 1) Temporal disaggregation of value added of branches GHI: Trade, transport and communication services, Eurozone12, SW dynamic factor model Monthly indicators for the disaggregation of the value added of the Financial services and business activities (codes J K)

6 9 Temporal disaggregation of value added of branches J-K: Financial services and business activities, Eurozone12, SW dynamic factor model Temporal disaggregation of value added of branches L-P: Other services, Eurozone12, SW dynamic factor model Temporal disaggregation of quarterly Taxes less subsidies, Eurozone12, SW dynamic factor model Temporal disaggregation of Final consumption expenditure, Eurozone12, SW dynamic factor model Temporal disaggregation of Gross capital formation, Eurozone12, SW dynamic factor model Temporal disaggregation of exports and imports, Eurozone12, SW dynamic factor model Monthly gross domestic product estimates for the euro area - Output approach Monthly gross domestic product estimates from the output side for the euro area Monthly gross domestic product estimates from the expenditure side for the euro area Combined monthly estimates of gross domestic product at market prices. 84

7 1 Introduction The availability of a representative, reliable and timely set of high frequency macroeconomic indicators is quintessential for the assessment of the state of the business cycle of the Eurozone economy and the conduct of monetary policy. With the purpose of satisfying the information requirements of policy makers, economic analysts, researchers and business cycle experts, Eurostat has organised a very comprehensive and representative number of monthly and quarterly time series in the Euro-IND database, accessible through the Euro-Indicators website. The latter contains time series observations on 80 macroeconomic variables for the Euro-zone, the European Union, as well as for Member States and EFTA countries, concerning the following domains: balance of payments; business and consumer surveys; external trade; industry, commerce and services; labour market; monetary and financial indicators; national accounts; consumer prices. Among this set, 19 indicators have been selected by the ECB and the Commission s Economic and Financial Affairs Directorate-General with the qualification of Principal European Economic Indicators (PEEI). In recent years there have been substantial advances in the methodology and the quality of infra-annual statistical information for the Eurozone, well accounted in the report Towards improved methodologies for Eurozone statistics and indicators, by the Commission of the European Communities (2002). In particular, the statistical methodology has made it possible to increase the length, coverage, and timeliness of short-term statistics for the Eurozone. Nevertheless, some of the PEEI are available at the quarterly frequency, whereas it would be desirable to have monthly estimates of the corresponding aggregates. These are listed below separately for three different classes: National Accounts indicators: Gross Domestic Product (GDP), Sector Accounts, Government finance statistics. Business indicators: Production in construction, Turnover index for other services, Corporate output price index for services. Labour Market Indicators: Job vacancy rate, Employment, Labour cost index. The relevance of these indicators and the need to make them available at higher (monthly) frequency provides the motivation for this task. It is one of the most relevant objectives of Eurostat to provide a more informative and timely assessment of cyclical stance by the construction of monthly indicators, making the most of the information already available. Gross domestic product (GDP) is perhaps the most relevant coincident economic indicator, as it provides a comprehensive measure of the level of economic activity. Currently, 2

8 its estimation requires a relevant flow of information that accrues to slowly to make it a timely measure for the general purposes of business cycle analysis. Quarterly national accounts are constructed according to a well established methodology and practices (see Eurostat, 1999), but yet monthly estimates are not made available officially. What motivates this project is the awareness that we may go beyond the quarterly estimates due to the availability of timely and reliable statistical information on related indicators at the monthly frequencies. Using the statistical methodology and the recent advances in the literature on temporal disaggregation we can take advantage of this information and produce indirect estimates of monthly GDP that are informative for short run analysis. As the monthly indicators represent measures of sectorial output (industrial production, retail turnover, number of passengers, etc.) or of sectorial input (employment, hours worked), we consider the breakdown of GDP into the value added of six branches of the NACE-Clio rev. 1 classification and proceed to the estimation of the monthly value added. The quarterly value added observed series will be distributed over the three months composing the quarters so as to preserve the quarterly aggregation constraint, that is ensuring that the sum of the three distributed values is consistent with the quarterly figure. The same approach is followed to estimate gross domestic product at market prices from the expenditure side, by using monthly indicators of the final demand. Finally the estimates of total GDP are reconciliated by combining the supply side and expenditure side estimates using optimal weights, which reflect the relative precision. The technical report is structured as follows. Section 2 discusses the information available. The methodology is dealt with by section 3, which is divided into two parts: the first (section 3.1, drawn from Proietti, 2004) is devoted to the traditional univariate disaggregation methods based on regression on the indicator variables, such as Chow-Lin, Litterman and Fernàndez. The second part deals with disaggregation methods based on multivariate models (section 3.3) and in particular on the dynamic factor model for the estimation of an index of coincident indicators proposed by Stock and Watson (1989) as a special case of the dynamic factor model introduced by Geweke (1977) and Sargent and Sims (1977). Section 4 discusses the aggregation of the monthly estimates of sectorial value added into GDP at basic and market prices and how chained-linked volume measures using 1995 as the reference year are obtained. Section 5 summarises the main estimation results of the two approaches as applied to the disaggregation of quarterly value added. Finally, in the conclusive section we put forward the agenda for future research. 3

9 2 The information set 2.1 Output side The construction of a monthly indicator of the euro area GDP is carried out indirectly through the temporal disaggregation of the value added of the six branches of the NACE Rev. 1 - Level A6 classification, using monthly indicators specific for each branch. The breakdown of total GDP is the following: GDP = Total gross value added + Taxes less subsidies on products Breakdown of Total gross value added Label Value added of branch A B Agriculture, hunting, forestry and fishing + C D E Industry, incl. Energy + F Construction + G H I Trade, transport and communication services + J K Financial services and business activities + L P Other services = Total Gross Value Added Quarterly observations on these components are available from the national accounts compiled by Eurostat made available in electronic form in the Euro-Indicators section of the Europa database. The complete series are available at the time of writing from the first quarter of 1995 to the third quarter of The statistical table is labelled naag_q (part of the table qb_a06_k of the Europa database) in the Euro-Indicators section database. The value added series are expressed in millions of euros at 1995 prices (na-b1g-kp-mio-eur). All the series are in seasonally adjusted form and refer to the Eurozone12. The series are relatively short. This is due to a major structural break in the series concerning the statistical allocation of Financial Intermediation Services Indirectly Measured (FISIM) and the changeover to chained volume measures. FISIM, which are the results of productive activity in which an institutional unit incurs liabilities on its own account for the purpose of acquiring financial assets by engaging in financial transactions on the market, are assimilated to intermediate consumption paid to financial service providers via the interest margin. Previously, FISIM was not allocated directly to the individual customers as intermediate consumption; as a result the value added of the individual branches was overstated and FISIM had to be deducted at total economy level only. 4

10 Secondly, in 2005 and 2006, most Eurozone member states have introduced chainlinking into their quarterly and annual national accounts to measure the development of economic aggregates in volume terms. This has important consequences on the estimation of monthly aggregate GDP for the Eurozone, since, as a result of chaining, additivity is lost. This innovation, introduced in the 3rd quarter of 2005, bears important consequences for the estimation of a monthly indicator of the Euro area gross domestic product at basic and market prices. The issue of aggregation of chain-linked volume measures is the topic of section 4. The monthly indicators available for the disaggregation of the value added of each branch are listed in table 1. A remarkable fact is that no indicator is available for the primary sector (AB). For Industry (CDE) and Construction (F) a core indicator is represented by the index of industrial production. For the remaining branches (services) the monthly variables tend to be less directly related to the economic content of value added. Table 3 describes the monthly series that are employed for estimating the monthly indicator of the Euro area GDP. 2.2 Expenditure side From the expenditure side the construction of a monthly indicator is carried out indirectly through the temporal disaggregation of the components of the demand by using monthly indicators specific for each component. The breakdown of total GDP is the following: GDP (market prices) = Final consumption expenditure + Gross capital formation + Exports of goods and services - Imports of goods and services. Quarterly observations on these components are available in the Europa database. The complete series are available at the time of writing from the first quarter of 1995 to the third quarter of The statistical table is labelled namq_gdp_k in the Euro- Indicators section database. The series are expressed in millions of euros at 1995 prices (na-b1g-kp-mio-eur). All the series are in seasonally adjusted form and refer to the Eurozone12. The monthly indicators available for the disaggregation of GDP through expenditure components are listed in table 2. For Final consumption expenditure some indicators of demand are available together with the production of consumer goods. For Gross capital formation a core indicator is the production index (both for industry and constructions), in addition to some specific variables for constructions. Unfortunately, monthly indices of new orders are available 5

11 only recently (from 1996), which prevents from using them for the present work. As far as the External Balance is concerned,a monthly volume index of Imports and Exports is provided by Eurostat, although with more than 1 month of delay. In addition we draw from Tendency Surveys to catch sentiments and expectations of economic agents. A finer disaggregation for the components, (e.g. household consumption and government consumption,investment in construction and other investment) may help in monthly indicator selection and separation. We leave to future developments this issue. 3 Methodology The construction of an indicator of monthly GDP, that is consistent with Eurostat s quarterly estimates is an exercise in temporal disaggregation. The aggregate series, concerning the quarterly totals of value added and other economic flows, such as taxes less subsidies, have to be distributed across the months, using related time series that are available monthly and timely. Our methodology for the estimation of monthly gross domestic product is based on versions of the dynamic factor model that was employed by Stock and Watson (1991) to estimate an index of coincident indicators for the U.S. economy. The model postulates that a multivariate time series is driven by a common, possibly nonstochastic, factor, which is responsible for the comovements. Each individual time series is also driven by idiosyncratic dynamics. The merits of this approach are discussed in section For the primary sector and taxes less subsidies, rather than for expenditure components, we use univariate disaggregation methods that are described in section 3.1 as they were implemented in state space form in Proietti (2004). The objective of this part is to provide an exhaustive documentation of the algorithms used for the estimation of the monthly GDP indicator. The algorithms and procedures were implemented in Ox, the matrix programming language by Doornik (2001), version 3.3, and are divided into two main programmes, TDSsm.ox, dealing with univariate methods, and MonthlyInd.ox, dealing with multivariate dynamic factor models. 3.1 Regression based univariate methods A very important class of temporal disaggregation methods is based on univariate dynamic regression models postulated at the monthly frequency. This class encompasses the most popular techniques for the distribution of economic flow variables, such as Chow- Lin, Fernández and Litterman. See Di Fonzo (2003) for a comprehensive review and 6

12 extensions. In this section we review the state space representation of such models. The state space methodology offers the generality that is required to address a variety of inferential issues that have not been dealt with previously, such as the role of initial conditions and the definition of a suitable set of real time diagnostics on the quality of the disaggregation and revision histories that support model selection. See e.g. Harvey (1989), Durbin and Koopman (2001), and Harvey and Proietti (2005) for general references in the state space methodology State space representation of the disaggregated time series models Let y t denote a monthly time series concerning an economic flow, such as value added, that is observed only at quarterly intervals, due to temporal aggregation. The time series models that we consider are specified at the disaggregate observation frequency (i.e. the monthly frequency) and admit the following state space representation: y t = z α t + x tβ, t = 1,..., n, α t = Tα t 1 + W t β + Hɛ t, t = 2,..., n α 1 = a 1 + W 1 β + H 1 ɛ 1, ɛ t NID(0, σ 2 ), β N(b, σ 2 V). (1) The first equation, known as the measurement equation, expresses the series as a linear combination of the m state elements in the vector α t, where z is a fixed m 1 vector, plus regression effects, x tβ. The second equation, known as the transition equation, expresses α t as a first order vector AR process with transition matrix T and disturbances Hɛ t, where H is a fixed matrix. The vectors x t and the matrices W t contain the values of related indicators. The initial state vector, α 1, is expressed as a function of fixed and known effects (a 1 ), random stationary effects (H 1 ɛ 1 ) and regression effects, W 1 β. Two assumptions can be made concerning β: (i) β is a fixed unknown vector (V 0); this is suitable if it is deemed that the transition process governing the states has started at time t = 1; (ii) β is a diffuse random vector, i.e. it has an improper distribution with a mean of zero (b = 0) and an arbitrarily large variance matrix (V 1 0). The diffuse case captures parameter uncertainty or the nonstationarity of a particular state component and entails marginalising the likelihood with respect to the parameter vector β. The most relevant special cases are the models proposed by Chow and Lin (1971, CL henceforth), Litterman (1983) and Fernàndez (1981). These are briefly reviewed in the following sections. 7

13 3.1.2 The Chow-Lin model The Chow-Lin disaggregation method enjoys wide popularity among practitioners. It rests on the assumption that y t can be represented by a linear regression model with AR(1) errors: y t = α t + x tβ, α t = φα t 1 + ɛ t, ɛ t NID(0, σ 2 ), (2) with φ < 1 and α 1 N(0, σ 2 /(1 φ 2 )). It must be remarked that when the aggregation period is even, which occurs when an annual time series is disaggregated into a quarterly time series, the sign of the parameter φ is not identifiable; usually the autoregressive parameter is restricted in the positive half of the stationary parameter space. The model is thus a particular case of (1), with scalar α t, system matrices z = 1, T = φ, H = 1. Since α t is a stationary zero mean AR(1) process, the initial distribution is α 1 N (0, σ 2 /(1 φ 2 )), which amounts to setting: a 1 = 0, W 1 = 0, and H 1 = (1 φ 2 ) 1/2. Deterministic components such as linear trends and interventions are handled by including appropriate regressors in the set x t, e.g. by setting x t = [1, t, x 3t,..., x kt ], and writing y t = µ + γt + j β jx jt + α t, with the first two elements of β being denoted µ and γ. α t Alternatively, they can be accommodated in the transition equation, which becomes = φα t 1 + m + gt + ɛ t. The state space form corresponding to this case features W t = [1, t, 0 ], for t > 1, whereas W 1 = [(1 φ) 1, (1 2φ)/(1 φ) 2, 0 ]. The first two elements of the vector of exogenous regressors x t are zero, since m and g do not enter the measurement equation. In fact, if it is assumed that the new transition model has applied since time immemorial, α 1 = ( m 1 φ + g 1 φ ) jφ j j=1 + ɛ 1 1 φl = m 1 φ + 1 2φ (1 φ) 2 g + ɛ 1 1 φl, recalling j=0 jφj = φ/(1 φ) 2 ; L is the lag operator, such that L j y t = y t j. Under fixed regression coefficients, the two alternative representations are exactly equivalent, with µ = 1 1 φ m φ (1 φ) g, γ = 1 g. (3) 2 1 φ However, an important difference arises with respect to the definition of the marginal likelihood when these coefficients are diffuse. 8

14 3.1.3 The Litterman and Fernández models The Litterman (1983) model is formulated as regression model with ARIMA(1,1,0) disturbances: y t = x tβ + u t, u t = φ u t 1 + ɛ t. (4) The Fernández (1981) model arises in the particular case when φ = 0 (u t is a random walk). The state space representation of (4) is obtained by defining the state vector and system matrices as follows: [ α t = u t 1 u t ], z = [1, 1], T = [ φ ], H = The Litterman initialisation assumes that the u t process has started off at time t = 0 with u 0 = u 0 = 0 (Litterman, 1983, last paragraph of page 170). This implies u 1 = u 0 + φ u 0 + ɛ 1 = ɛ 1, which is implemented casting: [ ] 0 a 1 = 0, W 1 = 0, H 1 =. 1 However, this is usually inadequate. Proietti (2006) proposed, as a more realistic alternative for the type of economic time series that are usually considered, to include u 1 in the vector β as its first element, in which case x t features a zero element in first position, and assuming that the stationary process has started in the indefinite past, the initial conditions are: [ ] [ ] [ ] W 1 =, H = +. 1 φ 2 φ 1 [ 0 1 ]. This follows from writing [ u 0 u 1 ] = [ u 1 0 ] + [ 1 φ ] u 0 + [ 0 1 ] ɛ 1, and taking u 0 N(0, σ 2 /(1 φ 2 )), ɛ 1 N(0, σ 2 ). The diffuse nature of u 1 arises from the nonstationarity of the model. It should be noticed that in this second setup we cannot include a constant in x t, since this effect is captured by u 1. Finally, the ARIMA(1,1,0) process can be extended to include a constant and a trend in u t = φ u t 1 + m + gt + ɛ t ; the parameters m and g are incorporated in the vector β and the matrices W 1 and W t are easily extended. 9

15 3.2 Temporal aggregation and State space form Due to temporal aggregation, y t is not observed. The data arise, instead, as the sum of s (equal to 3 in our case) consecutive values, s 1 y τs j, j=0 and are available at times τ = 1, 2,... [n/s], representing the quarters, where [k] denotes the integral part of k. In order to handle temporal aggregation, a new state space representation is derived, by augmenting the state vector of the original state space representation with a cumulator variable that is only partially observed. This approach, proposed by Harvey (1989, sec. 6.3), converts the disaggregation problem into a problem of missing values, that can be addressed by skipping certain updating operations in the filtering and smoothing equations. The state space representation of the disaggregated model is adapted to the observational constraint by defining the cumulator variable { yt c = ψ t yt 1 c 0, t = s(τ 1) + 1, τ = 1,..., [n/s] + y t, ψ t = 1, otherwise Temporal aggregation is such that only a systematic sample of every s-th value of y c t process is observed, y c τs, τ = 1,... [n/s], whereas all the remaining values are missing. The state space representation for y c t is derived as follows: first, replacing y t = z α t in (5) substituting from the transition equation, rewrite y c t = ψ t y c t 1 + z Tα t 1 + (z W t + x t)β + z Hɛ t ; then, appending y c t to the original state vector, and defining α t = [α t, y c t ] (5) y c t = z α t, α t = T t α t 1 + W t β + H ɛ t, α 1 = a 1 + W 1β + H 1ɛ 1, (6) with z = [0 1], and [ ] [ ] [ ] T t = T 0, W W t z t =, H H = T ψ t z W t + x t z H, [ ] [ ] [ ] a 1 = a 1, W W 1 z 1 =, H H 1 a 1 z W 1 + x 1 = 1 z H 1. 10

16 Equations (6) provide the new measurement and transition equations of the state space representation that is relevant to handle temporal aggregation of the original series, y t. Its system matrices are easily determined from the original ones. Notice that the regression effects are all contained in the transition equation, and the measurement equation has no measurement noise. Moreover, the new transition matrix is time varying. For the sake of illustration, we present the system matrices and relevant vectors of the CL model (2) under temporal aggregation: αt = [α t, yt c ], [ ] [ ] [ ] [ ] [ T φ 0 t =, Wt 0 =, H 1 0 t =, a 1 =, Wt = φ 1 0 ψ t x t 0 x 1 ], H 1 = [ 1 1 φ ] Statistical treatment of the univariate case The statistical treatment of the state space model (6) is based upon the augmented Kalman filter (KF) due to de Jong (1991), suitably modified to take into account the presence of missing values, which is easily accomplished by skipping certain updating operations. Hereby we provide a detailed description of the equations implemented in the routines TDSsm.ox. The algorithm enables exact inferences in the presence of fixed and diffuse regression effects; the parameters β can be concentrated out of the likelihood function, whereas the diffuse case is accommodated by simple modification of the likelihood. An alternative exact approach to the problem of initialisation under diffuse effects has been proposed by Koopman (1997). The usual KF equations are augmented by additional recursions which apply the same univariate KF to k series of zero values, where k is the dimension of the vector β, with different regression effect in the state equation, provided by the elements of W t. Using the initial conditions in (6), and further defining A 1 = W 1, P 1 = H 1H 1, q 1 = 0, s 1 = 0, S 1 = 0, the augmented Kalman filter consists of the following equations and recursions: if y c t is available, which occurs for t = 1,..., n and t = τs, τ = 1,..., [n/s]: v t = y c t z a t, V t = z A t, f t = z P t z, K t = T t P t z /f t a t+1 = T t+1a t + K t v t, A t+1 = Wt+1 + T t+1a t + K t V t, P t+1 = T t+1p t Tt+1 + H t H t K t K tf t, q t+1 = q t + vt 2 /f t, s t+1 = s t + V t v t /f t S t+1 = S t + V t V t/f t d t+1 = d t + ln f t (7) 11

17 Else, if y c t is missing, which occurs for t τs: a t+1 = T t+1a t, A t+1 = W t+1 + T t+1a t, P t+1 = T t+1p t T t+1 + H t H t, q t+1 = q t, s t+1 = s t, S t+1 = S t. (8) The symbol V t denotes a row vector with k elements. The quantities q t, S t, s t accumulate weighted sum of squares and cross-products that will serve the estimation of β via generalised least squares regression. As we shall see shortly, the quantities (21) are used to compute the innovations ṽ t = y c t E(y c t Y c t 1), where Y t denotes the information up to and including time t, and the one-step-ahead state predictions α t t 1 = E(α t Yt 1) c and the filtered estimates of the state vector, α t t = E(α t Yt c ), along with their conditional covariance matrices Maximum Likelihood Estimation Under the fixed effects model, as shown in Rosenberg (1973), maximising the likelihood with respect to β and σ 2 yields: The profile likelihood is ˆβ = S 1 n+1s n+1, Var(ˆβ) = S 1 n+1, ˆσ 2 = q n+1 s n+1s 1 [n/s] n+1s n+1, (9) L c = 0.5 [ d n+1 + [n/s] ( ln ˆσ 2 + ln(2π) + 1 )]. (10) The latter depends only on the AR parameter φ. Thus maximum likelihood estimation can be carried out via a grid search over the interval (-1,1). When β is diffuse (de Jong, 1991), the maximum likelihood estimate of the scale parameter is ˆσ 2 = q n+1 s n+1s 1 [n/s] k n+1s n+1 and the diffuse profile likelihood, denoted L, takes the expression: L = 0.5 [ d n+1 + ([n/s] k) ( ln ˆσ 2 + ln(2π) + 1 ) + ln S n+1 ]. (11) The diffuse likelihood is based on a linear transformation (e.g. first order differences) that makes the likelihood of the transformed data invariant to β. This yields an estimator of φ with better small sample properties, see Tunnicliffe-Wilson (1986) and Shephard and Harvey (1990). Some care about the parameterisation chosen should be exercised when φ is close to 1: this point can be illustrated with respect to the simple CL model with a constant term, 12,

18 which admits two representations: (A) y t = µ + ɛ t /(1 φl) and (B) y t = φy t 1 + m + ɛ t ; for both (A) and (B) assume y 1 = µ + (1 φ 2 ) 1/2 ɛ 1. As hinted in section 3.1.2, taking first differences yields a strictly noninvertible ARMA(1,1) process that does not depend on µ in case (A) and on m in case (B), except when φ = 1, for which y t = m + ɛ t and further differencing would be required to get rid of the nuisance parameter m. Hence, setting aside for a moment the temporal aggregation problem, L( y 2,..., y n ) cannot be made independent of m when φ = 1. Denoting by L µ and L m the two diffuse likelihood under (A) and (B), it can be shown (proof available from the author) that L µ = L m + 2 ln(1 φ), which has relevant implications if φ 1; all other inferences are the same. If it is suspected that φ = 1 is a likely occurrence, due to the nonstationarity of the series and the lack of cointegration with the indicators, only diffuseness under case (A) provides the correct solution. Hence, a difference arises in the definition of the diffuse likelihood according to the parameterisation of the deterministic kernel. The ambiguity arises from the fact that the transformation adopted (differencing) is dependent on the parameter φ. See Severini (2000, sec ) for a general treatment of marginal likelihoods based on parameter dependent transformations. The ambiguity is resolved anyway if (B) is reparameterised as y t = φy t 1 + (1 φ)µ + ɛ t, t > 1, in which case the two diffuse likelihood can be shown to be equivalent Diagnostic checking and disaggregated estimates Diagnostics and goodness of fit are based on the innovations, that are given by ṽ t = v t V ts 1 t s t, with variance f t = f t +V ts 1 t V t. The standardised innovations, ṽ t / f t can be used to check for residual autocorrelation and departure from the normality assumption. They also are a good indicator of the process of revision of the disaggregated estimates. The filtered, or real time, estimates of the state vector and their estimation error matrix are computed as follows: α t t = a t A t S 1 t s t + P t z ṽ t /f t, P t t = P t + A t S 1 t A t P t z z P t /f t. The smoothed estimates are obtained from the augmented smoothing algorithm proposed by de Jong (1988), appropriately adapted to hand missing values. Defining r n = 0, R n = 0, N n = 0, for t = n,..., 1, and t = τs, τ = 1,..., [n/s] (y c t is available): r t 1 = z v t /f t + (T t+1 K t z )r t, R t 1 = z V t/f t + (T t+1 K t z )R t, N t 1 = z z /f t + (T t+1 K t z )N t (T t+1 K t z ). Else, for t τs (y c t is missing), r t 1 = T t+1 r t, R t 1 = T t+1 R t, N t 1 = T t+1 N t T t+1. 13

19 The smoothed estimates are obtained as α t n = a t + A β t + P t (r t 1 + R t 1 β) P t n = P t + A t S 1 n+1a t P t N t 1 P t. The latter provide disaggregated estimates of the cumulator ỹ c t (corresponding to the last component of the state vector); the latter can be decumulated by inverting (5) so as to provide estimates of the disaggregated series; however, its estimation error variance is not available. To remedy this situation, the strategy is to augment the state vector by appending y t to it; the new transition equation is derived straightforwardly writing: y t = z α t + x tβ = z Tα t 1 + (x t + z W t )β + z Hɛ t Drawbacks of regression based methods There are two main related sources of criticism that arise with respect to the univariate disaggregation methods. The first deals with the exogeneity assumption, according to which the indicator is considered as an explanatory variable in a regression model. In general there is no causal relationship between, say, the monthly (deflated) turnover of the retail sector and its value added. Rather, the two phenomena share a common environment and are related measures of the level of economic activity of the branch. A multivariate framework is more realistic. The second is that the regression based methods assume that the indicators are measured without error. The consequence is that the information on the indicators is transmitted to the disaggregated series by a single regression coefficient and thus any outlying and purely idiosyncratic feature, such as trading day variation, is automatically attributed to the estimated series. 3.3 Multivariate methods Multivariate disaggregation methods move away from the criticism that affects the regression based methods. There are however several degrees of freedom as far as the specification of the model is concerned. There are relatively few examples in the literature. Harvey and Chung (2000) use a bivariate unobserved components model. Moauro and Savio (2005) have proposed multivariate disaggregation methods based on the class of Sutse models. Proietti and Moauro (2006) estimated monthly GDP for the U.S. and the Euro area using a direct approach by formulating a dynamic factor model proposed by Stock and Watson (1989), specified in the logarithms of the original variables and at 14

20 the monthly frequency. This poses a problem of temporal aggregation with a nonlinear observational constraint when quarterly time series are included (see also Proietti, 2006). The main methodological contribution was to provide an exact solution to this problem, that hinges on conditional mode estimation by iteration of the extended Kalman filtering and smoothing equations. Stock and Watson (1991, SW henceforth) developed an explicit probability model for the composite index of coincident economic indicators. They proposed a dynamic factor model featuring a common difference-stationary factor that defines the composite index. The reference cycle is assumed to be the value of a single unobservable variable, the state of the economy, that by assumption represents the only source of the co-movements of four time series: industrial production, sales, employment, and real incomes. On the other hand, GDP is perhaps the most important coincident indicator, although it is available only quarterly and it is subject to greater revisions than the four coincident series in the original SW model. This consideration motivated Mariano and Murasawa (2003) to extend the SW model with the inclusion of quarterly real GDP growth, proposing a linear state space model at the monthly observation frequency that entertains the presence of an aggregated flow. Although their model is formulated explicitly in terms of the logarithmic changes in the variables, the nonlinear nature of the temporal aggregation constraint is not taken into account. This is done in Proietti and Moauro (2006) The Stock and Watson dynamic factor model Let y t denote an N 1 vector of time series, that we assume to be integrated of order one, or I(1), so that y it, i = 1,..., N, has a stationary and invertible representation. The model is of course generalisable to higher orders of integration, but our applications concerned only the I(1) case. The dynamic factor model decomposes y t into a common nonstationary component and an idiosyncratic one, which is specific to each series. Although SW formulate their model in terms of y t we prefer to set up the model in the level of the variables. The advantages of this formulation are twofold: in the first place the mean square error of the estimated coincident index are immediately available both in real time (filtering) and after processing the full available sample (smoothing). Moreover, the treatment of the aggregation constraint in the log-levels is more transparent and efficient from the computational standpoint, in that it leads to a reduced state vector dimension. The level specification of the SW single index model expresses y t as the linear com- 15

21 bination of a common cyclical trend, that will be denoted by µ t, and an idiosyncratic component, µ t. Letting ϑ 0 and ϑ 1 denote N 1 vectors of loadings, and assuming that both components are difference stationary and subject to autoregressive dynamics, we can write: y t = ϑ 0 µ t + ϑ 1 µ t 1 + µ t + X t β, t = 1,..., n, φ(l) µ t = η t, η t NID(0, σ 2 η), D(L) µ t = δ + η t, η t NID(0, Σ η ), (12) where φ(l) is an autoregressive polynomial of order p with stationary roots: φ(l) = 1 φ 1 L φ p L p and the matrix polynomial D(L) is diagonal: D(L) = diag [d 1 (L), d 2 (L),..., d N (L)], with d i (L) = 1 d i1 L d ipi L p i and Σ η = diag(σ 2 1,..., σ 2 N ). The disturbances η t and η t are mutually uncorrelated at all leads and lags. The lag polynomial ϑ 0 + ϑ 1 L can also be rewritten as θ 0 + θ 1, where θ 0 = ϑ 0 + ϑ 1 and θ 1 = ϑ 1. The measurement equation can thus be reparameterised as y t = θ 0 µ t + θ 1 µ t + µ t + X t β. The model postulates that each series, in differences, y it, is composed of a mean term δ i, an individual AR(p ) process, d i (L) 1 η it, and a common AR(p) process, φ(l) 1 η t. Both µ t and µ t are difference stationary processes and the common dynamics are the results of the accumulation of the same underlying shock η t ; moreover, the process generating the index of coincident indicators is usually more persistent than a random walk and in the accumulation of the shocks produces cyclical swings. Notice that (12) assumes a zero drift for the single index and unit variance for its disturbances. These identification restrictions (we may alternatively restrict to unity one of the loadings in θ 0 and include a nonzero drift in the common index equation, provided we impose one linear constraint on β) can be removed at a later stage to enhance the interpretability of the estimated common index The advantages of the dynamic factor approach The dynamic factor model framework has several appealing features. First and foremost, it rationalises the practice of statistical offices which amounts to summarise the available 16

22 indicators in a unique summary (a weighted average, if a priori weights are available, or a statistical summary achieved through the use of static principal components analysis, or a simple combination with weights that are inversely proportional to the volatility of each indicator). The common indicator would then subject to smoothing and corrections for outliers and structural breaks. In our approach all these operations are carried out simultaneously in a model based framework, so that the common factor extracts the dynamics that are common to the indicators and that are relevant for the disaggregation of the quarterly flows. Furthermore, the approach is easily extended to include more factors. This is actually what was done for bringing in the model the business survey variables (see section 5.2.2). Finally, it has the flexibility of handling seasonal effects, calendar components and other effects affecting the level of the series (additive outliers, level shifts, etc.) simultaneously; in the regression approach adopted by statistical institutes these operations are carried out as preliminary corrections, which makes the disaggregation exercise more elaborate and less internally consistent. It should be noticed that the dynamic factor model formulated in the previous section is such that each of the component series is integrated of order 1, or I(1), and there is no cointegration among the series, unless more than one of the idiosyncratic variances were equal to zero. Under such circumstances, there exists a dynamic linear combination that is stationary. On the contrary, static cointegration is ruled out by the formulation of the model. Another way in which dynamic cointegration may arise is when the representation adopted for the idiosyncratic component is a stationary ARMA process. The original formulation of the model of coincident indicators by Stock and Watson specified an I(1) idiosyncratic component; the presence of cointegration was explicitly ruled out by pretesting. In our case, there are no theoretical and empirical arguments for assuming that say retail turnover, new car registrations and value added of the branch Trade, transport and communication services are cointegrated. In general, the dangers of assuming cointegration when it is not present are greater compared to the dangers of assuming it away when it is present. Roughly speaking, overdifferencing is a less costly form of dynamic misspecification compared to underdifferencing. It should be stressed, perhaps, that the dynamic factor model with idiosyncratic ARI- MA(1,1,0) dynamics is an unobserved components version of the Litterman model, where the common index µ t summarises the information that is common to a set of indicators. The ARIMA(1,1,0) specification is too rich for the quarterly temporally aggregated GDP series; essentially, this is so because the AR parameter is difficult to identify (see Proietti, 2006) on this point. Therefore, we will restrain the AR parameter to be equal to 0, so 17

23 that effectively we use a random walk specification for the idiosyncratic component of the temporally aggregated series. It must nevertheless be kept in mind that the monthly indicators are endogenous, which is another desirable feature of our approach (see section 3.2.4) State space representation In this section we cast model (12) in the state space form (SSF). We start from the single index, φ(l) µ t = η t, considering the SSF of the stationary AR(p) model for the µ t, for which: µ t = e 1pg t, g t = T µ g t 1 + e 1p η t, where e 1p = [1, 0,..., 0] and T µ = φ 1. φ p 1 I p 1 φ p 0. Hence, µ t = µ t 1 + e 1pg t = µ t 1 + e 1pT µ g t 1 + η t, and defining [ ] [ ] µ t 1 e α µ,t =, T µ = 1pT µ, g t 0 T µ the Markovian representation of the model for µ t becomes µ t = e 1,p+1α µ,t, α µ,t = T µ α µ,t 1 + H µ η t, where H µ = [1, e 1,p]. A similar representation holds for each individual µ it, with φ j replaced by d ij, so that, if we let p i denote the order of the i-th lag polynomial d i (L), we can write: µ it = e 1,p i +1α µi,t, α µi,t = T i α µi,t 1 + c i + H i ηit, where H i = [1, e 1,p i ], c i = δ i H i and δ i is the drift of the i th idiosyncratic component, and thus of the series, since we have assumed a zero drift for the common factor. Combining all the blocks, we obtain the SSF of the complete model by defining the state vector α t, with dimension i (p i + 1) + p + 1, as follows: α t = [α µ,t, α µ 1,t,..., α µ N,t]. (13) 18

24 Consequently, the measurement and the transition equation of SW model in levels is: y t = Zα t + X t β, α t = Tα t 1 + Wβ + Hɛ t, (14) where ɛ t = [η t, η1t,..., ηnt ] and the system matrices are given below: [ ] Z = θ 0,. θ diag(e p 1,..., e p N ), T = diag(t µ, T 1,..., T N ), H = diag(h µ, H 1,..., H N ). (15) The vector of initial values is written as α 1 = W 1 β + Hɛ 1, so that α 1 N(0, W 1 VW 1 + HVar(ɛ 1 )H ), Var(ɛ 1 ) = diag(1, σ1, 2..., σn 2 ). The first 2N elements of the vector β are the pairs {(µ 01, δ i, i = 1,..., N}, the starting values at time t = 0 of the idiosyncratic components and the constant drifts δ i. The regression matrix X t = [0, X t ] where X t is a N k matrix containing the values of exogenous variables that are used to incorporate calendar effects (trading day regressors, Easter, length of the month) and intervention variables (level shifts, additive outliers, etc.), and the zero block has dimension N 2N and corresponds to the elements of β that are used for the initialisation and other fixed effects. The 2N + k elements of β are taken as diffuse. For t = 2,..., n the matrix W is time invariant and selects the drift δ i for the appropriate state element: [ W = ] 0, C i = [0 pi +1,1.c i ], diag(c 1,..., C N ) whereas W 1 W 1 = [ 0 diag(c 1,..., C N ) ], C i = [ e 1,pi +1.c i ] Temporal aggregation Suppose that the set of coincident indicators, y t, can be partitioned into two groups, y t = [y 1t, y 2t], where the second block gathers the flows that are subject to temporal aggregation, so that δ 1 y2τ = y 2,τδ i, i=0 τ = 1, 2,..., [T/δ], 19

25 where δ denote the aggregation interval: for instance, if the model is specified at the monthly frequency and y 2t is quarterly, then δ = 3. The strategy proposed by Harvey (1989) consists of operating a suitable augmentation of the state vector (13) using an appropriately defined cumulator variable. In particular, the SSF (14)-(17) need to be augmented by the N 2 1 vector y2t, c generated as follows y2t c = ψ t y2,t 1 c + y 2t = ψ t y2,t 1 c + Z 2 Tα t 1 + [X 2t + Z 2 W t ]β + Z 2 Hɛ t where ψ t is the cumulator variable, defined as follows: { 0 t = δ(τ 1) + 1, τ = 1,..., [n/δ] ψ t = 1 otherwise. and Z 2 is the N 2 m block of the measurement matrix Z corresponding to the second set of variables, Z = [Z 1, Z 2] and y 2t = Z 2 α t + X 2 β, where we have partitioned X t = [X 1 X 2]. Notice that at times t = δτ the cumulator coincides with the (observed) aggregated series, otherwise it contains the partial cumulative value of the aggregate in the seasons (e.g. months) making up the larger interval (e.g. quarter) up to and including the current one. The augmented SSF is defined in terms of the new state and observation vectors: [ ] [ ] α t = α t y c 2t, y t = where the former has dimension m = m+n 2, and the unavailable second block of observations, y 2t, is replaced by y c 2t, which is observed at times t = δτ, τ = 1, 2,..., [n/δ], and is missing at intermediate times. The measurement and transition equation are therefore: y t = Z α t + X t β, α t = T α t 1 + W β + H ɛ t, (16) with starting values α 1 = W1β + H ɛ 1, and system matrices: [ ] [ ] [ ] [ Z Z 1 0 =, T T 0 =, W W =, H = 0 I N2 Z 2 T ψ t I Z 2 W + X 2 The state space model (16)-(17) is linear and, assuming that the disturbances have a Gaussian distribution, the unknown parameters can be estimated by maximum likelihood, using the prediction error decomposition, performed by the Kalman filter; given the parameter values, the Kalman filter and smoother will provide the minimum mean square 20 y 1t y c 2t I Z 2 ] (17) H.