100Econoic Papers Vol.11 No.1 Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model Ki-Ho Ki* Abstract Since GDP is announced on a quarterly basis, the data frequency of GDP is relatively lower than that of other ajor acro variables published on a onthly basis. Therefore, in actual fact, the utility of GDP in judging the econoy swiftly is not great. In ters of data utility, its utility is also low. If GDP is used, the other onthly data ust be converted into quarterly data, and the result is a reduction to one-third of the total nuber of observations. In this paper, onthly GDP is obtained by using onthly inforation data such as the IPI (Industrial Production Index) and the WRSI (Wholesale and Retail Sale Index), which are closely connected with GDP. To acquire onthly GDP, this paper suggests applying an unobserved coponent vector error correction odel that can utilize ixed frequency data in estiation directly, without extra procedures for data transforation (e.g., onthly to quarterly). In order to judge whether onthly GDP is estiated appropriately, this paper copares actual quarterly GDP with converted quarterly GDP based on the estiated onthly GDP. The RMSPE (root ean square percentage error) of this estiated quarterly GDP shows around 0.01. The final onthly GDP is produced by cobining two individual onthly GDP figures (one estiated using the IPI and the other by the WRSI) and then bencharking the cobined figure, and it is then copared to the actual quarterly GDP. The RMSPE of the final onthly GDP is close to 0. Meanwhile, the correlation coefficient and the lagged correlation coefficient between the final onthly GDP and the cyclical coponent of the CCI (Coincident Coposite Index) are high and significant, and the final onthly GDP leads CCI by approxiately one to three onths.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model101 JEL Classification Nuber: C31, C43, C51 Key Words: ixed frequency data, unobserved coponent vector error correction odel, onthly GDP I. Introduction GDP is a acro variable that indicates the overall econoic condition ost. Given that GDP is published on a quarterly basis, however, the data frequency of GDP is relatively longer than those of other acro variables such as inflation and uneployent. Thus the utility of GDP is substantially low in the arena of financial engineering that ainly utilizes data with short-ter data frequency relative to the data such as stock prices and interest rates. When GDP-related data is used, for exaple, since the GDP-concerned data is published on a quarterly basis, it is inevitable to convert other onthly data with shorter data observation periods than GDP into quarterly data for econoic analysis. If we use the data by converting all the other onthly variables except GDP into quarterly variables, substantial inforation loss arises because the nuber of observations is reduced to one-third of the total nuber of observations. When the odel is not changed into a quarterly odel where onthly data are converted into quarterly data, an Industrial Production Index (IPI) can be considered to be used as a proxy variable of GDP. When the onthly variables such as IPI is used as proxy variable, these variables need to sufficiently reflect the inforation pertaining to GDP. Although IPI posseses significant inforation in regard to the activities of anufacturing, ining, and electricity and gas industry, etc, it has liitations in not including any inforation at all related to the activities of the service industry 1), which has large proportion in GDP. These probles can be considerably resolved by cobining individual onthly GDP estiate coputed by using different inforation. Inforation probles arising fro the process of converting data can be solved by the ethod of utilizing the onthly and quarterly data as it is without * Econoist, Econoic Institutional Studies Tea, Institute for Monetary & Econoic Research, Bank of Korea (+82-2-759-5422, kihoki@bok.or.kr) The author would like to express his thanks to two anonyous referees, and Prof. Joon-Mo Yang at Yonsei University. The views expressed in this paper are those of the author and do not necessarily represent those of the Bank of Korea. All reaining errors are ine. 1) The service industry akes up 52.4 % of total industry based on gross value added as of present.
102Econoic Papers Vol.11 No.1 the data frequency conversion. For exaple, Zadrozny (1990) estiated the onthly GDP growth rate using his odel while suggesting the ethodology of resolving the proble arising fro the conversion of data frequency. Fro this perspective, this odel has an advantage that the proble of utilizing GDP quarterly data insufficiently can be considerably unraveled in the procedure of decision-aking in onetary policy. Since the odel of Zadrozny (1990) is founded on the assuption that the state vector is a stable variable, all nonstationary variables I(1) need to be first differenced, and consequently the cointegrating relationship existing aong variables can not be utilized. As a result this odel has liitations in not sufficiently utilizing long-run inforation pertaining to tie series. In addition, since Zadrozny (1990) converted all the variables into stationary variables by taking first differences, the onthly level variable can not be restored. This proble always takes place when using a differenced data. In order to overcoe the liitation in Zadrozny (1990), the defect that longrun cointegrating relationships are not considered due to the use of only differenced variables, this paper suggests the unobserved coponent VECM (UCVECM) by clearly adopting a long-run cointegrating relationship aong variables. In other words, the paper proposes a odel that can restore the level variable of onthly GDP considering it together with long-run inforation, while utilizing ixed frequency data without an additional process of data frequency conversion. And it copares actual quarterly GDP with the quarterly GDP estiate coputed fro estiated onthly GDP. Moreover, the analysis is conducted on whether the estiated onthly GDP can be eployed in a useful way in econoic analysis by exaining whether onthly GDP captures the oveents of the econoy well through a siple correlation coefficient between estiated onthly GDP and the coincident econoic index, and lagcorrelation coefficient analysis. The reaining part of this paper is organized as follows. In Chapter, we briefly introduce this research in regard to estiation of onthly GDP. Chapter explains the unobserved coponent vector error correction odel (UCVECM) proposed in this paper, focusing on the two siplest variables. In Chapter, we evaluate the accuracy of estiated onthly GDP, and attept to find the possibility of the utility of estiated onthly GDP. Lastly, Chapter suarizes and concludes.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model103 II. Outline of the previous research The standard ethod for the estiation of onthly GDP is interpolation, distribution and extrapolation using an estiator like the best linear unbiased estiator, which is based on ordinary least squares (Chow and Lin, 1971). However, these ethods have liitations in that they suppose that inforation on the variance and the covariance of the residuals are known. There are the ethods of Fernandez (1981) and Litteran (1983), which are fundaentally siilar to that of Chow and Lin (1971). These are different in the assuptions on residuals of the odel. However, Chow and Lin (1971) eploying a ethod that can be applicable only to cases where residuals basically satisfy the stationarity, Fernandez (1981) extended the odel that can be eployed to cases where residuals follow a rando walk. Considering that the ethod of Fernandez (1981) can not deal with the case where serial correlation exists in residuals, Litteran (1983) proposed a ethod that can be eployed in general cases where serial correlation exists in residuals. These three best linear unbiased estiators, however, are likely to have a proble when onthly variables are created by using all nonstationary I(1) variables. In other words, in order to apply these ethods, as Fernandez (1981) pointed out, the concerned tie series should be a stationary variable or the condition that serial correlation ust not exist in residuals should be satisfied. In order to eploy these ethods into nonstationary I(1) variables, therefore, variables need to be converted to have stationarity through differencing or the supposition that residuals follow a rando walk should be added. Meanwhile, no ethod aong the above three considers cointegrating relationships aong nonstationary tie seires. For exaple, when residuals have nonstationarity, there is a proble that the estiated coefficients do not indicate a cointegrating relationship. In regard to this, Fernandez (1981) took into account the case where residuals follow a rando walk. If the variables are I(1) variables, the fact that the residuals follow a rando walk eans that the basically estiated equation itself does not have a cointegrating relationship. As for the solution to the inforation loss entioned in the introduction, the ethod of utilizing ixed frequency data was developed by Zadrozny (1990). Chow and Lin (1971), Fernandez (1981), and Litteran (1983) did not use onthly and quarterly data at the sae tie. That is, these techniques do not use directly the dynaic interaction that exists between onthly data and quarterly data.
104Econoic Papers Vol.11 No.1 In contrast, Zadrozny (1990) estiated the Vector Auto-Regressive Moving Average odel without the additional frequency conversion of data by eploying the ethod of using ixed frequency data, i.e., the ethodology of utilizing siultaneously onthly and quarterly data. Zadrozny (1990) forecasted the onthly GNP of the US using this odel. Since the odel of Zadrozny (1990) was based on the assuption that the state vector follows a VARMA process, however, all I(1) variables need to be differenced. As cointegrating relationships existing aong variables are not considered, the odel has liitations in not sufficiently utilizing the long-run inforation of data, which is no different fro those of Chow and Lin (1971), Fernandez (1981), and Litteran (1983). Meanwhile, since Zadrozny (1990) converted all I(1) variables into I(0) variables through differencing, this odel has a proble that onthly level variables cannot be restored. III. A State Space Approach for the Estiation of Monthly GDP 1. State-space representation A. State-space representation This chapter exaines the unobserved coponent vector error correction odel (UCVECM) that will be utilized in estiating onthly GDP. When there is a two-diensional vector, X t, consisting of two onthly variables such as X 1t and X 2t, a onthly VAR odel e.g., AR(2) can be represented as follows: X ( 1t X 2t ) X 1,t-1 + 11 12 2 2 X 1,t-2 + 1,t ( )( X ) ( 21 22 2,t-1 2 2 ) ( X ) ( 2,t-2 ) 2,t = 11 12 1 1 21 22 1 1 When a cointegrating relationship exists between the two variables, equation (1) becoes reparaeterized into an error correction odel as equation (2): X 1t = 11 12 1 1 X 1,t-1 1 ( 1 2 ) X 1,t-2 + 1,t ( X 2t ) ( 21 22 1 1 ) ( X ) ( 2,t-1 2 ) ( X ) ( 2,t-2 ) 2,t The error correction odel of equation (2) can be expressed as a state-space representation, and this paper converts equation (2) into the state-space (1) (2)
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model105 representation coposed of the observation equation 2) transition equation in equation (4). 3) in equation (3) and the Y t = M t + t (3) Here Y t t = T t1 + Z t ( ) ( ) = X 1 0 0 0 0 0 0 0 0 0 0 1t, and M = 0 1 0 2 0 3 0 2 0 1 0 X 2t 1 /3 0 2 /3 0 1 0 2 /3 0 1 /3 0 0 (4) Matrix M as a atrix concerning the observation equation plays a role in connecting the state vector and the observed variable vector. Matrix M is explained in the latter part of this chapter. The state vector is defined as follows: t (X 1t, X 2t, X 1,t 1, X 2,t 1, ( 1 X 1,t 2 2 X 2,t 2)) (5) where X i,t (i = 1,2) and X t2 denote the first-differenced onthly state variable and cointegrating relationshipship aong onthly level state variables, respectively. And the transition atrix conforing with this state vector is as follows: 4) T = 11 12 1 1 1 1 1 2 1 21 22 T = 1 1 2 1 2 2 2 T = 1 0 0 0 0 (6) T = 0 1 0 0 0 T = 1 0 1 2 1 If this representation is extended to the general n-variable VAR(p) odel, the state vector is set up as follows: 5) t (X t, X t 1,, X tp, X tp) (7) 2) Refer to Proietti (1997), and Hendry and Mizon (1990). 3) In order for state-space representation to satisfy the stability condition, a root for I(np+r) Tz needs to be laid outside the unit circle. Refer to proposition 1 in Proietti (1997). 4) Refer to Appendix for all syste atrixes including atrix M and Z. 5) Given the fact that a constraint condition that the differenced value of quarterly variable should be equal to that of the onthly variable, the lag of the onthly differenced variable needs to turn out ore than 4. This signifies that the lag length of the VAR odel should be at least greater than 4.
106Econoic Papers Vol.11 No.1 where X t as (n1) vector denotes a first-differenced onthly state vector and X tp represents a cointegrating relationship aong onthly state variables. And the ((np+r) (np+r)) diensional transition atrix T conforable with the state vector is: T = 1 2 p T = I N 0 0 0 0 T = 0 I N 0 0 0 (8) T = 0 0 I N 0 0 T = 0 0 0 I r where i atrix is (nn)diensional short-run coefficient atrix, vector (nr)diensional adjustent coefficient vector, vector (rn) diensional cointegrating vector, I r atrix (rr)diensional identity atrix, and r cointegrating rank. 6) The observation equation for the error correction odel this paper considers is as follows. Since the paper uses ixed frequency data, i.e., ixed data of quarterly and onthly data, we need to consider that soe part of the state vectors are not observed in the onthly unit but observed only in quarterly unit. The paper eploys the concept of "flow" or "stock" of Zadrozny (1990). That is, a stock variable is defined as a variable observed every onth. And a flow variable is defined as a variable that cannot be observed every onth, but observed only in the for of teporally aggregated values in tie (eg, one to three onths). 7) The observation equation is constructed as follows. The observable variables are denoted as Y 1t, Y 2t, and Y1t, q respectively. Since X1t, the state variable is a 6) In a siilar way, the state vector and transition atrix can be defined in the error correction odel having lag variable in as level variable as follows, that is, the state vector is: t (X t 1, X t, X t 1,, X tp), transition atrix is set up: T = I r 0 0 0 T = ( 1) 2 p p T = 0 I N 0 0 0 T = 0 0 0 I N 0 7) According to this classification, uneployent rate as a flow variable is generally observed every onth and therefore is classified not as a flow variable but a stock variable. GDP observed only on a quarterly basis is a flow variable if it is defined as the su of onthly GDP.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model107 stock variable, i,e., observed every onth in t = 1,2,,T, these stock state variables have actual observations every onth. The actually-observed value of X 1t is expressed as Y1t. In the case of X 2t, a state and flow variable, state variables are supposed to exist every onth, but the actually-observed values can be observed in the for of quarterly values at the point of ultiples of 3, that is, t = 1,2,,T, the last onth of every quarter. The actual quarterly observation of the flow variable, X 2t, observed in such a way, is represented as Y1t. As X 1t is a stock variable, its observation exists every onth, the actual quarterly observations also exist in the last onth of every quarter. Zadrozny (1990) did not use quarterly stock variable (e.g., quarterly IPI) as observable quarterly data in the observation equation and utilized only observable onthly data(onthly IPI) in the odel. Since the observable quarterly variable(quarterly IPI) can be derived fro onthly stock state variable(state variable X 1t representing onthly IPI), however, there is no reason not to include the quarterly variable (quarterly IPI) in the observation equation. This case has an advantage in utilizing inforation of the quarterly inforation variable together with onthly inforation variable. Hence, this research adds the quarterly variable defined as the three-onth ean of observable onthly variables into the observation equation. This quarterly observation is represented as Y q 1t as entioned above. As Y q 1t is data in the for of an index, the quarterly value is observed. Thus, it differs fro Y q 2t defined as the su of three-onth onthly data. 8) The reason that the above-observed variables are expressed in the for of differences is that we attept to represent the actually-observed quarterly difference vector as the function of the onthly difference state vector as far as possible, considering the assuption that the onthly state vector is produced fro an error correction odel. The observed quarterly index type variable, Y1t, q has the following relationship with the onthly state vector, X1t: Y q 1t = ( 2 X 1,t-i)/3 (9) i=0 8) The quarterly value of index variables such as IPI is not defined as the su of the 3 onth onthly index, but as the ean of three-onth values. This has nothing to do with whether the index variable is a stock variable or a flow variable. In contrast, onthly GDP is not an index-type variable and therefore the quarterly value is defined as the su of the 3-onth onthly(flow) variable.
108Econoic Papers Vol.11 No.1 Siilarly, the observable quarterly variable, Y q 2t has the following relationship with the onthly state vector, X 2t : Y q 2t = 2 X 2,t-i (10) i=0 where t = 3,6,, T. Hence, the differenced values of Y q 1t, Y q 2t, Y q 1t and Y q 2t, respectively, are as follows: Y q 1t = ( 2 X 1,t-i i=0 i=0 Y q 2t = 2 X i=0 2,t-i i=0 2 X 1,t-(i+3))/3 (11) 2 X 2,t-(i+3) (12) Equations (11) and (12) denoting differenced quarterly variables Yi,t(i q = 1,2) can be represented using a differenced onthly state vector, X i,t(i = 1,2). Let the lag polynoial (L) be defined as (L) (1+2L+3L 2 +2L 3 +L 4 ), then Y q 1t and Y q 2t can be expressed as equations (13) and (14), respectively. Y q 1t =((L)X 1,t)/3 q 1,t (13) Y q 2t = (L)X 2,t q 2,t (14) The observation equation is ore concretely represented as follows. Since onthly stock variable, the actually-observed value Y 1t, is the sae as the onthly state variable, X 1t, Y 1t = X 1t (15) But the su(or ean) of onthly variables ay not be the sae as quarterly variables due to the easureent error or the difference in the data coverage used in the process of estiating actual quarterly variables, Y q 1t and Y q 2t. Reflecting these errors, this paper sets up the observation equation as follows: 9) Y q 1t =((L)X 1,t)/3 + q 1,t = q 1,t + q 1,t (16) Y q 2t = (L)X 2,t + q 2,t = q 2,t + q 2,t (17) 9) Since Zadrozny (1990) did not consider the error in estiating actual quarterly GDP, the error ter k, t(k=2,3) does not exist. For details with regard to this error ter, see the next chapter.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model109 The easureent atrix M is represented as: M = 1 0 0 0 0 0 0 0 0 0 0 M = 0 1 0 2 0 3 0 2 0 1 0 M = 1 /3 0 2 /3 0 1 0 2 /3 0 1 /3 0 0 Meanwhile, the nuber of data observations at a certain tie point cannot be the sae between the last onth of a quarter in which all onthly and quarterly data are observed and other onths of the quarter in which only onthly data are observed. Thus, M receives a constraint on the atrix diension depending on the tie when data are observed. Taking this into account, atrix M* is defined as follows: M* = St M, St = { So, if t = 1,2,4,5, (18) I s, if t = 3,6, where s of I s is the su of the nuber of observable onthly variables and the nuber of quarterly variables, and S o is a selection atrix coposed only of rows corresponding to the nuber of onthly variables. For exaple. M* turns out as the first row of M at the tie of t = 1, 2, 4, 5,, and is consistent with M at the tie of t = 3, 6,. Consequently, the observation equation is represented by the tie as in equation (19): Y t = S t M t + t (19) B. Assuptions on the State Vector The paper, in order to use state-space representation in an error correction odel, akes the following assuption about the state vector. The paper assues that a cointegrating relationship exists between onthly observable I(1) variables (e.g., IPI) and onthly state I(1) variables (for exaple, onthly GDP) as an unobserved coponent. In other words, it assues that a cointegrating relationship exists between the Industrial Production Index (IPI) as a onthly observable variable and unobserved onthly GDP. This is grounded in the following reasons. When the onthly variable is a generally nonstationary variable, even though this variable is converted into a quarterly variable 10), the 10) For exaple, onthly data can be converted into quarterly data by a ethod of calculating three-onth oving average for onthly variables, or using a onth-end balance.
110Econoic Papers Vol.11 No.1 converted variable becoes a nonstationary I(1) variable (Pierse and Snell(1995), Marcellino(1999)). In a siilar anner, when a stationary variable is converted in the sae way, the variable becoes a stationary quarterly variable (Wei and Stra, 1990). That is, when data frequency is changed fro high frequency to low frequency, the tie-series stationarity of the variable is not affected, Although onthly data are converted into quarterly data, the cointegrating relationship existing aong nonstationary variables is also not affected (Granger(1990), Marcellino(1996, 1999). 11) That is, the cointegrating vector is known to be unaffected by data frequency conversion. However, the results of stationarity of variables and cointegrating relationship exained so far hold under a situation where both onthly and quarterly data are observable, and therefore attention needs to be paid to this. Inversely, when quarterly data are converted into onthly data, we should investigate how the stationarity of variables is influenced. First, when quarterly data are converted into onthly data, it is not known in fact what distribution the onthly data takes because inforation on the variables within a quarter is unknown. 12) Thus it is not known how the stationarity of variables is affected in this case. When both onthly and quarterly data on certain variables exist in an observable for, as entioned above, the stationarity of onthly and quarterly data is not influenced. In other words, when onthly data derived by dividing stationary quarterly data becoe a nonstationary I(1)variable, quarterly data converted fro unstationary onthly data ust becoe I(0), conflicting with the result of Pierse and Snell (1999). Thus, if quarterly data are stationary, divided onthly data also need to be stationary. By a siilar logic, if onthly data derived fro dividing unstationary quarterly data becoe a stationary variable, the variable that adds a stationary variable with finite difference is quarterly data and therefore the quarterly variable should be a stationary variable. This also contradicts Pierse and Snell (1999). Consequently the stationarity of a variable should not be affected even when a quarterly variable is converted into a onthly variable. Considering this, when a quarterly variable is an observable and unstationary I(1) variable at the sae tie, even though the onthly state variable is unobservable, the onthly state variable can also be supposed as a nonstationary I(1) variable. Meanwhile, following a siilar logic, we suppose that when there exists a cointegrating relationship aong observable quarterly 11) Marcellino (1996) proved that even when the integration order is ore than two, the existence of cointegrating relationships is not affected by quarterly transforation. 12) In the case of GDP being observed on a quarterly basis, its oveent is not observed within a period shorter than Nyquist frequency, i.e., onthly fluctuation occurring within a quarter.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model111 variables, there also exists a cointegrating relationship between unobservable onthly variables and observable onthly variables. 13) This assuption is based on the fact that since change in a onthly variable occurring within a quarter cannot affect the long-run oveent of the variable, it also cannot affect the cointegrating relationship showing the existence of a long-run equilibriu relationship between variables. 14) In the eantie, if the data generating process (DGP) of onthly data can be guaranteed as 1:1, corresponding with a quarterly variable, the assuption would be fine as in the arguent so far. When there exists a possibility that the plural onthly data generating processes atch with one and the sae quarterly variable, attention needs to be paid to the difficulties of discerning the onthly data generating process. This is because either the ARMA process or the AR process, which are different fro each other, can be derived fro the sae quarterly process (Marcellino, 1996). Regarding this, Wei and Stra (1990) proposed an a priori sufficient condition in which high frequency tie series that are not aggregated correspond one to one with low frequency tie series derived fro aggregating the tie series. C. Restoring Level State Vector In this paper, all estiated state vectors are set up in the differenced for, X t1 (i = 1, 2,, p1) except state vector concerning cointegrating vector. As entioned previously, therefore, this deands the ethod of converting the differenced state variable into the state variable. This proble can be naturally resolved when onthly GDP is given at a certain tie point. Since actual onthly GDP does not exist in Korea, however, we need to recover onthly GDP. We can consider three kinds of ethod to restore level state variables. First, since the value of Y2, q X 2,t1: T, X 2,t2: T etc 15) is observed or estiated, the initial state can be set up as follows: 16) X 2,t1: r Y 2,X q 2,t2: 2X 2,t1: 3 (20) 13) The assuption, when soe observable variables fro aong onthly variables are known as I(1) variable, should proceed that a state variable (onthly GDP tie series) as an unobserved variable (onthly tie series of IPI) is nonstationary so that there can exist a cointegrating relationship between observable variables and unobserved onthly variables. 14) Monthly inforation variables in this paper, i.e., IPI and WRSI, are actually used in the process of estiating variables related to national incoe such as GDP. Therefore this assuption is reasonable. 15) In X 2,t1:, X 2,t2:, denotes specific quarter, and t i represents specific onth in the quarter. For exaple, X 2,1: 3 represents the first onth in the third quarter, that is, July in the third quarter, and X 2,3: 2 denotes the third onth in the second quarter, i.e., June in the second quarter.
112Econoic Papers Vol.11 No.1 Second, although it is soewhat arbitrary, we can consider a ethod in which the initial value of the quarterly variable divided by three can be used for the initial value of the onthly state variable. 17) Third, we can use a cointegrating vector. In this case, since the nuber of cointegrating vectors (r = 1) is saller (e.g., rn) than the nuber of variables (n = 1), level variables should be restored using only the r vector, X t-5. Thus, in the case of n = 2, r = 1 the only one (n 1), an additional equation is needed. In the state vector of equation (21), t = (X t, Xt-1,, Xt-4, ( X t-5)) (21) there is the only equation as a level variable: X t-5 = 1 X 1, t-5 + 2 X 2, t-5 (22) where X 1, t-5 and X 2, t-5 are onthly state variables. Hence, the unobserved onthly level state variable X 2 can be recovered as follows: 18) X 2, t-5 = (X t-5 1 X 2 1, t-5) = (X t-5 1 Y 1, t-5) 2 (23) Equation (23) uses X 1, t-5 = Y 1, t-5. Therefore, the onthly level state variable, X 2t can be recovered using the estiates of differenced state vectors such as X t, Xt-1,, X t-4. 16) Y2, q 3X 2,t1: X 2,t2: 2X 2,t1: or siply Y q 3X t1 X t2 2X t1. t1 denotes the first onth of the specific quarter, t 2 the second onth of the specific quarter, and t 3 the third onth of the specific quarter. Hence the following equation is established: Y q 3X 1 X 2 2X 1 3X 1 (X 2 X 3 ) 2(X 1 X 2 ) X 1 X 2 X 3 17) If X 1 X 2 X 3, 3X 1 3X 2 3X 3 X q is established since X 1 X 2 X 3 X q. Under such an assuption, the initial value is set up as X 1 X q 3 18) Since X 2, t-5, although it is a onthly level state variable (onthly GDP in this paper), is an unobserved variable, it can not be used for recovering a level variable. In contrast, X 1, t-5 can be used since there exists an observable actual value Y 1t
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model113 2. Cobined state variable and bencharking of state variable In the process of estiating a onthly state variable (here, onthly GDP) by the previous ethod, diverse variables can be included in onthly inforation data that can be utilized in disaggregating quarterly data. Thus, onthly state vector, i.e., onthly GDP can be also proposed with not one but any other variables. In this case, the onthly GDP estiate, being derived using only one specific onthly inforation variable, is liited in reflecting only the inforation held by the individual variable being used in the estiation. Hence, it is thought desirable to ultiately suggest a cobined estiate of total onthly GDP by putting together the inforation of any other individual onthly GDPs estiated on the basis of different inforation. A. Cobined state vector In general, a specific forecasting odel utilizes its own inforation. Fro this perspective, conducting a forecast by eploying a specific odel ends up with using biased inforation. Thus, it ay be necessary to attept to find a ethod to derive a new forecast through cobining various inforation. A cobined forecast ethod can be taken as a ethod that can be usefully eployed in cobining coputed forecasts utilizing heterogeneous inforation. This paper takes into account the cobined forecast ethod as a ethod of cobining the estiated ultiple individual onthly GDP figures. B. Bencharking of Monthly GDP Even though a cobined onthly GDP estiate X C 2t is derived by cobining individual onthly GDP estiates, the su of the three onthly GDP ay not necessarily be consistent with Y2t, q actual quarterly GDPs. This is because the constraint equation where the su of three onthly GDP as a onthly state variable is set up to be consistent with quarterly GDP ay not function as an identical equation due to the existence of error. Baseline data actually used in the process of estiating quarterly GDP, as discussed earlier, has disparity in coverage and target fro baseline data utilized in the estiation of annual GDP (Lee (2006), Ki (2004), Jeon (2001), etc). As a result, the su of quarterly GDP figures is not perfectly consistent with annual GDP. The disparity between the su of quarterly GDP figures and annual GDP can be interpreted as easureent error. In a siilar way, this logic can be applied between onthly GDP and quarterly GDP. This research represents the su of the three onthly
114Econoic Papers Vol.11 No.1 state variables as the su of the quarterly state vector and an error ter. Therefore, the su of onthly variables ay not perfectly correspond with the quarterly value. However, it is not desirable to have a discrepancy between the estiated quarterly and actual quarterly values. Hence, this research calculates the bencharked estiate of onthly GDP by aking the su of cobined estiates of onthly GDP exactly correspond with quarterly GDP, using a bencharking technique. This research eploys a proportional Denton ethod for calibrating the difference between the three-onth su of cobined estiates of onthly GDP and actual quarterly GDP. The proportional Denton ethod is a ethod that places constraints so that the quarterly su of onthly estiates corresponds with the quarterly estiated value, and then adjusts tie series to ake the quarterly benchark to indicator(bi) ratio 19) becoe the weighted average of the onthly BI ratio. In other words, under the constraint condition (equation (27)) that the su of high frequency data (e.g., onthly data) corresponds with low frequency data (e.g., quarterly data), the proportional Denton ethod derives a solution that iniizes the objective function (equation (26)), i.e., value of weighting differences between onthly data(x C ) before calibration and onthly data(x A ) after calibration. At this tie, the calibrated estiate of onthly GDP is derived as the solution of an optiization proble as follows: Min( X A X C )W( X A X C ) (26) s.t. CX A = X q (27) where X A and X C denoting a onthly variable vector with (3T1) diension, C representing (T3T) atrix, and X q standing for a quarterly variable vector with (T1) diension are defined, respectively as follows: X A =(X 1:1, X 2:1, X 3:1, X 1:2, X 2:2, X 3:2,, X 1:T, X 2:T, X 3:T) X q =(X 1, X 2,, X T) 19) Quarterly BI rate = Y q 2, 3 X 2,t j, j=1 Here, Y q 2,r and X 2,t j, r denote the quarterly estiate of quarter and onthly estiate of the jth onth in quarter, respectively.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model115 W is a atrix of (3T3T) diension. At this point, when the onthly variable 1 is a flow variable, C =, when it is an index, C = 3. The solution for the above optiization proble is as follows: X A * = X C W 1 C(CW 1 C) 1 ( X q CX C ) (28) where W = M 1 DDM, D =, M =, and D and M atrix have (3T3T) diension. The su of X A* tj : (tj = 1, 2, 3 ; = 1, 2,, T ) coputed in equation (28) corresponds with X (= q 1, 2,, T ) and thereby Y (X q A * 1: + X A * 2: + X 3:* A ) is established. That is, in this case, the quarterly value is consistent with the value of the su of the three sequential onthly values within a quarter. IV. Estiation Result 1. Model estiation result This chapter derives onthly GDP estiates using the cointegrating relationship with different data frequency exained earlier, and then yields quarterly GDP estiates through the three-onth su of the derived onthly GDP estiates. In order to do so, it is essential to select onthly inforation variables that have a strong link with GDP as well as sufficiently reflecting the oveents of GDP. This is because the credibility of estiated onthly GDP depends on which onthly inforation variable is used in the estiation since, even though arbitrary onthly data are used, onthly GDP estiates could be derived. With regard to this, Chow and Lin (1971) suggested that a reference series should be selected so that the estiated onthly GDP can fully reflect the reality. Considering this, this paper utilizes onthly inforation variables such as Industrial Production Index(all industries) and Wholesale and Retail Sales Index, which are used as the ajor references in the process of drawing up actual GDP. The reason for selecting IPI is to take into account the fact that the oveent of
116Econoic Papers Vol.11 No.1 IPI is substantially siilar to that of GDP. The correlation coefficient between the levels of the Industrial Production Index and GDP during the period of January, 1980 to Deceber, 2005 turns out to be 0.98, and the correlation coefficient between growth rates as 0.70, suggesting that the oveent of the two variables is closely connected. Looking at the correlation between quarterly GDP and quarterly IPI, it sees reasonable to utilize the all-industry industrial production index as the onthly reference variable for estiating onthly GDP. Using only IPI as an inforation variable, however, has liitations. In other words, the IPI has the liitation of not including the service industry, which accounts for the largest share in GDP. Exaining the shares of Korean GDP by econoic activity (gross value added), the service industry as of 2005 ade up 52.4% of total GDP, uch higher than agriculture and fishing, anufacturing or construction, which accounted for 3.9%, 32.4%, and 8.0%, respectively. There is a substantial need to ake active use of inforation on the activity of the service industry in estiating onthly GDP. Taking this into account, the paper additionally using the Wholesale and Retail Sales Index (WRSI) represents the level of the service industry's activity for a onthly inforation variable in conjunction with IPI. Looking at the correlation between WRSI and GDP during the period of Jan. 1980 to Dec. 2005, the correlation coefficient between level variables of the two variables turns out to be 0.99, with the correlation coefficient between the growth rates of the two variables being 0.79, deonstrating that the two variables ove in an intiate relationship. 20) Consequently, this research estiates onthly GDP by selecting two variables that have a close relationship with GDP oveent and include heterogeneous inforation, all-industry IPI and WRSI, and then by using each index separately. For integrating the heterogeneous inforation the two indexes hold, we derive onthly GDP by cobining the separately estiated onthly GDP figures. In regard to this, when two indexes are used together in one odel, an advantage arises in that it is not necessary to go to the inconvenience of cobining two individual estiates. Since the odel suggested in this research is based on the error correction odel and therefore can utilize two indexes at the sae tie, this ethod deserves substantial consideration. When variables are added to the VECM, however, the nuber of paraeters to be estiated increases drastically as the nuber of variable rises. In addition to this, in this case, we need to consider ultiple cointegrating vectors. Considering this, this paper does not attept to use two indexes in the onthly GDP estiation. 21) The onthly inforation variables being used are seasonally adjusted IPI (all
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model117 industry) and WRSI, and the nuber of observation coes to 312 fro Jan. 1980 to Dec. 2005. As for quarterly variables, seasonally adjusted GDP is used and the nuber of observations of GDP is 104 fro the first quarter of 1980 to the fourth quarter of 2005. The core precondition for this paper is that GDP is I(1) variable. Concerning this, we cannot exclude the possibility that Korean GDP is a I(0) variable having a structural change in its trend function before and after the currency crisis in 1998. We test that GDP is I(0) variable with a structural change in a trend function adopting Perron (1989). For the test, the following odels are estiated: Model A : y t = 1 + t + ( 2 1)DUt + e t Model B : y t = 1 + 1t + (2 1)DUt * + e t Model C : y t = 1 + 1t + ( 2 1)DUt + (2 1)DUt + e t Fro the above odels, residuals are derived as ~ y A t, ~ y B t, ~ y C t, respectively, where DUt = 1, if t T 1998, DTt * = t T 1998, if t T 1998 and DTt = t, if t T { 1998 0 other { 0 other { 0 other Using ~ y A t, ~ y B t, ~ y C t, T( ~ i 1) (i = A,B,C; t = 1,2,,T) is calculated fro the following equation: ~ y i t = ~ i ~ y i t1 + j=1 k cj ~ ytj ~ i ~ + e t, (i = A,B,C; t = 1,2,,T) 20) Since the WRS industry is only a part of the service industry, WSRI has liitations in representing the level of the overall service industry's activity. Thus, it is desirable to utilize a service activity index that directly indicates the activities of the overall service industry rather than WRSI. Meanwhile, the service activity index has been drawn up by the National Statistical Office only since 1999 and therefore has a proble in not having sufficient observations at this point. There is no proble to estiate using this index fro 1999 since onthly data are used. Given that the purpose of this research is the estiation of onthly GDP, however, it is thought to be necessary to extend the period, if possible. Taking this into account, this paper utilizes WRSI instead of service activity index for inforation variable of service industry. 21) With regard to this, when the industrial production index(wholesale and retail sales index) is used as an endogenous inforation variable, we can think of using a ethod in which WRSI(IPI) is regarded as a strongly exogenous variable and is utilized as such. This case has a liit that it is difficult to eploy the cobined forecast ethod whose purpose is cobining the individual estiates with different inforation. In this case, that is, each estiate already includes inforation of the inforation variables that it is attepted to be cobined.
118Econoic Papers Vol.11 No.1 Perron (1989) Test Result i According to the above test result, GDP cannot be regarded as a I(0) variable accopanying the structural change in the trend function as of 1998. This result verifies that this research satisfies the preise that GDP is a I(1) nonstationary variable. Table 1 Estiation Results of State-Space Error Correction Model (a) Industrial Production Index (b) Wholesale and Retail Sales Index The estiation results of the state-space odel is presented at (a) and (b) in Table 1 in which (a) uses the industrial production index as onthly inforation data, and (b) utilizes the wholesale and retail sales index as onthly inforation data. A considerable nuber of coefficients turn out to have very high t-values. As for D.W. statistics, in the case of the odel using the industrial production index, onthly IPI is 1.96 and quarterly GDP is 1.83. In the case of odel using WRSI, onthly WRSI is 1.95 and quarterly GDP is 1.99. Given that the likelihood of structural change in tie series is high in Korea, which underwent a currency crisis, a duy variable representing the currency
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model119 crisis is added to the odel for an explanatory variable. But the estiation result is less significant than was thought, and therefore the duy variable is excluded in the final estiation result. 2. Estiated onthly GDP This chapter exaines the onthly GDP estiate. We gain quarterly estiates fro the derived onthly state variables such as the industrial production index(ipi), wholesale and retail sales index(wrsi) and GDP, and investigate to what extent they approxiate to the estiations. And we copare the actual values with onthly estiates such as IPI and WRSI. According to the RMSPE 22) of estiates calculated by Kalan Soother over all period, RMSPEs of quarterly IPI (growth rate fro the previous quarter) and WRSI (growth rate fro the previous quarter) turn out to be 0.08%(0.11%p) and 0.47%(0.64%p), respectively. Figure 1 and Figure 2 show the actual and estiated values of quarterly IPI and WRSI, respectively. Figure 1 Coparison between Actual Values and Estiates of Quarterly Index: Industrial Production Index (1980.2/4~2005.4/4, trillion KRW) 22) RMSPE and RMSE are used to easure the degree of average difference between an actual value(oi) and the estiate (Fi), and are defined respectively as follows: T T 1 RMSE = 1 2 (Fi Oi) 2 Fi, RMSPE = Oi T i =1 T ( ) i =1 Oi RMSE, RMSPE. We use RMSE together with RMSPE because RMSPE can becoe excessively large as the growth rate variable gets close to zero.
120Econoic Papers Vol.11 No.1 Figure 2 Coparison between Actual Value and Estiate of Quarterly Index: Wholesale and Retail Sales Index (1980.2/4~2005.4/4, trillion KRW) Table 2 Correlation between Monthly GDP and Monthly Individual Indices Cross correlation coefficient between onthly GDP cyclical coponent and onthly cyclical coponent of coincident index. The onthly estiates and actual values are copared. RMSPE(RMSE)s on onthly IPI (growth rate) and onthly WRSI (growth rate) turn out as 0.04%(0.07%p) and 0.13%(0.29%p), respectively. We analyze procyclicality and correlation between the estiated onthly GDP and onthly IPI. The correlation between estiated onthly GDP and, IPI or WRSI turns out as 0.59 and 0.69, respectively. And individual onthly inforation variable and onthly GDP estiate are procyclical. We investigate whether onthly GDP can sufficiently restore the oveent of actual quarterly GDP. Two kinds of quarterly GDP estiates are derived fro two individual estiates obtained by using IPI and WRSI, respectively. Two individual onthly GDP estiates and the cobined estiate of the two individual estiates are copared with the actual value. In Figure 3 and Figure 4, we copare the actual value with the recovered
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model121 quarterly GDP estiate, i.e., the su of three-onth individual onthly GDP estiates derived by using individual indexes such as IPI and WRSI. The result of coparing the restored quarterly GDP and actual quarterly GDP shows that the RMSPE of the restored quarterly GDP derived by using IPI turns out as 0.23%, and the RMSE of the growth rate as 0.06%. Although we derive MAPE(ean absolute percentage error) and Theil's inequality coefficient (or Theil-U) for evaluating forecasting power, the result also indicates that there exists forecastability within the saple period. Figure 3 Estiated Quarterly GDP: Industrial Production Index (1980.2/4~2005.4/4)
122Econoic Papers Vol.11 No.1 Figure 4 presents restored quarterly GDP derived by using WRSI as a onthly inforation variable. While the RMSPE of restored quarterly GDP estiate and the RMSE of the growth rate estiate using WRSI turn out to be 1.17% and 0.04%, respectively, indicating that the level variable is high and growth rate is low relative to IPI. The result that the quarterly GDP converted fro onthly GDP estiate is coparable with actual GDP deonstrates that all restored GDP figures acquired by using IPI and WRSI show alost zero discrepancy. Figure 4 Restored Quarterly GDP: Wholesale and Retail Sales Index (1980.2/4~2005.4/4)
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model123 Table 3 In Saple, Forecasting Power of Restored Quarterly GDP It is necessary to consider to a certain extent the liitation that individual onthly GDP estiated by using IPI or WRSI alone as a onthly inforation variable does not reflect the overall oveents of the econoy. Taking this into account, we derive overall onthly GDP that can represent the activities of the overall econoy by cobining two individual onthly GDP estiates. In order to derive overall onthly GDP, we use a cobined forecast ethod. Figure 5 Total Quarterly GDP Estiate: Bates Method (1980.2/4~2005.4/4)
124Econoic Papers Vol.11 No.1 According to the result of using the Bates ethod, RMSPEs for level variable and for growth rate turn out to be 0.0003% and 0.0033%p, respectively during the entire saple period. 23) By contrast, the result of using the Raanathan ethod presents RMSPEs for level variable and for growth rate of 5.53% and 5.93%, respectively, during the entire saple period, showing a poorer outcoe relative to the Bates ethod. 24) Hence, this research cobines individual onthly GDP estiates to arrive at the overall onthly GDP using the Bates ethod. Three-onth sus of the overall onthly GDPs are copared with actual quarterly values. In the saple period, the overall onthly GDP cobined by the Bates ethod is evaluated to be greatly superior to individual onthly GDP.
Estiation of Korean Monthly GDP with Mixed-Frequency Data using an Unobserved Coponent Error Correction Model125 Table 4 In Saple, Forecasting Power of Restored Quarterly GDP Meanwhile, there exists a inute difference between the three-onth su of the overall onthly GDP and actual GDP and therefore they do not perfectly correspond. Thus, the final bencharked onthly GDP estiate is derived by bencharking the cobined onthly GDP estiate. The final bencharked onthly GDP is presented in Figure 6. Figure 6 Final Bencharked Monthly GDP Estiate (April. 1980~Dec. 2005, trillion KRW) Figure 7 shows the onthly GDP derived using the above onthly GDP, and the Denton and Chow-Lin ethods in Lee (2006). According to Figure 7, there is no big difference between the above final onthly GDP and the onthly GDP 23) K * turns out to be 0.391, and 0.609. 24) K * turns out to be 0.185, and 1.005.
126Econoic Papers Vol.11 No.1 derived using these three ethods. In other words, the onthly GDP derived in this paper shows a less than 1% difference with onthly GDP derived fro other ethods. According to the estiation results in this paper, there is no big difference in the trend of onthly GDP aong the results estiated by the ethod suggested in this paper, the Denton or the Chow-Lin ethod. The fact that the onthly GDPs estiated by each ethod show a difference within a 1% level is not judged as soething that can be siply passed over in consideration of the growth rate of onthly GDP. It is rather interpreted to reflect the estiation ethod's distinctive characteristics. Figure 7 Coparison of Monthly GDP Estiate Derived by Other Estiation Methods (April. 1980~Dec. 2005, trillion KRW) As seen in the estiation results of onthly GDP, quarterly GDP estiates restored fro onthly GDP estiates are alost copletely restored to actual quarterly GDP. In order to exaine the utility of onthly GDP estiates, we investigate whether onthly GDP estiates sufficiently reflect econoic oveents. For investigating whether or not the business cycle coincides with estiated onthly GDP, we conduct correlation analysis and lag correlation analysis between the cyclical coponents of onthly GDP and the cyclical coponents of the coincident coposite index(cci). During the period of June, 1980 to Dec, 2005, the siple correlation