تمذ ش ان اتح ان حه اإلخ ان انفهسط تاستخذاو ارج اال حذاس نهث ا اخ راخ انتشدداخ ان ختهطح

Transcription

1 تمذ ش ان اتح ان حه اإلخ ان انفهسط تاستخذاو ارج اال حذاس نهث ا اخ راخ انتشدداخ ان ختهطح Forecasting Palestinian Gross Doestic Product Using Mixed Data Sapling Regression Techniques Student Nae and Nuber: Mustafa M. Al-Qawasi / Discussion Date : June, 7, 2014 Coittee: Dr Tariq Sadiq Supervisor Dr Abd Al Hakee Eideh Dr Wael Al-Hashlaoon This Thesis was Subitted in Partial Fulfillent of the Requireents for the Master s Degree in Applied Statistics fro the Faculty of Graduate Studies at Birzeit University. Palestine

2 II تمذ ش ان اتح ان حه اإلخ ان انفهسط تاستخذاو ارج اال حذاس نهث ا اخ راخ انتشدداخ ان ختهطح Forecasting Palestinian Gross Doestic Product Using Mixed Data Sapling Regression Techniques Student Nae and Nuber: Mustafa M. Al-Qawasi / Discussion Date : June, 7, 2014 Coittee: Nae Dr Abd Al Hakee Eideh Dr Tariq Sadiq Dr Wael Al-Hashlaoon Signature

3 III Dedication To those who have supported e at all the tie.. To y other, To y Father, To y wife, To y son and daughter, To y Brothers and Sisters, To y Faily, To y colleagues. With all y love and respect

4 IV Acknowledgents I would like to express y deepest gratitude to y supervisor, Dr. Tariq Sadiq who suggested the subject of y thesis and who ade great effort to guide e throughout y studies. This project would not have been possible without his continuous guidance and support. I would like to express y deep appreciation and gratitude to both Dr. Abd Al Hakee Eideh and Dr. Wael Al-Hashlaoon for their contributions and encourageent. I gratefully acknowledgent the help of Mr. Eric Ghysels who have encouraged e to study ixed data sapling regressions and supported e his data and working files. I would like to thank and express y appreciation to y colleagues in work and study. I would like to express y appreciation to all doctors taught e during y study. Last but not least, y deep thanks go to y faily (specially y wife) for their endless encourageent and support and for everyone helped e or encouraged e.

5 V Table of Contents Table of Contents... V Abbreviations... VII Abstract... VIII...IX ملخص Chapter 1 : Introduction Background Literature Reviews Proble And Methodology... 5 Chapter 2 : MIDAS Regressions The Forecasting Environent And Notations Data For Tie Aggregation Tie-Averaging Step Weighting(U-MIDAS) MIDAS Regressions Siple R-MIDAS Regression Polynoial Specifications Exponential Alon Lag Polynoial Beta Lag function Alon Lag Polynoial of order (P) Lag Length Selection Criteria Testing Unit Root and Cointegration Unit Root Cointegration... 21

6 VI 2.7. Johansen s procedure Cointegration-MIDAS and ECM-MIDAS odels Distributed-Lag Models Estiation Methods The Method of Least squares The Method of Maxiu Likelihood Generalized Method Of Moents Estiation The Method of Nonlinear Least squares(gauss-newton ethod) Forecasting... 8 Chapter : Application Case Study Modeling The Palestinian GDP Discussion of Results and Conclusions Appendix A : Output Tables Appendix B : Tie Series Plots Appendix C : Testing The Flat Aggregation Schee(AGK test) Appendix D : The R-Codes Used In Analysis References Index... 70

7 VII Abbreviations MIDAS: Mixed Data Sapling. U-MIDAS: Unrestricted Mixed Data Sapling. R-MIDAS: Restricted: Mixed Data Sapling. ADL: Autoregressive Distributed Lag. ARIMA: Autoregressive Integrated Moving Averages. GDP: Gross Doestic Product. CIM: Cost of Iports. CPI: Consuer Price Index. EII: Eployent In Israel. EALP: Exponential Alon Lag Polynoial. GMM: General Method of Moents. NLS: Non-linear Least Squares. MLE: Maxiu-Likelihood Estiator. MSE: Mean Square Error. AIC: Aikaike s Inforation Criterion. BIC: Bayesian Inforation Criterion. ADF: Augented Dickey-Fuller. ECM: Error Correction Model. VECM: Vector Error Correction Models. OLS: Ordinary Least Squares. VAR: vector Autoregression. QMLE: Quasi Maxiu Likelihood Estiator. PMA: Palestine Monetary Authority. EIS: Eployent rate in Israel and Settleents. ARDL: Auto Regressive Distributed Lag. PCBS: Palestinian Central Bureau of Statistics.

8 VIII Abstract In the case of using classical linear regression odels for tie series, researchers usually deal with equal frequencies for all the variables. They cannot directly apply such odels to a ixed-frequency dataset. The Mixed Data Sapling (MIDAS) regression odels deal with this type of data; typically the econoic indicators fro those observed daily, onthly, quarterly to those yearly. In this study, we introduce MIDAS regression approach which is relatively assued as a new area. We will explain its ability of dealing with ixed frequency data, and its efficiency of iproving paraeters estiation and forecasting perforance in the presence of extree observations. To the best of author s knowledge, this is the first research that exaines the relationship between the real GDP in Palestine and other indicators using MIDAS regressions. The classical teporal aggregation ethod is copared to the two types of MIDAS regressions; the restricted and the unrestricted, to build both long-run and short-run relationships. The study results exhibited that both U- MIDAS and R-MIDAS were better than the classical Tie-Averaging ethod in reducing forecasting errors. The results exhibited that the quarterly Palestinian GDP in the long-run affected by its cost of iports in the second onth coputed in each quarter, also, the quarterly GDP has increasing general trend and affected by its first lag and the quarterly Eployent Rate In Israel and Settleents, and all these variables have significant positive relationship with GDP. In the short-run, the results showed that the quarterly Palestinian GDP is affecting by the second-onth and the third-onth of the cost of iports of Palestine.

9 IX ملخص ف حانح استخذاو ارج اال حذاس انخط انكالس ك ف دساسح انعاللاخ ت انسالسم انزي ح فئ انثاحث عادج يا تعايه يع تشدداخ يتسا ح ند ع ان تغ شاخ, ف ح أ ال ك تطث ك يثم ز ان ارج يثاششج ف حانح خ د يد عح ت ا اخ يتغ شاخ تشددات ا يختهطح غ ش يتسا ح. اعت ذخ ز انشسانح عه دساسح ع خاص ي ارج اال حذاس انت ك ا أ تتعايم يع يثم زا ان ع ي انث ا اخ انسالسم انزي ح راخ انتشدداخ ان ختهطح يثم ان ؤششاخ االلتصاد ح انت تى يعا ح تعع ا ي ا انثعط ا خش تى يعا ت ا ش ش ا أ ستع ا أ س ا. تعشض ز انذساسح ارج ا حذاس انث ا اخ راخ انتكشاساخ ان ختهطح ) ارج ي ذاس( انت تعتثش ي ان اظ ع انحذ ثح سث ا. ي خالل ز انذساسح س ف تى ششذ لذسج ز ان ارج عه انتعايم يع انث ا اخ راخ انتشدداخ ان ختهطح, ح ث س تى ششذ خصائص ا كفاءت ا ف تحس انتمذ ش انت ثؤ نه تغ شاخ االلتصاد ح, ك ا س تى يماس ح ز ان ارج ت ع ا )ان م ذ غ ش ان م ذ( يع أسه ب انتد ع انزي انكالس ك حسة األ ساغ رنك ي خالل ت اء ان ارج انت ت ثم انعاللاخ ت ان تغ شاخ االلتصاد ح عه ان ذ انط م انعاللاخ عه ان ذ انمص ش, ح ث أظ شخ تائح انذساسح تف ق ارج ي ذاس غ ش ان م ذج ان م ذج ف تمه م أخطاء انت ثؤ عه ارج انتد ع انزي انكالس ك حسة األ ساغ. حسة يعشفح ان ؤنف فئ زا انثحث األ ل انز ذسس انعاللح ت ان اتح ان حه اإلخ ان انحم م ف فهسط غ ش ا ي ان ؤششاخ االلتصاد ح تاستخذاو ارج ا حذاس انث ا اخ راخ انتكشاساخ ان ختهطح ي ذاس, ح ث أظ شخ تائح انذساسح خ د عاللح دانح إحصائ ا عه ان ذ انط م ت ان اتح ان حه اإلخ ان انشتع ف فهسط كم ي تكهفح ان اسداخ ان حس تح ف ثا ش ش ي كم ستع, يعذل انع انح ف إسشائ م ان ست ط اخ, ك ا أظ شخ ان تائح خ د دانح يتزا ذج نه اتح ان حه انشتع ف فهسط عه ان ذ انط م تأثش رات ا تم انساتمح أ ان تأخشج. ي اح ح أخش أظ شخ تائح انذساسح خ د عاللح دانح إحصائ ا عه ان ذ انمص ش ت ان اتح ان حه اإلخ ان انشتع ف فهسط تكهفح ان اسداخ ان حس تح ف أ ل ثا ش ش ي كم ستع.

10 1 Chapter 1 : Introduction 1.1 Background The analysis of tie series data is a pivotal issue. The ost of tie series are econoic or financial variables that affect and are affected by each other. Studying such variables is with a great deal of iportance. Both econoic and financial variables give us inforation about the econoic level that was reached by any country. Studying such variables help to investigate the effects caused by soe indicators on the econoic activity in general, then to assist the decision aking process. In order to study the relationships between these variables, the ost efficient widely used ethod is to build regression odels. Regression odels of tie series variables can be univariate or ultivariate odels. The univariate odels investigate the relationship between the variable and itself; in other words, with its tie delays(lags) like Autoregressive Integrated Moving Averages(ARIMA) odels. Fro the other hand, the ultivariate odels investigate the relationship between dependent (response) variable and independent variables(indicators or predictors). Multivariate regression odels ay investigate the effect of response lags with other predictors on the response, so in this case the odel is called autoregressive odel. In building odel stage, it is supposed that data(dependents or independents) are frequency-unified or have fixed frequency; this eans that each observation of the dependent variable is atched with an observation of the independent variable/s; in other words, each observation of the dependent corresponding to an observation of the independent/s. In this case, it is siply possible to build regression odels to investigate the relationship between these variables. Let's suppose that the gross doestic product(gdp) and the cost of iports(cim) in Palestine are coputed onthly by the Palestinian Central Bureau of Statistics; in this case each observation of GDP corresponds to an observation of CIM; the GDP in January is atched with the CIM in January; also, the GDP in February is atched with the CIM in February, and so on for the rest of onths and years. So we can easily construct regression odel to investigate the relationship between the GDP as dependent (response) and the CIM as independent(predictor). In fact, ost of research and statistics centers in the world cannot provide onthly inforation about GDP, but they can for CIM. They can just provide quarterly inforation about GDP, so in this case we have econoic variables with different frequencies(ixed frequencies). The GDP has quarterly frequency

11 2 observations, but the CIM has onthly frequency observations. It is clear that the frequency of the CIM observations is larger than the frequency of the GDP observations. We call the GDP as low-frequency variable and the CIM as highfrequency variable. We can iagine such ixed frequency data(ixed-data-sapled) as a ixture of observed and issing data; each onthly observation of CIM does not have corresponding observation of GDP since the GDP is quarterly sapled. So the for of data can be iagined by thinking of availability of CIM observations as three ties per quarter. In the first quarter of any year, the observation of CIM in January has no corresponding observation of GDP in January; also the observation of CIM in February has no corresponding observation of GDP in February, but the observation of CIM in March has corresponding observation of GDP in March. So we have two issing observations of GDP in both January and February but not in March, whereas there are no issing observations of CIM in the three onths. In the sae way we have issing observations of GDP in both April and May but not in June, whereas there are no issing observations of CIM in the three onths in the second quarter, and so on for the rest of onths and quarters of the year. Soeties, econoic and financial data and variables are available only on an annual basis. For exaple if we assue that the GDP is sapled annually in the end of each year, and the CIM is sapled onthly; so in this case, we have eleven issing observations of GDP fro January to Noveber; while the CIM observations are available copletely for the twelve onths in that year. So the question that arises here is how can we investigate the relationship between variables that are sapled with ixed frequencies like the GDP as dependent and the CIM as independent?. The availability of data sapled at different frequencies always presents a proble for researchers working with tie series data. If one deleted all cases containing the issing observations, then he would lose uch iportant inforation, because variables that are available at high frequency contain potentially valuable inforation[1,4]. The siplest idea that coes into ind after that is to aggregate the high-frequency observations by taking the average of the, and then, the highfrequency variable becoes as the low-frequency variable. For instance, taking the average of the three onthly observations of the CIM in each quarter to get one averaged observation atched with the quarterly observation of the GDP; but it is clear, by this way that we assign the sae iportance level for the partitions of the high-frequency variable. Therefore, we lose iportant onthly inforation, and the results of the analysis will be less accurate[18].

12 The previous explanations can be suarized by aggregating the highfrequency observations like the onthly observations of the CIM series, and converting it into a quarterly observations to be atched with the quarterly GDP data. Instead of this ethod, the coon solution in such cases is to estiate issing observations of the low-frequency variable like the quarterly GDP. This can be done by using the Kalan filter process, and these subjects are outside the fraework of this thesis[4,8,10,18]. Nowadays, Soe scientists have proposed ethods which can deal with variables sapled at different frequencies such as onthly and quarterly data. Such ethods have any advantages and properties coparing with old and coon ethods had known. We call these ethods Mixed Data Sapling(MIDAS) Regressions. 1.2 Literature Reviews We Always find statisticians interested in getting good properties and iproveent on estiations that can be yielded by traditional high level ethods such as ARIMA and regression odels that deal with tie series data. Ghysels, Santa-Clara and Valkanov in 2004 have recently proposed regression odels that directly accoodate variables that are sapled at different frequencies. These regression odels cobining for exaple onthly and quarterly data together, or quarterly with annually data. Their Mixed Data Sapling or MIDAS regressions represent siple, parsionious, and flexible class of tie series odels. They showed that their MIDAS regressions will always lead to ore efficient(less errorvariance) estiation than the classical approach that aggregate all series to be in the sae frequencies. Ghysels, Santa-Clara and Valkanov[1] used any functional fors dealing with few nuber of paraeters in order to estiate MIDAS odels. They exained the asyptotic properties of MIDAS regression estiation, and they discussed challenging econoetric issues. Cleents and Galvao[5, 22] investigated whether the MIDAS approach of Ghysels can be successfully adapted to the short ter forecasting of output growth, given that it has been used for forecasting financial variables with daily observations. They extended the distributed lag MIDAS specification to include an autoregressive ter (the MIDAS-AR). Sinko[2] discussed two lag paraeterizations of MIDAS regressions; the Exponential Alon and Beta. He introduced several new MIDAS specifications that include ore general ixeddata structures of unequally spaced observations. Klaus[4] in his thesis copared the forecasting success of two new ixed-frequency tie series odels: the ixedfrequency VAR and the MIDAS approach. During his work, he outlined the data

13 4 generating processes (DGP) that have been used for odel coparisons. Klaus listed soe conclusions about odel specification of ixed-frequency tie series odels. Due to hi, forecast perforance, lag length selection and restrictions of the weighting functions in the MIDAS fraework are interrelated. He concluded that whether the weighting functions should be restricted or not, depends on the data structure. Andreou, Ghysels and Kourtellos[8] covered MIDAS regressions and the relationship with the Kalan filter. They discussed DL-MIDAS(Distributed Lag Mixed Data Sapling) odels with autoregressive high-frequency predictors. Aresto, Engeann and Owyang[18] copared between Tie- Aggregation techniques which include Tie-Averaging, Step Weighting and MIDAS regressions. They also offered soe forecasting issues such as the End-Of-Period forecasting and the Intra-Period forecasting. Miguel[20] introduced robust procedure that pre-cleans the data by exponential soothing before the MIDAS regression constructed. His results showed that in the presence of outliers, utilizing the robust version of exponential soothing to clean the containated dataset prior to use MIDAS reduces forecast errors consistently. Chen and Tsay[9] presented a generalized autoregressive distributed lag (GADL) odel in order to conduct regression estiations that involve ixed-frequency data. They offer coparisons between the relative perforance of the OLS-based GADL specifications and the NLS-MIDAS(Nonlinear Least Squares Mixed Data Sapling odels). Foroni, Marcellino and Schuacher[6] discussed the unrestricted MIDAS(U-MIDAS) odels based on a siple linear lag polynoial. They showed that U-MIDAS work better than MIDAS ethod if the differences in sapling frequencies are sall. Goetz, Hecq and Urbain[21] proposed in their paper, ixed frequency error correction odel in order to odel non-stationary variables that are possibly cointegrated. Foroni, Marcillino & Shuacher[] studied the perforance of MIDAS odels which does not depend on the functional distributed lag polynoials. They copared U-MIDAS odels with MIDAS odels, and they showed that U-MIDAS perfors better than MIDAS in the case of sall differences in sapling frequencies. Foroni & Marcillino[] showed by siulation experients and actual data, that the use of ixed-frequency data, cobined with a proper estiation approach, can alleviate the teporal aggregation bias, itigate the identification issues, and yield ore reliable policy conclusions. Foroni and Marcellino[7] clarified that MIDAS odels appear to be ore robust than bridge equation odels and the state-space approaches in iss-specification, and coputationally sipler.

14 5 On the epirical side, Shi[17] studied the effect of lag nubers on MIDAS regressions. He concluded that there exists seasonal effect of the lag nubers in different horizons. He noted that the out-of -saple and in-saple results of realized volatility(rv) and realized absolute volatility(rav) are quite siilar, but in soeties, out-of saple perfors better. Shi stated several findings fro predictability of daily to onthly realized volatility of Chinese arket. Asiakopoulos, Paredes and Waredinger[19] concentrated on forecasting annual budgetary executions and their subcoponents issues. They assessed the effect of intra-annual fiscal data on the annual outturn of disaggregated series. They eployed MIDAS approach to analyze ixed frequency fiscal data. They concluded that tiely and good quality data are powerful tools to iprove forecasts. 1. Proble And Methodology In this thesis, we will study the proble of specifying the factors affect the Palestinian Gross Doestic Product(GDP) which is sapled quarterly, where soe of these probable factors are sapled onthly and the others are sapled quarterly, and we will forecast the future quarterly real GDP using these ixed sapled data using MIDAS regression approach without resorting to fix data frequencies or dealing with issing data. The MIDAS approach ay relate the observations of the low-frequency variable to the lagged observations of the high-frequency variable by distributed lag functions. The proper choice of the functional for, such as the exponential distributed lag polynoial, allows assigning few nuber of paraeters to large nuber of lags in order to obtain a parsionious odel which iplies to siply interpretation of results[18]. As an alternative to such paraeterization, we will discuss an unrestricted MIDAS(U-MIDAS), based on a siple linear lag polynoial. This specification has the advantage of a higher flexibility copared to the functional lag polynoials in the standard MIDAS approach[18]. However, U-MIDAS has disadvantage that it need to estiate a lot of paraeters if the lag order is large, this leads to difficulty of interpretations. On the other hand, U-MIDAS can be helpful when differences in sapling frequencies are sall[6]. For exaple, onthlyquarterly data; such as if quarterly GDP should be related to onthly cost of iports, and generally a sall nuber of lags is necessary to capture the dynaic relations. We will explain soe odels, and other ethods will be discussed in the study; such as the usage of MIDAS regressions in building Cointegrating relationships, and the equilibriu (error) correction odels, in order to study long-run and short-run

15 6 relationships between variables. We will iprove the discussion of all topics by applications and exaples supported by data tables, reports and suaries that can expand our understanding of the subject. In next sections, we discuss siple tie-averaging, step-weighting function(u- MIDAS), and the restricted MIDAS regressions(r-midas). We will copare between these ethods by odeling Palestinian GDP on soe iportant factors using the three ethods. We will use MIDAS regressions to forecast the future value of quarterly Palestinian GDP. The ain odel in this study is defined as: Q Q Q LGDP t = β0 + β 1 t + β 2 LGDP t 1 + β LEII t + Where: 1 () i=0 β i,5 LCIM i + u t t 1 βk,4 LCPI k + k=0 t LGDP t Q : the log of the Quarterly Gross Doestic Product as low-frequency Variable. t: tie as low-frequency Variable. Q LGDP t 1 : the log of the first lag of the Quarterly Gross Doestic Product as low-frequency Variable. LEII t Q : the log of the Quarterly Eployent in Israel and Settleents as lowfrequency Variable. LCPI k : the log of the Monthly Consuer Price Index as high-frequency t Variable. LCIM () i : the log of the Monthly Cost of Iports as high-frequency Variable. t u t : is the white Noise Process(independent identically distributed rando variables with zero ean and constant variance). Finally, we will use the sae frequencies of the high-frequency Variables that are onthly, we will not use weekly or daily high-frequency Variables.

16 7 Chapter 2 : MIDAS Regressions 2.1 The Forecasting Environent And Notations The proble of ixed sapling frequencies is exeplified in Figure (1) which shows quarterly real GDP and onthly cost of iports for the period 1999 to 2012 *. Fro the figure, we note that onthly cost of iports observations fluctuate between quarterly GDP observations(onthly observations are ore concentrated through quarterly observations into the sae tie period)[18]. When coparing across odeling environents, it is iportant to use coon notations. In the econoetric procedures that we will follow, our objective is to forecast a lower-frequency variable (y); sapled at periods denoted by tie index t. Past realizations of the lower-frequency variable are denoted by the lag operator, L. * The data used in plotting figure(1) were collected fro Palestinian Central Bureau of Statistics(PCBS).

17 8 For exaple, if y is the quarterly GDP, then the GDP one quarter prior would be the first lag of y t, Ly t = y t 1, two quarters prior would be L 2 y t = y t 2, and so on. In addition, we are interested in the inforation contents of the higher-frequency variable x which is sapled ties between the saple periods of y (between t 1 and t), so we will denote to x as x t. L k denotes the lag operator for the higherfrequency variable. If x t is the onthly cost of iports used to forecast quarterly GDP denoted by y t, then L k x t = xt k previous onth in the sae quarter[18]. Figure(2): Forecast Tie Line denotes the cost of iports of the k'th y t 2 y t 1 y t y t+1 x t 2 t-2 t-1 t t+1 x k t 1 x t 1 x k t x t x k t+1 x t+1 Figure (2) represents the forecast tieline, which for siplicity shows oneperiod-ahead forecasts. Assue that at tie t we are interested in forecasting y t+1 (the circled observation on the tieline). Standard forecasting experients would use data available up to tie t; this is depicted in the dashed section of the tieline. The dotted section of the tieline depicts inforation that becoes available during the t+1 period; which is called leads, these inforation ay be relevant for forecasting y t+1. Using MIDAS ethods, we can perfor intra-period forecasting experients using both data specified in the dashed section and that in the dotted section[18]. 2.2 Data For Before odeling high and low frequency data, it is iportant to know how is the for of these variables. Table (1) illustrates the ixed data schee by the help of the quarterly-onthly exaple. In the quarterly-onthly ixed data, we have (=), that is we have three observations of the high-frequency variable between the period t-1 and t. For two years, we have (n=24) onthly high-frequency observations of x (12 in each year), and we have (T=8) quarterly low-frequency observations of y (4 in each year), so we have n=*t=*8=24 known observations of x and only 8 known observations of y. If we assued that the first observation of x represents the value of

18 9 CIM(Cost of Iports) in January/1999 x 2, then the third observation of x, which 1 represents the value of CIM in March/1999 (x 1 ), will correspond to the first observation of y which represents the value of GDP in the first quarter y 1, and so on. For onthly-daily data, we have (=22) assuing fixed nuber of days per onth, and we can generalize the sae data schee or for for onthly-daily, annually-quarterly or any other for. The following table shows quarterly-onthly data structure. t Table (1): Quarterly-Monthly Data Structure for two years( ) Year/Quarter First onth x 2 t /(first quarter) x /(second quarter) x /(third quarter) x /(fourth quarter) x /(fifth quarter) x /(sixth quarter) x /(seventh quarter) x /(eighth quarter) x Tie Aggregation Second onth x 1 t x 1 1 x 1 2 x 1 x 1 4 x 1 5 x 1 6 x 1 7 x 1 8 Third onth x t x 1 = x1 x 2 = x 2 x = x x 4 = x4 x 5 = x5 x 6 = x6 x 7 = x7 x 8 = x8 In this section we will describe three ethods of Tie-Aggregation of the higher-frequency data that can be used in forecasting lower-frequency variables, that are: Tie-Averaging, Unrestricted MIDAS and Restricted MIDAS. y t y 1 y 2 y y 4 y 5 y 6 y 7 y 8

19 Tie-Averaging The first and the siplest ethod of Tie-Aggregation is Tie- Averaging[18]. The ethod of converting higher-frequency data to atch the observations of the lower-frequency data by coputing the siple average of the observations of the high-frequency variable x that occur between saples of the lower-frequency variable y such that: x t = 1 1 L k x t k=0 ; where L k x t = x k t ( ) Here, we are assign each lag of x by the sae weight or coefficient in each quarter which is (1/). For exaple by table (1) we have : x k t x 8 = 1 1 L k x t k=0 = 1 (x 8 + x8 1 + x8 2 ) Letting k=0 and for quarterly-onthly data(i.e. =), we have L k x t = =x t, which eans the last-onth observation of (x) in the quarter t. For k=1 we have L k x t = x 1 t, which eans the previous onth of the last-onth(i.e. the second onth) observation of (x) in the quarter t, and for k=2 we have L k x t = x 2 t, which eans the first-onth observation of (x) in the quarter t. The regression odel of y t on its own lags with x t and its lags(p lags for y and q lags of x ) becoes clearly identified as: y t = α + p i=1 β i L i y t q + γ j L j x t j =0 + u t (1) Where γ j are the coefficients of the tie-averaged x s, and u t is the white noise process (independent identically distributed rando variables with zero ean and constant variance). The odel in equation (1) called autoregressive distributed lag odel of order p and q (ADL(p,q))(see section 2.9)[8,51]. The ethod of tie-averaging is very siple and easy to conduct since it is depending on tie series regression approaches, but it is clear that this ethod ake a lot of potentially useful inforation discarded because of equal-weights assigning, thus rendering the relation between the variables will be less accurate[2,18]. Finally,

20 11 it is ore appropriate to take averages for stock variables, but for flow variables, the higher-frequency values are siply added[4] Step Weighting(U-MIDAS) The other ethod of tie aggregation is to give each lag of x different weight or coefficient in each quarter[18]. We call this ethod step-weighting or unrestricted ixed data sapling regression(u-midas). The autoregressive distributed lag U- MIDAS(ADL(p,)-UMIDAS) odel can be written as: y t = α + p i=1 β i L i y t + γ k L k x t k=0 + u t (2) Where u t is the white noise process as defined before. For the purposes of forecasting, the previous equation can be expanded for quarterly-onthly variables when = as: y t+1 = α + β 1 y t + β 2 y t β p y t p 1 + γ 1 x t + γ 2 x 1 t + γ x 2 t + u t Equation() represents forecasting equation for the period of t+1 of y. It is clear that this equation does not contain leads, but if it should, we can add lead ters, and those are: γ 1 x t+1 + γ2 x t 1 + γ x 2 t (). This eans that we have high-frequency observations now of the variable (x t ) in the period (t+1). For exaple, in our research, we analyzed quarterly-onthly data fro 1999 to 2012(56 quarterly observations of (y) and 168 onthly observations of (x)). We can forecast y at the first quarter in 201 directly by equation() without leads. But if we now in the third onth of the first quarter in 201; and the observations of x at January, February and March are available now; then we can insert the three lead ters in the equation to get benefits of leads inforation(see section 2.11 for ore details about forecasting)[6,18]. It is clear fro the equation () that despite step-weighting odel preserves the tiing inforation(because the high-frequency observations between the periods t and t-1 were taken into account and not just averaged), but it violates the parsiony principle. In the case of quarterly-onthly variables, it sees that there is no violation of the parsiony principle; because we need only three coefficients to estiate for the variable x. But for onthly-daily variables it is different; we will need 22 coefficients for the variable x alone, this is called a paraeter proliferation

21 12 proble. Also, Model () can be extended to ultiple lags, then the nuber of paraeters will becoe quite large[6,18]. 2.. MIDAS Regressions The odel which could solve the proble of paraeter proliferation while preserving soe tiing inforation is called MIDAS odel or the restricted ixed data sapling regression(r-midas)[18]. This odel was constructed by Ghysels, Santa-Clara, and Valkanov[1]. This odel can be written as: y t = α + p i=1 β i L i y t + γ B(k, θ) L k x t k=0 + u t (4) Where B(k, θ) is a polynoial function that deterines the weights for teporal aggregation, and the vector θ have a specific nuber of paraeters θ i, u t is white noise process[8,18]. Equation(4) represents an autoregressive distributed lag odel(adl-rmidas) because y t is regressed on its own lags with x t and its lags as entioned before. Returning to equation (2), we note that it is the sae as equation (4). Only the difference between the is that the weighting ters of the higher-frequency variable x t are {γ j } in equation (2) while they are {B(j, θ)} in equation (4). The iportant property of the weighting polynoial function B(j, θ) is that it contains liited nuber of paraeters(ghysels, Santa-Clara, and Valkanov suggested θ = θ 1, θ 2 for all j), while in equation (2), {γ j } are different paraeters. In equation (4), we need liited nuber of paraeters to estiate regardless of the nuber of lags of x t or the value of (), so the parsiony principle is satisfied here[8,18]. Now, for ore explanations, it is useful to introduce a siple R-MIDAS regression, and soe related notations. 2.4 Siple R-MIDAS Regression Suppose that the variable y t is available once between t 1 and t (say, onthly), another variable x t () is observed () ties in the sae period (say, daily ( or =22), and that we are interested in the dynaic relation between y t and x ) t. In other words, we want to project the left-hand side variable y t of the odel equation () on the history of lagged observations of x t j / [2,,4]. The latter ter denotes the higher-frequency variable and its tiing lags are expressed as fractions of the unit interval between (t-1) and (t). A siple R-MIDAS odel is :

22 1 y t = β 0 + β 1 B L 1 ; θ x t + u t (5) Where : B L 1 ; θ = K B k; θ L k k=0 ; K should be specified properly(see section 2.6 for lag specification criteria) and L k is the lag operator such that : L k x t = xt ( k Note that the suation started fro k=0 which guarantees x t to be in the odel, also x t values have corresponding yt values, but x t ( k issing y t values for k>0, u t is white noise process[2,,4]. ) ) values have corresponding the lag coefficients B k; θ of the corresponding lag operator L k are paraeterized as function of a sall-diensional vector of paraeters θ, usually the vector θ = θ 1, θ 2. To explain siple R-MIDAS regression odel (5), let us return to the quarterly-onthly variables. We have = and we assue that K=-1=2 *. By equation (5), our siple R-MIDAS regression will be: y t = β 0 + β 1 { K=2 k=0 B k; θ L k x t } + u t = β 0 + β 1 { Then, it can be expanded as the following: y t = β 0 + β 1 {B 0; θ x t + B 1; θ xt 1 K=2 k=0 + B 2; θ x 2 t B k; θ x t ( k ) } + u t } + u t (6) Fro equation (6), it is clear that the lower-frequency variable y t is regressed on each of the following: 1) The higher-frequency variable x t (x t ) whose values represent the third onth in each quarter; 2) The first lag of the higher-frequency variable x t (x 1 ) whose values represent the second onth in each quarter, and ) t The second lag of the higher-frequency variable x t (x 2 whose values represent the t first onth in each quarter. The ter y t and its lags can be added to the right hand side of this equation in order to forecast the future values(y t+1 ) in the left hand side[2,8].. * usually K = -1 such in quarterly-onthly data, in such data we have = so K is taken equal 2 since the suation starting fro zero.

23 14 t Again as we noted fro equation(), equation(6) can represent forecasting equation to the next period of t (t+1) and it does not contain leads. The lead ters can be added are: B 0; θ x t+1 + B 1; θ x 1 + B 2; θ x 2. These ters t+1 t+1 will be used when onthly observations of (x t ) in the period (t+1) are available[8]. Returning to our explanation exaple (that was shown in table (1) before), the following table(table (2)) shows the quarterly-onthly data structure of the siple MIDAS regression odel (6). Table (2): Two Years Quarterly-Monthly Data Structure for Eq(6) Quarterly-lowfrequency variable y t Third-Months variable x t =x t Second-Months variable x 1 t 1 y 1 x 1 = x1 x y 2 x 2 = x 2 x 2 1 y x = x x 1 4 y 4 x 4 = x4 x y 5 x 5 = x5 x y 6 x 6 = x 6 x y 7 x 7 = x7 x y 8 x 8 = x8 x Polynoial Specifications First-Months variable x 2 t x 2 1 x 2 2 x 2 x 2 4 x 2 5 x 2 6 x 2 7 x 2 8 Paraeterization of coefficients B k; θ in a parsionious fashion is one of the features of MIDAS approach[2]. We will discuss various ethods of specifications of lagged coefficients B k; θ of MIDAS regression odels Exponential Alon Lag Polynoial The first ethod for polynoial specification is to use the Exponential Alon Lag ethod to paraeterize B k; θ [2]. General Exponential Alon Lag function is defined as:

24 15 B k; θ = e θ 1k+θ 2k 2 + +θ Q k Q K e θ 1k+θ 2k 2 + +θ Q k Q k=1 Ghysels, Santa-Clara and Valkanov[1] used this functional for with two paraeters, i.e. θ = θ 1, θ 2, because this function is known to be quite flexible, and can take various shapes with only a few paraeters, so it will be in this for: B k; θ = e θ 1k+θ 2k 2 K e θ 1k+θ 2k 2 k=1 Figure () shows soe shapes of the Exponential Alon Lag weighting function with different values of two paraeters θ 1 and θ 2. We note that different values of paraeters obtain decreasing or hup-shaped weighting functions. It is clear fro the figure() that later lags of the higher-frequency variable x t are assigned to a lower weights[14].

25 16 We have soe notes and properties about this function: 1. When θ 1 = θ 2 = 0, then B k; θ represents the sae weights for all k. In the case of quarterly-onthly data, K==, and B k; θ = 1 for all k, which is the sae as the Tie-Averaging Aggregation. K 2. The su of B k; θ is up to one, i.e. k=1 B k; θ = 1.. Since B k; θ is a nonlinear function of paraeters θ 1 and θ 2, these paraeters can be estiated by the nonlinear least squares ethod (NLS)[2,4,8,2] Beta Lag function The second polynoial specification is to use the Beta function, so-called as Beta Lag Function which has only two paraeters such that: Where: B k; θ 1, θ 2 = f k K ;θ 1,θ 2 f x; θ 1, θ 2 = xθ x θ 2 1 Γ(θ 1 +θ 2 ) Γ(θ 1 )Γ(θ 2 ) K f k k=0 K ;θ 1,θ 2, Γ θ = e x x θ 1 0 The Beta lag Function has the sae properties stated for the Exponential Alon Lag function except that the Tie-Averaging Aggregation is obtained by the Beta lag Function when θ 1 = θ 2 = 1[2,4,8] Alon Lag Polynoial of order (P) The alon lag polynoial is used to specify β 1 B k; θ together or jointly in MIDAS regression odel (6) which is defined as: β 1 B k; θ 0, θ 1,, θ p = θ 0 + This polynoial can be written in atrix for as: β 1 B 0; θ B 1; θ B 2; θ B k; θ B K; θ = P p= P P P 1 k k 2 k P 1 K K 2 K P θ p k p θ 0 θ 1 θ P dx.

26 17 For exaple, by this polynoial, we have the following: β 1 B 0; θ =θ 0 β 1 B 1; θ =θ 0 + θ 1 + θ θ P β 1 B 2; θ =θ 0 + 2θ 1 + 4θ 2 + 8θ P θ P β 1 B ; θ =θ 0 + θ 1 + 9θ θ + + P θ P... β 1 B K; θ =θ 0 + Kθ 1 + K 2 θ 2 + K θ + + K P θ P For ore clarification, let us return to MIDAS odel (6) with vector θ = θ 0, θ 1, θ 2. Using weights of Alon-Lag-Polynoial of order (2), assuing K=- 1=2, equation (6) becoes: y t = β 0 + θ 0 x t + θ 0 + θ 1 + θ 2 x 1 t + (θ 0 + 2θ 1 + 4θ 2 ) x 2 t + u t (7) Where β 1 B 0; θ =θ 0, β 1 B 1; θ =θ 0 + θ 1 + θ 2, β 1 B 2; θ =θ 0 + 2θ 1 + 4θ 2. Arranging equation(7) will give the following equation: () y t = β 0 + θ 0 x t1 + θ 1 x t2 + θ 2 x t + u t (8) Where : x t1 = x t + xt 1 x t = x t 1 +4 x 2 t + x 2 t, x t2 = x 1 t +2 x 2 t Now, the paraeters θ 0, θ 1 and θ 2 could be estiated by ordinary least squares ethod(ols) such that : θ = (X T X) 1 (X T Y) Where The atrix X ={1, x t1, x t2, x t }, θ = β 0, θ 0, θ 1, θ 2, Y = y t, the sybol ( T ) here eans atrix transposition[2,15,16]. Ghysels, Santa-Clara, and Valkanov (2004) suggest that MIDAS odels can be estiated under general conditions via non-linear least squares (NLS), quasiaxiu-likelihood (MLE) or general ethod of oents (GMM)(see section 2.10)[4]. The other specification is to define MIDAS regression and estiating paraeters with step-function ethod[2,15,16]. and

27 Lag Length Selection Criteria As we saw in previous sections, there are nuber of lags selected to include in MIDAS odel, i.e. the MIDAS regression odel depends on K which is the axiu effective lag of the higher-frequency variable x should be taken. For siplicity, we have selected K= in our explanation exaples, but the deterination of lag length for any tie series is very iportant issue in all studies. In the literature of linear regression, there are various selection criteria used to deterine the best subset of independent variables should be included in the odel. Of course we always interested to get the odel with iniu ean square error(mse), but we can ake the MSE saller by adding another MA or AR ters in tie series odel(the sae thing in linear regression). But this ake a proble of getting coplicated odels with too any paraeters(violation of parsiony principle), so the use of inforation criterion techniques would help us to construct odels that fit the data well without having too any paraeters[25]. The sae idea is applied in tie series odels. We have various lag length selection criteria, such as: the Aikaike s inforation criterion(aic), Schwarz inforation criterion(sic), Hannan-Quinn criterion(hqc), Final Prediction Error(FPE) and the Bayesian inforation criterion(bic). Both AIC and BIC are the ost popular and they have been used widely by researchers[2,24,25]. Ghysels, Santa-Clara, and Valkanov(2004) stated that standard selection procedures such as the Akaike or Schwarz criterion can be applied to select the optial nuber of lags that should be included in MIDAS regression. Ghysels, Claudia, Klaus, Andreou and Kourtellos were concentrated on AIC and BIC. They also copared between the two criteria to get the optial lag length and to iprove results of MIDAS odels[,4]. In BIC which used uch ore, we choose the odel that has the iniu value of: BIC M = log σ 2 + M log T (9) T Where σ 2 is the estiated error-variance after fitting the odel(usually the su of squared residuals divided by the nuber of observations(t)), M is the nuber of estiated paraeters and T is the length of the tie series(low-frequency variable length) [4,20].

28 Testing Unit Root and Cointegration Unit Root Tie series data consist of observations, which are considered as a realization of rando variables that can be described by soe stochastic processes. The concept of stationarity is related to the properties of this stochastic processes. The concept of weak stationarity eans that the data are assued to be stationary if the eans, variances and covariances of the series are independent of tie, rather than the entire distribution. Nonstationarity in a tie series occurs when there is no constant ean, no constant variance or both of these properties. It can originate fro various sources but the ost iportant one is the unit root[54,42,4]. Any sequence that contains one or ore characteristic roots that are equal to one is called a unit root process. The siplest odel that ay contain a unit root is the AR(1) odel. Consider the autoregressive process of order one, AR(1), below: y t = φy t 1 + u t Where u t denotes a serially uncorrected white noise error ter with a ean of zero and a constant variance. If φ = 1, the above equation becoes a rando walk without drift odel, that is, a nonstationary process. When this happens, we face what is known as the unit root proble. This eans that we are faced with a situation of nonstationarity in the series. If, however, φ < 1, then the series y t is stationary. The stationarity of the series is iportant because correlation could persist in nonstationary tie series even if the saple is very large and ay result in what is called spurious (or nonsense) regression. The unit root proble can be solved, or stationarity can be achieved, by differencing the data set[54,42,4]. We have any ethods to stationarity, but the ost faous one is The augented Dickey-Fuller (ADF) test, The basic idea behind the ADF unit root test for nonstationarity is to siply regress y t on its (one period) lagged value y t 1 and find out if the estiated φ is statistically equal to 1 or not. The above AR(1) equation can be anipulated by subtracting y t 1 fro both sides to obtain Which can be written as y t y t 1 = (φ 1)y t 1 + u t Δy t = δy t 1 + u t Where δ = φ 1, and Δ is the first difference operator.

29 20 In practice, we shall estiate the last equation and test for the null hypothesis of δ = 0 against the alternative of δ 0. If δ = 0, then φ = 1, eaning that we have a unit root proble and the series under consideration is nonstationary. It should be noted that under the null hypothesis δ = 0, the t-value of the estiated coefficient of y t 1 does not follow the t-distribution even in large saples. This eans that the t- value does not have an asyptotic noral distribution. The decision to reject or not to reject the null hypothesis of δ = 0 is based on the Dickey-Fuller (DF) critical values of the τ(tau) statistic(dickey-fuller t-distribution). The DF test is based on an assuption that the error ters u t are uncorrelated[54,42,4]. However, in practice, the error ter in the DF test usually show evidence of serial correlation. To solve this proble, Dickey and Fuller have developed a test know as the Augented Dickey-Fuller (ADF) test. In the ADF test, the lags of the first difference are included in the regression equation in order to ake the error ter u t white noise and, therefore, the regression equation is presented in the following for: Δy t = δy t 1 + α i Δy t i + u t i=1 To be ore specific, the intercept ay be included, as well as a tie trend t, after which the odel becoes Δy t = β 0 + β 1 t + δy t 1 + α i Δy t i + u t where β 0 is a constant, β 1 the coefficient on a tie trend series, δ the coefficient of y t 1, is the lag order of the autoregressive process, Δy t = y t y t 1 are first difference of y t, y t 1 are lagged values of order one of y t, Δy t i are changes in lagged values, and u t is the white noise. The test procedure for unit roots is siilar to statistical tests for hypothesis, that is: (i ) Set the null and alternative hypothesis as H 0 : δ = 0 against H 1 : δ < 0. (ii) Deterine the test statistic using F τ = i=1 δ SE (δ) where SE(δ) is the standard error of δ. (iii) Copare the calculated test statistic F τ with the critical value fro Dickey-Fuller table to reject or not to reject the null hypothesis. (iv) The ADF test is a lower-tailed test, so if F τ is less than the critical value, then the null hypothesis of unit root is rejected and the conclusion is that the variable of the series does not contain a unit root and is stationary[54,42,4].

30 21 The DF and ADF tests are siilar since they have the sae asyptotic distribution. Finally, there are nuerous unit root tests, such as the Phillips-Perron test and the Schidt-Phillips test, but the ost notable and coonly used is the ADF test, which will be used in our study[54,42,4] Cointegration It is possible for two integrated series to ove together in a nonstationary way, so that their difference or any other linear cobination is stationary. These series are said to be cointegrated. Stationarity is like a rubber band pulling a series back to the fixed ean[51,52,54]. Cointegration is like a rubber band pulling the two series back to a fixed relationship with each other, even though both series are not pulled back to a fixed ean. If y and x are both integrated, we cannot rely on OLS standard errors or t statistics. By differencing, we can avoid spurious regressions: If y t = β 0 + β 1 x t + u t then y t = x t + u t. u t is stationary as long as u t is I(0) or I(1). Note that the differenced equation has no history. Now let us Suppose that u t is I(1), This eans that the difference u t = y t β 0 β 1 x t is not eanreverting and there is no long-run tendency for y to stay in the fixed relationship with x, so there is no cointegration between y and x, in this case u t is I(0), if y t is high relative to x t due to a large positive u t, then there is no tendency for y to coe back to x after t, thus estiation of differenced equation is appropriate[51,52,54]. Now Suppose that u t is I(0), that eans that the levels of y and x tend to stay close to the relationship given by the equation. Suppose that there is a large positive u t that puts y t about its long-run equilibriu level in relation to x t. With stationary u t, we expect the level of y to return to the long-run relationship with x over tie, and stationarity of u t iplies that corr(u t, u t+s ) 0 as s. Thus, future values of y should tend to be saller than those predicted by x in order to close the gap. In ters of the error ters, a large positive u t should be followed by negative u t values to return u t to zero if u t is stationary, and this is the situation where y and x are cointegrated. This is not reflected in the differenced equation, which says that future values of y are only related to the future x values, which eans there is no tendency to eliinate the gap that opened up at t [51,52,54]. In the cointegrated case, If we estiate in differences, we are issing the history of knowing how y will be pulled back into its long-run relationship with x. If we estiate in the original levels, we cannot rely on our test statistics because the variables (though not the error ter) are nonstationary[51,52,54].

31 22 The appropriate odel for the cointegrated case is the error-correction odel of Hendry and Sargan, ECM consists of two equations: The long-run cointegrating equation: y t = θ 0 + θ 1 x t + u t, where u t is I(0). The short-run (ECM) adjustent equation: y t = β 0 + p i=1 β i L i y t p + γ i L i x i=1 t + α y t 1 θ 0 θ 1 x t 1 + v t Note the presence of the error-correction ter with coefficient α in the ECM equation, this ter reflects the distance that y t 1 is fro its long-run relationship to x t 1. If α < 0, then y t 1 above its long-run level will cause y t to be negative when other factors are held constant, then it pulling y back toward its long-run relationship with x. Because both y and x are I(1), their differences are I(0). Because they are cointegrated with cointegrating vector θ 0, θ 1, the difference in the error correction ter is also I(0). It would not be if they weren t cointegrated and the ECM regression would be invalid. The ECM equation can be estiated by OLS without undue difficulty because all the variables are stationary[51,52,54]. In ultivariate cointegration, the concept of cointegration extends to ultiple variables. With ore than two variables, there can be ore than one cointegrating relationship (vector). Vector error-correction odels (VECM) allow for the estiation of error correction regressions with ultiple cointegrating vectors. In order to test for cointegration, the earliest test is Engle and Granger s extension of the ADF(Augented Dickey-Fuller) test has two steps: 1) Regressing the cointegrating regression by OLS, 2) Testing the residuals with an ADF test. Other, ore popular tests include the Johansen-Juselius test, which generalizes easily to ultiple variables and ultiple cointegrating relationships[51,52,5,54] Johansen s procedure Johansen's procedure builds cointegrated variables directly on axiu likelihood estiation instead of relying on OLS estiation. This procedure relies heavily on the relationship between the rank of a atrix and its characteristic roots. Johansen derived the axiu likelihood estiation using sequential tests for deterining the nuber of cointegrating vectors[54]. His ethod can be seen as a secondary generation approach in the sense that it builds directly on axiu likelihood instead of partly relying on least squares. In fact, Johansen's procedure is nothing ore than a ultivariate generalization of the Dickey-Fuller test. Consequently, he proposes two different likelihood ratio tests naely: the trace test and the axiu eigenvalue test. This procedure is a vector cointegration test