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- 1. UNIVERSITY OF ECONOMICS, HO CHI MINH CITYFACULTY OF DEVELOPMENT ECONOMICSTIME SERIES ECONOMETRICSCAUSALITY MODELSCompiled by Phung Thanh Binh1 (2010) You could not step twice into the same river; for other waters are ever flowing on to you. Heraclitus (540 380 BC)The aim of this lecture is to provide you with the keyconcepts of time series econometrics. To its end, youare able to understand time series based researches,officially published in international journals2 suchas applied economics, applied econometrics, and thelikes. Moreover, I also expect that some of you willbe interested in time series data analysis, and choosethe related topics for your future thesis. As the timethis lecture is compiled, I believe that the Vietnam1Faculty of Development Economics, University of Economics, HCMC. Email: ptbinh@ueh.edu.vn.2Selected papers were compiled by Phung Thanh Binh & Vo Duc Hoang Vu (2009). You can find themat the H library.1

2. time series data3 is long enough for you to conductsuch studies.Specifically, this lecture will provide youthefollowing points: An overview of time series econometrics Stationary versus non-stationary Unit roots and spurious regressions Testing for unit roots Vector autoregressive models Causality tests Cointegration and error correction models Optimal lag length selection criteria Basic practicalities in using Eviews 6.0 Suggested research topics1. AN OVERVIEW OF TIME SERIES ECONOMETRICSInthis lecture, wewill mainlydiscuss singleequation estimation techniques in a very different wayfrom what you have previously learned in the basiceconometricscourse.Accordingto Asteriou (2007),there are various aspects to time series analysis butthe most common theme to them is to fully exploit thedynamic structure in the data. Saying differently, wewill extract as much information as possible from the3The most important data sources for these studies can be World Banks World Development Indicators,IMF-IFS, GSO, and Reuters Thomson.2 3. past historyoftheseries.Theanalysis of timeseries isusuallyexplored within two fundamentaltypes, namely timeseries forecasting4 anddynamicmodelling. Pure time series forecasting, such as ARIMAmodels5,is often mentionedas univariateanalysis.Unlike most other econometrics, in univariate analysiswedonot concernmuchwithbuilding structuralmodels, understandingthe economyor testinghypothesis6, but what we really concern is developingefficient models, which are able to forecast well. Theefficient forecasting models can be evaluated usingvarious criteria such as AIC, SBC7, RMSE, correlogram,and fitted-actual value comparison8. In these cases,wetrytoexploitthedynamic inter-relationship,which exists over time for any single variable (say,sales, GDP, stock prices, ect). On the other hand,dynamic modelling, includingbivariate andmultivariate time series analysis, is still concernedwith understanding the structure of the economy andtesting hypothesis. However, this kind of modelling isbased on the view that most economic series are slowto adjustto anyshockand so to understandtheprocessmustfullycapturethe adjustment processwhich may be long and complex (Asteriou, 2007). Thedynamic modellinghas become increasingly popularthanks to the works of two Nobel laureates, namely,4See Nguyen Trong Hoai et al, 2009.5You already learned this topic from Dr Cao Hao Thi.6Both statistical hypothesis and economic hypothesis.7SBC and SIC are interchangeably used in econometrics books and empirical studies.8See Nguyen Trong Hoai et al, 2009.3 4. Clive W.J. Granger (for methods of analyzing economictime series with common trends, or cointegration) andRobert F. Engle (for methods ofanalyzingeconomictime series with time-varying volatility or ARCH)9. Upto now, dynamic modelling has remarkably contributedtoeconomicpolicy formulationinvarious fields.Generally, the key purpose of time series analysis isto capture and examine the dynamics of the data. In time series econometrics, it is equally importantthat the analysts should clearly understand the termstochastic process. According to Gujarati (2003)10, arandom or stochastic process is a collection of randomvariables ordered in time. If we let Y denote a randomvariable, and if it is continuous, we denote it aY(t), but if it is discrete, we denote it as Yt. Sincemost economic data are collected at discrete points intime, we usually use the notation Yt rather than Y(t).If we let Y represent GDP, we have Y1, Y2, Y3, , Y88,where the subscript 1 denotes the first observation(i.e., GDP for the first quarter of 1970) and thesubscript 88 denotes the last observation (i.e., GDPfor the fourth quarter of 1991). Keep in mind thateach of these Ys is a random variable. In what sense we can regard GDP as a stochasticprocess?Consider for instance the GDP of$2872.8billion for 1970Q1. In theory, the GDP figure for thefirst quarterof 1970 couldhave been anynumber,depending on the economic and political climate then9http://nobelprize.org/nobel_prizes/economics/laureates/2003/10 Note that I completely cite this from Gujarati (2003).4 5. prevailing. The figure of $2872.8 billion is just aparticular realization of all such possibilities. Inthiscase, we canthink ofthevalueof$2872.8billion as the mean value of all possible values ofGDP for the first quarter of 1970. Therefore, we cansay that GDP is a stochastic process and the actualvalues we observed for the period 1970Q1 to 1991Q4 areaparticularrealizationof thatprocess.Gujarati(2003)states that the distinctionbetween thestochastic process and its realization in time seriesdata is just like the distinction between populationand sample in cross-sectional data. Just as we usesample data to draw inferences about a population; intime series, we use the realization to draw inferencesabout the underlying stochastic process. The reason why I mention this term before examiningspecific models is that all basic assumptions in timeseriesmodels relate to thestochasticprocess(population).Stock&Watson(2007) saythattheassumption that the future will be like the past is animportant one in time series regression. If the futureis like the past, then the historical relationshipscan be used to forecast the future. But if the futurediffersfundamentally from the past,thenthehistorical relationships might not be reliable guidestothe future.Therefore,inthecontextof timeseriesregression, theidea thathistoricalrelationshipscan be generalizedto thefuture isformalized by the concept of stationarity.5 6. 2. STATIONARY STOCHASTIC PROCESSES2.1 DefinitionAccording to Gujarati (2003), a key concept underlyingstochastic process that has received a great deal ofattention and scrutiny by time series analysts is theso-called stationarystochastic process.Broadlyspeaking, a stochasticprocessis saidto bestationary if its mean and variance are constant overtime and the value of the covariance between the twoperiods depends only on the distance or gap or lagbetween the two time periods and not the actual timeatwhichthe covariance is computed.Inthetimeseries literature, such a stochastic process is knownas a weakly stationary, or covariance stationary, orsecond-order stationary,or widesense,stochasticprocess. By contrast, a timeseriesisstrictlystationary if all the moments ofitsprobabilitydistribution and not just the first two (i.e., meanand variance) are invariant over time. If, however,thestationary process is normal,the weaklystationarystochastic processis alsostrictlystationary, for the normal stochastic process is fullyspecified byits twomoments, themean and thevariance. For most practical situations, the weak typeof stationarity often suffices. According to Asteriou(2007), a time series is weakly stationary when it hasthe following characteristics: (a)exhibits mean reversion in that it fluctuatesaround a constant long-run mean;6 7. (b)has a finite variance that is time-invariant; and(c)has a theoretical correlogram that diminishes as the lag length increases.In its simplest terms a time series Yt is said to beweakly stationary (hereafter refer to stationary) if:(a) Mean:E(Yt) = (constant for all t);(b) Variance:Var(Yt) = E(Yt-)2 = 2 (constant for all t); and(c) Covariance: Cov(Yt,Yt+k) = k = E[(Yt-)(Yt+k-)]where k, covariance (or autocovariance) at lag k,isthe covariance between the values of Yt and Yt+k, thatis, between two Y values k periods apart. If k = 0, weobtain 0, which is simply the variance of Y (=2); ifk = 1, 1 is the covariance between two adjacent valuesof Y.Suppose we shift the origin of Y from Yt to Yt+m(say, from the first quarter of 1970 to the firstquarter of 1975 for our GDP data). Now, if Yt is to bestationary, the mean, variance, and autocovariance ofYt+m must be the same as those of Yt. In short, if atimeseriesisstationary,its mean, variance,andautocovariance (at various lags) remain the same nomatter at what point we measure them; that is, theyare time invariant. According to Gujarati (2003), suchtime series will tend to return to its mean (calledmeanreversion) and fluctuations around thismean(measuredbyits variance) willhavea broadlyconstant amplitude. 7 8. If a time series is not stationary in the sense justdefined, it is called a nonstationary time series. Inother words, a nonstationary time series will have atime-varying mean or a time-varying variance or both.Whyarestationarytimeseriesso important?According to Gujarati (2003), because if a time seriesis nonstationary, we can study its behavior only forthe time period under consideration. Each set of timeseriesdata willtherefore befor a particularepisode. As aconsequence,it isnotpossibletogeneralize it to other time periods. Therefore, forthe purpose of forecasting or policy analysis, such(nonstationary) time series may be of little practicalvalue. Saying differently, stationarity is importantbecause if the series is nonstationary, then all thetypical results of the classical regression analysisare notvalid. Regressionswithnonsta

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