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Instrument Generating Functionand Analysis of Persistent Economic Times Series: Theory and Application 21applying the nonlinear instruments generating function (IGF thereafter) method, proposedby Phillips et al. (2004), to measure the estimates of the half-life of shocks to the series. LikeAMU, the IGF estimator of Phillips et al. (2004) removes the downward bias of standard LSestimators. Moreover, the IGF based confidence intervals have slightly smaller coverageprobabilities than AMU, and the t-statistic is distributed as standard normal distributionasymptotically.Point and interval estimators are useful statistics for providing information to drawconclusions about the duration of shocks, unlike hypothesis testing, they are informativewhen a hypothesis is not rejected. Because the IGF estimator is shown to be asymptoticallystandard normal, the construction of confidence intervals is very straightforward.For AR(p) model, impulse-response approach is used, typical examples are Murray and Papell(2002) and Sekioua (2008). To further the study, based upon the FM asymptotics of Phillips(1995), we propose a FM-AR to directly estimate the coefficients of any AR(p) process.2. The Econometric methodology2.1 IGF estimator for DF-AR(1)Phillips et al. (2004) studies the properties of IGF estimator in which the instruments arenonlinear functions of integrated processes. Framework of Phillips et al. (2004) extends theanalysis of So and Shin (1999) 2 , providing a more general analysis of IV estimation inpotentially nonstationary autoregressions and showing that the Cauchy estimator has anoptimality property in the class of certain IV procedures.For (1), Phillips et al. (2004) consider the IV estimator of given by nt1nt 1F( y ) yt1FY ( ) ytt1 t1Here, is an IV estimator in which the instrument is generated by the IGF F. In its generalform, the class of IV estimators that can be represented by (2) includes, of course, theconventional OLS estimator as a special case with the linear IGF F(x) = x. However, thispaper will concentrate on IV estimators constructed with various nonlinear IGF’s.The bounded optimal IV estimator with asymptotic sign IGF has some nice properties thatthe conventional OLS estimator does not have. The estimator yields a t-ratio that has astandard normal limit distribution when =1, as well as when ||
22Advances in Econometrics - Theory and Applicationssectional dependency among currencies, we also follow Chang(2002) to insert a variable csuch that y t-1 Exp(-c|y t-1 |), and c is defined byIV EstimatorsInstrument generating functions, F()IVh1 sgn(y t-1 )IVh2y t-1 I{|y t-1 |K}+sgn(y t-1 )KI{| y t-1 |>K}IVh3 arctan(y t-1 )IVi1sgn(y t-1 ) I{|y t-1 |K}IVi2y t-1 I{|y t-1 |K}IVi3 y t-1 Exp(-|y t-1 |)Kc s( y ) Twhere s(y t ) denotes the standard deviation of y t , and K is a constant fixed at 3. In addition,the recursive de-meaning procedure 3 is also applied.Using the 0.05 and 0.95 quantile functions of estimate, we can construct two-sided 90%confidence intervals for the true . These confidence intervals can be used either to providea measure of the accuracy of or to construct the conventional exact one- or two-sided testsof the null hypothesis that = 0 . In this paper, we use such symmetric confidence intervalsonly to provide a measure of the accuracy of estimate.2.2 IGF estimator for ADF-AR(p)In addition, the presence of serial correlation (typical in economic time series) means that (1)will often not be appropriate. In such cases, (1) is augmented to be an AR(p) model byadding lagged first-order difference. Hence, the starting point of this analysis is thefollowing ADF regression:tpy y y(3)t t1k tk tk1Similarly, (3) is estimated by IGF. For augmented differenced lagged variables, instrumentsare themselves without IGF transformation. Subsequently, we then define the matricesbelowyp y1 p xp 1 p 1 y , y , X ,ε y y x T T1 T T collects the lagged difference terms. Then the augmented ARwhere xt yt1 ,...... ytpregression (3) can be written in matrix form as3 See eq.(25) in Phillips et al. (2004, p.231).
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