The Limits of Mathematics and NP Estimation in ... - Chichilnisky

28Advances in Econometrics - Theory and Applicationsof near-unit root bias and resulting the lack of distribution, the empirical studies generallyapply Andrews’ (1993) median unbiasedness method to estimate the AR(1) coefficient toinvestigate the persistence behavior.For AR(1) case, this paper contributes to the literature by applying the IGF approach ofPhillips et al. (2004) to estimate the coefficients of near unit root process, IGF estimator isproved to be normal asymptotically, hence it is very easy to construct confidence intervals.For AR(p) case, moreover, instead of recalculation, we propose a unrestricted FM-AR(p)model, a slight extension of Phillips’ (1995) FM-VAR, to estimate coefficients directly.Our empirical illustration of real effective exchange rate indicates that FM-AR(p) is a usefuland easy-to-use method to examine the econometric persistence.5. Appendix: The finite sample properties of FM-ARThis paper shows that a FM estimator can ameliorate the small sample biases that arise fromnear unit root bias. Our attention here is focused on the class of dynamic AR(p). A MonteCarlo simulation is used here to investigate the performance, our results indicate that FMestimator successfully reduces the small-sample bias.Assuming { yt} t 0 is governed by a AR(3) time series process generated from (, 2 ) = (0, 1),which satisfies the data-generating process specified below:y y y y (1)t 1 t1 2 t2 3 t3twhere t i.i.d.. and ’s represent the associated parameters, and is the standarddeviation. In our simulation, we generate an AR(3) process. The summation of the threecoefficients measures the degree of persistence, the vector is parameterized belowDegree of Persistence 1: { 1 , 2, 3 } = {0.90, 0.085, 0.015},Degree of Persistence 2: { 1 , 2, 3 } = {0.90, 0.050, 0.015}Degree of Persistence 3: { 1 , 2, 3 }= {0.85, 0.050, 0.015}andT{ 200, 400, 800, 1500, 3000}where T represents the vector of sampling size that are used in practice. The DGPs arecharacterized by a modest change in the innovation variance but allow for drastic changesin others.Table A1 reports the characteristics of the finite-sample distribution of both estimators of theelements of estimates. These include the deviation of the estimate from the true parametervalue, or bias, as well as measures of skewness and kurtosis. I compare the bias andnormality to illustrate the problem. There are several main results.Firstly, the biases are decreasing function of sample sizes. Even in small samples around 200and 400, the biases are in the range of 10 -2 . The bias for FM-AR is quite small.Secondly, the variance bias exhibits the similar conclusion. AR(3) is generated from (0,1), theempirical bias is in the range of 10 -3 , and is a decreasing function of sample size.Finally, the normality property of distribution is drawn from skewness and kurtosis.Unfortunately, no regular pattern is found among three parameter estimates and is relatedto the persistence of parameter vector designed; in general, skewness is close to zero whichgives normality an acceptable condition, although the excess kurtosis (>3) is found.As a result, FM-AR is a feasible estimator to directly estimate AR(p), whose empiricalapplications also calls for further studies in the future.