For each currency, there are 204 observations. JB(Prob.) is the JarqueBera statistic for normality with probability value in the parenthesis.Summary statistic
1. Introduction
The traditional approach to the persistence properties of time series is unit root tests, and because of the near unit root bias, the medianunbiased procedure of Andrews (1993) is widely used. In this article, we show that: To calculate half life from AR(1), the instrument generating function estimator of Phillips et al. (2004) is not only an asymptotically normal estimator, but also an easytouse alternative to Andrews (1993); to calculate half life from AR(p), we propose a FMAR model, which is a modified version of Phillips’ (1995) FMVAR.
There are two approaches to study the persistence/convergence property of a univariate time series, for example, real exchange rate: unit root test and half life. The unit root approach to the persistence properties models the series either as trendstationary, where innovations have no permanent effects, or difference stationary, implying that shocks have permanent effects
For example, Papell (1997).
. However, reliance on unit root tests does not provide a measure of uncertainty of the estimates of finiteness or permanence of innovations because a rejection of the unit root null could still be consistent with a stationary process with highly persistent shocks. In addition, an important pitfall in using the autoregressive (AR, thereafter) model to analyze the persistence of shocks to the data is that standard estimators, such as least squares, are significantlyLet
This model is the same as that used for testing whether there is a unit root in a time series— consequently, this model is often referred to as the Dickey–Fuller regression. The halflife of shocks, which is the time it takes for a unit shock to dissipate by 50%, is calculated from the AR parameter
Empirically, to compute halflife, one has to estimate coefficient α of (1), one problem of estimation is
As lower values of the AR parameter imply faster speeds of adjustment following a shock, this will also result in a downward bias to LSbased estimates of halflives of shocks. In addition to the inherent difficulties in distinguishing between the stationary and random walk processes for the real interest rate.
In a nutshell, conventional procedure estimates α to characterize the persistence of time series has two main disadvantages: (i) the least squares estimates of the AR parameter in unit root regressions will be biased toward zero in small samples (Orcutt, 1948); and (ii) they have low power against plausible trend stationary alternatives (De Jong et al., 1992). The downward bias in LS estimates of the AR parameter arises because there is an asymmetry in the distribution of estimators of the AR parameter in AR models (the distribution is skewed to the left, resulting in the median exceeding the mean). As a result, the median is a better measure of central tendency than the mean in least squares estimates of AR models.
The medianunbiased procedure proposed by Andrews (1993) and Andrews and Chen (1994) is usually suggested in literature to estimate (1), which combines unbiasedness with the use of point and interval estimators in achieving a more accurate estimate of the persistence of shocks to economic time series. Andrews’ (1993) median unbiased estimator (MU thereafter) uses a bias correction method which delivers an impartiality property to the decision making process because there is an equal chance of under or over estimating the AR parameter in the unit root regression. MU is widely used in empirical studies. For example, Murray and Papell (2002), Cashin
Recently, Phillips et al. (2004) proposed an instrument generating function(IGF thereafter) estimator to estimate (1), which is also an
In contrast to this stark dichotomy between whether shocks to the series are mean reverting (finite persistence) or not (permanent), this paper characterizes the extent of reversion by applying the nonlinear instruments generating function (IGF thereafter) method, proposed by Phillips et al. (2004), to measure the estimates of the
Point and interval estimators are useful statistics for providing information to draw conclusions about the duration of shocks, unlike hypothesis testing, they are informative when a hypothesis is not rejected. Because the IGF estimator is shown to be asymptotically standard normal, the construction of confidence intervals is very straightforward.
For AR(p) model, impulseresponse 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 FMAR to directly estimate the coefficients of any AR(p) process.
2. The Econometric methodology
2.1. IGF estimator for DFAR(1)
Phillips et al. (2004) studies the properties of IGF estimator in which the instruments are nonlinear functions of integrated processes. Framework of Phillips et al. (2004) extends the analysis of So and Shin (1999)
In recent work, So and Shin (1999) suggested the use of the Cauchy estimator, which uses the sign function as an instrumental variable, in place of the ordinary least squares (OLS) estimator in autoregressions that included both stationary and nonstationary cases.
, providing a more general analysis of IV estimation in potentially nonstationary autoregressions and showing that the Cauchy estimator has an optimality property in the class of certain IV procedures.For (1), Phillips et al. (2004) consider the IV estimator of
Here, α is an IV estimator in which the instrument is generated by the IGF
The bounded optimal IV estimator with asymptotic sign IGF has some nice properties that the conventional OLS estimator does not have. The estimator yields a
Because of singularity problem, in this article, we report the IVi3 results for our empirical study, where
where
See eq.(25) in Phillips et al. (2004, p.231).
is also applied.Using the 0.05 and 0.95 quantile functions of α estimate, we can construct twosided 90% confidence intervals for the true α. These confidence intervals can be used either to provide a measure of the accuracy of α or to construct the conventional exact one or twosided tests of the null hypothesis that α = α_{0}. In this paper, we use such symmetric confidence intervals only to provide a measure of the accuracy of α estimate.
2.2. IGF estimator for ADFAR(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(
Similarly, (3) is estimated by IGF. For augmented differenced lagged variables, instruments are themselves without IGF transformation. Subsequently, we then define the matrices below
where
where
where
Under the null, we have
and the variance of
For differenced lagged variables, themselves are used as the instruments without IGF transformation, details are explained in Chang(2002). The halflife calculated from the value of (1) assumes that shocks to the data decay monotonically, which is inappropriate for ADF regressions represented by (3), since in general shocks to an AR(
From the IGF estimates of (3), coefficients of (4) can be recalculated as follows:
When the coefficients are obtained, the
(5) is widely used, for example, Murray and Papell (2002), however, not only is it subject to strict restriction, but also bias if some of righthandside variables of (4) are cointegrated. Therefore, instead of using (3) and (5) to indirectly calculate the coefficients of (4), we propose a FullyModified AR (p) to estimate (4) directly, the model is derived from FullyModified VAR of Phillips (1995). Section below continues the study.
2.3. Unrestricted fullymodified AR(p)
Phillips’ (1995) FMVAR is a level system regression with/without error correction terms (differenced terms). FMVAR of Phillips (1995) generalized the asymptotics of Phillips and Hansen (1990), allowing full rank I(1) regressors and possible cointegration existed among lagged dependent variables. Mostly important, the asymptotics of FMVAR is normal, or mixed normal, which allows us to construct confidence intervals and half life. The methodology for FMAR(
where
Here
Here
The first step is to correct for the serial correlation to the LS estimator
In (8),
leads us to estimate the equation below
Phillips (1995, Pp. 10331034) shown that this transformation reduces to the ideal correction
Secondly, the serial correction term has the form
where
Combining the endoneneity and serial corrections we have the FM formula for the parameter estimates
For (9), the limit theory of FM estimates of the stationary components of the regressors is equivalent to that of LS, while the FM estimates of the nonstationary components retain their optimality properties as derived in Phillips and Hansen (1990); that is, they are asymptotically equivalent to the MLE estimates of the cointegrating matrix.
For the finite sample property of FMAR, Appendix offers a simulation study illustrating a good performance of this estimator in calculating halflife of near I(1) process.
3. Empirical results
To illustrate these approaches, BIS real effective exchange rates (REER thereafter) of four economies (Germany, Japan, UK, and USA) are used, samples range from 1994M1 to 2010M12. BIS’ REER is CPIbased and is a broad index of monthly averages, with 2005=100. Figure 1 plots the time series plot of them, with a horizontal line of the base year index. The Yaxis also illustrates the histogram to depict the distribution property of these data. The common feature of them is an apparent behavior of stochastic trend.
In Table 1, the upper panel DFAR(1) presents the results of (1); the lower panel ADFAR(p) summarizes those of (5), and the BIC criterion returns unanimously 2 lags. Comparing both results, taking Germany as an example, expressed in years, its half life takes 12.55 for DFAR(1) and substantially drops to 0.916 for ADFAR(2); similar finding is also found in Japan. UK and USA does not have finite bounds. Apparently, the dynamic lag structure is substantial.
Table 2 of FMAR(2) tabulates more results. Unrestricted FMAR(2) yields finite bounds of halflives for all four economies, most of them are roughly less than two years. To be more informative, Figure 2 graphs the impulse response plot of Germany, expressed in month; clearly, the impulse response function of Germany has a rather wide confidence interval.
GERMANY  JAPAN  UK  USA  
Mean  101.40  108.02  95.97  99.52 
Median  100.91  106.55  99.93  98.83 
Maximum  116.79  151.11  107.81  116.00 
Minimum  90.91  79.68  75.88  86.53 
Std. Dev.  5.50  14.98  8.91  7.93 
Skewness  0.63  0.42  0.68  0.31 
Kurtosis  3.09  2.99  2.04  2.10 
JB(Prob.)  13.71( 0.001)  6.073(0.048)  23.72(0.000)  10.18(0.006) 
DFAR(1) 

Std  mean HL  HL, 90% CI  



Germany  0.995  0.028  12.55  0.95  ∞  
Japan  0.984  0.028  3.58  0.78  ∞  
UK  1.025  0.014  ∞  15.59  ∞  
USA  1.014  0.014  ∞  4.29  ∞  
ADFAR(2) 


mean HL  HL, 90% CI  



Germany  1.180  0.253  0.916  0.541  ∞  
Japan  1.272  0.330  1.132  0.624  ∞  
UK  1.162  0.159  ∞  ∞  ∞  
USA  1.332  0.340  ∞  1.911  ∞ 


mean HL  HL, 90% CI  



Germany  1.121  0.156  1.789  0.376  ∞ 
Japan  1.292  0.335  1.525  0.356  ∞ 
UK  1.158  0.191  1.926  0.258  ∞ 
USA  1.270  0.298  2.253  0.406  ∞ 
4. Conclusion
The empirical study of time series persistence uses two main approaches: unit root tests and halflife. However, reliance on unit root tests does NOT provide a measure of uncertainty of the estimates of finiteness or permanence of innovations because a rejection of the unit root null could still be consistent with a stationary process with highly persistent shocks. Because of nearunit root bias and resulting the lack of distribution, the empirical studies generally apply Andrews’ (1993) median unbiasedness method to estimate the AR(1) coefficient to investigate the persistence behavior.
For AR(1) case, this paper contributes to the literature by applying the IGF approach of Phillips et al. (2004) to estimate the coefficients of near unit root process, IGF estimator is proved 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 FMAR(p) model, a slight extension of Phillips’ (1995) FMVAR, to estimate coefficients directly.
Our empirical illustration of real effective exchange rate indicates that FMAR(p) is a useful and easytouse method to examine the econometric persistence.
5. Appendix: The finite sample properties of FMAR
This paper shows that a FM estimator can ameliorate the small sample biases that arise from near unit root bias. Our attention here is focused on the class of dynamic AR(p). A Monte Carlo simulation is used here to investigate the performance, our results indicate that FM estimator successfully reduces the smallsample bias.
Assuming
where
Degree of Persistence 1: {
Degree of Persistence 2: {
Degree of Persistence 3: {
where
Table 4 reports the characteristics of the finitesample distribution of both estimators of the elements of estimates. These include the deviation of the estimate from the true parameter value, or bias, as well as measures of skewness and kurtosis. I compare the bias and normality to illustrate the problem. There are several main results.
Firstly, the biases are decreasing function of sample sizes. Even in small samples around 200 and 400, the biases are in the range of 10^{2}. The bias for FMAR is quite small.
Secondly, the variance bias exhibits the similar conclusion. AR(3) is generated from (0,1), the empirical 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 related to the persistence of parameter vector designed; in general, skewness is close to zero which gives normality an acceptable condition, although the excess kurtosis (>3) is found.
As a result, FMAR is a feasible estimator to directly estimate AR(p), whose empirical applications also calls for further studies in the future.
{DGP coefficients}  Estimates biases  Skewness  Kurtosis  Variance biases σ^{2}=1 

T  β_{1}  β_{2}  β_{3}  β_{1}  β_{2}  β_{3}  β_{1}  β_{2}  β_{3}  
{0.9, 0.085, 0.015}  
200  0.027  0.036  0.007  0.194  0.099  0.301  5.48  2.88  0.451  0.029 
400  0.023  0.030  0.011  0.186  0.174  0.320  5.27  3.25  4.502  0.024 
800  0.020  0.027  0.015  0.122  0.25  0.314  5.45  3.88  4.923  0.018 
1500  0.019  0.024  0.018  0.117  0.337  0.365  5.79  4.56  5.328  0.014 
3000  0.019  0.023  0.021  0.108  0.399  0.445  6.27  5.36  5.847  0.011 
{0.9, 0.05, 0.015}  
200  0.020  0.024  0.016  0.060  0.19  0.058  5.10  3.12  4.25  0.029 
400  0.020  0.017  0.014  0.030  0.27  0.027  5.50  3.45  4.73  0.019 
800  0.017  0.015  0.014  0.009  0.336  0.0005  5.63  4.06  5.01  0.012 
1500  0.016  0.014  0.013  0.038  0.376  0.025  5.94  4.80  5.40  0.0073 
3000  0.015  0.013  0.012  0.049  0.432  0.057  6.49  5.67  5.92  0.0042 
{0.85, 0.05, 0.015}  
200  0.020  0.0067  0.040  0.340  0.17  0.09  5.6  3.04  4.74  0.02 
400  0.016  0.0079  0.038  0.190  0.15  0.124  5.4  3.29  4.72  0.015 
800  0.013  0.0076  0.038  0.110  0.16  0.12  5.4  3.76  5.12  0.009 
1500  0.011  0.0060  0.038  0.077  0.198  0.108  5.6  4.36  5.54  0.0045 
3000  0.010  0.0058  0.038  0.056  0.233  0.111  6.1  5.10  6.08  0.0015 
References
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Murray C. J. Papell D. H. 2002 The purchasing power parity persistence paradigm.  7.
Orcutt G. H. 1948 A study of the autoregressive nature of the time series used for Tinbergen’s model of the economic system of the United States 19191932,  8.
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Phillips P. C. B. 1995 Fullymodified least squares and vector autoregression.  12.
So B. S. Shin D. W. 1999 Recursive mean adjustment in timeseries inferences.
Notes
 For example, Papell (1997).
 In recent work, So and Shin (1999) suggested the use of the Cauchy estimator, which uses the sign function as an instrumental variable, in place of the ordinary least squares (OLS) estimator in autoregressions that included both stationary and nonstationary cases.
 See eq.(25) in Phillips et al. (2004, p.231).