The literature of unit roots and structural breaks has produced numerous tests that follow nonstandard asymptotic distributions. This paper by fitting a seminonparametric model to them proposes a new simple way of calculating the p-values

Adda, Jerome

Gonzalo, Jesús

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P-values for non-standard distributions with an application to the DF test

The literature on unit roots and structural breaks has produced numerous tests that follow non-standard asymptotic distributions. By fitting a semi-non-parametric model to these distributions, this paper proposes a new simple way to calculate the p-values.Publicad

Economics Letters

P-Values for non-standard distributions with an application to the DF test

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Working Paper 96-06 Departamento de Estadística y Econometría Statistics and Econometrics Series 02 Universidad Carlos III de Madrid January 1996 Calle Madrid, 126 28903 Getafe (Spain) Fax (341) 624-9849 P-VALUES FOR NON-STANDARD DISTRIBUTIONS WITH AN APPLICATION TO THE DF TEST Jerome Adda and Jesús Gonzalo· Abstract _ The literature of unit roots and structural breaks has produced numerous tests that follow non-standard asymptotic distributions. This paper by fitting a seminonparametric model to them proposes a new simple way of calculating the p-values. Key Words DF Tests; non-standard distributions; p-values; SNP Approximations; Unit roots. ·Adda, INRA and Department of Economics, Boston University; Gonzalo, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid. ·1--- ------1 1 .-_._._..._.._-----------------.,---------------------------------1 Introduction Most of the tests for structural breaks and for unit roots follow non-standard asymptotic distributions. These distributions are summarized by reporting a set of critical values (CV) (1 % , 5 % and 10 %). With this information applied researchers can not calculate the p-values and therefore can not set their own level of significance. This paper presents a new way of approximating the p-values (p(x)) by using the semi-nonparametric (SNP) techniques introduced by Gallant and Nychka (1987) and Fenton and Gallant (1994). Instead of reporting a set of CV's, we propose to report a set of parameters that describe a polynomial approximation to the non-standard empirical density. Simple in-tegration of that polynomial produces the approximate p-values. This can be a very efficient way for econometrics software packages to store the empirical distributions of most of the , unit root tests and structural break tests. Previous attempts to approximate p-value functions for non-standard test statistics in econometrics were made by Hansen (1992, 19951) and Mackinnon (1994). These works basically approximate the p-value functions by H(J(x, O)) were f(x) is a simple polynomial in x with parameters Oand H(.) is the Normal distribution function in Mackinnon (1994), a transformation of the Chi-square distribution in Hansen (1995) and the identity function in Hansen (1992). In Hansen (1992) and Mackinnon (1994) the parameters Oare estimated by a least squares regression of H- 1(p) on a set of quantiles (x). The approach in Hansen (1995) differs in the sense of using an U norm, for r large, to measure the distance Ip(x) - pi. This paper follows an approach that can be seen as more general than the one in the previous literatU:re, in the sense of not being problem specific. vVe do not need to assume any ad-hoc H(.) function depending on whether the test is a unit root test or a structural break test. We can also fit the whole p value function with one model, instead of doing the fitting by regions like the previous literature. This paper is organized as follows. Section 2 presents the general framework and method-ology. Section 3 shows with the particular example of the DF test for unit roots how to approximate in practice the p-values of a non-standard test. Section 4 conc1udes. 1 \Ve received a copy oí Hansen (1995) aíter this research was done. 1 2 The General Framework and Methodology Our goal is to approximate a function p-value = p(Tn ) which maps an observed test statistic Tn into the appropriate value in the range [0,1]. This test statistic has an asymptotic distribution F(x) with the corresponding density function f( x). We approximate the p-value function p( x) = F(x), by integrating a polynomial approximation of the density function. In order to do that, we simply put the density equal to a Hermite series following the SNP approach introduced by Gallant and Nychka (1987) andFenton and Gallant (1994). This approach consists on approximating the density funetion of Tn by k] k fn(x, O) = [t; OiWi((X - m)j{j) 2 j t; ol (1) where Wo(x) = (y'2;tl/2e-x2/4 Wl (x) = (y'2;tl/2xe-x2 /4 (2) Wi(X) = [XWi-l(X) - J""'-(i---1) Wi-2(X)]fVi. The parameters O= (m, (j, Oi) can be estimated by maximum likelihood like in Fenton and Gallant (1994). Instead of doing that, this paper proposes to estimate Oby minimizing the following 1055 function (3) where the first element of O, 00 is set to one and CVaj is the critical value at the exi % significance level. According to our experience the minimization problem (3) produces better approximations for the p-values and converges fas ter than the equivalent maximum likelihood problem. The tuning parameters in (3) are • (i) the q p-values to match (Pl%,P2%'''',Pq%) • (ii) the number k of parameters ()i. The next section shows how to choose those tuning parameters in a particular example. 2 - -------------------------,----------------------3 An Example: The DF Test The methodology introduced in section 2 is valid íor obtaining approximate p-values oí any non-standard distribution. In this section we apply our method to the DF distribution. Suppose we wish to test the null hypothesis that the variable Yt has a unit root. The simplest and most commonly used unit root tests are the DF tests. These tests are based on ordinary least squares estimates oí any oí the íollowing regressions: R(l) : ~Yt = >'Yt-l +E~=l 8¡DYt_¡ + et R(2) : ~Yt = Po +>'Yt-l +Ef=l 8¡DYt_¡ + et R(3) : ~Yt = Po +p1t +>'Yt-l +E~=l 8¡DYt_¡ +et where ~Yt =Yt - Yt-l, t is a linear deterministic time trend, et is an error term, and >. is a parameter that equals zero under the null hypothesis oí a unit root. The DF(i) test considered in this paper is the t statistic íor >. to equal O in the regression R(i). This test has a well known non-standard asymptotic distribution (see Dickey and Ful1er (1979)). In order to calculate the approximated p-values oí DF(i) (i=l, 2, 3) we íol1ow five simple steps: • (i) Calculate the quantiles (x) írom the empirical distribution íunction oí DF(i) test constructed írom 50,000 independent draws írom their (approximate) asymptotic dis-tributions. In each draw the sample size is set to n=2000. • (ii) Standardize x by estimating the mean (m) and the standard deviation (O') by their sample counterparts. This does not change the results but increases the speed oí convergence oí the minimization problem (3). • (iii) Decide which q p-values to match in (3). In this application we have chosen only sixteen p-values (.01, .02, .03, .05, .06, .08, .10, .12, .15, .17, .20, .25, .30, 040, .50, .60). There is a higher concentration in the leít tail because oí the nature oí the DF test. • (iv) Minimize the íunetion dk ((}) íor different values oí k. Plot dk ((}¡) on k, like Figure 1, and choose k accordingly. From Figure 1, k has been set equal to six. 3 4 [Figure 1 about here] • (v) Calculate the approximate p-values by simple integration oí In(x, ()) írom -00 to evo;. Once the parameters ()i are estimated, the integration can be done by any standard software (Gauss, Mathematica, Matlab, ... ). Table 1 shows the estimated parameters oí the polynomial approximation In (x, O) and Table 2 the approximate p-values and the absolute error oí the approximation. Notice that although only sixteen p-values are chosen to be matched in (3), the approximation works uniform1y very well íor all the p-values írom .01 to .60. In íact íor the case we íocus in this application (DF(3)) the p(x) are very close to the true ones in the whole range [O, 1]. [Tables 1 and 2 about here] Similar results are available íor different sample sizes. They are not reported here to make the note short. The calculations have been done using the Optimum library in GAUSS. The programs are available upon request. Conclusion This note shows a simple and a very general way oí approximating the p-values oí non-standard distributions. 4 References Dickey, D.A. and Fuller, W.A., 1979, Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association 74, 427-431. Fenton, V.M. and R.A. Gallant, 1994, Qualitative and asymptotic performance of SNP den-sity estimator. University of North Caroline, mimeo. Gallant, R.A. and D.W. Nychka, 1987, Seminonparametric maximum likelihood estimation, Econometrica 55, 363-390. Hansen, B.E., 1992, Test for parameter instability in regressions with I(1) processes, Journal of Business and Economic Statistics 10, 321-335. Hansen, RE., 1995, Approximate asymptotic p-values for structural change tests, Boston College, mimeo. Mackinnon, J.G., Approximate asymptotic distribution functions for unit-root and cointe-gration tests, Journal of Business and Economic Statistics 12, 167-176. 5 .....".----------.------------r-----------------------------Table 1: Approximate empirical densitity of the DF t test for p :S .60 f.(·,O) = ¡t,0¡W,«. -ID)/uf /t,0) wo(x) = (-/2ii)-1/2e-x2 /4 WI(X) = (-/2ii)-1/2 xe-x2 /4 Wi(X) =[XWi-I(X) - vT=l Wi-2(X)]/Ji parameter DF(1) DF(2) DF(3) m -0.42483 -1.53206 -2.17745 (j 0.97505 0.83702 0.75122 61 -0.30044 -0.65441 -0.04769 6z 0.02413 -0.61767 -0.09269 63 0.50388 -0.23093 -0.10166 64 0.51716 0.12300 -0.09983 6s 0.28817 0.17911 -0.06281 66 0.02567 0.12666 -0.06952 ..._._... _._--------------,--------;-------------Table 2: Approximate p-values for the DF t test Approximate p-values Absolute Error p-values DF(l) DF(2) DF(3) DF(l) DF(2) DF(3) 1% 1.00319 0.99984 1.00320 .00003 .00000 .00003 2% 1.98983 2.00171 1.99539 .00010 .00002 .00005 3% 2.98137 2.99925 2.95448 .00019 .00001 .00046 4% 4.04239 3.99781 4.07692 .00042 .00002 .00077 5% 5.04151 4.96833 5.09336 .00042 .00032 .00093 6% 6.02473 6.05821 6.05094 .00025 .00058 .00051 7% 7.01476 6.99862 7.06335 .00015 .00001 .00063 8% 7.99055 7.96124 7.96878 .00009 .00039 .00031 9% 9.01392 8.94637 8.93215 .00014 .00054 .00068 10 % 10.02747 9.97407 9.94182 .00027 .00026 .00058 11% 11.00684 11.08123 10.92839 .00007 .00081 .00072 12 % 11.97744 12.05579 11.96005 .00023 .00056 .00040 13 % 12.98543 13.05882 12.99713 .00015 .00059 .00003 14 % 13.99103 14.04091 13.99807 .00009 .00041 .00002 15 % 14.98372 15.05265 14.93359 .00016 .00053 .00066 16 % 15.97357 15.93928 15.95400 .00026 .00061 .00046 17% 16.94230 16.90459 16.93799 ;00058 .00095 .00062 18 % 17.96399 17.94639 17.98578 .00036 .00054 .00014 19 % 18.96067 18.96677 19.02162 .00039 .00033 .00022 20 % 19.97044 20.00598 20.07539 .00030 .00006 .00075 21 % 21.01644 21.01417 21.05782 .00016 .00014 .00058 22 % 22.06442 21.99547 22.07475 .00064 .00005 .00075 23 % 23.03593 22.97540 23.03562 .00036 .00025 .00036 24 % 24.00016 24.00336 24.06584 .00000 .00003 .00066 25 % 24.98652 25.02584 25.05570 .00013 .00026 .00056 30 % 29.98828 29.97709 30.03994 .00012 .00023 .00040 35 % 35.02969 34.99348 34.96467 .00030 .00007 .00035 40 % 40.03144 39.99644 40.01450 .00031 .00004 .00015 45 % 45.03378 44.88271 45.08477 .00034 .00117 .00085 50 % 50.23323 50.03430 50.11525 .00233 .00034 .00115 55 % 55.25007 55.08260 54.96449 .00250 .00083 .00036 60 % 59.83218 59.98085 59.83991 .00168 .00019 .00160 65 % 63.89082 64.84299 64.91459 .01109 .00157 .00085 70 % 67.10778 69.25286 69.65490 .02892 .00747 .00345 '''',.._--,--,---_._-------,--,-----------------------0.4 0.8 1.2 1 .6 2.0 2.4 2.8 '1 \.O e, (1) -.>. 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http://dx.doi.org/10.1016/0165-1765(95)00741-5

P-values for non-standard distributions with an application to the DF test

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