EXTENDED NEYMAN SMOOTH GOODNESS-OF-FIT TESTS, APPLIED TO COMPETING HEAVY-TAILED DISTRIBUTIONS. J. Huston McCulloch and E. Richard Percy, Jr.

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1 EXTENDED NEYMAN SMOOTH GOODNESS-OF-FIT TESTS, APPLIED TO COMPETING HEAVY-TAILED DISTRIBUTIONS J. Huston McCulloch and E. Richard Percy, Jr.* J. Econoetrics Special Issue Subission: Sept. 4, 200 First Revision: Feb. 5, 20 Second Revision: April 2, 20 Third Revision: May 6, 20 Fourth Revision: June 2, 20 J. Huston McCulloch E. Richard Percy, Jr. Departent of Econoics 3503 Treehouse Lane Ohio State University Canal Winchester, OH 430 JEL Classification Codes: C2 (Hypothesis Testing), C6 (Specific Distributions) Keywords: Stable distribution, Student t distribution, Generalized Error Distribution, Lagrange Multiplier Test * The authors are grateful to two anonyous referees for their very helpful and patient coents and suggestions.

2 ABSTRACT A siplified version of the Neyan (937) Sooth goodness-of-fit test is extended to account for the presence of estiated odel paraeters, thereby reoving overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyan greatly siplifies the calculations. Polynoials, splines, and the step function of Pearson s test are copared as alternative perturbations to the theoretical unifor distribution. The extended tests have negligible size distortion and ore power than standard tests. The tests are applied to copeting syetric leptokurtic distributions with US stock return data. These are generally reected, priarily because of the presence of skewness.

3 . Introduction A siplified version of the Neyan (937) Sooth goodness-of-fit test (GFT) is extended to adust for the presence of estiated odel paraeters. The effect is to reduce size distortion and to increase the power of the test. As is well known (e.g. Bai 2003), standard tests that do not account for the estiation of odel paraeters tend to under-reect the distribution assued by the null hypothesis. The heavy tails of ost financial return series often lead to an easy reection of the Gaussian distribution. A paraetric distribution that is consistent with the data would allow the ean to be estiated ore efficiently than would a nonparaetric procedure. Furtherore, paraetric distributions allow out-of-saple extree tail probabilities to be estiated. These tail probabilities are iportant for the pricing of options and contingent clais, for Value at Risk calculations, and for non-financial applications such as the risk of floods and other extree events. When heavy tails are present, Mandelbrot (963), Faa (965), Saorodnitsky and Taqqu (994), and McCulloch (996) have suggested the use of stable distributions. Blattberg and Gonedes (974), Hageran (978), Perry (983), and Boothe and Glassan (987) propose the Student t distributions. Nelson (99) investigates the generalized error distribution (GED), while Praetz (972) and Clark (973) ipleent a ixture of noral distributions. This paper is aied at developing an appropriate GFT that can work well with all the above distributions. Although this paper assues identically and independently distributed (i.i.d.) errors, Percy (2006) addresses other odels including ARCH and GARCH odels of volatility clustering.

4 Section 2 introduces a siplified Lagrange Multiplier (LM) version of the Neyan Sooth GFT. Section 3 extends this test to accoodate estiated paraeters. Section 4 illustrates the extended test for the proposed heavy-tailed distributions with stock return data. Section 5 reports on the power of the test to discriinate between the candidate distributions, and Section 6 concludes. This paper draws heavily upon Percy (2006), which provides ore theory and details than are available herein. 2. A siplified Neyan Sooth GFT with known paraeters We can develop an LM test for any distribution with known or unknown paraeters by first constructing an LM test for the unifor distribution. 2.. An LM test for unifority For perturbation functions φ ( z),, that are linearly independent, bounded, and, α,,, define integrate to zero on the unit interval and a vector of coefficients g z; : φ z α, z [0, ]. For α in a sufficiently sall neighborhood N of the origin in, g ;α is a probability density function, with cuulative distribution function denoted by G ;α. This distribution nests the standard unifor distribution U(0,) when α 0. u u,, Consider a rando saple n H : u i ~ i.i.d. ;α u n. We wish to test H 0: u i ~ i.i.d. U(0,) vs. G with α 0. The log likelihood function of interest is Such tests are also called efficient score tests, score tests, or soeties Rao score tests in honor of C.R. Rao, who first proposed this type of test. 2

5 3 n i i u φ log ) ; L( log u α, and the score vector of first derivatives with respect to the s is n i h i h h i u u φ log L/ : ) ; ( u α s. When evaluated at the null, this siplifies to n i u i φ 0;u s. () Letting ;α G ~ Z, the Fisher inforation atrix with respect to the perturbation paraeters is,, ' I α α I where dz z z z Z Z I h h h 0 φ φ φ ; log g ; log g E : α α α. At the null hypothesis, this siplifies to dz z z, 0 φ φ 0 I. (2) The LM statistic is then n LM / ; ; u 0 s 0 I u 0 s. (3) Since the null hypothesis is a single internal point in the set N, the asyptotic distribution of this statistic is 2 with degrees of freedo under the null.

6 2.2. Functional for and basis Although any set of linearly independent bounded functions integrating to 0 ay be used in theory, there are nuerical considerations in choosing a basis so that tractable results can be obtained. We investigated polynoials, orthogonal polynoials, splines, and B-splines (Judd 999, Percy 2006). For sall values of, the deeaned standard polynoials φ ( z) z /( ) are an obvious choice. For larger values of, however, these standard polynoials are ill-conditioned for the atrix inversion required to copute the Fisher inforation atrix. Neyan-Pearson orthogonal polynoials are atheatically equivalent but coputationally better conditioned in this respect. However, rounding errors still ay arise in the evaluation of the score as there is a large disparity in the orders of agnitude between coefficients, even for idsize values of : Experience with polynoials derived by truncating [Taylor] series, [especially in their use with estiating transcendental functions,] ay islead one into thinking that the use of high order polynoials does not lead to coputational difficulties. However it ust be appreciated that truncated [Taylor] series are not typical of polynoials in general. [Truncated Taylor series] have the special feature that the ters decrease rapidly in size for values of x in the range for which they are appropriate. A tendency to underestiate the difficulties involved in working with general polynoials is perhaps a consequence of one s experience in classical analysis. There it is natural to regard a polynoial as a very desirable function since it is bounded in any finite region and has derivatives of all orders. In nuerical work, however, polynoials having coefficients which are ore or less arbitrary are tiresoe to deal with by entirely autoatic procedures. (Wilkinson 963, p. 38) Polynoials have the further drawback that fitting one region better ay require fitting other distant regions ore poorly. Cubic splines, which are piecewise cubic functions with discontinuities in the third derivative at selected knotpoints, are ore pliable and better able to fit a particular interval without affecting ore distant intervals as uch. Siple spline basis functions can create siilar nuerical probles with the inversion of the inforation atrix but the so-called B-spline basis solves this 4

7 issue, while generating the sae space of functions. Thus, for uch of the analysis, we used B- splines as a basis but also investigated orthogonal polynoials with a oderate nuber of paraeters. s have soe nuerical properties that are ore desirable than polynoials, while the tradeoff in the other properties is not severe. The classic Pearson GFT arises as a special case of the LM test when each perturbation function is a step function integrating to zero, with discontinuities at selected points. Piecewise linear and quadratic spline perturbation functions are also considered below. The proposed LM GFT is closely related to the Neyan (937) Sooth test, so naed because the alternative distributions vary soothly away fro the null hypothesized distribution rather than with discontinuities as in the Pearson test. However, Neyan s test, which he called the 2 test (as contrasted with Pearson s 2 test), was based on a likelihood ratio statistic rather than an LM statistic and hence required estiating the odel both under the null and the perturbed alternative. Furtherore, in order to prevent negative densities, his perturbations were exponentiated polynoials, nuerically constrained to integrate to zero. As noted already by Rayner and Best (989), the LM siplification of the Neyan test eployed here only needs to be estiated under the null, a uch sipler calculation. Furtherore, since the alternative only needs to be a proper density in a neighborhood of the null hypothesis, polynoial or spline perturbations, which are easily constrained to integrate to zero, ay be used directly, without exponentiation LM test for a copletely specified distribution The LM test for any continuous distribution with known paraeters is essentially the sae as that for the unifor distribution developed above. 5

8 Consider a rando saple n y y,, y n. One would like to test: H 0: y i ~ i.i.d. F(), where F() is a copletely specified continuous distribution with density f()=f (), vs. H : y i ~ i.i.d. F G ;α with 0 α. Define u F i y i. Then, as is well known, y i ~ F() if and only if u i ~ U(0,). Under the alternative, the density is f g F ; α. The log-likelihood function is then n ; y log gu i ; α log f y i n log L α. i i Since the second suation does not depend on α, the derivatives necessary to calculate the LM statistic are identical to those for the test for the unifor distribution. Thus one can siply use the transfored observations u i with the test for the unifor distribution (3). All tables and critical values that are suitable for the test of unifority are also suitable for a general distribution Finite saple properties with a copletely specified distribution As noted, the LM statistic has a liiting asyptotic distribution that is chi-squared with degrees of freedo equal to, the nuber of perturbation paraeters. Siulations reported in Percy (2006) indicate that for saple size n 30, 5, and test size = 0.05, the convergence to the liiting distribution is quite rapid. At the indicated values of, n, and test size, the 95 th percentile of the siulated distributions was generally within sapling error of the 95 th percentile of a chi-squared rando variable. 6

9 Difference 0.05 Cubic Size Distortion (=6) ` Cuulative Probability n=30 n=00 n=300 n=000 95% CI Figure. Size distortion of cubic spline test ( = 6). Figure shows results of the siulations for 9999 replications with a cubic spline perturbation and = 6. The solid lines indicate the difference between the theoretical chi-squared distribution and the epirical distribution of the siulations. The dashed lines indicate 95% confidence intervals for the quantiles of an epirical CDF of rando draws fro the chi-squared distribution. Except for very sall saples with n = 30, the correspondence is as close as could be expected at all quantiles. See Percy (2006) for illustrations of size distortion for other paraeter sizes and basis functions. 7

10 3. The LM test with estiated odel paraeters We now extend the LM test to the typical situation in which the values of the odel paraeters are not known and so ust be estiated. Our null hypothesis is now H 0 : y i ~ i.i.d. F ;θ, where F is a continuous distribution with density ;θ f that depends on the k vector of paraeters θ. This vector lies in the interior of a convex set Θ, but is otherwise unknown. The Fisher inforation atrix is assued to be positive definite and finite for all interior points of this set. Let the axiu likelihood (ML) estiate of θ under the null hypothesis be θˆ and set u ˆ uˆ,, uˆ where ˆ F y ;θˆ n u. i i For ost of the copeting heavy-tailed distributions considered here, naely the syetric stable, Student t, and generalized error distributions, the unknowns consist of a location, scale, and shape paraeter, so that k = 3. For a ixture of two Gaussian distributions with a coon ean, two variances, and an unknown ixing probability, k = 4. More generally, the location of each y i could be deterined by a possibly non-linear function h(x i ;), where x i is a vector of known constants or of exogenous rando variables and is a vector of unknown coefficients to be estiated by ML. However, the present paper illustrates the extended test only in cases involving a coon location paraeter. Our alternative hypothesis is now H : y i ~ i.i.d. F defined as above. The log likelihood is now n, α; y log gfy i ; θ; α log f y ; θ i n log L θ. The full score vector i i G ;θ ;α, with 0 α and G ;α s θ, α; y s s θ α θ, α; y θ, α; y log L/ log L/ h hk 8

11 is now (k+), but θ ˆ, 0; y 0 by the ML first order conditions, while θ, 0; y s 0; uˆ (), since g ( ; α) 0 when α 0. The full Fisher inforation atrix I θ, α I I θθ θα s θ θ, α Iθαθ, α θ, α I θ, α is now of diension (k+) (k+), with I I I αα θ ˆ, α I θθθ ˆ, αhh h, hk θθ, θα θ ˆ, α I θαθ ˆ, αh hk, θ ˆ, α I ααθ ˆ, α, αα. Letting Z ~ F I I I θθ θα αα G ;θ ;α, we have log g F Z; θ θˆ, 0 : E hh' f ; αf Z; θ log gfz ; θ; αf Z; θ z; θ f z; θ h h h log g F Z; θ θˆ, 0 : E h f z; θ / f z; θ dz, θθˆ ; αf Z; θ log gfz ; θ; αf Z; θ h φ log g F Z; θ θˆ, 0 : E φ so that θ, 0 I0 I αα 0 h Fz; θ dz, θθˆ ; αf Z; θ log gfz ; θ; αf Z; θ zφ z dz, ˆ as in (2). h s α θθˆ, α0 θθˆ, α0 θθˆ, α0 ˆ as in 9

12 LM s s Using the partitioned atrix inverse forula, the extended LM test statistic becoes θˆ, 0; yi θˆ, 0sθˆ, 0; y/ n 0, uˆ I0 I θˆ, 0I θˆ, 0I θˆ, 0 s0, uˆ / n. θα θθ θα By standard theory, the asyptotic distribution of this statistic under the null is again 2 with degrees of freedo. The subtraction of I I I inside the inverse increases the extended test θα θθ θα statistic by ust enough to prevent overacceptance of the null hypothesis when û is used in place of u. Although I 0 can readily be coputed in closed for for polynoial, spline, or step perturbation functions, calculation of the reaining inforation coponents of the extended LM statistic requires nuerical integrations that ay depend on the distribution in question, its paraeters, and the for and nuber of the perturbation functions. DuMouchel (975) has tabulated I θθ θ, 0for the stable distributions. See Percy (2006) for coputational details and tables for the distributions and perturbation functions considered here. In the extended test, finite saple critical values depend on the specific odel and true paraeter values. Siulations reported in Percy (2006) indicate ore size distortion than with known paraeters. However, size distortion can still largely be ignored, even for relatively sall saple sizes. 4. The extended test on stock arket returns The extended LM test is illustrated in this section using continuously copounded percent real onthly returns, including dividends, on the CRSP value-weighted stock arket index, during the 40-year period fro /53-2/92 (480 observations), as eployed in McCulloch (997). Percy (2006) also considers the 50-year period ending in 2/2002. However, volatility clustering is ore 0

13 apparent in the longer series. Although estiated ARCH or GARCH paraeters ay be incorporated into the extended test, the present paper assues i.i.d. errors, and hence this section uses only the shorter period. The extended test was perfored with the following syetric distributions: () Gaussian, (2) syetric stable, (3) Student t with a ean and scale paraeter, and (4) the generalized error distribution. We also used various perturbation functions: a step function that extends the traditional Pearson test, Neyan-Legendre polynoials, and linear, quadratic, and cubic B-splines. All the splines, as well as the extended Pearson test, used equidistant knotpoints. Table reports axiu likelihood estiates under a Gaussian null hypothesis. Table 2 reports the extended LM test statistics and asyptotic p values for the hypothesis that these returns are Gaussian, with up to twelve perturbation paraeters in the alternative hypotheses. Gaussian ML Estiates Estiate Standard Error z-score Probability Std. Deviation Mean Log L Table. Gaussian axiu likelihood estiates (n = 480).

14 Pearson Lagrange Multiplier Test Statistics Neyan- Legendre Linear Quadratic Cubic Copleent of chi square inverse of test statistic Pearson Neyan- Legendre Linear Quadratic e-07.8e-05.3e-07 Cubic e e e e e-0 5.7e-08.2e e e-0.6e e-0 3.5e-0 6.2e-06.2e-0.4e-08.6e e e-06.7e-2.5e-08.4e e- 8 3.e e-3 4.4e e-0 7.e e-5 4.6e e- 2.0e e-7 6.8e e-2 2.6e e-7 5.3e e-2 2.8e e-7 6.6e-0.6e-2.3e-4 Table 2. LM Test Statistics and p values for Gaussian null hypothesis. The Jarque-Bera (987) statistic for i.i.d. norality is 89.9 with this data. The p value for this statistic with the appropriate 2 degrees of freedo is 0-4. Many of the p values in Table 2 are zero to several places, but none is as sall as for the Jarque-Bera statistic. Jarque-Bera is designed to be sensitive to departures of skewness fro 0 and of kurtosis fro 3, and is unquestionably the best test to use for departures fro norality that show up strongly in these oents. The 2

15 extended Neyan LM statistic, on the other hand, will be sensitive to other departures fro norality, depending on the nuber and type of basis functions, and, with sufficiently large saple size and sufficiently large values of, could eventually detect distributions that were not Gaussian even if they had zero skewness and kurtosis equal to 3. Drawing attention to the Pearson statistics oentarily, one can see that the Pearson statistics with low paraeter nubers are not particularly adept at identifying the non-gaussian nature of this data set. For =, the significance level is and for = 2, the significance level is Since Pearson takes no account of where an error resides within the bin to which it is assigned, it will often not be very sensitive to departures fro the posited distribution, so we recoend against use of this statistic whenever the null is a continuous distribution, even when it is corrected, as here, for estiated paraeters. In Table 3, we estiate the syetric stable distribution paraeters by axiu likelihood using the density approxiation of McCulloch (998). As in McCulloch (997), the estiated stable characteristic exponent (not to be confused with the perturbation coefficients of the preceding sections) is.845. The algorith fits the natural logarith of the scale c, so that the asyptotic standard errors apply to log c rather than c itself. Although.845 is 2.63 asyptotic standard errors fro the value of 2 corresponding to a Gaussian distribution, and the likelihood ratio (LR) statistic for the hypothesis of norality is 26.66, these statistics do not have their usual N(0,) and 2 () distributions, because 2 is on the boundary of the perissible paraeter space. Nevertheless, the siulations of McCulloch (997) deonstrate that the 5% critical value for the LR statistic is less than.2 for this saple size, and that norality can be reected with p <<.004. Table 4 reports the extended LM test of the null that these real returns are syetric stable, with the sae alternatives as considered in Table 2. 3

16 Syetric Stable ML Estiates Estiate Standard Error z-score Probability Log scale Scale c 2.7 Stable a Mean Log L Table 3. Syetric stable axiu likelihood estiates (n = 480). Pearson Lagrange Multiplier Test Statistics Neyan- Legendre Linear Quadratic Cubic Copleent of chi square inverse of test statistic Pearson Neyan- Legendre Linear Quadratic Cubic Table 4. LM Test Statistics and p values for syetric stable null hypothesis. 4

17 Turning our attention to the Neyan-Legendre polynoial tests, the first three tests do not reect the syetric stable distribution at the 5% test size. However, when we add the fourth basis function, we begin to obtain significant reections. The first three Neyan-Legendre statistics are necessarily identical to the first statistic for the linear, quadratic, and cubic splines, respectively. For the cubic spline with ore than 3 paraeters and therefore at least one internal knotpoint, the results are siilar to those for the Neyan-Legendre statistics. The step-function Pearson results are far ore dependent on than are the sooth alternatives. Tables 5 and 6 report the sae tests for the Student t distribution. Infinite Student t degrees of freedo (DOF) correspond to a proper, Gaussian distribution, so the search was paraeterized in ters of reciprocal DOF rather than DOF directly. Tables 7 and 8 report these tests for the GED. An infinite GED power paraeter leads to a proper, U(-, ) liit, so this search was also paraeterized in ters of the reciprocal of the shape paraeter. In order for the Fisher inforation atrix to be finite, it is necessary to restrict the power to be strictly greater than, so as to ust exclude the Laplace distribution. However, this restriction was not binding. Percy (2006) reports siilar test results for a ixture of two Gaussian distributions. 5

18 Student t ML Estiates Estiate Standard Error z-score Probability Log Scale Scale c 3.53 Reciprocal DOF Deg. of Freedo Mean e-5 Log L Table 5. Student t axiu likelihood estiates (n = 480) Pearson Lagrange Multiplier Test Statistics Neyan- Legendre Linear Quadratic Cubic Copleent of chi square inverse of test statistic Pearson Neyan- Legendre Linear Quadratic Cubic Table 6. LM Test Statistics and p values for Student t null hypothesis. 6

19 Generalized Error Distribution (GED) ML Estiates Estiate Standard Error z-score Probability Log Scale Scale c 4.65 Recip. power e-4 GED power.404 Mean e-5 Log L Table 7. GED axiu likelihood estiates (n = 480) Pearson Lagrange Multiplier Test Statistics Neyan- Legendre Linear Quadratic Cubic Copleent of chi square inverse of test statistic Pearson Neyan- Legendre 7 Linear Quadratic Cubic Table 8. LM Test Statistics and p values for GED null hypothesis.

20 The ost easily reected of these three distributions is the GED. These distributions have relatively thin tails copared to the others. As a general rule, the presence of ore than the expected value of outliers (or even one extree outlier) often allows for the reection of null hypotheses of thin-tailed distributions, while the absence of outliers does not allow for as easy of a reection of null hypotheses of heavy-tailed distributions. Of course as the saple size grows, eventually a test for a heavy-tailed distribution will decrease its p value if outliers do not eventually appear. For purely coputational reasons, the present paper considers only syetric distributions. Although all three syetric distributions considered in the present section encounter frequent reections (with the notable exception of the Student t for between 6 and 9), it is likely that any of these reections would be reversed if syetry were not iposed. The nuerical approxiation of Nolan (997) perits skew-stable distributions to be fit by ML, while the Student t distribution can be generalized to incorporate skewness in a natural way (Ki and McCulloch 2007). It is also possible to skew the GED, albeit in an ad hoc anner, siply by expanding the horizontal axis on one side of the origin while copressing it on the other. With liited data, densities that are siilar over uch of their support cannot be distinguished fro one other very easily. This does not allow one to ake very strong stateents about the extree tail probabilities, where the densities ay differ considerably. Although this data set has only 480 onthly returns, 40 years of daily returns would yield about 0,000 observations. When daily data is used, however, the returns often becoe less independent and less identically distributed since there is ore apparent volatility clustering, day-of-the-week effects in both ean and scale, holiday effects, end-of-year effects, and other coplications. However, the extended 8

21 Neyan Sooth GFT developed here allows these additional effects to be estiated without size distortion. 5. Size distortion and power with financial data paraeter values This section investigates criteria for deterining the best test to use in ters of size distortion and power against the syetric leptokurtic alternatives considered above, and also a ixture of two norals with a coon ean. With the extended test, size distortion and power ay depend on both the distribution and true paraeter values in question. For this purpose, following Percy (2006), we use paraeter estiates for percent log real onthly returns for the 50- year period January, 953, through Deceber, The ML paraeter estiates for this longer period were as follows: Syetric stable: =.862, log scale =.024, Student t: degrees of freedo = 6.864, log scale =.293, ean = 0.64 Generalized error distribution: power =.49, log scale =.568, ean = Mixture of two Gaussians: probability of saller standard deviation = 0.906, log(saller st. dev.) =.308, log(larger st. dev.) = 2.25, ean = These values were used in siulations for both null hypotheses and data-generating processes. It is iportant to note that the conclusions to be drawn using these paraeter values ay not be applicable with other values. The size distortion and power against each of the other distributions considered were investigated for each of the four leptokurtic distributions. All sizes and powers are based on siulations using 000 saples, described in ore detail in Percy (2006). All data is generated by one of the four hypothesized distributions with the indicated paraeters. There are two test sizes 9

22 investigated for each scenario, 0.0 and Fro to 20 basis paraeters are tested, or 3 to 20 for the cubic splines. Six saple sizes were considered: n = 32, 00, 36, 000, 362 and 0,000 (0 k, k = 3 / 2, 2, 5 / 2, 3, 7 / 2, 4). We tested the size distortion for each extended goodness-of-fit test and did the sae using conventional tests without the adustent for estiated paraeters. This produces 8 power tests per null hypothesis, based on 6 possible saple sizes with 3 possible alternative hypotheses. For each category there is a size-adusted power for the corrected tests and non-adusted powers for both the corrected tests and the conventional tests. Preliinary tests indicated that the Neyan-Legendre polynoial basis and the Cubic basis generally outperfored the other bases investigated (Pearson, the Quadratic and the Linear ). Accordingly, only those two bases are copared here. 5.. Size distortion There is treendous size distortion with the uncorrected conventional tests in every instance for every value of. This distortion does not go away as the saple size increases fro 32 to 0,000. It diinishes soewhat as the nuber of basis paraeters increases, but this is still not very helpful. The distortion is, as expected, in the direction of over-acceptance. Figure 2 illustrates the size iproveent fro the extended test for the syetric stable distribution with saple size 36 and test size 0.0. It shows that the size of the extended test lies ostly within 95% confidence liits of the intended size, while that of the conventional tests lies copletely outside this interval. This exaple is typical of the full set of figures reported in Percy (2006). For the corrected tests, there is soe size distortion for saller saple sizes, but generally uch saller than with the conventional tests. This size distortion vanishes, within sapling error, for oderate saple sizes. 20

23 Percent Reect Null 0.4 Actual Size and 95% Conf Liits around 0.0 Stable, 36 observations, 0.0 test Ext. Poly Ext. Polynoial 95% CI Paraeters Figure 2. Size Iproveent due to Extended Test, syetric stable null, n = 36, test size.0, with 95% CI bracketing noinal test size. For the ixture distribution and as few as 00 observations, with up to = 6 perturbation paraeters, the extended test size distortion is undetectable. For 36 observations and the level 0.0 test, the results are ust barely in the low end of the confidence interval for ost values of. For 000 and ore observations, there is no significant distortion. Distortion gradually disappears for the stable null fro n = 36 to 000 as well, although it is sall for saller saple sizes with sall nubers of basis paraeters as well. For Student nulls, distortion dies out at only 00 observations. GED nulls require around 36 observations Power and recoendations The over-acceptance caused by the size distortion in the conventional tests contributes to poor power against the chosen nulls. With the conventional tests, fitting the odel paraeters 2

24 biases against reecting even false hypotheses. Hence, practitioners ay all too often istakenly conclude that, since their test does not reect an alternative hypothesis, they are ustified in accepting the validity of the assuptions in their study. The uncorrected tests have power even less than the test size for the largest saple sizes! The corrected tests do have ore power than the conventional tests. However, it is quite difficult to tell the leptokurtic distributions under consideration fro one another when the saple size is sall, even with the corrected tests. When the saple size is high enough for there to be reasonable levels of power (4 to 5 ties the test size), there is negligible size distortion. However, there are often significant power gains to be ade by adapting the functional for and nuber of perturbation paraeters to the specific null in question. Accordingly, recoendations are ade below that are specific to the null, yet robust to the actual data-generating distribution, using paraeters as fit above to stock arket data. With a stable null: a GED distribution can start to be detected with as few as 36 observations. A pattern eerges which has a power peak at only = 2 polynoial test paraeters, or 3 for the cubic spline. This power peak also works well for detecting a Student distribution. The ixture of norals, with its extra paraeter, is quite difficult to identify, requiring ore than 3000 observations even to get odest 30% power levels. At 0,000 observations the ost power coes with a large nuber of paraeters. But at 0,000 observations, any value of (ore than 2) will have the sae (00%) power levels for the GED and Student distributions. At this saple size, it is recoended to use the cubic spline, perhaps with = 5 test paraeters, because it has fewer nuerical difficulties than the polynoial as increases. See Percy (2006) for graphs illustrating these findings. 22

25 Under a Student t null hypothesis, a stable distribution can be seen alost half the tie with 36 observations. Alost any value of greater than 2 will work equally effectively. GED and noral ixtures are still concealed for the ost part at this saple size, but a test with 2 or 3 paraeters has the best chance of finding the. So, if Student t is the null, one ay proceed with 2 or 3 test paraeters, regardless of saple size. With a GED null hypothesis, the advisable nuber is 4 paraeters. Using a basis with only three paraeters has very low power but there is a treendous increase at 4, with sall increases after that. With a ixture of two Gaussians as the null hypothesis, there is little power to detect the other three distributions, unless one has at least 3000 observations. The pattern is a bit unusual, however, in that higher values of generally yield ore power. For this reason, the cubic spline is recoended here to avoid nuerical inaccuracy, with about 5 paraeters. At 0,000 observations, even with the possible inaccuracies, the test with 8 paraeters has fairly low distortion levels, so it would be safe to use against the chosen alternatives. For further discussion, see Percy (2006). 6. Conclusion The extended Neyan Sooth test can be used to analyze econoic and financial data to probe the distribution underlying the generation of the data. Soe parsionious paraetric distributions ay be found that will aid inferences about levels of and relationships between econoic variables. Thus, asyptotically consistent estiates of paraeters are possible without either presuing norality of error ters or using solely nonparaetric techniques. In that regard, these new procedures can offer new answers to old questions. 23

26 Unlike any goodness-of-fit tests, unknown odel paraeters can be estiated with the extended Neyan tests without preudicing the tests. Since these tests rely on axiu likelihood techniques, they asyptotically eet the conditions of the Neyan-Pearson lea against any siple alternative hypothesis in its paraeter space. Tests with one-sided alternatives that eet these criteria qualify as Uniforly Most Powerful (UMP) tests for arbitrary significance levels. odels are ore tractable than polynoial odels with existing double precision software, and it does not appear that this tractability is obtained at the cost of lower power in tests of interest. Size distortion that is present in the conventional tests is lowered considerably with the extended tests, even for odest saple sizes. A related benefit is increased test power. REFERENCES Bai, J., 2003, Testing paraetric conditional distributions of dynaic odels. Review of Econoics and Statistics 85, Blattberg, R., and N. Gonedes, 974, A coparison of the stable and Student distributions as statistical odels for stock prices. Journal of Business 47, Boothe, P., and D. Glassan, 987, The statistical distribution of exchange rates. Journal of International Econoics 22, Clark, P.K., 973, A subordinated stochastic process odel with finite variance for speculative prices. Econoetrica 4, DuMouchel, W.H., 975, Stable distributions in statistical inference: 2. Inforation fro stably distributed saples. Journal of the Aerican Statistical Association 70, Faa, E.F., 965, The behavior of stock prices. Journal of Business 38, Hageran, R.L., 978, More evidence on the distribution of security returns. Journal of Finance 33,

27 Jarque, C.M., and A.K. Bera, 987, A test for norality of observations and regression residuals. International Statistical Review 55, Judd, K.L., 999, Nuerical ethods in econoics. MIT Press, Cabridge. Ki, Y.I., and J.H. McCulloch, 2007, The skew-student distribution with application to U.S. stock arket returns and the equity preiu. Ohio State University. Mandelbrot, B., 963, The variation of certain speculative prices. Journal of Business 36, McCulloch, J.H., 996, Financial applications of stable distributions, in: G.S. Maddala and C.R. Rao, (Eds.), Handbook of statistics, Vol. 4: Statistical ethods in finance, Elsevier Science, Asterda. McCulloch, J.H., 997, Measuring tail thickness to estiate the stable index A critique. Journal of Business & Econoic Statistics 5, McCulloch, J.H., 998, Nuerical approxiation of the syetric stable distribution and density, in: R. Adler, R. Feldan, and M.S. Taqqu (Eds.), A practical guide to heavy tails, Burkhäuser, Boston. Nelson, D.B., 99, Conditional heteroskedasticity in asset returns: A new approach. Econoetrica 59, Neyan, J., 937, Sooth test for goodness of fit. Skandinavisk Aktuarietidskrift 20, Nolan, J.P., 997, Nuerical calculation of stable densities and distribution functions. Counications in Statistics Stochastic Models 3, Percy, E.R., Jr., 2006, Corrected LM goodness-of-fit tests with application to stock returns, < Perry, P.R., 983, More evidence on the nature of the distribution of security returns. Journal of Financial and Quantitative Analysis 8, Praetz, P.D., 972, The distribution of share price changes. Journal of Business 45, Rayner, J.C.W., and D.J. Best, 989, Sooth tests of goodness of fit. Oxford University Press, Oxford. Saorodnitsky, G., and M.S. Taqqu, 994, Stable non-gaussian rando processes: Stochastic processes with infinite variance. Chapan and Hall, New York. Wilkinson, J.H., 963, Rounding Errors in algebraic processes. Prentice-Hall, Englewood Cliffs, NJ. 25

28 TYPOGRAPHIC NOTES The JE ay wish to use a double stroke font such as LaTeX Bbold or Blackboard Bold for the expectations operator E in the equation following (), as well as on p. 9. This is unfortunately not available in MS Word Equation Editor. Likewise, the JE ay wish to use a cursive font such as Script MT Bold (I) for the inforation atrix I in the equation following (), as well as in (2), (3), and several places on p. 9. However, since the identity atrix is not used in this paper, no confusion arises fro using I. Again, this font is not available in MS Word Equation Editor. Every attept has been ade to use bold face for vectors and atrices. If there is any abiguity, please contact the authors. Vectors and atrices, when defined in ters of their eleents, are consistently defined using large parentheses, as in Sion and Blue, Matheatics for Econoists. A subscript defines the range of the subscripts, as requested by Referee. If the JE would prefer a different notation, such as square brackets as in Greene s Econoetric Analysis, the authors will be happy to work with the production editor on this. 26