By

Mukhtar M. Ali

Department of Economics

University of Kentucky

Lexington, KY 40506

E-mail: MMALI1@POP.UKY.EDU

Preliminary. Not to be quoted without February, 1996

permission from the author. Revised Oct., 1996

SUMMARY

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exists a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.

JEL Classification: C13, C22

Key Words: Unit Root; Saddlepoint Approximation; Asymptotic expansion.

Consider the first-order autoregressive model

(1.1) y_t = ßy_t-1 + e_t, t = 1, 2, ..., n

where ß is a real constant and the errors e_t's are identically independently distributed normal variables each with mean = 0 and variance = s_e². The initial observation, y₀ can be treated as fixed or stochastic. When y₀ is stochastic, it is assumed to have a normal distribution with mean 0 and variance s_e²/(1 - ß²), ½ß½ < 1, and is independent of e_t's. For ease of exposition, these cases are identified with three models, each corresponding to a different nature of y₀. When y₀ is fixed at 0, it is named Model A; and Model B when y₀ is fixed at a non-zero real constant; and Model C when y₀ is N(0, s_e²/(1 - ß²)) and ½ß½ < 1. When y₀ is stochastic, the model is stationary. When y₀ is fixed, if ½ß½ < 1, the model is stationary asymptotically. If ½ß½ = 1, this is a well-known random walk model and if ½ß½ > 1, the model is explosive. The random walk model implies unit root hypothesis. Recently, there has been an enormous interest in testing for unit root. They include, among others, Dickey (1976), Dickey and Fuller (1979, 1981), Evans and Savin (1981, 1984), Fuller (1976), Hasza and Fuller (1979), Perron and Phillips (1987), Phillips and Perron (1988), Schwert (1987) and Stock and Watson (1989). Diebold and Nerlove (1990) have given an excellent survey of works in this area.

The unknown parameter ß is customarily estimated by its least squares estimator

(1.2) .

Under the assumption that e_t's are normally distributed, this is also the maximum likelihood estimate of ß. The distribution of has been studied extensively. Unfortunately, the exact distribution of , in closed form is unknown. Asymptotically, it has a normal distribution (Mann and Wald, 1943) if ½ß½ < 1, a Cauchy distribution (White, 1958) if ½ß½ > 1 and a non-standard distribution (White, 1958, Rao, 1978) if ½ß½ = 1. These asymptotic distributions can be used to approximate the finite sample distribution of . Such an approximation results in a nonsmooth transformation from a normal distribution to a non-standard distribution to a Cauchy distribution. Because the exact distribution of is continuous for all values of , this suggests that the asymptotic distribution would not approximate adequately the finite sample distribution especially near the discontinuity point of = 1. In fact it has been found that, unless ß is close to zero, these asymptotic distributions do not approximate well the distribution in finite samples (Evans and Savin, 1981; Tsui, 1989). The non-standard limiting distribution when ½ß½ = 1 (Rao, 1978) seems to approximate well the finite sample distribution when ½ß½ is close to 1, but it is too complicated for practical use. An accurate approximation to this limiting distribution can, however be obtained from the asymptotic expansion given by Abadir (1993) .

The distribution of is not known in closed form but it can be obtained numerically. The exact distribution has been numerically computed by Phillips (1977, 1978), Evans and Savin (1981) for selected values of sample size n and the autoregressive coefficient ß, and comprehensively for a wide range of (n, ß) by Tsui (1989) and Tsui and Ali (1994). However, these numerical approaches are often computationally demanding and expensive even with modern high speed computers. Besides the numerical methods, several authors (Dickey, 1976; Fuller, 1976) have performed Monte Carlo experiments to tabulate the distribution in the case of ß = 1 and y₀ = 0 (special case of Model A). These distributions can be used to test for unit root hypothesis but are not much of use for further inference about ß.

As an alternative to numerically computing (or simulating) the exact distribution, several authors have attempted to obtain convenient approximations to the distribution of in finite samples. Phillips (1977, 1978), Satchell (1984), and Tsui and Ali (1992) have examined approximations by Edgeworth expansion and found them, except for ß close to 0 and only at the center of the distribution, to be unsatisfactory. Tsui and Ali (1992) examined also approximations by Cornish-Fisher-type (Cornish and Fisher, 1937; Fisher and Cornish, 1960; Hill and Davis, 1968) expansions and the four-parameter Pearson distributions. Accuracy of these approximations was found to depend substantially on sample size and the values of the autoregressive coefficient. None of these approximations was found to be reliable when the autoregressive coefficient is moderately large and the sample size is small. In the case of Model C, Phillips (1978) derived a saddlepoint approximation to the probability density function. An approximation to the distribution can be obtained by numerically integrating this approximate density function. He found this approximation to be exceptionally accurate, certainly for sample sizes as large as 30. Unfortunately, the approximation was not defined over a sizable region of the tail for values of the autoregressive parameter greater than 0.4. For the same case of Model C, Wang (1992) derived a saddlepoint approximation to the distribution function which is available over the entire range of . Wang's approximation was found to be exceptionally accurate. Lieberman (1994) derived an alternative saddlepoint approximation to the probability density of which is also available over the entire range of . From an illustrative check for the accuracy of the distribution function derived from this approximate density, the approximation was found, for the Model C, to be excellent for both n = 10 and n = 30 and for all four ß values of .2, .4, .6 and .8 that were examined. The approximation seems promising but its accuracy has been tested for only a few values of n and of ß and for only the model C. Furthermore, this approximation can be computationally demanding (even to the point of being impractical), especially for large sample sizes. This is because a crucial step in the implementation of the approximation is to obtain a solution of a highly non-linear equation which requires either a computation of eigenvalues of an nxn matrix or repeated inversion of nxn matrices. Moreover, expensive numerical integration may be needed to obtain the distribution function from the approximate density function.

Phillips (1988) considered a near-integrated random process where the autoregressive parameter is defined by

(1.3) = exp(c/n).

c is a real constant measuring the deviation from the unit root case. Cavanagh (1986), Nabeya and Tanaka (1990) and Perron (1989) examined , in the cases of Model A and Model B, the limiting distribution of under (1.3) as an approximation to the finite sample distribution of . They claim the approximation to be quite good in the case of Model A. It is computationally demanding but this limiting distribution can be computed numerically and it has been tabulated by these authors. Larsson (1995) has, however provided a convenient approximation to this limiting distribution in the case of Model A. This approximation may serve as an approximation to the finite sample distribution of . In search of further approximation to the distribution of , Perron (1991a, b) considered the continuous time Ornstein-Uhlenbeck process dy_t = y_tdt + dw_t, where w_t is a Wiener process. Let be the continuous time maximum likelihood (conditional upon y₀) estimator of based upon a single path of data of length T. He then advanced the exact distribution of as an approximation to the finite sample distribution of . Again, it is computationally demanding but the distribution of can be computed numerically. Perron (1991a, b) have provided some selected percentage points of this distribution.

In summary, it seems, with all the shortcomings, only viable approximations available in the literature are those by Phillips (1978), Perron (1991a), Wang (1992) and Lieberman (1994) in the case of Model C, those by Perron (1991b), Lieberman (1994) and Larsson (1995) in the case of Model A and that by Perron (1991b) in the case of Model B.

In this paper, an uniform asymptotic expansion for the distribution (not the density function) of the least squares estimator of ß is obtained. This expansion is applicable for all the three models, Model A, Model B and Model C. An alternative expression of the joint characteristic function of the numerator and denominator of as a product in trigonometric functions avoids the costly computation of eigenvalues or repeated inversion of high order matrices that requires in implementing Lieberman (1994) approximation. Accuracy of approximation to the distribution by a first few terms of this expansion is investigated for a wide range of values of n and of ß and for all the three models. It is found that the very first term of this expansion approximates well the distribution, especially at the extreme tails. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. The accuracy improves by including further term beyond the first term of this expansion in the approximation. But occasionally the accuracy of such an approximation with additional term(s) deteriorates and sometimes it deteriorates to the point of being useless giving probability value which is outside the range of (0, 1). The first-term approximation compares favorably with the approximations given by Phillips (1978), Wang (1992) and Lieberman (1994) in the case of Model C, is as good as and in general, better than the approximations given by Larsson (1995) in the case of Model A , Perron (1991b) in the case of Model A and Model B and Perron(1991a) in the case of Model C.

The plan of this study is as follows. The uniform asymptotic expansion for the distribution of is obtained in section 2. The accuracy of the expansion to the distribution is examined in section 3. Accuracy of the first-term approximation is then compared with those given by Phillips (1978), Perron (1991a), Wang(1992) and Lieberman (1994) in the case of Model C; Perron (1991b), Lieberman (1994) and Larsson (1995) in the case of Model A; and Perron(1991b) in the case of Model B. These comparisons are also reported in section 3. Examples illustrating the use of the results of this paper are reported in section 4. Some concluding remarks are given in section 5.

2. ASYMPTOTIC EXPANSION

There seems to have been some confusion in the literature in regards to the sample size n. Following Hurwicz (1950), the sample size is taken as the number of stochastic y_t in (1.1). Thus, the sample size is n in Models A and B where y₀ is fixed and it is (n+1) in Model C where y₀ is stochastic. Defining

(2.1) P = and Q = ,

the least squares estimator for all the three models (A, B and C) is = P/Q.

Let y(iu, iv) = E(e^iuP+ivQ) be the joint characteristic function of P and Q. Then, by Theorem 1 of Gurland (1948, p. 229), the cumulative distribution function of is given as

(2.2) G(w) = Pr(< w) = Pr(P/Q < w)

=

A change of variable (new u = old ui) in (2.2) leads to

(2.3) G(w) =

where L is a path of integration made up of two segments: from -¥i to -di, and from di to ¥i. It can be shown that the integrand has a simple pole at u = 0 and y(0, 0) = 1, so that, by Cauchy's integral formula,

(2.4)

where C is any closed curve encircling no singularity other than u = 0, in the positive (counterclockwise) direction. Suppose that the curve C is a circle of radius d, then we can rewrite (2.3) as

(2.5) G(w) = ,

where the new path of integration R is obtained by adding that half of the circle C for which Re(u) > 0 to the original path L. It can be shown that the integrand is analytic for Re(u) > 0. Thus, by the well known theorem of Cauchy, the path of integration R can be modified to obtain

(2.6) G(w) = , c > 0.

It can be seen that the integrand in (2.6) has a simple pole at u = 0 and following Lieberman (1994), it can be shown that the integrand has only one saddlepoint. Defining, h(u) = -(1/n)lny(u, -uw), the integral in (2.6) is, then exactly in the form of that in equation (65) in Rice (1968). Hence, from equation (68) of Rice (1968), the uniform asymptotic expansion for G(w) is given by

(2.7) G(w) = ,

where F(.) and f(.) are, respectively the distribution and probability density function of a standard normal variable,

p_j = , j ³ 0,

u₁ is the saddlepoint, h(0) = 0, h^(r)(u) = , r ³ 0, h⁽⁰⁾ = h, h₁^(r) = h^(r)(u₁), h₁ = h(u₁), h⁽¹⁾(u₁) = 0,

x = (-h₁)^(1/2)u₁/½u₁½, z = u₁(h₁⁽²⁾/2)^(1/2), (q)₀ = 1, (q)_j = q(q + 1) ... (q + j - 1), b₀₀ =1, b_0i = 0, for i ³ 1, and

b_k+1,i+1 = ,

d_s = -2h₁^(s+2)/[(s+2)!h₁⁽²⁾], s ³ 1.

In deriving the expansion in (2.7), it is assumed that the origin, at which there is a simple pole for the integrand in (2.6), does not coincide with the saddlepoint. If the saddlepoint is at the origin, a classical saddlepoint analysis (Lugannani and Rice, 1980, p. 479) can be applied to obtain an asymptotic expansion for G(w) as

(2.8)

where q_i = h⁽ⁱ⁾(0)/(i![h⁽²⁾(0)]^(i/2)), i = 3, 4, ....

3. ACCURACY OF THE EXPANSION

The distribution can be computed numerically from (2.7) (or (2.8)). A major problem is the evaluation of y(u, -uw) and hence of the function h(u) that appears in the integrand in (2.6). Lieberman (1994) obtained y(u, -uw) as a determinant of an nxn matrix depending on w. He also expressed it as an elementary function of eigenvalues of the same nxn matrix. For either case, as it requires determinant or eigenvalues of an nxn matrix for each w, the computation becomes prohibitively time consuming as the sample size n becomes large, say larger than 200. From White (1958), an expression for y(u, -uw) can be derived in closed form. It, however involves in raising some (real) expressions to the power of sample size n. Thus, for large sample sizes, it becomes problematic to maintain reasonable numerical accuracy. Alternatively, following White (1961), y(u, -uw) can be expressed as a polynomial of degree n, the sample size. Unfortunately, this creates computational problems in large sample sizes because this polynomial of high degree often contains a large number of terms each of which is negligible individually but significant collectively. The expression for y(u, -uw) that is found to be most convenient is the one obtained by Tsui (1989) and Tsui and Ali (1994). This expression is in trigonometric functions and is given as

(3.1) y(u, -uw) = D_n(u, -uw)^-1/2, for Model A

= D_n(u, -uw)^-1/2, for Model B

= Ö(1 - ß²)[D_n+1(u, -uw) - ß²D_n(u, -uw)]^-1/2, for Model C

where a = y_0/s_e,

(3.2) D_n(u, -uw) = S_n-1(u, -uw) - (ß + u)²S_n-2(u, -uw), and

(3.3) S_n(u, -uw) = .

Using expression (3.1), it is a routine matter to compute h(u) and its various derivatives that are required to obtain the distribution function from (2.7) (or, (2.8)). In this computations, the basic function to evaluate is S_n(u, -uw) and its derivatives. As S_n(u, -uw) = Sgn[S_n(u, -uw)]½S_n(u, -uw)½, where Sgn(x) = 1, if x ³ 0 and = -1, if x < 0, the function S_n(u, -uw) and its derivatives can be obtained from those of ½S_n(u, -uw)½ and its derivatives. In turn, the function ½S_n(u, -uw)½ and its derivatives may be evaluated utilizing the relation

(3.3) ln½S_n(u, v)½ = .

It may be noted that the saddlepoint u₁ is a solution to the equation

(3.4) h⁽¹⁾(u₁) = 0.

The equation (3.4) can be solved by the iterative Newton-Raphson method starting at u₁ = 0 and modifying, if necessary the step size at each iteration so that y(u, -uw) is positive.

The expansion (2.7) (or, (2.8)) can be used to compute the distribution function, G(w). Unfortunately, most asymptotic expansions are nonconvergent, with the magnitude of successive terms tracking a J curve of initial decline followed by a steep rise. Fortunately, often a few of the beginning terms provide adequate approximation. To check for the accuracy of the approximation to the distribution of by a few terms in the expansion (2.7) (or, (2.8)), we consider three approximations, LEAD, LEAD2 and LEAD3 which are obtained by truncating the expansion (2.7) (or (2.8)) at the leading term, the second term and the third term, respectively. More specifically,

(3.5) LEAD: expansion (2.7) truncated at j = 0 (or, expansion (2.8) truncated to the terms at most of order O(n^-1/2))

(3.6) LEAD2: expansion (2.7) truncated at j = 1 (or, expansion (2.8) truncated to the terms at most of order O(n^-3/2))

(3.7) LEAD3: expansion (2.7) truncated at j = 2 (or, expansion (2.8) truncated to the terms at most of order O(n^-5/2))

To check for the accuracy of these approximations, the exact distributions obtained by evaluating numerically the integral in (2.2) (see Tsui (1989) and Tsui and Ali (1994)) are compared with these approximations. A fortran program is written to implement these approximations and can be obtained on request. For a thorough investigation, the distributions and the approximations were computed comprehensively for Models A, B and C. In all cases, we take various sample size n = 10, 20, 30, 40, 50, 75, 100, 150, 200, 250, 300 and 500. For Models A and B, choices of autoregressive coefficient, ß include 0.4, 0.6, 0.8, 0.9, 0.95, 0.99, 1.0, 1.01. For Model B, the values of the parameter a are: 1 and 4. As for Model C, choices of autoregressive coefficient, ß are: 0.4, 0.6, 0.8, 0.9, 0.95, 0.975, 0.99. In each case, the distribution G(w) is computed at 33 percentile points, w = x/g(n) + ß (x = -16.0, -12.0, -8.0, -6.0, -4.0, -3.5, -3.0, -2.8, -2.6, -2.4, -2.2, -2.0, -1.8, -1.6, -1.4, -1.2, -1.0, -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 and 4.0) where g(n) = , if ½ß½ < 1, = , if ½ß½ = 1.0 and = , if ½ß½ > 1. It may be worth observing that in this experimentation, we have taken only positive values of ß. This is because the distribution of for a given ß is the mirror-image of that for - ß (see Cryer, Nankervis and Savin, 1989). To save space, Tables 1, 2 and 3 reproduce some selected but representative results for Models A, B and C, respectively. All the exact and approximate distributions that are computed can be obtained upon request.

Comparing the approximations with the exact, some definite conclusions can be drawn. In particular, it is found that, for all cases considered, LEAD approximates exceptionally well, especially at the tails of the distributions where most of the inferential interests lie. For example, for sample size as small as 10, the LEAD approximations of 0.0223, 0.0177 and 0.0357 to Pr(g(n)( - ) < x) for x = -6.0 and = 1.0 in Model A, for x = -6.0, = 1.0 and = 1 in Model B and x = -6.0, = 0.95 in Model C , respectively match quite closely to the corresponding exact values of 0.0208, 0.0165 and 0.0331 (see Tables 1, 2 and 3). As can be seen from the entries in Tables 1, 2, and 3, LEAD approximation matches, in almost all the cases to the second decimal place and quite often to the third decimal place to the corresponding exact value. Almost without exception, the LEAD approximates exceptionally well at the tails of the distribution (where most of the inferential interests lie) matching the exact value at the third decimal place. Quite often LEAD2 or LEAD3 provides more accurate approximation than that given by LEAD. Thus, for example, while the exact value of Pr(g(n)( - ) < x) for n = 10, x = -6.0, = 1.0 in Model A is 0.0208, the approximate values from LEAD, LEAD2 and LEAD3 are, respectively 0.0223, 0.0203 and 0.0211 (see Table 1). Unfortunately, in a number of cases the accuracy of LEAD2 and LEAD3 deteriorates to the point of being useless giving probability outside the range of (0, 1). For example, LEAD approximation of 0.6428 to Pr(g(n)( - ) < x) for x = -0.4, = 1.01, n = 10 in Model A (see Table 1) matches well the exact value of 0.6396 but both approximations LEAD2 and LEAD3 provide values which are outside the range of (0, 1).

Preceding analysis suggests that, for all practical purposes, LEAD provides excellent approximation to Pr(g(n)( - ) < x) for all three Models A, B and C. To shed further light on its accuracy, in what follows we compare its accuracy with several other approximations that are available in the literature, namely the approximations by Perron (1991b), Lieberman (1994) and Larsson (1995) in the case of Model A; Perron (1991b) in the case of Model B; and Phillips (1978), Perron (1991a), Wang (1992) and Lieberman (1994) in the case of Model C.

In the case of Model A, Perron (1991b) provides an approximation to the distribution of the least squares estimator and tabulates a selected set of percentage points as approximations to the percentage points of the distribution of . These percentage points were tabulated for a number of parameter values of c where = exp(c/n). At these percentage points, we evaluate the Lieberman (1994), Larsson (1995) and our LEAD approximations to the the distribution function for . We also obtain "exact" evaluation of the distribution function at these percentage points by simulation with one-half of one million replications. A selected but representative set of these evaluations of the distribution function is reported in Table 4. Lieberman (1994) approximation matches the LEAD to the fourth decimal place throughout our comparison and hence it is not shown in this table. It can be seen from the entries in this table that Larsson (1995) approximation is not available for a part of the region of the distribution function but whenever it is available it provides an accurate approximation to the Perron (1991b) approximation; and that LEAD approximates well and matches the distribution function, almost always to the second decimal place. Accuracy of Perron (1991b) or Larsson (1995) approximation is rather disappointing. For example, for the case when c = -5.0 (see Panel A in the table), at the percentage point of -10.4633, while the exact evaluation of the distribution function, at n = 10, = 0.6065 is 0.0039, Perron and Larsson approximations are , respectively 0.050 and 0.0580. The accuracy of both Perron and Larsson approximations improves as the sample size and/or is increased. Thus, in the above example, at n = 50, = 0.9048, the exact evaluation of the distribution function is 0.0358 which is closer to the Perron and Larsson approximations of 0.050 and 0.0580 than it was when n = 10, = 0.6065. Even then, for a sample of size as large as 50, the accuracy of both Perron and Larsson approximations is questionable. However, it may be of some comfort to know that both Perron and Larsson provide more accurate approximation to the distribution at the right tail than at the left tail and both approximations are reasonably accurate when and/or n are sufficiently large. Thus, in the case of c = 2.0, n = 50 and = 1.041, at the percentage point of 0.7585, the exact evaluation of the distribution function of 0.9425 is reasonably close to the Perron approximation of 0.95 and also to the Larsson approximation of 0.9332.

In the case of Model B, Perron (1991b) provides an approximation to the distribution of the least squares estimator and tabulates a selected set of percentage points as approximations to the percentage points of the distribution of . These percentage points were tabulated for a number of parameter values of c and where = exp(c/n) and = n^1/2. At these percentage points, we evaluate our LEAD approximation to the distribution function for . We also obtain "exact" evaluation of the distribution function at these percentage points by simulation with one-half of one million replications. A selected but representative set of these evaluations of the distribution function is reported in Table 5. Examining the entries, it is clear that, at every percentage point, the LEAD approximation outperforms the Perron approximation. While the LEAD approximation matches the exact evaluation, almost always to the third decimal place and often to the fourth decimal place, the Perron approximation fails often to match even to the second decimal place. Accuracy of the Perron approximation improves as the sample size is increased but it has a questionable accuracy even for a sample of size as large as 50. For example, for c = -5.0, = 0.5, n = 50, = 0.9048, = 3.54, at the percentage point of -8.7550, while (see Table 5, Panel A) the exact evaluation of the distribution is 0.0369, the Perron approximation is 0.050.

In the case of Model C, Perron (1991a) provides an approximation to the distribution of the least squares estimator and tabulates a selected set of percentage points as approximations to the percentage points of the distribution of . These percentage points were tabulated for a number of parameter values of c where = exp(c/n). At these percentage points, we evaluate our LEAD approximation to the distribution function for . We also obtain "exact" evaluation of the distribution function at these percentage points by simulation with one-half of one million replications. A selected but representative set of these evaluations of the distribution function is reported in Table 6. Examining the entries, it is clear that, at every percentage point, the LEAD approximation outperforms the Perron approximation. As in the case of Model B, while the LEAD approximation matches the exact evaluation, almost always to the third decimal place and often to the fourth decimal place, the Perron approximation fails often to match even to the second decimal place except at the right tail of the distribution. Accuracy of the Perron approximation improves as the sample size is increased but it has a questionable accuracy even for a sample of size as large as 50. For example, for c = -10.0, n = 50, = 0.8187, at the percentage point of -11.834, while (see Table 6, Panel A) the exact evaluation of the distribution is 0.0278, the Perron approximation is 0.050.

There are several other approximations in the case of Model C. Phillips (1978), Wang (1992) and Lieberman (1994) have provided approximations to the Pr. For a number of sample sizes of n , a variety of values of , and at a number of percentage points, x, Phillips (1978) tabulates his approximation (wherever it is available) to this probability along with its exact value. We evaluate our LEAD approximation for each of these combinations of n, and x. A selected but representative set of these evaluations is reported in Table 7. Lieberman (1994) approximation matches the LEAD to the fourth decimal place throughout our comparison and hence it is not shown in the table. Note, however that the approximations reported by Lieberman (1994, Table 1) do not match the corresponding LEAD approximations but they match the LEAD approximations to the left-tailed probability, namely the Pr. It is clear from the entries in Table 7 that all four approximations, Phillips (1978, whenever it is available), Wang (1992), Lieberman (1994) and the LEAD are highly accurate and they are all of comparable accuaracy.

Preceding analysis suggests that, for all practical purposes, the distribution of the least squares estimator , G(w) = Pr( < w) can be well approximated by the first term in the expansion (2.7) (or, (2.8)), specifically by

(4.1) AG(w) = F() - f()(), if the saddlepoint u₁ ¹ 0

= 0.5 + (2pn)^-½q₃, if the saddlepoint u₁ = 0

where u₁ is the saddlepoint satisfying h⁽¹⁾(u) = 0, h(u) is as defined preceding the equation (2.7), h₁ = h(u₁), x = (-h₁)^(1/2)u₁/½u₁½, z = u₁(h₁⁽²⁾/2)^(1/2), and q₃ = h⁽³⁾(0)/(6[h⁽²⁾(0)]^(3/2)). The approximation is applicable for all the Models A, B and C and for all possible values of the parameter ß (½ß½ < 1, ½ß½ = 1, or ½ß½ > 1).

The approximation AG(w) can then be used to make inference about the parameter ß. In most applications, it is of interest to know whether ½ß½ < 1, or, ½ß½ = 1, or ½ß½ > 1. Such inference can be made from an appropriate confidence interval estimate for ß. These confidence interval estimates can be obtained using the approximation AG(w). It can be shown that the lower and upper limit of the central 100(1 - g)% confidence interval are, respectively the solutions of

(4.2) AG() = 1 - g/2, and

(4.3) AG() = g/2

where is the least squares estimate of ß. These equations can be solved by any of a variety of successive approximation methods (see Abramowitz and Stegun, 1970, p. 18). A fortran program which can be obtained on request is written to solve these equations using the Newton's Rule. To start the approximation method, both the lower and upper limit can be taken to be the estimate .

Confidence interval is a convenient tool to make inference. However, if there is a specific inference needs to be made, it may be more convenient to make such inference by testing specific hypothesis. Thus, if one is interested to know whether there is a root = 1 in the autoregressive polynomial or not, one may test the hypothesis H₀: ß = 1, against the alternatives H_a: ß < 1. We would reject H₀ if the p-value given by Pr( < sample ½ ß = 1) is smaller than the level of significance. Using our approximation AG(w) to the distribution of , this p-value can be computed.

To illustrate the use of these results, we have analyzed three sets of data generated from the Model A. In each case, we have chosen n = 25 and s_e = 1. The first data set is generated from the model with ß = .95 (asymptotically stationary model), the second with ß = 1 (random-walk model) and the third with ß = 1.05 (explosive model). The three data sets, estimate based on these data and the central 95% confidence interval estimates are found as follows.

Data Set 1: 0.86, 1.26, 2.39, 2.60, 2.81, 4.15, 3.36, 1.25, 1.17, 0.16, -0.09, 0.54, -0.57

-2.62, -3.10, -1.30, 0.19, 1.56, 1.60, 1.49, 3.62, 3.96, 3.03, 2.49, 3.64

= 0.930; 95% confidence interval: [0.772, 1.186]

The p-value, in testing H₀: ß = 1 against H_a: ß < 1, is 0.371

Data Set 2: 0.86, 1.31, 2.50, 2.82, 3.16, 4.64, 4.06, 2.12, 2.11, 1.15, 0.91, 1.54, 0.45,

-1.62, -2.23, -0.59, 0.83, 2.21, 2.33, 2.30, 4.51, 5.03, 4.29, 3.91, 5.18

= 0.991; 95% confidence interval: [0.874, 1.204]

The p-value, in testing H₀: ß = 1 against H_a: ß < 1, is 0.626

Data Set 3: 0.86, 1.35, 2.61, 3.06, 3.56, 5.22, 4.89, 3.20, 3.34, 2.56, 2.44, 3.19, 2.27

0.31, -0.29, 1.33, 2.83, 4.35, 4.68, 4.89, 7.34, 8.23, 7.91, 7.91, 9.58

= 1.066; 95% confidence interval: [0.989, 1.222]

The p-value, in testing H₀: ß = 1 against H_a: ß < 1, is 0.965

Based on these confidence intervals and p-values, one would not reject the hypothesis of unit root in all these three cases. However, for data set 2, unit root is located near to the center of this interval; for data set 1, it is close to the upper limit of the interval and for data set 3, it is close to the lower limit of the interval. This evidence may be interpreted to mean that the model generating the data set 1 is likely to have root (the parameter ß) less than 1, that generating the data set 2 is likely to have unit root and that generating the data set 3 to have root larger than 1.

5. CONCLUDING REMARKS

Autoregressive models have been found to be most prominent among models in describing time movement of a time series variable. Inference on the parameters of these models are then of utmost interest to the practitioners. Often, such inference has been based on the least squares estimators. As the exact distributions of these least squares estimators are rarely known, the practitioners have been forced to rely on the known asymptotic distributions for inferential purposes. There are at least two disadvantages to such procedures. First, the asymptotic distributions are either too complicated for practical use or provide impractical poor approximations to exact distributions, especially for sample sizes that are usually available in practice. Second, the form of the asymptotic distributions depend upon the parameter values. More specifically, asymptotic distribution that is appropriate when all the roots of the autoregressive polynomial are less than 1 in magnitude is different from the one that is appropriate if at least one of the roots is equal to 1, and these asymptotic distributions are, in turn different from the one appropriate when some of the roots are greater than 1 in magnitudes. This nonsmooth transformation of the asymptotic distribution from the case when the roots are less than 1 to the case when some roots are equal to 1 to the case when some roots are greater than 1 is, to say the least counterintuitive and it makes it essential prerequisite to have the knowledge of any root of the autoregressive polynomial to be equal to 1 or greater than 1 before one can use the appropriate asymptotic distribution for inference.

This paper considers the autoregressive model of order one. An uniform asymptotic expansion for the distribution of the least squares estimator for the autoregressive coefficient is derived. This is a valid expansion irrespective of the size of the root of the autoregressive polynomial. Unfortunately, this asymptotic expansion is, like most asymptotic expansions nonconvergent. Fortunately, however, it is found, after a detailed investigation that the leading term of the expansion provides excellent approximation to the exact distribution. Thus, this approximation can be used, for all practical purposes to make inference. To implement this approximation, one is required to evaluate a certain characteristic function and its first two derivatives. A convenient closed-form expression for the characteristic function is given which should facilitate such computations.

There are a number of alternative approximations to the distribution of the least squares estimator. Of these approximations, the most promising ones seem to be those given by Perron (1991b), Lieberman (1994) and Larsson (1995) in the case of Model A; Perron (1991b) in the case of Model B; and Phillips (1978), Perron (1991a), Wang (1992) and Lieberman (1994) in the case of Model C. A thorough comparison of accuracy of these approximations with that of our approximation (LEAD), we find that the LEAD is exceptionally accurate, and its accuracy is far superior to those given by Perron (1991b) and Larsson (1995) in the case of Model A, Perron (1991b) in the case of Model B, and Perron (1991a) in the case of Model C; and is as accurate or better than those given by Lieberman (1994) in the case of Model A and Phillips (1978), Wang (1992) and Lieberman (1994) in the case of Model C. In short, in the case of Model B, approximation from Perron (1991b) has questionable accuracy and there is no rival to the LEAD approximation. In the case of Model A, approximations from Perron (1991b) and Larsson (1995) have questionable accuracy, and both the LEAD and the approximation from Lieberman (1994) provide excellent accuracy. Thus, in this case of Model A, there are two excellent approximations. However, on the ground of computational convenience, the LEAD is preferable to that given by Lieberman (1994), especially if the sample size is large. Finally, in the case of Model C, approximation from Perron (1991a) provide questionable accuracy but the LEAD and the remaining three approximations from Phillips (1978), Wang (1992) and Lieberman (1994) provide excellent accuracy. Of the four viable alternative approximations, the approximation from Phillips (1978) is not available for a sizable region in the tail of the distribution where most of the inferential interest lie; the computational burden to implement the approximation from Lieberman (1994) can be prohibitive, especially if the sample size is large; and computationally, the approximation from Wang (1992) is as convenient as the LEAD. Thus, in the case of Model C, the LEAD and the approximation from Wang (1992) are preferable to all the others and there is not much reason to prefer Wang (1992) to the LEAD . In summary, the LEAD provides the most desirable approximation in all the cases of Model A, Model B and Model C and only in the case of Model C, the approximation from Wang (1992) can be as desirable as the LEAD.

Table 1: Exact and Approximations (LEAD, LEAD2, LEAD3) to Pr(g(n)( - ) < x)

x ß = 1.0 ß = 1.01

EXACT LEAD LEAD2 LEAD3 EXACT LEAD LEAD2 LEAD3

-16.0 0.0000 0.0000 0.0000 0.0000 0.2160 0.2333 0.2169 0.2034

-12.0 0.0001 0.0001 0.0001 0.0001 0.2861 0.3045 0.2930 0.2755

-8.0 0.0047 0.0050 0.0046 0.0047 0.3800 0.3943 0.3906 0.3754

-6.0 0.0208 0.0223 0.0203 0.0211 0.4382 0.4482 0.4480 0.4344

-4.0 0.0730 0.0791 0.0705 0.0739 0.5032 0.5096 0.5121 0.5012

-3.5 0.0975 0.1059 0.0942 0.0975 0.5205 0.5263 0.5293 0.5187

-3.0 0.1293 0.1405 0.1257 0.1265 0.5382 0.5435 0.5471 0.5381

-2.8 0.1445 0.1570 0.1410 0.1399 0.5455 0.5506 0.5543 0.5469

-2.6 0.1614 0.1753 0.1584 0.1545 0.5528 0.5577 0.5616 0.5569

-2.4 0.1802 0.1954 0.1780 0.1710 0.5602 0.5649 0.5689 0.5689

-2.2 0.2011 0.2175 0.2002 0.1897 0.5678 0.5723 0.5763 0.5849

-2.0 0.2243 0.2417 0.2252 0.2115 0.5754 0.5797 0.5838 0.6092

-1.8 0.2501 0.2683 0.2533 0.2370 0.5831 0.5873 0.5870 *

-1.6 0.2789 0.2974 0.2845 0.2669 0.5909 0.5948 0.5956 *

-1.4 0.3111 0.3292 0.3191 0.3013 0.5988 0.6025 0.6035 *

-1.2 0.3471 0.3640 0.3571 0.3401 0.6068 0.6103 0.6101 *

-1.0 0.3876 0.4022 0.3984 0.3828 0.6149 0.6182 0.6131 *

-0.8 0.4328 0.4441 0.4433 0.4292 0.6230 0.6261 0.6031 *

-0.6 0.4826 0.4903 0.4920 0.4812 0.6313 0.6342 0.5265 *

-0.4 0.5361 0.5414 0.5450 0.5357 0.6396 0.6428 * *

-0.2 0.5939 0.5976 0.5989 * 0.6480 0.6558 * *

0.0 0.6566 0.6586 0.6646 0.6550 0.6565 0.6585 0.6645 0.6550

0.2 0.7227 0.7226 0.7304 0.6487 0.6650 0.6624 * *

0.4 0.7883 0.7859 0.7946 0.7874 0.6736 0.6745 * *

0.6 0.8475 0.8433 0.8531 0.8468 0.6823 0.6833 0.7737 *

0.8 0.8952 0.8904 0.8999 0.8944 0.6910 0.6917 0.7169 *

1.0 0.9300 0.9255 0.9336 0.9292 0.6997 0.7002 0.7116 *

1.2 0.9536 0.9500 0.9562 0.9530 0.7085 0.7087 0.7148 0.9275

1.4 0.9691 0.9664 0.9709 0.9687 0.7173 0.7172 0.7191 *

1.6 0.9792 0.9772 0.9804 0.9790 0.7260 0.7256 0.7338 0.6746

1.8 0.9858 0.9843 0.9866 0.9857 0.7348 0.7340 0.7422 0.7135

2.0 0.9902 0.9891 0.9907 0.9901 0.7435 0.7424 0.7508 0.7339

4.0 0.9995 0.9995 0.9995 0.9995 0.8265 0.8224 0.8327 0.8260

g(n) = , if ½ß½ < 1, = , if ½ß½ = 1 and = , if ½ß½ =1; EXACT is the exact probability computed by numerical integration (Tsui, 1989); LEAD, LEAD2 and LEAD3 are, respectively the approximations from the leading one, two and three terms in the expansion (2.7) or (2.8).

* the number is outside the range of (0, 1).

Table 2: Exact and Approximations (LEAD, LEAD2, LEAD3) to Pr(g(n)( - ) < x)

x ß = 1.0 ß = 1.01

EXACT LEAD LEAD2 LEAD3 EXACT LEAD LEAD2 LEAD3

-16.0 0.0000 0.0000 0.0000 0.0000 0.1967 0.2130 0.1969 0.1839

-12.0 0.0001 0.0001 0.0001 0.0001 0.2659 0.2839 0.2727 0.2533

-8.0 0.0035 0.0037 0.0034 0.0035 0.3612 0.3754 0.3730 0.3548

-6.0 0.0165 0.0177 0.0162 0.0167 0.4216 0.4311 0.4331 0.4163

-4.0 0.0622 0.0672 0.0601 0.0631 0.4901 0.4956 0.5008 0.4872

-3.5 0.0845 0.0916 0.0816 0.0850 0.5085 0.5133 0.5192 0.5055

-3.0 0.1140 0.1239 0.1106 0.1124 0.5274 0.5317 0.5382 0.5254

-2.8 0.1284 0.1395 0.1249 0.1251 0.5352 0.5392 0.5459 0.5341

-2.6 0.1445 0.1570 0.1413 0.1390 0.5431 0.5469 0.5538 0.5437

-2.4 0.1625 0.1763 0.1599 0.1544 0.5511 0.5546 0.5617 0.5545

-2.2 0.1827 0.1979 0.1812 0.1719 0.5592 0.5625 0.5697 0.5678

-2.0 0.2054 0.2218 0.2056 0.1923 0.5674 0.5705 0.5778 0.5862

-1.8 0.2308 0.2482 0.2333 0.2166 0.5758 0.5787 0.5859 0.6157

-1.6 0.2594 0.2774 0.2646 0.2455 0.5842 0.5869 0.5903 *

-1.4 0.2916 0.3095 0.2998 0.2796 0.5928 0.5953 0.5994 *

-1.2 0.3281 0.3450 0.3387 0.3190 0.6015 0.6037 0.6075 *

-1.0 0.3696 0.3842 0.3817 0.3630 0.6103 0.6123 0.6129 *

-0.8 0.4166 0.4276 0.4287 0.4114 0.6192 0.6210 0.6092 *

-0.6 0.4689 0.4760 0.4801 0.4642 0.6282 0.6298 0.5588 *

-0.4 0.5257 0.5301 0.5364 0.5235 0.6373 0.6390 0.0750 *

-0.2 0.5879 0.5905 0.5943 * 0.6465 0.6522 * *

0.0 0.6561 0.6568 0.6656 0.6542 0.6558 0.6564 0.6655 0.6541

0.2 0.7284 0.7270 0.7371 0.6866 0.6652 0.6620 * *

0.4 0.7995 0.7961 0.8065 0.7990 0.6747 0.6742 * *

0.6 0.8618 0.8570 0.8674 0.8611 0.6842 0.6838 0.7505 *

0.8 0.9094 0.9045 0.9138 0.9087 0.6937 0.6931 0.7149 *

1.0 0.9420 0.9379 0.9451 0.9414 0.7033 0.7024 0.7142 0.2491

1.2 0.9629 0.9597 0.9651 0.9625 0.7130 0.7118 0.7187 *

1.4 0.9760 0.9737 0.9774 0.9757 0.7226 0.7210 0.7318 0.6456

1.6 0.9842 0.9826 0.9852 0.9841 0.7322 0.7304 0.7410 0.7039

1.8 0.9894 0.9883 0.9900 0.9893 0.7418 0.7396 0.7503 0.7301

2.0 0.9928 0.9920 0.9932 0.9927 0.7514 0.7489 0.7596 0.7464

4.0 0.9997 0.9996 0.9997 0.9997 0.8405 0.8355 0.8470 0.8397

g(n) = , if ½ß½ < 1, = , if ½ß½ = 1 and = , if ½ß½ =1; EXACT is the exact probability computed by numerical integration (Tsui, 1989); LEAD, LEAD2 and LEAD3 are, respectively the approximations from the leading one, two and three terms in the expansion (2.7) or (2.8).

* the number is outside the range of (0, 1).

Table 3: Exact and Approximations (LEAD, LEAD2, LEAD3) to Pr( )

x ß = 0.95 ß = 0.99

EXACT LEAD LEAD2 LEAD3 EXACT LEAD LEAD2 LEAD3

-16.0 0.0001 0.0001 0.0001 0.0001 0.0086 0.0094 0.0083 0.0088

-12.0 0.0012 0.0013 0.0012 0.0012 0.0209 0.0232 0.0199 0.0215

-8.0 0.0123 0.0131 0.0120 0.0124 0.0505 0.0566 0.0481 0.0495

-6.0 0.0331 0.0357 0.0322 0.0336 0.0803 0.0899 0.0790 0.0744

-4.0 0.0852 0.0923 0.0824 0.0854 0.1332 0.1450 0.1376 0.1261

-3.5 0.1077 0.1168 0.1046 0.1062 0.1530 0.1642 0.1592 0.1469

-3.0 0.1365 0.1479 0.1336 0.1313 0.1771 0.1868 0.1848 0.1720

-2.8 0.1502 0.1625 0.1477 0.1430 0.1882 0.1971 0.1965 0.1835

-2.6 0.1654 0.1787 0.1637 0.1560 0.2003 0.2082 0.2092 0.1961

-2.4 0.1822 0.1964 0.1817 0.1708 0.2136 0.2203 0.2231 0.2099

-2.2 0.2009 0.2159 0.2019 0.1877 0.2283 0.2335 0.2382 0.2252

-2.0 0.2218 0.2373 0.2248 0.2075 0.2446 0.2481 0.2550 0.2423

-1.8 0.2451 0.2609 0.2505 0.2306 0.2627 0.2643 0.2736 0.2616

-1.6 0.2713 0.2868 0.2793 0.2575 0.2832 0.2825 0.2945 0.2835

-1.4 0.3008 0.3154 0.3114 0.2884 0.3063 0.3031 0.3181 0.3086

-1.2 0.3341 0.3470 0.3470 0.3235 0.3328 0.3266 0.3449 0.3376

-1.0 0.3719 0.3820 0.3866 0.3628 0.3634 0.3539 0.3757 0.3712

-0.8 0.4146 0.4209 0.4304 0.4064 0.3992 0.3855 0.4113 0.4104

-0.6 0.4627 0.4644 0.4790 0.4554 0.4408 0.4226 0.4523 0.4564

-0.4 0.5165 0.5131 0.5328 0.5110 0.4891 0.4659 0.4994 0.5177

-0.2 0.5753 0.5673 0.5892 0.9894 0.5439 0.5157 0.5467 *

0.0 0.6385 0.6267 0.6553 0.6353 0.6037 0.5709 0.6117 0.6289

0.2 0.7041 0.6896 0.7207 0.6758 0.6654 0.6294 0.6773 0.0854

0.4 0.7683 0.7526 0.7836 0.7581 0.7246 0.6874 0.7321 0.7474

0.6 0.8261 0.8109 0.8395 0.8169 0.7776 0.7411 0.7858 0.8027

0.8 0.8738 0.8605 0.8846 0.8658 0.8222 0.7881 0.8316 0.8443

1.0 0.9101 0.8995 0.9185 0.9039 0.8583 0.8277 0.8687 0.8760

1.2 0.9365 0.9283 0.9426 0.9322 0.8868 0.8601 0.8977 0.8997

1.4 0.9551 0.9490 0.9594 0.9523 0.9092 0.8864 0.9199 0.9174

1.6 0.9680 0.9635 0.9710 0.9663 0.9266 0.9075 0.9367 0.9311

1.8 0.9770 0.9737 0.9790 0.9759 0.9404 0.9244 0.9495 0.9420

2.0 0.9832 0.9809 0.9847 0.9826 0.9512 0.9380 0.9592 0.9510

4.0 0.9988 0.9987 0.9989 0.9988 0.9912 0.9891 0.9924 0.9905

EXACT is exact the probability computed by numerical integration (Tsui, 1989); LEAD, LEAD2 and LEAD3 are, respectively the approximations from the leading one, two and three terms in the expansion (2.7) or (2.8).

* the number is outside the range of (0, 1).

Table 4: Exact and Approximations (PERRON, LARSSON, LEAD) to

n = 10, = .6065 n = 25, = .8187 n = 50, = .9048

x PERRON LARSSON EXACT LEAD EXACT LEAD EXACT LEAD

-16.7378 0.010 0.0110 0.0000 0.0000 0.0020 0.0020 0.0050 0.0051

-13.1743 0.025 0.0282 0.0004 0.0004 0.0083 0.0088 0.0156 0.0160

-10.4633 0.050 0.0580 0.0039 0.0038 0.0238 0.0250 0.0358 0.0369

-7.7242 0.100 0.1216 0.0242 0.0246 0.0627 0.0655 0.0808 0.0831

2.4561 0.900 NA 0.9127 0.9137 0.9068 0.9095 0.9038 0.9068

3.0637 0.950 NA 0.9573 0.9561 0.9551 0.9556 0.9532 0.9541

3.5239 0.975 NA 0.9761 0.9998 0.9773 0.9768 0.9766 0.9765

4.0342 0.990 NA 0.9869 0.9862 0.9899 0.9893 0.9902 0.9898

n = 10, = 1.0 n = 25, = 1.0 n = 50, = 1.0

x PERRON LARSSON EXACT LEAD EXACT LEAD EXACT LEAD

-13.6919 0.010 0.0115 0.0011 0.0011 0.0049 0.0054 0.0076 0.0078

-10.4399 0.025 0.0297 0.0079 0.0082 0.0169 0.0181 0.0210 0.0221

-8.0383 0.050 0.0616 0.0262 0.0276 0.0392 0.0424 0.0445 0.0478

-5.7133 0.100 0.1307 0.0724 0.0773 0.0871 0.0952 0.0941 0.1014

0.9280 0.900 0.8505 0.8618 0.8578 0.8850 0.8825 0.8931 0.8903

1.2854 0.950 0.9379 0.9151 0.9110 0.9376 0.9340 0.9442 0.9409

1.6122 0.975 0.9711 0.9473 0.9436 0.9656 0.9628 0.9705 0.9682

2.0325 0.990 0.9889 0.9710 0.9687 0.9844 0.9825 0.9875 0.9860

n = 10, = 1.221 n = 25, = 1.083 n = 50, = 1.041

x PERRON LARSSON EXACT LEAD EXACT LEAD EXACT LEAD

-12.0557 0.010 0.0125 0.0040 0.0043 0.0071 0.0079 0.0085 0.0093

-8.8762 0.025 0.0351 0.0174 0.0193 0.0214 0.0242 0.0234 0.0260

-6.5587 0.050 0.0890 0.0438 0.0501 0.0470 0.0537 0.0489 0.0551

-4.3431 0.100 0.9664 0.1002 0.1117 0.0988 0.1116 0.1003 0.1119

0.5191 0.900 0.8376 0.8445 0.8219 0.8788 0.8576 0.8901 0.8692

0.7585 0.950 0.9332 0.9034 0.8860 0.9334 0.9192 0.9425 0.9292

1.0067 0.975 0.9694 0.9398 0.9282 0.9631 0.9550 0.9699 0.9624

1.3550 0.990 0.9884 0.9675 0.9611 0.9831 0.9793 0.9870 0.9839

c is a reparameterization of (see Perron, 1991b) with = exp(-c/n); EXACT is the exact probability computed by simulation with one-half of one million replications; PERRON, LARSSON and LEAD are, respectively the approximations from Perron (1991b), Larsson (1995) and the leading term in the expansion (2.7) or (2.8).

NA: not available

Table 5: Exact and Approximations (PERRON, LEAD) to

n = 10, = .6065, n = 25, = .8187, n = 50, = .9048,

= 1.58 = 2.50 = 3.54

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-13.8955 0.010 0.0000 0.0000 0.0021 0.0021 0.0050 0.0052

-10.9832 0.025 0.0003 0.0003 0.0090 0.0091 0.0160 0.0161

-8.7550 0.050 0.0039 0.0039 0.0253 0.0253 0.0369 0.0371

-6.4899 0.100 0.0250 0.0251 0.0656 0.0660 0.0825 0.0832

2.2695 0.900 0.9244 0.9249 0.9106 0.9129 0.9061 0.9082

2.8257 0.950 0.9684 0.9677 0.9591 0.9598 0.9547 0.9561

3.2413 0.975 0.9855 0.9846 0.9808 0.9807 0.9779 0.9786

3.6788 0.990 0.9937 0.9933 0.9924 0.9921 0.9913 0.9913

n = 10, = 1.00, n = 25, = 1.00, n = 50, = 1.00,

= 1.58 = 2.50 = 3.54

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-10.9570 0.010 0.0011 0.0011 0.0056 0.0055 0.0076 0.0078

-8.3535 0.025 0.0081 0.0083 0.0177 0.0183 0.0213 0.0222

-6.4315 0.050 0.0264 0.0278 0.0405 0.0426 0.0456 0.0479

-4.5709 0.100 0.0731 0.0755 0.0898 0.0954 0.0951 0.1015

0.7643 0.900 0.8774 0.8717 0.8904 0.8870 0.8951 0.8920

1.0481 0.950 0.9303 0.9261 0.9423 0.9392 0.9460 0.9433

1.3042 0.975 0.9607 0.9577 0.9698 0.9673 0.9722 0.9703

1.6338 0.990 0.9814 0.9800 0.9872 0.9857 0.9884 0.9875

n = 10, = 1.2214, n = 25, = 1.0833, n = 50, = 1.0408,

= 1.58 = 2.50 = 3.54

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-9.2554 0.010 0.0040 0.0043 0.0075 0.0080 0.0088 0.0095

-6.7080 0.025 0.0174 0.0194 0.0220 0.0246 0.0236 0.0265

-4.8444 0.050 0.0438 0.0496 0.0477 0.0540 0.0488 0.0556

-3.0206 0.100 0.0986 0.1038 0.1001 0.1068 0.0999 0.1078

0.3756 0.900 0.8646 0.8438 0.8859 0.8663 0.8929 0.8737

0.5401 0.950 0.9232 0.9091 0.9402 0.9271 0.9449 0.9327

0.7143 0.975 0.9579 0.9490 0.9689 0.9612 0.9717 0.9649

0.9680 0.990 0.9811 0.9771 0.9869 0.9835 0.9881 0.9855

c and are reparameterizations of and = y_0/ (see Perron, 1991b) with = exp(-c/n) and = n^1/2; EXACT is the exact probability computed by simulation with one-half of one million replications; PERRON and LEAD are, respectively the approximations from Perron (1991b), and the leading term in the expansion (2.7) or (2.8).

Table 6: Exact and Approximations (PERRON, LEAD) to

n = 10, = 0.3679 n = 25, = 0.6703 n = 50, = 0.8187

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-18.478 0.010 0.0000 0.0000 0.0006 0.0005 0.0032 0.0032

-14.728 0.025 0.0000 0.0000 0.0037 0.0037 0.0111 0.0114

-11.834 0.050 0.0002 0.0002 0.0129 0.0136 0.0278 0.0286

-8.846 0.100 0.0045 0.0044 0.0426 0.0441 0.0688 0.0698

3.793 0.900 0.9553 0.9563 0.9267 0.9278 0.9143 0.9154

4.729 0.950 0.9861 0.9861 0.9704 0.9709 0.9613 0.9621

5.436 0.975 0.9947 0.9945 0.9879 0.9882 0.9828 0.9896

6.154 0.990 0.9979 0.9977 0.9961 0.9961 0.9939 0.9965

n = 10, = 0.9048 n = 25, = 0.9608 n = 50, = 0.9802

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-11.848 0.010 0.0009 0.0009 0.0049 0.0051 0.0073 0.0075

-8.942 0.025 0.0066 0.0068 0.0161 0.0169 0.0203 0.0213

-6.831 0.050 0.0222 0.0229 0.0375 0.0395 0.0434 0.0459

-4.818 0.100 0.0634 0.0657 0.0842 0.0889 0.0922 0.0970

0.971 0.900 0.8864 0.8806 0.8940 0.8892 0.8976 0.8921

1.322 0.950 0.9357 0.9327 0.9447 0.9411 0.9475 0.9439

1.639 0.975 0.9628 0.9609 0.9709 0.9683 0.9730 0.9707

2.042 0.990 0.9814 0.9803 0.9873 0.9858 0.9888 0.9875

n = 10, = 0.9900 n = 25, = 0.9960 n = 50, = 0.9980

x PERRON EXACT LEAD EXACT LEAD EXACT LEAD

-8.168 0.010 0.0037 0.0041 0.0074 0.0077 0.0086 0.0092

-5.634 0.025 0.0154 0.0165 0.0214 0.0227 0.0234 0.0249

-3.906 0.050 0.0380 0.0416 0.0459 0.0496 0.0481 0.0523

-2.405 0.100 0.0880 0.0957 0.0959 0.1042 0.0978 0.1071

0.495 0.900 0.8864 0.8616 0.8942 0.8685 0.8981 0.8708

0.727 0.950 0.9377 0.9233 0.9446 0.9302 0.9474 0.9325

0.971 0.975 0.9654 0.9576 0.9713 0.9636 0.9730 0.9655

1.312 0.990 0.9837 0.9802 0.9877 0.9846 0.9886 0.9859

c is a reparameterization of (see Perron, 1991a) with = exp(-c/n); EXACT is the exact probability computed by simulation with one-half of one million replications; PERRON and LEAD are, respectively the approximations from Perron (1991a), and the leading term in the expansion (2.7) or (2.8).

Table 7: Exact and Approximations (PHILLIPS, WANG, LEAD) to

= 0.2 = 0.8

x EXACT PHILLIPS WANG LEAD EXACT PHILLIPS WANG LEAD

1.0 0.2987 0.2926 0.2987 0.2988 0.3444 NA 0.3582 0.3572

1.4 0.1400 0.1283 0.1400 0.1400 0.2280 NA 0.2371 0.2372

1.8 0.0551 0.0274 0.0551 0.0550 0.1565 NA 0.1628 0.1629

2.2 0.0191 NA 0.0189 0.0190 0.1083 NA 0.1126 0.1126

2.6 0.0060 NA 0.0057 0.0056 0.0745 NA 0.0775 0.0774

3.0 0.0018 NA 0.0015 0.0015 0.0507 NA 0.0522 0.0525

EXACT is the exact probability as reported in Phillips (1978); PHILLIPS is the saddlepoint approximation given in Phillips (1978); WANG is the approximation given in Wang (1992); LEAD is the approximation from the leading term of the expansion (2.7) or (2.8).

NA: not available; the approximate density is not defined.

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