Monte Carlo Methodology and the Finite Sample Properties of Statistics for Testing Nested and Non-Nested Hypothesis

International Finance Discussion Papers

Number 317

December 1987

MONTE CARLO METHODOLOGY AND THE FINITE SAMPLE PROPERTIES OF STATISTICS FOR TESTING NESTED AND NON-NESTED HYPOTHESES

by

Neil R. Ericsson

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

Using recently developed Monte Carlo methodology, this paper investigates the effect of dynamics and simultaneity on the finite sample properties of maximum likelihood and instrumental variables statistics for testing both nested and non-nested hypotheses. Numerical-analytical approximations (response surfaces) to the unknown finite sample size and power functions of those statistics are obtained for dynamic one- and two-equation models. The results illustrate the value of asymptotic theory in interpreting finite sample properties and certain limitations for doing so. Two practical finite sample results arise: the F form of the Wald statistic is strongly favored over its chi-squared form; and the effects of "large-o" and a small effective sample size are particularly pronounced for Sargan’s (1958) instrumental variables statistic and Ericsson's (1983) Cox-type instrumental variables statistic. Re-examining Pesaran and Deaton's (1978) empirical example illustrates the additional information

gained from the instrumental variables statistics.

Keywords and phrases: asymptotic distributions. dynamics. econometrics. encompassing. evaluation criteria. finite sample properties. _infefence. Monte Carlo. non-nested hypotheses. power. response surfaces. simultaneity. simulation. test statistics.

Monte Carlo Methodology and the Finite Sample Properties of Statistics For Testing Nested and Non-nested Hypotheses

by

Neil R. Ericsson-

1. Introduction

Statistical inference has profoundly influenced econometric methodology and practice, both with regard to estimation and with regard to hypothesis testing. Mann and Wald (1943b), Haavelmo (1943, 1944), and Koopmans, Rubin, and Leipnik (1950) systematically exposit the framework for applying both aspects to the modeling of systems of economic relationships. Although the former (estimation) often has taken the more important role in econometrics, extensive testing of econometric models is becoming more common. Several reasons for that include a clearer understanding of the relationships between various econometric estimators, a marked reduction in the computing costs of estimating econometric models (those costs often having been a motivation for deriving different estimators), and a more widespread appreciation of the weaknesses of untested models. (E.g., see Hausman (1975) and Hendry (1976), Hendry and Srba (1980), and Hendry and Mizon

(1978) and Sargan (1980a,b,d).) More extensive testing is of some comfort

Ithe author is a staff economist in the International Finance Division. This research was supported in part by grants from the Social Science Research Council to the Programme in Methodology, Inference and Modelling in Econometrics at the London School of Economics and to the project "The Modelling and Evaluation of Dynamic Econometric Systems" (HR8789) at Nuffield College, Oxford while I was at those institutions. The paper was prepared in part during a visit to the Center for Operations Research and Econometrics, Louvain-la-Neuve. I am grateful for the financial assistance from the $.S.R.C. although the views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting those of the S.S.R.C., the Board of Governors of the Federal Reserve System, or other members of their staffs. Helpful suggestions and comments from Denis Sargan, David Hendry, Julia Campos, Raymond Chapman, Gary Chamberlain, Robert Summers, Ralph Tryon, and two anonymous referees are gratefully acknowledged. I am indebted to Heidi Lyss for preparing the graphs and to Frank Srba and David Hendry for the use of and their help in modifying their computer programs DAGER, GIVE, NAIVE, and PC-GIVE (Hendry and Srba (1977b, 1978, 1979, 1980) and Hendry (1987)). All numerical results were obtained using those programs.

to users of econometric models, particularly in light of the ease with which

seemingly highly significant but nevertheless spurious regression results can be obtained with time series (nb. Yule (1926), Granger and Newbold (1974), Hendry (1980), and Phillips (1986b)). In recognition of the importance of hypothesis testing in econometric modeling, Pesaran (1982), Ericsson (1983), and Godfrey (1983) derive and analyze asymptotic properties of various statistics for testing nested and non-nested hypotheses in systems of economic relationships. However, both analytical and Monte Carlo studies indicate that the presence of dynamics and simultaneity may substantially influence the finite sample properties of statistics for testing nested hypotheses (cf. Phillips (1977, 1980), Sargan (1980c), Mizon and Hendry (1980), and Hendry (1984)); and even for simple static models, considerable discrepancies may exist between the finite sample and asymptotic properties of Cox's statistic for testing non-nested hypotheses (cf. Pesaran (1974, 1982)).

Using recently developed Monte Carlo methodology, this paper investigates the effects of dynamics and simultaneity on the finite sample properties of the statistics discussed in Pesaran (1982) and Ericsson (1983), including the Wald, Cox and F statistics. Section 2 describes the statistics and their asymptotic properties; Section 3, the class of econometric models to be investigated. Monte Carlo studies in econometrics often have been highly imprecise in estimating the underlying finite sample properties and specific to the particular parameter values and sample sizes chosen, so making any conclusions very tentative at best. Hendry (1984) presents a methodology reducing both imprecision and specificity and which aims to obtain “numerical-analytical formulae [response surfaces] which jointly summarize the experimental findings and known analytical results in order to help interpret empirical evidence and to compute outcomes at other

points within the relevant parameter space" (p. 944). That methodology

affects all aspects of Monte Carlo experimentation: design, simulation, and post-simulation analysis. Section 5 considers each of those aspects in turn for a Monte Carlo study of the properties of various instrumental variables test statistics in a dynamic simultaneous two-equation model, following a review in Section 4 of the role of response surfaces in analyzing the Monte Carlo simulations themselves. Section 6 provides a

brief empirical example illustrating the potential practical value of these

statistics. 2. The Test Statistics and Their Asymptotic Properties

This section summarizes existing analytical results for the statistics of interest.” Consider the two non-nested hypotheses

Ho: y = XpBo + Uo Ug ~ D(0,0§*1,) (1) and

H,: y = X,A, + uy uy ~ D(0,of+1,) (2) and the comprehensive hypothesis

Ho: y = Xgfoq + Ug Uy ~ D(0,0$°I1,,) (3) where the dependent variable y is Txl, T being the econometric sample size; x; isa Txk, matrix of regressors and B; the corresponding k,x1 vector of coefficients (i=0,1,2); X, includes all the non-redundant variables in (X9:X1) with £. conformable; and u; is a Txl vector of disturbances distributed with mean zero and variance oFely (i=0,1,2). Two approaches have been suggested for testing H, against H,: "direct" comparison of the non-nested hypotheses and comparison of each non-nested hypothesis with the comprehensive model. For the former, Pesaran (1974) proposes evaluating a

modified likelihood ratio statistic for Hy and H, when Xg and X, are

2 so 4s ; . : : The statistics and most of their analytical properties appear in Pesaran

(1974) and Ericsson (1983). Godfrey's (1983) statistic G, is a linear function of tg, so only one (tg) is considered. *

See MacKinnon, White, and Davidson (1983) for regularity conditions and MacKinnon (1983) for a review of the.subject.

predetermined, following Cox (1961, 1962). Alternatively, because those hypotheses are nested in Hz, the restrictions implied by going from Hz to Hg (or H,) can be tested using the F or Wald statistics.° Those Cox, F, and Wald statistics for testing H) are denoted Dy, f2, and cg. Under Hg and under H, as a local alternative, they each are asymptotically distributed with the central and non-central distributions given in Table 1.4

When simultaneity is present, Cox's test may be inconsistent unless the entire systems of equations from which Hg and H, are drawn are specified and estimated (see Pesaran and Deaton (1978)); likewise, F and Wald tests using the least-squares estimator may be inconsistent. Instrumental variables (IV) statistics provide a convenient alternative. Sargan (1958, 1980c) proposes a xy*-statistic (cg) and the corresponding F-statistic (f,) for testing the specification of an equation after estimation by instrumental variables: cg is the criterion function for the IV estimator, and it and fy, are asymptotically distributed as x?(m-ky,*) and F(m-ky,T-m,*), as given in

5

* Table I, with m being the number of instrumental variables Z . IV

generalizations of f, and cy may be used: those IV statistics also have the

See. Cox (1961, pp. 105-106, 120-122), Dhrymes et al. (1972, pp. 316-317), and Cox and Hinkley (1974, pp. 327-328, 331-337) on the Cox statistic and Fisher (1922), Wald (1943, pp. 469, 479), Stroud (1971), Silvey (1975, pp. 115-116), and Phillips (1986a) on the F and Wald statistics.

Ericsson (1983) presents explicit formulae for inter alia the asymptotic mean and variance of Dp under H, (49, wg) and the asymptotic non-centrality of f, and cg (A,). A second and minor approximation is made (i.e., in addition to the asymptotic one) to obtain those formulae. For brevity’s sake, those approximate asymptotic distributions are referred to as "asymptotic" as well.

Pesaran'’s (1982) derivation using a different local alternative uses only the usual asymptotic approximation but requires at least as many regressors (total) under Hy as regressors in H, but not in Hy. In many cases, that restriction is not satisfied and so Pesaran'’s formulae are not computable. For Pesaran’s (1974) model, the two approximations are numerically similar; cf. Ericsson (1986) and Pesaran (1982, 1987).

The statistics cg and fy and various of their properties are described in Sargan (1958, pp. 401-404; 1959, pp. 93-94, 99-100; 1964, pp. 28-29; 1976b, p. 19; 1980c, pp. 1124, 1136). See also Kiviet (1987, Chapter V).

4a

Table I.

Asymptotic Properties of Statistics for Testing Nested and Non-nested Hypotheses

a

Statistic Asymptotic Distribution Conditions for Asymptotic Equiyalences . b under Hy and H, Name Type Sources Ho Hy

I

Do ML Cox (1961), N(O,1) N(-o ,&o) ‘ Pesaran (1974) X, CZ te, IV Ericsson (1983) N(0,1) N (pig 6) under t, IV Ericsson (1983) N(0,1) N(jb4 4) Ho only Co IV Sargan (1958) x? (m-ky , 0) x? (m-Kg , AQ) c always fy IV Sargan (1980c) F(m-ky ,T-m,0) F(m-ky ,T-m, A) m = ky c, -¢ Wald (1943) x2 (ky “Ky , 0) x? (Ky -Ky 2) J always f, - Fisher (1922) F(kg-Kg,T-k2,0) F(Kg-Ko,T-kg,A2)

I

Notes: a. Two statistics are said to be "asymptotically equivalent" under a given hypothesis if, when rescaled to be O (1) (but not o_(1)) and possibly after some nonlinear transformatibns, they differ only by a scale factor plus terms of o_(1) (nb. Mann and Wald’s (1943a,

p. 218 notation). P The asymptotic properties of the statistics under H, when H, is neither H, nor H, are discussed in Ericsson (1983).

b. The arguments py, 9, Ha, 4, Hes 6, and \, are each a positive rational function of the parameters of H, and of the population second moments of the data.

c. The statistic f, is [co/(m-kg) ]*[(T-m)/(T-kg) ]/[1-¢9/(T-Ko) |}; which is c,/(m-k)) with finite-sample adjustments arising from the finite-sample boundedness of cy. fy is exactly distributed as an F-ratio when Z* = X, and X, is fixed.

d. The Wald statistic and its F-transformation are applicable to

testing hypotheses using a broad class of estimators, including ML and IV; cf. Stroud (1971).

e. The statistic f, is c,/(k,-ko), which is the classical F statistic for testing the exclusion from H, of those variables in X, but not

in X,. f2 is exactly distributed as an F-ratio when Z* = X, and X, is fixed.

asymptotic distributions given in Table I, but with A, a more complicated function of the parameters and second moments. Using the IV criterion function cy in place of the likelihood function, Ericsson (1983) obtains IV statistics (denoted t, and t,) resembling Cox's statistic Dy and which are asymptotically equivalent to it under Hy when the set of instrumental variables includes all the regressors and the regressors are predetermined. However, t, and tg are valid in the presence of simultaneity whereas Dg may not be. Given a suitable set of instrumental variables, the statistics t, and tg are asymptotically distributed as normal variates: standardized under H), and with non-zero means and non-unit variances under H, as a local alternative. The final column of Table I gives conditions for asymptotic equivalences between the various test statistics. Whether they are equivalent or not, the asymptotic powers of t4, tg, Co, C2, f,, f., and (if applicable) Dy can be numerically calculated from the formulae in Table I, given a particular data generation process.° Similar statistics exist for testing the specification of (2) and are denoted by ts, t7, Ci, Cs, f,, f3, and D, (i.e., with incremented subscripts). Finally, the statistics for testing non-nested hypotheses may be interpreted as "variance-encompassing" test statistics or, equivalently, statistics for testing a certain scalar

nonlinear restriction on the hypothesis H,./

3. The Data Generation Process

Using Monte Carlo techniques, Hendry and Harrison (1974) investigate

the properties of single-equation estimators in the context of a dynamic

In this paper, "the power of the statistic cy" means "the power of an appropriate test based on cg", and likewise for the other test statistics. This is done for brevity’s sake, and no ambiguity should arise therefrom.

For extensive discussion on the encompassing approach, see Davidson, Hendry, Srba and Yeo (1978), Davidson and Hendry (1981), Hendry and Richard (1982), Hendry (1983), Mizon (1984), and Mizcn and Richard (1983, 1986).

: . 8 : . . -simultaneous two-equation model. Their model provides a convenient

framework for analyzing the statistics discussed above: it is

ye bY, + cZ, + dy, 4 + ou, u. (4) ~ NID(O,5)

yy = ay, +h we + ve ve (5)

w= AW, 4 + Ve vu 7 IID(0,) (6)

where Cy, ¥,)' and w, are 2x1 and 4x1 vectors of endogenous and exogenous

variables at time t (t=l,...,T); the i,j) element of = is O55 that of 2

is w..; h’ = 1j

diagonal matrix diag(p,:p2:p3:P4); the latent root of (4)-(5) po

(hg, :hee:hgg:hgq), wi (Z,tW5 Wa, Wy.) and h,, = 0; Aisa

(= d/(1-ab)) and all the latent roots of A lie within the unit circle; and

9 , ¢ = * E(uvin) = EQ vig) QO for all t and t*.

The structure studied herein is the dynamic simultaneous two-equation model defined by (4)-(5) with non-zero a, b, d, and 049.9 In order to study size as well as power, one of the non-nested hypotheses is assumed to be correctly specified and, without loss of generality, it is Hj. Thus,

Hy: Y= bY, + eZ, + dy. 4 + U5, > (7)

Bor. Hendry and Srba (1977a), Hendry (1979a), Maasoumi and Phillips (1982), Hendry (1982), and Kiviet (1985). Also, see Mizon and Hendry (1980) and Hendry (1984, pp. 971-972) on the influence of dynamics on the finite sample properties of the Wald statistic in a single-equation context, and Pesaran (1974, 1982), Godfrey and Pesaran (1983), and King and McAleer (1987) on the properties of the F and Cox statistics for a single static equation. See Cox (1962, pp. 414-415, 422-423), Jackson (1968), Atkinson (1970, p. 338), and Pereira (1977, 1978) for Monte Carlo analyses of the Cox statistic in the statistics literature.

equations (4)-(6) above correspond to equations (2)-(5) in Hendry and Harrison (1974), with some slight changes in notation.

Hendry and Harrison's model allows for autoregressive errors on the structural equations whereas (4)-(5) does not. However, noting that autoregressive errors imply a common factor restriction, enough equation dynamics are sufficient to account for such errors; cf. Appendix B.

Except for the inclusion of a constant term, the data generation process for Pesaran’s (1974) Monte Carlo study is a particular case of the model in (4)-(6) with b = d = o,, = 0 and Ps = O for i=1,...,4.

10

A dynamic single-equation model served as a pilot study and is described in Appendix A.

in keeping with the notation of (1) and where (4) is the equation of interest. Although non-nested alternatives to (7) might involve mis-specification of dynamics or simultaneity, falsely included (or excluded) exogenous variables, or any combination thereof, attention is restricted to the (false) hypothesis that

Hi: ¥, = bY, + hi2Wo, + Uy (8) with 7 = corr(Z, ,wo,) > 0. For y close to unity, it may be difficult to

detect which of the two exogenous variables, Z enters the correct

and Wo

t t’

specification. The comprehensive hypothesis (3) is

Ho: y, = bY, + cZ, + Wiow, + dy, yt Yop (9) and will be used for constructing the Wald and F statistics.

The data generation process (or DGP) defined by (4)-(6) and the relationships of interest in (7)-(9) have certain implications for the properties of the statistics being examined. In (8), the exogenous variable Wo, is falsely included and Z, is falsely excluded (as in the pilot study and in Pesaran (1974)), but also the lagged dependent variable Yeel is falsely excluded (hence mis-specified dynamics). Pesaran’s study and the one above differ also in the degrees of freedom for each statistic. In the former, the F statistic (asymptotically equivalent to the Wald statistic) has one degree of freedom in the numerator. In the latter, the Wald statistic has two degrees of freedom for (8) but only one for (7). The degrees of freedom for Sargan’s statistic depend upon the number of instruments selected; but, for instance, with (Ye. ¥p) as instruments (i.e., two-stage least-squares), it is asymptotically distributed as a x?(2) for (7) and as a non-central x?(3) for (8). In addition to affecting the asymptotic powers of those statistics, the degrees of freedom may have a significant effect on their finite sample properties. Consistent estimation

of the parameters in (7) requires some simultaneous equations estimation

technique, as would be true for those in (8) if (8) were the DGP, so only the IV statistics are considered for that structure.

Various finite sample properties of the statistics might be analyzed (e.g., their means and variances; see Mizon and Hendry (1980, p. 40)), but their powers and sizes are viewed as being of primary importance, and as providing a simple way of summarizing their properties. Before turning to the experimental design, simulation, and results of this Monte Carlo study,

I discuss the analysis of Monte Carlo data on powers and sizes.

4, Response Surface Methodology

Cox (1970, Chapters 3 and 6), in his discussion of the empirical logistic transform, implicitly provides the basis for developing response surfaces of estimated finite sample probabilities, including both estimated finite sample powers and estimated finite sample probabilities of type I error.) Consider a binary response variable for which the probability of "success" (or, later, acceptance or rejection by a particular test) is m

(0 < x < 1) and on which there are N observations (N > 1), S being the

number of "successes". -Letting A = [S(N-S)]/(N-1) , (10) ue) = AMfin |S 0o<¢ <1, (11) 1-¢ and -1 icc) = al/in |S = ON guy tee <1-cayt, 9 (a2) 1-¢- ant it can be shown that (s,m) = L(s)- Li) x NOL (13)

ee Cochran and Cox (1957, pp. 335ff), Cox (1958, pp. 113-128), and the references in Cochran and Cox (1957, p. 369) on the use of response surfaces in statistical analyses. Their use in econometrics is relatively recent although Summers (1959) proposes using them; cf. Summers (1965) and Sowey (1973). Ericsson (1986) describes and uses response surfaces and other techniques for post-simulation analysis of Pesaran’s (1974) Monte Carlo study of nested and non-nested hypothesis test statistics.

P P : : 12 where s = S/N and 5 denotes "converges in distribution to, asN7o™". In

the context of Monte Carlo studies of power, N is the number of replications in a particular experiment, S the number of replications for which the value of the test statistic lies in the critical region, and a the finite sample (i.e., finite econometric sample T) probability of the test statistic lying in the critical region. Below, m is treated as if it were "power" although all that is said applies equally for size.

Typically, is some unknown function g(@,T) (say) where @ is the vector of all parameters (except T) which define the model generating the binary random variable of interest, and the aim of finite sample research (whether using analytical or Monte Carlo techniques, or both) is to obtain a close approximation to it,23 Even though it is unknown, g(#,T) is implicitly defined by the computer program generating the Monte Carlo data. Further, approximations to g(@,T) may be found and the accuracy of those approximations may be tested. As a first step to approximating g(@,T), it is helpful to solve as much of the problem as possible analytically in order

to minimize the imprecision and specificity arising from simulation. With

that in mind, let

_- | = _ 3 | expte"(9,T)) (14) l1-fn l-fa T a

without loss of generality, where m (= E(S/N)) is subscripted by T so as to emphasize that it is a function of the econometric sample size; x, is the (local) asymptotic (i.e., as T + ~) power of the test; and ct(e,+) is some

appropriate function. By assumption, unt > m, as T+, so ct(e,-) is o(1).

Thus, (14) splits Tp into two components, an asymptotic term and a term

12 .

That differs from the sense of "asymptotic" elsewhere in this paper, where it means "as T+". Unless otherwise noted, "asymptotic" and "finite sample" refer to T, not N. Ericsson (1986, Appendix) gives a proof of (13). 13

If g(*,*) were known, the exact finite sample probability (of "success", rejection) for any particular value of (6,T) could be calculated directly,

obviating any need for conducting Monte Carlo experiments to estimate To

10

involving the deviation between the finite sample and asymptotic

distributions. Because m, can be calculated analytically for any (8,T), the

problem of directly simulating Tp (of O(1)) simplifies to one of simulating only G*(+,*) (of 0(1), and quite possibly o(t”1/*)).44 In the analysis of an estimator’s properties, an analogous partition is between its asymptotic

value (its plim) and its finite sample bias (the deviation of the estimator

from its plim) .1°

Using (14), (13) may be rewritten as * L'(s) - Lim) = a/*.ct(o,t) + « © x N(O,1) , (15) providing a stochastic relationship between a feasible and unbiased

estimator of Top (i.e., s) and the known quantities mo) 6, and T. However,

the functional form of ct(+,*) remains unknown. From asymptotic theory, one expects that

etie,ty) = tT 2%e¢9,7° 72) (16) where G(9,T 1/7) is 0(T°) (cf. Phillips (1977, p. 474; 1982), Sargan (1980c, p. 1120)). Thus, G(+*,*) might be expanded in powers of q l/2 (about T = ~) and of the elements of @. Truncating the series for cca, Tt t/2) , the coefficients of the powers and cross-products of @ and qi? may be estimated by least squares, correcting for heteroscedasticity using the

1/2 i.e., from estimating

weight A L(s) - un,) = ala l/?u¢g 71/2) 4 (17)

-1/2, . : . : -1/2

where H(@,T ) is the weighted least squares approximation to G(§,T )

and the error terme is the combination of e« (the error from estimating Ton

I4i¢ an analytical approximation to x,, better than 7_ is available (e.g., an Edgeworth expansion), it could appear in (14) in place of m_, further reducing the order of the term being simulated; cf. Phillips (1982).

The functions G (*,+*), G(*,*), and H(*,*) in this section differ slightly from those identically labeled in Ericsson (1986). The change lends itself to a clearer exposition.

See Campos (1986a) for a discussion on response surfaces for estimator biases and standard deviations and estimated asymptotic standard errors.

11

by s) and alle 1/2060. 6) -H(*, +)} (the error from approximating G(*,+*) by H(*,*)). The parameterization of @ is not unique and, before expanding G(e,*), it may be worthwhile transforming "natural" parameters of the model into parameters which span the same range as L"(s) and which have econometrically interesting interpretations. For instance, it may be convenient to reparameterize @ to include a function of ms such as In{x/(1-1,)}. For the experimental design adopted in Section 5.1 below in which m is a design variable, that seems particularly appropriate.

A response surface like (17) summarizes a possibly vast array of Monte Carlo simulations in a relatively simple formula which may account for much of the variation in s across experiments and may be useful for predicting Top at points within the parameter space of the experimental design (denoted ®xI; see Section 5 below) but not included in the simulations. Further, the response surface may adequately approximate the underlying finite sample distribution function. One primary source of information exists for inferring how "good" a response surface like (17) is:

€ x NID(O,1) . (18) Using (18), many testable implications follow from the null hypothesis that H(*,*)=G(+,°).

(A) o2 =1.° If H(e,*)#G(*,¢), then o2 > 1 because ¢ is uncorrelated with aleg l/2igc. y-H(e,«)). The hypothesis a2 = 1 may be tested by noting that, under the null, the residual sum of squares from (17) is distributed as a x? random variate with its degrees of freedom equal to the number of experiments less the number of regressors, provided N is large. Power under the alternative is directly related to the magnitude of at’ 4G(+,+)-H(+,«))?2 over the experiments.

(B) The error e does not include any terms of o(t 1/2) involving @ and

-1/2 . T / . By using OLS, e can not include any of the terms in H(*,¢).

However, if H(*,*)G(*,*), e contains terms of a higher order than those

12

included in H(+,*) (cf. Maasoumi and Phillips (1982, p. 198) and Hendry (1982, p. 210)). By initially specifying a general formulation for H(-,+) and simplifying, one can use an F statistic to test for the presence of such factors in the e’s of the final specification.

(C) The error e does not include any terms of O(T°) involving 6. By construction from (14), to /26(., 6) does not. However, if H(+,*)*G(*,°), regressors of 0(T°) in (17) such as a constant term or In{m,/(1-,)) may be "statistically significant", thereby revealing the mis-specification of H(*,*). This hypothesis is particularly noteworthy, given the importance of the insignificance of In{x,/(1-x,)) in (17) vis-a-vis the analytical properties of the response surface.

(D) The error e is normally distributed.

(E) The e's are serially independent for any ordering of experiments specified prior to simulation. That follows from the independence of « across experiments. If H(+,+*)#G(*,+*) and experiments are ordered to be (e.g.) increasing in values of @ and T, terms in e involving @ and pie may induce serial correlation and/or heteroscedasticity in the e's.

(CF) HCe,*) is constant over regions of the parameter space which were not included in the estimation of (17).

Table II lists most of the test statistics reported below; the convention used is that €;(q) and 1; (q4,P) denote statistics which have central x?(q) and F(q,p) distributions respectively under a common null and against the jth alternative. Thus, €9(q) and ny 9(q,K-m-q) both test for qr? - order residual autocorrelation. There are K experiments and n regressors in the response surface under the null hypothesis.

The extent to which (A)-(F) are not satisfied reflects the degree of approximation of the response surface to the underlying conditional probability formula (response function) although the power of tests of

(A)-(F) depends crucially on the number of replications per experiment, on

12a

Table II. Criteria for Evaluating Response Surfaces

Null Alternative Statistic® Sources

(A) o2 > 1 €5(K-n) Theil (1971, pp. 137-138) e

(B) q invalid parameter n3(q,K-n-q) Johnston (1963, p. 126) restrictions

(B) q¢? order RESET na(q,K-n-q) Ramsey (1969)

(c) px? ns(1,K-n-1) Hendry (1984, p. 962)

(D) skewness (SK) and &6(2) Jarque and Bera (1980) excess kurtosis (EK)

(D) heteroscedasticity n7(q,K-n-q-1) White (1980a, p. 825), quadratic in regressors Nicholls and Pagan (1983) (q quadratic terms)

(E) q@P-order ARCH €3(q), Engle (1982)

(E) first-order residual dw Durbin and Watson (1950, autocorrelation 1951), Farebrother (1980)

(E) qt} - order residual €9(q); Box and Pierce (1970); autocorrelation ng(q,K-n-q) Godfrey (1978), Harvey

(1981, p. 173)

(F) H’(+,+) not constant N19 ((j-1)n,K-jn) Fisher (1922), over j subsamples Chow (1960, pp. 595ff)

(F) predictive failure €1(q); Hendry (1979b, p. 222); over a subset gf q n1(q,K-n-q) Chow (1960, pp. 594-595) observations ~’

Notes: a. The value of q may differ across statistics, as may the number of

regressors n and the number of experiments K across response surfaces and Monte Carlo studies.

* . ~ is the coefficient on L (x_) if the latter is included on the

right-hand side of the respofise surface (17).

. The Chow statistic is labeled n,(q,K-n-q). The covariance test

statistic 1 9((j-1)n,K-jn) is often (and confusingly) referred to as the "Chow statistic" although Chow (1960, p. 592) was well aware of its presence in the literature.

. Constancy may be tested using Chow's statistic, the covariance

statistic, or the usual x? statistic based upon the forecast errors. Often, an even more stringent test may be constructed by substituting unity for the estimated value of o? in the relevant statistic, thereby testing the "absolute" accuracy of the response surface. Such statistics are designated as those above, but with a prime added, e.g., €,(q) becomes €/(q).

13

the experimental design (i.e., the points in oxr examined), and on the choice of DGP and @xI. Finally, even if any of (A)-(F) are rejected, the response surface still has certain desirable properties as an approximation to the unknown function G(*,*) (White (1980b, pp. 155-157)), and it still may account for (and so summarize) much of the inter-experiment variation.

Instead of estimating response surfaces of the form (17), econometricians sometimes have estimated ones like:

s = h(6,T) + e (19) where h(@,T) is the least squares approximation to g(é@,T) and e is the residual. Unlike (17), (19) does not account for the heteroscedasticity of s, conditional upon (@,T), nor does it bound the range of h(6,T), e.g., h(@,T) could go outside the unit interval. Even so, White's standard errors are consistent; the response surface h(@,T) is a least squares approximation to the underlying response function g(9,T) and has the desirable properties that that entails; for this Monte Carlo study at least, some very simple response surfaces of the form (19) do very well at approximating Trp (as measured by the magnitude of the deviations s-h(@,T)); and, in so doing, those response surfaces succinctly summarize a large number of simulations. The more sophisticated response surfaces of the form (17) are more appealing theoretically and, with enough terms, can explain much of the remaining prediction error of the naive response surfaces, but the former lose in terms of summarizing the Monte Carlo results because of their complexity. For convenience, "naive" response surfaces (of the form (19)) are called

type A; "sophisticated" ones (of the form (17)) are called type p16

5. Simulation Evidence: A Two-equation Model

This section describes the experimental design, simulation, and

post-simulation analysis of a Monte Carlo study of the nested and non-nested

16. 5e B response surfaces need not. be complex, nor type A response surfaces

simple. However, the latter’s appeal lessens if they are complex.

14

hypothesis test statistics discussed in Section 2 for the dynamic simultaneous two-equation model in Section 3. At each stage, particular attention is given to techniques which will obtain as precise and general results as possible on the finite sample properties of those statistics. 5.1. Experimental Design

Following Hendry’s (1984, p. 940) notation and terminology, the Monte

Carlo design variables for the econometric model given in (4)-(6) are

6 = (b, c, d, a, hh’, 043, O42, O22, (vecA)', (vecn)'’)! e @= (6 | lp; |<1,i=0,...,4; |2Z|>0; |[Q]>0; = and Q symmetric} (20) and Toe T= [T,,T] (21)

where [ is pre-assigned with tT. and Ty being the smallest and largest econometric sample sizes considered. Equations (4)-(6) are the data generation process (DGP); ®xI is the parameter space; equations (7)-(9) are the relationships of interest; and the objective of the Monte Carlo study is to determine the finite sample distributions of the statistics Co, Cy, fo, fi, Cg, C3, fo, fg, ty, ts, tg, and t, as defined by the relationships of interest, within the specified parameter space of the DGP. More modestly, letting r be any of those statistics and 6 be the critical value associated with a test based on 1, the objective is to find the finite sample rejection

frequency Tr = prob(|r|25). That probability depends upon § and T and can

be expressed as a conditional probability formula:

Tp, = prob(|r|26 | 6,T) = g(6,T) , (22) where Tn and g(6,T) are precisely the probability and function discussed in Section 4,7 Thus, we wish to know (or obtain a good approximation to)

g(@,T) over xT, focusing on the effect on the statistics’ finite sample

power and type I error of dynamics, simultaneity, sample size, and (in the

Tmplicitly, g(*,*) is a function of 6 as well. However, because § is held constant for each of the statistics examined, its presence in g(+,+) is ignored in the analysis below.

15

case of finite sample power) asymptotic power. Hence the key parameters in

the experimental design are d, b, T, and mo

First, consider the other parameters. As in Hendry and Harrison (1974,

pp. 164-166), the matrices A and Q are held constant across experiments. In the present set of experiments, the diagonal elements of A and Q are

(P1 > Po: Pg: Pg) = (.8 : .7 2 4: 22 ) (23)

(W141! Woo: Wggi Wag) = (.52: 152: 172: 172), 2 (24) implying that var(w;.) is virtually constant across i. A isa diagonal matrix (as in Hendry and Harrison (1974)); but Q is not, with w,. (=w2,) chosen such that y = .925, and all other 5; = 0 (i¥j). That implies that Wit (= Z.) and Wor combined have a variance about twice that of M34 and Wat combined. All the exogenous variables are stochastic, i.e., varying across replications as well as across experiments. The parameters in the second equation are fixed across experiments, with a = .3, h = (O: 1:1: 1)’, and 922 = 1.0. The error covariance o,2 is chosen such that corr(u, ,v,) = .5; and c ~ 1.0 (without loss of generality).

The values of the key parameters b, d, and T cover a range typical of econometric models estimated with actual data: b = (-.5, .3), d= (-.4, .2, .7), and T = (20, 40, go) .18 The number of replications N varies inversely with the econometric sample size (N = 4000 for T = 20, N = 2000 for T = 40, N = 1000 for T = 80), keeping computational costs virtually constant across sample sizes and giving more precise information on finite sample powers at smaller sample sizes (where the asymptotic approximations would be expected to provide less information about those © finite sample powers). The error variance o,, is the final parameter in the

experimental design. Rather than assign it somewhat arbitrary values,

possibly implying very high (or very low) finite sample powers, o,, is set

Boe, Hendry and Harrison (1974, p. 166) who chose a similar range for b, d,

and T. See Klein (1969) and Hendry (1974) inter alia for estimated values of such parameters in empirical macro-economic models.

16

to obtain certain values of asymptotic power mo thus controlling to some

extent the values of finite sample power n So, o,, is chosen such that

T i (.25, .5, .75, .90) : (25) which seems a relevant range of powers, and one which ought to avoid having observed rejection frequencies too close to unity. 1? However, because several statistics are being considered, some having different asymptotic powers, it remains undecided as to which statistic the asymptotic power x in (25) corresponds. Because of the degrees of freedom involved and because H, is in fact the DGP, it is conjectured that (in general) t7, would be most powerful, followed by cg and the asymptotically equivalent f£,, followed by c,, with the placement of ts uncertain. So, to avoid any of the statistics having consistently high (or consistently low) power for all experiments, the asymptotic power of cz is the To in (25). Even so, the asymptotic power of t, implied by those values of b, d, T, and o,, is always greater than one-half, and that for t, always less than .17, highlighting the difficulties of designing a Monte Carlo study for statistics with different asymptotic powers. (Note, however, that in the response surfaces below, mo and 2 (= In(m,/(1-x,)}) are for whatever statistic is being examined, and not just for cg.)

Given the choices of b, d, T, and mW a full factorial design is adopted, with 72 experiments in all. Three randomly selected experiments are retained from each econometric sample size for prediction. Estimation is by two-stage least squares.

5.2. Simulation and Computational Aspects

Noting the similarity between evaluating (for instance) Dg under Hy and

H, and evaluating both Dy and D, under Hy only, and that the latter is

computationally more efficient in these studies, only simulations under Hy

19 : See Appendix A, Mizon and Hendry (1980, p. 34), and Hendry (1984, p. 971)

for counter-examples; cf. Poskitt and Tremayne (1981, pp. 266, 268).

17

were considered. Those Monte Carlo simulations were carried out with a modified version of Hendry and Srba’s (1979, 1980) computer program NAIVE on the University of London’s CDC 7600 computer. For a given set of parameters @ defining the DGP in (4)-(6), each Cu...) was generated as a rescaled pair of normal pseudo-random numbers using Box and Muller’s (1958) transformation on two uniform pseudo-random numbers . 2° Each Vi_ Was generated as an appropriately rescaled sum of twelve pseudo-random uniform numbers from RNDM, which very closely approximates a pseudo-random normal number; see Hammersley and Handscomb (1964, pp. 39-40), The series for Yer Yee and we

were determined from those for Ue MEP and vy using (6) and the reduced form

of (4)-(5). The relevant statistics were then calculated for each of N such

replications.?1

For a particular experiment, N replications were generated, of which $ (dependent upon the statistic) were "successes" (e.g., the number of rejections; see Section 4 above). The fraction of successes s (=S/N) is an unbiased Monte Carlo estimator of the (unknown) finite sample rejection frequency Ts and from those estimates, numerical-analytical approximations to x, were obtained by estimating response surfaces as described in Section 4. To calculate the asymptotic powers of the statistics, the moments of the DGP for each experiment were obtained using Hendry and Srba's (1977b, 1980) program DAGER, from which the non-centrality of c,, f,, ¢

3?

and f,; and the asymptotic means and variances of D,, ts, and t, were

20c¢. Hendry and Harrison (1974, p. 153). The random number generators are

Carrier, Atkins, and Taylor's (1969) mixed-congruential generator RNDM

and NAg’s (1977) multiplicative-congruential generator GOS5SCAF. Different random number generators were used for each number in the pair of uniform pseudo-random numbers in order to avoid potential difficulties with Box and Muller's transformation: see Neave (1973). Nb. Hammersley and Handscomb (1964, Chapter 3), Kennedy and Gentle (1980, pp. 136ff). See Sowey (1972, 1973, 1978, 1986) and Sahai (1979) for bibliographies.

21ohe initial value for (y YEW.) in each replication was its unconditional mean, so the first thirty obsérvations generated for each replication were discarded to ensure stationarity of the series used for estimation and testing (cf. Hendry and Harrison (1974, p. 153)).

18

determined, using formulae in Ericsson (1983). The asymptotic powers of D,, ts, and t; were calculated assuming a symmetric two-sided test with critical values of 41.96. The asymptotic powers of c, (f,) and cg (£3) were calculated assuming critical values corresponding to the 5% level, and approximating the non-central y? (singly non-central F) by a central ;?

(central F) 22

5.3. Post-simulation Analysis

This subsection examines how well various analytical and numericalanalytical formulae approximate the underlying finite sample properties of the test statistics. To organize presentation, discussion centers around four formulae: the asymptotic (m,), the F-adjusted asymptotic (m,, explained below), and type A and B response surfaces (Rp and Tr) - These are in order of increasing accuracy of approximation (and complexity) and are examined in that order. The value s is an unbiased estimate of Tp, SO a natural measure of the degree of approximation of these formulae is their deviation from s. Graphs portray all this information concisely: results for (cg, cs), (f2, £3), (cg, c,), and (tg, t,) appear in Figures la-d, 2a-d, 3a-d, and

4%

4a-d, respectively. For a given pair of statistics, Figures a and b graph the results for power and size, and Figures c and d plot the corresponding deviations of the formulae from 5.23 In the figures and elsewhere, the data

are ordered by increasing values of T, then of mo of d, and of b. The

remainder of this section describes the approximations obtained, interprets

22 ee Patnaik (1949), Johnson (1959), Kendall and Stuart (1973, pp. 237-240,

262-263), and Mizon and Hendry (1980, pp. 32-33) for further details. 230m all experiments, the observed rejection frequencies of t, are small (as are its asymptotic values), so no response surfaces are given for it. Although t, does not appear particularly useful in testing non-nested hypotheses when one of those hypotheses is the DGP, it shows promise for testing non-nested hypotheses when neither is the DCP (see Section 6 and Ericsson (1983, p. 294)).

The statistics fy and f, were not calculated in the Monte Carlo study, but there was little need to do so, given the results for m, below.

19

them in light of existing finite sample theory, and proposes directions for further research.

The Asymptotic Approximation mo: Figure la displays the asymptotic power (w,) and the unbiased estimate of finite sample power (s) for c3. As T increases, s converges to mo with the latter usually an upper bound of the former. The estimated finite sample size of cy generally is somewhat larger than its nominal value (.05) for small T, but tends to .05 as T increases (Figure lb). The largest estimated finite sample size is 7.53%, occurring at the smallest sample size and with substantial dynamics and simultaneity present (T=20, b=-.5, d=.7). The properties of f, and f3; resemble those of cy, and cz, except that discrepancies between s and m, are typically smaller. That is particularly noticeable for the size, with s infrequently lying outside the interval [.035, .05], indicating how useful the F transformation is in small samples, even in the presence of dynamics and simultaneity. By contrast, the estimated finite sample sizes of cy and tg are almost always greater than 5%, and often exceed 15% (Figures 3b and 4b). The estimated finite sample power of c, generally exceeds its asymptotic power substantially: however, if its size were properly adjusted, its finite sample power would be considerably less, quite possibly bringing it more closely in line with its asymptotic values. The estimated finite sample power of t, deviates only slightly from its asymptotic power: adjusting its size would reduce its finite sample power considerably but generally increase deviations from its asymptotic values although the magnitude of those changes is difficult to predict.

The F-adjusted Asymptotic Approximation ™- In certain circumstances, f. is exactly distributed as an F-ratio although c, remains only asymptotically x? by failing to account for the variability in the estimated error variance used in its calculation. Because of the analytical

relationship between c, and f,, it is possible to calculate the size of the

19a

Figure 1a. Asymptotic and Estimated Finite Sample Powers of c 5

NH WA OA AN DO O O

On Ot @ it OR @ i Ol Cl On Onn)

O 12 24 56 48 60 72 Experiment Number

Figure ib. Asymptotic and Estimated Finite Sample Sizes of c, .

. 100

.O075

.025

.000

0 12 24 36 48 60 72 Experiment Number

19b

Figure 1c. Prediction Errors for Calculated Powers ofc .

3

Experiment Number

2

Figure 1d. Prediction Errors for Calculated Sizes ofc .

0 fo) | w i H ' 4

3

.0

2

6.0

Experiment Number

19c

Figure 2a. Asymptotic and Estimated Finite Sample Powers of f

Oo

WwW fr Oo DN WO OO

No

OR On Olen Oe On Ol OOO ®)

O 12 24 56 48 60 72 Experiment Number

Figure 2b. Asymptotic and Estimated Finite Sample Sizes of f.

. 100

.O075

.050

.025

.000

0 12 24 36 48 60 72

19d Figure 2c. Prediction Errors for Calculated Powers off .

3

Experiment Number

2

Figure 2d. Prediction Errors for Calculated Sizes of f .

.03

0.02

—-0.03

Experiment Number

—

oo 0ClUmUODmUlUCCOCOmUCOUCODCCODCOCOD NW FF oO AN DO O O

19e

Figure 3a. Asymptotic and Estimated Finite Sample Powers of c ,

0 12 24 36 48 60 72 Experiment Number

Figure 3b. Asymptotic and Estimated Finite Sample Sizes of cy"

0 12 24 36 48 60 72 Experiment Number

19f

Figure 3c. Prediction Errors for Calculated Powers ofc .

1

-------S— Ng

Ns

—_

re rg ER

OMDORrRONTMNN TH OT NNT T PODS DPD FR FOO © I tot

Experiment Number

ie)

Figure 3d. Prediction Errors for Calculated Sizes ofc .

=

ee

__.

o | \7

|, ay

wy NAN

“ v

‘\ "

x

iN

\

voev

¥

80

72

60

Experiment Number

—s

0OoO0o0OoOUmUCOUCUCOUMUCOUmUCOUWUCUCOlWUlUlO

©

19g

Figure 4a. Asymptotic and Estimated Finite Sample Powers of t ;

.O 9 8 . 7 .6 fs) 4 5 2

1 0 -

O 12 24 56 48 60 72

Experiment Number Figure 4b. Asymptotic and Estimated Finite Sample Sizes of te Fd OS _—-— ft, 3 Too Ay S)

2

1

T=40 O 12 24 56 48 60 72

Experiment Number

19h

Figure 4c. Prediction Errors for Calculated Powers of t .

Experiment Number

6

Figure 4d. Prediction Errors for Calculated Sizes of t .

-----S 05

err

“s

Litisteas

T=80

40 56

20

24 48 60 72 Experiment Number

12

20

test using cz (called the F-adjusted asymptotic size) that would result from taking that variability into account. A similar adjustment can be made for the asymptotic power: both are plotted in Figures la-b. Although typically TMs for c, is small relative to the remaining discrepancy S-T,, the F-correction captures a dominant finite sample term (of oct!) in the

distribution of c,, as is apparent in Figure lb. Because f, is not exactly

an F-ratio in these experiments, the F-adjusted asymptotic approximation need not improve upon the regular asymptotic approximation although it is useful because it does .2* Indeed, a similar correction for cg (and c,) appears unimportant relative to remaining fluctuations in s.2> Those fluctuations and similar ones for tg exhibit a pronounced pattern, inspiring

the following digression.

The Effective Sample Size T" and Large-o Effects. The concepts of effective sample size and small-o asymptotics are valuable for interpreting the finite sample fluctuations of all the statistics, and particularly those of co and te¢.

Using a concept from Sims (1974), Hendry (1979a, 1984) develops the notion of an "effective sample size", that is, one which accounts for the lagged (and hence redundant) information accrued by each new observzetion of a dynamic process. For example, in an AR(1) process with autoregressive

* coefficient p, the effective sample size T is T(1-p?) where T is the number

e4viviet (1986) studies a variety of mis-specification test statistics for a

dynamic single equation and also finds the F form preferred to the y? form. 25the F-adjustment to cg does aid analysis of further finite sample terms; cf. Table IV. Also, fy may be preferred over cy for other related reasons. Cy is bounded from above by (T-ky), regardless of the DGP. Although that by itself may not induce large finite sample effects into the distribut:ion of Co under Hy, it easily can under H, when A, is sizable. Additionally, the numerator and denominator of cy, in Ericsson (1983, eq. (10)) are positively correlated in finite samples, and that may lead to the finite sample distribution of cy deviating significantly from the x? distribution.

Nb. Sargan (1980c, pp. 1135-1137).

of ooservations. The greater the dynamics, the less new information on the process is gained with each additional observation, e.g., relative to a white-noise process: that is reflected in the measure T", In general, T" involves all the latent roots of the dynamic system generating the data. However, for (4)-(6) and the experimental design in Section 5.1, Po is the only latent root that changes, so T is defined as T(1-p2). In static models, terms of oct 1/2 are often important, so in dynamic models the focus is on terms involving (rt) V2 denoted T for convenience . 7°

Most asymptotic results in econometrics are "large-sample", i.e., large T. Kadane (1970, 1971) proposes an alternative approach, small-ao asymptotics, in which T is held fixed and the equation error variance o? is let to approach zero. Anderson (1977) discusses the relationship between large-T and small-o asymptotics. Just as small-sample phenomena may appear when T is small enough, “large-o" phenomena may exist for large enough o. In (4), the error variance is o,,, so Vox is denoted oa.

The dominant finite sample term for Co appears to be To: that can be “seen in several ways. In the cross-plot of s and To, their correlation is striking (Figure 5a). Even with no constant term and no correction for heteroscedasticity, the least-squares regression of (s-.05) on To is:

(m- .05) = ,0890Ta R? = .964 o, = 2.388% (26)

(.0020)

[ .0032] where (*) and [*] denote conventionally calculated and White's (1980a, pp. 820-821; 1980b, p. 156) heteroscedasticity-consistent standard errors, R? is the unadjusted squared multiple correlation coefficient, o, is the

square root of the residual variance, and ~ denotes the least squares

26n150, Phillips (1977) shows that the first Edgeworth correction term to

the anormal distribution is O(T) for the t-ratio of the coefficient in an AR(1) process.

2la

Figure 5a. Cruss—plot of s (for e ) and Tc.

Ss 0.4

-__oo Least-squares Line O. QO. O. QO.

0 1 2 3 4 To

Figure 5b. The Nominal Size (.05), Unbiased Estimated Finite Sample

Size (s),and a Simple Approximation to the Finite Sample Size (% ) ofc . T i)

Experiment Nur®

22

estimate in this very simple type A response surface. */ The estimated finite sample size s and the fitted values Tr from (26) appear in Figure 5b, along with the asymptotic value (.05). Although the standard error of the prediction errors is still 2.4%, it is small relative to the size of the fluctuations in Tp and is a remarkable reduction from 9.2%, the standard error of s-x,. Individually, neither T nor o approximate Tr -O5 nearly so well. Together they conveniently summarize effects of the sample size, dynamics, simultaneity, and goodness-of-fit, i.e., To = /(o11/(T{1-[d/(1-ab) ]2})]. Based upon this rather suggestive evidence and upon the analytics for effects from T and o, @ is reparameterized to include T and o in the response surfaces.

Before turning to the response surfaces, it is valuable to consider why T and (especially) o so dominate this Monte Carlo study and not others. Two aspects of the experimental design are responsible. First, T" and o range widely, over [8, 78] and [.424, 20.2] respectively. For T that arises because |p 9| spans [.17, .77] and T, [20, 80]; o is used to control m of Cg, given the values of (b,d,T) selected. Second, as the parameter for controlling m4 O effectively is one of the experimental design parameters, and one which varies over all experiments. Other investigators typically have normalized on o, used it to control the population R?, or included it explicitly as a design parameter but with a small number of values. Each approach has its merits, but none would be likely to elicit large-o effects to the extent that the design in Section 5.1 does.

Type A Response Surface Approximations (%

ape A Kesponse Surface Approximations (my). type A response surfaces for co, cg, fz, f3, cg, Cy, tg, and t,;. They

Tables III and IV give

involve simple terms in T: T itself and/or T interacting multiplicatively

with o, «x (= |po|), and (for powers) m, OY ™,. In simplifying from a

Because there is no constant term in the regression, R? may lie outside the unit interval. However, o_, not R?, is the appropriate measure of the goodness-of-fit for response surfaces. :

22a

Table III. Estimates of Type A Response Surfaces for Finite Sample Sizes and Powers

CO OO

Test Statistic r and Dependent Variable®

Regressors ae and Size —___— Power Diagnostic Statistics £, Co f, C3 s-.05 S- S-m S-T, eee T -1.89 -2.29 -32.5 ~36.4 (.66) (.70) (9.5) (10.9) [.67] [.75] (7.3] [8.3] To -1.70 -1.87 -8.02 -7.82 (.28) (.30) (1.74) (1.80) [.21] [.21] [1.54] [1.67] TKo 2.08 2.23 17.8 oo 17.7 (.39) (.42) (2.3) (2.4) [.29] [.31] [2.0] [2.1] Tx_ (Tx, for cs) - .459 -.381 . (.125) (.134) {.114] {.117] R? 661 -698 . 868 .870 a . 743 797 4.315 4.329 e

s 4.04: 4.86 47.36 51.51 (.81) (1.16) (23.25) (22.87)

s-n - 96 -.14 -8.86 - -8.48 a (.81) (1.16) (7.42) (7.83) S-7 - -1.12 ; - -9.03 x (.88) (7.42)

S- ity - .06 -.05 20 .18 (.73) (.78) (4.22) (4.23)

S-Ttep -.03 -.04 - 08 - 08 (.46) (.49) (1.88) (2.07)

S-Ty (@) (@) 0 0 (.47) (.50) (1.06) (1.06)

Notes: a. The dependent variable (and x and m when they appear on the right-hand side) are rescaled y 100° to make them percentages and to achieve a reasonable scaling of coefficients. That implies that o. is a percentage; oa, however, is in its original units.

b. Noting that s-x, has zero mean and variance n,,(1-2,,)/N, for each experiment the latter is approximated as s(1-S)/N, with the values given above being based on averages across experiments.

Regressors

Test Statistic r

22b

Table IV.

Estimates of Type A Response Surfaces for Finite Sample Sizes and Powers

nn

Power

-25,

[4.

3 4) 5] 7 11 7) ( 9] [ 64. (10. [8. 6 -12. 8) (1 2] [1 -74. (17. [13. 963 221 1

Size

.35

.957

.876

and Dependent Variable®

and Size Diagnostic Statistics Co S-" x T -5.75 (1.29) [ .98] To 14.24 (.54) [.55] Tk Tko -7.05 (.77) [.85] T2 Tac a R2 -988 a 1.457 e Means and Standard Deviations s 13.70 (9.15) S-1, 8.70 # (9.15) S-T,, 9.31 . (9.06) S-,, -.07 ~ (1.43) S-T,, -.09 ~ (1.31) S-T,, 0 . (.83)

27 12 18) (5 70 7 08) (5 24

60)

31

11) (1 .03 - .63) (1 0

86) (

.08 .25)

.08 .25)

.03 .84)

. LO -21)

0

.79)

32 48)

.17 .39)

.54

. 36)

. 26

.06)

.73)

———————

Notes:

See the notes for Table III.

23

regression in simple products of these parameters to a given response surface in Table III or IV, the primary criteria were parsimony and a small prediction error (not necessarily complementary criteria).

The size and power of the Wald statistic strongly resemble those of the F-statistic, once netted of its F-adjusted asymptotic approximation mt rather than the simple asymptotic approximation m4! the similarities are apparent both from the estimates for the respective response surfaces (Table III) and from the resulting predictions and actual values of s (Figures la-d and 2a-d). The terms in the response surfaces for the size of cg and f2 are highly significant statistically, but the lower portion of Table III shows that they achieve only a moderate reduction in the residual standard deviation. That arises because the variances of (s-m,) and (s-7,) (for co, f.) are close to their theoretical minima, i.e., the variances arising exclusively from the sampling of s, equivalent to the square of the residual standard deviation obtained if g(@,T) were known (estimated by the last row. in the table). That reflects the observation above for Figures 1b and 2b that estimated sizes for these statistics stay close to 5% over the entire range of sample sizes, dynamics, and simultaneity. The reduction is more substantial for powers, approximately threefold in the residual variance.

The type A response surfaces for cg, c,, tg, and t, (Table IV) are similar in form to those for cg, cg, fz, and f,, but the magnitude and statistical significance of the estimated coefficients of the former are far greater than those of the latter and the signs of the estimated coefficients generally are reversed. For instance, the strong positive biases from To for cg, Cy, tg, and ty are five to ten times the magnitude of comparable negative biases for cy, c3, fz, and £4; observed negative biases in the finite sample size of cy and tg are negligible, but positive ones are large and frequent. Because of the analytical relationships between cz, c3, fo,

and f,;, and, to a lesser extent, between cy, c,, tg, and ty, similarities in

24

propeties of statistics of either set are expected. However, neither the form, magnitude, nor sign of the finite sample effects was anticipated, nor was the considerable discrepancy between properties of statistics in

different sets.

Type B Response Surface Approximations (mp) - Unrestricted type B —— response surfaces are estimated with regressors involving T, being multi dlicative combinations of T, ao, Po, and (for powers) £2, and combinations of powers thereof. Because of the experimental design, the factors T, o, po, and 2, appear up to (and including) powers of three, two, one, and one, respectively. All combinations except for To7p,2, T2072 pf, T8o2p,2 are included initially: that implies thirty-three regressors in the unrestricted response surfaces of estimated finite sample powers and eighteen in the unrestricted response surfaces of estimated finite sample type I errors.7° Restricted versions of all response surfaces are presented in Tables I and II of Appendix C. No sets of restrictions are rejected at the 5% significance level. Unlike those for the type A response surfaces, the primary criteria here are those listed in Table I. Parsimony is not central in this framework, and so the response surfaces are more complex than those in Tables III and IV, while capturing more of the deviations between the finite sample and asymptotic properties of the test statistics. In the response surfaces for c, and f,, o. is insignificantly different from unity, leaving little residual variation beyond that inherent from estimating Tp by s. With the additional complexity of these response

surfaces, the standard deviations of the prediction errors for f, and Co

fall from .73 and .78 (for S-Tp) to .46 and .49 (for S-7y) versus the

8 : . . 2 There are fewer regressors in the latter because the unit vector is

collinear with 2, which is in that case 1n(.05/.95). Cf. Cochran and Cox (1957, pp. 148ff, 342ff) and Cox (1958, pp. 113-117) on factorial design and response surfaces.

To? pol, T?a7%pgl, T3a7p gf are not included due to limitations in the number of regressors in PC-GIVE.

25

estimated theoretical lower bounds of .47 and .50. Similar moderate reductions in variation are achieved for cy and tg although substantial explainable variation remains unexplained for even type B response surfaces.

Type B response surfaces fare better for the powers of these tests, although still at the expense of greater complexity. Graphs c and d for Figures 1-4 illustrate the reductions in prediction error across the different predictors (m5 Ts Tp and Tr) with the most marked improvements generally being between the purely analytical formulae (m) ) and the response surfaces (ip Tr) - That does not discount the value of analytical formulae: to the contrary, the response surfaces serve to augment whatever analytical results are available. Further, the analytical formulae for powers frequently explain much of the unconditional inter-experiment variation in s, as is apparent both from Figures la-4a and from the first two rows of the lower half of Tables III and 1v.2? Also, although the finite sample terms explain over 95% of the variation in L’ (s)-L(m,) for c, and t,, the value of oe implies that under 25% of the remaining (residual) variation is due to sampling fluctuations (i.e., in estimating To: by s), with over 75% being due to additional finite sample components. Better analytical approximations would be of considerable value here.

Remarks. In retrospect, several features of the experimental design are notable.

(a) Even though many of the response surfaces appear mis-specified, White’s standard errors are consistent under the sorts of mis-specification present and the coefficient estimates are still useful for prediction within the population being investigated: see Hendry (1982, pp. 210-211) and White (1980b, pp. 155-157). In fact, both type A and type B response surfaces

track the simulation estimates of the finite sample size and power

29 ss Hendry (1973) convincingly argues the merits of analytical formulae in interpretiig Monte Carlo studies.

26

remarkably well, in spite of mis-specification. That apparent contradiction has the following explanation. For a given set of experiments and their assoc:ated rejection frequencies, the estimates of the coefficients in a response surface are essentially invariant to the number of replications N,

whereas (a2 - 1) is proportional to N times the square of any unexplained

finite sample fluctuation. °° The large values of N, both in Section 5 and Appendix A, magnify the effect on a, of discrepancies between uy and Tr (or tr) although those discrepancies themselves are insensitive to N and appear

quite small in general. Hence, a larger number of experiments with fewer replications per experiment would have been preferable.>!

(b) For a third of the experiments, the largest latent root of (4)-(5) is .77, resulting in considerable dynamics affecting the statistics. Although that latent root is smaller for the other experiments, the largest latent root of the entire system (4)-(6) is always p, (=.8). Also, the actual sample size is only twenty for a third of the experiments, equaling

the smallest sample size in Pesaran’s (1974) experiments”*: the effective

sample size is sometimes as small as eight.>? In light of that, the

experimental design may be over-ambitious.

3°Note that A = (S(N-S))/(N-1) ~ Ns(1-s) and that the rescaling factor al/? is applied to all variables.

sleontrol variates for the estimated finite sample power might have been used to achieve more efficient Monte Carlo estimates of a,,: see Sargan (1976a, pp. 444-448) and Rothery (1982). However, their Torivation appears practiically intractable for most dynamic models: cf. Nankervis and Savin (1985).

52 The Cox statistic departs significantly from its asymptotic properties in some of those experiments even though no dynamics or simultaneity is Present and fewer instruments are used: see Ericsson (1986).

Hendry and Neale’s (1987) recently developed recursive Monte Carlo techniques permit rapid graphical analysis of estimator’s properties for every feasible sample size up to the largest: parallel techniques for statistics would permit far more extensive analysis of their T- and T*-dependent properties than currently feasible.

27

(c) There is a clear need to limit the range of the (unknown) finite sample power of a statistic over experiments so as to avoid generating experiments with uninformative estimated powers (e.g., unity). Setting o,, such that the asymptotic power ms takes particular values goes some way to achieving that, although there is an inherent difficulty present when statistics with different asymptotic powers are being examined. Jurther control over the range of Tp may be possible, e.g., by using different critical values for statistics with different asymptotic powers.

(d) Small sample adjustments to statistics are a long-run objective of studying their finite sample properties and could take many forms in addition to the F-adjustment. Under the hypothesis (7), co has more degrees of freedom than any of the other statistics (likewise, c, has more than any other, under the hypothesis (8)); and that may be partly responsible for the apparent relative poorness of the asymptotic approximation for cg (and for c,).°4 Sargan’s (1980c) transformation of the IV criterion may aneliorate that effect. Also, the properties of tg may improve from using tnrat criterion rather than cy in constructing tg. Godfrey and Pesaran (1983) propose bias adjustments to the numerator and denominator of the Cox statistic and thereby design a Cox-type statistic with a better finite sample size. Similar adjustments may be possible for the IV statistics, at least when Zz" = X,. Finally, further analytical results on the statistics’ finite sample properties, even for simple models, could be of value for deriving finite sample adjustments.

To summarize, the. finite sample properties of the Wald and F statistics are quite closely in line with their asymptotic properties, with the F-adjustment capturing a dominant finite sample term of ocr}y in the

distribution of c,. The behavior of f, (versus that of c.) favors use of

34 See Sargan (1958, pp. 393, 400, 409, 414-415) and Sargan and Mikhail (1971, pp. 156-158) on similar considerations for the distributions of econometric estimators.

28

the F form of the Wald statistic rather than its x? form, even for situations in which the Wald statistic has no known exact sampling distribution. A similar correction for Co appears unimportant relative to remaining fluctuations in Tp. The finite sample sizes of both cg and tg are typically strongly and positively biased: To accounts for much of that bias. That contrasts with small and negative biases by To for c, and f,. Biases in tne finite sample power of c, and t, are even larger, with T and o being primary explanatory factors. Because of the analytical relationships between cy, cs, f2, and f, and, to a lesser extent, between Co, Cy, tg, and t;, :9roperties of statistics within either set are generally similar: that

helps to unify the results.

6. An_ Empirical Example

Pesaran and Deaton (1978) consider several (hypothesized) non-nested economic relationships between consumers’ expenditure and income to demonstrate the application of Cox’s maximum likelihood (ML) statistics in econometric modeling. This section re-examines their two linear models to illustrate the use of the IV statistics. Those models are:

Hy: C. = Boo + BorX, + BooW, + Up, Upp ~ NID(0,02) (27)

H,: Cc. = Bio + BiiX, + Bi2C. 5 tu, uy, ~ NID(0,o7) (28) where the data are quarterly, seasonally adjusted series in constant 1958 dollars for the United States (1954ii-1974iii) for consumers’ personal expenditure (C), personal disposable income (Y), and personal wealth (W), with wealth measured at the beginning of each period. Throughout their analysis, Pesaran and Deaton assume that conditioning upon current income does not affect inference about the B;,'s: IV estimation allows relaxation of that assumption. The IV statistics are calculated for several possible

sets of instruments, namely:

29

: P 35 where « varies from one to eight.

Tables Va and Vb summarize the results for all values of 41. To focus discussion, consider the statistics and estimated coefficients for 1=5:

Ho: CC, = 23.2 + .862y, + .00652W, (30) (10.3) (.038)* —(.00601)

o§ = 18.078 t, = -32.84 tg, = 86.41 cy = 51.74 cy = 50.18

H,: C. = 2.65 + -067Y, + .930C, 5 (31) (2.13) (.116) (.129) of = 12.974 ty = -7.59 tg =-.19 cy = 21.42 c, = .04

where ~ denotes IV estimation and the values in parentheses are the IV estimated standard errors. The estimated coefficients in (30) are very similar to those obtained by ML, but those in (31) are not: that for Y. is no longer significantly different from zero, consistent with simultaneity bias in the ML estimates, and that for C._1 is now essentially unity.

The values of ty, tg, co, and C2 in (30) all point to the mis-specification of Hy. cg indicates that Chiy is significant if added to (30). co shows that the instruments used are not independent of the residuals, most likely because inter alia Chey is an instrument end is not included in the specification of Hy. t, and t, are so significart because (31) markedly variance-dominates (30) (and that, because Cg is significant).

With H, as the null hypothesis, only Co and ty, appear significant: Sargan's statistic indicates that some of the instrumental variables are not valid (i.e., they are correlated with the residuals), suggesting that H, incorrectly omits certain lagged values of C and Y. The statistic t, also appears to detect that (although tg does not), possibly because Hy excludes all lagged values of Y and C. The results for 1=5 are typical of all values

of 4 except s=l1, in which case none of the tests detect mis-specification.

3Sthe behavior of the IV statistics is determined both by the number of

instruments (here, 21+2) and by the lags at which variables appear in the instrument set: this analysis makes no attempt to separate those effects.

29a

Table Va.

Values of the statistics for Hy: C= Boo + Bort, + BooW, + Uoe

Estimation t, te [Do] Co Co (=f,) method b ML 2.1 36.5 29.6 46.8 {-47.1] Ivo v= 3.3 157.7 46.5 45.7 w=2 -13.8 107.1 46.9 50.1 o=3 -23.4 102.6 49.3 49.9 u=4 -24.1 101.7 51.6 51.1 u=5 -32.8 86.4 51.7 50.2 t=6 -35.3 82.7 51.9 49.7 t=] -40.8 81.9 53.6 49.3 t=8 -56.4 80.5 58.1 47.9

Notes: See the notes for Table Vb.

Table Vb.

Valves of the statistics for H,: C. = Big + Pi, + Pi2e, 4 + Uy,

Estimatiion ty - te, [Do] Co cy (=f,) methoc! b ML -.40 -.37 .16 .16 { .37] Ivo v=. -.03 -.03 00 00 c=? -3.22 -.13 8.08 .02 u=3 -5.11 -.14 13.19 .02 u=dy -5.31 -.14 14.07 .02 w=5) -7.59 -.19 21.42 04 -b=6 -8.26 -.21 23.66 .05 t=] -9.45 -.21 27.11 .05 - p=B -12.77 -.22 36.38 .06

Notes: a, Under the null hypothesis (H, for Table Va, H, for Table Vb), the statistics t,, tg, and Dy are asymptotically distributed as N(0,1); cg is asymptotically distributed as y?(2e-1) for IV (x?(1) for ML); and cy is asymptotically distributed as y?(1).

b. The instruments for ML are {1, Wee Chi4: ar

c. The instruments for IV are. {1, Wee ((C. gs Ypiq)> i=l,...,e)}.

30

These two models illustrate the potential value of both nested and non-nested hypothesis test statistics in practice. The IV statistics point to the possible importance of additional lags on income and consumers’ expenditure in the equation for the determinants of consumers’ expenditure, thus establishing a basis for the re-specification of that equation. >” In general, IV statistics complement ML statistics by allowing for situations in which the specification of a complete set of simultaneous dynamic economic relationships is undesirable or impractical, as is typical in many

existing econometric models.

7. Concluding Remarks

The finite sample properties of test statistics often deviate markedly from their asymptotic ones. Exact analytical results typically are not available for precisely those situations which are most interesting from a practical standpoint, e.g., dynamic, simultaneous, mis-specified models. This paper presents and implements an approach for obtaining numericalanalytical formulae (response surfaces) which integrate existing analytical knowledge with Monte Carlo (experimental) results. Response surfaces can help summarize and interpret Monte Carlo simulations, and may reasonably approximate the unknown finite sample conditional probability formulae of the statistics evaluating the relationships of interest, for the DGP considered. Cox (1970) provides the basis for assessing the closeness of that approximation and, more generally, for conducting inference about response surfaces. Applying this approach, this paper investigates the effect of dynamics and simultaneity on the finite sample properties of maximum likelihood and instrumental variables statistics for testing both

nested and non-nested hypotheses for dynamic one- and two-equation models.

However, the results themselves are more elucidative than substantive: both estimated equations and the comprehensive model exhibit considerable

residual autocorrelation and parameter non-constancy, so further inferences are dubious.

31

The results demonstrate the value of asymptotic theory in interpreting finite sample properties and certain limitations for doing so. Response surfaces summarize the Monte Carlo results conveniently and provide simple formulae for obtaining reasonably accurate and computationally inexpensive predicticns of finite sample rejection frequencies within a sizable parameter space. Two practical finite sample results arise. First, transforming the x? Wald statistic to its F form eliminates a dominant term of oct}y. In fact, under the null hypothesis the resulting statistic is approximately an F-ratio and is virtually invariant to the degree of dynamics and simultaneity considered; under the alternative it is only moderately affected by those factors. Second, "large-o" and a small effective sample size strongly affect the finite sample properties of Sargan’s (1958) instrumental variables statistic and Ericsson’s (1983) Cox-type instrumental variables statistic. Additional analytical results could he.p in specifying the functional dependence on o and T and for deriving finite sample adjustments. Re-examination of Pesaran and Deaton’s (1978) empirical example illustrates the additional information gained from the instir-umental variables statistics. Although Monte Carlo experimentation can not eplace analysis, the two can complement each other effectively to

provide convenient formulae for interpreting empirical findings.

32

Appendix A. Simulation evidence: A _single-equation model

This appendix describes the experimental design, simulation, and post-simulation analysis of a pilot Monte Carlo study of the Cox and Wald statistics for a dynamic single-equation model with autocorrelated regressors, used to assess the potential value of the asymptotic formilae in Section 2 and of the response surface methodology in Section 4. The model is defined by the restrictions b = 9,2 = 0, so (4) and (5) are recursive. The correctly specified hypothesis Hy is

Hy: y= eZ, + dy, 4 + Up, . (Al) The following (falsely specified) non-nested alternative is considered:

Hi: ¥, = hioWy, + dye.) + YE (A2) with 7 = corr(Z, ,Wo,) * 0, so the comprehensive hypothesis Hy, is

Ho: Y¥, = eZ, + hi 2Wo, + dy, 4 + Uy, . (A3) The pair of non-nested hypotheses above is similar to that in Pesaran (1974) in that both have competing sets of exogenous variables. However, his DGP has no dynamics, whereas dynamics enter (Al) both directly (d ¥ 0) and indirectly (CP; x 0 for i=1, 2). A.1. Experimental design

The Monte Carlo design variables of this study are

C8’, T) = (ce, dy O11, Pir Par 11» M22, Y, T) (A4) where w,. is chosen to give selected values of y. Three parameters are normalized without loss of generality: o,, = 1 and w,, = wgg = 1/12. Rather arbitrarily, p, = pg and N= 1000. The remaining parameters span rarges

similar to those in Hendry and Harrison (1974, Section 6.1): c = (1., 4.)

d= (.2, .7), pg = (.3, .9), y = (.8, .9, .95), and T = (20, 50, 80) with a full factorial design of seventy-two experiments.

A.2. Simulation

Given this study's exploratory nature and in order to minimize

computational expenses, the exogenous variables 2. and Wo, are

33

non-stochastic (i.e., constant across replications, but not across experiments), and the Vi, are uniform. Estimation is by OLS and only the Cox statistic and the y? Wald statistic are evaluated. All other computational aspects are as in Section 5. A.3. Post-simulation analysis

Cursory examination of the Monte Carlo results reveals that the Cox test generally rejects H, more frequently than the Wald test, parallelling Pesairan’s (1974, 1982) Monte Carlo results for static models. It is unclear which test is more powerful because the estimated sizes of both tests are almost always larger than the asymptotic 5% level, with the Cox test the worse of the two. To evaluate powers as such, some control of the size would be necessary, e.g., by estimating the finite sample size by an order statistic based on the test statistic values from a Monte Carlo experiment when the assumed null hypotheses is true, or by the techniques discussed in Mehta (1979). Even without such adjustments, the simulation results are worthwhile evaluating because they use critical values which an applied econometrician typically would employ. In estimating response surfaces, all experiments for which the asymptotic power or the rejection frequency of H, is greater than .998 for either test, are excluded. Of the remaining fifty-two experiments, six randomly selected experiments are retained for the Chow test. In the spirit of Mizon and Hendry (1980), Type B response surfaces are estimated with 2, 2/T, A3/T, 7? ple and vl

for powers; 2, ple vl at 1/2

as regressors -1 . 37

, and dI ~ as regressors for sizes.

Parsimonious representations of those more general response surfaces appear

in (A5)-(A8) with a selection of evaluation criteria.

375 the order of the second approximation for the asymptotic distribution of D,, is also included in the unrestricted response surface for D,, but proved insignificant. The data are ordered by sample size, increasing in i, within each group. Detailed Monte Carlo results for Appendix A and Section 5 are available from the author upon request.

34

The restricted response surfaces for the type I error of the Wald (c2)

and Cox (Dg) statistics are

L*(s)-1(.05) = 1.92a)/2p71/2 (AS) (.16) [.20}

R2 = .43 a. = 1.659 n,(6,45) = .62 €,(51) 140.4 dw= 1.95

73(4,47) = 2.05 n7(1,49) = 10.23

u*(s)-L(.05) = 3.16A (.20) [.24]

1/2,-1/2 (A6)

R?2 = .57 a, = 2.240 1,(6,45) = .84 &,(51) = 255.9 dw = 1.06

n3(4,47) = .93 7(1,49) = 11.58

respectively. The response surfaces for the finite sample powers of the

Wald (c,) and Cox (D,) statistics are

L'(s) = .96L(m,) - 6.65L(,)/T - 1.81a/2(.,/T) (A7) (.11) ? (3.01) (.48) [ 08] [2.22] [.51]

R2 = .78 a. = 7.490 ,(6,43) = 2.03 €o(49) = 2749.1 dw = 2.11

73(3,46) = 2.75 7(6,42) = 2.34

L"(s) = 1.08L(2_) - 11.68L(«,)/T (A8) (.15) #4 (3.82) [.11] [3.83]

R? = .70 a, = 8.069 7,(6,44) = 1.90 €5(50) = 3255.4 dw = 2.17

n3(5,45) = .41 97(3,46) = 2.74 .

These response surfaces are in general agreement with the theory discussed earlier: the coefficients of L(.05) and L(a,) are statistically insignificantly different from unity, and a, is close to unity fo the response surfaces of type I errors. The coefficient on ql/2 in (A6) (versus that in (A5)) captures the larger positive finite sample bias in

size fer the Cox statistic. Also, the coefficients for £/T and X,/T are

35

negative in (A7), in line with Mizon and Hendry'’s (1980, pp. 35, 42) response: surfaces for the Wald test of a common factor. However, although

R? is large in (A7) and (A8), the corresponding values of o, and n7, suggest

that much more variation present in L*(s) could be explained by additional terms in those response surfaces. Re-using random numbers across experiments could aid in estimating the response surface coefficients more precisely and in reducing the values of oe. but it also might create spurious correlation between experiments; cf. Mizon and Hendry (1980, pp. 34-37 and footnote 4) and Hendry (1984, p. 971). The lack of control over the range of Top (or even m,) and the resulting loss. of one quarter of the experiments motivate the experimental design in Section 5.

The asymptotic formulae in Section 2 explain many features of the Monte Carlo simulations across experiments, with response surfaces providing a

concise, useful method of analyzing the relationship between observed

fluctuations, asymptotic approximations, and finite sample effects.

36

Appendix B. Hendry and Harrison's (1974) model

This appendix briefly describes Hendry and Harrison's (1974) model and its relationship to the model used in this paper. Their model can be expressed as

By, + Cz, + DY, -1 = UL ur = Ruy +e,

Ze AZ. 4 = VE (t=1,...,T) (B1) where Ye and Zz, are 2x1 and 4x1 vectors of endogenous and exogenous variables at time t; the normalizations are b,, = bog = -1.; B, C, D, R, and A are matrices of dimension 2x2, 2x4, 2x2, 2x2, and 4x4, respectively, and all the latent roots of Bp, R, and A lie within the unit circle;

e. 7 NID(O,=) and v7 IID(0,Q); and E(e.v’) = 0 for all t and s. Hendry and Harrison (1974, pp. 153-154) restrict C, D, and R such that the first

line in (Bl) may be written extensively as

Yae ~ Pa2e¥oe + Cra2%ye F 41Y) ga + de Yye 7 Tai4y ey + 1 4

with c,,c,, = 0 and with an implicit change in sign of the disturbances.

Rewriting (B2) in a simpler notation (and with Ye and ¢€, now denoting

t scalars), Ye bY, + eZ, + dy. 4 + uy ue = TUL te (B3) Y, = ay, + f'w. te, (B4) where £' = (C91:Cg9iCo3:Co4) and we in (B4) equals Z in (Bl). Thus, w= AW, 4 + ve . (B5)

Equations (4)-(6) in Section 3 are equivalent to (B3)-(B5) with r = 0 and correspond to equations (2)-(5) in Hendry and Harrison (1974), but with some slight changes in notation: Ps (rather than As) denotes the jth diagonal element of A; h’ and hy, are their f’ and C453 and the disturbances on the structural equations are u. and ve rather than ey and e,- Those changes are

made to avoid confusion with other notation in this paper.

37

Hendry and Harrison's model allows for autoregressive errors on the structural equations whereas (4)-(5) do not. At first blush, including autoregressive disturbances (r ~ 0) might appear an interesting extension. Pesaran (1974, pp. 164-169) derives the Cox statistic for models with fixed regressors and first-order autoregressive errors; and Sargan’s statistic cy genel‘alizes to allow for models nonlinear in their parameters (and in particular for linear models with autoregressive errors, see Sargan (1959, pp. 101-105; 1964, pp. 25-29) and Campos (1986b)), from which an IV statistic for testing non-nested nonlinear hypotheses could be constructed. However, autoregressive errors can be regarded as arising from a common factor restriction on a more general dynamic model with serially uncorrelated errors, a restriction which is often invalid in empirical studies. Hence, the restriction r = 0 is imposed in both Monte Carlo studies, but with d = 0. Note, though, that for that particular dynamic specification, (4) does not include a first-order autoregressive error as a

special case except trivially so when b = c = 0.

38 ror both theoretical and empirical discussions of the common factor

restriction, see Sargan (1959, pp. 91-92, 101-105), Durbin (1960a,

pp. 150-153; 1960b, pp. 235-238), Sargan (1964, pp. 27, 39-41), Hendry (1974), Hendry and Mizon (1978), Mizon and Hendry (1980), Sargan (1980d), and Hendry, Pagan and Sargan (1984, pp. 1045-1047, 1078-1080).

38

Appendix GC. Type B Response Surfaces * The dependent variable in each response surface is L (s)-L(m,) where nm, is .05 for f, and te, m. for f, and t,, and m™ for cy, Cs, Co, andc,. All

right-hand side variables represent effects present only in finite samples.

Table C.I. Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

Regressors Test Statistic 7 and : Diagnostic Statistics f. Co f, C3 T 11.8 9.7 8.2 6.1 (3.2) (3.0) (2.8) (2.9) [3.1] [2.9] [2.4] (2.6] T? -130. -106. -77. -54. (32.) (30.) (26.) (28.) [32.] [29.] (21. ] [24.] TS 328. 268. 140. 89. (81.) (76.) (63.) (68.) [79.] [71.] {[50.] [58.] To -2.10 -1.76 -2.44 -2.17 (.48) (.46) (.35) (.36) {.38] [.35] {[.42] [.38] T?0 16.3 © 12.5 6.9 4.8 (5.0) (4.7) (1.6) (1.6) [3.7] (3.4] [1.6] [1.6] T8o -36. -27. (13.) (12.) [ 9.] [ 9.] To? .0197 .0200 .0400 .0378 (.0046) (.0044) (.0192) (.0204) [ .0044] { .0040] [ .0166] {.0179] T?o? -59 .59 (.13) (.14) {.10] [.12] TPo -17.2 -13.7 -26.5 -24.0 (5.3) (5.0) (6.1) (6.6) [3.5] (3.2] [4.7] [5.5] T? po 185. 148. 184. 153. (50.) (46.) (54.) (58.) [34.] [30.] [41.] [48.]

39

Table C.I. (cont. Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

—_— ee

Regressors Test Statistic +r and Diagnos<zic Statistics f, Co f; C3 T3p5 -443. -352. -366. -283. (118.) (109.) (121.) (129.) [86.] [75.] [91.] [107. ] Tpoo 2.15 1.60 5.32 4.74 (.66) (.62) (.81) (.81) [.45] [.44] [.68] [ .68] T? poo -21.3 -15.6 -29.8 -24.1 (6.9) (6.5) (5.7) (5.6) [4.5] [4.5] [4.6] [4.6] TS poo 48.4 34.7 44.3 33.8 (17.4) (16.3) (10.7) (10.7) [11.4] [11.2] {8.3] [ 8.8] Tpgo? -.127 -.122 (.017) (.017) [.017] [.016] Th -3.33 -3.30 (.49) (.51) [.52] [.53] T22 . 7.74 6.61 (1.97) (2.06) [2.02] {2.10] Tho 542 .543 (.063) (.066) [.085] [ .082] T3 Lo? .530 .559 (.092) (.108) [ .067] { .090) Tpola -.504 - .485 (.077) (.082) [ .093] [.092]

40

Table C.1I. (concl.)

Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

rn

Regressors Test Statistic r and Diagnostic Statistics f. Co f, C3 R2 .850 . 866 985 985 oy 1.080 1.092 2.069 2.256 n 13 13 19 19 Chow n,(9,63-n) .85 .93 .88 L.04 RSS €5(72-n) 68.8 70.4 226.9 269.8 Parsimony n3(q,/2-n-q) .50 1.23 .40 . 88 {5} {6} {14} (14) Functional form n3(q,/71-n-q) 46 .39 1.26 75 {29} (29) (28) {30} RESET n4(4,68-n) 2.92 3.32 1.01 .90 Unit coefficient 1.02 | 1.36 1.11 1.13 (.37) (.34) (.22) (.23) Unit coefficient n5(1,71-n) .00 1.10 .26 .35 Normality €,(2) 24 .60 7.23 3.82 Heteroscedasticity n7(q,/1-n-q) -46 45 48 -49 {26} {26} {38} {38} ARCH ng (12,59-n) .59 60 15 .27 dw 2.31 2.25 1.86 1.86 AR residuals Ng (12,60-n) .87 1.02 .81 .85 Note; The value of the degrees of freedom q appears in curly brackets {+}.

41

Table C.I1. Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

Regressors Test Statistic r Regressors Test Statistic r and and Diagnostic Diagnostic Statistics Co Cy Statistics tes ty T -.57 qi/2 -38.5 -109. (.13) (5.0) (11.) [.10] [5.4] [10. ] T? -17.90 T 35.9 113. (.84) (4.7) (11.) [.92] [5.0] [11.] To 1.39 -.72 rr 1/2 -28.9 (.04) (.33) (5.6) [.04] [.35] [5.5] T20 - 50 15.9 Tl/2, 17.2 6.3 (.18) (2.7) (2.3) (1.4) [.15] [2.8] [2.7] [1.1] T3o -38.7 To -14.1 -6.1 (6.1) (2.1) (1.2) [6.9] [2.5] [1.0] To? .207 ql/2o2 -.90 (.026) (.15) [.028] [.19] T?202 -.153 - .080 To? 82 (.016) (.126) (.14) [.016) [.114] [.17] To 1.85 rr i/242 - 520 (.70) (.055) [.60] [.072] T2p, -3.01 -45.0 ql/2,, 44.5 146. (2.17) (8.3) (8.5) (26.) [1.92] [7.5] [8.7] [25.] Tp, 128. Tpo -44.4 -167. (33.) (8.2) (25.) [30.] [9.1] (25.) Tpoe -.155 qr l/2,, 25.3 137. (.040) (12.5) (36.) [.037] [15.2] [33.] 2 -1/2 Tpoo -.077 T Poo -17.5 1.53 (.027) (3.0) (.52) [ .030] [3.5] [.44] T? poo? .63 Tpoo 17.0 (.15) (2.8) [.15] [3.3] Te -2.00 qr /2n 6 -7.6 (.13) (2.5) [.15] [2.6]

42

Table C.II. (cont, ) Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

Regressors Test Statistic r Regressors Test Statistic 7 and até aan Diagnostic Diagnostic Statistics Co Cy Statistics te ty SSS Tlo .34 ql/2, 0? .96 (.10) (.19) {.12] [.23] Tpo2 -13.8 Tp 0? -.98 (2.6) (.18) [2.4] [.21] T2 p52 95. mr /2, 92 87 (18.) (.17) [16.] [.18] T3p 2 -166, ql/2, 30.3 (32.) (4.8) [26.] [3.8] Tpoka -46 TL -30.8 (.11) (4.6) {.11] (3.7] TLo? 081 tv29, -7.38 (.014) (.93) [.015] [.74] TLlo 7.35 (.91) [.75] tr 1/29, -4.21 (.48) {.39] ee -36.1 (11.3) { 9.2] Tpof 45.9 (11.0) [9.0] mr l/2, op -53.1 (16.8) [14.6] Tpoka -.88.

Lf a

Table C.II. (concl.) Estimates of Type B Response Surfaces for Finite Sample Sizes and Powers

Regressors Test Statistic +r

and and Diagnostic

Statistics Co Cy

R2 .9958 -9947 oe 1.489 2.097 n 7 17 Chow 7,(9,63-n) .90 81 RSS €2(72-n) 144.0 241.9 Parsimony n3(q,72-n-q) 73 94 {11} {16} Functional form n3(q,/71-n-q) .83 .90 {33} {31} RESET n4(4,63-n) 34 3.91 Unit coefficient . .97 .93 (.03) (.07) Unit coefficient ns(1,71-n) 1.26 1.11 Normality &,6(2) 1.00 1.34 Heteroscedasticity "7(q,7l-n-q) 1.08 45 {14} {34} ARCH ng(12,59-n) 37 57 dw 2.19 2.39 AR residuals Mg (12,50-n) 2.24 1.23

Regressors

Diagnostic Statistics te

R2 .9923 a 1.703 e n 16 Chow 7,(9,63-n) .82 RSS €2(72-n) 162.4 Parsimony n3(q,/2-n-q) .28 {2} Functional form n3(q,71-n-q) 1.27 {30} RESET n4(4,68-n) 7.74 Unit coefficient 1.02 (.04) Unit coefficient n5(1,71-n) .19 Normality €,6(2) 1.97 Heteroscedasticity n7(q,71-n-q) 2.59 {31} ARCH ng (12,59-n) 32 dw 2.13 AR residuals Ng (12,60-n) .59

Note: The value of the degrees of freedom q appears in curly brackets

Test Statistic r

—_—--———————————

eee

- 66 504.3 .60 {6} 41 (33) 4.84

1.09 (.19)

.22 1.42

54 {35}

.39

1.33

{

° ~

44

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52

International Finance Discussion Papers

IFDP NUMBER TITLES AUTHOR (s) 1987

317 Monte Carlo Methodology and the Finite Neil R. Ericsson Sample Properties of Statistics for Testing Nested and Non-Nested Hypotheses

316 The U.S. External Deficit: Its Causes Peter Hooper and Persistence Catherine 1. Mann

315 Debt Conversions: Economic Issues for Lewis S. Alexander Heavily Indebted Developing Countries

314 Exchange Rate Regimes and Macroeconomic David H. Howard Stabilization in a Developing Country

313 Monetary Policy in Taiwan, China Robert F. Emery

312 The Pricing of Forward Exchange Rates Ross Levine

311 Realignment of the Yen-Dollar Exchange Bonnie E. ‘Loopesko Rate: Aspects of the Adjustment Process Robert E. Johnson in Japan

310 The Effect of Multilateral Trade Ellen E. Meade Clearinghouses on the Demand for International Reserves

309 Protection and Retaliation: Changing Catherine L. Mann the Rules of the Game

308 International Duopoly with Tariffs Eric O'N. Fisher

Charles A. Wilson

307 A Simple Simulation Model of International Henry S. Terrell Bank Lending Robert S. Dohner

306 A Reassessment of Measures of the Dollar's B. Dianne Pauls Effective Exchange Value William L. Helkie

305 Macroeconomic Instability of the Less David F. Spigelman Developed Country Economy when Bank Credit is Rationed

304 The U.S. External Deficit in the 1980s: William L. Helkie

An Empirical Analysis

Peter Hooper

Please address requests for copies to International Finance Discussion

Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, D.C. 20551.

LFDP NUMBER

303

302

301

300

299

297

296

295

294

293

292

291

290

53 International Finance Discussion Papers

TITLES

An Analogue Model of Phase-Averaging Procedures

A Model of Exchange Rate Pass-Through

The Out-of-Sample Forecasting Performance of Exchange Rate Models When Coefficients are Allowed to Change

Financial Concentration and Development: An Empirical Analysis of the Venezuelan Case

Deposit Insurance Assessments on Deposits at Foreign Branches of U.S. Banks

1986 The International Debt Situation

The Cost Competitiveness of the Europaper Market

Germany and the European Disease

The United States International Asset and Liability Position: A Comparison of Flow of Funds and Commerce Department

An International Arbitrage Pricing Model with PPP Deviations

The Structure and Properties of the FRB Multicountry Model

Short-term and Long-term Expectations of the Yen/Dollar Exchange Rate: Evidence from Survey Data

Anticipated Fiscal Contraction: The Economic Consequences of the Announcement of Gramm-Rudman-Hollings

Tests of the Foreign Exchange Risk Premium Using the Expected Second Moments Implied by Option Pricing

AUTHOR(s)

Julia Campos

Neil R. Ericsson David F. Hendry Eric O'N. Fisher Garry J. Schinasi P.A.V.B. Swamy

Jaime Marquez Janice Shack-Marquez

Jeffrey C. Marquardt

Edwin M. Truman Rodney H. Mills John Davis

Patrick Minford

Guido E. van der Ven John E. Wilson

Ross Levine

Hali J. Edison Jaime R. Marquez Ralph W. Tryon Jeffrey A. Frankel Kenneth A. Froot

Robert A. Johnson

Richard K. Lyons