Finite Sample Limited

We propose nite sample tests and on den e sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables methods. The validity of the pro edures is una e ted by the presen e of identi ation problems or \weak instruments", so no dete tion of su h problems is required. We study two distin t approa hes for various models onsidered by Pagan (1984). The rst one is an instrument substitution method whi h generalizes an approa h proposed by Anderson and Rubin (1949) and Fuller (1987) for di erent (although related) problems, while the se ond one is based on splitting the sample. The instrument substitution method uses the instruments dire tly, instead of generated regressors, in order to test hypotheses about the \stru tural parameters" of interest and build on den e sets. The se ond approa h relies on \generated regressors", whi h allows a gain in degrees of freedom, and a sample split te hnique. For inferen e about general possibly nonlinear transformations of model parameters, proje tion te hniques are proposed. A distributional theory is obtained under the assumptions of Gaussian errors and stri tly exogenous regressors. We show that the various tests and on den e sets proposed are (lo ally) \asymptoti ally valid" under mu h weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general, they outperform the usual asymptoti inferen e methods in terms of both reliability and power. Finally, the te hniques suggested are applied to a model of Tobin's q and to a model of a ademi performan e. 1. Introdu tion A frequent problem in e onometri s and statisti s onsists in making inferen es on models whi h ontain unobserved explanatory variables, su h as expe tational or latent variables and variables observed with error; see, for example, Barro (1977), Pagan (1984, 1986) and the survey of Oxley and M Aleer (1993). A ommon solution to su h problems is based on using instrumental variables to repla e the unobserved variables by proxies obtained from auxiliary regressions (generated regressors). It is also well known that using su h regressors raises diÆ ulties for making tests and on den e sets, and it is usually proposed to repla e ordinary least squares (OLS) standard errors by instrumental variables (IV) based standard errors; see Pagan (1984, 1986) and Murphy and Topel (1985). In any ase, all the methods proposed to deal with su h problems only have an asymptoti justi ation, whi h means that the resulting tests and on den e sets an be extremely unreliable in nite samples. In parti ular, su h diÆ ulties o ur in situations involving \weak instruments", a problem whi h has re eived onsiderable attention re ently; see, for example, Nelson and Startz (1990a, 1990b), Buse (1992), Maddala and Jeong (1992), Bound, Jaeger, and Baker (1993, 1995), Angrist and Krueger (1995), Hall, Rudebus h, and Wil ox (1996), Dufour (1997), Shea (1997), Staiger and Sto k (1997) and Wang and Zivot (1998) and Zivot, Startz, and Nelson (1998) [for some early results relevant to the same issue, see also Nagar (1959), Ri hardson (1968) and Sawa (1969)℄. In this paper, we treat these issues from a nite sample perspe tive and we propose nite sample tests and on den e sets for models with unobserved and generated regressors. We also onsider a number of related problems in the more general ontext of linear simultaneous equations. To get reliable tests and on den e sets, we emphasize the derivation of truly pivotal (or boundedly pivotal) statisti s, as opposed to statisti s whi h are only asymptoti ally pivotal; for a general dis ussion of the importan e of su h statisti s for inferen e, see Dufour (1997). We study two distin t approa hes for various models onsidered by Pagan (1984). The rst one is an instrument substitution method whi h generalizes an approa h proposed by Anderson and Rubin (1949) and Fuller (1987, Se tion 1.4) for di erent (although related) problems, while the se ond one is based on splitting 1 the sample. The instrument substitution method uses the instruments dire tly, instead of generated regressors, in order to test hypotheses and build on den e sets about \stru tural parameters". The se ond approa h relies on \generated regressors", allowing a gain in degrees of freedom, and a sample split te hnique. Depending on the problem onsidered, we derive either exa t similar tests (and on den e sets) or onservative pro edures. The hypotheses for whi h we obtain similar tests (and orrespondingly similar on den e sets) in lude: (a) hypotheses whi h set the value of the unobserved (expe ted) variable oeÆient ve tor [as in Anderson and Rubin (1949) and Fuller (1987)℄; (b) analogous restri tions taken jointly with general linear onstraints on the oeÆ ients of the (observed) exogenous variables in the equation of interest; and ( ) hypothesis about the oeÆ ients of \surprise" variables when su h variables are in luded in the equation. Tests for these hypotheses are based on Fisher-type statisti s, but the on den e sets typi ally involve nonlinear (although quite tra table) inequalities. For example, when only one unobserved variable (or endogenous explanatory variable) appears in the model, the on den e interval for the asso iated oeÆ ient an be omputed easily on nding the roots of a quadrati polynomial. Note that Anderson-Rubin-type methods have not previously been suggested in the ontext of the general Pagan (1984) setup. The general setup we onsider here in ludes as spe ial ases the ones studied by Pagan (1984), Fuller (1987) and Zivot, Startz, and Nelson (1998), allowing for stru tural equations whi h in lude more than one endogenous \explanatory" variable as well as exogenous variables, so the hypotheses of type (a) we onsider and the asso iated on den e sets are in fa t more general than those onsidered by Fuller (1987, Se tion 1.4) and Zivot, Startz, and Nelson (1998). In parti ular, for the ase where the stru tural equation studied in ludes one endogenous explanatory variable, we extend the range of ases where lose-form quadrati on den e intervals [similar to those des ribed by Fuller (1987), Dufour (1997), and Zivot, Startz, and Nelson (1998)℄ are available. Further, problems su h as those des ribed in (b) and ( ) above have not apparently been onsidered at all from this perspe tive in the earlier literature. In the ase of the instrument substitution method, the tests and on den e sets so obtained an be interpreted as likelihood ratio (LR) pro edures (based on appropriately hosen redu ed form alternatives), or equivalently as pro le likelihood te hniques [for fur2 ther dis ussion of su h te hniques, see Bates and Watts (1988, Chapter 6), Meeker and Es obar (1995) and Chen and Jennri h (1996)℄. The exa t distributional theory is obtained under the assumptions of Gaussian errors and stri tly exogenous regressors, whi h ensures that we have well-de ned testable models. Although we stress here appli ations to models with unobserved regressors, the extensions of Anderson{Rubin (AR) pro edures that we dis uss are also of interest for inferen e in various stru tural models whi h are estimated by instrumental variable methods (e.g., simultaneous equations models). Furthermore, we observe that the tests and on den e sets proposed are (lo ally) \asymptoti ally valid" under mu h weaker distributional assumptions (whi h may involve non-Gaussian errors and weakly exogenous instruments). It is important to note that the on den e sets obtained by the methods des ribed above, unlike Wald-type on den e sets, are unbounded with non-zero probability. As emphasized from a general perspe tive in Dufour (1997), this is a ne essary property of any valid on den e set for a parameter that may not be identi able on some subset of the parameter spa e. As a result, on den e pro edures that do not have this property have true level zero, and the sizes of the orresponding tests (like Wald-type tests) must deviate arbitrarily from their nominal levels. It is easy to see that su h diÆ ulties o ur in models with unobserved regressors, models with generated regressors, simultaneous equations models, and di erent types of the error-in-variables models. In the ontext of the rst type of model, we present below simulation eviden e that strikingly illustrates these diÆ ulties. In parti ular, our simulation results indi ate that tests based on instrument substitution methods have good power properties with respe t to Wald-type tests, a feature previously pointed out for the AR tests by Maddala (1974) in a omparative study for simultaneous equations [on the power of AR tests, see also Revankar and Mallela (1972)℄. Furthermore, we nd that generated regressors sample-split tests perform better when the generated regressors are obtained from a relatively small fra tion of the sample (e.g., 10% of the sample) while the rest of the sample is used for the main regression (in whi h generated regressors are used). An apparent short oming of the similar pro edures proposed above, and probably one of the reasons why AR tests have not be ome widely used, is the fa t that they are restri ted to testing hypotheses whi h spe ify the values of the oeÆ ients of all the endogenous 3 (or unobserved) explanatory variables, ex luding the possibility of onsidering a subset of oeÆ ients (e.g., individual oeÆ ients). We show that inferen e on individual parameters or subve tors of oeÆ ients is however feasible by applying a proje tion te hnique analogous to the ones used in Dufour (1989, 1990), Dufour and Kiviet (1996, 1998)and Kiviet and Dufour (1997). We also show that su h te hniques may be used for inferen e on general possibly nonlinear transformations of the parameter ve tor of interest. The plan of the paper is as follows. In Se tion 2, we des ribe the main model whi h may ontain several unobserved variables (analogous to the \anti ipated" parts of those variables), and we introdu e the instrument substitution method for this basi model with various tests and on den e sets for the oeÆ ients of the unobserved variables. In Se tion 3, we propose the sample split method for the same model with again the orresponding tests and on den e sets. In Se tion 4, we study the problem of testing joint hypotheses about the oeÆ ients of the unobserved variables and various linear restri tions on the oeÆ ients of other (observed) regressors in the model. Se tion 5 extends these results to a model whi h also ontains error terms of the unobserved variables (the \unanti ipated" parts of these variables). In Se tion 6, we onsider the problem of making inferen e about general nonlinear transformations of model oeÆ ients. Then, in Se tion 7, we dis uss the \asymptoti validity" of the proposed pro edures proposed under weaker distributional assumptions. Se tion 8 presents the results of simulation experiments in whi h the performan e of our methods is ompared with some widely used asymptoti pro edures. Se tion 9 presents appli ations of the proposed methods to a model of Tobin's q and to an e onomi model of edu ational performan e. The latter explains the relationship between students' a ademi performan e, their personal hara teristi s and some so io-e onomi fa tors. The rst example illustrates inferen e in presen e of good instruments, while in the se ond example only poor instruments are available. As expe ted, on den e intervals for Tobin's q based on the Wald-type pro edures largely oin ide with those resulting from our methods. On the ontrary, large dis repan ies arise between the on den e intervals obtained from the asymptoti and the exa t inferen e methods when poor instruments are used. We on lude in Se tion 10. 4 2. Exa t inferen e by instrument substitution In this se tion, we develop nite sample inferen e methods based on instrument substitution methods for models with unobserved and generated regressors. We rst derive general formulae for the test statisti s and then dis uss the orresponding on den e sets. We onsider the following basi setup whi h in ludes as spe ial ases Models 1 and 2 studied by Pagan (1984): y = Z Æ +X + e ; (2.1) Z =WB + U ; Z = Z + V (2.2) where y is a T 1 ve tor of observations on a dependent variable, Z is a T G matrix of unobserved variables, X is a T K matrix of exogenous explanatory variables in the stru tural model, Z is a T G matrix of observed variables,W is a T q matrix of variables related to Z ; while e = (e1; : : : ; eT )0; U = [u0 1; : : : ; u0 T ℄0 and V = [v0 1; : : : ; v0 T ℄0 are T 1 and T G matri es of disturban es. The matri es of unknown oeÆ ients Æ; ; and B have dimensions respe tively G 1; K 1 and q G: In order to handle ommon variables in both equations (2.1) and (2.2), like for example the onstant term, we allow for the presen e of ommon olumns in the matri es W and X. In the setup of Pagan (1984), U is assumed to be identi ally zero (U = 0); et and v t are un orrelated [E(etv t) = 0℄; and the exogenous regressors X are ex luded from the \stru tural" equation (2.1). In some ases below, we will need to reinstate some of the latter assumptions. The nite sample approa h we adopt in this paper requires additional assumptions, espe ially on the distributional properties of the error term. Sin e (2.2) entails Z =WB+V where V = U + V ; we will suppose the following onditions are satis ed: X andW are independent of e and V ; (2.3) 5 rank (X) = K ; 1 rank (W ) = q < T ; G 1 ; 1 K +G < T ; (2.4) (et; v0 t)0 ind N [0; ℄ ; t = 1; : : : ; T ; (2.5) det( ) > 0 : (2.6) If K = 0, X is simply dropped from equation (2.1). Note that no assumption on the distribution of U is required. Assumptions (2.3) { (2.6) an be relaxed if they are repla ed by assumptions on the asymptoti behavior of the variables as T ! 1. Results on the asymptoti \validity" of the various pro edures proposed in this paper are presented in Se tion 7. Let us now onsider the null hypothesis: H0 : Æ = Æ0 : (2.7) The instrument substitution method is based on repla ing the unobserved variable by a set of instruments. First, we substitute (2.2) into (2.1): y = (Z V )Æ +X + e = ZÆ +X + (e V Æ) : (2.8) Then subtra ting ZÆ0 on both sides of (2.8), we get: y ZÆ0 =WB(Æ Æ0) +X + u (2.9) where u = e V Æ0+U (Æ Æ0): Now suppose that W and X have K2 olumns in ommon (0 K2 < q) while the other olumns of X are linearly independent of W : W = [W1; X2℄ ; X = [X1; X2℄ ; rank [W1; X1; X2℄ = q1 +K < T (2.10) where W1, X1 and X2 are T q1, T K1 and T K2 matri es, respe tively (K = K1+K2, 6 q = q1 +K2). We an then rewrite (2.9) as y ZÆ0 =W1Æ1 +X + u (2.11) where Æ1 = B1(Æ Æ0), 2 = 2+B2(Æ Æ0); = ( 01; 02 )0; Bi is a Ki G matrix (i = 1; 2) and B = [B0 1; B0 2℄0: It is easy to see that model (2.11) under H0 satis es all the assumptions of the lassi al linear model. Furthermore, sin e Æ1 = 0 when Æ = Æ0, we an test H0 by a standard F -test of the null hypothesis H0 : Æ1 = 0 : (2.12) This F -statisti has the form F (Æ0; W1) = (y ZÆ0)0P (M(X)W1) (y ZÆ0)=q1 (y ZÆ0)0M([W1; X℄) (y ZÆ0)=(T q1 K) (2.13) where P (A) = A(A0A) 1A0 andM(A) = IT P (A) for any full olumn rank matrixA:When Æ = Æ0, we have F (Æ0; W1) F (q1; T q1 K); so that F (Æ0; W1) > F ( ; q1; T q1 K) is a riti al region with level for testing Æ = Æ0; where P [F (Æ0; W1) F ( ; q1; T q1 K)℄ = 1 : The essential ingredient of the test is the fa t that q1 1, i.e. some instruments must be ex luded from X in (2.1). On the other hand, the usual order ondition for \identi ation" (q1 G) is not ne essary for applying this pro edure. In other words, it is possible to test ertain hypotheses about Æ even if the latter ve tor is not ompletely identi able. It is then straightforward to see that the set CÆ( ) = fÆ0 : F (Æ0; W1) F ( ; q1; T q1 K)g (2.14) is a on den e set with level 1 for the oeÆ ient Æ. The tests based on the statisti F (Æ0; W1) and the above on den e set generalize the pro edures des ribed by Fuller (1987, pp. 16-17), for a model with one unobserved variable (G = 1); X limited to a onstant variable (K = 1) and two instruments (q = 2; in luding a onstant), and by Zivot, Startz, 7 and Nelson (1998) for a model with one unobserved variable (G = 1), no exogenous variables and an arbitrary number of instruments (q 1): Consider now the ase where Z is a T 1 ve tor and X is a T K matrix. In this ase, the on den e set (2.14) for testing H0 : Æ = Æ0 has the following general form: CÆ( ) = Æ0 : (y ZÆ0)0A1(y ZÆ0) (y ZÆ0)0A2(y ZÆ0) 2 q1 F (2.15) where F = F ( ; q1; T q1 K) and 2 = T q1 K and the matri es A1 = P (M(X)W1); A2 =M([W1; X℄): Sin e ( 2=q1) only takes positive values, the inequality in (2.15) is equivalent to the quadrati inequality: aÆ20 + bÆ0 + 0 (2.16) where a = Z 0CZ; b = 2y0CZ; = y0Cy; C = A1 G A2 and G = (q1= 2)F : Again, the above quadrati on den e intervals may be viewed of generalizations of the quadrati on den e intervals des ribed by Fuller (1987, page 55) and Zivot, Startz, and Nelson (1998). 2 In empiri al work, some problems may arise due to the high dimensions of the matri es M(X) and M([W1; X℄). A simple way to avoid this diÆ ulty onsists in using ve tors of residuals from appropriate OLS regressions. Consider the oeÆ ient a = Z 0CZ. We may repla e it by the expression Z 0A1Z G Z 0A2Z and then rewrite both terms as follows: Z 0A1Z = (Z 0M(X)) (M(X)W1) [(M(X)W1)0(M(X)W1)℄ 1(M(X)W1)0(M(X)Z) ; Z 0A2Z = Z 0M([W1; X℄)Z = [M([W1; X℄)Z℄0[M([W1; X℄)Z℄ : In the above expressions, M(X)Z is the ve tor of residuals obtained by regressing Z on X; M(X)W1 is the ve tor of residuals from the regression ofW1 onX; and nallyM([W1; X℄)Z is a ve tor of residuals from the regression of Z on X and W1. We an pro eed in the same way to ompute the two other oeÆ ients of the quadrati inequality (2.16). This will require only two additional regressions: y on X; and y on both X and W1: 8 Table 1 Confiden e sets based on the quadrati inequality aÆ20 + bÆ0 + 0 0 < 0 (real roots) ( omplex roots) a > 0 [Æ1 ; Æ2 ℄ Empty a < 0 ( 1; Æ1 ℄ [ [Æ2 ; 1) ( 1; +1) a = 0 b > 0 1; b b < 0 b ; 1 b = 0; > 0 Empty b = 0; 0 ( 1; +1) It is easy to see that the on den e set (2.16) is determined by the roots of the se ond order polynomial in (2.16). The shape of this on den e set depends on the signs of a and = b2 4a : All possible options are summarized in Table 1 where Æ1 denotes the smaller root and by Æ2 the larger root of the polynomial (when both roots are real). Note that the on den e set CÆ( ) may be empty or unbounded with a non-zero probability. Sin e the redu ed form for y an be written y =W1 1 +X1 12 +X2 22 + vy (2.17) where 1 = B1Æ; 21 = 1; 22 = 2 + B2 and vy = e + U Æ; we see that the ondition 1 = B1Æ may be interpreted as an overidentifying restri tion. Jointly with Æ = Æ0; this ondition entails the hypothesis H0 : B1(Æ Æ0) = 0 whi h is tested by the statisti F (Æ0; W1): Thus an empty on den e set means the ondition B1(Æ Æ0) = 0 is reje ted for any value of Æ0 and so indi ates that the overidentifying restri tions entailed by the stru tural model (2.1) (2.2) are not supported by the data, i.e. the spe i ation is reje ted. However, if the model is orre tly spe i ed, the probability of obtaining an empty on den e set is not greater than : On the other hand, the possibility of an unbounded on den e set is a ne essary hara teristi of any valid on den e set in the present ontext, be ause the stru tural parameter Æ may not be identi able [see Dufour (1997)℄. Unbounded on den e sets are most likely to o ur when Æ is not identi ed or lose to being unidenti ed, for then all values of Æ are almost observationally equivalent. Indeed an unbounded on den e set obtains when a < 0 or (equivalently) when F ( 1 = 0) < F ; where F ( 1 = 0) is the 9 F -statisti for testing 1 = 0 in the regression Z =W1 1 +X + vZ : (2.18) In other words, the on den e interval (2.15) is unbounded if and only if the oeÆ ients of the exogenous regressors in W1 [whi h is ex luded from the stru tural equation (2.1)℄ are not signi antly related to Z at level : i.e:, W1 an be interpreted as a matrix of \weak instruments" for Z: In ontrast, Wald-type on den e sets for Æ are typi ally bounded with probability one, so their true level must be zero. Note nally that an unbounded on den e set an be informative: e.g., the set ( 1; Æ1 ℄ [ [Æ2 ; 1) may ex lude e onomi ally important values of Æ (Æ = 0 for example). 3. Inferen e with generated regressors Test statisti s similar to those of the previous se tion may alternatively be obtained from linear regressions with generated regressors. To obtain nite sample inferen es in su h ontexts, we propose to ompute adjusted values from an independent sample. In parti ular, this an be done by applying a sample split te hnique. Consider again the model des ribed by (2.1) to (2.6). In (2.9), a natural thing to do would onsist in repla ing WB by WB̂, where B̂ is an estimator of B. Take B̂ = (W 0W ) 1W 0Z; the least squares estimate of B based on (2.2). Then we have: y ZÆ0 =WB̂(Æ Æ0) +X + [u+W (B B̂) (Æ Æ0)℄ = ẐÆ0 +X + u (3.1) where Æ0 = Æ Æ0 and u = e V Æ0+[U +W (B B̂) ℄(Æ Æ0). Again, the null hypothesis Æ = Æ0 may be tested by testing H0 : Æ0 = 0 in model (3.1). Here the standard F statisti for H0 is obtained by repla ing W1 by Ẑ in (2.13), i.e. F (Æ0; Ẑ) = (y ZÆ0)0P (M(X)Ẑ) (y ZÆ0)=G (y ZÆ0)0M([Ẑ; X℄) (y ZÆ0)=(T G K) ; (3.2) 10 ifK = 0 [noX matrix in (2.1)℄, we onventionally setM(X) = IT and [Ẑ; X℄ = Ẑ: However, to get a null distribution for F (Æ0; Ẑ), we will need further assumptions. For example, in addition to the assumptions (2.1) to (2.6), suppose, as in Pagan (1984), that e and V U + V are independent. (3.3) In this ase, when Æ = Æ0 = 0; Ẑ and u are independent and, onditional on Ẑ, model (3.1) satis es all the assumptions of the lassi al linear model (with probability 1). Thus the null distribution of the statisti F (0; Ẑ) for testing Æ0 = 0 is F (G; T G K). Unfortunately, this property does not extend to the more general statisti F (Æ0; Ẑ) where Æ0 6= 0 be ause Ẑ and u are not independent in this ase. A similar observation (in an asymptoti ontext) was made by Pagan (1984). To deal with more general hypotheses, suppose now that an estimate ~ B of B su h that ~ B is independent of e and V (3.4) is available, and repla e Ẑ =WB̂ by ~ Z =W ~ B in (3.1). We then get y ZÆ0 = ~ ZÆ0 +X + u (3.5) where u = e V Æ0+ [U +W (B ~ B)℄ (Æ Æ0). Under the assumptions (2.1) { (2.6) with Æ = Æ0 and onditional on ~ Z (or ~ B), model (3.5) satis es all the assumptions of the lassi al linear model and the usual F -statisti for testing Æ0 = 0; F (Æ0; ~ Z) = (y ZÆ0)0P (M(X) ~ Z)(y ZÆ0)=G (y ZÆ0)0M([ ~ Z; X℄)(y ZÆ0)=(T G K) (3.6) where the usual notation has been adopted, follows an F (G; T G K) distribution. Consequently, the riti al region F (Æ0; ~ Z) > F ( ; G; T G K) has size : Note that ondition (3.3) is not needed for this result to hold. Furthermore ~ CÆ( ) = fÆ0 : F (Æ0; ~ Z) F ( ; G; T G K)g (3.7) 11 is a on den e set for Æ with size 1 : For s alar Æ (G = 1), this on den e set takes a form similar to the one in (2.15), ex ept that A1 = P (M(X) ~ Z) and A2 =M([ ~ Z; X℄): A pra ti al problem here onsists in nding the independent estimate ~ B. Under the assumptions (2.1) { (2.6), this an be done easily by splitting the sample. Let T = T1+T2, where T1 > G + K and T2 q, and write: y = (y0 (1) ; y0 (2))0; X = (X 0 (1) ; X 0 (2))0; Z = (Z 0 (1) ; Z 0 (2))0; W = (W 0 (1) ; W 0 (2))0; e = (e0(1) ; e0(2))0; V = (V 0 (1) ; V 0 (2))0 and (U 0 (1) ; U 0 (2))0; where the matri es y(i); X(i) ; Z(i) ; W(i) ; e(i) ; V (i) and U (i) have Ti rows (i = 1; 2): Consider now the equation y(1) Z(1)Æ0 = ~ Z(1)Æ0 +X(1) + u(1) (3.8) where ~ Z(1) = W(1) ~ B; ~ B = [W 0 (2)W(2)℄ 1W 0 (2)Z(2) is obtained from the se ond sample, and u(1) = e(1) V (1)Æ0 + [U (1) +W(1)(B B̂) ℄(Æ Æ0): Clearly ~ B is independent of e(1) and V (1); so the statisti F (Æ0; ~ Z(1)) based on equation (3.8) follows a F (G; T1 K G) distribution when Æ = Æ0. A sample split te hnique has also been suggested by Angrist and Krueger (1995) to build a new IV estimator, alled Split Sample Instrumental Variables (SSIV) estimator. Its advantage over the traditional IV method is that SSIV yields an estimate biased toward zero, rather than toward the probability limit of the OLS estimator in nite sample if the instruments are weak. Angrist and Krueger show that an unbiased estimate of the attenuation bias an be al ulated and, onsequently, an asymptoti ally unbiased estimator (USSIV) an be derived. In their approa h, Angrist and Krueger rely on splitting the sample in half, i.e., setting T1 = T2 = T2 when T is even. However, in our setup, di erent hoi es for T1 and T2 are learly possible. Alternatively, one ould sele t at random the observations assigned to the ve tors y(1) and y(2): As we will show later (see Se tion 8) the number of observations retained for the rst and the se ond subsample have a dire t impa t on the power of the test. In parti ular, it appears that one an get a more powerful test on e we use a relatively small number of observations for omputing the adjusted values and keep more observations for the estimation of the stru tural model. This point is illustrated below by simulation experiments. Finally, it is of interest to observe that sample splitting 12 te hniques an be used in onjun tion with the Boole-Bonferroni inequality to obtain nitesample inferen e pro edures in other ontexts, su h as seemingly unrelated regressions and models with moving average errors; for further dis ussion, the reader may onsult Dufour and Torr es (1998). 4. Joint tests on Æ and The instrument substitution and sample split methods des ribed above an easily be adapted to test hypotheses on the oeÆ ients of both the latent variables and the exogenous regressors. In this se tion, we derive F -type tests for general linear restri tions on the oeÆ ient ve tor. Consider again model (2.1) { (2.6), whi h after substituting the term (Z V ) for the latent variable yields the following equation: y = (Z V )Æ +X + e = ZÆ +X + (e V Æ) : (4.1) We rst onsider a hypothesis whi h xes simultaneously Æ and an arbitrary set of linear transformations of : H0 : Æ = Æ0; R1 = 10 where R1 is a r1 K xed matrix su h that 1 rank(R1) = r1 K: The matrix R1 an be viewed as a submatrix of a K K matrix R = [R0 1; R0 2℄0 where det(R) 6= 0, so that we an write R = 264 R1 R2 375 = 264 R1 R2 375 = 264 1 2 375 : (4.2) Let XR = XR 1 = [XR1 ; XR2 ℄ where XR1 and XR2 are T r1 and T r2 matri es (r2 = K r1). Then we an rewrite (4.1) as y = ZÆ +XR1 1 +XR2 2 + (e V Æ) : (4.3) 13 Subtra ting ZÆ0 and XR1 10 on both sides, we get y ZÆ0 XR1 10 = [W1B1 +X2B2℄ (Æ Æ0) +XR1( 1 10) +XR2 2 + [e V Æ0 + U (Æ Æ0)℄ : (4.4) Suppose now thatW andX have K2 olumns in ommon (with 0 K2 < q); while the other olumns of X are linearly independent of W as in (2.10). Sin e X = [X1; X2℄ = XRR = XR1R1 +XR2R2 ; we an write X = [X1; X2℄ = [XR1R11 +XR2R21; XR1R12 +XR2R22℄ ; where R1 = [R11; R12℄, R2 = [R21; R22℄ and Rij is a ri Kj matrix (i; j = 1; 2): Then repla e X2 by XR1R12 +XR2R22 in (4.4): y ZÆ0 XR1 10 =W1Æ 1 +XR1 1 +XR2 2 + u (4.5) where Æ 1 = B1(Æ Æ0), 1 = R12B2(Æ Æ0) + ( 1 10), 2 = R22B2(Æ Æ0) + 2, and u = e V Æ0 + U (Æ Æ0): Consequently, we an test H0 by testing H 0 0 : Æ 1 = 0; 1 = 0;in (4.5), whi h leads to the statisti : F (Æ0; 10; W1; XR1) = fy (Æ0; 10)0P (M(XR2)WR1) y (Æ0; 10)=(q1 + r1)g fy (Æ0; 10)0M([W1; X℄) y (Æ0; 10)=(T q1 K)g (4.6) where y (Æ0; 10) = y ZÆ0 XR1 10 andWR1 = [W1; XR1 ℄; if r2 = 0; we setM(XR2) = IT: : Under H0; F (Æ0; 10; W1; XR1) F (q1 + r1; T q1 K) and we reje t H0 at level when F (Æ0; 10; W1; XR1) > F ( ; q1 + r1; T q1 K). Correspondingly, f(Æ00; 010)0 : F (Æ0; 10; W1; XR1) F ( ; q1 + r1; T q1 K)g is a on den e set with level 1 for Æ and 1 = R1 1. Suppose now we employ the pro edure with generated regressors using an estimator ~ B independent of u and V . We an then pro eed in the following way. Setting ~ Z = W ~ B and V̂ = Z ~ Z; we have:y ZÆ0 XR1 10 = ~ ZÆ 1 +XR1 1 +XR2 2 + u (4.7) where Æ 1 = Æ Æ0, 1 = 1 10 and u = e V Æ0 + [U +W (B ~ B)℄(Æ Æ0). In this 14 ase we will simply test the hypothesis H0 : Æ 1 = 0; 1 = 0. The F statisti for H0 takes the form: F (Æ0; 10; ~ Z; XR1) = fy (Æ0; 10)0P (M(XR2) ~ ZR1) y (Æ0; 10)=(G+ r1)g fy (Æ0; 10)0M([ ~ Z; X℄) y (Æ0; 10)=(T G K)g (4.8) where y(Æ0; 10) = y ZÆ0 XR1 10; and ~ ZR1 = [ ~ Z; XR1 ℄: Under H0, F (Æ0; 10; ~ Z; XR1) F (G + r1; T G K): The orresponding riti al region with level is given by F (Æ0; 10; ~ Z; XR1) > F ( ; G + r1; T G r1) ; and the on den e set at level 1 is thus f(Æ00; 010)0 : F (Æ0; 10; ~ Z; XR1) F ( ; G+ r1; T G Kg: 5. Inferen e with a surprise variable In many e onomi models we en ounter soalled \surprise" terms among the explanatory variables. These re e t the di eren es between the expe ted values of latent variables and their realizations. In this se tion we study a model whi h ontains the unanti ipated part of Z [Pagan (1984, model 4)℄ as an additional regressor beside the latent variable, namely: y = Z Æ + (Z Z ) +X + e = ZÆ + V +X + e V Æ ; (5.1) Z = Z + V =WB + (U + V ) =WB + V ; (5.2) where the general assumptions (2.3) { (2.6) still hold. The term (Z Z ) represents the unanti ipated part of Z. This setup raises more diÆ ult problems espe ially for inferen e on . Nevertheless we point out here that the pro edures des ribed in the pre eding se tions for inferen e on Æ and remain appli able essentially without modi ation, and we show that similar pro edures an be obtained as well for inferen e on provided we make the additional assumption (3.3). Consider rst the problem of testing the hypothesis H0 : Æ = Æ0. Applying the same 15 pro edure as before, we get the equation: y ZÆ0 =WB(Æ Æ0) +X + V + (e V Æ0) (5.3) hen e, assuming that W and X have K2 olumns in ommon, y ZÆ0 =W1B1(Æ Æ0) +X1 1 +X2 2 + e+ V ( Æ0) =W1Æ1 +X + u (5.4) where Æ1 = B1(Æ Æ0), 2 = 2+B2(Æ Æ0), = ( 01; 20)0 and u = e+V ( Æ0). Then we an test Æ = Æ0 by using the F -statisti for Æ10 = 0: F (Æ0; W1) = (y ZÆ0)0P (M(X)W1) (y ZÆ0)=q1 (y ZÆ0)0M [X(W1)℄ (y ZÆ0)=(T q1 K) : (5.5) When Æ = Æ0, F (Æ0; W1) F (q1; T q1 K): It follows that F (Æ0; W1) > F ( ; q K2; T q1 K) is a riti al region with level for testing Æ = Æ0 while fÆ0 : F (Æ0; W1) F ( ; q1; T q1 K)g is a on den e set with level 1 for Æ. Thus, the pro edure developed for the ase where no surprise variable is present applies without hange. If generated regressors are used, we an write: y ZÆ0 =WB̂(Æ Æ0) +X + e+ V ( Æ0) + V̂ (Æ Æ) : (5.6) Repla ing WB̂ by ~ Z =W ~ B, where ~ B is an estimator independent of e and V , we get y ZÆ0 = ~ ZÆ +X + u (5.7) where Æ = Æ Æ0 ; u = e+V ( Æ0)+ ~ V (Æ Æ0) and ~ V = Z ~ Z: Here the hypothesis Æ = Æ0 entails H 0 0 : Æ = 0. The F -statisti F (Æ0; ~ Z) de ned in (3.6) follows an F (G; T G K) distribution when Æ = Æ0. Consequently, the tests and on den e set pro edures based on F (Æ0; ~ Z) apply in the same way. Similarly, it is easy to see that the joint inferen e pro edures des ribed in Se tion 4 also apply without hange. 16 Let us now onsider the problem of testing an hypothesis on the oeÆ ient of the surprise term, i.e. H0 : = 0. In this ase, it appears more diÆ ult to obtain a nite-sample test under the assumptions (2.1) { (2.6). So we will assume that the following onditions, whi h are similar to assumptions made by Pagan (1984) setup, hold: a) U = 0 ; b) e and V are independent. (5.8) Then we an write: y = Z Æ + (Z Z ) +X + e = Z +W1Æ 1 +X + e : (5.9) Subtra ting Z 0 on both sides yields y Z 0 = Z +W1Æ1 +X + e (5.10) where = 0. We an thus test = 0 by testing = 0 in (5.10), using F ( 0; Z) = (y Z 0)0P (M([W1; X℄)Z) (y Z 0)=G (y Z 0)0M([W1; Z; X℄) (y Z 0)=(T G q1 K) : (5.11) When = 0, F ( 0; Z) F (G; T G q1 K) so that F ( 0; Z) F ( ; G; T G q1 K) is a riti al region with level for = 0 and f 0 : F ( 0; Z) F ( ; G; T G q1 K)g (5.12) is a on den e set with level 1 for . When is a s alar, this on den e set an be written as: 0 : (y Z 0)0D(y Z 0) (y Z 0)0E(y Z 0) 2 1 F (5.13) where 1 = G = 1, 2 = T G q1 K, D = P (M([W1; X℄)), E =M([W1; Z; X℄): Sin e the ratio 2= 1 always takes positive values, the on den e set is obtained by nding the 17 values 0 that satisfy the inequality a 20 + b 0 + 0 ; where a = Z 0LZ ; b = 2Z 0Ly ; = y0Ly ; L = D H E and H = ( 1= 2)F . Finally it is straightforward to see that the problem of testing a joint hypothesis of the type H0 : = 0; R1 = 10 an be treated by methods similar to the ones presented in Se tion 4. 6. Inferen e on general parameter transformations The nite sample tests presented in this paper are based on extensions of Anderson{Rubin statisti s. An apparent limitation of Anderson{Rubin type tests omes from the fa t that they are designed for hypothesis xing the omplete ve tor of the endogenous (or unobserved) regressor oeÆ ients. In this se tion, we propose a solution to this problem whi h is based on applying a proje tion te hnique. Even more generally, we study inferen e on general nonlinear transformations of Æ in (2.1), or more generally of (Æ0; 01)0 where 1 = R1 is a linear transformation of ; and we propose nite sample tests of general restri tions on subve tors of Æ or (Æ0; 01)0: For a similar approa h, see Dufour (1989, 1990)and Dufour and Kiviet (1998). Let = Æ or = (Æ0; 01)0 depending on the ase of interest. In the previous se tions, we derived on den e sets for whi h take the general form C ( ) = f 0 : F ( 0) F g (6.1) where F ( 0) is a test statisti for = 0 and F is a riti al value su h that P [ 2 C ( )℄ 1 : If = 0, we have P [ 0 2 C ( )℄ 1 ; P [ 0 = 2 C ( )℄ : (6.2) Consider a (possibly nonlinear) transformation = f( ) of . Then it is easy to see that C ( ) f 0 : 0 = f( ) for some 2 C ( )g (6.3) 18 is a on den e set for with level at least 1 ; i.e. P [ 2 C ( )℄ P [ 2 C ( )℄ 1 (6.4) hen e P [ = 2 C ( )℄ : (6.5) Thus, by reje ting H0 : = 0 when 0 = 2 C ( ), we get a test of level . Further 0 = 2 C ( ), 0 6= f( 0) ; 8 0 2 C ( ) (6.6) so that the ondition 0 = 2 C ( ) an be veri ed by minimizing F ( 0) over the set f 1( 0) = f 0 : f( 0) = 0g and he king whether the in mum is greater than F . When = f( ) is a s alar, it is easy to obtain a on den e interval for by onsidering variables L = inff 0 : 0 2 C ( )g and U = supf 0 : 0 2 C ( )g obtained by minimizing and maximizing 0 subje t to the restri tion 0 2 C ( ): It is then easy to see that P [ L U ℄ P [ 2 C ( )℄ 1 (6.7) so that [ L; U ℄ is a on den e interval with level 1 for : Further, if su h on den e intervals are built for several parametri fun tions, say i = fi( ); i = 1; ::: ;m; from the same on den e set C ( ); the resulting on den e intervals [ iL; iU ℄; i = 1; ::: ;m; are simultaneous at level 1 ; in the sense that the orresponding m dimensional on den e box ontains the true ve tor ( 1; ::: ; m) with probability (at least) 1 ; for further dis ussion of simultaneous on den e sets, see Miller (1981), Savin (1984) and Dufour (1989). When a set of on den e intervals are not simultaneous, we will all them \marginal intervals". Consider the spe ial ase where = Æ = (Æ1; Æ02)0 and = Æ1; i.e. is an element of Æ: Then the on den e set C ( ) takes the form: C ( ) = CÆ1( ) = fÆ10 : (Æ10; Æ02)0 2 CÆ( ); for some Æ2g: (6.8) 19 Consequently we must have: P [Æ1 2 CÆ1( )℄ 1 ; P [Æ10 = 2 CÆ1( )℄ : (6.9) Further if we onsider the random variables ÆL1 = inffÆ10 : Æ10 2 CÆ1( )g and ÆU1 = supfÆ10 : Æ10 2 CÆ1( )g obtained by minimizing and maximizing Æ10 subje t to the restri tion Æ10 2 CÆ1( ), [ÆL1 ; ÆU1 ℄ is a on den e interval with level 1 for Æ1: The test whi h reje ts H0 : Æ1 = Æ10 when Æ10 = 2 CÆ1( ) has level not greater than . Furthermore, Æ10 = 2 CÆ1( ), F (Æ010; Æ02)0 > F ;8Æ2 : (6.10) Condition (6.10) an be he ked by minimizing the F (Æ010; Æ02)0 statisti with respe t to Æ2 and omparing the minimal value with F . The hypothesis Æ1 = Æ10 is reje ted if the in mum of F (Æ010; Æ02)0 is greater than F . In pra ti e, the minimizations and maximizations required by the above pro edures an be performed easily through standard numeri al te hniques. Finally, it is worthwhile noting that, even though the simultaneous on den e set C ( ) for may be interpreted as a on den e set based on inverting LR-type tests for = 0 [or a pro le likelihood on den e set [see Meeker and Es obar (1995) or Chen and Jennri h (1996)℄, proje tion-based on den e sets, su h as C ( ); are not (stri tly speaking) LR on den e sets. 7. Asymptoti validity The methods des ribed in previous se tions are exa t in nite samples under relatively spe i assumptions, it is easy to see that they remain asymptoti ally valid (as the number of observations goes to in nity) under mu h under weaker assumptions. Consider again the model des ribed by (2.1) { (2.6) and (2.10), whi h yields the following equations: y = ZÆ +X + u ; Z =W1B1 +X2B2 + V (7.1) 20 where u = e V Æ. If we are prepared to a ept a pro edure whi h is only asymptoti ally \valid", we an relax the nite-sample assumptions (2.3) { (2.6) sin e the normality of error terms and their independen e are no longer ne essary. For example, onsider the statisti F (Æ0; W1) de ned in (2.13). Then, under general regularity onditions, we an show that: a) under the null hypothesis Æ = Æ0 the F -statisti in (2.13), follows a 2q1=q1 distribution asymptoti ally (as T !1); b) under the xed alternative Æ = Æ1, provided B1(Æ1 Æ0) 6= 0; the value of (2.13) tends to get in nitely large as T in reases, i.e. the test based on F (Æ0; W1) is onsistent. In parti ular, the following onditions are suÆ ient for the latter properties to hold: u0u T ; u0V T ; V 0V T ! 2u; uV ; V ; (7.2) X 0X T ; X 0W1 T ; W 0 1W1 T ! ( XX ; XW1 ; W1W1) ; (7.3) (T 12 X 0u; T 1 2 W 0 1u; T 1 2 X 0V; T 12 W 0 1V )) ( Xu; W1u; XV ; W1V ) ; (7.4) where ! and ) denote respe tively onvergen e in probability and onvergen e in distribution as T ! 1, and the joint distribution of the random variables in is multinormal with the ovarian e matrix of ( 0Xu; 0W1u)0 given by = V 264 Xu W1u 375 = 264 2 XX XW1 0XW1 W1W1 375 where XW1 = 0W1X and det( ) 6= 0: For further details, the reader may onsult Dufour and Jasiak (1993) and Staiger and Sto k (1997). It is easy to prove similar asymptoti results for the other tests proposed in this paper. 21 8. Monte Carlo study In this se tion, we present the results of a small Monte Carlo experiment omparing the performan e of the exa t tests proposed above with other available (asymptoti ally justi ed) pro edures, espe ially Wald-type pro edures. A total number of one thousand realizations of an elementary version of the model (2.1){(2.2), equivalent to Model 1 dis ussed by Pagan (1984), were simulated for a sample of size T = 100: In this parti ular spe i ation, only one latent variable Z is present. The error terms in e and V (where e and V are ve tors of length 100) are independent with N(0; 1) distributions. We allow for the presen e of only one instrumental variable W in the simulated model, whi h was also independently drawn from a N(0; 1) distribution. Following Pagan's original spe i ation, there is no onstant term or any exogenous variables in luded. The explanatory power of the instrumental variable W depends on the value of the parameter B. Hen e, we let B take the following values: 0, 0.05, 0.1, 0.5 and 1. When B is lose or equal to zero, W has little or no explanatory power, i.e. W is a bad instrument for the latent variable Z: For ea h value of B we onsider ve null hypotheses: H0 : Æ = Æ0 ; for Æ0 = 0; 1; 5; 10 and 50 ; ea h one being tested against four alternative hypotheses of the form H1 : Æ = Æ1 ; for Æ1 = Æ0 + p I(Æ0) : The alternative H1 is onstru ted by adding an in rement to the value of Æ0 where p = 0, 0.5, 1, 2 and 4, and I(Æ0) = 1 for Æ0 = 0; and I(Æ0) = Æ0 otherwise. Table 2 summarizes the results. In the rst 3 olumns, we report the values of B; Æ0 and the alternative Æ1: When the entries in olumns II and III are equal, we have Æ0 = Æ1; and the orresponding row reports the levels of the tests. The next three olumns (IV, V and VI) show the performan e of the Wald-type IV-based test [as proposed by Pagan (1984)℄, whi h onsists in orre ting the understated standard errors of a two stage pro edure by 22 repla ing them by a 2SLS standard error. We report the orresponding results in olumn IV [asymptoti (As.)℄. In ases where the level of Pagan's test ex eeds 5%, we onsider two orre tion methods. The rst method is based on the riti al value of the test at the 5% level for spe i values of Æ0 and B in ea h row of the table [lo ally sizeorre ted tests; olumn V (C.L.)℄. The riti al value is obtained from an independent simulation with 1000 realizations of the model. Another independent simulation allows us to ompute the riti al value at 5% level in an extreme ase when the instrumental variable is very bad, i.e. by supposing B = 0 also for ea h value of Æ0 [globally sizeorre ted tests; olumn VI (C.G.)℄. This turns out to yield larger riti al values and is thus loser to the theoreti ally orre t riti al value to be used here (on the assumption that B is a tually unknown). In olumn VII, we present the power of the exa t test based on the instrument substitution method. In the following four olumns (VIII to XI) we show the performan e of the exa t test based on splitting the sample, where the numbers of observations used to estimate the stru tural equation are, respe tively, 25, 50, 75 and 90 over 100 observations. Finally, we report the level and power of a naive two-stage test as well as the results of a test obtained by repla ing the latent variable Z in the stru tural equation by the observed value Z. Let us rst dis uss the reliability of the asymptoti pro edures. The level of the IV test proposed by Pagan ex eeds 5% essentially always when the parameter B is less then 0.5, sometimes by very wide margins. The tests based on the two-stage pro edure or repla ing the latent variable by the ve tor of observed values are both extremely unreliable no matter the value of the parameter B. The performan e of Pagan's test improves on e we move to higher values of the parameter B, i.e. when the quality of the instrument in reases. The improvement is observed both in terms of level and power. It is however important to note that Pagan's test has, in general, the same or less power than the exa t tests. The only ex eption is the sample split test reported in olumn VIII, where only 25 observations were retained to estimate the stru tural equation. For B higher then 0.5, the two other asymptoti tests are still performing worse then the other tests. They are indeed extremely unreliable. In the same range of B, the exa t tests behave very well. They show the best power properties ompared to the asymptoti ally based pro edures and in general outperform the other tests. 23 Table 2 Simulation study of test performan e for a model with unobserved regressors Parameter Values Reje tion Frequen ies B Æ0 Æ1 Wald-type IS Split-sample 2S OLS As. C.L. C.G. 25 50 75 90 I II III IV V VI VII VIII IX X XI XII XIII 0.00 0.0 0.0 0.1 5.1 5.1 6.1 5.2 5.4 5.1 0.00 0.0 0.5 0.0 4.7 5.1 4.4 4.1 3.9 4.7 0.00 0.0 1.0 0.0 5.6 4.8 5.5 5.7 5.4 5.6 0.00 0.0 2.0 0.0 4.2 4.5 4.5 3.8 4.5 4.2 0.00 0.0 4.0 0.0 5.2 5.3 5.9 4.3 5.0 5.2 0.00 1.0 1.0 7.3 5.1 5.1 5.0 4.6 4.9 4.8 5.2 15.7 4.7 0.00 1.0 1.5 6.8 5.5 5.5 4.4 4.8 4.4 5.4 6.1 15.7 6.8 0.00 1.0 2.0 7.6 5.9 5.9 5.0 4.3 4.8 4.8 5.1 17.9 6.5 0.00 1.0 3.0 8.6 6.6 6.6 6.3 5.0 4.9 5.0 5.8 19.9 7.0 0.00 1.0 5.0 6.6 4.9 4.9 4.4 4.3 4.6 5.5 4.6 18.1 5.1 0.00 5.0 5.0 54.1 5.5 5.5 5.1 5.5 4.2 5.2 4.9 70.5 69.3 0.00 5.0 7.5 52.8 5.4 5.4 4.9 6.1 4.9 5.1 4.6 69.7 69.0 0.00 5.0 10.0 56.5 5.7 5.7 4.8 4.5 6.1 5.0 4.8 71.7 71.5 0.00 5.0 15.0 50.7 4.6 4.6 4.8 4.5 4.3 4.5 3.8 66.6 67.0 0.00 5.0 25.0 52.7 5.2 5.2 4.6 4.5 4.6 5.6 5.0 67.8 68.8 0.00 10.0 10.0 69.0 4.5 4.5 4.9 5.3 6.0 4.9 5.1 84.5 85.0 0.00 10.0 15.0 68.4 5.7 5.7 5.9 4.7 5.0 5.6 4.5 84.3 83.9 0.00 10.0 20.0 68.6 5.0 5.0 5.7 4.3 4.9 4.7 5.2 84.6 84.3 0.00 10.0 30.0 70.2 4.9 4.9 4.5 5.4 5.2 5.0 5.2 85.4 84.4 0.00 10.0 50.0 68.7 5.3 5.3 4.8 4.2 5.1 5.6 5.0 83.6 83.1 0.00 50.0 50.0 86.5 6.4 6.4 5.4 4.4 5.0 5.1 5.4 96.9 96.5 0.00 50.0 75.0 85.2 6.7 6.7 6.2 3.9 5.0 6.6 6.7 95.1 96.1 0.00 50.0 100.0 87.4 5.2 5.2 4.6 6.5 5.0 4.5 5.5 96.8 96.4 0.00 50.0 150.0 85.8 6.5 6.5 5.8 5.0 5.3 5.9 5.9 97.1 97.1 0.00 50.0 250.0 86.7 6.8 6.8 5.9 4.8 6.0 6.2 5.8 97.1 97.3 0.05 0.0 0.0 0.0 4.8 5.0 3.6 3.6 5.3 4.8 0.05 0.0 0.5 0.2 4.9 5.1 5.5 4.8 5.2 4.9 0.05 0.0 1.0 0.0 7.4 5.4 5.7 6.2 7.6 7.4 0.05 0.0 2.0 0.3 16.6 8.7 11.7 14.7 15.7 16.6 0.05 0.0 4.0 1.0 47.8 16.4 26.9 38.1 44.0 47.8 24 Table 2 ( ontinued) 0.05 1.0 1.0 6.9 5.2 5.6 4.7 4.8 4.4 4.8 5.5 16.9 7.9 0.05 1.0 1.5 6.0 4.6 4.7 5.4 6.0 6.0 5.4 5.2 16.9 7.5 0.05 1.0 2.0 4.7 3.9 3.9 5.3 5.7 4.6 5.1 5.2 18.1 7.6 0.05 1.0 3.0 4.0 2.7 2.7 9.9 6.3 7.4 8.4 10.5 25.3 7.4 0.05 1.0 5.0 2.6 2.1 2.1 27.0 9.0 14.9 23.2 25.4 51.1 5.6 0.05 5.0 5.0 33.8 4.6 1.6 4.6 5.8 5.3 5.2 4.8 71.7 72.7 0.05 5.0 7.5 21.0 2.3 0.2 6.3 4.8 4.6 5.3 6.0 69.7 71.4 0.05 5.0 10.0 12.4 0.4 0.1 8.7 4.8 5.6 7.6 8.5 71.9 69.9 0.05 5.0 15.0 5.1 0.1 0.0 14.8 6.1 8.6 11.7 13.2 81.2 66.9 0.05 5.0 25.0 3.9 0.0 0.0 47.1 15.3 26.2 39.1 43.0 93.6 59.0 0.05 10.0 10.0 34.9 7.6 0.2 6.3 6.6 6.3 6.4 6.5 84.8 84.0 0.05 10.0 15.0 22.9 1.3 0.0 6.4 4.4 5.8 5.8 5.9 85.8 78.9 0.05 10.0 20.0 14.1 0.6 0.0 8.6 5.1 6.1 6.7 7.6 88.9 79.0 0.05 10.0 30.0 5.1 0.0 0.0 14.5 6.7 10.4 13.3 13.9 90.0 74.2 0.05 10.0 50.0 4.4 0.1 0.0 52.5 18.6 30.1 40.8 49.1 97.5 62.2 0.05 50.0 50.0 32.7 5.1 0.0 4.7 4.7 6.0 5.2 4.5 97.5 92.0 0.05 50.0 75.0 21.2 1.7 0.0 6.4 4.5 4.9 5.3 6.2 96.9 89.2 0.05 50.0 100.0 14.3 0.6 0.0 8.5 5.8 7.0 7.2 7.3 97.7 86.5 0.05 50.0 150.0 6.4 0.3 0.0 17.6 7.0 11.1 15.1 15.8 97.0 79.8 0.05 50.0 250.0 3.2 0.0 0.0 51.3 16.0 28.3 38.7 46.1 99.8 65.3 0.10 0.0 0.0 0.0 4.8 4.2 4.9 4.5 5.0 4.8 0.10 0.0 0.5 0.2 8.2 6.8 7.1 6.9 7.4 8.2 0.10 0.0 1.0 0.1 15.8 7.1 8.9 13.9 13.5 15.8 0.10 0.0 2.0 2.4 49.4 16.9 29.3 40.7 46.0 49.4 0.10 0.0 4.0 8.8 97.1 47.7 78.9 93.2 95.9 97.1 0.10 1.0 1.0 7.3 4.4 5.6 4.7 5.3 5.1 4.5 4.7 15.2 14.0 0.10 1.0 1.5 4.4 2.9 3.8 6.6 4.4 5.6 6.3 6.2 19.8 16.2 0.10 1.0 2.0 3.0 1.9 2.3 10.6 6.6 7.3 9.5 10.0 25.8 14.3 0.10 1.0 3.0 0.9 0.7 0.9 28.3 9.3 18.7 23.8 26.6 49.5 10.9 0.10 1.0 5.0 0.6 0.3 0.5 80.1 26.4 49.4 66.1 74.1 92.4 7.4 0.10 5.0 5.0 17.4 4.6 0.6 5.2 5.2 4.7 4.8 5.4 71.5 78.9 0.10 5.0 7.5 5.8 1.1 0.0 7.2 6.0 6.4 7.4 7.5 73.7 74.4 0.10 5.0 10.0 2.3 0.2 0.0 16.5 7.9 11.1 14.0 16.0 81.6 73.0 0.10 5.0 15.0 1.0 0.0 0.0 50.5 15.4 27.2 38.7 45.7 94.8 65.2 0.10 5.0 25.0 0.4 0.0 0.0 97.0 45.5 76.6 89.4 95.0 100.0 46.9 25 Table 2 ( ontinued) 0.10 10.0 10.0 17.1 5.6 0.0 4.7 4.6 4.7 6.0 5.7 84.6 86.0 0.10 10.0 15.0 6.0 1.5 0.0 7.0 6.4 7.0 8.0 6.7 85.0 84.8 0.10 10.0 20.0 2.7 0.1 0.0 14.1 6.5 10.4 11.3 13.2 90.7 79.4 0.10 10.0 30.0 0.8 0.0 0.0 51.9 18.0 28.8 40.9 47.9 97.8 68.9 0.10 10.0 50.0 0.5 0.1 0.0 96.5 49.5 77.6 91.6 94.1 100.0 49.3 0.10 50.0 50.0 19.8 4.8 0.0 5.9 4.5 5.1 5.1 4.8 97.0 89.6 0.10 50.0 75.0 6.5 0.8 0.0 7.7 5.5 5.7 6.6 6.6 97.4 86.1 0.10 50.0 100.0 3.5 0.5 0.0 17.7 9.4 12.3 15.7 17.3 97.7 82.2 0.10 50.0 150.0 0.9 0.0 0.0 45.9 16.4 27.7 39.5 43.5 99.6 73.1 0.10 50.0 250.0 0.8 0.0 0.0 97.2 48.9 78.5 94.0 95.6 100.0 49.7 0.50 0.0 0.0 2.7 4.6 5.4 4.3 4.8 4.4 4.6 0.50 0.0 0.5 60.3 67.7 24.1 41.8 55.0 63.8 67.7 0.50 0.0 1.0 98.8 99.9 68.7 92.8 99.1 99.6 99.9 0.50 0.0 2.0 99.6 100.0 98.4 100.0 100.0 100.0 100.0 0.50 0.0 4.0 99.0 100.0 100.0 100.0 100.0 100.0 100.0 0.50 1.0 1.0 5.3 4.8 4.2 5.0 4.7 5.1 4.9 4.6 17.6 98.4 0.50 1.0 1.5 8.5 5.2 2.6 41.4 15.5 24.4 32.4 39.3 64.4 92.8 0.50 1.0 2.0 68.0 58.1 47.4 93.4 39.7 68.6 84.3 90.6 98.4 62.6 0.50 1.0 3.0 98.7 98.2 97.5 100.0 90.3 99.8 100.0 100.0 100.0 1.7 0.50 1.0 5.0 99.8 99.7 99.6 100.0 100.0 100.0 100.0 100.0 100.0 0.1 0.50 5.0 5.0 7.4 5.6 0.0 5.1 4.2 5.0 4.4 5.3 69.6 100.0 0.50 5.0 7.5 9.7 1.7 0.0 66.6 18.4 39.4 54.5 61.6 97.7 99.9 0.50 5.0 10.0 92.6 69.1 0.0 99.7 63.9 90.5 97.9 99.4 100.0 99.2 0.50 5.0 15.0 99.1 97.9 0.0 100.0 98.8 100.0 100.0 100.0 100.0 5.4 0.50 5.0 25.0 99.6 99.1 0.0 100.0 100.0 100.0 100.0 100.0 100.0 0.1 0.50 10.0 10.0 6.9 5.2 0.0 5.1 5.5 5.2 4.2 5.6 83.5 100.0 0.50 10.0 15.0 8.6 1.0 0.0 67.9 21.7 39.9 55.4 62.0 99.6 99.7 0.50 10.0 20.0 92.1 74.2 0.0 99.7 66.6 93.2 98.7 99.8 100.0 99.1 0.50 10.0 30.0 99.5 99.0 0.0 100.0 99.4 100.0 100.0 100.0 100.0 5.6 0.50 10.0 50.0 99.5 99.1 0.0 100.0 100.0 100.0 100.0 100.0 100.0 0.0 0.50 50.0 50.0 8.3 6.7 0.0 4.6 3.9 4.5 4.4 4.5 96.3 100.0 0.50 50.0 75.0 8.9 3.7 0.0 69.8 21.8 39.1 56.1 64.7 99.9 100.0 0.50 50.0 100.0 94.3 88.8 0.0 99.6 63.2 92.3 98.5 99.5 100.0 99.4 0.50 50.0 150.0 98.8 98.3 0.0 100.0 99.4 100.0 100.0 100.0 100.0 5.2 0.50 50.0 250.0 99.5 99.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 0.3 26 Table 2 ( ontinued) 1.00 0.0 0.0 5.1 5.6 4.9 5.0 5.6 5.8 5.6 1.00 0.0 0.5 99.5 99.5 64.9 91.2 98.5 99.2 99.5 1.00 0.0 1.0 100.0 100.0 99.2 100.0 100.0 100.0 100.0 1.00 0.0 2.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 1.00 0.0 4.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 1.00 1.0 1.0 6.8 7.2 3.8 6.3 5.4 7.0 6.9 6.8 17.9 99.7 1.00 1.0 1.5 87.9 89.2 82.2 93.3 39.5 68.3 84.7 90.1 98.1 33.7 1.00 1.0 2.0 100.0 100.0 100.0 100.0 89.9 99.8 100.0 100.0 100.0 0.7 1.00 1.0 3.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 57.3 1.00 1.0 5.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.1 1.00 5.0 5.0 4.8 4.4 0.0 4.1 5.5 4.4 4.7 4.8 67.2 100.0 1.00 5.0 7.5 98.8 98.3 0.0 99.6 62.5 91.5 98.0 99.4 100.0 67.6 1.00 5.0 10.0 100.0 100.0 0.0 100.0 99.0 100.0 100.0 100.0 100.0 1.3 1.00 5.0 15.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 65.9 1.00 5.0 25.0 100.0 100.0 7.3 100.0 100.0 100.0 100.0 100.0 100.0 98.3 1.00 10.0 10.0 5.1 4.4 0.0 6.0 6.2 5.8 6.9 6.3 85.3 100.0 1.00 10.0 15.0 98.8 98.5 0.0 99.6 63.1 91.1 97.7 99.4 100.0 69.5 1.00 10.0 20.0 100.0 100.0 0.0 100.0 99.0 100.0 100.0 100.0 100.0 0.6 1.00 10.0 30.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 66.5 1.00 10.0 50.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 99.2 1.00 50.0 50.0 5.2 5.0 0.0 5.5 5.5 5.3 5.2 6.9 96.8 100.0 1.00 50.0 75.0 99.0 98.7 0.0 99.9 65.8 91.4 98.3 99.3 100.0 68.1 1.00 50.0 100.0 100.0 100.0 0.0 100.0 98.8 100.0 100.0 100.0 100.0 0.6 1.00 50.0 150.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 67.0 1.00 50.0 250.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0 100.0 100.0 99.0 Notes: I: value of parameter B; VIII: sample split test using 25 observations II: null hypothesis; for the stru tural equation; III: alternative hypothesis; IX: sample split using 50 observations; IV: Pagan's test; X: sample split using 75 observations; V: Pagan's test lo ally sizeorre ted XI: sample split using 90 observations; (B known); XII: two-stage test (2S); VI: Pagan's test globally sizeorre ted XIII: test with latent variable repla ed by (B = 0); observed ve tor (OLS). VII: instrument substitution test (IS); 27 9. Empiri al illustrations In this se tion, we present empiri al results on inferen e in two distin t e onomi models with latent regressors. The rst example is based on Tobin's marginal q model of investment [Tobin (1969)℄, with xed assets used as the instrumental variable for q. The se ond model stems from edu ational e onomi s and relates students' a ademi a hievements to a number of personal hara teristi s and other so ioe onomi variables. Among the personal hara teristi s, we en ounter a variable de ned as \self{esteem" whi h is viewed as an imperfe t measure of a latent variable and is instrumented by measures of the prestige of parents' professional o upation. The rst example is one where we have good instruments, while the opposite holds for the se ond example. Consider rst Tobin's marginal q model of investment [Tobin (1969)℄. Investment of an individual rm is de ned as an in reasing fun tion of the shadow value of apital, equal to the present dis ounted value of expe ted marginal pro ts. In Tobin's original setup, investment behavior of all rms is similar and no di eren e arises from the degree of availability of external nan ing. In fa t, investment behavior varies a ross rms and is determined to a large extent by nan ial onstraints some rms are fa ing in the presen e of asymmetri information. For those rms, external nan ing may either be too ostly or not provided for other reasons. Thus investment depends heavily on the rm's own sour e of nan ing, namely the ash ow. To a ount for di eren es in investment behavior implied by nan ial onstraints, several authors [Abel (1979), Hayashi (1982, 1985), Abel and Blan hard (1986), Abel and Eberly (1993)℄ introdu ed the ash ow as an additional regressor to Tobin's q model. It an be argued that another explanatory variable ontrolling the pro tability of investment is also required. For this reason, one an argue that the rm's in ome has to be in luded in the investment regression as well. The model is thus Ii = 0 + ÆQi + 1CFi + 2Ri + ei (9.1) where Ii denotes the investment expenses of an individual rm i, CFi and Ri its ash ow and in ome respe tively, while Qi is Tobin's q measured by equity plus debt and 28 approximated empiri ally by adding data on urrent debt, long term debt, deferred taxes and redit, minority interest and equity less inventory; Æ and = ( 0; 1; 2)0 are xed oeÆ ients to be estimated. Given the ompound hara ter of Qi; whi h is onstru ted from several indexes, xed assets are used as an explanatory variable forQi in the regression whi h ompletes the model: Qi = 0 + 1Fi + vi : (9.2) For the purpose of building nite-sample on den e intervals following the instrument substitution method, the latter equation may be repla ed (without any hange to the results) by the more general equation ( alled below the \full instrumental regression"): Qi = 0 + 1Fi + 3CFi + 4Ri + vi : (9.3) Our empiri al work is based on \Sto k Guide Database" ontaining data on ompanies listed at the Toronto and Montreal sto k ex hange markets between 1987 and 1991. The re ords onsist of observations on e onomi variables des ribing the rms' size and performan e, like xed apital sto k, in ome, ash ow, sto k market pri e, et . All data on the individual ompanies have previously been extra ted from their annual, interim and other reports. We retained a subsample of 9285 rms whose sto ks were traded on the Toronto and Montreal sto k ex hange markets in 1991. Sin e we are interested in omparing our inferen e methods to the widely used Waldtype tests, we rst onsider the approa h suggested by Pagan (1984). Sin e usual estimators of oeÆ ient varian es obtained from the OLS estimation of equation (9.1) with Qi repla ed by Q̂i are in onsistent [for a proof, see Pagan (1984)℄, Pagan proposed to use standard twostage least squares (2SLS) methods, whi h yield in the present ontext (under appropriate regularity onditions) asymptoti ally valid standard errors and hypothesis tests. For the 2SLS estimation of model (9.1){(9.2), the dependent variable Ii is rst regressed on all the exogenous variables of the system, i.e., the onstant, CFi; Ri and Fi; where Fi is the identifying instrument for Qi; and then the tted values Q̂i are substituted for Qi in the 29 Table 3 Tobin's Q model, N = 9285 A) 2SLS estimators of investment equation (9.1 ) Dependent variable: INVESTMENT (I) Explanatory Estimated Standard t statisti p-value variable oeÆ ient error Constant 0.0409 0.0064 6.341 0.0000 Q 0.0052 0.0013 3.879 0.0001 CF 0.8576 0.0278 30.754 0.0000 R 0.0002 0.0020 0.109 0.9134 B) Instrumental OLS regressions Dependent variable: Q Full instrumental regression Equation (9.2) Regressor Estimated Stand. t p-value Estimated Stand. t p-value oeÆ ient error oeÆ ient error Constant 0.6689 0.0919 7.271 0.0000 1.0853 0.1418 7.650 0.0000 F -2.7523 0.0527 -52.195 0.0000 2.4063 0.0400 60.100 0.0000 CF 21.2102 0.3188 66.517 0.0000 R 1.2273 0.0291 42.111 0.0000 C) Con den e intervals Marginal on den e intervals for Æ Proje tion-based simultaneous on den e intervals (instrument substitution) Method Interval CoeÆ ient Interval 2SLS [0:0026 ; 0:0078℄ 0 [0:0257 ; 0:0564℄ Augmented two-stage [0:0025 ; 0:0079℄ Æ [0:0037 ; 0:0072℄ Two-stage [ 0:0091 ; 0:0029℄ 1 [0:7986 ; 0:9366℄ Instrument substitution [0:0025 ; 0:0078℄ 2 [0:0033 ; 0:0042℄ Sample split 50% [0:0000 ; 0:0073℄ Sample split 75% [0:0017 ; 0:0077℄ Sample split 90% [0:0023 ; 0:0078℄ 30 se ond stage regression. The results are summarized in Tables 3A, while the instrumental OLS regressions appear in 3B. From the latter, we see that the identifying instrument for Q is strongly signi ant and so appears to be a \good" instrument. Table 3C presents 95% (marginal) on den e intervals for Tobin's q parameter based on various methods, as well as proje tion-based simultaneous on den e intervals for the oeÆ ients of equation (9.1). The three rst intervals are obtained from, respe tively, 2SLS, two-stage and augmented two-stage methods by adding or subtra ting 1.96 times the standard error to/from the estimated parameter value.3 Below we report the exa t on den e intervals (instrument substitution and sample split) based on the solution of quadrati equations as des ribed in Se tions 2 and 3. Reall that the pre ision of the on den e intervals depends, in the ase of the sample split method, on the number of observations retained for the estimation of the stru tural equation. We thus show the results for, respe tively, 50%, 75% and 90% of the entire sample (sele ted randomly). The simultaneous on den e intervals for the elements of the ve tor = ( 0; Æ; 1; 2)0 are obtained by rst building a simultaneous on den e set C ( ), with level 1 = 0:95 for a ording to the instrument substitution method des ribed in Se tion 4 and then by both minimizing and maximizing ea h oeÆ ient subje t to the restri tion 2 C ( ) [see Se tion 6℄. The program used to perform these onstrained optimizations is the subroutine NCONF from the IMSL mathemati al library. The orresponding fourdimensional on den e box has level 95% (or possibly more), i.e: we have simultaneous on den e intervals (at level 95%): From these results, we see that all the on den e intervals for Æ; ex ept for the two-stage interval (whi h is not asymptoti ally valid), are quite lose to ea h other. Among the nitesample intervals, the ones based on the instrument substitution and the 90% sample split method appear to be the most pre ise. It is also worthwhile noting that the proje tionbased simultaneous on den e intervals all appear to be quite short. This shows that the latter method works well in the present ontext and an be implemented easily. Let us now onsider another example where, on the ontrary, important dis repanies arise between the intervals based on the asymptoti and the exa t inferen e methods. Montmarquette and Mahseredjian [Montmarquette and Mahseredjian (1989), Montmar31 quette, Houle, Crespo, and Mahseredjian (1989)℄ studied students' a ademi a hievements as a fun tion of personal and so ioe onomi explanatory variables. Students' s hool results in Fren h and mathemati s are measured by the grade, taking values on the interval 0 100: The grade variable is assumed to depend on personal hara teristi s, su h as age, intelle tual ability (IQ) observed in kindergarten and \self{esteem" measured on an adapted hildren self{esteem s ale ranging from 0 to 40. Other explanatory variables in lude parents' in ome, father's and mother's edu ation measured in number of years of s hooling, the number of siblings, student's absenteeism, his own edu ation and experien e as well as the lass size. We examine the signi an e of self{esteem, whi h is viewed as an imperfe tly measured latent variable to explain the rst grader's a hievements in mathemati s. The self esteem of younger hildren was measured by a Fren h adaptation of the M Daniel{Piers s ale. Noting the measurement s ale may not be equally well adjusted to the age of all students and due to the high degree of arbitrariness in the hoi e of this riterion, the latter was instrumented by Blishen indi es re e ting the prestige of father's and mother's professional o upations in order to take a ount of eventual mismeasurement. The data stem from a 1981{1982 survey of rst graders attending Montreal fran ophone publi elementary s hools. The sample onsists of 603 observations on students' a hievements in mathemati s. The model onsidered is: LMATi = 0 + Æ SEi + 1 IQi + 2 Ii + 3 FEi + 4MEi + 5 SNi + 6Ai + 7ABPi + 8 EXi + 9EDi + 10ABSi + 11 CSi + ei (9.4) where (for ea h individual i) LMAT = `n(grade/(100 grade)), SE = `n(self esteem test result/(40 self esteem test result)), IQ is a measure of intelligen e (observed in kindergarten), I is parents' in ome, FE andME are father's and mother's years of s hooling, SN denotes the sibling's number, A is the age of the student, ABP is a measure of tea her's absenteeism, EX indi ates the years of student's work experien e, ED measures his eduation in years, ABS is student's absenteeism and CS denotes the lass size. Finally, the 32 instrumental regression is: SEi = 0 + 1 FPi + 2MPi + vi (9.5) where FP and MP orrespond to the prestige of the father and mother's profession expressed in terms of Blishen indi es. We onsider also the more general instrumental regression whi h in ludes all the explanatory variables on the right-hand side of (9.4) ex ept SE: The 2SLS estimates and proje tion-based simultaneous on den e are reported in Table 4A while the results of the instrumental regressions appear in Table 4B. Standard (bounded) Wald-type on den e intervals are of ourse entailed by the 2SLS estimation. For Æ however, the instrument substitutionmethod yields the on den e interval de ned by the inequality: 31:9536 Æ20 84:7320 Æ0 850:9727 0 : Sin e the roots of this se ond order polynomial are omplex and a < 0; this on den e interval a tually overs the whole real line. Indeed, from the full instrumental regression and using t-tests as well as the relevant F -test (Table 4B), we see that the oeÆ ients of FP and MP are not signi antly di erent from zero, i.e. the latter appear to be poor instruments. So the fa t that we get here an unbounded on den e interval for Æ is expe ted in the light of the remarks at the end of Se tion 2. The proje tion-based on den e intervals (Table 4A) yield the same message for Æ; although it is of interest to note that the intervals for the other oeÆ ients of the model an be quite short despite the fa t that Æ may be diÆ ult to identify. As in the ase of multi ollinearity problems in linear regressions, inferen e about some oeÆ ients of a model remains feasible even if the ertain parameters are not identi able. 10. Con lusions The inferen e methods presented in this paper are appli able to a variety of models, su h as regressions with unobserved explanatory variables or stru tural models whi h an be estimated by instrumental variable methods (e.g., simultaneous equations models). They may be onsidered as extensions of Anderson-Rubin pro edures where the major improvement onsists of providing tests of hypotheses on subsets or elements of the parameter ve tor. 33 Table 4 Mathemati s a hievement model N = 603 2SLS estimators of a hievement equation (9.4) Proje tion-based Dependent variable: LMAT 95% on den e intervals Explanatory Estimated Standard t statisti p-value (instrument substitution) variable oeÆ ient error Constant -4.1557 0.9959 -4.173 0.0000 [-4.8601 , -3.7411℄ SE 0.2316 0.3813 0.607 0.5438 ( 1;+1) IQ 0.0067 0.0015 4.203 0.0000 [0.006600 , 0.006724℄ I 0.0002 0.3175 0.008 0.9939 [-0.09123 , 0.10490℄ FE 0.0015 0.0089 0.172 0.8636 [-0.00914 , 0.01889℄ ME 0.0393 0.0117 3.342 0.0009 [0.02868 , 0.05762℄ SN -0.0008 0.0294 -0.029 0.9767 [-0.1546 , 0.1891℄ A 0.0144 0.0070 2.050 0.0408 [0.01272 , 0.01877℄ ABP -0.0008 0.0005 -1.425 0.1548 [-0.003778 , 0.000865℄ EX -0.0056 0.0039 -1.420 0.1561 [-0.01307 , 0.00333℄ ED -0.0007 0.0206 -0.035 0.9718 [-0.0123 , 0.2196℄ ABS -0.0001 0.0002 -0.520 0.6033 [-0.0001764 , 0.0000786℄ CS -0.0184 0.0093 -1.964 0.0500 [-0.03003 , -0.009790℄ Marginal 95% quadrati on den e interval for Æ ( 1;+1) Instrumental OLS regressions Dependent variable: SE Full instrumental regression Equation (9.5) Regressor Estimated Stand. t p-value Estimated Stand. t p-value oeÆ ient error oeÆ ient error Constant -1.2572 1.0511 -1.1960 0.232 0.8117 0.1188 6.830 0.0000 FP 0.5405 0.3180 1.7000 0.090 0.5120 0.2625 1.951 0.0516 FM 0.3994 0.3327 1.2004 0.230 0.6170 0.2811 2.194 0.0286 IQ 0.003822 0.000611 6.2593 0.000 I 0.02860 0.03161 0.9049 0.366 F -statisti for signi an e of FP and FE -0.01352 0.01136 -1.1899 0.235 FM in full instrumental regression: ME -0.004028 0.01517 -0.2655 0.791 F (2; 589) = 2:654 (p-value = 0.078) SN -0.01439 0.03325 -0.4326 0.665 A 0.003216 0.008161 0.3941 0.694 ABP 0.000698 0.000577 1.2108 0.226 EX -0.002644 0.004466 -0.5920 0.554 ED -0.02936 0.02080 -1.4117 0.159 ABS 0.000426 0.000194 2.1926 0.029 CS 0.01148 0.009595 1.1966 0.232 34 This is a omplished via a proje tion te hnique allowing for inferen e on general possibly nonlinear transformations of the parameter ve tor of interest. We emphasized that our test statisti s, being pivotal or at least boundedly pivotal fun tions, yield valid on den e sets whi h are unbounded with a non-zero probability. The unboundedness of on den e sets is of parti ular importan e when the instruments are poor and the parameter of interest is not identi able or lose to being unidenti ed. A ordingly, a valid on den e set should over the entire set of real numbers sin e all values are observationally equivalent [see Dufour (1997) and Gleser and Hwang (1987)℄. Our empiri al results indi ate that inferen e methods based on Wald-type statisti s are unreliable in the presen e of poor instruments sin e su h methods typi ally yield bounded on den e sets with probability one. The results in this paper thus unders ore another short oming of Wald-type pro edures whi h is quite distin t from other problemati properties, su h as non-invarian e to reparameterizations [see Dagenais and Dufour (1991)℄. In general, non-identi ability of parameters results either from low quality instruments or, more fundamentally, from a poor model spe i ation. A valid test yielding an unbounded on den e set be omes thus a relevant indi ator of problems involving the e onometri setup. The power properties of exa t and Wald-type tests were ompared in a simulationbased experiment. The test performan es were examined by simulations on a simple model with varying levels of instrument quality and the extent to whi h the null hypotheses di er from the true parameter value. We found that the tests proposed in this paper were preferable to more usual IV-based Wald-type methods from the points of view of level ontrol and power. This seems to o ur despite the fa t that AR-type pro edures involve \proje tions onto a high-dimensional subspa e whi h ould result in redu ed power and thus wide onden e regions" [Staiger and Sto k (1997, p. 570)℄. However, it is important to remember that sizeorre ting Wald-type pro edures requires one to use huge riti al values that an easily destroy power. Wald-type pro edures an be made useful only at the ost introdu ing important and omplex restri tions on the parameter spa e that one is not generally prepare to impose; for further dis ussion of these diÆ ulties, see Dufour (1997, Se tion 6). It is important to note that although the simulations were performed under the normality assumption, our tests yield valid inferen es in more general ases involving non-Gaussian 35 errors and weakly exogenous instruments. This result has a theoreti al justi ation and is also on rmed by our empiri al examples. Sin e the inferen e methods we propose are as well omputationally easy to perform, they an be onsidered as a reliable and a powerful alternative to more usual Wald-type pro edures.