where b = (X'X)Y'X'y is the least squares estimator of fl. It is easily shown that z is N(6, o_2 V) where V = R (X'X) 'R'. The usual unbiased estimator of o.2 is s2= (y -Xb)'(y -Xb)/(Tk). In situations in which we wish only to decide whether H is true or not we can use a direct test of H such as an F test. It is perhaps more common that when H is rejected we want to know which components of 6 are different from zero and of the non-zero components which are positive and which negative. In this situation we have a multiple decision problem and a natural solution is to use an induced test. As an example suppose in the case q = 2 that we wish to test the hypothesis H: 01 =02 = 0. Since H is true if and only if the separate hypotheses H1: 01 = 0 and H2: 02= 0 are both true, this suggests a sequence of separate tests which will induce a test of H. Testing the two hypotheses H1 and H2 where we are interested in whether 01 or 62 or both are different from zero induces a multiple decision problem in which the four possible decisions are

Silvey [10]. For a model with nonstochastic regressors we show that a systematic inequality relation exists among the test statistics; namely, the value of the Wald statistic is greater than or equal to that of the LR statistic which, in turn, is greater than or equal to that of the LM statistic. When the null hypothesis is true, we find that the Wald, LR, and LM test statistics have identical limiting chi-square distributions. Since for a large sample test the three procedures employ the same critical region, the inequality relation among the test statistics implies that there exists a significance level such that the tests will produce conflicting inferences. These results are parallel to those obtained by Berndt and Savin [2] in the context of a multivariate regression model with independent disturbance vectors. We also consider the Wald and LR tests for a model with a lagged dependent variable. In this case the Wald statistic is not the same as in the nonstochastic regressor case with the result that the inequality between the Wald and LR test statistics no longer holds. We conclude the paper with an empirical example which illustrates the relation among the test statistics.

[In this paper we generalize the family of single equation k-class estimators to systems of equations. The systems k-class estimator with k = 1 is the 3SLS estimator. After developing the asymptotic properties we introduce a further member of the systems k-class, the systems LVR estimator. A systems version of Basmann's identifiability test statistic is also considered.]

IN NONLINEAR MODELS the power function is often approximated by asymptotic methods. The most common approach is to consider the asymptotic local power function. The local power function is monotonic and it has essentially the same shape as the power function in the classical normal linear regression model. However, the accuracy of the approximation can be poor at nonlocal alternatives. This note examines the exact powers of the Wald test in the case of a one parameter nonlinear regression model with normal errors. The model is based on the exponential response function f( x, O) = exp( Ox). The results show that the exact power function of the Wald statistic can be nonmonotonic. For selected designs the exact powers of the Wald test first increase and then eventually decline as the distance between the hypothesized and the true values of the parameter increases. The exponential structure appears in many nonlinear models; see Gallant (1975, 1987) and Bates and Watts (1988). This suggests that nonmonotonicity of the Wald test is a feature of a wide class of nonlinear models. Indeed, Nelson and Savin (1988) show that it arises in standard logit, probit, and Tobit models as well. The focus here on the nonlinear regression model is for expository convenience. While the existence of nonmonotonic power is not new, the surprising results are that this phenomenon occurs in very simple nonlinear models and that it can be quite severe. In such cases the asymptotic local power approximation provides a very poor guide to the performance of alternative tests.

The exact distribution of the regression t statistic for testing the value of the AR parameter in a Gaussian first ord er autoregressive model is investigated by Monte Carlo methods. The S tudent's t distribution is not a satisfactory approximation for sampl es typical in economic applications. The main problem is the location of the distribution of the t statistic rather than the shape. Once t he t statistic is adjusted so that it has the same mean and standard deviation as Student's t, the distribution of the adjusted t statisti c is accurately approximated by Student's t. Techniques are presented for mak-ing these adjustments in practice.

[This paper considers procedures for testing for autocorrelation when there are missing observations on both the dependent and explanatory variables. These procedures include Durbin-Watson type tests given the vector of residuals, tests based on a set of uncorrelated residuals, and large sample likelihood ratio and Wald tests.]

Recent studies by Durbin and Watson [5], L'Esperance and Taylor [10], Koerts and Abrahamse [8], Tillman [15], Vinod [16], Savin and White [14] and others have shown increasing interest in the test of autocorrelation based on the d statistic proposed by Durbin and Watson [3 and 4]. The focus of these papers has been the computation of the exact distribution of d and the power of the test based on d. The exact distribution of d has been developed by Imhof [7] and Pan Jie-Jian [12]. However, few of the generally available computer programs for regression analysis incorporate these methods,2 possibly because of computational costs, particularly for large samples. With the Durbin and Watson [4] tables the bounds test is restricted to time series regressions with 15 to 100 observations and a maximum of 5 regressors in addition to unity. Often regression studies do not meet these restrictions since samples with less than 15 observations commonly occur with annual time series and regressions with more than 5 regressors are often found in the context of simultaneous equations and of distributed lags.3 In this paper we present extended tables for the bounds test. Our tables can be used for samples with 6 to 200 observations and for as many as 20 regressors.

[This paper investigates the exact sampling distribution of the least squares estimator of β in the model y"t = @m + @by"t"-"1 + @u"t where the @u"t are independently N(0, @s extasciicircum2). The distribution is calculated for the case where y"0 is a known constant and where y"0 is a random variable. Given y"0 is a constant we prove a small @s asymptotic result and compute the exact powers of nonsimilar tests of the random-walk hypothesis β = 1 and of the stability hypothesis β = 0.9. The exact powers of a test of the stability hypothesis are calculated for the case where y"0 is random. The accuracy of the standard normal approximation is examined for both start-up regimes.]