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{\bf Abstract}
Rao:1948 introduced the score test statistic as an alternative to the likelihood ratio and Wald test statistics. In spite of the optimality properties of the score statistic shown in RaoPoti:1946, the Rao score (RS) test remained unnoticed for almost 20 years. Today, the RS test is part of the “Holy Trinity" of hypothesis testing and has found its place in Statistics and Econometrics textbooks and related software. Reviewing the history of the RS test we note that remarkable test statistics proposed in the literature earlier or around the time of Rao:1948 mostly from intuition, such as Pearson:1900 goodness-fit-test, Moran:1948 I test for spatial dependence and DW1950 test for serial correlation, can be given RS test statistic interpretation. At the same time, recent developments in the robust hypothesis testing under certain forms of misspecification, make the RS test an active area of research in Statistics and Econometrics. From our brief account of the history of the RS test we conclude that its impact in science goes far beyond its calendar starting point with promising future research activities for many years to come.
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Key words: Applications to Econometrics and Statistics; Hypothesis testing; Rao's score; Robust tests; Sequential testing. \\
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C. R. Rao’s work was always inspired by some practical problems. In 1946, he was deputed from the Indian Statistical Institute (ISI), Calcutta, to work on an anthropometric project in the Museum of Anthropology and Ethnology at the Cambridge University, U.K. While at Cambridge, Rao took the opportunity to contact R. A. Fisher, then the Belfour Professor of Genetics, and registered for a Ph.D. degree in Statistics under Fisher’s guidance. As recollected in Rao:2001, Fisher agreed under the condition that Rao spends time in the Genetics Laboratory where Fisher was breeding mice to map their chromosomes. Rao started by mating mice of different genotypes to collect the necessary data and additionally, he was trying to develop appropriate statistical methodology to analyze the experimental data. The problem was estimation of linkage parameters (recombination probabilities in the various segments of the chromosomes) using data sets from different experiments, designed in such a way that each data set had information on the same parameters. It was thus necessary to test whether the parameters in different experiments are the same or not.
Rao wrote and published two papers based on this work. The first paper, Rao:1948, deals with the general problem of testing simple and composite hypotheses concerning a vector parameter. The test was based on the scores, derivatives of the log-likelihood function with respect to the individual parameters. The paper was published in the Proceedings of the Cambridge Philosophical Society, where he termed the test principle as a score test. In this paper, we will refer to it as the Rao score (RS) test. The other paper, Rao:1950, contains the detail steps for \textit{analyzing the data} involving the segregation of several factors in mating of different genotypes. And it used the RS test for the meta-analysis of testing the equality of parameters coming from different experimental data sets. That paper was published in Fisher’s new journal \textit{Heredity}. For more, see Rao:2001.
The rest of the paper is organized as follows. In Section 2, we start with the first principle of testing, namely the Neyman-Pearson Lemma and derive the simplest version of RS test and then discuss it in its full generality. There, we also provide RS test interpretation to some of the classic tests in Econometrics and Statistics, such as the quintessential Pearson:1900 goodness-fit-test, which was suggested mostly by pure intuition, but its theoretical foundation can be buttressed by RS test principle. In Sections 3 and 4, we list a (somewhat incomplete) catalogue of RS tests in Econometrics and Statistics. In Section 5, we outline some of the possible ways an assumed probability model can be misspecified, and discuss how the various RS tests can be robustified to make them valid under misspecification. We close the paper in Section 6 (Epilogue) with some concluding remarks. At the outset let us mention that while compiling the 75 years (from 1948 to 2023) history of the RS test, we have included here some of our own past historical accounts and cited accordingly. Our aim is to have a comprehensive review as far as possible at one place, like a one-stop-shopping for the RS test history.
We start by introducing some notation and concepts. Suppose we have $n$ independent observations $y_1, y_2,..., y_n$ on a random variable $Y$ with density function $f(y; \theta)$, where $\theta$ is a $p\times 1$ parameter vector with $\theta \in \Theta \subseteq \mathbb{R}^p$. It is assumed that $f(y; \theta)$ satisfies the regularity conditions stated in Rao:1973 and Serfling:1980. The likelihood function is given by
where y $= (y_1, y_2, \ldots , y_n)$ denotes the sample. Suppose we want to test $H_0: \theta = \theta_0$ against $H_1: \theta \neq \theta_0$ based on the sample y.
The foundation of the theory of hypothesis testing was laid by NeymanPearson:1933 fundamental lemma. This lemma provides a way to find the most powerful (MP) and uniformly most powerful (UMP) tests. According to the Neyman-Pearson (N-P) Lemma, the MP critical region for testing $H_0 : \theta = \theta_0$ versus $H_1 : \theta = \theta_1$ having size $\alpha$, is given by
where $\kappa$ is such that $\Pr[{\omega(\textbf{y}) | H_0}] = \alpha$.
If an MP test maximizes powers uniformly in $\theta_1 \in \Theta_1 \subseteq \Theta$, the test is called UMP test. Unfortunately, an UMP test rarely exists, and when it does not, there is no single critical region best for all alternatives. We, therefore, try to find a critical region that is good for alternatives close to the null hypothesis, called local alternatives, hoping that the region will also be good for alternatives away from the null. Lehmann:1999 advocated for such critical region when the sample size $n$ is large, stating, “if the true value is at some distance from $\theta_0$, a large sample will typically reveal this so strikingly that a formal test may be deemed unnecessary."
For a critical region $\omega(\textbf{y})$, let us define the power function as
Assuming a scalar $\theta$ and that $\gamma$($\theta$) admits Taylor series expansion, we have
where $\theta^*$ is a value in between $\theta$ and $\theta_0$. If we consider local alternatives of the form $\theta = \theta_0 + n^{-\frac{1}{2}}\delta, 0 < \delta < \infty$, the third term will be of the order O($n^{-1}$). Therefore, from (4), to obtain the highest power, we need to maximize
for $\theta > \theta_0$, assuming the regularity conditions that allow differentiation under the sign of intergration.
Using the generalized N-P Lemma given in NeymanPearson:1936, it is easy to see that the locally most powerful (LMP) critical region for $H_0 : \theta = \theta_0$ versus $H_1 : \theta > \theta_0$, is given by
or
where $l(\theta)$ denotes the log-likelihood function and $\kappa$ is a constant such that the size of the test is $\alpha$. The quantity $S(\theta) = {\partial l(\theta)}/{\partial \theta}$ is called the Fisher-Rao score function. The above result in (6) was first discussed in RaoPoti:1946, who stated that a LMP critical region for $H_0 : \theta = \theta_0$ is given by
where $\kappa_2$ is so determined that the size of the test is equal to a preassigned value $\alpha$ with $\kappa_1$ as $+1$ or $-1$, respectively, for alternatives $\theta > \theta_0$ and $\theta < \theta_0$.
Let us define the Fisher information as
The result that under $H_0$, $S(\theta_0)$ is asymptotically distributed as normal with mean zero and variance $\mathcal{I}(\theta_0)$, led RaoPoti:1946 to suggest a test based on $S(\theta_0)/\sqrt{\mathcal{I}(\theta_0)}$ as standard normal [or $S^2(\theta_0)/\mathcal{I}(\theta_0)$ as $\chi^2_1$], for large $n$.
RaoPoti:1946 can be viewed as a precursor to Rao:1948. Generalization of the LMP test in (7) to the multiparameter case ($p \ge 2$) is not trivial. There will be scores for each individual paramter, and the problem is to combine them in an “optimal" way. Let $H_0: \theta = \theta_0$, where now $\theta = (\theta_1, \theta_2,...,\theta_p)^\prime$ and $\theta_0 = (\theta_{10}, \theta_{20},...,\theta_{p0})^\prime$. Consider a $\textit{scalar}$ linear combination
where $\delta = (\delta_1, \delta_2,..., \delta_p)^\prime$ is a fixed vector and test the hypothesis $H_{0 \delta}$ : $\delta^\prime \theta = \delta^\prime \theta_0$ against $H_{1 \delta}$ : $\delta^\prime \theta \neq \delta^\prime \theta_0$, $\delta \in \mathbb{R}^p$.
We rewrite the Fisher information in (8) as
Asymptotically, under $H_0$, $\delta^\prime S(\theta_0)$ is distributed as normal, with mean zero and variance $\delta^\prime \mathcal{I}(\theta_0)\delta$. Thus if $\delta$'s were known, a test could be based on
which under $H_0$ will be distributed as $\chi^2_1$ as in RaoPoti:1946. Note that our $H_0 : \theta = \theta_0$ for $p \ge 2$ can be expressed as $H_0 \equiv \underset{\delta \in \mathbb{R}^p}{\bigcap} H_{0\delta}$, i.e., the multiparameter testing problem can be decomposed into a series of single-parameter problems. To obtain a linear function like (9), Rao:1948 maximized (11) with respect to $\delta$. Using the Cauchy-Schwarz inequality
where $u$ and $v$ are column vectors and $A$ is a non-singular matrix, we have
In (13), the supremum reaches at $\delta = \mathcal{I}(\theta_0)^{-1} S(\theta_0)$ and this provides an optimal linear combination of scores.
Roy1953 used Rao's maximization technique (13) to develop his union-intersection (UI) method of testing. Let $H_0 \equiv \underset{j \in J}{\bigcap} H_{0j}$, where $J$ is an index set. Roy's UI method gives the rejection region for $H_0$ as the union of rejection regions for all $H_{0j}$, $j \in J$. Consider testing $H_{0\delta}:$ $\delta^\prime \theta = \delta^\prime \theta_0$ against $H_{1\delta}:$ $\delta^\prime \theta \neq \delta^\prime \theta_0$, $\delta \in \mathbb{R}^{p}$. Let $H_0 = \underset{\delta \in \mathbb{R}^p}{\bigcap} H_{0\delta}$ and $H_1 \equiv \underset{\delta \in \mathbb{R}^p}{\bigcap} H_{1\delta}$. If $T_\delta$ is the likelihood ratio (LR) statistic for testing $H_{0\delta}$ against $H_{1\delta}$, then
is Roy's LR statistic for testing $H_0$ against $H_1$. This is the same principle that was used by Rao:1948 to convert a “multivariate" problem into a series of “univariate" ones, as we have seen in equation (13).
When the null hypothesis is composite, like $H_0:h(\theta) = c$, where $h(\theta)$ is an $r\times 1$ vector function of $\theta$ with $r \leq p$, the general form of the RS test statistic is
where $\widetilde{\theta}$ is the restricted maximum likelihood estimator (MLE) of $\theta$, i.e., $h(\tilde{\theta}) = c$. Asymptotically, under $H_0$, the RS test statistic is distributed as {$\chi^{2}_{r}$}. Therefore, we observe two optimality principles behind the RS test; first, in terms of LMP test as given in (6), and second, in deriving the “optimal" direction for the multiparameter case, as in (13).
Rao:1948 suggested the score test as an alternative to the Wald:1943 statistic, which for testing $H_0: h(\theta) = c$ is given by
where $\hat{\theta}$ is the unrestricted MLE of $\theta$, and $H(\theta) = {\partial h(\theta)}/{\partial \theta}$ is a $r\times p$ matrix with full column rank $r$. Rao:1948 stated that his test (15), “besides being simpler than Wald's has some theoretical advantages." For more on this see Bera:2000 and Bera:2001.
NeymanPearson:1928 suggested their LR test as
Their suggestion did not come from any search procedure satisfying an optimality criterion. It was purely based on intuitive grounds; as Neyman:1980 stated, “The intuitive background of the likelihood ratio test was simply as follows: if among the contemplated admissible hypotheses there are some that ascribe to the facts observed probabilities much larger than that ascribed by the hypothesis tested, then it appears `reasonable' to reject the null hypothesis."
The three statistics RS, W, and LR, given respectively in (15), (16), and (17) are referred to as the “Holy Trinity." These tests can be viewed as three different distance measures between $H_0$ and $H_1$. When $H_0$ is true, we should expect the restricted and unrestricted MLEs of $\theta$, namely $\widetilde{\theta}$ and $\hat{\theta}$ to be close, and likewise the log-likelihood functions $l(\widetilde{\theta})$ and $l(\hat{\theta})$, respectively. The LR statistic in (17) measures the distance through the log-likelihood function and is based on the difference $l(\hat{\theta}) - l(\widetilde{\theta})$. To see the intuition behind the RS test, note that $S(\hat{\theta}) = 0$ by construction, and thus we should expect $S(\widetilde{\theta})$ to be cloes to zero if $H_0$ is true. Therefore, the basis of the RS test is $S(\widetilde{\theta}) - S(\hat{\theta}) = S(\widetilde{\theta})$, distance between $\widetilde{\theta}$ and $\hat{\theta}$ measured through the function $S(\theta)$. Finally to test $H_0: h(\theta) = c$, W considers the distance directly in terms of $h(\theta)$, and is based on $\left[h(\hat{\theta}) - c\right] - \left[h(\widetilde{\theta}) - c\right] = h(\hat{\theta}) - c$, where $h(\widetilde{\theta}) = c$ by construction, as we see in expression (16). It is interesting to note the similarity between the Wald and the RS tests based on $h(\hat{\theta})$ and $S(\widetilde{\theta})$, respectively. Therefore the RS test statistic is closer to W than LR. Therefore, it makes sense that Rao:1948 mentioned his test as an alternative to W.
The interrelationships among these three tests can be brought home to the students of Statistics through the following amusing story [see Bera:chap2001]: Once around 1946 Ronald Fisher invited Jerzy Neyman, Abraham Wald, and C.R. Rao to his Cambridge University lodge for afternoon tea. During their conversation, Fisher mentioned the problem of deciding whether his dog, who had been going to an “obedience school” for some time, was disciplined enough. Neyman quickly came up with an idea: leave the dog free for some time and then put him on leash. If there is not much difference in his behavior, the dog can be thought of as having completed the course successfully. Wald, who lost his family in the concentration camps, was adverse to any kind of restrictions and simply suggested leaving the dog free and seeing whether it behaved properly. Rao, who had observed the nuisances of stray dogs in Calcutta streets, did not like the idea of letting the dog roam freely, and suggested keeping the dog on a leash at all times and observing how hard it pulls on the leash. If it pulled too much, it needed more training. That night when Rao was back in his Cambridge dormitory after tending Fisher’s mice at the genetics laboratory, he suddenly realized the connection of Neyman and Wald’s recommendations to the Neyman–Pearson LR and Wald tests, respectively. He got an idea and the rest, as they say, is history.
At this stage, it will be instructive to provide a geometric illustration highlighting the fundamental connections and contrasts among the three tests [see Bera:1983Phd]. For simplicity, let us consider the case of scalar $\theta$, i.e., $p = 1$, and that the null hypothesis is $H_0: \theta = \theta_0$. In Figure 1, we plot the score function $S(\theta) = {d l(\theta)}/{d \theta}$ against $\theta$, the solid curved line. The unrestricted MLE $\hat{\theta}$ is obtained by setting $S(\hat{\theta}) = 0$, i.e., at the point D. From the Figure 1, it is easily seen that
Therefore,
For our particular case, $h(\theta) = \theta - \theta_0, H(\theta) = 1$ and $c = 0$. Thus, W in (16) can be expressed as
$\mathcal{I}(\hat{\theta})$ can be obtained from $-{d^2l(\theta)}/{d\theta^2} = -{dS(\theta)}/{d\theta}$, evaluated at $\theta = \hat{\theta}$, i.e., from $\tan\phi_{\hat{\theta}} = {CG}/{CD}$. Therefore,
On the other hand, the RS test will be based on $S(\theta)$ at $\theta_0$, i.e., on the distance CF.
The variance of $S(\theta_0)$ can be estimated by $-{dS(\theta_0)}/{d\theta} = \tan\phi_{\theta_0} = {CF}/{CE}$. Hence,
From above, we can note the following features of the three tests. First, since the tests are based on three different areas in general, they will yield conflicting inference if the same critical value is used [see Savin:1977]. Second, the RS tests depends only on $S(\theta)$ and the slope of $S(\theta)$ at $\theta_0$. We can draw many curved lines through F with the same slope at F, and the dotted line $A^\prime FD^\prime$ is an example. This implies that there may be other $S(\theta)$ functions, i.e., other likelihood function representing different alternative hypothesis, with the same slope at $\theta_0$, giving rise to the same RS test statistic. In the literature, this property is known as invariance property of the RS test principle [see Godfrey:1988]. Finally, for both the RS and W tests, the variances can be calculated in a number of ways which are asymptotically equivalent. This can lead to different versions of the test statistics. It is not clear which versions will give better results in finite samples.
Example 1: Let us start with a simple example where $y_i \sim IID N(\theta,1), \ i = 1, 2,\ldots, n$, and we test $H_0: \theta = \theta_0=0$ against $\theta > 0$. Here the log-likelihood and score functions are respectively
where $\bar{y}=\sum_i y_i/n$. Note that here $S(\theta)$ is linear in $\theta$, and thus from Figure 1, all the three tests LR, W, and RS will be identical. Given that $S(\theta_0) = n\bar{y}$ with $Var[S(\theta_0)] = n$, we will reject $H_0$, if $\sqrt{n}\Bar{y} > \textbf{z}_\alpha$, where $\textbf{z}_\alpha$ is the upper $\alpha$ percent cut-off point of standard normal distribution. For fixed $n$, the power of this test goes to 1 as $\theta \rightarrow \infty$. Hence the score test $\sqrt{n}\Bar{y} > \textbf{z}_\alpha$ is not only LMP, but also UMP for all $\theta > 0$.
Example 2: [Ferguson:1967] Consider testing for the median of a Cauchy distribution with density
Since here $\mathcal{I}(\theta)=n/2$, the RS test will reject $H_0: \theta = \theta_0$ against $H_1: \theta > \theta_0$, if
As $\theta \rightarrow \infty$ with $n$ remaining fixed, $\min(y_i - \theta_0) \overset{p}{\rightarrow} \infty$, and ${S(\theta_0)}/{\sqrt{\mathcal{I}(\theta_0)}} \overset{p}{\rightarrow} 0$. Thus the power of the test tends to zero as $\theta \rightarrow \infty$. Therefore what works for local alternatives may not work for not-so-local alternatives. This is in contrast to Example 1 where the LMP test is also the UMP.
In the example below we illustrate one of the most famous tests in the Statistics literature that was suggested long before 1948 and the theoretical foundation of which can be buttressed by the RS test principle.
Example 3: [Pearson:1900 Goodness-of-fit test]. Consider a multinomial distribution with $p$ classes and let the probability of an observation belonging to the $j$-th class be $\theta_j (\ge 0)$, $j = 1, 2, \ldots, p$, so that $\sum^p_{i = 1}\theta_j = 1$. Denote the observed frequency of the $j$-th class by $n_j$ with $\sum^p_{j = 1} n_j = n$. We are interested in testing $\theta_j = \theta_{j0}$, $j = 1,2, \ldots, p$, where $\theta_{j0}$ are known constants. Pearson:1900 suggested the statistic
where $O$ and $E$ denote respectively, the observed and expected frequencies. Given the profound importance of $P$ in almost all branches of science, we demonstrate the theoretical underpinnings of $P$ based on the RS test principle. The log-likelihood function, score and information matrix are respectively, given by [see Bera:2001]
and
where 1 $= (1, 1, \ldots,1)^\prime$ is a $(p-1) \times 1$ vector of ones. We end up with effectively $(p-1)$ parameters since $\sum^p_{j=1}\theta_j = \sum^p_{j=1}\theta_{j0} = 1$. Using the above expressions, it is easy to see that
where $\theta_0 = (\theta_{10}, \theta_{20},..., \theta_{p0})^\prime$ [see Rao:1973 and CoxHinkley:1974]. The coincidence that $P$ is same as the RS test, is an amazing result. Pearson:1900 suggested his test mostly based on intuitive grounds almost 50 years before Rao:1948.
RS test was well ahead of its time. It went unnoticed for very many years. It is fair to say that econometricians can claim major credit in recognizing its importance and applying the RS test in several useful contexts and coming up with closed form, neat test statistics. Rao himself acknowledged this fact by writing [see Rao2005] “I am gratified to see the large number of papers contributed by econometricians on the application of the score statistic to problems in econometrics and the extensions and improvements they have made." More recently, statisticians are catching up with innovative applications. To obtain a quantitative perception of the influence of Rao:1948, we plot the yearly citations for the last 75 years in Figure 2. The corresponding cumulative citations are depicted in Figure 3. First thing to note is that the total number of citations in the last 75 years is only 980, apparently a very low number for such a seminal paper. Of course, we need to take into consideration of the fact that there are many papers, especially in the Statistics literature, that use the score test without making any reference to Rao:1948. Second, there are only a handful citations during the first thirty years, i.e., until around 1978. That was the time econometricians recognized the usefulness of the Rao test principle, and used it in developing several model specification tests. There was another surge in its use after another 30 years, i.e., around 2008, in both the Statistics and Econometrics literature. Finally, from both Figures 2 and 3, it is clear that overall, the number of citations is still going up at an increasing rate, indicating continuing influence of Rao:1948, as far as the citation numbers go.
Byron:1968 was probably the first to apply the RS test in Econometrics. He used Silvey:1959 Lagrange multiplier (LM) version along with the LR statistic for testing homogeneity and symmetry restrictions in the demand system. In the Econometrics literature, the RS test is known as the LM test - the terminology came from Silvey:1959. Note that the restricted MLE $\widetilde{\theta}$ under the restriction $H_0: h(\theta) = c$ can be obtained from the first order condition of the Lagrangian function
where $\lambda$ is an $r \times 1$ vector of Lagrange multipliers. The first order conditions are
where $H(\theta) = {dh(\theta)}/{d\theta}$. Therefore, from (32) we have $S(\widetilde{\theta}) = H(\widetilde{\theta})\widetilde{\lambda}$. Given that $H(\theta)$ has full rank, $S(\widetilde{\theta})$ = 0 is equivalent to $\widetilde{\lambda} = 0$. These multipliers can be interpreted as the implicit cost (shadow prices) of imposing the restrictions $h(\theta) = c$. It can be shown that
i.e., the multipliers give the rate of change of the maximum attainable value of the log-likelihood function with respect to the change in constraints. If $H_0: h(\theta) = c$ is true and $l(\widetilde{\theta})$ gives the optimal value, $\widetilde{\lambda}$ should be close to zero. Given this “economic" interpretation in terms of Lagrange multipliers, it is not surprising that econometricians prefer the term LM rather than RS. In terms of Lagrange multipliers, (15) can be expressed as
After Byron:1968, it took another decade for econometricians to realize the potential of the RS test. The earlier notable contributions include Savin:1976, Savin:1977, Breusch:1978, Breusch:1979 and Godfrey78a, Godfrey78b, Godfrey78c. Possibly Breusch:1980 had been the most influential. They collected relevant research reported in the Statistics literature, presented the RS test in a general framework in the context of evaluating various econometric models, and discussed many applications. In a full length research monograph, Godfrey:1988 provided a comprehensive account of most of the available RS tests in Econometrics. Bera:1991 and Bera:2001 demonstrated that many of the commonly used specification tests could be given a score-test interpretation. For the last two-score years the RS tests had been the most common items in econometricians' kit for testing tools. It is not hard to understand the popularity of the score test principle in economics. In most cases, the algebraic forms of W and LR tests can hardly be simplified beyond their original formulae (16) and (17). On the other hand, in the majority of the cases the RS test statistics can explicitly be reduced to neat and elegant explicit formulae enabling its easy incorporation into computer software.
We will not make any attempt to provide a comprehensive list of applications of the RS test in Econometrics, for there are far too many. For instance, consider the workhorse of basic econometric modeling, the linear regression model:
where $y_i$ is the $i$-th observation on the dependent variable, $x_i$ is the $i$-th observation on $k$ exogenous variables and $\epsilon_i \sim IID N(0, \sigma^2), \ i = 1, 2, \ldots, n$. The ordinary least squares (OLS) estimation and the related hypotheses tests are based on the four basic assumptions: correct linear functional form; the assumptions of disturbance normality; homoskedasticity; and serial independence. Just to name some of the uses of the RS test principle, test for normality was derived by Bera:1981 and JB:1987; BreuschPagan79 proposed a test for homoskedasticity; and Godfrey78a, Godfrey78b developed tests for serial independence which are very close to the earlier DW1950 test.
To see the attractiveness of the RS test, let us briefly consider the popular Jarque and Bera (JB) test for normality. Bera:1981 started with the Pearson:1895 family of distributions for the disturbance term $\epsilon_i$ in (36). That means if the pdf of $\epsilon_i$ is $f(\epsilon_i)$, we can write
where $c_1$ and $c_2$ are constants. The null hypothesis of normality can be stated as $H_0: c_1 = c_2 = 0$ in (37). Given the complexity of ML estimation of $\sigma^2$, $c_1,$ and $c_2$ in the Pearson family of distributions, W and LR tests are ruled out from a practical point of view. However, the score functions corresponding to $c_1$ and $c_2$ in (37), evaluated under the normality assumption, are given respectively by
and
where $\sqrt{b_1} = {m_3}/{m^{3/2}_2} $ and $b_2 = {m_4}/{m^2_2}$ with $m_j = \frac{1}{n}\sum^n_{i = 1} \tilde{\epsilon}_i^j$, $\tilde{\epsilon}_i = y_i - x^\prime_i\tilde{\beta}$ as OLS residuals, $j = 2, 3, 4$. For large $n$, under normality
and they are asymptotically normally distributed. Thus, a simple test statistic for normality is given by
which is asymptotically distributed as $\chi^2_2$. It turns out that this test was mentioned by Bowman:1975 but was hardly used in practice due to its lack of theoretical underpinnings. The RS test principle uncovered the theoretical justification of (42), ensuing the asymptotic optimality of the test. As it is obvious, JB is based on the two moments, third and fourth. One could have started with these two moments directly without going through the full derivations. From that point of view this RS test has a moment test interpretation.
It is quite common to express specification tests in Econometrics as moment tests. In a way “any" moment test can be obtained as a RS test under a suitably defined density function. To see this, let us write the $r$ moment restrictions as
where $E_f[\cdot]$ means that (43) is true only when $f(y; \theta)$ is the correct pdf. A test for the hypothesis $H_0: E_f[m(y; \theta)] = 0$ can be based on the estimate of the sample counterpart of $E_f[m(y; \theta)]$, namely,
Now consider an auxiliary density function
where $\phi(\theta, \gamma) = \ln\int \exp[\gamma^\prime m(y; \theta)]f(y;\theta)dy$, with $\gamma$ as $(r\times 1)$ parameter vector.
Note that if $f(y; \theta)$ is the correct pdf, then $\gamma = 0$ in ((ref)). The log-likelihood function under the alternative hypothesis is
Therefore, the score function for testing $\gamma = 0$ in ((ref)) is given by
and it provides the identical moment test as in (44). This interpretation of the moment test as a score test was first noted by White:1994. It is easy to see that there are many choices of auxiliary pdf $f^*(y; \theta, \gamma)$ and the score test will be invariant with respect to these choices, as depicted in Figuire 1. The LR and W tests, however, will be sensitive to the forms of $f^*(y; \theta, \gamma)$. This ends our coverage of the use of the RS test in Econometrics.
Rao:1950seq proposed a sequential test of null hypotheses based on the score statistic and his work on locally most powerful (LMP) tests in the case of one-sided alternative hypotheses. His proposal was a reaction to Wald:1945 sequential probability ratio test (SPRT) which was based on the idea of likelihood ratio test for the fixed sample case.
Wald's SPRT statistic was devised to discriminate between different alternative hypotheses for the value of the unknown parameter $\theta$. On the other hand, Rao:1950seq seeks to test a null hypothesis $H_0\!: \theta=\theta_0$ against a one-sided alternative hypothesis $H_1\!: \theta>\theta_0$ with a test statistic that depends only on the null value.
For the fixed sample case, with sample of size $N$, the LMP test suggested by RaoPoti:1946 is defined by [also see equation (6)]
where $P_N^\prime(\theta_0)$ is the first derivative of $P_N(\theta)$ at $\theta=\theta_0$, with $\mu$ chosen so as to maintain Type-I error at a predetermined level. Motivated by this result, Rao:1950seq proposes a sequential test of the form
with $n\le N$, $A(N)$ a properly determined constant depending on the overall level of significance, and $N$ being the upper limit to the number of observations. According to this sequential testing scheme, the sampling stops with rejection of the null hypothesis, at the smallest value of $n$ for which the inequality ((ref)) holds true. If by the $N$th sampled unit ((ref)) is not realized, the null is not rejected.
Berk:1975 proved that the sequential score tests against a one-sided alternative, where the stopping rule is the first time a certain random walk exceeds a bounded interval, are LMP tests asymptotically.
Sequential testing procedures that perform interim analyses during the evolution of the experiment, with the goal of obtaining the result earlier than the termination time suggested by the fixed sample analysis due to time or monetary cost considerations or ethical reasons, are easier validated with the use of score-based test statistics rather than the analogues of LR statistics. We may refer to chapters 9-11 of Sen1981book for the role of score processes in sequential nonparametrics, where it is mentioned (p.339) “it is comparatively simpler to verify these regularity conditions [i.e., for the score] than those for the likelihood function.”
Lombardi:1951, in a thesis on how to select a panel of judges for taste testing and quality evaluation using scientifically sound methods, appears to be one of the first applications of the sequential testing using Rao:1950seq methodology and its comparison with Wald's approach.
To diagnose the potential ability of candidate judges and to decide on the selection of a taste panel, each candidate judge is required to perform a prespecified number of sample comparisons. The number of sample comparisons that should be performed by each candidate judge before reaching to a decision on who to include in the taste panel is always a concern.
Bradley:1953, in conjunction with Lombardi, adapted Rao's method to binomial distribution, and is an early effort to communicate these statistical procedures for the selection of a taste panel to food technologists.
What makes the Rao procedure relatively more appropriate than that of Wald is that a limit to the testing of any one potential judge may be set. In this application, $N$ denotes the maximum number of tests to be given to any judge. As it is noted by Bradley:1953: “The theory of the procedure needs further investigation since its properties are not well known. However, when it is applied to sequences of triangle tests, apparently satisfactory results are obtained."
In another context, time-series researchers use sequential analysis to determine and test for structural breaks. In a recent application, Bucci:2024 proposes a sequentially computed score statistic to test for the number of regimes in multivariate nonlinear models.
Another context where the RS test statistic has found fruitful applications for inference is the analysis of survival data. The semiparametric proportional hazards model proposed by Cox:1972 is a standard tool of analysis for time-to-an-event data met in medical, engineering and economic applications. The parameter estimation using the partial likelihoood of Cox:1975 initiated an intense research activity for the validation of inference. The majority of the test statistics are special cases of weighted score statistics for different weighting functions and different type of covariates.
The partial likelihood score statistic has a natural martingale characterization. By rewriting the model within the counting process framework, AndersenGill:1982 were able to obtain a general asymptotic theory of the score statistic and the associated estimator. In a research related to sequentially computed score test statistic for repeated significance tests, Tsiatis:1981 established the joint asymptotic normality of efficient scores test for the proportional hazards model calculated over time. In a fundamental breakthrough, Sellke:1983 showed that the score process (over time) of the partial likelihood is approximated by a suitable martingale and thus behaves asymptotically like Brownian motion.
Bilias:2000 offered an application of a repeated significance test in a retrospective analysis of the Pennsylvania `Reemployment Bonus' controlled experiments conducted by the US Department of Labor. Their main purpose was to determine whether the offer of a bonus amount to the unemployment insurance (UI) claimants, provided that they find a job with some required permanence within a given period of time, can act as an incentive for more intensive job-seeking with subsequent reduction of the unemployment spells. The response of primary interest is the length of insured unemployment spell and it is assumed that it follows a proportional hazard regression model. The statistic for measuring the effect of the various bonus packages on the duration of insured unemployment relative to the existing scheme is the partial likelihood score statistic. In carrying out sequential analysis, the score statistic is evaluated repeatedly, at different points in chronological time, each time with the available data. The retrospective sequential analysis concluded that the experiment could be concluded earlier than the fixed sample analysis with gains in time and monetary savings.
As we have narrated in the previous sections the success of the RS test had been phenomenal. However the main problem in these specification tests is that they are developed under the assumption that the underlying probability model is correctly specified. When the assumed model is misspecified, it is well known that the RS test loses its local optimal properties.
While discussing the problem in statistical hypothesis testing, Haavelmo:1944 stated, “Whatever be the principles by which we choose a “best" critical region of size $\alpha$, the essential thing is that a test is always developed with respect to a given fixed set of possible alternatives $\Omega^0$." Haavelmo called $\Omega^0$, the a priori admissible hypotheses and according to him, a test is not robust if we shift our attention to another admissible set $\Omega^\prime$ (that may be obtained by extending $\Omega^0$ to include new/different alternatives), for which the proposed test has poor size and power properties.
Very often it is difficult to interpret the results of a test applied to a misspecified model. For instance, while testing the significance of some of the regression coefficients in the linear regression models, the results are not easily interpretable when a nonlinear model is the appropriate one [see, White:1980IER, BeraByron:1983 and Byron:1983]. In the Statistics and Econometrics literature, most emphasis has been put on the minimization of type-I and type-II error probabilities. There are, however, only a few works that seriously consider the consequences and suggest remedies of misspecifying the a priori admissible hypothesis -- which can be called the type-III error.
Note that the model under our a priori admissible hypothesis could be misspecified in a variety of ways. Here we consider only two kinds: distributional and parametric. In the former case, the assumed probability density function differs from the true data generating process (DGP). Kent:1982 and White:1982 analyzed this case and suggested a modified version of the RS test that involves adjustment of the variance of the score function. In the parametric misspecification case, the dimension of the assumed parameter space does not match with the true one. Bera:1993 developed a modified RS test that is valid under the local parametric misspecification.
Let the true DGP be described by the unknown density $g(y)$ and $f(y;\theta)$ be our assumed distribution. The RS test statistic given in (15) is not valid when $g(y)$ and $f(y; \theta)$ differ. This is because some of the standard results breakdown under distributional misspecification. For instance, consider the information matrix (IM) equality:
where $E_f[\cdot]$ denotes expectation under $f(y;\theta)$. Let us now define
where $\theta_g$ minimizes the Kullback-Leibler information criterion [see White:1982]
One can easily see that $J(\theta_g) \neq K(\theta_g)$, in general.
Example 4: Suppose we take $f(y; \theta) \equiv N(\mu,\sigma^2)$, and let the DGP $g(y)$ satisfy $E_g[y] = \mu$, $E_g[y - \mu]^2 = \sigma^2$, $E_g[y - \mu]^3 = \mu_3$ and $E_g[y - \mu]^4 = \mu_4$. Then it is easy to show that
and
Hence, $J(\theta_g) = K(\theta_g)$ if and only if $\mu_3 = 0$ and $\mu_4 = 3\sigma^4$. We can clearly see the connection of these conditions and the JB test for normality given in (42).
Due to this divergence between $J$ and $K$, and noting that we defined the information matrix $\mathcal{I}(\theta)$ in (10) by taking expectation under $f(y; \theta)$ instead of under the DGP $g(y)$, the standard RS test in (15) is not valid. Let us define an estimator of $\theta$ by maximizing a likelihood function based on the misspecified density $f(y; \theta)$ in place of the unknown DGP $g(y)$. Such an estimator is called quasi-MLE (QMLE). An early reference to QMLE can be found in Koopmans:chap [for more on this see BBY2020]. We will denote QMLE of $\theta$ (under $H_0$) by $\tilde{\theta}$. Kent:1982 and White:1982 suggested the following robust form of the RS test statistic for testing the $H_0:h(\theta)=c$:
where $H(\theta)=\partial h(\theta)/\partial \theta$, $B(\theta) = K(\theta)^{-1}J(\theta)K(\theta)^{-1}$ and the notation $RS^*(D)$ is used to signify robust RS test statistic under distributional misspecification.
Under $H_0: h(\theta) = c$, $RS^*(D)$ is asymptotcally distributed as $\chi^2_r$ even under distributional misspecification, that is, when the assumed density $f(y; \theta)$ does not coincide with the true DGP $g(y)$. This approach of finding the asymptotically correct formula for variance has its origin in Koopmans:chap; for more on this see Bera:2021. Expression (56) can be simplified if the parameter vector $\theta$ $(p\times 1)$ can be partitioned as $\theta=(\gamma^\prime, \psi^\prime)^\prime$ where $\gamma$ and $\psi$ have dimensions $m$ and $r$, respectively, $m+r=p$, and we test $H_0: \psi=\psi_*$ (say). Let us also partition the score function $S(\theta)$ and $J(\theta)$ [similarly $K(\theta)$] as
and
While testing $H_0: \psi=\psi_*$, under this setup $h(\theta)=\psi - \psi_*$ and $H(\theta)=[0_{r\times (p-r)}, I_{(r\times r)}]$, and we can express $RS^*(D)$ in (56) as [see also BBY2020]
where $\tilde{\theta}=(\tilde{\gamma}^\prime, \psi_*^\prime)^\prime$, the restricted MLE under $H_0: \psi=\psi_*$.
Example 5: In (37) the parameters $c_1$ and $c_2$ of the Pearson family of distributions can be treated, respectively, as the “skewness" and “kurtosis" parameters. Suppose we test the symmetry ignoring the (excess) kurtosis. Then we can start with the system (37) with $c_2 = 0$, that is,
After some derivation, it can be shown that the standard RS test for $c_1 = 0$ is given by
which is essentially the first part of JB in (42). If $f(\epsilon_i)$ in (60) is not the true DGP, $RS_{c_1}$ will not be valid; in particular, the asymptotic variance formula used in (61), $Var(\sqrt{n}b_1) = 6$ is incorrect [see also equation (40)]. For instance in the presence of excess kurtosis, there will be proportionately more outliers, resulting in higher variance, and thus “6" will underestimate the true variance of $\sqrt{n}b_1$. After incorporating the variance correction as in $RS^*_\psi(D)$ in (59) the robust RS test statistic can be written as [for further details see Bera:2017]:
where $m_j = \frac{1}{n}\sum^n_{i = 1}\tilde{\epsilon}_i^j$, $j = 2, 4, 6$. From (62) we can write the population counterpart of the $Var(\sqrt{n}b_1)$ as
where $\mu_j$ denotes the $j$-th population moment of $\epsilon$. Therefore, the construction of the robust RS test statistic $RS^*_{c_1}(D)$ indicates that the true variance of $\sqrt{n}b_1$ that is valid under excess kurtosis is given by (63). If we impose normality, $\mu_6 = 15\sigma^6$ and $\mu_4 = 3\sigma^4$, then with $\mu_2 = \sigma^2$, (63) reduces to $Var(\sqrt{n}b_1) = 9 + 15 - 6 \times 3 = 6$, as in (61).
Example 6: Consider the test for homoskedasticity under the regression framework of (36), where we now explicitly specify the heteroskedastic structure as $Var(\epsilon_i) = \sigma^2_i = \sigma^2 + \delta^\prime z_i$, where $\delta$ is a $r \times 1$ vector and $z_i$'s are fixed exogenous variables, $i = 1, 2, \ldots ,n$. Assuming normality of $\epsilon_i$ the RS statistic for testing homoskedasticity hypothesis $H_0: \delta = 0$ is given by [see BreuschPagan79]
where $\nu_i = \widetilde{\epsilon}^2_i - \widetilde{\sigma}^2$, $\nu = (\nu_1, \nu_2, \ldots, \nu_n)^\prime$ and $Z = (z_1, z_2, \ldots, z_n)^\prime$. The factor “$2\widetilde{\sigma}^4$" is the consequence of the normality assumption, and therefore, the test in (64) will not be valid even asymptotically if $\epsilon_i$'s are not distributed as normal. Using (61), the robust form of RS test statistics can be derived as
This is the same modification suggested by Koenker:1981. Note that the modification amounts to replacing $Var(\epsilon^2_i) = \mu_4 - \mu^2_2 = 3\sigma^4 - \sigma^4 = 2\sigma^4$ (derived under normality) by a robust estimate, namely, by, $\frac{1}{n}\sum^n_{i = 1}(\widetilde{\epsilon}_i^2 - \widetilde{\sigma}^2)^2 = ({\nu^\prime \nu})/{n}$. For other applications of $RS^*(D)$ see for instance, Lucas:1998 and Bera:2017. In a similar fashion the Wald statistic in (16) can be robustified as [see Kent:1982, White:1982, and PaceSalvan:1997]:
and asymptotically it has $\chi^2_r$ distribution under the null hypothesis $H_0: h(\theta) = c$. Thus, robust $RS^*$ and $W^*$ are obtained by robustifying the variance expressions, respectively, of $S(\widetilde{\theta})$ and $h(\hat{\theta})$. However, similar robustification of LR statistic in (17) is not possible. Kent:1982 showed that under distributional misspecification LR statistic is asymptotically distributed as a weighted sum of $r$ independent $\chi^2_1$ variables, and thus no obvious “variance" adjustment is possible.
Consider a general statistical model represented by the log-likelihood function $l(\gamma, \psi, \phi)$ where $\gamma, \psi, \text{and }\phi$ are parameter vectors with dimensions $(m \times 1)$, $(r \times 1)$ and $(q \times 1)$, respectively. Thus our $(p\times 1)$ parameter vector is $\theta = (\gamma^\prime, \psi^\prime, \phi^\prime)^\prime$ and $p = m + r + q$. Suppose an investigator sets $\phi = 0$ and tests $H_0: \psi = 0$ using the log-likelihood function $l_1(\gamma, \psi) = l(\gamma, \psi, 0)$. We will denote the RS statistic for testing $H_0$ in $l_1(\gamma, \psi)$ by $RS_\psi$. Let us also denote $\widetilde{\theta} = (\widetilde{\gamma}^\prime, 0, 0)^\prime$, where $\widetilde{\gamma}$ is MLE of $\gamma$ when $\psi = 0$ and $\phi = 0$. The score vector and the information matrix are defined, respectively, as
If $l_1(\gamma, \psi)$ were correctly specified, then the RS test statistic of (15), in the current context can be written as
where $\mathcal{I}_{\psi\cdot\gamma} = \mathcal{I}_\psi - \mathcal{I}_{\psi\gamma}\mathcal{I}_{\gamma}^{-1}\mathcal{I}_{\gamma\psi}$ and it will be asymptotically distributed as central $\chi^2_r$. Under this set-up, asymptotically $RS_\psi$ will have the correct size and will be locally optimal. Let us now consider the case of parametric misspecification. Suppose the true log-likelihood function is $l_2 = (\gamma, \phi) = l(\gamma, 0, \phi)$, so that the alternative $l_1(\gamma, \psi)$ becomes misspecified. Using the sequence of local DGP $\phi = {\delta}/{\sqrt{n}}$, Davidson:1987 and Saikkonen:1989 showed that under $l_2(\gamma, \phi)$ with $\phi = {\delta}/{\sqrt{n}}$, $RS_\psi$ in (69), under $H_0: \psi = 0$ will be distributed as non-central $\chi^2_r$ with noncentrality parameter,
with $\mathcal{I}^\prime_{\phi\psi \cdot \gamma} = \mathcal{I}_{\psi\phi \cdot \gamma} = \mathcal{I}_{\psi\phi} - \mathcal{I}_{\psi\gamma}\mathcal{I}^{-1}_\gamma \mathcal{I}_{\gamma\phi}$. Owing to the presence of this non-centrality parameter, $RS_\psi$ will reject the null hypothesis $H_0: \psi = 0$ more often than allowed by the preassigned size of the test, even when $\psi=0$. Therefore, under parametric misspecification, $RS_\psi$ will have an excessive size. For the expression of $\lambda(\delta)$ in (70), we note that the crucial quantity is $\mathcal{I}_{\psi\phi \cdot \gamma}$, which can be interpreted as the conditional covariance between the scores $S_\psi$ and $S_\phi$ given $S_\gamma$. If $\mathcal{I}_{\psi\phi \cdot \gamma} = 0$, then the local presence of the misspecified parameter $\phi = {\delta}/{\sqrt{n}}$ will have no effect on the performance of $RS_\psi$. Using the expression in (70), Bera:1993 suggested a modification to $RS_\psi$ so that the resulting test is robust to the presence of $\phi$. The modified statistic is given by
Here the notation $RS^*(P)$ is used to signify robust RS test statistic under parametetric misspecification. Under $H_0: \psi = 0$, $RS^*_\psi(P)$ is asymptotically distributed as central $\chi^2_r$, i.e., $RS^*_\psi(P)$ has the same asymptotic distribution as of $RS_\psi$ in (69) based on the correct specification. Thus, $ RS^*_\psi(P)$ provides an asymptotically correct-size test under the locally misspecified alternative $l_2(\gamma, \phi) $. $RS^*_\psi(P)$ essentially adjusts the asymptotic mean and variance of standard (unadjusted) $RS_\psi$. Another way to look at $RS^*_\psi(P)$ is to view the quantity, $\mathcal{I}_{\psi\phi \cdot \gamma}(\widetilde{\theta})\mathcal{I}_{\phi \cdot \gamma}(\widetilde{\theta})^{-1}S_\phi(\widetilde{\theta})$ as the prediction of $S_\psi(\widetilde{\theta})$ by $S_\phi(\widetilde{\theta})$. Here $S_\phi(\widetilde{\theta})$ is the score function of the parameter vector $\phi$ whose effect we want to take into account in constructing the robust version of the test. Therefore, the net score $S^*_\psi(\widetilde{\theta})=S_\psi(\widetilde{\theta}) - \mathcal{I}_{\psi\phi \cdot \gamma}(\widetilde{\theta})\mathcal{I}^{-1}_{\phi \cdot \gamma}(\widetilde{\theta})S_\phi(\widetilde{\theta})$ is the part of $S_\psi(\widetilde{\theta})$ that remains after eliminating the effect of $S_\phi(\widetilde{\theta})$. In summary, $S^*_\psi(\widetilde{\theta}\perp S_\phi(\widetilde{\theta})$, though $S_\phi(\widetilde{\theta})$ has “peer" effect on $S_\psi(\widetilde{\theta})$. Three more things regarding $RS^*_\psi(P)$ are worth noting. First, $RS^*_\psi(P)$ requires estimation only under the joint null, namely for the constrained model in which both $\psi = 0$ and $\phi = 0$. Given the full specification of the model $l(\gamma, \psi, \phi)$, it is of course possible to derive a RS test for $H_0: \psi = 0$ in the presence of $\phi$. However, that requires the MLE of $\phi$, which could be difficult to obtain in some cases. \textit{Second}, when $\mathcal{I}_{\psi\phi \cdot \gamma} = 0$, $RS^*_\psi(P) = RS_\psi$. This is a simple condition to check in practice. As mentioned earlier, if this condition is true, $RS_\psi$ is an asymptotically valid test in the local presence of $\phi$. \textit{Finally}, Bera:1993 showed that for local misspecification $RS^*_\psi(P)$ is asymptotically equivalent to Neyman:1959 $C(\alpha)$ test, and therefore, shares its optimality properties.
Example 7: To illustrate the usefulness of the robust score statistic $RS^*_\psi(P)$, we now consider the tests developed in Anselin:1996 for the mixed regressive - spatial autoregressive (SAR) model with a SAR disturbance
In this model, $y$ is an $(n \times 1)$ vector of observations on a dependent variable recorded at each of $n$ locations, $X$ is an $(n \times m)$ matrix of exogenous variables, and $\gamma$ is a $(m \times 1)$ vector of parameters, $\phi$ and $\psi$ are scalar spatial parameters and $W$ is an observable spatial weight matrix with positive elements, associated with the spatially lagged dependent variable and SAR disturbance $u$. This spatial weight matrix represents “degree of potential interactions" among neighboring locations and are scaled so that the sum of the each row elements of $W$ is equal to one. The conventional RS statistic for testing $H_0: \psi = 0$ is given by
where $\widetilde{u} = y - X\widetilde{\gamma}$ are the OLS residuals, $\widetilde{\sigma}^2 = {\widetilde{u}^\prime \widetilde{u}}/{n}$ and $T = tr[(W^\prime + W)W]$. One very interesting observation here is that $RS_\psi$ is essentially same as the widely used Moran:1948 I test. Let us now consider testing $H_0$ under the local presence of $\phi$. First, the crucial quantity to consider is $\mathcal{I}_{\psi\phi \cdot \gamma}$ which is equal to $T$ and that can never be zero. Therefore robustification of $RS_\psi$ is needed. Anselin:1996 derived the robust test as
where
with $M = I- X(X^\prime X)^{-1}X^\prime$. A comparison of (73) and (74) clearly reveals that $RS^*_\psi(P)$ modifies the standard $RS_\psi$ by correcting the asymptotic mean and variance of the score function $S_\psi$.
In a similar way we can find $RS_\phi$ and $RS^*_\phi(P)$ which are given, respectively, by
and
where $\mathcal{I}_{\phi\cdot\gamma} = \mathcal{I}_\phi - \mathcal{I}_{\phi\gamma}\mathcal{I}_{\gamma}^{-1}\mathcal{I}_{\gamma\phi}$ using the submatrices of the partinioned form of $\mathcal{I}(\theta)$ given in (68). Anselin:1988 derived a joint RS test for $H_0: \psi = \phi = 0$ under the framework of (72) and that takes the following form
This statistic is asymptotically distributed $\chi^2_2$. It is easy to verify that [see BBY2020]
In other words, the directional RS test for $\psi$ and $\phi$ can be decomposed into sum of the unadjusted one-directional test for one type of alternative and the adjusted form for the other alternative. Equalities in (79) can facilitate computations of the adjusted (robust) RS tests after having the unadjusted versions which are easy to obtain and are reported in most of the spatial software.
Anselin:chap and Anselin:1996 provided simulation results on the finite sample performance of the unadjusted and adjusted RS tests and some related tests. The adjusted tests $RS^*_\psi(P)$ and $RS^*_\phi(P)$ performed remarkably well. Those had very reasonable empirical sizes, remaining within the confidence intervals in all cases. In terms of power they performed exactly the way they were supposed to.
Now we combine the results of Sections 5.1 and 5.2 and develop robust tests $RS^*_\psi (DP)$ which provides a two-way protection against both types of misspecifications, distributional $(D)$ and parametric $(P)$. As we have noted in (59), $RS^*_\psi (D)$ involves both the $J(\theta)$ and $K(\theta)$ matrices in the variance expression of $S_\psi(\tilde{\theta})$. While to account of the parametric misspecification, as we did in (71), $\mathcal{I}_{\psi\phi \cdot \gamma}(\widetilde{\theta})\mathcal{I}^{-1}_{\phi \cdot \gamma}(\widetilde{\theta})S_\phi(\widetilde{\theta})$ must be subtracted from $S_\psi(\tilde{\theta})$ to center its mean to zero. The expression for $RS^*_\psi (DP)$ is given by [for details see BBY2020]:
where
similarly $B_{\phi\cdot\gamma}$ and
and similarly $B_{\phi\psi\cdot\gamma}$. Expressions for the general forms of $J(\theta)$ and $K(\theta)$ are given in (51) and (52) and here we are using their partitioned forms for $\theta=(\gamma^\prime, \psi^\prime, \phi^\prime)^\prime$. Under $H_0:\psi=0$, the $RS^*_\psi(DP)$ test statistic will be asymptotically distributed as $\chi^2_r$ in the presence of both distributional and parametric misspecifications. Although $RS^*_\psi(DP)$ has rather a lengthy expression as in (80), it is actually easy to compute requiring only $\widetilde{\theta}=(\widetilde{\gamma}^\prime, 0^\prime, 0^\prime)^\prime$. It can be easily seen that under no distributional misspecification, i.e., when $f(y; \theta)\equiv g(y)$, resulting in $K(\widetilde{\theta})=J(\widetilde{\theta})$,
and similarly under no parametric misspecification, i.e., when $\delta=0$ in $\phi=\delta/\sqrt{n}$,
Finally, trivially when $K=J$ and $\delta=0$,
as given in (69).
Example 8: Let us briefly go back to Example 7 and now introduce distributional misspecification along with the presence of parametric misspecification. This case has been rigorously considered by Fang:2014 and they demostrated both analytically and through extensive simulations that $RS^*_\psi(P)$ and $RS^*_\phi(P)$ as given, respectively in (74) and (77) are valid under non-normality. Therefore, $RS^*_\psi(DP)=RS^*_\psi(P)$ and $RS^*_\phi(DP)=RS^*_\phi(P)$. This is a somewhat unsual situation. For this model as given in (72), information matrix equality does not hold, i.e., $J(\theta_g)\neq K(\theta_g)$ [see equations (51)-(53)]. However, still $J^{-1}K J^{-1}=J^{-1}$. This is a serendipitous situation, since no additional adjustment is needed for the distributional misspecification. The intuition behind this serendipity is that the hypotheses $\psi=0$ and $\phi=0$ relate to the conditional mean (first moment) of $y$ in (72) (conditional on the neighborhood as captured by the $W$ matrix). However, in general, only tests for variance (second moment) and higher moments get affected by non-normality. A similar case appeared in BBY2020 where they considered testing for random effects and serial correlation within an error component model. Extensive simulation results are also given in Koley2022, Koley2024 demonstrating the robustness of the RS tests under non-normality in finite sample in spatial regression model set up.
We started this survey paper by stating that C.R.\ Rao's work was always inspired by some practical problems. In his 2003 Econometric Theory (ET) Interview [see Bera2003ET], on the RS test, Rao had the following to say, “The test evolved in a natural way while I was analyzing some genetic data. As I recall, the problem was the estimation of a linkage parameter using data sets from different experiments designed in such a way that each data set had information on the same linkage parameter. It was, however, necessary to test whether such an assumption could be made because of unforeseen factors affecting the experiments. This required a test for consistency of estimates derived from different experimental data sets." Thus we had a new statistical test principle, after LR and W, motivated by a practical problem in genetics. However, as we have narrated here, the resulting RS test principle has a far reaching influence even beyond the Statistics, and in particular, it has become one of the most useful model misspecification testing tools in the Econometrics literature. In the history of any scientific field, once in a while there comes a moment for a major breakthrough. Appearance of Rao:1948 was such a historical moment. In fact, after that we have not witnessed any new test principle, beyond the trinity, LR, W and RS.
To keep our exposition simple and to be close to the spirit of Rao:1948, we have sticked to the likelihood framework. However, it is easy to extend the RS test and its various ramifications to the generalized method of moments (GMM) and estimating functions (EF) frameworks [for more on these, see for instance Basawa:1991 and Bera:2010]. We have also largely confined ourselves to the asymptotic properties and distributions of the tests. However there is a huge literature on the investigation of the finite sample performance of the RS, particularly, in relation to that of LR and W and finding bootstrap critical values; for example see, Mukerjee:1990, Mukerjee93:chap and Horowitz:1997.
To conclude, we can only speculate what is stored in the future. Given the current vastness of the field we have lost “sharp moments of birth", like that of Rao:1948. However, considering that 75 years have already been passed, it might be a time for a brand new equally good test principle.
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We are profoundly thankful to an anonymous referee for her/his careful reading and offering many pertinent comments which led to improvement of the paper. An earlier version of the paper was presented at the Invited Memorial Session for Professor C.\ R.\ Rao, Joint Statistical Meetings (JSM), Portland, Oregon, August 3-8, 2024. We are thankful to the participants of that conference for their comments, especially to the organizer of the Session, Professor Ronald L. Wasserstein, Executive Director of the American Statistical Association (ASA), for giving us the opportunity to present our paper. This paper was also presented at the Department of Statistics, University of Illinois at Urbana-Champaign (UIUC). We would like to thank the attendees of the UIUC seminar for their constructive feedback that further helped in producing an improved version. We are grateful to our research assistant (RA) Anirudh Adhikary. Without his assistance this paper wouldn't have taken off. We are also grateful to our other RAs, Rong Yuwen, Tiancheng Guo, Scarlett He, Yice Zhang and Wenqi Zeng for their diligent work on compiling the citation numbers and a very careful analysis. We thank Professor Osman Do\u{g}an for reading an earlier version of the paper with great care and thoroughness, and providing many pertinent comments with detail suggestions for improvements. We wish we could incorporate all his suggestions. Thanks are also due to Dr.\ Malabika Koley for comments that improved the exposition of the paper. At the publication stage, the Chair Editor, Professor V.\ K.\ Gupta paid painstaking attention to the finest details. Indeed, his suggestions, we believe, greatly improved the final presentation. We, however, retain the responsibility of any remaining errors. Part of this work was completed when the second author visited UIUC. \vskip0.3cm