A test for bivariate normality with applications in microeconometric models

Abstract

In this paper, we propose a test for bivariate normality in imperfectly observed models, based on the information matrix test for censored models with bootstrap critical values. In order to evaluate its properties, we run a comprehensive Monte Carlo experiment, in which we use the bivariate probit model and Heckman sample selection model as examples. We find that, while asymptotic critical values can be seriously misleading, the use of bootstrap critical values results in a test that has excellent size and power properties even in small samples. Since this procedure is relatively inexpensive from a computational viewpoint and is easy to generalise to models with arbitrary censoring schemes, we recommend it as an important and valuable testing tool.

Appendices

Appendix A: Information matrix tests in two-equation limited dependent variable models

Consider the latent variable model¹⁷:

$$\begin{aligned} y_1^{*}&= x_1'\beta _1+ v_1 \end{aligned}$$

(11)

$$\begin{aligned} y_2^{*}&= x_2'\beta _2+ v_2 \end{aligned}$$

(12)

where $v_1$ and $v_2$ have the following bivariate normal distribution:

$$\begin{aligned} \left( \begin{array}{cc} v_{1} \\ v_{2} \end{array} \right) \sim N \left[ \left( \begin{array}{cc} 0 \\ 0 \end{array} \right) ; \left( \begin{array}{cc} \omega _1^{2} &{}\quad \omega _{12} \\ \omega _{12} &{}\quad \omega _2^{2} \end{array} \right) \right] \end{aligned}$$

(13)

where $x_1$ and $x_2$ are vectors of exogenous variables and $\beta _1$, $\beta _2$ are the parameter vectors. Since $v_1$ can be written as $v_1 = \rho v_2 + u$, where $\rho = \omega _{12}/\omega _2^{2}$, the model for $y_1^{*} | y_2^{*}$ is:

$$\begin{aligned} y_1^{*}&= x_1'\beta _1 + \rho v_2 + u \end{aligned}$$

(14)

$$\begin{aligned} y_2^{*}&= x_2'\beta _2+ v_2 \end{aligned}$$

(15)

with $u|x_1,v_2 \sim N(0, \omega _{11.2}^{2})$ where $\omega _{11.2}^{2}= \omega _1^{2}-\omega _{12}/ \omega _2^{2}$.

The log-likelihood for the latent variables $\ell ^{*}$ can be split into conditional $\ell _{12}^{*}$ and marginal $\ell _{2}^{*}$ log-likelihoods so that:

$$\begin{aligned} \ell ^{*} (y_1^{*},y_2^{*};\theta ) = \ell _{12}^{*}(y_1^{*}|y_2^{*};\theta ) + \ell _2^{*}(y_2^{*};\theta _2) \end{aligned}$$

(16)

where $\theta = (\theta _1', \theta _2')'$, $\theta _1 = (\beta _1',\rho )$ and $\theta _2 = (\beta _2', \omega _2^2)$. For an iid sample $(y_i, x_i)$, the observational rules for $y_1^{*}$ and $y_2^{*}$ are assumed to be independent from the parameters.

The following are the key results on the likelihood function (Gourieroux et al. 1984) crucial to the presentation of the test statistic for a model subject to an arbitrary censoring scheme. The score and Hessian matrix elements for observables can be derived quite easily from the score and Hessian matrix for the unobservables as follows:¹⁸

$$\begin{aligned} \frac{\partial \ell }{\partial \theta } = E\left[ \frac{\partial \ell ^{*}}{\partial \theta } \Bigg | y\right] \end{aligned}$$

(17)

and the Hessian matrix as:

$$\begin{aligned} \frac{\partial ^2 \ell }{\partial \theta \partial \theta '} = E\left[ \frac{\partial ^2 \ell ^{*}}{\partial \theta \partial \theta '} \Bigg | y\right] +V\left[ \frac{\partial \ell ^{*}}{\partial \theta } \Bigg | y\right] \end{aligned}$$

(18)

These quantities can be shown to be functions of the generalised error product of order (r,s) ($\hbox {GEP}(r,s)$), introduced by Smith, defined as:

$$\begin{aligned} \overline{\varepsilon ^{r}\xi ^{s}} = E(\varepsilon ^{r}\xi ^{s}| y) - E(\varepsilon ^{r}\xi ^{s}) \end{aligned}$$

(19)

where $\varepsilon = u/\omega _{11.2}$ and $\xi = v_2/\omega _2$. $E(\varepsilon ^{r}\xi ^{s}| y)$ is the expectation conditional on the censoring scheme, that is the relevant region of integration. In (Smith 1985, Sect. 1) some examples are given. The sample counterpart of the GEP($r,s$) is the generalised residual product of order ($r,s$), GRP($r,s$), which is the GEP($r,s$) evaluated at $\hat{\theta }_{ML}$ the ML estimator of model (14)–(15).

The IM test statistic is based on the moment conditions

$$\begin{aligned} C_i= \hbox {vech} \left[ \frac{\partial ^2 \ell _i}{\partial \theta \partial \theta '} + G_i G_i' \right] . \end{aligned}$$

evaluated at $\theta = \hat{\theta }_{ML}$. The contributions to the Hessian matrix and to the OPG are derived using (17) and (18) and are linear functions of the GRP($r,s$), $r+s \le 4$. Expressions for the model (14)–(15) are given in (Smith 1985, Appendix 1 and 2), for both unobservables and observables.

Appendix B: Rank analysis

1.1 B.1 Rank analysis for the sample selection model

As mentioned in 3.1, in the the case of the sample selection model we are only able to prove that the upper bound for $df$ is equal $k(k+1)/2 - 1$ since the last moment condition $C_i^{\psi ,\psi }$ is always dropped in the OPG regression. This happens when different sets of regressors without constant terms for the two equations are considered. The lower bound is more difficult to determine, and will be derived only for a specific setup as an example.

Let us now consider two sets of completely different regressors $x_{is}$ with $s=1,\ldots ,m$ and $w_{ir}$ with $r=1,\ldots ,h$ without constant terms. All the moment conditions that are going to be considered are non-zero only for uncensored observations, so the $d_i$ index will be dropped to simplify the notation (see also Table (1)). $C_i^{\psi ,\psi }$ can be written as a linear combination of other columns of matrix $M$. For this purpose, the following expressions are developed. Consider first the generic $r$-condition $C_i^{\gamma _r,\sigma }$ also as a function of $u_i$ and $b_i$:

$$\begin{aligned} C_i^{\gamma _r,\sigma }= \frac{1}{\sigma }\mu _i \left[ s_{\psi }c_{\psi }^2 u_ib_i + s_{\psi }^2 c_{\psi }u_i^2 + c_{\psi }u_i^2-c_{\psi }\right] w_{ir} \end{aligned}$$

We now multiply $C_i^{\gamma _r,\sigma }$ by $\sigma \gamma _r$ and then sum across the $r$ moment conditions obtaining

$$\begin{aligned} \sigma \sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\sigma } = \mu _i \left[ s_{\psi }c_{\psi }^2 u_ib_i + s_{\psi }^2 c_{\psi }u_i^2 + c_{\psi }u_i^2-c_{\psi }\right] \sum _{r=1}^{h}\gamma _rw_{ir} \end{aligned}$$

which gives

$$\begin{aligned} \sigma \sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\sigma } = \mu _i \left[ s_{\psi }c_{\psi }^2 u_ib_i^2 + c_{\psi }^3 u_i^2 b_i-c_{\psi }b_i \right] \end{aligned}$$

(20)

since $\sum _{r=1}^{h}\gamma _rw_{ir}=b_i$ and $c_{\psi }s_{\psi }^2 + c_{\psi }= c_{\psi }^3$. We will later need also

$$\begin{aligned} t_{\psi }^2\sigma \sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\sigma } = \mu _i \left[ s_{\psi }^3 u_ib_i^2 + s_{\psi }^2 c_{\psi }u_i^2 b_i - \frac{s_{\psi }^2}{c_{\psi }}b_i \right] \end{aligned}$$

(21)

Similar transformations applied to $C_i^{\gamma _r,\psi }$ yield

$$\begin{aligned} t_{\psi }\sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\psi } = \mu _i \left[ -s_{\psi }^2c_{\psi }b_i^3 -s_{\psi }c_{\psi }^2 u_i b_i^2 - s_{\psi }^3 u_i b_i^2 - s_{\psi }^2c_{\psi }u_i^2b_i + \frac{s_{\psi }^2}{c_{\psi }}b_i \right] \nonumber \\ \end{aligned}$$

(22)

Let us now write $C_i^{\sigma ,\psi }$ as a function of $u_i$ and $b_i$. We get

$$\begin{aligned} \sigma t_{\psi }C_i^{\sigma ,\psi } = \mu _i \left[ -2s_{\psi }u_i - \frac{s_{\psi }^2}{c_{\psi }}b_i + s_{\psi }c_{\psi }^2 u_i^3 + s_{\psi }^3 u_i b_i^2 +2s_{\psi }^2c_{\psi }u_i^2 b_i \right] \end{aligned}$$

(23)

and, finally, we need

$$\begin{aligned} t_{\psi }G_i^{\psi } = \mu _i \left[ \frac{s_{\psi }^2}{c_{\psi }}b_i +s_{\psi }u_i \right] \end{aligned}$$

(24)

Since $C_i^{\psi ,\psi }$, written as a function of $u_i$ and $b_i$ is

$$\begin{aligned}&C_i^{\psi ,\psi } = \mu _i \left[ a_i(1-c_i^2)\right] \nonumber \\&\quad \quad \quad \quad =\mu _i \left[ -s_{\psi }c_{\psi }^2 u_i^3 -s_{\psi }^2c_{\psi }b_i^3 - c_{\psi }^3 u_i^2 b_i -s_{\psi }^3 u_i b_i^2 \right. \nonumber \\&\quad \quad \quad \quad \quad \left. -2s_{\psi }^2c_{\psi }u_i^2 b_i -2s_{\psi }c_{\psi }^2 u_i b_i^2 + c_{\psi }b_i +s_{\psi }u_i \right] \end{aligned}$$

(25)

it can now be expressed as a linear combination of (20), (21), (22), (23) and (24)

$$\begin{aligned} C_i^{\psi ,\psi } = - \sigma (1-t_{\psi }^2)\sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\sigma } + t_{\psi }\sum _{r=1}^{h}\gamma _r C_i^{\gamma _r,\psi } - \sigma t_{\psi }C_i^{\sigma ,\psi } -t_{\psi }G_i^{\psi } \end{aligned}$$

While it makes sense to consider the case of the same sets of regressors for the rank analysis in the bivariate probit model and therefore to choose the limiting case of only two constants to study $df$’s lower bound, the choice of the case study for the Heckman selection model needs further discussion. First of all, it is not possible to consider only two constants since the model would not be identified. Secondly, it is quite common to see applications with two at least slightly different sets of regressors. Therefore we believe that the simplest form of a reasonable setup is one containing a constant term and a continuous regressor $w$ in the selection equation, and only a constant term in the main equation. The vector of parameters of the model just described is ($\beta _0, \gamma _0, \gamma _1, \sigma , \psi$). Six of the fifteen moment conditions are dropped in the OPG regression. Naturally

$$\begin{aligned} C_i^{\beta _0,\beta _0}&= d_i\frac{1}{\sigma }\left[ G_i^{\sigma }- \frac{s_{\psi }c_{\psi }}{\sigma }G_i^{\psi } \right] \\ C_i^{\beta _0,\gamma _0}&= d_i\frac{c_{\psi }^2}{\sigma }G_i^{\psi } \end{aligned}$$

are dropped as they are linear combinations of the score elements. Also $C_i^{\gamma _0,\gamma _0}$ is a linear combination of $G_i^{\gamma _0}, G_i^{\gamma _1}, G_i^{\psi }$. First notice that in this specific setup

$$\begin{aligned} b_i = \gamma _0 + w_i\gamma _1 \end{aligned}$$

and

$$\begin{aligned} a_i = c_{\psi }\gamma _0 + c_{\psi }w_i\gamma _1 + s_{\psi }u_i; \quad c_i = s_{\psi }\gamma _0 + s_{\psi }w_i\gamma _1 + c_{\psi }u_i \end{aligned}$$

then write

$$\begin{aligned} C_i^{\gamma _0,\gamma _0} = d_i \mu _i \left[ -c_{\psi }^3\gamma _0 -c_{\psi }^3 w_i\gamma _1 -s_{\psi }c_{\psi }^2 u_i\right] - (1-d_i)\mu _i \left[ \gamma _0 + w_i\gamma _1 \right] \end{aligned}$$

Given the following transformations, only for uncensored observations,

$$\begin{aligned} -c_{\psi }^2 \gamma _0 G_i^{\gamma _0} = -c_{\psi }^3\gamma _0 \mu _i; \quad -c_{\psi }^2 \gamma _1 G_i^{\gamma _1} = -c_{\psi }^3 w_i\gamma _1 \mu _i \end{aligned}$$

and

$$\begin{aligned} -s_{\psi }c_{\psi }G_i^{\psi }= -\mu _i \left[ -c_{\psi }^3\gamma _0 -c_{\psi }^3w_i\gamma _1 -c_{\psi }^2s_{\psi }u_i \right] , \end{aligned}$$

$C_i^{\gamma _0,\gamma _0}$ can be written as

$$\begin{aligned} C_i^{\gamma _0,\gamma _0} = -d_i\left[ c_{\psi }s_{\psi }G_i^{\psi } + \gamma _0 G_i^{\gamma _0} + \gamma _1 G_i^{\gamma _1} \right] - (1-d_i)\left[ \gamma _0 G_i^{\gamma _0} + \gamma _1 G_i^{\gamma _1}\right] . \end{aligned}$$

$C_i^{\gamma _0,\sigma }$ is also a linear combination of $G_i^{\psi }$ and of the moment conditions $C_i^{\beta _0, \gamma _1}$ and $C_i^{\beta _0,\psi }$.

Rearranging algebraically $C_i^{\gamma _0,\sigma }$ and $C_i^{\beta _0,\psi }$ we get

$$\begin{aligned} C_i^{\gamma _0,\sigma } = C_i^{\beta _0,\psi } -\mu _i \frac{1}{\sigma } \left[ s_{\psi }^2c_{\psi }b_i + s_{\psi }c_{\psi }^2 u_i \right] (\gamma _0 + \gamma _1w_i). \end{aligned}$$

(26)

Applying the following transformations

$$\begin{aligned}&\frac{s_{\psi }c_{\psi }\gamma _0}{\sigma }G_i^{\psi }= \frac{\mu _i}{\sigma }\left[ s_{\psi }^2c_{\psi }b_i + s_{\psi }c_{\psi }^2 u_i \right] \gamma _0\\&\frac{s_{\psi }}{c_{\psi }}\gamma _i C_i^{\beta _0, \gamma _1} = \frac{\mu _i}{\sigma }\left[ s_{\psi }^2c_{\psi }b_i + s_{\psi }c_{\psi }^2 u_i \right] \gamma _1 w_i \end{aligned}$$

we can rewrite (26) as

$$\begin{aligned} C_i^{\gamma _0,\sigma } = C_i^{\beta _0,\psi } - \frac{s_{\psi }c_{\psi }\gamma _0}{\sigma }G_i^{\psi } - \frac{s_{\psi }}{c_{\psi }}\gamma _1 C_i^{\beta _0, \gamma _1}. \end{aligned}$$

(27)

The OPG regression also drops

$$\begin{aligned} C_i^{\gamma _0,\psi } = -\mu _i (c_{\psi }a_i c_i - s_{\psi }). \end{aligned}$$

(28)

Considering

$$\begin{aligned} -\sigma \frac{c_{\psi }}{s_{\psi }}C_i^{\beta _0,\psi } = -\mu _i \left( c_{\psi }a_i c_i + \frac{c_{\psi }}{s_{\psi }}u_i c_i -\frac{c_{\psi }^2}{s_{\psi }}\right) \end{aligned}$$

and

$$\begin{aligned} \frac{\sigma }{s_{\psi }c_{\psi }}C_i^{\gamma _0,\sigma }= \mu _i \left( c_{\psi }b_i u_i + \frac{c_{\psi }^2}{s_{\psi }}u_i^2 - \frac{1}{s_{\psi }}\right) \end{aligned}$$

we can rewrite (28) as

$$\begin{aligned} C_i^{\gamma _0,\psi } = -\sigma \frac{c_{\psi }}{s_{\psi }}C_i^{\beta _0,\psi } + \frac{\sigma }{s_{\psi }c_{\psi }}C_i^{\gamma _0,\sigma } \end{aligned}$$

(29)

and substituting (27) in (29) we finally get

$$\begin{aligned} C_i^{\gamma _0,\psi } = -\sigma \frac{s_{\psi }}{c_{\psi }} C_i^{\beta _0, \psi } - \gamma _0 G_i^{\psi } - \frac{\sigma }{c_{\psi }^2}\gamma _1 C^{\beta _0, \gamma _1} \end{aligned}$$

Finally the last column to be dropped is $C_i^{\psi ,\psi }$ as we discussed earlier. So in this simple setup the number of moment conditions are $k(k+1)/2 = 15$ of which only $9$ are kept for testing.

1.2 B.2 Rank analysis for the bivariate probit model

Let us start from the extreme case in which in each equation the only regressor is a constant; then,

$$\begin{aligned} - x_{1i}\beta _1 = a_{i} = \bar{a} \qquad - x_{2i}\beta _2 = b_{i} = \bar{b} \qquad P_i = \bar{P} \end{aligned}$$

are also constant across observations with differences depending only on the observational rule. Therefore the three score elements $G_i$ are also constant across observations

$$\begin{aligned} G_i^{a_i} = \bar{S}^{\bar{a}} \qquad G_i^{b_i} = \bar{S}^{\bar{b}} \qquad G_i^{\psi } = \bar{S}^{\psi } \end{aligned}$$

This makes every moment condition $C_i$ a linear combination of the score elements (compare Tables 2 and 9), which means that all these conditions are collinear to the score matrix and, as a consequence, do not contribute to the rank of $M$, as defined in Sect. 3.1.

Table 9 Moment conditions for the bivariate probit model with only two constant terms as regressors

Let us turn to the opposite extreme case, with non-overlapping sets of regressors.¹⁹ It is possible to prove that the last moment condition, $C_i^{\psi ,\psi }$ is always collinear to the rest of the columns of $M$, even though no suspicious redundancy is apparent.

Dropping the $i$ index for clarity, consider $x_{1r} \ne x_{2s}$ for every $r=1,\ldots ,k_1$ and $s=1,\ldots ,k_2$; then

$$\begin{aligned} a = \sum _{r=1}^{k_1} x_{1r}\beta _{1r} \qquad b = \sum _{s=1}^{k_2} x_{2s}\beta _{2s} \end{aligned}$$

The generic condition associated with cross-derivatives of $\beta _{1r}$ and $\beta _{2s}$ (see also Table (2)), may be written as:

$$\begin{aligned} C^{\beta _{1r}, \beta _{2s}} = c_{\psi }^2 S^{\psi }x_{1r}x_{2s} \end{aligned}$$

(30)

Now write the moment condition associated with the cross-derivative of $\beta _{1t}$ and $\psi$ as:

$$\begin{aligned} C^{\beta _{1t}, \psi } = - \left[ c_{\psi }\left( \sum _{r=1}^{k_1} x_{1r}\beta _{1r} \right) x_{1t} - s_{\psi }\left( \sum _{s=1}^{k_2} x_{2s}\beta _{2s} \right) x_{1t} \right] c_{\psi }S^{\psi } \end{aligned}$$

By using (30), the previous expression becomes

$$\begin{aligned} C^{\beta _{1t}, \psi } = -c_{\psi }^2 S^{\psi } \sum _{r=1}^{k_1} x_{1t} x_{1r}\beta _{1r} + t_{\psi }\sum _{s=1}^{k_2}C^{\beta _{1t},\beta _{2s}}\beta _{2s} \end{aligned}$$

(31)

By symmetry, the moment condition associated with the cross-derivative of $\beta _{2m}$ and $\psi$ can be written as

$$\begin{aligned} C^{\beta _{2m}, \psi } = -c_{\psi }^2 S^{\psi } \sum _{s=1}^{k_2} x_{2m} x_{2s}\beta _{2s} +t_{\psi }\sum _{r=1}^{k_1}C^{\beta _{1r},\beta _{2m}}\beta _{1r} \end{aligned}$$

(32)

Let us now rewrite $C^{\psi ,\psi }$ as follows:

$$\begin{aligned} C^{\psi ,\psi }&= S^{\psi }[c_{\psi }^2 ab + s_{\psi }^2 ab - s_{\psi }c_{\psi }a^2 - s_{\psi }c_{\psi }b^2 - t_{\psi }]\\&= c_{\psi }^2 S^{\psi } \sum _{r=1}^{k_1}\sum _{s=1}^{k_2} x_{1r} x_{2s}\beta _{1r}\beta _{2s} +s_{\psi }^2 S^{\psi }\sum _{r=1}^{k_1}\sum _{s=1}^{k_2} x_{1r} x_{2s}\beta _{1r}\beta _{2s} \\&- s_{\psi }c_{\psi }S^{\psi }\sum _{r=1}^{k_1}\sum _{t=1}^{k_1} x_{1r}x_{1t}\beta _{1r}\beta _{1t} - s_{\psi }c_{\psi }S^{\psi }\sum _{s=1}^{k_2}\sum _{m=1}^{k_2} x_{2s}x_{2m}\beta _{2s}\beta _{2m} - S^{\psi }t_{\psi }. \end{aligned}$$

Note that

$$\begin{aligned} c_{\psi }^2 S^{\psi }\sum _{r=1}^{k_1}\sum _{s=1}^{k_2} x_{1r} x_{2s}\beta _{1r}\beta _{2s}&= \sum _{r=1}^{k_1}\sum _{s=1}^{k_2} C^{\beta _{1r}\beta _{2s}}\beta _{1r}\beta _{2s} \end{aligned}$$

(33)

$$\begin{aligned} s_{\psi }^2 S^{\psi }\sum _{r=1}^{k_1}\sum _{s=1}^{k_2} x_{1r} x_{2s}\beta _{1r}\beta _{2s}&= t_{\psi }^2 \sum _{r=1}^{k_1}\sum _{s=1}^{k_2}C^{\beta _{1r}\beta _{2s}}\beta _{1r} \beta _{2s}. \end{aligned}$$

(34)

and that by multiplying (31) by $\beta _{1t}t_{\psi }$ one gets

$$\begin{aligned} - s_{\psi }c_{\psi }S^{\psi }\sum _{r=1}^{k_1}\sum _{t=1}^{k_1} x_{1r}x_{1t}\beta _{1r}\beta _{1t} = t_{\psi }\sum _{t=1}^{k_1} C^{\beta _{1t},\psi }\beta _{1t} -t_{\psi }^2 \sum _{t=1}^{k_1}\sum _{s=1}^{k_2}C^{\beta _{1t},\beta _{2s}} \beta _{1t},\beta _{2s}; \nonumber \\ \end{aligned}$$

(35)

similarly, (32) may be multiplied by $\beta _{2m}t_{\psi }$ to obtain

$$\begin{aligned} - s_{\psi }c_{\psi }S^{\psi }\sum _{s=1}^{k_2}\sum _{m=1}^{k_2} x_{2s}x_{2m}\beta _{2s}\beta _{2m} \!=\! t_{\psi }\sum _{m=1}^{k_2}C^{\beta _{2m},\psi }\beta _{2m} -t_{\psi }^2\sum _{r=1}^{k_1}\sum _{m=1}^{k_2}C^{\beta _{1r}, \beta _{2m}}\beta _{1r},\beta _{2m}. \nonumber \\ \end{aligned}$$

(36)

Finally, after rearranging (33), (34), (35) and (36), we can rewrite $C^{\psi ,\psi }$ as

$$\begin{aligned} C^{\psi ,\psi }&= (1-t_{\psi }^2)\sum _{r=1}^{k_1}\sum _{s=1}^{k_2} C^{\beta _{1r},\beta _{2s}}\beta _{1r}\beta _{2s} \\&+\, t_{\psi }\sum _{r=1}^{k_1} C^{\beta _{1r},\psi } \beta _{1r} + t_{\psi }\sum _{s=1}^{k_2} C^{\beta _{2s},\psi } \beta _{2s} - t_{\psi }S^{\psi }, \end{aligned}$$

that is, a linear combination of elements of other columns of $M_i$.

Different combinations of constant and duplicated regressors across equations lead to intermediate cases. We are particularly interested in studying the case (which often occurs in practice) in which we have the same set of regressors for both equations including constant terms. Other than the $C^{\psi ,\psi }$ element, we now prove that in this particular case other moment conditions are always collinear in the OPG regression. Moreover, in this special case an explicit formula to determine a priori the rank of $M$ can be obtained.

Consider $x_{1} = x_{2} = x$ and $k_1 = k_2 = q$ so (again, the $i$ index is dropped) $x' = (1, x_{2}, \ldots , x_{q})$ and

$$\begin{aligned} a = \sum _{r=1}^{q}x_{r}\beta _{1r} \qquad b = \sum _{r=1}^{q}x_{r}\beta _{2r}. \end{aligned}$$

Similarly, $G^{\beta _{j}}$ has $q$ elements

$$\begin{aligned}{}[G^{\beta _{j1}}, G^{\beta _{j2}},\dots , G^{\beta _{jq}} ] \end{aligned}$$

for $j=1,2$, such that

$$\begin{aligned} G^{\beta _{1r}} = S^{a}x_{r} \qquad G^{\beta _{2r}} = S^{b}x_{r} \end{aligned}$$

for $r=1,\ldots ,q$ (see also Sect. 4.2).

To begin with, the three moment conditions associated to the two constant terms drop because

$$\begin{aligned} C^{\beta _{11},\beta _{11}}&= - \left[ \left( \sum _{r=1}^{q}x_{r}\beta _{1r}\right) S^{a} + s_{\psi }c_{\psi }S^{\psi } \right] = - \left[ \sum _{r=1}^{q}G^{\beta _{1r}}\beta _{1r} + s_{\psi }c_{\psi }S^{\psi } \right] \\ C^{\beta _{21},\beta _{21}}&= - \left[ \left( \sum _{r=1}^{q}x_{r}\beta _{2r}\right) S^{b} + s_{\psi }c_{\psi }S^{\psi } \right] = - \left[ \sum _{r=1}^{q}G^{\beta _{2r}}\beta _{2r} + s_{\psi }c_{\psi }S^{\psi } \right] \end{aligned}$$

Consider now the $q^2$ elements associated with cross derivatives of $\beta _{1r},\beta _{2s}$

$$\begin{aligned} C^{\beta _{1r},\beta _{2s}} = c_{\psi }^2 S^{\psi }x_{r}x_{s} \end{aligned}$$

with $s=1,\ldots ,q$. Since the sets of regressors are the same, the number of elements dropped due to collinearity will be $q^2-q(q+1)/2$ plus the condition associated with the cross derivative of the constant terms

$$\begin{aligned} C^{\beta _{11},\beta _{21}} = c_{\psi }^2 S^{\psi }. \end{aligned}$$

There are also $2q$ elements, collinear to other columns of $M$, associated with the cross derivatives of regressors with $\psi$ since

$$\begin{aligned} C^{\beta _{1t},\psi }&= - \left[ c_{\psi }\left( \sum _{r=1}^{q} x_{r}\beta _{1r} \right) - s_{\psi }\left( \sum _{r=1}^{q} x_{r}\beta _{2r} \right) \right] c_{\psi }S^{\psi }x_{t} \\&= -c_{\psi }^2 S^{\psi } \sum _{r=1}^{q}x_{t}x_{r}\beta _{1r} + s_{\psi }c_{\psi }S^{\psi }\sum _{r=1}^{q}x_{t}x_{r}\beta _{2r}\\&= - \sum _{r=1}^{q}C^{\beta _{1r},\beta _{2t}}\beta _{1t} + t_{\psi }\sum _{r=1}^{q}C^{\beta _{1r},\beta _{2t}}\beta _{2r} \end{aligned}$$

and as well

$$\begin{aligned} C^{\beta _{2t},\psi }&= - \left[ c_{\psi }\left( \sum _{r=1}^{q}x_{r}\beta _{2r} \right) - s_{\psi }\left( \sum _{r=1}^{q}x_{r}\beta _{1r} \right) \right] c_{\psi }S^{\psi }x_{t}\\&= - \sum _{r=1}^{q}C^{\beta _{1r},\beta _{2t}}\beta _{2r} + t_{\psi }\sum _{r=1}^{q}C^{\beta _{1r},\beta _{2t}}\beta _{1r} \end{aligned}$$

Finally, as shown earlier in this section, $C^{\psi ,\psi }$ is always a linear combination of other columns of $M$. So, in this setup, the number of degrees of freedom amounts to

$$\begin{aligned} df = \frac{k(k+1)}{2} -2 -q^2 + \frac{q(q+1)}{2} -1 -2q -1 \end{aligned}$$

and since $k = 2q +1$ we have

$$\begin{aligned} df = 3\left[ \frac{q(q+1)}{2}-1 \right] . \end{aligned}$$