Specification analysis of corporate equity financing decision: a conditional residual approach

Abstract

This article revisits the debate on the nature of private placements by specifying that informed insiders make trading decisions in the secondary market and equity issuance decision in the primary equity market (Lee and Wu (2008)). This article uses conditional residuals from the insider trading regression (abnormal insider trades) and conditional residuals from equity financing choice regression (unexpected equity financing choice) to measure private information. An important advantage of conditional correlation coefficient approach over the two-stage approach (Lee and Wu 2008) in testing the presence of asymmetric information is that the former is bounded by −1 and 1 and thus permits cross-sectional comparisons the relatedness between abnormal insider trades and unexpected equity financing choice.

Appendices

Appendix 1: The derivation of Eq. 3

Getting the two equations in Eq. 2 into reduced form and expressing them as matrix notation, we write

$$ \begin{aligned} \left[ \begin{array}{l} p_{i} \\ c_{i} \end{array} \right] &=d_{k}\tilde{e}_{i,t}^{T}+\pi_{k},\hbox{ where} \\ d_{k} &=\left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right] =\left[ \begin{array}{c} \gamma_{2} \\ \beta_{2} \end{array} \right], \hbox{ and }\pi_{k}=\left[ \begin{array}{c} \pi_{1} \\ \pi_{2} \end{array} \right] =\left[ \begin{array}{c} x_{obs,i,t}^{T}(\gamma_{1}+A^{T}\gamma_{2})+\lambda _{t}+\tilde{u}_{i,t} \\ x_{obs,i,t}^{T}(\beta_{1}+A^{T}\beta_{2})+\lambda _{t}+\tilde{v}_{i,t} \end{array} \right] \end{aligned} $$

Our estimation approach is to use p _i as a proxy for the private information $\tilde{e}_{i,t}^{T},$

$$ \tilde{e}_{i,t}^{T}=\frac{p_{i}-\pi_{1}}{d_{1}}=\frac{ p_{i}-x_{obs,i,t}^{T}(\gamma_{1}+A^{T}\gamma_{2})-\lambda _{t}-\tilde{u}_{i,t}}{\gamma_{2}}, $$

so that,

$$ \begin{aligned} c_{i} &=d_{2}\tilde{e}_{i,t}^{T}+\pi_{2}=d_{2}\left(\frac{ p_{i}-x_{obs,i,t}^{T}(\gamma_{1}+A^{T}\gamma_{2})-\lambda _{t}-\tilde{u}_{i,t}}{\gamma_{2}}\right)+\pi_{2} \\ &=\beta_{2}\left(\frac{p_{i}-x_{obs,i,t}^{T}(\gamma_{1}+A^{T}\gamma _{2})-\lambda_{t}-\tilde{u}_{i,t}}{\gamma _{2}}\right)+x_{obs,i,t}^{T}(\beta _{1}+A^{T}\beta_{2})+\lambda_{t}+\tilde{v}_{i,t} \\ &=x_{obs,i,t}^{T}(\beta_{1}+A^{T}\beta_{2})+\frac{\beta_{2}}{\gamma _{2}} \left(p_{i}-x_{obs,i,t}^{T}(\gamma_{1}+A^{T}\gamma_{2})-\lambda _{t}\right)+\left(\tilde{v}_{i,t}-\frac{\beta_{2}}{\gamma_{2}}\tilde{u}_{i,t}\right) \\ &=x_{obs,i,t}^{T}(\beta_{1}+A^{T}\beta_{2})+\frac{\beta_{2}}{\gamma _{2}} (\tilde{e}_{i,t_k}^{T}\gamma _{2}+\tilde{u}_{i,t})+(\tilde{v}_{i,t}-\frac{\beta _{2}}{\gamma_{2}}\tilde{u}_{i,t}) \end{aligned} $$

We can simplify the above procedure and rewrite the equity-selling mechanism choice equation,

$$ c_{i}=\beta_{3}({\tilde{e}_{i,t_k}}^{T}\gamma _{2}+\tilde{u}_{i,t})+x_{obs,i,t}^{T}(\beta_{1}+A^{T}\beta _{2})+\tilde{w}_{i,t},\hbox{ with }\beta_{3}=\frac{\beta _{2}}{\gamma_{2}}\hbox{ and }\tilde{w}_{i,t}=(\tilde{v}_{i,t}-\frac{\beta_{2}}{\gamma _{2}}\tilde{u}_{i,t}). $$

Appendix 2: The derivations of the conditional correlation coefficient and its test statistic

$$ \begin{aligned}&p_{i,t}^{\ast } =x_{obs,i,t}^{T}(\gamma _{1}+A^{T}\gamma_{2})+\lambda _{t}+\tilde{e}_{i,t}^{T}\gamma_{2}+\tilde{u}_{i,t}, \\ &c_{i,t}^{\ast } =x_{obs,i,t}^{T}(\beta_{1}+A^{T}\beta_{2})+\lambda _{t}+ \tilde{e}_{i,t}^{T}\beta_{2}+\tilde{v}_{i,t},\end{aligned} $$

The conditional correlation coefficient is given by

$$Corr(\tilde{e}_{i,t}^{T}\gamma_{2}+\tilde{u}_{i,t},\tilde{e}_{i,t}^{T}\beta_{2}+\tilde{v}_{i,t}\left\vert x_{obs,i,t}\right)$$

$$ \begin{aligned} \,&=\frac{Cov(\tilde{e}_{i,t}^{T}\gamma _{2}+\tilde{u}_{i,t},\hbox{ } \tilde{e}_{i,t}^{T}\beta _{2}+\tilde{v}_{i,t}\left\vert x_{obs,i,t}\right) }{\sqrt{Var(\tilde{e}_{i,t}^{T}\gamma_{2}+\tilde{u}_{i,t}\left\vert x_{obs,i,t}\right) }\sqrt{Var(\tilde{e}_{i,t} ^{T}\beta_{2}+\tilde{v}_{i,t}\left\vert x_{obs,i,t}\right) }} \\ &=\frac{\gamma_{2}^{T}\sum_{e}\beta_{2}}{(\gamma_{2}^{T}\sum_{e}\gamma _{2}+\sigma_{u}^{2})^{1/2}(\beta_{2}^{T}\sum_{e}\beta_{2}+\sigma _{v}^{2})^{1/2}}. \end{aligned} $$

The necessary and sufficient condition for $Corr(\tilde{e}_{i,t}^{T}\gamma_{2}+\tilde{u}_{i,t},\tilde{e}_{i,t}^{T}\beta _{2}+ \tilde{v}_{i,t}\left\vert x_{obs,i,t}\right)$ to be positive is $\gamma_{2}^{T}\sum_{e}\beta_{2} > 0.$ Two special cases can be summarized as follows:

• If dim $(\sum_{e})=p > 1,$ with $\sum_{e} =\hbox{diag}\{\sigma _{e,1}^{2},\ldots, \sigma_{e,p}^{2}\}$ and $\beta_{2,j}\gamma_{2,j} > 0$ for j = 1,..., p, then $\gamma_{2}^{T}\sum_{e}\beta_{2} > 0.$

• If dim $(\sum_{e})=1, \gamma_{2}^{T}\sum_{e}\beta_{2} > 0$ if and only if $\gamma_{2}\beta_{2} > 0.$

  The direction of the conditional correlation depends on the sign of the numerator, $\gamma_{2}^{T}\sum_{e}\beta_{2}.$ In the case where the off-diagonal elements of $\sum\limits_{e}$ are zero and each element of the two column vectors, β₂ and γ₂, is of the same sign (except for the zero vector), then $\gamma_{2}^{T}\sum\limits_{e}\beta_{2}$ is greater than zero. If, instead of being multidimensional, x _unobs,i can be fully captured in a variable, then $\beta_{2}\sigma _{e}^{2}\gamma_{2}$ is greater than zero if and only if γ₂β₂ is greater than zero.

We compute the generalized residuals, $(\hat{e}_i^t \hat{r}_2+\hat{u}_{i})$ and $( \hat{e}_{i}^{t}\hat{\beta}_{2}+\hat{v}_{i}),$ given by the conditional means, $\widehat{E}((\hat{e}_{i}^{t}\hat{r}_{2}+\hat{u} _{i})\left\vert p_{i}\right)$ and $\widehat{E}((\hat{e} _{i}^{t}\hat{\beta}_{2}+\hat{v} _{i})\left\vert c_{i}\right),$ respectively. We then define a test statistic given by

$$ \widehat{\rho}=\frac{\frac{1}{n}\sum\limits_{i=1}^{n}(\hat{e}_{i}^{t}\hat{r}_{2}+\hat{u} _{i})(\hat{e}_{i}^{t}\hat{\beta }_{2}+ \hat{v}_{i})}{\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}(\hat{e}_{i}^{t}\hat{r}_{2}+ \hat{u}_{i})^{2}}\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n} (\hat{e}_{i}^{t}\hat{\beta }_{2}+\hat{v}_{i})^{2}}}\mathop{\approx}\limits^{d}N(\rho,\frac{1}{n}), $$

$$ n(\widehat{\rho }-\rho)^{2}\mathop{\approx}\limits^{d}\chi^{2}(1), $$

Under the null of iid of the generalized residuals, $\sqrt{n}(\widehat{\rho}-\rho)$ is distributed asymptotically as a standard normal distribution. Consequently, $n(\widehat{\rho}-\rho)^{2}$ is distributed asymptotically as a chi-square (χ²) distribution with one degree of freedom. Compared with the test statistic Ψ developed by Gouriéroux et al. (1987) and employed by Chiappori and Salanie (2000) for testing asymmetric information in the French automobile insurance market,

$$ \Uppsi =\frac{(\sum_{i=1}^{n}(\hat{e} _{i}^{t}\hat{r}_{2}+\hat{u}_{i})(\hat{e}_{i}^{t}\hat{\beta}_{2}+ \hat{v}_{i}))^{2}}{\sum_{i=1}^{n}(\hat{e} _{i}^{t}\hat{r}_{2}+\hat{u}_{i})^{2}(\hat{e}_{i}^{t}\hat{\beta }_{2}+\hat{v}_{i})^{2}}\mathop{\approx}\limits^{d}\chi^{2}(1), $$

there are two important advantages of adopting $\widehat{\rho }$ instead of Ψ. First, in addition to the common assumption that observations are cross-sectional independently and identically distributed (i.e., iid), their test statistic is only valid under one more assumption: the pair of generalized residuals are conditional independent, i.e., $Cov(\tilde{e}_{i}^{T}\gamma _{2}+\tilde{u}_{i},\tilde{e}_{i}^{T}\beta_{2}+ \tilde{v}_{i}\left\vert x_{obs,i}\right) =0.$ Second, Ψ fails to retain information on both the direction and the magnitude of correlation. Since the key objective of this article is to detect the information content of equity-selling mechanisms based on the direction and the magnitude of the conditional association between private placements and insider trading, we adopt $\widehat{\rho }$ as a test statistic instead of Ψ.

Appendix 3: The non-parametric χ² test for independence

Here, we also use the non-parametric χ² test to examine if the estimated impact of abnormal insider purchases (sales) is sensitive to the methodological approach. The proposed grouping is to let x _i be a vector of m (number of independent variables) dummies (either 0 or 1). Firms having the same values for m dummies are grouped by 2^m. In each grouping, a two-by-two classification table generated by the values of c _i and p _i is computed. Let j = 0,1.

	p i  = 0	p i  = 1
c i = 0	n 00	n 01
c i = 1	n 10	n 11

Define n _j. = n _j0 + n _j1,n _.j  = n _0j  + n _1j , and n _.. = n _.0 + n _.1. Now consider the test statistic $V_{m}=\sum_{j,k=0,1}\frac{\left[ n_{jk}-(n_{j.}n_{.k}/n_{..})\right]^{2}}{n_{jk}},$ which is the well-known χ² test for independence. If c _i and p _i are independently distributed, then V _m is χ²(1). The probability of rejecting independence in each grouping is assumed to be 0.05. That is, if the value of V _m for a certain grouping is greater than 3.84, the 5% critical value for χ²(1), the null hypothesis of independence is rejected. Furthermore, if V _m is assumed to be independent between groupings, then the probability of rejecting independence for the 2^m groupings has a binomial distribution with the number of trials equals 2^m and the probability of success equals 0.05, the probability of rejecting independence in each grouping. The statistic is generated by adding up the V _m values for 2^m cells, which is distributed as a χ²(2^m).

To implement this test, we first need to do the groupings. It is infeasible to use all observable variables (x _obs ), since it will result in too small number of observations in each cell. We use four variables that are highly significant in the regressions for equity-selling mechanism choices and the intensive insider trading: (1) Age ₂, (2) Spread(%), (3) ln (asset), and (4) Option.

Appendix 4: The impact of the omitted variables

The true covariance $Cov(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs,i}\right),$ given the complete set of observable variables x _obs,i , equals $E(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs,i}\right) -E(c_{i}^{\ast }\left\vert x_{obs,i}\right) E(p_{i}^{\ast }\left\vert x_{obs,i}\right).$ For the purpose of the analysis in Sect. 5.2.1., we assume that $Cov(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs,i}\right) > 0.$ Let x _obs1,i and x _omit,i denote, the control and omitted variables, respectively. By definition, x _obs,i  = x _obs1,i  + x _omit,i ; Given the control variables x _obs1,i , the true covariance $Cov(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs,i}\right)$ equals

$E(E(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right) -E(E(c_{i}^{\ast }\left\vert x_{obs,i}\right) E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right) =$

$$ E(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs1,i}\right) -E(E(c_{i}^{\ast }\left\vert x_{obs,i}\right) E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right) > 0. $$

(1)

Now, the estimated covariance $Cov(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs1,i}\right),$ equals

$$ E(c_{i}^{\ast },p_{i}^{\ast }\left\vert x_{obs1,i}\right) -E(c_{i}^{\ast }\left\vert x_{obs1,i}\right) E(p_{i}^{\ast }\left\vert x_{obs1,i}\right) < 0 \hbox{ or}\approx 0, $$

(2)

comparing (1) and (2) shows that

$$ E(c_{i}^{\ast }\left\vert x_{obs1,i}\right) E(p_{i}^{\ast }\left\vert x_{obs1,i}\right) > E(E(c_{i}^{\ast }\left\vert x_{obs,i}\right) E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right). $$

(3)

We can rewrite the left-hand side of (3) as $E(E(c_{i}^{\ast}\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right) E(E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right)$ and the left-hand side of (2) as $E(c_{i}^{\ast},p_{i}^{\ast }\left\vert x_{obs1,i}\right) -E(E(c_{i}^{\ast}\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right) E(E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \left\vert x_{obs1,i}\right).$ Let $E(c_{i}^{\ast }\left\vert x_{obs,i}\right) \equiv \overline{c_{i}^{\ast }}=x_{obs,i}^{T}\beta,$ and $E(p_{i}^{\ast }\left\vert x_{obs,i}\right) \equiv \overline{p_{i}^{\ast }}=x_{obs,i}^{T}\gamma.$ Now, the left-hand side of (3) equals $E(\overline{c_{i}^{\ast }}\left\vert x_{obs1,i}\right) E(\overline{p_{i}^{\ast }}\left\vert x_{obs1,i}\right)$ and the right-hand side of (3) becomes $E(\overline{c_{i}^{\ast }},\overline{p_{i}^{\ast }} \left\vert x_{obs1,i}\right).$ For the inequality (3) to hold, it must be that $E(\overline{c_{i}^{\ast }}, \overline{p_{i}^{\ast }}\left\vert x_{obs1,i}\right) < E(\overline{ c_{i}^{\ast }}\left\vert x_{obs1,i}\right) E(\overline{p_{i}^{\ast }} \left\vert x_{obs1,i}\right),$ i.e., $Cov(\overline{c_{i}^{\ast }}, \overline{p_{i}^{\ast }}\left\vert x_{obs1,i}\right) < 0,$ or $Cov( x_{obs,i}^{T}\beta,x_{obs,i}^{T}\gamma \left\vert x_{obs1,i}\right) < 0.$ Therefore, $Cov(x_{obs,i}^{T}\beta,x_{obs,i}^{T}\gamma \left\vert x_{obs1,i}\right) < 0$ as $Cov(x_{omit,i}^{T}\beta,x_{omiti}^{T}\gamma \left\vert x_{obs1,i}\right) < 0.$