Example: Takeover Bids

Jaggia and Thosar (1993) model the number of bids received by 126 U.S. firms that were targets of tender offers during the period from 1978 through 1985 and were actually taken over within 52 weeks of the initial offer. The dependent

Table 5.1. Takeover bids: actual frequency distribution

Count 0 1 2 3 4 5 6 7 8 9 10

Frequency 9 63 31 12 6 1 2 1 0 0 1

Relative frequency .071 .500 .246 .095 .048 .008 .016 .008 .001 .000 .008

Table 5.2. Takeover bids: variable definitions and summary statistics

Standard

Variable Definition Mean deviation

NUMBIDS Number of takeover bids 1.738 1.432

LEGLREST Equals 1 if legal defense by lawsuit .429 .497

REALREST Equals 1 if proposed changes in asset structure .183 .388

FINREST Equals 1 if proposed changes in ownership structure .103 .305

WHITEKNT Equals 1 if management invitation for friendly .595 .493 third-party bid

BIDPREM Bid price divided by price 14 working days before bid 1.347 .189

INSTHOLD Percentage of stock held by institutions .252 .186

SIZE Total book value of assets in billion of dollars 1.219 3.097

SIZESQ SIZE squared 10.999 59.915

REGULATN Equals 1 if chronic condition limiting activity .270 .446

count variable is the number of bids after the initial bid (NUMBIDS) received by the target firm. These data are also analyzed at the end of section 5.3.4.

Data on the number of bids are given in Table 5.1. Less than 10% of the firms received zero bids, one half of the firms received exactly one bid (after the initial bid), a further one quarter received exactly two bids, and the remainder of the sample received between three and ten bids. The mean number of bids is 1.738 and the sample variance is 2.050. This is only a small amount of overdispersion (2.050/1.738 = 1.18), which can be expected to disappear as regressors are added.

The variables are defined and summary statistics given in Table 5.2. Regres- sors can be grouped into three categories: (1) defensive actions taken by man- agement of the target firm: LEGLREST, REALREST, FINREST, WHITEKNT; (2) firm-specific characteristics: BIDPREM, INSTHOLD, SIZE, SIZESQ; and (3) intervention by federal regulators: REGULATN. The defensive action vari- ables are expected to decrease the number of bids, aside from WHITEKNT, which may increase bids as it is itself a bid. With greater institutional hold- ings it is expected that outside offers are more likely to be favorably received, which encourages more bids. As size of the firm increases there are expected to be more bids, up to a point where the firm gets so large that few others are capable of making a credible bid. This is captured by the quadratic in firm size. Regulator intervention is likely to discourage bids.

Table 5.3. Takeover bids: PoissonPMLEwith NB1standard errors and t ratios

Poisson PMLE

Variable Coefficient Standard errors t statistic

ONE .986 .461 2.14 LEGLREST .260 .130 2.00 REALREST −.196 .166 −1.18 FINREST .074 .187 .40 WHITEKNT .481 .137 3.51 BIDPREM −.678 .326 −2.08 INSTHOLD −.362 .367 −.99 SIZE .179 .052 3.44 SIZESQ −.008 .003 −2.81 REGULATN −.029 .139 −.21 −lnL 185.0

and t statistics assuming anNB1variance function. The estimated value of the overdispersion parameter α is 0.746, which is considerably less than unity. At the same time, a formal test of underdispersion using theLMtest does not reject the null hypothesis of no overdispersion, leading Jaggia and Thosar (1993) to prefer the Poisson estimator.

The defensive action variables are generally statistically insignificant at 5% except for LEGLREST, which actually has an unexpected positive effect. While the coefficient of WHITEKNT is statistically different from zero at 5%, its coefficient implies that the number of bids increases by 0.481 × 1.738  .84 of a bid. This effect is not statistically significantly different from unity. (If a white-knight bid has no effect on bids by other potential bidders we expect it to increase the number of bids by one.) The firm-specific characteristics with the exception of INSTHOLD are statistically significant with the expected signs. BIDPREM has a relatively modest effect, with an increase in the bid premium of 0.2, which is approximately one standard deviation of BIDPREM, or 20%, leading to a decrease of 0.2 × 0.677 × 1.738  .24 in the number of bids. Bids first increase and then decrease as firm size increases. Government-regulator intervention has very little effect on the number of bids.

Summary statistics for different definitions of residuals from the same Pois- sonPMLestimates are given in Table 5.4. These residuals are the raw, Pearson, deviance, and Anscombe residuals defined in, respectively, (5.1), (5.2), (5.4), and (5.6), small-sample corrected or studentized Pearson and deviance residuals (5.13) and (5.14) obtained by division by√1− hii, and the adjusted deviance

residual (5.15).

The various residuals are intended to be closer to normality, that is, with no skewness and kurtosis equal to 3, than the raw residual if the data areP[µi].

Table 5.4. Takeover bids: descriptive statistics for various residuals

Standard

Residual Mean deviation Skewness Kurtosis Minimum 10% 90% Maximum r .00 1.23 1.4 7.4 −3.22 −1.30 1.27 5.57 p .00 .83 1.1 4.9 −1.61 −.96 .99 3.03 p∗ −.00 .89 1.1 5.1 −1.87 −1.02 1.02 3.11 d −.05 .96 .7 9.4 −3.65 −.95 .56 4.07 d∗ −.05 1.03 .7 9.7 −3.80 −1.06 .58 4.28 dadj .09 .96 .6 9.3 −3.55 −.86 .69 4.19 a −.10 .85 .2 3.9 −2.41 −1.16 .89 2.41

Note: r, raw; p, Pearson; p∗, studentized Pearson; d, deviance; d∗, studentized deviance; dadj, adjusted deviance; a, Anscombe residual.

Table 5.5. Takeover bids: correlations of various residuals

Residual r p p∗ d d∗ dadj a r 1.000 p .976 1.000 p∗ .983 .998 1.000 d .919 .917 .918 1.000 d∗ .925 .913 .918 .998 1.000 dadj .920 .917 .919 1.000 .988 1.000 a .951 .980 .977 .934 .928 .934 1.000

Note: r, raw; p, Pearson; p∗, studentized Pearson; d, deviance; d∗, studentized de- viance; dadj, adjusted deviance; a, Anscombe residual.

the deviance residual, which has quite high kurtosis. The Anscombe residual is clearly preferred on these criteria. Studentizing makes little difference. It is expected that it will make little difference for most observations, because the average hii = 10/126 = .079 leading to a small correction. For this sample

even the second largest value of hii = .321 only leads to division of Pearson

and deviance residuals by .82, not greatly different from unity.

The similarity between the residuals is also apparent from Table 5.5, which gives correlation amongst the various residuals. The correlations between the residuals are all in excess of 0.9, and small-sample corrected residuals have correlation of 0.998 or more with the corresponding uncorrected residual.

We conclude that for this sample the various residuals should all tell a sim- ilar story. We focus on the Anscombe residual ai, since this is the closest to

normality. Various residual plots are presented in Figure 5.2.

Panel 1 of Figure 5.2 plots the Anscombe residual against the dependent variable. This shows the expected positive relationship, explained earlier. It is better to plot against the predicted mean, which is done in panel 2. It is difficult to visually detect a relationship.

Figure 5.2. Takeover bids: residual plots.

A normal score plot of the Anscombe residual, that is, a plot of the residual against the prediction (5.11) if the residual is normally distributed, is given in panel 3 of Figure 5.2. The relationship is close to linear, although the high values of the Anscombe residuals are above the line, suggesting higher-than-expected residuals for large values of the dependent variable.

The hat matrix defined in (5.12) can be used for detecting influential ob- servations. A plot of the ith _{diagonal entry h}

ii against observation number is

given in panel 4 of Figure 5.2. For this sample there are six observations with hii > 3k/n = .24. These are observations 36, 80, 83, 85, 102, and 126 with

hii of, respectively, .28, .32, .70, .32, .28, and .30. If instead we use theOLS

leverage measures, H= X(XX)−1X, the corresponding diagonal entries are .18, .27, .45, .58, .18, and .16, so that one would come to similar conclusions.

On dropping these six observations the coefficients of the most highly statis- tically significant variables change by around 30%. The major differences are a change in the sign of SIZESQ, and that both SIZE and SIZESQ become very statistically insignificant. Further investigation of the data reveals that these six observations are for the six largest firms, and that the size distribution has a very fat tail with the kurtosis of SIZE equal to 31, explaining the high leverage of these observations. The leverage measures very strongly alert one to the prob- lem, but the solution is not so clear. Dropping the observations with large SIZE

is not desirable if one wants to test the hypothesis that, other things being equal, very large firms attract fewer bids than medium-size firms. Different functional forms for SIZE might be considered, such as log(SIZE) and its square, or an indicator variable for large firms might be used.

For this example there is little difference in the usefulness of the various standardized residuals. The sample size with 126 observations regressors is relatively small for statistical inference based on asymptotic theory, especially with 10 regressors, yet is sufficiently large that small-sample corrections made virtually no difference to the residuals. Using the hat matrix to detect influential observations was useful in suggesting possible changes to the functional form of the model.