Question: Manipulating rates of return with stock splits. Some firms have been accused of using stock splits to manipulate their stock prices before being acquired by another firm. An article in Financial Management (Winter 2008) investigated the impact of stock splits on long-run stock performance for acquiring firms. A simplified version of the model fit by the researchers follows:

$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$

where

y = Firm’s 3-year buy-and-hold return rate (%)

x₁ = {1 if stock split prior to acquisition, 0 if not}

x₂ = {1 if firm’s discretionary accrual is high, 0 if discretionary accrual is low}

a. In terms of the β’s in the model, what is the mean buy and- hold return rate (BAR) for a firm with no stock split and a high discretionary accrual (DA)?

b. In terms of the β’s in the model, what is the mean BAR for a firm with no stock split and a low DA?

c. For firms with no stock split, find the difference between the mean BAR for firms with high and low DA. (Hint: Use your answers to parts a and b.)

d. Repeat part c for firms with a stock split.

e. Note that the differences, parts c and d, are not the same. Explain why this illustrates the notion of interaction between x₁ and x₂.

f. A test for H₀: β₃ = 0 yielded a p-value of 0.027. Using α = .05, interpret this result.

g. The researchers reported that the estimated values of both β₂ and β₃ are negative. Consequently, they conclude that “high-DA acquirers perform worse compared with low-DA acquirers. Moreover, the underperformance is even greater if high-DA acquirers have a stock split before acquisition.” Do you agree?

In terms of β’s in the model, the mean buy-and-hold return rate (BAR) for a firm with no stock split and a high discretionary accrual (DA) can be computed when the value of x₁= 0 and x₂ = 1;

Therefore, in terms of β’s in the model, mean BAR = (β₀ + β_2­)

In terms of β’s in the model, the mean buy-and-hold return rate (BAR) for a firm with no stock split and a low discretionary accrual (DA) can be computed when the value of x₁= 0 and x₂ = 0, this essentially is the base level;

Therefore, in terms of β’s in the model, the mean BAR = β₀

For firms with no stock split, the difference between the mean BAR for firms with high and low DA can be computed by subtracting the mean BAR level for firms with no stock split and high DA with the firms with mean BAR level with no stock split and low DA;

mean BAR for firms with no stock split and high DA = (β₀ + β_2­)

mean BAR for firms with no stock split and low DA = β₀

Therefore, mean Bar level for firms with high and low DA = (β₀ + β_2­) - β₀= β₂

For firms with no DA, the difference between the mean BAR for firms with stock split and no stock split can be computed by subtracting the mean BAR level for firms with stock split and no DA with the firms with mean BAR level with no stock split and no DA;

mean BAR for firms with stock split and no DA = (β₀ + β_1­)

mean BAR for firms with no stock split and low DA = β₀

Therefore, mean Bar level for firms with stock split and no stock split = (β₀ + β_1­) - β₀= β₁

The answers in part c and part d are different, this might indicate towards some interaction amongst the two variable of stock split and discretionary accrual. Interaction exists in a model when there is some correlation amongst the two variable.

H₀: β₃ = 0 while H_a: β₃ ≠ 0 yielded a p-value of 0.027. Using α

Here, H₀ is rejected if p-value < α. Since p-value = 0.027 and α = 0.05

There’s sufficient evidence to reject H₀ at 95% confidence level

Therefore, β₃ = 0 meaning there’s no interaction between x₁ and x_2.

Negative sign of the β₂ value denotes that there’s an inverse relation between BAR and discretionary accrual meaning that for the value of x₂ = 1, the base level of x₂ = 0 will have higher value.

Similarly, for Negative sign of the β₃ value denotes that there’s an inverse relation between BAR and the interaction between discretionary accrual and stock split indicating that high DA acquirers with stock splits underperforms than low DA acquirers with stock split.