In-Sample Goodness-of-Fit

As Wooldridge (2004) notes, forecasting models require a measure of goodness-of-fit within the sample used to obtain the parameter estimates. This is usually the adjusted R2, which is reported widely in the existing research literature, and here an F-test and Voung‟s likelihood ratio test are also used. The computation of these statistics is explained in greater detail below.

Adjusted R2

In order to determine the goodness of fit of regression models, the present research study reports adjusted R2 as an in-sample measure of the degree of association between future operating cash flow and the current cash flow, the current earnings or the disaggregation of earnings into cash flow, accruals and their components. The adjusted R2 is computed as follows:

= 1-

(3.5)

where:

RSS = Residual sum of squares TSS = Total sum of squares

n = Number of observations

k = Number of independent variables plus intercept

It is worth noting that, in least squares regression, R2 increases weakly with the number of regressors used in the model. Thus, R2 cannot be used alone as a meaningful comparison of models with different numbers of independent variables; an F-test should be carried out on the residual sum of squares, as discussed below.

F- Statistic

As noted, whilst the model with the highest adjusted R2 amongst other models should be selected as the best model, adding a variable to a model may increase adjusted R2 without reducing the residual sum of squares (Gujarati, 2004). The following F-test is therefore recommended when adding a variable to a regression model:

F=

where:

m = the number of new regressors n = the number of observations

k = the number of parameters in the new model, Gujarati (2004, p.263)

A significant F-statistic indicates that the added variable increases explanatory power, and is used to compare nested models. A different approach is required to test non-nested models, as discussed in the next section.

Voung‟s (1989) Likelihood Ratio test

According to Gujarati (2004), there are two main approaches to testing non-nested models, broadly characterised as the „discriminating‟ approach and the „discerning‟ approach, which are defined as follows:

(1) the discrimination approach, where given two or more competing models, one chooses a model based on some criteria of goodness of fit [such as the adjusted R2], and (2) the discerning approach where, in investigating one model, we take into account information provided by other models.

In fact, there are several of the latter tests of model selection in the econometrics literature, such as the Davidson-MacKinnon J-test, Cox‟s test, and the Mizon-Richard test (see Gujarati, 2004: p. 536). It should be noted that, to select the best model in this case, each of these tests will consider the attributes of opponent models. Accounting research into cash flow prediction has used Voung‟s Likelihood Ratio test for model selection (e.g. Dechow 1994 and Barth et al 2001). Davidson-MacKinnon‟s J-test is sometimes suggested to overcome problems in the non-nested F testing method, (see Gujarati, 2004, P.533); however, Dechow (1994) has noted that when the explanatory

power of variables is incremental, the J-test may not be powerful and cannot make a distinction between the competing models. Hence, Voung‟s test is a more powerful test than the J-test.

In investigating the role of accounting accruals in the measurement of firm performance, and in order to compare competing models, Dechow (1994) explained Voung‟s Likelihood Ratio test of non-nested model selection as follows:

[Voung‟s (1989)] has provided a likelihood ratio test for model selection to test the null hypothesis that the two models are equally close to explaining the ‘true data generating process’ against the alternative that one model is closer to the ‘true data generating process’. Therefore, the Voung test allows rejection of cash flows in favour of earnings in situations where ambiguous results would otherwise be obtained.

With the Voung test, a model is superior to another model when the log likelihood is higher than the log likelihood for the model(s) considered. Following Dechow (1994), the Likelihood Ratio test is computed as follows:

 First, the differences in log-likelihoods between the two models is calculated as: LR= log [L (Ma)] - log [L (Mb)] (3.7)

 In a second step, the variance of LR, ω2

is estimated by the following equation:

ω2 2 2 2 2 2 2 2 1 ( ) ( ) 1 1 1 1 1 1 log( ) log( ) LR 2 2 2 2 n bi ai b a b a i e e n _{e e} n   _{  }       _{  }  _{  }  



e = estimated residuals under either model

 Finally, based on the following equation, Voung‟s Z-statistic is calculated as: Z =

(3.9) This Z-statistic can be interpreted such that, if it is significantly positive, model

b has higher explanatory power with respect to model a. If the Z-statistic is significantly

negative, it indicates that model a should be selected, and a non-significant Z-statistic

implies that there is no difference in explanatory power between the two models. Accordingly, this thesis employs Voung‟s (1989) Likelihood Ratio test to identify the explanatory power of the four cash flow prediction models that have been introduced above.