Interpreting the fitted logistic regression model

In the logistic regression model in Table 5.1, the slope coefficient of an independent variable

represents the change in the logit corresponding to a change of one unit in the independent

variable. An increase of one unit in a continuous variable, however, is too large for some

continuous variables (e.g., financial ratios) and yet too small for other continuous variables

(e.g., age) to be meaningful. To indicate how the risk of failure changes with the explanatory

variables, I have calculated the effect of an arbitrary change of “c” unit in the continuous

variables in Table 5.2. For a change of “c” unit in a continuous variable, the change in the log-

odds ratio is . The associated odds ratio is and the endpoints

Variable Unit SE 95% confidence interval of odd ratio Age 10 -0.010 0.904 0.000 (0.903, 0.913) Size 1 -0.111 0.895 0.004 (0.887, 0.902) Cash/total assets 0.25 -1.440 0.698 0.044 (0.683, 0.713) Creditors/total liabilities 0.25 1.011 1.288 0.022 (1.274, 1.302) EBIT/total assets 0.25 -0.648 0.851 0.037 (0.835, 0.866)

Fixed assets/total assets 0.25 -1.047 0.770 0.030 (0.759, 0.781) Loans/total assets 0.25 0.301 1.078 0.032 (1.061, 1.095)

Total liabilities/total assets 0.25 0.339 1.088 0.026 (1.075, 1.102) Working capital/total assets 0.25 -0.636 0.853 0.026 (0.842, 0.864) Agriculture, forestry and fishing

Mining 1 0.140 1.150 0.126 (0.899, 1.471)

Construction 1 0.406 1.501 0.063 (1.327, 1.698)

Manufacturing 1 0.649 1.914 0.063 (1.693, 2.164)

Transportation & communications 1 0.458 1.581 0.066 (1.389, 1.800)

Wholesale trade 1 0.263 1.301 0.063 (1.149, 1.472)

Retail trade 1 0.423 1.526 0.064 (1.347, 1.729)

Finance, insurance and real estate 1 0.492 1.636 0.063 (1.445, 1.852)

Services 1 0.344 1.411 0.063 (1.247, 1.596)

Public administration 1 0.817 2.263 0.156 (1.666, 3.073) Western Europe

Northern Europe 1 -0.172 0.842 0.042 (0.775, 0.915)

Southern Europe 1 1.395 4.036 0.015 (3.918, 4.157)

South Eastern Europe 1 -0.147 0.863 0.038 (0.800, 0.931)

Eastern Europe 1 0.917 2.503 0.030 (2.358, 2.657)

International reserves per head 1 -1.544 0.213 0.010 (0.209, 0.218) Trade balance (% of GDP) 1 0.181 1.198 0.002 (1.194, 1.202)

Unemployment rates 1 -0.268 0.765 0.002 (0.761, 0.768)

Table 5.2 suggests that for an increase of 10 years in firm age, the odds of failure are estimated

to decrease by 0.904 times and the decrease ranges from 0.903 to 0.913 times at a 95%

confidence interval. For one unit increase in natural logarithm of total assets, the odds of failure

decrease by 0.895 times and the decrease can be as little as 0.902 times or much as 0.887 times.

total assets reduces the odds of failure by 0.851, 0.770 and 0.853 times respectively. Table 5.2

provides the 95% confidence intervals for these estimated odds ratios. While the ratios of cash

to total assets, EBIT to total assets, fixed assets to total assets and working capital to total assets

reduce the risk of failure, other financial ratios increase the likelihood of failure. An increase

of 0.25 in the ratio of creditors to total liabilities will lead to an increase in the odds of failure

by 1.288 times, and at a 95% confidence interval, the increase is between 1.274 and 1.302

times. Similarly, an increase of 0.25 in the ratios of loans to total assets and total liabilities to

total assets increases the odds of failure by 1.078 and 1.088 times respectively. Table 5.2 also

reports the effect of one unit increase in every macroeconomic variable on the risk of failure.

An increase of USD 1000 in international reserves per head reduces the odds of failure by 0.213

times, and the decrease is between 0.209 and 0.218 times at a 95% confidence interval. With

an increase of 1% in trade balance (% of GDP), the odds of failure increase by 1.198 times and

the increase ranges from 1.194 to 1.202 times at a 95% confidence interval. For an increase of

1% in unemployment rates, the odds of failure decrease by 0.765 times, and the decrease is

between 0.761 and 0.768 times at a 95% confidence interval.

Table 5.2 also presents the odds ratios for the dummy variables for industry and region. The

independent variable of region has five categories: Western Europe, Northern Europe, Southern

Europe, and South Eastern Europe, and Eastern Europe. According to the odds ratio of

, , the odds of failure in Northern European firms

are 0.842 times the odds of failure in Western European firms. At a 95% level of confidence,

the odds of failure in Northern European firms could be as little as 0.915 times or much as

0.775 times the odds of failure in Western European firms. One can interpret the odds ratios

services, and 10) public administration. The estimated coefficients for the industry dummy

variables except mining are statistically significant at the 1% level. The odds ratio of

OR services, agriculture, forestry &fishing , for example, is 1.411, suggesting that the odds of failure among firms in services are 1.411 times the odds of failure among firms in

agriculture, forestry and fishing. The 95% confidence interval indicates that the odds of failure

among firms in services could be as little as 1.247 times or much as 1.596 times the odds for

those firms in agriculture, forestry and fishing. A similar interpretation can be made of the odds

ratios for the dummy variables for other industries and their 95% confidence intervals.

Because of the above simple relationship between the estimated coefficients and the odds

ratios, logistic regression has proved to be a very highly interpretable tool. An alternative way

to assess the effect of an independent variable on the probability of failure is to examine its

marginal effect. The marginal effect (also called the partial change in the probability) measures

the expected instantaneous change in the event probability as a function of a change in an

independent variable when all other independent variables are held constant. The marginal

effect is contingent on the values of all predictors and their estimated coefficients; therefore,

one has to decide on the levels of the independent variables in the computation of the marginal

effect. Long (1997, pp. 72-74) compares two approaches. The first approach is to compute the

marginal effect at every observation and then calculate the sample average. The second

approach is to compute the marginal effect at the mean of the predictors. The second approach

has several problems. First is the difficulty in the translation of a marginal effect into the change

in the probability in case of a discrete change in an independent variable. Secondly, mean

the marginal effect of -0.0006 indicates that an increase of one unit in age reduces the

probability of failure by 0.0006. The marginal effect of size is -0.0062, suggesting that an

increase of one unit in size decreases the probability of failure by 0.0062. In a similar way, the

marginal effects of financial ratios and macroeconomic variables indicate that an increase of

one unit in the ratios of cash to total assets, EBIT to total assets, fixed assets to total assets and

working capital to total assets, international reserves per head, and unemployment rates

decreases the probability of failure by 0.0800, 0.0360, 0.0582, 0.0353, 0.0858 and 0.0149

respectively. In contrast, an increase of one unit in the ratios of creditors to total liabilities,

loans to total assets and total liabilities to total assets, and trade balance (% of GDP) increases

the probability of failure by 0.0562, 0.0167, 0.0188 and 0.0100 respectively. The marginal

effect of the construction dummy variable is 0.0247, suggesting that firms in construction is

0.0247 more likely to fail than firms in agriculture, forestry and fishing (the reference industry).

A similar interpretation can be made of the marginal effects of the dummy variables for other

industries except that the marginal effect of the mining dummy variable is not statistically

significant at the 10% level. Interestingly, firms in all other industries are more likely to fail

than firms in the agriculture, forestry and fishing industry. For the dummy variables for

Northern Europe and South Eastern Europe, their marginal effects suggest that firms in

Northern Europe and South Eastern Europe are 0.0090 and 0.0078 respectively less likely to

fail than firms in Western Europe (the reference category). In contrast, firms in South Europe

and Eastern Europe are 0.0784 and 0.0664 respectively more likely to fail than firms in Western

Europe.