Assessing the size of effect of management practices on financial performance

In a second step, a regression model was set up in order to assess the magnitude of the impact of these practices on financial performance, following the classical equation structure:

𝑌 = 𝛽₀+ ∑ 𝛽_𝑖𝑗 𝐾

𝑖=1

𝑋_𝑖𝑗+ 𝜀_𝑖

with 𝑌 being the independent variable, 𝛽₀ the intercept with the Y-axis, 𝑋_𝑖𝑗 the independent variables, 𝛽𝑖𝑗 the parameters and 𝜀 the error terms.

The purpose of a regression analysis is to estimate relationships between variables. It is often applied to predict or forecast outcomes and/or effects of events. The econometric model used to link management practices and financial performance can be specified with the following equation:

𝐹𝑃𝐼_𝑖 = 𝛽₀+ ∑ 𝛽_𝑖𝑗 𝐾

𝑖=1

𝑋_𝑖𝑗 + 𝜀_𝑖

Where 𝐹𝑃𝐼𝑖 is the financial performance indicator for farm 𝑖 and 𝐹𝑃𝐼𝑖 ∈ {𝑅𝑜𝑆𝑖, 𝐴𝑇𝑂𝑖 , 𝑅𝑜𝐴𝑖,𝑅𝑜𝐸𝑖}; 𝑋𝑖𝑗 are the explanatory variables such as formal business planning, cash

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flow planning, benchmarking, education, accessing advice, size, tenure, age and having a working spouse9; 𝛽_𝑖𝑗 are the coefficients to be estimated; and 𝐾 is the number of coefficients.

There are several econometric models that can be used for this study (Carter-Hill et al., 2011). When the dependent variable is limited (binary, categorical or ordered), discrete choice models need to be used such as:

 The probit and logit model, which is used when the dependent variable takes only two values (0 or 1) – this is the case when the farms would for example be classified in successful/unsuccessful categories, and the regression would be run on this categorisation (e.g. top 50% and bottom 50% of the sample);

 The multivariate probit model, which estimates the joint relationship between several dependent variables and independent variables; and

 Ordered probit/logit model, which is used when the independent variable has more than 2 options - which would be the case if the ratios are classified into categories (for example 20% percentiles or 5 categories).

When the dependent variable is continuous, there are, among others, the following options:

 Simple linear regression, when there is only one independent variable;

 Multiple linear regression, when there are at least 2 independent variables; and

 Tobit (Heckit) model, when the dependent variable is truncated.

The section below describes the model used in this research, namely a multiple linear regression model, as the dependent variable is continuous and not truncated and there are several independent variables. The discrete models were tested, and rejected as too much of the detail was lost with the conversion of financial performance to successful and unsuccessful farms, resulting in models with few significant variables. For RoS, RoA and RoE, a multiple linear regression model was estimated, as the ratios are continuous and there are several independent variables. For ATO, the lower limit of the data is 0, as sales are generally not negative. This data is, however, not truncated or censored, so a multiple linear regression model is also appropriate for use.

61 3.3.4. The multiple linear regression model

There are six assumptions underpinning the multiple linear regression model (Carter-Hill et al., 2011):

1. There is a linear relationship between the dependent variable and independent variables10_:

𝑦 = 𝛽₀+ ∑ 𝛽_𝑖𝑥_𝑖𝑗+ 𝑒_𝑖 𝐾

𝑖𝑗

2. The expected value of the error term is 0 (exogeneity):

𝐸(𝑒) = 0 ↔ 𝐸(𝑦) = 𝛽₀+ ∑ 𝛽_𝑖𝑥_𝑖𝑗 𝐾

𝑖𝑗

3. The variance of the error term is constant (homogeneity of variance or homoscedasticity):

𝑣𝑎𝑟(𝑒) = 𝜎2_𝐼

𝑛 = 𝑣𝑎𝑟 (𝑦)

with 𝐼_𝑛 a n x n identity matrix.

4. The covariance between errors is zero – the errors of one observation are not correlated with the errors of other observations (independence or not correlated):

𝑐𝑜𝑣( 𝑒𝑖, 𝑒𝑗) = 𝑐𝑜𝑣 (𝑦𝑖, 𝑦𝑗) = 0

5. The variables 𝑥_𝑡𝐾 are not random, and are not exact linear functions of the other explanatory or independent variables (no linear dependence or collinearity)

6. The values of 𝜀 are normally distributed about their mean: 𝑒 ~ 𝑁( 0, 𝜎2)

10 _{If a non-linear relationship exists, transformation of the dependent and/or independent variables is sometimes}

feasible in order to achieve a linear relationship. These include a log-log model, a log-linear model and a linear- log model, where either the dependent and/or independent variables are transformed.

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If the assumptions stated above hold, the estimation of the parameters is based on the ordinary least squares (OLS) principle that minimises the sum of squared residuals, resulting in the best linear unbiased estimators (BLUE) of the parameters 𝛽, calculated with the following formula:

if 𝑆𝑆𝐸 = ∑𝑁_𝑖=1𝑒̂_𝑖2 and 𝑆𝑆𝐸∗ = ∑𝑁_𝑖=1𝑒̂_𝑖∗2 then 𝑆𝑆𝐸 < 𝑆𝑆𝐸∗

𝛽̂ = (𝑋′_𝑋)−1_𝑋′𝑌

with 𝑆𝑆𝐸 being the sum of squares due to the error, and 𝛽̂ the best linear unbiased according to the Gauss-Markov theorem (Carter-Hill et al., 2011).

When the variance of the errors is not constant (there is heteroskedasticity) or when the errors are correlated, resulting in a violation of the assumptions mentioned above, parameters can be estimated through the Generalised Least Squares (GLS) method (Cameron & Trivedi, 2010):

𝑉(𝑒|𝑋) = 𝜎2Ω

With Ω being a symmetric, positive definite n x n matrix. In this case, the Gauss-Markov theorem does not hold anymore, and the estimated 𝛽 are inefficiently estimated, even though they are unbiased.

The GLS method will overcome this issue, and will result in efficient estimates, with smaller standard errors and larger t-statistics:

𝛽̂ = (𝑋𝑔𝑙𝑠 ∗′𝑋∗)−1𝑋∗′𝑌∗ = (𝑋′Ω−1𝑋)𝑋′Ω−1𝑌

When Ω is equal to 𝐼_𝑛, then 𝛽̂_𝑔𝑙𝑠 will be equal to 𝛽̂.

Special cases of GLS are weighted least squares (WLS) or feasible generalised least squares (FGLS). When there is heteroskedasticity present in the model, but no correlation between the errors, WLS is used; while FGLS when there is heteroskedasticity and serial correlation (Greene, 2003).

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Goodness of fit of the model is tested with R-squared (𝑅2), which indicates how much of the variation in the sample is explained by the regression model:

𝑅2 = ∑(𝑦̂ − 𝑦̅)𝑖 2

∑(𝑦𝑖 − 𝑦̅)2

= 1 −𝑆𝑆𝐸

𝑇𝑆𝑆

with 𝑇𝑆𝑆 the total sum of squares for 𝑦 and 𝑆𝑆𝐸 the sum of squares due to the residuals. When none of the variance in the dataset is explained by the model, the R-squared is 0; while an R- squared of 1 means that the model is a perfect fit (all variance in the sample is explained by the model). The rule, therefore, is the higher the R-squared, the better.

Measuring goodness of fit can also be assessed through Aikake’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC), when the R-squared is not available, and as there are issues with the interpretation of Pseudo R-squared calculations (Williams, 2015). They allow for a comparison between different model estimates instead of generating the absolute deviation of observed data in a model.

They are calculated as follows:

𝐴𝐼𝐶 = 𝐷𝐸𝑉𝑀 + 2𝑃

and

𝐵𝐼𝐶 = 𝐷𝐸𝑉_𝑀 + ln(𝑁) ∗ 𝑃

with P being the number of estimated parameters, N the number of observations, and 𝐷𝐸𝑉_𝑀 is equal to −2 ∗ the log-likelihood of the model (𝐿𝐿_𝑀). For both criteria the rule is the smaller the result, the better the fit.