Interest rate spreads the likelihood function

What interest rate spreads are, and how to effectively model them, are some of the challenges that have been recognised by many authors on this topic.116 For example, Birchwood (2004) and Brock and Franken (2002) acknowledged this problem and all expressed dissatisfaction at the lack of an agreed general framework in the literature on how to model interest rate spreads. We find that empirical analyses based on the theory of the firm had skewed micro level analysis while neglecting the role of macro and financial fundamentals in interest rate spreads. For example, the Marshal-Lerner condition equation emphasised the microeconomic factors without accounting for the contribution of macroeconomic fundamentals realised. Another common definition used to model interest rate spreads covers the interest earned and interest rate paid accounting identity – this is called net interest rate. Net interest rate spread is the difference between interest earned on loans, securities and other interest-earning assets and the interest paid on deposits and other interest-bearing liabilities. We disagree with the use of accounting identity on the following grounds. Firstly, because it is an identity, therefore it implies that the identity conditions must hold at any time irrespective of whatever is happening.117 Thus, modelling this as an equation without formulating the assumptions that turns the identity into a

116 _{It has been recognised that both ex ante and ex post definitions of interest rate spreads have their}

weaknesses.

stochastic equation is methodologically wrong. Secondly, the interest rate earnings minus interest rate income identity emphasises the balance sheet variables more than relating it to other factors that do not feature directly in the balance sheets of financial firms. Thus, it is very difficult to relate variables that do not directly link firm balance sheets to this accounting identity. Taking into account all these challenges we decided to use the implicit formulation by Woodford and Curdia (2009) and the linear model used by Classens et al. (2006) to come up with the likelihood function of how to model interest rate spreads. As in Curdia and Woodford (2009) and Classens et al. (2006) we proposed that interest rate spreads is a function of the average cost of originating the loan, the price of credit from central bank, perceived risk, income, and other macroeconomic and financial fundamental realised in the country under study. This relation should also include structural dummies to take into account structural changes and transitions from one regime to another over sample period. We expressed this economic relation as follows:

∆(c− ) =

p¡jžM jM + ∆Ojp¡€ + ∆r€pOV€k Ož¢ + ∆( p¡&¼Oj jpO ~ ¼Ejk €jM ~ž) +

OjM€€žM kO¼¼€€jMO ~ + žMEpME ~ kEO€ž + €¡ M€ (3.8).

In order to take into account the unit root process and endogenous structural breaks manifested in some of the spread variables we will start from the first difference with pulse dummy variables. It is important that we highlight some of the challenges usually encountered in estimating a single equation such as this and how we propose to overcome them. The first immediate challenge relates to uncertainty around the true functional form of interest rate spreads. This refers to whether the true relationship between changes in the spreads and the determinants should be treated as a linear or non-linear relationship. Similarly, since the dependent variable is a time series the functional form should take into account the dynamic structure of the dependent variable and the length of memory in the average spreads; that is, the lag length to address autocorrelation. The next challenge concerns how to deal with regime shifts and the unit root process with structural breaks as has been observed in some of the interest rate spreads in Namibia. Lastly, we need to determine the list of covariates to include in the model while avoiding over fitting.

Firstly, the issue of linear or non-linear functional form can be handled very well by using a Generalised Method of moments while other tests such as the reset test can be used to check the adequacy of the model. Holly and Turner (2012, p. 21) pointed out that

‘the main advantage of GMM estimator is that we don’t have to write down a conventional regression relationship. Instead we can specify an implicit relationship between variables.’ In addition, this allows techniques to minimise the problem of multicollinearity through instruments while bearing in mind that the model is less restrictive on the data generating process. However, we must highlight other methods that are adequate in analysing the problem of interest rate spreads. These methods include the Stepwise least squares method, the smooth transition auto-regressive (STAR) model and the logistic smoothed transition model (LSTAR). STAR and LSTAR methods estimate simultaneously estimate the linear and non-linear part of the dependent variable with the ability to identify whether the non- linear part is statistically significant. In addition, the STAR model accounts for structural breaks and the transition function, and whether the transition is governed by logistic or exponential functions from one regime to another. However, STAR models do fail when both linear and non-linear parts exhibit the unit root process with structural breaks. This seems to be the case with the interest rate spreads in Namibia whereby even after identifying and including the endogenous structural break in the dependent variables (i.e. spreads) these still do not pass the unit root test with structural breaks. This implies that augmenting the process with structural breaks does not make the time series variable stationary. Thus, we differenced the variables involved in the regression and used the impulse dummy rather than the level shift dummy.118

Further we adjust the single equation to account for the timing effects and the memory of the dependent variable. It is therefore essential to append some macroeconomic fundamentals to lags the in the list of independent variables. The long memory in the dependent variable will require the use of the Auto-regression Integrated Moving Average model with exogenous variables (ARIMAX). However, the ARIMAX model is limited to how many exogenous variables should enter because too many with their lags make it very difficult to establish which should be in or out even with a Granger causality test. In the case of GMM and related estimators this task is simplified by omitted variables and redundant variable tests to avoid over fitting the model. After establishing that there is no serial correlation, insignificant independent variables can be assessed jointly and individually in regard to whether they are redundant in the final regression. Finally, we used two unconditional inflation and interest rate volatilities measures, which summarise the factor effects of macroeconomic and financial instability on interest rate spreads. Other

118 _{Another alternative method for regime shifts is the stepwise regression method. Although this method}

may partially address the problem of regime shift, it is argued that it is too subjective and the outcomes are either over fitted regressions with less optimum results.

factors such as the changes in income, interest rate, financial depth and perceived risk are theoretically suggested by Woodford and Curdia (2009) and Groth (2012, p. 1) as factors that contribute to the variation of interest rate spreads.

In all, we aimed to overcome the quandary about modelling spreads as we opt for the Generalised Method of Moments to investigate the factors that seem to explain interest rate spreads in Namibia. GMM requires less information about the exact mathematical relations of the problem that needs to be examined. Therefore, in a situation whereby we have less information about the likelihood functions (that is, an explicit linear or non-linear function that describes interest rate spreads), the GMM approach is an appropriate tool to estimate the partially specified economic models and the results from a single equation can be examined for consistency when results from two closely related estimators OLS and TSLS are estimated alongside.119