5.5 Variable Transformations

5.5.1	The principles of variable transformation articulated in the MMS also apply to macro models. Variable transformations have a material impact on macro models and on ECL. Therefore, institutions should test, choose and document the most appropriate transformations applied to both the macro variables and to the dependent variable.
5.5.2	Stationarity: Modelled economic relationship should be stable over time. In the context of time series regression model, variables should be stationary in order to construct robust and meaningful econometric models. Stochastic trends, seasonality and structural breaks are most common sources of non-stationarity. This property should be tested for both the dependent and independent variables, according to the following principles:
	(i)	Macroeconomic variables should meet stationarity criteria prior to be used for modelling. The absence of stationarity has adverse material consequences on macro models because it often leads to spurious correlations. Macro variables that are not stationary should either be transformed to obtain stationary series or should be excluded from the modelling process.
	(ii)	Even after transformations, in some circumstances full stationarity is challenging to obtain if series are short and data is scarce. In this case, institutions should use judgement and critical thinking to balance economic significance and stationarity requirement in order to assess if modelling can proceed. In this assessment, institutions should pay particular attention to the presence of trends, that often leads to spurious correlations.
	(iii)	To test for stationarity, standard unit root test may be used, including the Augmented Dickey-Fuller test, the Phillips-Perron test, the Kwiatkowski-Phillips-Schmidt-Shin (KPSS test). In case there is evidence of the presence of stochastic trend, standard transformations can be applied such as quarter-on-quarter or year-on-year log differencing.
	(iv)	Seasonality may be checked using X12 or X13 seasonal adjustment algorithms. Year-on-year differencing could also be used to remove stable seasonal patterns. Formal structural breaks tests (e.g. Chow test) may be employed if there is visual evidence of break in the series.
	(v)	Common stochastic trends between two variables may be explicitly modelled using the rigorous application of standard co-integration models (e.g. Engle-Granger two step method or Johansen approach).
	(vi)	The absence of stationarity of the dependent variable can also be addressed by a first order time differencing or by autoregressive models. However, this can potentially lead to further complexity in implementation. Institutions should use judgement in this choice provided that it is justified and clearly documented.
5.5.3	Differencing: Time differencing should be based upon the following principle. Let X_t be a time series of the macroeconomic variable X at regular time steps t. Formally we can define two types of changes: (i) backward looking returns that estimate the change of the variable over a previous horizon h and (ii) forward looking returns that estimate the change of the variable over a coming horizon h. Formally:
	It is recommended to build macro models based on backward looking returns as these are more intuitive to interpret and easier to implement. It is also recommended to compute backward looking default rates in order to ensure that both the dependent and independent variables are homogeneous.
5.5.4	Horizon of differencing: Institutions should choose carefully the horizon of return applied to macro variables, i.e. the period used to compute the change of a variable through time. Institutions should take notes of the following principles:
	(i)	For macroeconomic models, common return horizons include quarterly, half-yearly and yearly. The most appropriate return horizon should be chosen to maximize the explanatory power of the macro variables.
	(ii)	Note that the return horizon is not necessarily equal to the granularity of the time series. For instance, rolling yearly returns can be computed on quarterly time steps.
	(iii)	Institutions should be aware of the degree of noise in high frequency data. Consequently judgement should be used when using high frequency returns.
5.5.5	Lags: Variable lags should be considered in the modelling process to capture delayed effects of macro drivers. The use of large lags (more than 6 quarters) should be justified since long lags delay the impact of macro shocks on the dependent variable. For each macro variable, the choice of the most appropriate lag should be based on its statistical performance and economic meaning.
5.5.6	Smoothing: This means reducing the presence of spikes and outliers in times series. This is commonly addressed by the usage of moving average. Such practice is permitted but should be employed with caution and documented. The right balance of smoothing needs to be found. No smoothing (too much noise) in time series can lead to weak models. Alternatively, too much smoothing can dilute the strength of correlations. Smoothing techniques should be documented when applied.
5.5.7	Standard and intuitive transformations should be used. For example, the growth rate of a variable that can be zero or negative is not a meaningful measure.