The Forecast of Credit Risk of Romania
Simona Andreea DOSPINESCU1
Abstract:In this paper we propose to forecast the credit risk in Romania using Box-Jenkins metholody. In this purpose, we used as variable the rate of credit risk for the period 4thquarter of 1995 - 1stquarter of 2014. The treatment of the data is done with the EViews software. The results obtained in this paper show what will happen in the future and can be used as arguments in taking the adequate decisions for prevent or diminuate the consequences of this phenomenon.
Keywords:time series; ARIMA; Box-Jenkins metholody.
JEL Classification:B22, C22, C53, C87, E51.
1 Introduction
The forecast of the credit risk is based on using the indicator the rate of credit risk. In this paper we propose using Box-Jenkins metholody for forecasting the credit risk in Romania for a period of 3 quarters.
The random nature of the credit risk requires as forecasting method the Box-Jenkins methodology.
The Box-Jenkins methodology refers to a systematic method for identify, estimate, test and use of models for time series integrated autoregressive and moving average (ARIMA). It is suitable for medium and long time series length (more than 50 observations).
The data used in this analysis have been extracted from the official data of National Bank of Romania.
The analysis was carried out for the period 4th quarter of 1995 – 1st quarter of 2014.
2. Method
For realizing the forecast of the analyzed time series we will use classical methods, but also modern methods, such as Box-Jenkins methodology.
2.1. General Elements
Any time seriesX can be, according to Box-Jenkins methodology, as a combination of values and / or errors from the past,X and / oret.
X = ∅ + ∅ X + ∅ X + ⋯ + ∅ X + e −θ e −θ e − ⋯ −θ e
In order to model the real time series using the equation above are necessary four steps. First, the original series Xt must be transformed to become stationary around its own mean and variance.
Second, the values of p and q must be correctly calculated. Third, the values of the parameters
∅ , ∅ , … , ∅ și / sauθ ,θ , … ,θ must be estimated using non-linear optimization procedures that
1Alexandru Ioan Cuza University of Iași, Romania,[email protected]
minimizes the sum of squared errors. Finally, must find practical ways of modeling the seasonal series and calculate the values of the corresponding parameters.
2.2. Steps of Box&Jenkins Methodology
For applying the Box-Jenkins methodology we have to follow the next steps:
analysis of the series;
identification of the model. This step has the purpose to detect seasonality and to identify the order of seasonal autoregressive terms and seasonal moving average terms. In this stage is calculated the estimated autocorrelation function (FAC) and the estimated partial autocorrelation function (FACP).
These functions measure the statistical dependence between observations of data outputs;
estimation of ARIMA’s parameters. The estimation of ARIMA’s parameters is achieved by nonlinear least squares method. The values of the model coefficients are determined in relation to a particular criterion, one of this may be the maximum likelihood criterion. It can be shown that the likelihood function associated with a correct ARIMA model, used to determine the estimates of maximum likelihood of the parameters, contains all the useful information from data series about the model's parameters (Popescu T. and Demetriu S., (1991));
diagnostic checking. In this stage it is assumed that the errors represent a stationary process and the residues are an white noise type (or independent if the distribution is normal) with a normal distribution with mean and variance constant. The tests used to validate the model are based on the estimated residues. It is checked that the components of this vector are autocorrelated. If there is autocorrelation, the checked model is not correctly specified. In this case the dependencies between the components series are specified in an incomplete manner and we have to return to the model identification step and try another model. Otherwise, the model is good and can be used to make predictions for a given time horizon (Tudorel, A. (2003)).
forecasting.
3. The empirical analysis
We consider the time series provided by the National Bank of Romania (www.bnro.ro) for the rate of credit risk in Romania (RRC), between 4th quarter of 1995 and 1st quarter of 2014.
3.1. Analysis of The Series
The rate of credit risk is presented in the following figure. During the analyzed period, 4th quarter of 1995 – 1st quarter of 2014, the rate of credit risk presents an increasing trend. This evolution is due to the economic crisis established in July 2007. In 2008 the crisis worsened as stock markets arround the world collapsed and became unstable and the credit risk began to increase.
Figure 1 The evolution of credit risk in Romania, between 2007 and 2014
As we see in the 1st figure, the series admits a deterministic trend and may be non-stationary.
Numerically, this is demonstrated by the results using Augmented Dickey – Fuller test.
The results obtained in the following table confirm that the analyzed variable is not stationary and is, at least, integrated of order 1.
Table 1 Testing the rate of credit risk stationarity using Augmented Dickey-Fuller test
Variable\Tested model Intercept Trend and intercept None
ADF -1.424964 -1.068683 1.954179
Probability 0.5531 0.9138 0.9850
Akaike 2.851005 2.897140 3.196326
Schwarz 2.998262 3.093482 3.294497
Source: Data processed using the statistical program EView
To transform the variable in a stationary one we have to differentiate for order 1 the variable the rate of credit risk (DRRC) (Chirilă V. (2013)) and then we test its stationarity with the Augmented Dickey- Fuller test.
Table 2 Testing differentiated variables stationarity using Augmented Dickey-Fuller test
Source: Data processed using the statistical program EViews 0
5 10 15 20 25 30 35
2008 2009 2010 2011 2012 2013 2014 RATA_RISC_CRED
Variable\Tested Model Intercept Trend and intercept None
ADF -2.711668 -3.165746 -1.131128
Probability 0.0873 0.1157 0.2268
Akaike 2.917406 2.857373 3.084438
Schwarz 3.065514 3.054850 3.183177
Because for the analyzed variable, DRRC, the Augmented Dickey-Fuller test don’t offer the results that confirm us its stationarity, we include another 2 tests, Philips-Perron test and KPSS test (Chirilă, V. (2012)). The results obtained are presented in the Annex 1 and confirm the stationarity of the variable.
3.2. Identification of The Model
To determine whether the rate of credit risk in Romania (RRC) is the autocorrelated is performed the correloram of this variable.
Figure 1 The credit risk correlogram
The probabilities associated with Ljung-Box test (Q-Stat) show that the variable analyzed is autocorrelated.
3.3. Estimation of ARIMA’s Parameters
The graphical representation of the rate of credit risk indicate the existence of a deterministic trend, that can be linear or parabolic. So, we have to determine which is the best model of deterministic trend to use for forecast. Also, the rate of credit risk is an autocorrelated variable and we have to estimate the parameters of the autoregressive model.
The model with linear trend (Jaba, E. (2002)) takes the form:
0 1
t t
Y t The model with parabolic trend takes the form:
0 1 2 2
t t
Y t t
The estimation results of the two types of trends are shown in the Annex 2.
The estimated model which include the linear trend, but also the offset variable, is:
Because the estimation of the parameter, 1, is greater than zero, we deduce that the credit risk has an increasing trend.
The estimated model which include the parabolic trend, but also the offset variable, is:
2 1
0,008 2,039 0,029 0,516
t t t
Y t t Y e
Because the estimation of the parameters,2, from the parabolic trend is negative indicate us that this curve admits a peak.
3.4. Diagnostic Checking
The estimated regression models respect the assumptions regarding the errors. The results of testing the hypothesis of errors normality, the lack of autocorrelation, homoscedasticy and the error mean is zero.
The estimated models are statistically significant because the correlation ratio of each model is significantly different from zero. The estimators models are also significantly different from zero.
3.5. Forecast
The series was modeled taking into account, as suggested schedule a linear trend and a parabolic trend.
Therefore we should choose the best model to achieve forecast.
The results obtained (Eviews program) to forecast the model including linear trend are shown in the table below.
Table 5 The forecast of credit risk rate using the linear trend
0 10 20 30 40 50
2008 2009 2010 2011 2012 2013 2014
RRCF1 ± 2 S.E.
Forecast: RRCF1 Actual: RRC
Forecast sample: 2007Q4 2014Q4 Adjusted sample: 2008Q1 2014Q4 Included observations: 25
Root Mean Squared Error 1.406597 Mean Absolute Error 1.146372 Mean Abs. Percent Error 10.64083 Theil Inequality Coefficient 0.032231 Bias Proportion 0.020416 Variance Proportion 0.201249 Covariance Proportion 0.778335
Source: Data processed using the statistical program EViews
In summary, the table bellow presents some statistics which allow us to choose the best model of the rate of credit risk evolution, that will be used for the forecast. Since there are two models of the same variable we can use to compare the models also the indicators built on Akaike and Schwartz information theory. Since the two models have different numbers of parameters, we use the ratio of adjusted determination.
Table 6 Choosing the model of the rate of credit risk evolution The Statistic Indicator Model - Linear trend Model - Parabolic trend
Root Mean Squared Error 1.406597 0.883161
Mean Absolute Error 1.146372 0.751370
Mean Abs. Percent Error 10.64083 5.367873
Theil Inequality Coefficient 0.032231 0.020040
Akaike 2.827134 2.536816
Schwartz 2.973399 2.731836
Adjusted R-squared 0.989483 0.992392
Source: Data processed using the statistical program EViews
The minimum values of the mean square error of adjustment, medium absolute error, medium error as percent, Theil's inequality coefficient, Akaike criterion and Schwartz criterion indicates the best model. The maximum ratio of adjusted determination indicates the best model.
All indicators selected to determine the best forecasting model indicates the model with parabolic trend. The expected values that takes into account the parabolic trend (RRCF2) obtained using Eviews are shown in Figure 3 and Annex 3.
0 5 10 15 20 25 30 35 40
2007 2008 2009 2010 2011 2012 2013 2014
RRCF1 RRCF2
Figure 3 The forecasted values of the rate of credit risk using the model with linear trend (RRCF1) and parabolic trend (RRCF2)
The figure of the forecasted values for the rate of credit risk reveals that the model with parabolic trend shows an increasing evolution, but lower than the model with linear trend.
The predicted values for the rate of credit risk are: for the 2nd quarter of 2014 - 34.61 (model with linear trend), respectively, 32.95 (parabolic trend model), for the 3rd quarter of 2014 - 35.72 (model with linear trend), respectively, 33.42 (parabolic trend model) and for the 4th quarter - 36.82 (model with linear trend), respectively, 33.82 (parabolic trend model).
4. Conclusions
The Box-Jenkins methodology gives the possibility to forecast the rate of the credit risk in Romania and, by this, helps taking the adequate decisions for prevent or diminuate the consequences of this phenomenon.
During the period 4th quarter of 2007 and 1st quarter 2014, the rate of credit risk shows an increasing trend. This evolution is mainly due the economic crisis that was established in July 2007. In 2008 crisis has worsened as stock markets around the world collapsed and became unstable, and credit risk began to rise.
The predicted values for the model with linear trend and parabolic trend model are higher than those registered, which means that the rate of credit risk continue to rise.
5. References
Andrei, T. (2003).Statistică şi econometrie/Statistics and econometrics, Bucharest, Economic Publishing
Chirilă, V. & Chirilă C. (2012), Relation Between Expected Return and Volatility at Bucharest Stock Exchange, on Business Cycle Stages”,Annales Universitatis Apulensis, Series Oeconomica,Nr. 14, Vol. 1, pp. 149-163.
Chirilă, V. (2013)Analiza econometrică a burselor de valori şi a ciclurilor de afaceri/Econometric analysis of stock markets and business cycles, ASE Publishing, Bucureşti
Jaba, E. (2002) Statistica, ediţia a 3-a, Editura Economică, Bucureşti
Popescu, S., (1991), Practica modelării şi predicţiei seriilor de timp/Practical modeling and prediction of time series, Bucharest, Technical Publishing
Raport Statistic BNR (2014). Web page. Retrieved from http://www.bnr.ro/Raport-statistic.html, date: 09.09.2014
Annex 1. Testing stationarity of the variable DRRC using Phillips-Perron test and KPSS test Table 1 Testing stationarity of the variable using Phillips-Perron test
Table 2 Testing stationarity of the variable using KPSS test
Annex 2. The estimation of the models with linear trend and parabolic trend for the rate of credit risk
Table 1 Estimation of linear deterministic trend Dependent Variable: RRC
Method: Least Squares
Sample (adjusted): 2008Q1 2014Q1 Included observations: 25 after adjustments Convergence achieved after 4 iterations
Variable Coefficient Std. Error t-Statistic Prob.
C 6.111996 4.393740 1.391069 0.1781
@TREND 1.097452 0.218863 5.014336 0.0001
AR(1) 0.845773 0.137295 6.160265 0.0000
R-squared 0.990359 Mean dependent var 20.12360 Adjusted R-squared 0.989483 S.D. dependent var 9.169730 S.E. of regression 0.940395 Akaike info criterion 2.827134 Sum squared resid 19.45555 Schwarz criterion 2.973399 Log likelihood -32.33917 Hannan-Quinn criter. 2.867702 F-statistic 1129.968 Durbin-Watson stat 1.872430 Prob(F-statistic) 0.000000
Source: Data processed using the statistical program EViews
Variable\Tested model Intercept Trend and intercept None
PP -4.795952 -4.996850 -2.222521
Probability 0.0009 0.0027 0.0280
Akaike 2.859970 2.869339 3.273054
Schwarz 2.958141 3.016596 3.322139
Variable\Tested model Intercept Trend and intercept
KPSS 0.173794 0.085376
Critical value 0.463000 0.146000
Akaike 2.758863 2.802907
Schwarz 2.807618 2.900417
Table 2 Estimation of parabolic deterministic trend Dependent Variable: RRC
Method: Least Squares
Sample (adjusted): 2008Q1 2014Q1 Included observations: 25 after adjustments Convergence achieved after 4 iterations
Variable Coefficient Std. Error t-Statistic Prob.
C 0.008902 1.575119 0.005652 0.9955
@TREND 2.039890 0.235335 8.668024 0.0000
@TREND^2 -0.029717 0.007817 -3.801547 0.0010
AR(1) 0.516693 0.158561 3.258632 0.0038
R-squared 0.993343 Mean dependent var 20.12360 Adjusted R-squared 0.992392 S.D. dependent var 9.169730 S.E. of regression 0.799832 Akaike info criterion 2.536816 Sum squared resid 13.43435 Schwarz criterion 2.731836 Log likelihood -27.71020 Hannan-Quinn criter. 2.590906 F-statistic 1044.492 Durbin-Watson stat 1.943140 Prob(F-statistic) 0.000000
Source: Data processed using the statistical program EViews
Annex 3 The forecasted values for the rate of credit risk
Table 1. The forecasted values for the rate of credit risk using the model with linear trend (RRCF1) and the model with parabolic trend (RRCF2)
Quarters RRCF1 RRCF2
2007Q4 NA NA
2008Q1 5.423179 4.081249
2008Q2 6.796121 5.035326
2008Q3 8.126576 6.411664
2008Q4 9.421095 7.977456
2009Q1 10.68522 9.612413
2009Q2 11.92364 11.25438
2009Q3 13.14032 12.87125
2009Q4 14.33861 14.44642
2010Q1 15.52135 15.97133
2010Q2 16.69094 17.44154
2010Q3 17.84940 18.85476
2010Q4 18.99845 20.20981
2011Q1 20.13954 21.50608
2011Q2 21.27390 22.74326
2011Q3 22.40257 23.92118
2011Q4 23.52643 25.03975
2012Q1 24.64621 26.09893
2012Q2 25.76255 27.09871
2012Q3 26.87598 28.03907
2012Q4 27.98694 28.91999
2013Q1 29.09582 29.74149
2013Q2 30.20293 30.50356
2013Q3 31.30856 31.20619
2013Q4 32.41292 31.84939
2014Q1 33.51622 32.43316
2014Q2 34.61862 32.95749
2014Q3 35.72025 33.42239
2014Q4 36.82124 33.82786
