PPNR Modeling – OLS, Co-Integration and ARIMAX
The objective of this article is to evaluate different techniques for time series forecasting. These techniques include OLS model, Co-integration model and ARIMAX model Business problem: To forecast the different components of PPNR. These components include Non-interest Income and Non-interest Expense. Proposed solution OLSModelCo-IntModelARIMAXModelNotesPreference HighMedium Low Complexity Low Medium High Dependent variable is stationaryOLS should be used ARIMAX should be used For ARIMAX both (dependent and independent variables) should be stationary togetherIndependent variable is stationaryDependent variable is non-stationaryCo-Int should be used ARIMAX should be used For ARIMAX both (dependent and independent variables) should be non-stationary togetherIndependent variable is non-stationaryAuto-correlation DW test close to 2DW test close to 2DW test close to 2If for OLS or Co-integration DW fails then ARIMAX should be usedVariable significance p-value < 0.05p-value < 0.05p-value < 0.05For ARIMAX the AR, MA and exogenous terms should be significant Multi co-linearityVIF < 5VIF < 5VIF < 5Residual is stationary ADF test should passADF test should passADF test should passResidual is non-stationaryFor all the three approaches, the residual should be stationary Normality and homoscedasticity of residual Should pass Should pass Should pass OLSAdvantages – easy to develop / test and easy to explain Disadvantages– difficult to finding strong correlation between dependent and independent variables Co-IntegrationAdvantages – easy to find strong correlations between dependent and independent variablesDisadvantages – difficult to pass all the tests / assumptions of co-integration ARIMAXAdvantages – very powerful modeling technique to overcome the shortcomings of OLS and co-integration models Disadvantages – complex to develop as there are two stages. In stage 1 OLS model is developed and in stage 2 ARIMAX model is developed post identification of AR and MA terms Introduction PPNR Pre-provision net revenue (PPNR), under the Federal Reserve’s Comprehensive Capital Analysis and Review (CCAR), measures net revenue forecast from asset-liability spreads and non-trading fees of banks.Pre-provision Net Revenue (PPNR) = Net Interest Income + Non-interest Income – Non-interest Expense Interest Income: Loans and SecuritiesInterest Expense: Deposits and BondsNon-Interest Income: Credit Related Fees and Non-Credit RelatedNon-Interest Expense: Employee Compensation, Processing / Software, Occupancy, Credit / Collections and Residential Mortgage Repurchase 2.2 Modeling Approaches If the dependent and independent variables are stationary ADF test is done on the independent variables. Only those variables are kept, those are stationary.Correlation between independent variables and dependent variable is done. Only those variables are kept, those have high correlation with dependent variable. OLS Model is developed. If the dependent and independent variables are non-stationary ADF test is done on the independent variables. Only those variables are kept, those are non-stationary Co-integration between independent variables and dependent variable is done. Only those variables are kept, those are co-integrated with dependent variable.Correlation between independent variables and dependent variable is done. Only those variables are kept, those have high correlation with dependent variable. OLS Model is developed 2.3 Independent Variables RawDiff QoQDiff YoYPct Diff QoQPct Diff YoYLags 0, 1 and 2Lags 0, 1 and 2Lags 0, 1 and 2Lags 0, 1 and 2Lags 0, 1 and 2GDP growthYesNoNoNoNoIncome growthYesNoNoNoNoCPI growthYesNoNoNoNoUnemp rateYesYesYesNoNo3mT rateYesYesYesNoNo5yT rateYesYesYesNoNo10yT rateYesYesYesNoNoBBB rateYesYesYesNoNoPrime rateYesYesYesNoNoHPIYesNoNoYesYes 2.4 Model Outputs Time Period Historical – 44 data points (from 2005Q1 to 2015Q4)Forecasted – 13 data points (from 2016Q1 to 2019Q1) Non-interest Income and Non-interest Expense are modeled Non-interest Expense is modeled using the stationary model developed approach Non-interest Income is modeled using the non-stationary model developed approach Non-interest Expense(Stationary model developed approach)Non-interest Income(Non-stationary model developed approach) 2.5 Model Tests Stationarity of dependent and independent variables: ADF test is done If the p-value <= 0.10 then the series is stationary If the p-value > 0.10 then the series is non-stationary Multi co-linearity: Correlation matrix is used to test multi co-linearity If the correlation between variables is less than 0.30 or more than -0.30 then there is low multi co-linearity If the correlation between variables is more than 0.70 or less than -0.70 then there is high multi co-linearity Significance: The p-value <= 0.05 then the coefficient is statistically significant The p-value > 0.05 then the coefficient is statistically insignificant Auto correlation:Durbin-Watson test is doneIf DW statistics is less than 1 then there is positive auto correlationIf DW statistics is close to 2 then there is no auto correlation If DW statistics is more than 3 then there is negative auto correlation Stationarity of residual: ADF test is done If the p-value <= 0.10 then the series is stationary If the p-value > 0.10 then the series is non-stationary 3. Stationary Series 3.1 Process ADF test is done on the independent variables. Only stationary variables are kept (23 out of 72 variables are selected). Correlation between independent variables and dependent variable is done. Only those variables are kept, that have high correlation with dependent variable (2 out of 23 variables are selected).OLS Model is developed, checks on multi co-linearity, significance of the variable and stationary of the residuals are done (2 out of 2 variables are selected). 3.2 Dependent Variables It is observed that the dependent variables (Non-Interest Income 1st Difference and Non-Interest Expense 1st Difference) are stationary Non-Interest Income 1st Diff = Non-Interest Income (t) – Non-Interest Income (t-1)Non-Interest Expense 1st Diff = Non-Interest Expense (t) – Non-Interest Expense (t-1) VarADFPvalNonInt Inc diff-5.200.00NonInt Exp diff-5.980.00 3.3 Independent Variables It is observed that out of 72 independent variables, 23 independent variables are stationary. If the p-value <= 0.10 then the series is stationary If the p-value > 0.10 then the series is non-stationary It is observed that no macro-economic variable has high correlation with Non-Interest Income 1st Diff. However, few macro-economic variables have high correlation with Non-Interest Expense 1st Diff.If correlation is more than 0.30 or less than -0.30 then it is marked as highIt is observed that out of 23 independent variables, 2 independent variables have high correlation with Non-Interest Expense 1st Diff. NonInt Exp diffCPI growth0.31GDP growth 20.43 3.4 Model Development It is observed that the model has low R-Sq and Adj R-Sq. No. Obs:43.00R-squared:0.29Df Model:2.00Adj. R-squared:0.26 There are 2 variables in the model. CPI growth and GDP growth (lag 2) The p-value for both the variables is less than 0.05 coefstd errtP>|t|const-377,300.00134,000.00-2.820.01CPI growth86,320.0035,500.002.430.02GDP growth 2122,800.0036,600.003.360.00 It is observed that there is very low multi co-linearity in the model Correlation between variables is less than 0.30 or more than -0.30 CPI growthGDP growth 2CPI growth-0.04GDP growth 2-0.04 It is observed that there is no auto-correlation in the model and the residual is stationaryDW test statistics is close to 2 The p-value of the ADF test is less than 0.10 Durbin-Watson:2.36 Var:ADF:Pval:RESI-8.300.00 3.5 Projection The projection is done for 13 Quarters  If t = 1: Predicted Non-Interest Expense (t) = Actual Non-Interest Expense (t)If t > 1: Predicted Non-Interest Expense (t) = Predicted Non-Interest Expense (t-1) + Predicted Non-Interest Expense 1st Diff (t)The severely adverse projection is done for forecasted period 4. Non-stationary Series 4.1 Process ADF test is done on the independent variables. Only non-stationary variables are kept (49 out of 72 variables are selected). Co-integration between independent variables and dependent variable is done. Only those variables are kept, those are co-integrated with dependent variable (6 out of 49 variables are selected).OLS Model is developed, checks on multi co-linearity, significance of the variable and stationary of the residuals are done (1 out of 6 variables is selected). 4.2 Dependent Variables It is observed that the dependent variables are non-stationary VarADFPvalNonInt Inc-2.140.23NonInt Exp-1.490.54 4.3 Independent Variables It is observed that out of 72 independent variables, 49 independent variables are non-stationary. If the p-value <= 0.10 then the series is stationary If the p-value > 0.10 then the series is non-stationary It is observed that no macro-economic variable is co-integrated with Non-Interest Expense. However, few macro-economic variables are co-integrated with Non-Interest Income.If the p-value <= 0.10 then the series is co-integrated If the p-value > 0.10 then the series is not co-integrated VarCoint_IncPval_Inc3mT rate dyoy-3.370.053mT rate dyoy 1-3.240.065yT rate dyoy-3.210.075yT rate dyoy 1-3.380.04Prime rate dqoq 2-3.390.04Prime rate dyoy-3.310.05 4.4 Model Development It is observed that the model has high R-Sq and Adj R-Sq. No. Obs:44.00R-squared:0.66Df Model:1.00Adj. R-squared:0.65 There is 1 variable in the model. 3mT rate (difference YoY) The p-value for the variable is less than 0.05 coefstd errtP>|t|const7,656,000.00249,000.0030.800.003mT rate dyoy1,786,000.00199,000.008.990.00 It is observed that there is positive auto-correlation in the model and the residual is stationaryDW test statistics is less than 1 The p-value of the ADF test is less than 0.10 Durbin-Watson:0.85 Var:ADF:Pval:RESI-3.330.01 Since there is positive auto-correlation in the model, ARIMAX model is developed The ACF and PACF plots are generated for the OLS residual Based on the ACF and PACF plot, AR(1) model is developed Reference: Time Series Modeling and Forecasting—An Application to Bank’s Stress Testing, SAS Global Forum 2015, Paper 3338-2015 ARIMAX model specifications P, D, Q = 1, 0, 0 X = 3mT rate dyoyWhen AR(2) term was introduced in the model, it was found to be insignificant, hence higher lags for AR are not included in the model No. Obs:44.00AIC1,380.03Sample:0.00BIC1,375.54 There are 2 variables in the model. AR(1) term and 3mT rate (difference YoY) The p-value for both the variables is less than 0.05The sigma2 in the coefficients table is the estimate of the variance of the error term. coefstd errtP>|t|const7,656,000.00563,000.0013.590.003mT rate dyoy1,786,000.00264,000.006.760.00ar.L10.560.134.240.00sigma21.75E+120.171.05E+130.00 It is observed that there is no auto-correlation in the model and the residual is stationaryDW test statistics is close to 2 The p-value of the ADF test is less than 0.10 Durbin-Watson:1.77 Var:ADF:Pval:RESI-5.740.00 4.5 Projection The projection is done for 13 Quarters The dip in 2008-2009 is captured well by the model The severely adverse projection is done for forecasted period GraphPredicted (Blue line) – OLS model Forecasted (Red line) – ARIMAX model
The objective of this article is to evaluate different techniques for time series forecasting. These techniques include OLS model, Co-integration model and ARIMAX model  Business problem: To forecast the different components of PPNR. These components include Non-interest Income and Non-interest Expense. Proposed solution OLSModel Co-IntModel ARIMAXModel Notes Preference  High Medium  Low   Complexity  Low  Medium  […]
The objective of this article is to evaluate different techniques for time series forecasting. These techniques include OLS model, Co-integration model
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["API", "TPU", "programming_languages:R", "AI", "R"]
["AI", "TPU", "R", "API", "programming_languages:R"]
Rohit Garg has close to 7 years of work experience in field of data analytics and machine learning. He has worked extensively in the areas of predictive modeling, time series analysis and segmentation techniques. Rohit holds BE from BITS Pilani and PGDM from IIM Raipur.