Econometric Forecasting of Short term Interest Rates in India
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The document is a project report that aims to forecast short term yields in India using linear regression and ARIMA models. It analyzes the dependence of short term yields on various macroeconomic factors like inflation, money supply, exchange rates, and interest rates. Linear regression models are used to find the relationship between yields and these explanatory variables. ARIMA models are also employed to forecast short term interest rates over the next 20 months based on past yield data from 2001 to 2011. Key findings and conclusions from the analysis are presented.
Econometric Forecasting of Short term Interest Rates in India
- 1. Financial Econometric<br />Forecasting the Short term yield & finding its dependence on the Macro factors 2011Submitted To :Dr. Kakali KanjilalProfessorIMT GhaziabadIMT Ghaziabad<br />Institute Of Management Technology, Ghaziabad<br />Project Report <br />Financial Econometrics<br />Under the guidance of Dr.Kakali Kanjilal, Professor<br /> IMT - Ghaziabad<br />Submitted By:<br />Sumit Chugh (10DCP-042)<br />Vatan Lunia (10DCP-046)<br />Akash Jauhari (10DCP-056)<br />Alok Mishra (10DCP-057)<br />Ankit Bhardwaj (10DCP-060)<br />Raghav Agarwal (10DCP-087) <br />Table of Contents TOC \o \" 1-3\" \h \z \u 1. Abstract PAGEREF _Toc303550173 \h 32. Introduction: Short term T-bill yields in India PAGEREF _Toc303550174 \h 42.1 Fluctuations in Security yields PAGEREF _Toc303550175 \h 53. Data PAGEREF _Toc303550176 \h 54. Methodology PAGEREF _Toc303550177 \h 65. Linear regression Models PAGEREF _Toc303550178 \h 65.1 Short term yield dependent on Macro Factors PAGEREF _Toc303550179 \h 75.2 Short term yield dependent on growth variables PAGEREF _Toc303550180 \h 85.3 Short term yield dependent on Macro factors-with Differencing PAGEREF _Toc303550181 \h 115.4 Reverse Model – M3 on interest rates -with Differencing PAGEREF _Toc303550182 \h 135.5 Reverse Model: WPI Growth dependent on Short term yield & Repo rate PAGEREF _Toc303550183 \h 156. ARIMA Models…………………………………………………………………………………………………………………….…16 6.1 ARIMA for Log-Short term yield data………………………………………………………………………………...16 6.2 ARIMA output for Log-short term yield - with trend differencing…………………………….…………17 6.3 ARIMA output for Log-short term yield - with seasonal differencing…………………………………..186.4 Forecasting ARIMA for Log- Short term Yield PAGEREF _Toc303550184 \h 207. Key findings and Conclusion PAGEREF _Toc303550185 \h 21<br />1. Abstract<br />Key words: Short term yields, Linear Regression, ARIMA modeling<br />Interest rate is a key economic indicator for a country. It affects bank lending rates, foreign investments, exchange rates and stock returns. In a fast growing economy like India, appropriate interest rates are even more important as they are a vital balance between money supply, inflation and growth. However Indian 91 day T-bill yields have been quite volatile in past few years, stretching from a high of 9.1% in August 2008 to a low of about 3.2% in May 2009.<br />We attempt to understand the dependence of Short Term Yields on Macro factors and create a model to forecast yield, focusing on the 91-day T bill, based on Linear Multiple Regression and ARIMA modeling. We also intend to understand and study the reverse relationship i.e. any dependence of Macro factors like money supply and Inflation on the interest rates. Possible explanatory variables are – Repo rate, WPI, IIP, Money supply, Stock Indices, Global Oil prices, Exchange rate for rupee. Such a forecasting model can be decisive for banks and corporates in planning their operations as well as capital structure.<br />2. Introduction: Short term T-bill yields in India<br />Treasury Bills, which are money market instruments, are short term debt instruments issued by the Government of India and are presently issued in three tenors viz. 91 day, 182 day and 364 day. Treasury Bills are zero coupon securities and pay no coupon. They are issued at a discount and redeemed at the face value at maturity. For example, a 91 day Treasury Bill of Rs.100/- (face value) may be issued at a discount of say, Rs.1.80, that is Rs.98.20 and redeemed at the face value of Rs.100/-. The return to the investors is, therefore, the difference between the maturity value or face value (i.e., Rs.100) and the issue. Currently, the notified amount for issuance of 91 day and 182 day Treasury Bills is Rs.500 crore each whereas the notified amount for issuance of 364 day Bill is higher at Rs.1000 crore.<br />2.1 Fluctuations in Security yields<br />The price of a Government security, like other financial instruments, keeps fluctuating in the secondary market. The price is determined by demand and supply of the securities. Specifically, the prices of Government securities are influenced by the level and changes in interest rates in the economy and other macro-economic factors, such as, expected rate of inflation, liquidity in the market etc. Developments in other markets like money, foreign exchange, credit and capital markets also affect the price of the government securities. Further, developments in international bond markets, specifically the US Treasuries affect prices of Government securities in India. Policy actions by RBI (e.g. announcements regarding changes in policy interest rates like Repo Rate, Cash Reserve Ratio, Open Market Operations etc.) can also affect the prices of government securities.<br />Source: CMIE database.<br />High of about 9.1% in August 2008, low of about 3.2% in May 2009.
- 2. Standard deviation of about 1.42 units considering data for Jan 2001 to Jun 2011.3. Data <br />Following Variables have been taken for the research:-
- 3. Short term yield: 91 day T-bill yield
- 4. WPI (Wholesale Price Index): Inflation
- 5. Money Supply (M-3, RBI data)
- 6. IIP (Index for Industrial Production)
- 7. Exchange Rate (Rs per $, average)
- 8. Repo rate (issued by RBI)
- 9. Oil Prices (WTI rate in $ per barrel)
- 10. Import & Exports for India (in million $)
- 11. Sensex (proxy for stock markets in India)
- 12. The period of data is from January 2001 to June 2011. Frequency of data taken is monthly average.
- 13. Source of the data have been CMIE database.4. Methodology<br />In this process we take short term yield as the dependent variable and try to regress <br />Linear Regression: Short term yield on Macro factors
- 14. First we attempt to find the dependence of Short term yield on Macro factors like WPI, M-3 and Exchange rate etc. We used SAS for getting various outputs.
- 15. Based on the principles and issues of linear regression like autocorrelation, multi-co linearity, hetroscedasticity etc, we tried to improve our model at every step. Techniques like differencing, taking growth figures, log data were used.
- 16. Linear Regression: Reverse Model
- 17. We also tried to find any dependence of short term yield and Repo rate on Money supply and Inflation. Again linear regression though SAS was used.
- 18. ARIMA Modeling:
- 19. We tried to find out best estimated model based on Univariate ARIMA modeling. Based on the model we tried to forecast short term interest rates for next 20 months. Also to check our model we applied ex-post test on available data.
- 20. 5. Linear regression Models<br />5.1 Short term yield dependent on Macro Factors<br />Dependent Variable: Short term yield<br />Independent Variables: Repo rate, BOP, WPI, IIP, Exchange rate (Rs/$), Money Supply (M-3)/ Oil Prices ($/barrel).<br />Analysis of VarianceSourceDFSum ofMeanF ValuePr > FSquaresSquareModel7212.550530.36436152.95<.0001Error10721.241490.19852 Corrected Total114233.792 Root MSE0.44555R-Square0.9091 Dependent Mean5.84243Adj R-Sq0.9032 <br />The model seems to be good as R-square is high and F- Value is significant.Parameter Estimate Table -<br />VariableLabelDFParameterStandardt ValuePr > |t|Variance95% Confidence LimitsEstimateErrorInflation InterceptIntercept1-5.602741.67195-3.350.00110-8.91718-2.2883repo_rateRepo Rate11.201940.0476325.23<.00012.114691.107521.29636IIPIIP10.008170.003612.260.025822.430340.0010.01533exch_rateExch rate1-0.045120.02984-1.510.13353.57453-0.104260.01403SensexSensex1-0.00015985.25E-05-3.040.000112.99939-0.00026-5.6E-05m3_oil_prcM3/Oil prc1-1.573E-054.18E-06-3.760.00032.67557-2.4E-05-7.5E-06imp___expImp – Exp1-9.099E-053.45E-05-2.640.00969.91017-0.00016-2.3E-05WPIWPI10.026820.009132.940.00442.355110.008730.04492<br />The table shows that there is multi-co linearity in WPI and IIP. Also exchange rate is found to be insignificant. Let us look at the correlation matrix.Correlation of EstimatesVariableLabelInterceptrepo_rateIIPexch_rateSensexm3_oil_prcimp___expWPIInterceptIntercept1-0.56880.2543-0.6138-0.07740.48240.6601-0.5802repo_rateRepo Rate-0.56881-0.0649-0.0619-0.24460.0448-0.51640.4894IIPIIP0.2543-0.064910.0265-0.3711-0.17590.341-0.6626exch_rateExch rate-0.6138-0.06190.026510.5312-0.5905-0.0152-0.182SensexSensex-0.0774-0.2446-0.37110.531210.04550.0774-0.2598m3_oil_prcM3/Oil prc0.48240.0448-0.1759-0.59050.045510.2064-0.0783imp___expImp – Exp0.6601-0.51640.341-0.01520.07740.20641-0.7796WPIWPI-0.58020.4894-0.6626-0.182-0.2598-0.0783-0.77961<br />Only WPI and BOP seem to have high correlation. Hence either we can take ratio or can drop one of the variable.Test of First and SecondMoment SpecificationDFChi-SquarePr > ChiSq3527.880.7983Durbin-Watson D1.017 Number of Observations115 1st Order Autocorrelation0.491 <br />The lower and upper limit for d- values are found out to be 1.52 and 1.82. Hence d=.49 is a clear sign of positive correlation.
- 21. There is no hetroscedasticity, as P.0.79 and thus we accept the null hypothesis is absence of hetroscedasticity.
- 22. To improve the above model we go for –
- 23. Consider growth of variables WPI,IIP and Sensex
- 24. Differencing5.2 Short term yield dependent on growth variables<br />Dependent Variable: Short term yield<br />Independent Variables: Repo rate, BOP, WTI Oil price ($/brl), WPI growth, IIP growth, Exchange rate (Rs/$), Money Supply (M-3)<br />Analysis of VarianceSourceDFSum ofMeanF ValuePr > FSquaresSquareModel8200.860425.10755110.18<.0001Error10523.927790.22788 Corrected Total113224.7882 Root MSE0.47737R-Square0.8936 Dependent Mean5.81623Adj R-Sq0.8854 Coeff Var8.20758 <br />The model seems to be good as R-square is high and F-Value is significantParameter EstimatesVariableLabelParameterStandardt ValuePr > |t|VarianceEstimateErrorInflationInterceptIntercept-1.19141.34475-0.890.37770repo_rateRepo Rate1.013760.0509319.91<.00011.99394imp___expImp – Exp-5.7E-053.3E-05-1.710.097.83425wti_oil_prc____bbl_WTI Oil prc ($/bbl)0.020440.005054.05<.00018.55729wpi_growthWPI growth-0.314830.07623-4.13<.00011.3709iip_growthIIP growth-0.003280.00791-0.410.6791.08034exch_rateExch rate-0.021630.02646-0.820.41552.446sensex_growthSensex growth-0.004220.00566-0.750.45751.14073M3M31.10E-079.12E-081.20.23257.23784<br />The model has no multi-collinearity. Also, only repo rate, WTI oil price and WPI growth are significant at 5% level of significance.Correlation of EstimatesVariableLabelrepo_rateimp___expwti_oil_prc____bbl_wpi_growthiip_growthM3InterceptIntercept-0.25270.3501-0.65140.0810.0730.0315repo_rateRepo Rate1-0.2078-0.23610.2130.0430.5983imp___expImp - Exp-0.20781-0.52770.24670.1972-0.5741wti_oil_prc____bbl_WTI Oil prc ($/bbl)-0.2361-0.52771-0.414-0.1595-0.2839wpi_growthWPI growth0.2130.2467-0.41410.22090.0711iip_growthIIP growth0.0430.1972-0.15950.22091-0.0749exch_rateExch rate-0.0683-0.20850.7011-0.1307-0.0754-0.2968sensex_growthSensex growth0.23740.00710.10290.01460.0021-0.028M3M30.5983-0.5741-0.28390.0711-0.07491<br />The correlation matrix does not show a strong positive or negative correlation between any two variables.Test of First and SecondMoment SpecificationDFChi-SquarePr > ChiSq4437.650.7392Durbin-Watson D0.882 Number of Observations114 1st Order Autocorrelation0.553 <br />Since the model is suffering from positive autocorrelation, we applied the 1st order differencing. It resulted in a negative R-square and all variables were insignificant. Hence the model is void and no dependence is proved. <br />5.3 Short term yield dependent on Macro factors-with Differencing<br />To remove autocorrelation from above model we go for trend differencing by one. Here are the outputs.<br />Number of Observations Read127 Number of Observations Used125 Number of Observations with Missing Values2 Analysis of VarianceSourceDFSum ofMeanF ValuePr > FSquaresSquareModel76.429420.918496.61<.0001Error11716.25310.13892 Corrected Total12422.68252 Root MSE0.37271R-Square0.2835 Dependent Mean-0.0048Adj R-Sq0.2406 Coeff Var-7764.86 <br />Here as we can find, F value is significant. However, R-square has dropped to 24%, which is expected when we are dealing with differencing case.VariableLabelParameterStandardt ValuePr > |t|SquaredSquaredVarianceEstimateErrorPartialPartialInflation Corr Type ICorr Type II InterceptIntercept0.008530.034010.250.8024..0d1___repo_rateD1 - Repo Rate0.737730.134215.5<.00010.220980.205241.09572d1___imp___expD1 - Imp – Exp1.68E-051.81E-050.930.35640.002970.007281.27191d1___wpiD1 - WPI-0.002610.00143-1.820.0710.020510.02761.01595d1___iipD1 - IIP0.002380.002031.170.24410.017140.011581.14287d1___exch_rateD1 - Exch rate0.062590.053141.180.24130.021930.011721.44721d1___sensexD1 - Sensex-5.8E-057.71E-05-0.750.45680.002360.004741.34529d1___m3_oil_prcD1 - M3/Oil prc-9E-066.12E-06-1.460.14670.017910.017911.15338<br />Correlation Matrix:<br />VariableLabelrepo_rateimp___expd1___wpid1___iipexch_ratesensexm3_oil_prcInterceptIntercept0.07-0.090.06-0.09-0.04-0.13-0.06d1___repo_rateD1 - Repo Rate1.00-0.08-0.01-0.030.110.030.21d1___imp___expD1 - Imp - Exp-0.081.00-0.050.250.280.200.20d1___wpiD1 - WPI-0.01-0.051.00-0.11-0.04-0.010.00d1___iipD1 - IIP-0.030.25-0.111.00-0.12-0.090.13d1___exch_rateD1 - Exch rate0.110.28-0.04-0.121.000.480.00d1___sensexD1 - Sensex0.030.20-0.01-0.090.481.000.15d1___m3_oil_prcD1 - M3/Oil prc0.210.200.000.130.000.151.00<br />Test of First and SecondMoment SpecificationDFChi-SquarePr > ChiSq3531.580.6338Durbin-Watson D2.181 Number of Observations125 1st Order Autocorrelation-0.094 <br />Now we have eliminated autocorrelation. Also multi-co linearity and hetroscedasticity are not present.
- 25. However, 5 of the 7 variables have become insignificant.
- 26. This signals that the short term interest rates are not determined on Macro factors or market forces. Rather it is highly regulated and determined by RBI and other bodies of Ministry of Finance.5.4 Reverse Model – M3 on interest rates -with Differencing<br />We ran a linear regression with M3 as the dependent variable and short term yield, including other factors, as independent variables. From the model, we observed the following-
- 27. 1. The model is suffers from positive autocorrelation as the d value is very close to 0.
- 28. 2. Also there seems to be hetroscedasticity as we are rejecting hypothesis of absence of hetroscedasticity.
- 29. 3. To improve the model, we go for differencing.
- 30. Dependent Variable: Money Supply (M-3)
- 31. Independent variables: Short term yield, Repo rateNumber of Observations Read127 Number of Observations Used125 Number of Observations with Missing Values2 Analysis of VarianceSourceDFSum ofMeanF ValuePr > FSquaresSquareModel26.28E+083.14E+080.170.843Error1222.24E+111.84E+09 Corrected Total1242.25E+11 Root MSE42867R-Square0.0028 Dependent Mean43277Adj R-Sq-0.0136 <br />Parameter Estimate<br />VariableLabelDFParameterStandardt ValuePr > |t|VarianceEstimateErrorInflationInterceptIntercept1432803847.01511.25<.00010d1___shrt_trm_bd_yD1 - Shrt trm bd Y1-5634.6110198-0.550.58161.28366d1___repo_rateD1 - Repo Rate11513.917167080.090.9281.28366<br />Correlation of EstimatesVariableLabelInterceptd1___shrt_trm_bd_yd1___repo_rateInterceptIntercept1-0.02810.0809d1___shrt_trm_bd_yD1 - Shrt trm bd Y-0.02811-0.4701d1___repo_rateD1 - Repo Rate0.0809-0.47011<br />Test of First and SecondMoment SpecificationDFChi-SquarePr > ChiSq53.130.6795Durbin-Watson D1.661 Number of Observations125 1st Order Autocorrelation0.167 <br />After taking the difference R-square has reduced to 0%.
- 32. Both the independent variables are found to be insignificant.
- 33. Hence we do not find any substantial dependence of interest rates- short term yield and repo rate on the Money Supply.5.5 Reverse Model: WPI Growth dependent on Short term yield & Repo rate<br />Dependent Variable: WPI growth<br />Independent Variable: Short term yield, Repo rate<br />Number of Observations Read127 Number of Observations Used114 Number of Observations with Missing Values13 Analysis of VarianceSourceDFSum ofMeanF ValuePr > FSquaresSquareModel21.274630.637321.350.264Error11152.489390.47288 Corrected Total11353.76402 Root MSE0.68766R-Square0.0237 Dependent Mean0.44531Adj R-Sq0.0061 Coeff Var154.42132 <br />VariableLabelDFParameterStandardt ValuePr > |t|VarianceEstimateErrorInflation InterceptIntercept10.958090.376042.550.01220yield_on_short_termyield on short term1-0.045470.09865-0.460.64584.62648repo_rateRepo rate1-0.036270.11175-0.320.74614.62648<br />It is clearly seen from table that R-square is very low and F- value is also insignificant. Hence no dependence of WPI growth can be proved on change in short term yield & Repo rate.6. ARIMA MODELLING TO FORECAST SHORT TERM YIELD<br />6.1 ARIMA for Log- Short term yield data<br />Autocorrelation Check for White NoiseTo LagChi-SquareDFPr > ChiSqAutocorrelations6421.096<.00010.9290.8450.7640.6750.5880.50512473.6412<.00010.420.3370.2490.1640.0790.00218500.2218<.0001-0.063-0.107-0.14-0.179-0.22-0.25424586.6824<.0001-0.28-0.301-0.305-0.314-0.317-0.313<br />There is no white noise present as we are rejecting null hypothesis of presence of white noise.
- 34. There is a definite pattern in ACF, which shows presence of non stationary data.
- 35. Hence we apply trend differencing.
- 36. 6.2 ARIMA output for Log-Short term yield – with Trend DifferencingAutocorrelation Check for White NoiseTo LagChi-SquareDFPr > ChiSqAutocorrelations68.5360.20160.2030.0620.1180.058-0.0110.061211.32120.50170.020.0660.053-0.003-0.04-0.1051814.76180.67860.111-0.0450.081-0.045-0.0250.0182420.25240.68220.037-0.1710.025-0.005-0.056-0.042<br />After taking trend differencing, we find that white noise creeps into model. As we cannot forecast a random data, we will not forward.
- 37. Alternatively, we will apply seasonal differencing (12) only.6.3 ARIMA Output for Log- Short term Yield- with Seasonal Differencing<br />Autocorrelation Check for White NoiseTo LagChi-SquareDFPr > ChiSqAutocorrelations6351.886<.00010.9330.8360.7340.630.5270.42412380.1312<.00010.3190.2090.089-0.023-0.134-0.2318446.6918<.0001-0.268-0.281-0.283-0.289-0.298-0.3022449424<.0001-0.297-0.282-0.25-0.216-0.182-0.154<br />Augmented Dickey-Fuller Unit Root TestsTypeLagsRhoPr < RhoTauPr < TauFPr > FZero Mean0-5.53320.1035-1.590.1042 1-10.55330.0223-2.220.0261 2-11.810.0154-2.270.0231 Single Mean0-5.4340.3875-1.560.49981.480.6934 1-10.39480.1143-2.190.21242.550.4221 2-11.58930.0842-2.230.19572.650.3963Trend0-6.02940.7342-1.670.75721.430.8916 1-11.18010.3419-2.280.44222.60.6574 2-12.41120.2735-2.320.42042.70.6387<br />After taking seasonal differencing, there is no white noise in the system.
- 38. Also using Dickey Fuller test, we can say that data is now stationary.
- 39. Hence we can use this for forecasting. We will try to estimate the model through the ACF & PACF outputs.
- 40. ACF is declining exponentially and there is a spike in Non seasonal part of PACF above the X-axis. This indicates an AR – 1 component.
- 41. Also around 12, we find a spike in PACF on the seasonal part, which indicates a SAR – 1 component.
- 42. A MA-1 component as there is a spike in PACF on negative X-axis.
- 43. Hence the model is estimated to be (1,0,1)*(1,1,0)6.4 Forecasting ARIMA for Log- Short term Yield - with Seasonal Differencing<br />Forecasts for variable log_shortObsForecastStd Error95% Confidence Limits1280.91170.0470.81971.00381290.94590.07310.80271.08911300.93390.08970.75811.10971310.95460.10190.75491.15431320.96130.11140.74311.17961330.96970.11890.73661.20291340.96650.12510.72121.21171350.95890.13020.70371.21411360.95540.13450.69191.2191370.95510.1380.68461.22571380.98780.1410.71141.26421390.99370.14360.71221.27511400.98470.16060.66991.29951411.0140.17910.6631.3651420.99740.19360.61791.3771431.01390.20540.61131.41641441.01670.2150.59531.43811451.02150.2230.58441.45851461.01480.22970.56461.46491471.0040.23530.54291.4652<br />7. Key findings and CONCLUSIONS<br />Running Linear Regression Model of Short term bond yield on Macro factors, and eliminating autocorrelation and multi-co linearity, we found the explanatory variables to be insignificant.
- 44. Hence we conclude that we do not find a substantial dependence of short term yields on Macro factors, for the given time period data.
- 45. This re-confirms the point that interest rates in India are highly regulated and are controlled by RBI and the Ministry of Finance.
- 46. Even the attempt to find the reverse dependence, i.e. of Macro factors like Money Supply and WPI on yields & repo rates, did not find any evidence of a strong relationship.
- 47. Applying Univariate ARIMA model to Short term yield, we found that there is some seasonality in the time series data, but no trend. The model was estimated to be (1,0,1)*(1,1,0).
- 48. Forecasted value show that short term yield to fluctuate between 8.15% and 9.5% for next twenty months, starting from July 2011.