econometric forecasting of short term interest rates in india

Submitted To :Dr. Kakali Kanjilal

ProfessorIMT Ghaziabad

IMT Ghaziabad1 Institute of Management Technology Financial

Econometrics

Financial Econometric

Institute Of Management Technology, Ghaziabad

Project Report

Financial Econometrics

Under the guidance of Dr.Kakali Kanjilal, Professor

IMT - Ghaziabad

Submitted By:

Sumit Chugh (10DCP-042)

Vatan Lunia (10DCP-046)

Akash Jauhari (10DCP-056)

Alok Mishra (10DCP-057)

Ankit Bhardwaj (10DCP-060)

Raghav Agarwal (10DCP-087)

2 Institute of Management Technology Financial Econometrics

TABLE OF CONTENTS

1. Abstract...................................................................................................................................................................................... 3

2. Introduction: Short term T-bill yields in India..........................................................................................................4

2.1 Fluctuations in Security yields.................................................................................................................................5

3. Data.............................................................................................................................................................................................. 5

4. Methodology............................................................................................................................................................................ 6

5. Linear regression Models...................................................................................................................................................6

5.1 Short term yield dependent on Macro Factors..................................................................................................7

5.2 Short term yield dependent on growth variables............................................................................................8

5.3 Short term yield dependent on Macro factors-with Differencing..........................................................11

5.4 Reverse Model – M3 on interest rates -with Differencing.........................................................................13

5.5 Reverse Model: WPI Growth dependent on Short term yield & Repo rate........................................15

6. 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…………………………………..18

6.4 Forecasting ARIMA for Log- Short term Yield ...............................................................................................20

7. Key findings and Conclusion..........................................................................................................................................21


1. ABSTRACT

Key words: Short term yields, Linear Regression, ARIMA modeling

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.

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.


2. INTRODUCTION: SHORT TERM T-BILL YIELDS IN INDIA

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.

2.1 FLUCTUATIONS IN SECURITY YIELDS

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.

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Yield on 91-day T-bill in India

Shrt trm bd Y

Source: CMIE database.


High of about 9.1% in August 2008, low of about 3.2% in May 2009. Standard deviation of about 1.42 units considering data for Jan 2001 to Jun 2011.

3. DATA

Following Variables have been taken for the research:- Short term yield: 91 day T-bill yield WPI (Wholesale Price Index): Inflation Money Supply (M-3, RBI data) IIP (Index for Industrial Production) Exchange Rate (Rs per $, average) Repo rate (issued by RBI) Oil Prices (WTI rate in $ per barrel) Import & Exports for India (in million $) Sensex (proxy for stock markets in India)

The period of data is from January 2001 to June 2011. Frequency of data taken is monthly average.

Source of the data have been CMIE database.

4. METHODOLOGY

In this process we take short term yield as the dependent variable and try to regress

Linear Regression: Short term yield on Macro factorsFirst 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.

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.

Linear Regression: Reverse ModelWe 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.

ARIMA Modeling:


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.

5. LINEAR REGRESSION MODELS

5.1 SHORT TERM YIELD DEPENDENT ON MACRO FACTORS

Dependent Variable: Short term yield

Independent Variables: Repo rate, BOP, WPI, IIP, Exchange rate (Rs/$), Money Supply (M-3)/ Oil Prices ($/barrel).

Analysis of Variance

Source DFSum of Mean

F Value Pr > FSquares SquareModel 7 212.550

530.36436 152.95 <.0001

Error 107 21.24149

0.19852

Corrected Total 114 233.792 Root MSE 0.44555 R-

Square0.9091

Dependent Mean 5.84243 Adj R-Sq 0.9032

The model seems to be good as R-square is high and F- Value is significant.

Parameter Estimate Table -

Variable Label DF

Parameter

Standard

t Value Pr > |t|

Variance95% Confidence

LimitsEstimate Error Inflation

Intercept Intercep

t1 -5.60274 1.67195 -3.35 0.0011 0 -

8.91718-2.2883

repo_rate Repo Rate

1 1.20194 0.04763 25.23 <.0001 2.11469 1.10752 1.29636

IIP IIP 1 0.00817 0.00361 2.26 0.0258 22.43034

0.001 0.01533

exch_rate Exch rate

1 -0.04512 0.02984 -1.51 0.1335 3.57453 -0.10426

0.01403

Sensex Sensex 1 -0.000159

8

5.25E-05 -3.04 0.0001 12.99939

-0.00026

-5.6E-05


m3_oil_prc M3/Oil prc

1 -1.573E-05

4.18E-06 -3.76 0.0003 2.67557 -2.4E-05

-7.5E-06

imp___exp Imp – Exp

1 -9.099E-05

3.45E-05 -2.64 0.0096 9.91017 -0.00016

-2.3E-05

WPI WPI 1 0.02682 0.00913 2.94 0.004 42.35511

0.00873 0.04492

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 Estimates

Variable Label Interceptrepo_rat

e IIP exch_rate Sensex m3_oil_prcimp___ex

p WPIIntercept Intercept 1 -0.5688 0.2543 -0.6138 -

0.07740.4824 0.6601 -

0.5802repo_rate Repo

Rate-0.5688 1 -

0.0649-0.0619 -

0.24460.0448 -0.5164 0.4894

IIP IIP 0.2543 -0.0649 1 0.0265 -0.3711

-0.1759 0.341 -0.6626

exch_rate Exch rate

-0.6138 -0.0619 0.0265 1 0.5312 -0.5905 -0.0152 -0.182

Sensex Sensex -0.0774 -0.2446 -0.3711

0.5312 1 0.0455 0.0774 -0.2598

m3_oil_prc

M3/Oil prc

0.4824 0.0448 -0.1759

-0.5905 0.0455 1 0.2064 -0.0783

imp___exp Imp – Exp

0.6601 -0.5164 0.341 -0.0152 0.0774 0.2064 1 -0.7796

WPI WPI -0.5802 0.4894 -0.6626

-0.182 -0.2598

-0.0783 -0.7796 1

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 Specification

DF

Chi-Squar

ePr > ChiS

q35 27.88 0.7983

Durbin-Watson D 1.017 Number of Observations

115

1st Order Autocorrelation

0.491


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.

There is no hetroscedasticity, as P.0.79 and thus we accept the null hypothesis is absence of hetroscedasticity.

To improve the above model we go for –a. Consider growth of variables WPI,IIP and Sensexb. Differencing

5.2 SHORT TERM YIELD DEPENDENT ON GROWTH VARIABLES

Dependent Variable: Short term yield

Independent Variables: Repo rate, BOP, WTI Oil price ($/brl), WPI growth, IIP growth, Exchange rate (Rs/$), Money Supply (M-3)



F Value Pr > FSquares SquareModel 8 200.860

425.1075

5110.18 <.0001

Error 105 23.92779

0.22788

Corrected Total 113 224.7882

Root MSE 0.47737

R-Square

0.8936

Dependent Mean

5.81623

Adj R-Sq 0.8854

Coeff Var 8.20758

The model seems to be good as R-square is high and F-Value is significant

Parameter Estimates

Variable Label

Parameter

Standard

t Value

Pr > |t|

Variance

Estimate ErrorInflatio

nIntercept Intercept -1.1914 1.3447 -0.89 0.377 0


5 7repo_rate Repo Rate 1.01376 0.0509

319.91 <.000

11.9939

4imp___exp Imp – Exp -5.7E-05 3.3E-05 -1.71 0.09 7.8342

5wti_oil_prc____bbl_

WTI Oil prc ($/bbl)

0.02044 0.00505

4.05 <.0001

8.55729

wpi_growth WPI growth -0.31483 0.07623

-4.13 <.0001

1.3709

iip_growth IIP growth -0.00328 0.00791

-0.41 0.679 1.08034

exch_rate Exch rate -0.02163 0.02646

-0.82 0.4155

2.446

sensex_growth Sensex growth

-0.00422 0.00566

-0.75 0.4575

1.14073

M3 M3 1.10E-07 9.12E-08

1.2 0.2325

7.23784

The model has no multi-collinearity. Also, only repo rate, WTI oil price and WPI growth are significant at 5% level of significance.


Variable Labelrepo_ra

teimp___e

xpwti_oil_prc____b

bl_wpi_grow

thiip_grow

th M3Intercept Intercept -0.2527 0.3501 -0.6514 0.081 0.073 0.031

5repo_rate Repo

Rate1 -0.2078 -0.2361 0.213 0.043 0.598

3imp___exp Imp - Exp -0.2078 1 -0.5277 0.2467 0.1972 -

0.5741

wti_oil_prc____bbl_

WTI Oil prc ($/bbl)

-0.2361 -0.5277 1 -0.414 -0.1595 -0.283

9wpi_growth WPI

growth0.213 0.2467 -0.414 1 0.2209 0.071

1iip_growth IIP

growth0.043 0.1972 -0.1595 0.2209 1 -

0.0749


exch_rate Exch rate -0.0683 -0.2085 0.7011 -0.1307 -0.0754 -0.296

8sensex_growth Sensex

growth0.2374 0.0071 0.1029 0.0146 0.0021 -0.028

M3 M3 0.5983 -0.5741 -0.2839 0.0711 -0.0749 1

The correlation matrix does not show a strong positive or negative correlation between any two variables.


DF

Chi-Squar

ePr > ChiS

q44 37.65 0.7392


114


0.553

Since the model is suffering from positive autocorrelation, we applied the 1 st

order differencing. It resulted in a negative R-square and all variables were insignificant. Hence the model is void and no dependence is proved.

5.3 SHORT TERM YIELD DEPENDENT ON MACRO FACTORS-WITH DIFFERENCING

To remove autocorrelation from above model we go for trend differencing by one. Here are the outputs.

Number of Observations Read 127 Number of Observations Used 125 Number of Observations with Missing Values

2




F Value Pr > FSquares SquareModel 7 6.42942 0.91849 6.61 <.0001Error 117 16.2531 0.13892 Corrected Total 124 22.6825

2

Root MSE 0.37271 R-Square

0.2835

Dependent Mean -0.0048 Adj R-Sq 0.2406 Coeff Var -

7764.86

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.

Variable Label

Parameter

Standard

t Value

Pr > |t|

Squared

Squared

Variance

Estimate Error Partial PartialInflatio

n

Corr

Type ICorr

Type II Intercept Intercept 0.00853 0.03401 0.25 0.8024 . . 0d1___repo_rate

D1 - Repo Rate

0.73773 0.13421 5.5 <.0001 0.22098

0.20524

1.09572

d1___imp___exp

D1 - Imp – Exp

1.68E-05 1.81E-05

0.93 0.3564 0.00297

0.00728

1.27191

d1___wpi D1 - WPI -0.00261 0.00143 -1.82 0.071 0.02051

0.0276 1.01595

d1___iip D1 - IIP 0.00238 0.00203 1.17 0.2441 0.01714

0.01158

1.14287

d1___exch_rate

D1 - Exch rate

0.06259 0.05314 1.18 0.2413 0.02193

0.01172

1.44721

d1___sensex D1 - Sensex

-5.8E-05 7.71E-05

-0.75 0.4568 0.00236

0.00474

1.34529

d1___m3_oil_prc

D1 - M3/Oil prc

-9E-06 6.12E-06

-1.46 0.1467 0.01791

0.01791

1.15338

Correlation Matrix:

Variable Labelrepo_rat

eimp___ex

pd1___w

pid1___ii

pexch_rat

esense

xm3_oil_p

rcIntercept Interce

pt0.07 -0.09 0.06 -0.09 -0.04 -0.13 -0.06


d1___repo_rate

D1 - Repo Rate

1.00 -0.08 -0.01 -0.03 0.11 0.03 0.21

d1___imp___exp

D1 - Imp - Exp

-0.08 1.00 -0.05 0.25 0.28 0.20 0.20

d1___wpi D1 - WPI

-0.01 -0.05 1.00 -0.11 -0.04 -0.01 0.00

d1___iip D1 - IIP -0.03 0.25 -0.11 1.00 -0.12 -0.09 0.13d1___exch_rate

D1 - Exch rate

0.11 0.28 -0.04 -0.12 1.00 0.48 0.00

d1___sensex D1 - Sensex

0.03 0.20 -0.01 -0.09 0.48 1.00 0.15

d1___m3_oil_prc

D1 - M3/Oil prc

0.21 0.20 0.00 0.13 0.00 0.15 1.00


DF

Chi-Squar

ePr > ChiS

q35 31.58 0.6338


125


-0.094

Now we have eliminated autocorrelation. Also multi-co linearity and hetroscedasticity are not present.

However, 5 of the 7 variables have become insignificant.

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


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-1. The model is suffers from positive autocorrelation as the d value is very close to 0.2. Also there seems to be hetroscedasticity as we are rejecting hypothesis of absence of hetroscedasticity.3. To improve the model, we go for differencing.

Dependent Variable: Money Supply (M-3)Independent variables: Short term yield, Repo rate


2



F Value Pr > FSquares SquareModel 2 6.28E+0

83.14E+0

80.17 0.843

Error 122 2.24E+11

1.84E+09

Corrected Total 124 2.25E+11

Root MSE 42867 R-Square

0.0028

Dependent Mean 43277 Adj R-Sq -0.0136

Parameter Estimate

Variable Label DF

Parameter

Standard

t Value

Pr > |t|

Variance

Estimate ErrorInflatio

nIntercept Intercept 1 43280 3847.01

511.25 <.000

10

d1___shrt_trm_bd_y

D1 - Shrt trm bd Y

1 -5634.61 10198 -0.55 0.5816

1.28366

d1___repo_rate D1 - Repo Rate

1 1513.917 16708 0.09 0.928 1.28366



Variable LabelInterce

ptd1___shrt_trm_bd

_yd1___repo_ra

teIntercept Intercept 1 -0.0281 0.0809d1___shrt_trm_bd_y

D1 - Shrt trm bd Y

-0.0281 1 -0.4701

d1___repo_rate D1 - Repo Rate

0.0809 -0.4701 1


DF

Chi-Squar

ePr > ChiS

q5 3.13 0.6795


125


0.167

After taking the difference R-square has reduced to 0%. Both the independent variables are found to be insignificant. 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

Dependent Variable: WPI growth

Independent Variable: Short term yield, Repo rate


13



F Value Pr > FSquares SquareModel 2 1.27463 0.63732 1.35 0.264Error 111 52.4893

90.47288

Corrected Total 113 53.76402

Root MSE 0.68766 R-Square

0.0237

Dependent Mean 0.44531 Adj R-Sq 0.0061 Coeff Var 154.42132

Variable Label DF

Parameter Standard

t ValuePr > |

t|

VarianceEstimate Error Inflation

Intercept Intercept 1 0.95809 0.37604 2.55 0.0122 0yield_on_short_term yield on short

term1 -0.04547 0.09865 -0.46 0.6458 4.62648

repo_rate Repo rate 1 -0.03627 0.11175 -0.32 0.7461 4.62648

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

6.1 ARIMA for Log- Short term yield data

Autocorrelation Check for White Noise

To Lag Chi-Square DF Pr > ChiSq Autocorrelations6 421.09 6 <.0001 0.929 0.845 0.764 0.675 0.588 0.505

12 473.64 12 <.0001 0.42 0.337 0.249 0.164 0.079 0.00218 500.22 18 <.0001 -

0.063-

0.107-0.14 -

0.179-0.22 -

0.25424 586.68 24 <.0001 -0.28 -

0.301-

0.305-

0.314-

0.317-

0.313

There is no white noise present as we are rejecting null hypothesis of presence of white noise. There is a definite pattern in ACF, which shows presence of non stationary data.


Hence we apply trend differencing.

6.2 ARIMA output for Log-Short term yield – with Trend Differencing


To Lag Chi-Square DF Pr > ChiSq Autocorrelations6 8.53 6 0.2016 0.203 0.062 0.11

80.058 -

0.0110.06

12 11.32 12 0.5017 0.02 0.066 0.053

-0.003

-0.04 -0.105

18 14.76 18 0.6786 0.111 -0.045

0.081

-0.045

-0.025

0.018

24 20.25 24 0.6822 0.037 -0.171

0.025

-0.005

-0.056

-0.042

After taking trend differencing, we find that white noise creeps into model. As we cannot forecast a random data, we will not forward.

Alternatively, we will apply seasonal differencing (12) only.


6.3 ARIMA Output for Log- Short term Yield- with Seasonal Differencing


To Lag Chi-Square DF Pr > ChiSq Autocorrelations6 351.88 6 <.0001 0.933 0.836 0.734 0.63 0.527 0.424

12 380.13 12 <.0001 0.319 0.209 0.089 -0.023

-0.134

-0.23

18 446.69 18 <.0001 -0.268

-0.281

-0.283

-0.289

-0.298

-0.302

24 494 24 <.0001 -0.297

-0.282

-0.25 -0.216

-0.182

-0.154

Augmented Dickey-Fuller Unit Root Tests

Type Lags RhoPr < Rho Tau

Pr < Tau F Pr > F

Zero Mean

0 -5.5332 0.1035 -1.59 0.1042

1 -10.553

3

0.0223 -2.22 0.0261

2 -11.81 0.0154 -2.27 0.0231 Single Mean

0 -5.434 0.3875 -1.56 0.4998 1.48

0.6934

1 -10.394

8

0.1143 -2.19 0.2124 2.55

0.4221

2 -11.589

3

0.0842 -2.23 0.1957 2.65

0.3963

Trend 0 -6.0294 0.7342 -1.67 0.7572 1.43

0.8916

1 -11.180

1

0.3419 -2.28 0.4422 2.6 0.6574

2 -12.411

2

0.2735 -2.32 0.4204 2.7 0.6387


After taking seasonal differencing, there is no white noise in the system. Also using Dickey Fuller test, we can say that data is now stationary. Hence we can use this for forecasting. We will try to estimate the model through the ACF &

PACF outputs.

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.

Also around 12, we find a spike in PACF on the seasonal part, which indicates a SAR – 1 component.

A MA-1 component as there is a spike in PACF on negative X-axis.

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

Forecasts for variable log_short

Obs Forecast Std Error95% Confidence

Limits128 0.9117 0.047 0.8197 1.0038129 0.9459 0.0731 0.8027 1.0891130 0.9339 0.0897 0.7581 1.1097131 0.9546 0.1019 0.7549 1.1543132 0.9613 0.1114 0.7431 1.1796133 0.9697 0.1189 0.7366 1.2029134 0.9665 0.1251 0.7212 1.2117135 0.9589 0.1302 0.7037 1.2141136 0.9554 0.1345 0.6919 1.219137 0.9551 0.138 0.6846 1.2257138 0.9878 0.141 0.7114 1.2642139 0.9937 0.1436 0.7122 1.2751140 0.9847 0.1606 0.6699 1.2995141 1.014 0.1791 0.663 1.365142 0.9974 0.1936 0.6179 1.377143 1.0139 0.2054 0.6113 1.4164144 1.0167 0.215 0.5953 1.4381145 1.0215 0.223 0.5844 1.4585146 1.0148 0.2297 0.5646 1.4649147 1.004 0.2353 0.5429 1.4652


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

2

4

6

8

10

12

Forecast for Short term yields-for next 20 months, starting from July 2011

Forecast

7. KEY FINDINGS AND CONCLUSIONS

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.

Hence we conclude that we do not find a substantial dependence of short term yields on Macro factors, for the given time period data.

This re-confirms the point that interest rates in India are highly regulated and are controlled by RBI and the Ministry of Finance.

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.

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).


Forecasted value show that short term yield to fluctuate between 8.15% and 9.5% for next twenty months, starting from July 2011.


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