Description
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.
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Institute of Management Technology Financial Econometrics
Forecasting the Short term yield &
finding its dependence on the
Macro factors
2011
Submitted To :
Dr. Kakali Kanjilal
Professor
IMT Ghaziabad
IMT Ghaziabad
Financial Econometric
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Institute of Management Technology Financial Econometrics
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)
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TABLE OF CONTENTS
1. Abstract ................................................................................................................................................................................... 4
2. Introduction: Short term T-bill yields in India ....................................................................................................... 5
2.1 Fluctuations in Security yields ............................................................................................................................... 5
3. Data ........................................................................................................................................................................................... 6
4. Methodology ......................................................................................................................................................................... 6
5. Linear regression Models ................................................................................................................................................ 7
5.1 Short term yield dependent on Macro Factors ............................................................................................... 7
5.2 Short term yield dependent on growth variables .......................................................................................... 9
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
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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.
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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.
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.
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Yield on 91-day T-bill in India
Shrt trm bd Y
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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 $)
? Sensex (proxy for stock markets in India)
? The period of data is from January 2001 to June 2011. Frequency of data taken is monthly
average.
? Source of the data have been CMIE database.
4. METHODOLOGY
In this process we take short term yield as the dependent variable and try to regress
? Linear Regression: Short term yield on Macro factors
First we attempt to find the dependence of Short term yield on Macro factors like WPI, M-3 and
Exchange rate etc. We used SAS for getting various outputs.
? Based on the principles and issues of linear regression like autocorrelation, multi-co linearity,
hetroscedasticity etc, we tried to improve our model at every step. Techniques like differencing,
taking growth figures, log data were used.
? Linear Regression: Reverse Model
We also tried to find any dependence of short term yield and Repo rate on Money supply and
Inflation. Again linear regression though SAS was used.
? ARIMA Modeling:
We tried to find out best estimated model based on Univariate ARIMA modeling. Based on the
model we tried to forecast short term interest rates for next 20 months. Also to check our model we
applied ex-post test on available data.
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5. LINEAR REGRESSION MODELS
5. 1 SHORT TERM YIELD DEPENDENT ON MACRO FACTORS
Dependent Variable: Short term yield
Independent Variables: Repo rate, BOP, WPI, IIP, Exchange rate (Rs/$), Money Supply (M-3)/ Oil Prices
($/barrel).
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 7 212.5505 30.36436 152.95 <.0001
Error 107 21.24149 0.19852
Corrected Total 114 233.792
Root MSE 0.44555 R-Square 0.9091
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|
Variance
95% Confidence
Limits
Estimate Error Inflation
Intercept Intercept 1 -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 -
0.0001598
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.
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Correlation of Estimates
Variable Label Intercept repo_rate IIP exch_rate Sensex m3_oil_prc imp___exp WPI
Intercept Intercept 1 -0.5688 0.2543 -0.6138 -
0.0774
0.4824 0.6601 -
0.5802
repo_rate Repo
Rate
-0.5688 1 -
0.0649
-0.0619 -
0.2446
0.0448 -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 Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
35 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 Sensex
b. Differencing
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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/$), Money Supply (M-3)
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 8 200.8604 25.10755 110.18 <.0001
Error 105 23.92779 0.22788
Corrected Total 113 224.7882
Root MSE 0.47737 R-Square 0.8936
Dependent
Mean
5.81623 Adj R-Sq 0.8854
Coeff Var 8.20758
? The model seems to be good as R-square is high and F-Value is significant
Parameter Estimates
Variable Label
Parameter Standard
t Value Pr > |t|
Variance
Estimate Error Inflation
Intercept Intercept -1.1914 1.34475 -0.89 0.3777 0
repo_rate Repo Rate 1.01376 0.05093 19.91 <.0001 1.99394
imp___exp Imp – Exp -5.7E-05 3.3E-05 -1.71 0.09 7.83425
wti_oil_prc____bbl_ WTI Oil prc
($/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.
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Correlation of Estimates
Variable Label repo_rate imp___exp wti_oil_prc____bbl_ wpi_growth iip_growth M3
Intercept Intercept -0.2527 0.3501 -0.6514 0.081 0.073 0.0315
repo_rate Repo Rate 1 -0.2078 -0.2361 0.213 0.043 0.5983
imp___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 -0.2839
wpi_growth WPI
growth
0.213 0.2467 -0.414 1 0.2209 0.0711
iip_growth IIP growth 0.043 0.1972 -0.1595 0.2209 1 -0.0749
exch_rate Exch rate -0.0683 -0.2085 0.7011 -0.1307 -0.0754 -0.2968
sensex_growth Sensex
growth
0.2374 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.
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
44 37.65 0.7392
Durbin-Watson D 0.882
Number of
Observations
114
1st Order
Autocorrelation
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.
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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
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 7 6.42942 0.91849 6.61 <.0001
Error 117 16.2531 0.13892
Corrected Total 124 22.68252
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
Paramete
r
Standar
d
t Valu
e
Pr > |t
|
Square
d
Square
d
Varianc
e
Estimate Error Partial Partial
Inflatio
n
Corr
Type I
Corr
Type II
Intercept Intercept 0.00853 0.03401 0.25 0.8024 . . 0
d1___repo_rate D1 - Repo
Rate
0.73773 0.13421 5.5 <.0001 0.2209
8
0.2052
4
1.09572
d1___imp___ex
p
D1 - Imp –
Exp
1.68E-05 1.81E-
05
0.93 0.3564 0.0029
7
0.0072
8
1.27191
d1___wpi D1 - WPI -0.00261 0.00143 -1.82 0.071 0.0205
1
0.0276 1.01595
d1___iip D1 - IIP 0.00238 0.00203 1.17 0.2441 0.0171
4
0.0115
8
1.14287
d1___exch_rate D1 - Exch
rate
0.06259 0.05314 1.18 0.2413 0.0219
3
0.0117
2
1.44721
d1___sensex D1 -
Sensex
-5.8E-05 7.71E-
05
-0.75 0.4568 0.0023
6
0.0047
4
1.34529
d1___m3_oil_p
rc
D1 -
M3/Oil
prc
-9E-06 6.12E-
06
-1.46 0.1467 0.0179
1
0.0179
1
1.15338
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Correlation Matrix:
Variable Label repo_rate imp___exp d1___wpi d1___iip exch_rate sensex m3_oil_prc
Intercept Intercept 0.07 -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 0.13
d1___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
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
35 31.58 0.6338
Durbin-Watson D 2.181
Number of
Observations
125
1st Order
Autocorrelation
-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.
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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
Number of Observations Read 127
Number of Observations Used 125
Number of Observations with Missing
Values
2
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 2 6.28E+08 3.14E+08 0.17 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 Error Inflation
Intercept Intercept 1 43280 3847.015 11.25 <.0001 0
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
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Correlation of Estimates
Variable Label Intercept d1___shrt_trm_bd_y d1___repo_rate
Intercept Intercept 1 -0.0281 0.0809
d1___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
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
5 3.13 0.6795
Durbin-Watson D 1.661
Number of
Observations
125
1st Order
Autocorrelation
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.
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5. 5 REVERSE MODEL: WPI GROWTH DEPENDENT ON SHORT TERM YIELD &
REPO RATE
Dependent Variable: WPI growth
Independent Variable: Short term yield, Repo rate
Number of Observations Read 127
Number of Observations Used 114
Number of Observations with Missing
Values
13
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 2 1.27463 0.63732 1.35 0.264
Error 111 52.48939 0.47288
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 Value
Pr > |t
|
Variance
Estimate Error Inflation
Intercept Intercept 1 0.95809 0.37604 2.55 0.0122 0
yield_on_short_term yield on short
term
1 -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.
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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 Autocorrelations
6 421.09 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 0.002
18 500.22 18 <.0001 -
0.063
-
0.107
-0.14 -
0.179
-0.22 -
0.254
24 586.68 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.
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6.2 ARIMA output for Log-Short term yield – with Trend Differencing
Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 8.53 6 0.2016 0.203 0.062 0.118 0.058 -
0.011
0.06
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.
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6.3 ARIMA Output for Log- Short term Yield- with Seasonal Differencing
Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 351.88 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 Rho
Pr <
Rho Tau
Pr <
Tau F Pr > F
Zero
Mean
0 -5.5332 0.1035 -1.59 0.1042
1 -
10.5533
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 -
10.3948
0.1143 -2.19 0.2124 2.55 0.4221
2 -
11.5893
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 -
11.1801
0.3419 -2.28 0.4422 2.6 0.6574
2 -
12.4112
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.
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? 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)
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6. 4 FORECASTING ARIMA FOR LOG- SHORT TERM YIELD - WITH SEASONAL
DIFFERENCING
Forecasts for variable log_short
Obs Forecast Std Error
95% Confidence
Limits
128 0.9117 0.047 0.8197 1.0038
129 0.9459 0.0731 0.8027 1.0891
130 0.9339 0.0897 0.7581 1.1097
131 0.9546 0.1019 0.7549 1.1543
132 0.9613 0.1114 0.7431 1.1796
133 0.9697 0.1189 0.7366 1.2029
134 0.9665 0.1251 0.7212 1.2117
135 0.9589 0.1302 0.7037 1.2141
136 0.9554 0.1345 0.6919 1.219
137 0.9551 0.138 0.6846 1.2257
138 0.9878 0.141 0.7114 1.2642
139 0.9937 0.1436 0.7122 1.2751
140 0.9847 0.1606 0.6699 1.2995
141 1.014 0.1791 0.663 1.365
142 0.9974 0.1936 0.6179 1.377
143 1.0139 0.2054 0.6113 1.4164
144 1.0167 0.215 0.5953 1.4381
145 1.0215 0.223 0.5844 1.4585
146 1.0148 0.2297 0.5646 1.4649
147 1.004 0.2353 0.5429 1.4652
0
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8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Forecast for Short term yields-for next 20 months, starting from July 2011
Forecast
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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.
doc_977723096.docx
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.
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Forecasting the Short term yield &
finding its dependence on the
Macro factors
2011
Submitted To :
Dr. Kakali Kanjilal
Professor
IMT Ghaziabad
IMT Ghaziabad
Financial Econometric
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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)
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TABLE OF CONTENTS
1. Abstract ................................................................................................................................................................................... 4
2. Introduction: Short term T-bill yields in India ....................................................................................................... 5
2.1 Fluctuations in Security yields ............................................................................................................................... 5
3. Data ........................................................................................................................................................................................... 6
4. Methodology ......................................................................................................................................................................... 6
5. Linear regression Models ................................................................................................................................................ 7
5.1 Short term yield dependent on Macro Factors ............................................................................................... 7
5.2 Short term yield dependent on growth variables .......................................................................................... 9
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
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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.
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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.
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.
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Yield on 91-day T-bill in India
Shrt trm bd Y
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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 $)
? Sensex (proxy for stock markets in India)
? The period of data is from January 2001 to June 2011. Frequency of data taken is monthly
average.
? Source of the data have been CMIE database.
4. METHODOLOGY
In this process we take short term yield as the dependent variable and try to regress
? Linear Regression: Short term yield on Macro factors
First we attempt to find the dependence of Short term yield on Macro factors like WPI, M-3 and
Exchange rate etc. We used SAS for getting various outputs.
? Based on the principles and issues of linear regression like autocorrelation, multi-co linearity,
hetroscedasticity etc, we tried to improve our model at every step. Techniques like differencing,
taking growth figures, log data were used.
? Linear Regression: Reverse Model
We also tried to find any dependence of short term yield and Repo rate on Money supply and
Inflation. Again linear regression though SAS was used.
? ARIMA Modeling:
We tried to find out best estimated model based on Univariate ARIMA modeling. Based on the
model we tried to forecast short term interest rates for next 20 months. Also to check our model we
applied ex-post test on available data.
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5. LINEAR REGRESSION MODELS
5. 1 SHORT TERM YIELD DEPENDENT ON MACRO FACTORS
Dependent Variable: Short term yield
Independent Variables: Repo rate, BOP, WPI, IIP, Exchange rate (Rs/$), Money Supply (M-3)/ Oil Prices
($/barrel).
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 7 212.5505 30.36436 152.95 <.0001
Error 107 21.24149 0.19852
Corrected Total 114 233.792
Root MSE 0.44555 R-Square 0.9091
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|
Variance
95% Confidence
Limits
Estimate Error Inflation
Intercept Intercept 1 -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 -
0.0001598
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.
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Correlation of Estimates
Variable Label Intercept repo_rate IIP exch_rate Sensex m3_oil_prc imp___exp WPI
Intercept Intercept 1 -0.5688 0.2543 -0.6138 -
0.0774
0.4824 0.6601 -
0.5802
repo_rate Repo
Rate
-0.5688 1 -
0.0649
-0.0619 -
0.2446
0.0448 -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 Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
35 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 Sensex
b. Differencing
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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/$), Money Supply (M-3)
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 8 200.8604 25.10755 110.18 <.0001
Error 105 23.92779 0.22788
Corrected Total 113 224.7882
Root MSE 0.47737 R-Square 0.8936
Dependent
Mean
5.81623 Adj R-Sq 0.8854
Coeff Var 8.20758
? The model seems to be good as R-square is high and F-Value is significant
Parameter Estimates
Variable Label
Parameter Standard
t Value Pr > |t|
Variance
Estimate Error Inflation
Intercept Intercept -1.1914 1.34475 -0.89 0.3777 0
repo_rate Repo Rate 1.01376 0.05093 19.91 <.0001 1.99394
imp___exp Imp – Exp -5.7E-05 3.3E-05 -1.71 0.09 7.83425
wti_oil_prc____bbl_ WTI Oil prc
($/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.
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Correlation of Estimates
Variable Label repo_rate imp___exp wti_oil_prc____bbl_ wpi_growth iip_growth M3
Intercept Intercept -0.2527 0.3501 -0.6514 0.081 0.073 0.0315
repo_rate Repo Rate 1 -0.2078 -0.2361 0.213 0.043 0.5983
imp___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 -0.2839
wpi_growth WPI
growth
0.213 0.2467 -0.414 1 0.2209 0.0711
iip_growth IIP growth 0.043 0.1972 -0.1595 0.2209 1 -0.0749
exch_rate Exch rate -0.0683 -0.2085 0.7011 -0.1307 -0.0754 -0.2968
sensex_growth Sensex
growth
0.2374 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.
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
44 37.65 0.7392
Durbin-Watson D 0.882
Number of
Observations
114
1st Order
Autocorrelation
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.
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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
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 7 6.42942 0.91849 6.61 <.0001
Error 117 16.2531 0.13892
Corrected Total 124 22.68252
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
Paramete
r
Standar
d
t Valu
e
Pr > |t
|
Square
d
Square
d
Varianc
e
Estimate Error Partial Partial
Inflatio
n
Corr
Type I
Corr
Type II
Intercept Intercept 0.00853 0.03401 0.25 0.8024 . . 0
d1___repo_rate D1 - Repo
Rate
0.73773 0.13421 5.5 <.0001 0.2209
8
0.2052
4
1.09572
d1___imp___ex
p
D1 - Imp –
Exp
1.68E-05 1.81E-
05
0.93 0.3564 0.0029
7
0.0072
8
1.27191
d1___wpi D1 - WPI -0.00261 0.00143 -1.82 0.071 0.0205
1
0.0276 1.01595
d1___iip D1 - IIP 0.00238 0.00203 1.17 0.2441 0.0171
4
0.0115
8
1.14287
d1___exch_rate D1 - Exch
rate
0.06259 0.05314 1.18 0.2413 0.0219
3
0.0117
2
1.44721
d1___sensex D1 -
Sensex
-5.8E-05 7.71E-
05
-0.75 0.4568 0.0023
6
0.0047
4
1.34529
d1___m3_oil_p
rc
D1 -
M3/Oil
prc
-9E-06 6.12E-
06
-1.46 0.1467 0.0179
1
0.0179
1
1.15338
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Correlation Matrix:
Variable Label repo_rate imp___exp d1___wpi d1___iip exch_rate sensex m3_oil_prc
Intercept Intercept 0.07 -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 0.13
d1___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
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
35 31.58 0.6338
Durbin-Watson D 2.181
Number of
Observations
125
1st Order
Autocorrelation
-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.
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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
Number of Observations Read 127
Number of Observations Used 125
Number of Observations with Missing
Values
2
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 2 6.28E+08 3.14E+08 0.17 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 Error Inflation
Intercept Intercept 1 43280 3847.015 11.25 <.0001 0
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
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Correlation of Estimates
Variable Label Intercept d1___shrt_trm_bd_y d1___repo_rate
Intercept Intercept 1 -0.0281 0.0809
d1___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
Test of First and Second
Moment Specification
DF
Chi-
Square Pr > ChiSq
5 3.13 0.6795
Durbin-Watson D 1.661
Number of
Observations
125
1st Order
Autocorrelation
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.
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5. 5 REVERSE MODEL: WPI GROWTH DEPENDENT ON SHORT TERM YIELD &
REPO RATE
Dependent Variable: WPI growth
Independent Variable: Short term yield, Repo rate
Number of Observations Read 127
Number of Observations Used 114
Number of Observations with Missing
Values
13
Analysis of Variance
Source DF
Sum of Mean
F Value Pr > F Squares Square
Model 2 1.27463 0.63732 1.35 0.264
Error 111 52.48939 0.47288
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 Value
Pr > |t
|
Variance
Estimate Error Inflation
Intercept Intercept 1 0.95809 0.37604 2.55 0.0122 0
yield_on_short_term yield on short
term
1 -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.
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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 Autocorrelations
6 421.09 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 0.002
18 500.22 18 <.0001 -
0.063
-
0.107
-0.14 -
0.179
-0.22 -
0.254
24 586.68 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.
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6.2 ARIMA output for Log-Short term yield – with Trend Differencing
Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 8.53 6 0.2016 0.203 0.062 0.118 0.058 -
0.011
0.06
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.
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Institute of Management Technology Financial Econometrics
6.3 ARIMA Output for Log- Short term Yield- with Seasonal Differencing
Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 351.88 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 Rho
Pr <
Rho Tau
Pr <
Tau F Pr > F
Zero
Mean
0 -5.5332 0.1035 -1.59 0.1042
1 -
10.5533
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 -
10.3948
0.1143 -2.19 0.2124 2.55 0.4221
2 -
11.5893
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 -
11.1801
0.3419 -2.28 0.4422 2.6 0.6574
2 -
12.4112
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.
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Institute of Management Technology Financial Econometrics
? 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)
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Institute of Management Technology Financial Econometrics
6. 4 FORECASTING ARIMA FOR LOG- SHORT TERM YIELD - WITH SEASONAL
DIFFERENCING
Forecasts for variable log_short
Obs Forecast Std Error
95% Confidence
Limits
128 0.9117 0.047 0.8197 1.0038
129 0.9459 0.0731 0.8027 1.0891
130 0.9339 0.0897 0.7581 1.1097
131 0.9546 0.1019 0.7549 1.1543
132 0.9613 0.1114 0.7431 1.1796
133 0.9697 0.1189 0.7366 1.2029
134 0.9665 0.1251 0.7212 1.2117
135 0.9589 0.1302 0.7037 1.2141
136 0.9554 0.1345 0.6919 1.219
137 0.9551 0.138 0.6846 1.2257
138 0.9878 0.141 0.7114 1.2642
139 0.9937 0.1436 0.7122 1.2751
140 0.9847 0.1606 0.6699 1.2995
141 1.014 0.1791 0.663 1.365
142 0.9974 0.1936 0.6179 1.377
143 1.0139 0.2054 0.6113 1.4164
144 1.0167 0.215 0.5953 1.4381
145 1.0215 0.223 0.5844 1.4585
146 1.0148 0.2297 0.5646 1.4649
147 1.004 0.2353 0.5429 1.4652
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Forecast for Short term yields-for next 20 months, starting from July 2011
Forecast
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Institute of Management Technology Financial Econometrics
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.
doc_977723096.docx