Summer Internship Project

Submitted to:

Professor Gopal Naik,

Chairperson, Centre for Public Policy

Indian Institute of Management, Bangalore

By:

Dinkar Pawan

Mayank Nagpal

M.sc Economics

Indira Gandhi Institute of Development Research

“Is the Pepper Commodity Market Efficient? “

A Cointegration Analysis

Abstract

In this paper, we attempt to examine the short term and long term dynamics, as well as the efficiency of the Pepper Commodity Market by analyzing the daily time series of the Spot and Futures prices of pepper. The econometric technique used for this purpose is the Cointegration Analysis, through the Engle Granger methodology. A significant relationship between the futures (or spot) prices and the lagged futures as well as lagged spot prices, is testimonial of market inefficiency. Moreover, in an efficient market, in the long run, we expect a one–on–one relationship between the two market prices. We find that this is indeed the case in the pepper market. We also attempt to forecast spot prices and compare our estimates with the futures prices in the Appendix.

Introduction

The Spot price or Spot rate of a commodity, a security or a currency is the price that is quoted for immediate (spot) settlement (payment and delivery). Spot settlement is normally one or two business days from trade date. This is in contrast with the Forward price established in a Forward contract or Futures contract, where contract terms (prices) are set now, but delivery and payment will occur at a future date. Alternatively, Futures price refers to the price at which parties to a Futures contract agree to transact upon a future settlement date.

The Spot-Futures parity is a condition that should theoretically hold for the markets to be efficient. In the case where this condition fails, arbitrage opportunities will exist, indicating an inefficient market. In plain English, if I can purchase a good today for price ‘S’ and conclude a contract to sell it one month from today for price ‘F’, the difference in these two prices should be no greater than the cost of using money minus any expenses from holding the asset. If the difference is greater, an opportunity to buy and sell the "spots" and "futures" for a risk-free profit would exist.

In theory, the equilibrium futures price should be equal to the spot price plus the cost of carry, which is defined as the sum of the cost of storage plus the interest rate forgone. In other words, it is the cost of carrying the contract to the future date. As the costs of storage are fairly constant within the storage season, spot and futures prices of the commodity should move together, i.e. an increase or a decrease in any one of them should cause a change of a similar magnitude in the other. The failure of such a relationship between the two would make us believe of unexplained inefficiencies in the market.

The Market Mechanism

The relationship between the two prices indicates the state of the market or the state of the immediate demand and supply. A case where the spot prices exceed the futures price (Backwardation) signals a shortage of the physical commodity in the market. The opposite condition of Contango, where the futures price exceeds the spot price, signals an excess of the commodity in the market. Anything that hinders the steady flow of the commodity tends to drive the market into Backwardation. As the futures contract approaches its maturity date, the difference between the two prices gets smaller as the cost of carry becomes smaller. At maturity, the difference between the two diminishes to zero because spot and futures prices converge. While spot and futures prices might significantly diverge over the life of the futures contract, futures prices have to converge to spot prices once the contract expires. This theoretical connection between the spot and the futures prices is a long-run, rather than a short-run, concept.

In the short-run, there might be deviations between spot prices and futures prices caused by thin trading, lags in information transmission, insufficient inventory levels, seasonal patterns of consumption and many such factors, which may cause the markets to function inefficiently. But, in the long-run, spot and futures prices are driven by the same fundamentals, such as interest rates, macroeconomic variables and oil reserves, because futures prices represent nothing but the expectations of the future spot prices of the physical commodity. Thus we should expect spot and futures prices for any commodity to be linked through a long-run equilibrium relationship.

Data and Methodology

We study this relationship by the econometric concept of Cointegration. Two time-series variables (in our case, spot prices and futures prices) are said to be cointegrated if there exists a long-run equilibrium relationship between the two. We also attempt to fit in an Error Correction Model (ECM), because it corrects for deviations from equilibrium in the past time period and explains the short term dynamics of the market. The fundamental aim of the cointegration analysis is to detect any common stochastic trends in the data, and to use these common trends for an analysis of the data. For our analysis we have used the data available on the National Commodity & Derivatives Exchange Limited (NCDEX) website. We have used daily data for ’Average spot price’ and ‘Closing futures price’ for the period of January 2005 to April 2009 for our analysis. The econometric tests and analysis were performed using the Econometric software STATA.

Data Description

6000

800010000120001400016000

1st Jan 2005 29th April 2009Time (Daily)

Spot Price Futures Price

Spot Price v/s Futures Price (Rs./Quintal)

The figure above shows the time series plots of both the variables, while the table gives the summary statistics. Both the series seem to have a stochastic trend rather than a deterministic trend when observing the plots with the naked eye. This indicates the presence of a unit root. Tests suggested to detect the presence of unit roots are the Augmented Dickey-Fuller Test (ADF) and the Phillips–Perron (PP) test. The ADF test results for the ‘chosen model’ for both the series are as follows.

The ‘spot’ series

Null Hypothesis: ‘Spot’ has a unit root; Model: No constant, No trend

Lag Length: 98; Test Statistic Z(t): 0.445

Level of Significance 1% 5% 10%

Critical Value -2.58 -1.95 -1.62

The ‘futures’ series

Null Hypothesis: ‘Futures’ has a unit root; Model: No constant, No trend

Lag Length: 98; Test Statistic Z(t): 0.463

Level of Significance 1% 5% 10%

Critical Value -2.58 -1.95 -1.62

Price (Rs / Quintal)

No. of Observations Mean Standard Deviation Min Max

Spot 1287 10778.79 3278.363 6003.68 15771.25

Futures 1287 10672.55 3228.99 6032 16344

Since the test statistic is not more negative than the critical value at all levels of significance, we do not reject the null hypothesis of the presence of at least one unit root in both the series. This is a clear case of stochastic volatility which can be done away with through first differencing of the data. The following are the first differenced time plots of the spot and futures series. The figures show random fluctuations around the value 0, over time, indicating the absence of unit roots in the two differenced series.

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Time Plot of First Differenced Spot Prices

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Time Plot of First Differenced Futures Prices

To test for the presence of a higher order of non stationarity we apply the ADF test to the first differenced values. The results are as follows:

The ‘First differenced spot’ series

Null Hypothesis: ‘First Differenced Spot’ has a unit root; Model: No constant, No trend

Lag Length: 98; Test Statistic Z(t): -3.698*

Level of Significance 1% 5% 10%

Critical Value -2.58 -1.95 -1.62

The ‘First differenced futures’ series

Null Hypothesis: ‘First Differenced Futures’ has a unit root; Model: No constant, No trend

Lag Length: 98; Test Statistic Z(t): -3.865*

Level of Significance 1% 5% 10%

Critical Value -2.58 -1.95 -1.62

In both the cases, the value of the test statistic is more negative than the critical value at all the levels of significance. Thus we reject the null hypothesis of the presence of a unit root in the first differenced series, implying that these series are stationary. According to definition, if a series, y

t must be differenced d times before it becomes stationary, then it is said to be integrated of order

d. This would be written as y t ~ I(d). So, if yt ~ I(d), then, ∆d yt ~ I(0). The latter piece of terminology states that applying the difference operator, ∆, d times, leads to an I(0) process, with no unit roots. The majority of financial and economic time series contain a single unit root. In our case, we needed to difference, both the ‘spot’ and the ‘futures’ series once to make them stationary, therefore, both are integrated of order 1 or simply, I(1).

The Cointegration Analysis

A set of variables is defined as cointegrated if a linear combination of them is stationary. Many time series are non-stationary but ‘move together’ over time – that is, there exists some influences on the series (for example, market forces), which imply that the two series are bound by some relationship in the long run. A cointegrating relationship may also be seen as an equilibrium phenomenon. In the Spot and Futures market for a commodity (or an asset), market forces arising from no arbitrage conditions suggests that there should be an equilibrium relationship between the series concerned. Spot and futures prices of a commodity are expected to be cointegrated as they are prices for the same commodity at different points in time, and hence will be affected in very similar ways by given pieces of information. If a set of series are cointegrated in levels, they would also be cointegrated in log-levels.

The Engle Granger 2 Step Method – A Residual Based Approach

This is a single equation technique which is conducted as follows. The first step is to ensure that all the variables are I(1). Then, we estimate the cointegrating regression using Ordinary least Squares and store the estimated residuals. Test these residuals to be I(0). If they are I(0), the variables concerned are said to be cointegrated. The stationary linear combination of the non-stationary variables is also known as the cointegrating vector. Finally, we fit in the Error Correction Models (ECMs) to determine the short term dynamics of the market and its efficiency.

The Model

Ft = α + β (St) + Ut

Here, Ft, represents the futures prices and St, represents the spot prices. The constant α represents the cost of carry charges or the difference between the two prices. The parameter β measures the responsiveness of the futures price as a result of a unit increase in the spot price. Ut represents the error term (residuals). The estimated regression results are as follows:

Parameter �� ��

Estimated Value 101.787 0.9807

Since it is not possible to perform any hypothesis tests on these estimated parameter values as both series are non-stationary, the corresponding standard errors and the respective p-values have not been reported. The following figures give us the time plot of the estimated residuals and the plot of the residuals against its own lagged values.

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Residuals

1st Jan 2005 29th April 2009Time (Daily)

Time Plot of Residuals

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Residuals

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Lagged Residuals

Residuals v/s Lagged Residuals

For the ‘spot’ and the ‘futures’ series to be cointegrated, the estimated residuals should be

stationary, or integrated of order 0, i.e. I(0). Since this is a test on residuals of a model, ∧

U t, then the critical values are different from those used in a DF or an ADF test on a series of raw data.

We can also use the Durbin-Watson (DW) test statistic to test for non-stationarity of ∧

U t. If the DW test is applied to the residuals of the potentially cointegrating regression, it is known as the Cointegrating Regression Durbin Watson (CRDW). Unfortunately, this test only works when the residuals follow a first-order autoregressive process (no higher-order of Auto-Regression in the residuals). This makes the CRDW unsuitable in all but very limited situations. A look at the Correlograms of the estimated residuals indicates the possibility of the order of AR being greater than 1.

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LagBartlett's formula for MA(q) 95% confidence bands

Autocorrelations of Residuals

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Lag95% Confidence bands [se = 1/sqrt(n)]

Partial autocorrelations of Residuals

For our purpose we use the Phillips-Perron Test to test for the stationarity of the estimated residuals.

Null Hypotheses, H0: Absence of long term equilibrium between the ‘spot’ and ‘futures’ prices

∧

U t ~ I(1)

Alternative Hypotheses, H1: Convergence of both series in the long term

∧

U t ~ I(0)

Thus, under the null hypotheses there is a unit root in the potentially cointegrating regression residuals, while under the alternative, the estimated residuals are stationary. Hence if the null hypothesis is rejected, there is cointegration between the spot and futures prices. The results of the Phillips-Perron test for stationarity on the estimated residuals are as follows:

Lag Length: 98; Test Statistic Z(t): -19.912*

Test Statistic Z(Rho): -302.478*

Level of Significance 1% 5% 10%

Critical Value Z(Rho) -13.800 -8.100 -5.700

Critical Value Z(t) -2.580 -1.950 -1.620

As evident from the values of the test statistic, both of which are more negative than the respective critical values at all levels of significance, we conclude that the residuals are stationary which in turn implies that there exists a series of linear combinations of the spot and the futures prices which is stationary, even though the two themselves are not individually stationary. Alternatively, we say that the series of spot and futures prices are cointegrated with each other, i.e. an increase or a decrease in any one of them causes a similar change in the other variable.

The Error Correction Models (ECM)

These are models that use combinations of first differenced and lagged levels of cointegrated variables. These are also known as the Equilibrium Correction Models.

∆Ft = ø1 ∆St + ø2 (∧

U t-1) + Vt

Regression Results

First Differenced Futures Coefficient Standard Error t-stat P > t 95% Confidence Interval

First Differenced Spot*

1.035045

0.0500673

20.67

0.000

0.9368225

1.133268

Lagged Residuals* -0.2464013 0.0210106 -11.73 0.000 -0.2876201 -0.2051824

Here, ∧

U t-1= Ft-1 - �� – ��(St-1). It is now valid to perform tests of significance (standard procedures for statistical inference) in the second-stage regression, i.e. concerning the parameters since all variables in these two regressions are stationary. It is also possible to include a constant term in the regression, but on the basis of financial theory we prefer a model without an intercept.

Interpretation of an Error Correction Model

F is expected to change between t-1 and t as a result of changes in the value of the explanatory variable, S, between t-1 and t, and also in part to correct for any disequilibrium that existed during the previous period. The error correction term {Ft-1 - �� – ��(St-1)} appears with a lag as this becomes part of the information set at the beginning of the time period t.

The parameter β (the slope coefficient of the initial regression) defines the long run relationship between F and S, while ø1 defines the short run relationship between changes in S and changes in F. Broadly, ø2 describes the speed of adjustment back to equilibrium. It measures the proportion of the last period’s equilibrium that is corrected for.

The coefficients on the lagged gap, (ø2) as well as on the first differenced spot series (ø1) are highly significant. (ø1) is approximately equal to 1 implying that a current change in the change in the spot prices is fully incorporated in forming future expectations by changing the change in the futures prices by an equal amount. Moreover, ø2 is negative. This indicates that if the actual

value of F fell short of the equilibrium value in t-1, or if Ft-1 < �� + ��(St-1). ) or ∧

U t-1< 0 then, “error correction” would cause F to increase (i.e. ∆Ft > 0), ceteris paribus. This can only happen if ø2 is negative, which it is. The idea is that, the dependent variable should converge to its equilibrium level. “Ceteris Paribus” can be taken to mean: barring changes in S and other disturbances (Vt). Alternatively, it can be taken to mean: setting ∆St and Vt to zero.

Lead-lag and Long Term relationships between Spot and Futures Markets

If the markets are frictionless and functioning efficiently, changes in the futures price and its corresponding changes in the spot prices of a commodity (or a financial asset) would be expected to be contemporaneously correlated and not be cross-autocorrelated. Mathematically, these notions can be represented as

corr(∆Ft, ∆St) ≈ 1

corr(∆Ft, ∆St-k) ≈ 0 for all k > 0

corr(∆Ft-j, ∆St) ≈ 0 for all j > 0

In other words, changes in spot prices and changes in futures prices are expected to occur at the same time. The current change in futures price is also expected not to be related to previous changes in the spot price, and the current change in the spot price is expected not to be related to previous changes in the futures price. We test the second condition by running the following model:

∆Ft = γ1 (∆St-1) + γ2 (∆Ft-1) + γ3 (Ut-1) + Zt

Regression Results

First Differenced Futures Coefficient Standard Error t-stat P > t 95% Confidence Interval

Lagged Differenced Spot

.071474

.055919

1.28

0.201

-0.0382287

0.1811768

Lagged Differenced Futures .0364093 .0343118 1.06 0.289 -0.0309041 0.1037227

Lagged Residuals* -.0475927 .0255063 -1.87 0.062 -0.0976314 0.0024459

The coefficient on the lagged differenced spot price (γ1) as well as the coefficient on the lagged differenced futures price (γ2) is statistically significant at a 5% level of significance. This indicates an absence of autocorrelation in the futures price, on an average, and the fact that the spot market does not lead the futures market, respectively, as lagged changes in spot prices do not lead to a change in the subsequent futures price. The critical value at a 5% level of significance for a one left-tailed test is -1.76. Since the t-statistic on the coefficient on lagged residuals is -1.87 (which is < -1.76), the coefficient (γ3) is negative and statistically significant.

Conclusion

Based on our cointegration analysis we observe an approximately one-to-one relationship between the futures price and the spot price, measured by the parameter β (0.9807). The parameter ø1 (1.035045) which measures the short run relationship between the two markets is statistically significant and very close to the ideal value of 1, which theory predicts. The insignificance of γ1 and γ2 shows that the information available about time period t-1 is being fully used when forming expectations about future at time period t. Hence, we conclude that the Spot and Futures price series for the commodity Pepper from the period 1st January, 2005 to 29th April, 2009 are cointegrated implying a long term relationship between the two. The cointegrating vector is given by [1, 0.9807]. In terms of efficiency, when analysed as two wings of one market, they come out to be efficient.

References

• Introductory Econometrics for Finance, Second Edition, Chris Brooks

• http://faculty.chicagobooth.edu/john.cochrane/research/Papers/time_series_book.pdf

• http://www.econ.uiuc.edu/~econ472/tutorial9.html

• http://www.econ.queensu.ca/students/phds/byrne/853/Stata/coint.do

• http://ricardo.ecn.wfu.edu/~cottrell/ecn215/error_corr_2004.pdf

• http://futures.tradingcharts.com/tafm/

• http://www.nuffield.ox.ac.uk/politics/papers/2005/Keele%20DeBoef%20ECM%20041213.pdf

• http://www.tau.ac.il/cc/pages/docs/sas8/ets/chap2/index.htm

• http://homepages.wmich.edu/~corder/PLS%20692%20Class%204.pdf

• http://www.ncdex.com/product/Agro_product.aspx?comm=PEP

APPENDIX

Here, we analyze the Spot price series and test the ability of a univariate, in-sample, one-step ahead forecasting rule by comparing it with the pepper futures market price, which provides an alternative forecast of the pepper prices determined by the interaction of numerous buyers and sellers in the market. We test the hypothesis that the futures price is a more accurate predictor of pepper spot prices than our forecasting rule.

Setting up a Forecasting Model

We set up a forecasting model for the spot price series, predicting values for the next day, using basic Time Series analysis. Knowing the series to be I (1) from our previous analysis, we try to set up an Autoregressive Integrated Moving Average (ARIMA) model, with order of differencing 1. To determine the order of the Autoregressive (AR) and the Moving Average (MA) parts, we plot the Autocorrelation and Partial Autocorrelation for various Lags for the Differenced series. As evident below, the two graphs do not point to a clear AR or MA order for the series.

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LagBartlett's formula for MA(q) 95% confidence bands

Autocorrelations of First Differenced Spot Prices

Thus, to come up with a forecasting rule we set up ARIMA models of various orders, which include ARIMA(1, 1, 0), ARIMA(2, 1, 0), ARIMA(3, 1, 0), ARIMA (1, 1, 1), ARIMA (1, 1, 2), ARIMA(2, 1, 1) and ARIMA(2, 1, 2), and tested them for a White Noise error term. An absence of a white noise or random error term would indicate a misspecified model. We also compared these models using Akaike information Criterion (AIC) and Schwarz-Bayesian Information Criterion (SBIC), to finally come up with an ARIMA (2, 1, 0) model as the one which best describes the data. Portmanteau (Q) White Noise Test results for the Selected ARIMA (2, 1 ,0) model clearly show that the white noise error hypothesis is accepted at a 95% confidence interval.

Portmanteau (Q) White Noise Test for ARIMA (2,1,0) Model

Portmanteau (Q )statistic = 53.5437

Prob > chi2(40) = 0.0744

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Lag95% Confidence bands [se = 1/sqrt(n)]

Partial Autocorrelations of First Differenced Spot Prices

Information Criterion Results for ARIMA (2,1,0) Model

Model AIC BIC

ARIMA(2,1,0) 15970.68 15986.1

We estimated the following ARIMA (2, 1, 0) model for the spot series to come up with the results given in the table below.

��� � ���� � ���� � ��

Using the model and the above results we get the following forecasting rule, where the values of ��� and ����are as given by the table above. Here the next day’s forecast of the spot price depends on today’s spot price and the spot prices of the previous two days.

�� � � � �������� � ���� �� �������� � ��������

Or,

�� � �� ������� ��� � ����!�����" ��� � �������#"� ����

∆�� Coefficient Standard

Error z P > z 95% Confidence Interval

∆��� ���������* .0180829 14.64 0.000 .2292679 .3001516

∆��� ������#"�* .0206844 3.38 0.001 .0293981 .1104796

Comparing the Accuracy of Forecasting model and the Futures Prices To test for the efficiency of this rule we compared the forecasted spot prices with the lagged futures prices, which represent the expected value of the spot prices for the next day. As our rule is a one-step ahead forecasting rule we compare the lagged futures price and the forecasted spot price of only the date of expiry of a contract. This is done to make sure that when we compare the two forecasts, we do so at a point where both the forecasts use the same information available. The contract expires on the 20th day of the delivery month and as a new contract is formed every month, we have a single expiry date for each month. If 20th happens to be a holiday, a Saturday or a Sunday then the due date is the immediately preceding trading day of the Exchange, which is other than a Saturday. Thus for our complete data set we have 52 observations of the forecasted spot price which we compare with lagged futures price or the expected value of the spot price in the futures market for the same date. We compared the sum of the absolute differences between the actual spot price and the forecasted spot price and the sum of the absolute differences between the actual spot price and the lagged futures price or the expected spot price. Results The table below shows the extent of aberration of the predicted price from the actual spot price for each expiry date for both the futures market prediction and the forecasting rule prediction. The dates highlighted in grey represent the dates for which the price predicted by the forecasting rule is more accurate than that predicted by the futures market.

Date of Forecasted

Value Actual Spot

Price

Expected

Spot Price in Futures Market

Forecasted Price (Using Forecasting

Rule)

Error in Futures Forecast

Forecast Error (Using

Forecasting Rule)

01/20/05 6916.075 6930 6960.483528 13.925 44.40852787 02/18/05 6474.375 6380 6400.559122 94.375 73.8158776 03/18/05 6976.55 6865 6851.23104 111.55 125.3189595 04/20/05 6743.075 6951 6788.738089 207.925 45.66308871 05/20/05 6467.4 6553 6477.521938 85.6 10.12193791 06/20/05 6335.575 6415 6336.61706 79.425 1.042059603 07/20/05 6164.2 6238 6185.396707 73.8 21.19670669 08/18/05 6391.1 6206 6405.969902 185.1 14.8699016 09/20/05 6384.125 6231 6394.572619 153.125 10.44761917 10/20/05 6316.3 6070 6324.149183 246.3 7.849182788 11/18/05 6602.125 6516 6593.405568 86.125 8.719431525 12/20/05 7425.975 7268 7367.58356 157.975 58.39144044

01/20/06 6798.6 6489 6815.130292 309.6 16.53029203 02/20/06 7038.1 6855 6975.647819 183.1 62.45218108 03/20/06 7122.75 6941 7162.342266 181.75 39.5922663 04/20/06 7195.075 7037 7209.179558 158.075 14.10455807 05/19/06 6950.4 6502 6967.888593 448.4 17.48859317 06/20/06 6865.85 6567 6832.688718 298.85 33.16128239 07/20/06 7960.875 7889 7933.237792 71.875 27.63720825 08/18/06 10091.575 10024 10299.42248 67.575 207.8474761 09/20/06 13666.275 13749 13720.42462 82.725 54.1496202 10/20/06 12344.025 12230 12424.06071 114.025 80.03571299 11/20/06 10955.6 10443 11031.95563 512.6 76.35562865 12/20/06 10043.7 9774 10097.12516 269.7 53.42516312 01/19/07 11323.25 11138 11326.62987 185.25 3.379873783 02/20/07 12486.6 12491 12549.597 4.4 62.99699616 03/20/07 12030.225 12102 12026.00026 71.775 4.224739537 04/20/07 15008.625 14759 15036.92124 249.625 28.29624443 05/18/07 15089.925 14378 14813.50958 711.925 276.4154163 06/20/07 14218.55 13677 14136.26731 541.55 82.28269394 07/20/07 14842.275 14819 14902.22728 23.275 59.95227762 08/20/07 13494.25 12948 13471.33567 546.25 22.91433251 09/20/07 12592.2 12060 12695.58845 532.2 103.3884457 10/19/07 14000.425 13839 14083.39677 161.425 82.97176829

11/20/07 13966.05 13795 13941.92686 171.05 24.12313871 12/20/07 13096.825 12734 13100.96988 362.825 4.144877683 01/18/08 14820.65 15023 14630.76564 202.35 189.884361 02/20/08 14228.325 14066 14196.92852 162.325 31.39648043 03/20/08 14965.075 14728 15054.5712 237.075 89.4961971 04/17/08 14315.2 13801 14275.81225 514.2 39.38774768 05/20/08 14491.55 14280 14408.66306 211.55 82.88693795 06/20/08 14417 14017 14325.26466 400 91.73534489

07/18/08 14548.925 14282 14441.1042 266.925 107.8207965 08/20/08 14430.325 14057 14420.44673 373.325 9.878272605 09/19/08 13187.65 12676 13149.32086 511.65 38.32913522 10/20/08 13334.075 12542 13258.6349 792.075 75.44009688 11/20/08 11983.475 12043 12189.25343 59.525 205.7784292 12/19/08 10384.025 9981 10306.00434 403.025 78.02065705 01/20/09 12209.025 12272 12019.69732 62.975 189.3276753 02/20/09 11175.925 10962 11163.57784 213.925 12.34715568 03/20/09 10858.275 10574 10844.30766 284.275 13.96733626 04/20/09 13250.9 13491 13342.0441 240.1 91.14409629

Sum of Absolute Differences between the Actual Spot Price and the Expected Price

Using Futures Forecast 12690.35 Using Forecasting Rule 3206.55

As the sum of absolute differences comes out to be less in case of the forecasted price than that in case of the expected price in the futures market we can consider our forecasted prices to be a better estimate of the spot price than the prices determined in the futures market. But we must also state that our analysis is very preliminary and is based on a very small data set with only 52 observations. Naturally the results of the analysis may change, as more data is made available.

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Futures Market Expectations Forecasting Rule

Absolute Error In Forecasting