Kapchenko M.M.

Simulation of an empirical relationship between Dow Jones

Industrial Average and S&P 500 index

Consider the sampling with the value of the index S&P 500 and Dow Jones Industrial Average. The values were recorded every minute during the auction. Here are some theoretical reference:

The S&P 500 (Standard & Poor's 500), is an American stock market index based on the market capitalizations of 500 large companies having the largest capitalization. Dow Jones Industrial Average is a stock market index created by Wall Street Journal editor and Dow Jones & Company co-founder Charles Dow. This index was created to monitor the development of the industrial component of the US stock markets. For its calculation used the average scalable - the amount of the price is divided by the divisor, which changes whenever stocks included in the index is subject to fragmentation (split) or association (consolidation).

Let's identify the relationship between them.

At the beginning consider the matrix scatterplot(Var1-SP, Var2-DJ):11.png

As we can see in the diagram there is a clear relationship between these indices. To check it, let's conduct the corresponding regression:

12.png

 

Thus, the model has a rather high coefficient of determination = 0.91743188

Model addiction, adjusted by the method of least squares is:

DJ ≈ -855.525 + 9.364*SP

Let’s do analysis of residues:

123.png

The scatter plot Predicted - Observed values shows that the prediction formula captures the basic trend observations.

Now Consider the scatter plot Predicted - Residual scores:121.png

On diagram clear dependence are not observed. The points are scattered randomly, indicating the inability to improve the prognosis of the available data. Let's verify the normality of the distribution of errors.

To do this, we construct the diagram quantile against quantile (Q–Q plot):321.png

We can see that errors follow a normal distribution.

Now let's construct confidence intervals for the regression coefficients. For this we use the following formula:

N = 391, we believe that α = 0,05 then using a probability calculator we have:

= 1,967930

-855.525 òîä³  = -855.525 ± 1,967930 * 256.9112

[-1361.1082;  -349.9417]

9.364 òîä³   = 9.364 ± 1.967930 * 0.1422

[9.0842;  9.6438]

Thus, we have obtained a regression model  DJ ≈ -855,525 + 9,364*SP with a relatively high coefficient of determination, which can be applied in practice to calculate the Dow Jones Industrial Average  through the S&P 500 index.

References

1.   KARTASHOV M. Probability, Process, Statistics. Publishing and Printing Center Kyiv University' 2008.

2.   Majboroda R. Regression: linear models. TViMS, 2004 - 283 p.

3.   Leonenko N., Tinsel Y., Parkhomenko V. and Yadrenko M. Theoretical-probability and stochastic methods in financial mathematics and econometrics. K .: Informtehnika, 1995. - 380 p.