Constructing and testing the model of the us stock market

To estimate this model we need to conduct regression analysis. To do this we apply for the special function in Microsoft. The Excel calculates everything and presents the result in several successive tables. This model is estimated according to data which was taken from

Now the coefficients and the whole model should be checked according three basic tests: t-test, R 2 -test and F-test. Mind that t statistics for a0 and al are also given in the tables of regression analysis (numbers in square brackets in the system)

Interpreting the coefficients of model

Coefficients of regression show how the dependent variable (regressand) of the model will change if the independent variables (regressors) change within the certain model.

Coefficient a 1 before X 1 (apple shares quotes) means that if Apple Shares quotes for example increase by 1 $ Dow-Jones industrial average (Y) will increase by 14,34 points. This coefficient shows the certain dependence of Y from X 1 in point of view math. So does the coefficient a 2 but concerning Google shares quotes (X 2 ) and Dow-Jones industrial average. If X 2 increases by 1$ Y will increase by 11,85 points.

Coefficient a 0 is the value of Dow-jones industrial average (Y) in case if a 1 and a 2 are both equal to 0.

t-Test

As through the whole investigation, these tests are supposed to be held due to Microsoft Excel functions («СТЬЮДРАСПОБР», «F.ОБР.ПX»). In case of t-test it is necessary to apply for «СТЬЮДРАСПОБР» function which let us define the t critical. It considers two parameters in order to calculate t critical:

T critical calculation      Table 1

a=	0,05	t crit=	2,10
a=	0,01	t crit=	2,88
a=	0,1	t crit=	1,73

The sense of t-test states that if the absolute value of t statistics of each parameter is more than t crit obtained above:

│t│>t crit (1)

then we may conclude that parameter is significant for the model.

In case of this model, it is obvious that all coefficients a0, a1 and a2 are significant under the circumstance that α=0,05 within this model.

R2-Test

Concerning this test everything is rather simple and not go beyond the analysis of R2. According to this test if R2 is close to 1 means that specification is constructed in very good way because this parameter shows x variables influences the y variable by linear regression. In this case R2=0,76 and it means specification is quite good:  76% of variances X1 and X2 describe variance Y by linear regression model within this model.

F-Test

This test also requires to calculate the F crit and to compare it with F given in regression analysis.

The function we are going to apply for has already been mentioned -«F.ОБР.ПX».

F critical calculation        Table 2

a=	0,05	F crit=	3,55
a=	0,01	F crit=	6,01
a=	0,1	F crit=	2,62

This tests checks the whole specification whether its quality is high or low and if R² is random variable or not. If Fcrit is more than F of a model:

Fcrit>F, then the quality is low and R2 is random. Otherwise, vice versa.

In this case according to the table above under a = 0,05 F_crit2 is not random within this model.

So the form of specification will be:

r  Yt = 8670,58 + 14,34X1t + 11,85X2t + £t

(1148,42)   (6,23)   (1,67)  (44,01)

tstat     [7,63]      [2,30]     [7,11]

-                  R2 = 0,758

F = 28,12 tcrit = 2,10; a = 0,05 <             Fcrit = 3,55; a = 0,05

Goldfield – Quandt Test

According to Goldfield - Quandt test, we assume that & ( ^G ) = const . As a result of this test, we find out, if the residuals are homoscedastic or not and if we may use ordinary square to estimate parameters.

GQ coefficient calculation         Table 3

GQ=	1,17
1/GQ=	0,85
Fcrit=	3,44

Basing on these two inequalities, we compare our figures: GQcrit and 1/GQcrit so we can conclude the residuals are homoscedastic and we may use ordinary square to estimate parameters or coefficients of the model.

Durbin-Watson test

The next step is to carry out Durbin-Watson test to check if there exists correlation between residuals. To calculate DW constant there is the formula

DW =

∑(et - et-1)2

^∑e t ² DW=1,56

Defining the intervals means to find du and di in the table. In this model there are two regressors and sample size 21. Using table of values for Durbin-Watson criteria we find values d l = 1,125 and d u =1,538. Then make a table.

Intervals:

Picture 1

Positive correlation no correlation Negative correlation 0 dl du 2 4-du 4-dl 4 0 1,125 1,538 2 2,462 2,875 4 dw 1,56

It is obvious that value of DW got into interval from du to 4-dl, that means that within this model there is no correlation between residuals. The third precondition of Durbin–Watson theory is valid so residuals are homoscedastic and coefficients are said to be exact.

Confidence interval and adequacy of the model

Finally, it is important to estimate the adequacy of the model. For that, it is necessary to construct confidence interval. Interval is calculated:

(Y - tcrit ∗ σ; Y + tcrit ∗ σ)

To estimate DJIA (Y^̂) for 01.10.2014 we use coefficients a 0 , a 1 and a 2 and values X 1 and X 2.

• Y^̂ = a 0 + a 1 ∗ X 1 + a 2 ∗ X 2 = 16915,19

Baring in mind when calculating tcrit, level of significance taken is equal to 0,01.

Confidence Interval (16788,5;17041,87)

Now DJIA 01.10.2014 = 16804,71 is covered by confidence interval. Forecasts obtained are said to be correct with the probability of p=100 - α=99 (α=1%) and model is adequate.

Conclusion: despite the fact that these two companies are not considered in DJIA, the model constructed proves that they should be included in the list and reflect the trend of the whole market.