elasticity of demand - regression to estimate demand

Starting with the date from Problem 6 and the data on the price of a related commodity for the years 1986 to 2005 given below, we estimated the regression for the quantity demanded of a commodity (which we now relabel Q Ì?_x), on the price commodity which we now label P_x, consumer income ( which we now label Y), and the price related commodity (P_(z) ), and we obtain the following results.

Year 1986 1987 1988 1989 1990
Pz ($) 14 15 15 16 17
Year 1991 1992 1993 1994 1995
Pz ($) 18 17 18 19 20
Year 1996 1997 1998 1999 2000
Pz($) 20 19 21 21 22
Year 2001 2002 2003 2004 2005
Pz($) 23 23 24 25 25

Q Ì?x=121.86-9P_x+0.04Y-2.21P_z
(-5.12) (2.18) (-0.68)
R^(2 )= .9633 F=167.33 D-W=2.38

(a) Explain why you think we have chosen to include the price commodity Z in the above regression. (b) Evaluate the above regression results. (c) What type of commodity is Z? Can you be sure?

**15(b) is to evaluate the above regression results in terms of the signs of
the coefficients, the statistical significance of the coefficients and the
explanatory power of the regression (R2) The number in parentheses
below the estimated slope coefficients refer to the estimated t values.
The rule of thumb for testing the significance of the coefficients is if the
absolute t value is greater than 2, the coefficient is significant, which
means the coefficient is significantly different from zero. For example,
the absolute t value for Px is 5.12 which is greater than 2, therefore, the
coefficient of Px, (-9.50) is significant. In order words, Px does affect
Qx. If the price of the commodity X increases by $1, the quantity
demanded (Qx) will decrease by 9.50 units.

15(c) X and Z are complementary or substitutes?

INFORMATION FROM PROBLEM 6
. From the following table giving the quantity demanded of a commodity (Y), its price (X1), and the consumers income (X2) from 1986 to 2005, a) estimate the regression equation of Y on X1 and X2, b) test at the 5 % level for the statistical significance of the slope parameters, c) find the unadjusted and the adjusted coefficients of determination and d) test at the 5 % level for the overall statistical significance of the regression. Show all your results to three decimal places.
Year Y X1 X2
1986 72 $10 $2,000
1987 81 9 2100
1988 90 10 2,210
1989 99 9 2,305
1990 108 8 2,407
1991 126 7 2,500
1992 117 7 2,610
1993 117 9 2,698
1994 135 6 2,801
1995 135 6 2,921
1996 144 6 3,000
1997 180 4 3,099
1998 162 5 3201
1999 171 4 3,308
2000 153 5 3,397
2001 180 4 3,501
2002 171 5 3,689
2003 180 4 3,800
2004 198 4 3,896
2005 189 4 3,989
Solution:
The regression equation is given as follows:
y=-9.470x1+0.029x2+114.074
where x_1 represents the independent variable 'price'
x_2 represents the independent variable 'consumer^' s income^'
and,y represents the dependent variable 'quantity demanded^'
Let us test the significance for the slope parameters of the independent variables.
The null hypothesis,H_0 states that there is no significant correlation, or the correlation coefficient ?=0.
Decision Rule:
Reject H_0 if the p-value < 0.05 (significance level,alpha)

Independent Variables p- value Null Hypothesis Decision Conclusion
x_1 0.00007 ?_1=0 Reject H_0 x_1 has a significant contribution in the regression model
x_2 0.0003 ?_1=0 Reject H_0 x_2 has a significant contribution in the regression model

The unadjusted and the adjusted coefficients of determination are given as 0.984 and 0.964 respectively.

The null hypothesis,H_0 states that there is no significant correlation, or the correlation coefficient ?=0.
Significance Level, ? = 0.05

Decision Rule:
Reject H_0 if the Significance F( p-value) < 0.05 (significance level,alpha)

From the ANOVA table, we find that the Significance F=1.856Ã-ã?-10ã?-^(-13) , which is much less than 0.05. Therefore, we reject the null hypothesis that there is no significant correlation and conclude that, according to the overall test of significance, the regression model is valid.