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Why Choose Us? Accuracy: We are a company employed with highly qualified Economics experts to ensure fast and accurate homework solutions aimed at any difficult homework – both Micro economics and Macro economics. Econometrics Assignment Sample Questions and Answers: Question 1: From the following data from the regression equations, Ye = a + bX
and
Xe = a + bY
Use the normal equation method: X: Y:
1 15
3 18
5 21
7 23
9 22
Also, estimate the value of Y when X = 4, and the value of X when Y = 24.
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Solution: Formulation of the regression equations X
Y
X2
1 3 5 7 9
15 18 21 23 22 Y = 99
1 9 25 49 81 2 X = 165
Y2
225 324 441 529 484 2 Total X = 25 Y = 2003 (i) Regression equation of X on Y. This is given by
XY 15 54 105 161 198 XY = 533 N=5
Xe =a + bY To find the values of the constant a and b in the above formula, the following two normal equations are to be simultaneously solved: X = Na + b Y XY = a Y + b Y 2 Substituting the respective values in the above formula we get, 25 = 5a + 99b 533 = 99a + 2003b Multiplying the equation (i) by 99 and eqn. (ii) by 5 and presenting them in the form of a subtraction we get, 2475 = 495a + 9801b (-)2665 = 495a + 10015b/-190 = -214b or
214b = 190
∴
b = 190/214 = .888 approx.
Putting the above values of b in the eqn. (i) we get, 25 = 5a + 99(.888) or
5a = 25 – 87.912 = -62.912
∴
a = -62.912/5 = -12.5824
Thus,
a = -12.5824 and b = 01.888.
Substituting the above values of the constants a and b, we get the regression equation of X on Y as, Xe = -12.5824 + 0.888 Y Copyright © 2012-2015 Economicshelpdesk.com, All rights reserved
Thus, when
Y = 24,
Xe = -12.5824 + 0.888 (24)
= -12.5824 + 21.312 = 8.7296.
(ii) Regression equation Y on X. This is given by Ye = a + bX To find the values of the constants a and b in the above formula, the following two normal equations are to be simultaneously solved as under: đ?‘Œ = Na + b đ?‘‹ đ?‘‹đ?‘Œ = a đ?‘‹ + b đ?‘‹
‌.(i) 2
‌.(ii)
Substituting the respective values in the above formula we get, 99 = 5a + 25b ‌.(i) 533 = 25a + 165b
‌.(ii)
Multiplying the equation (1) by 5 and getting the same subtracted from the equation (ii) we get,
Thus,
533 = 25a + 165b
‌.(ii)
(-)495 = 25a + 125b
‌.(iii)
38 = 40b Copyright Š 2012-2015 Economicshelpdesk.com, All rights reserved
∴
b = 38/40 = 0.95
Putting the above values of b in the equation (i) we get, 99 = 5a + 25(0.95) or
5a = 99 = 23.75 = 75.25
∴
a = 75.25/5 = 15.05
Thus,
a = 15.05 and b = 0.95
Substituting the above values of the two constants a and b we get the regression equation of Y on X as,
Ye = 15.05 + 0.95X Thus, when X = 4,
Ye = 15.05 + 0.95(4)
= 15.05 + 3.80 = 18.85. Note. It may be noted that the above normal equation method of formulating the two regression equations is very lengthy and tedious. In order to do away with such difficulties, any of the following two methods of deviation may be used advantageously. 2. Method of deviation from the Means Under this method, the two regression equation are developed in a modified form from the deviation figures of the two variables from their respective actual Means rather than their actual values. For this, the two regression equations are modified as under: (i) Regression equation of X on Y. This is given by X = X + bxy (Y - Y),
or X - X = bxy (Y - Y)
(ii) Regression equation of Y on X. This is given by Y = Y + byx (X - X), or Y - Y = byx (X - X) In the above formulae, X = given value of the X variable Y = given value of the Y variable X = arithmetic average of the X variable, Y = arithmetic average of the Y variable, bxy = regression coefficient of X on Y i.e., r ∴
bxy = r
đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
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Where,
đ?œŽđ?‘Ľ = standard deviation of X variable
đ?œŽđ?‘Œ = standard deviation of Y variable and
byx = regression coefficient of Y on X i.e., r
∴
byx = r
Question 2: below find.
đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
Using the method of deviations from the actual Means from the data given
(i) the two regression equations (ii) the correlation coefficient and (iii) the most probable value of Y when X = 30 X: Y:
25 43
28 46
35 49
32 41
31 36
36 32
29 31
38 30
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34 33
32 39
Solution: Determination of the regression equations by the method of deviation from the Means X
Y
25 28 35 32 31 36 29 38 34 32 đ?‘‹ = 320
(X – 32) x -7 -4 3 0 -1 4 -3 6 2 0 �=0
43 46 49 41 36 32 31 30 33 39 đ?‘Œ = 380
(Y – 38) Y 5 8 11 3 -2 -6 -7 -8 -5 1 đ?‘Œ=0
X2 49 16 9 0 1 16 9 36 4 0 2 đ?‘Ľ = 140
Y2 25 64 121 9 4 36 49 64 25 1 2 đ?‘Œ = 398
(a)(i) Regression equation of X on Y This is given by đ?œŽđ?‘Ľ
X=X+r where
X=
đ?‘Œ đ?‘
đ?œŽđ?‘Ľ =
đ?œŽđ?‘Ś = r=
(Y - Y)
X/N = 320/10 = 32
đ?‘Œ=
and
đ?œŽđ?‘Ś
= 380/10 = 38 đ?‘Ľ2 đ?‘ đ?‘Ś2 đ?‘
đ?‘Ľđ?‘Ś đ?‘ đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
=
140
=
398
10
10
= 3.74 approx.
= 6.31 approx.
= -93/10 Ă— 3.74 Ă— 6.31
= -93/10 × 23.5994 = -93/235.99 = -0.394 Putting the respective values in the above equation we get, X = 32 + -0.394 × -93/235.99 = -0.394 Putting the respective values in the above equation we get, X = 32 + -0.394 × 3.74/6.31 (Y – 38) = 32 – 0.2337 (Y – 38) = 32 + 8.8806 – 0.2337 Y Copyright Š 2012-2015 Economicshelpdesk.com, All rights reserved
Xy -35 -32 33 0 2 -24 21 -48 -10 0 đ?‘Ľđ?‘Ś = -93
X = 40.8806 – 0.2337Y Aliter Substituting the regression coefficient of X on Y i.e.,
đ?œŽ
r đ?œŽ đ?‘Ľ by đ?‘Ś
đ?‘Ľđ?‘Ś đ?‘Ś
2
we get, X=đ?‘‹+
đ?‘Ľđ?‘Ś đ?‘Ś
2
(Y - đ?‘Œ)
= 32 + -93/398 (Y – 38) = 32 – 0.2337 (Y – 38) = 32 + 8.8806 – 0.2337Y = 40.8806 – 0.2337Y (ii) Regression equation of Y on X This is given by Y = đ?‘Œ + r
đ?œŽđ?‘Ś đ?œŽđ?‘Ľ
(X - X)
Substituting the respective values in the above we get, Y = 38 + -0.394 Ă— 6.31/3.74 (X – 32) = 38 – 0.6643 (X – 32) = 38 + 21.2576 – 0.6643X = 59.25876 – 0.6643X ∴
Y = 59.2576 – 0.643 X
Aliter Replacing the formula of regression coefficient. r
đ?œŽđ?‘Ś đ?œŽđ?‘Ľ
by
y=đ?‘Ś+
đ?‘Ľđ?‘Ś đ?‘Ľ
2
đ?‘Ľđ?‘Ś đ?‘Ľ
2
we get, (X - X)
= 38 + -93/140 (X – 32) = 380 – 0.6643 (X – 32) = 38 + 21.2576 – 0.6643X ∴
Y = 59.2576 – 0.6643X
Thus, the two regression equations are: X on Y : X = 40.8806 – 0.2237 Y and
Y on X : Y = 59.2576 – 0.6643 X Copyright Š 2012-2015 Economicshelpdesk.com, All rights reserved
(b) Coefficient of Correlation The coefficient of correlation between the two variables, X and Y is given by đ?‘Ľđ?‘Ś
rxy =
đ?‘ đ?œŽđ?‘Ľ đ?œŽđ?‘Ś
= -93/10 Ă— 3.74 Ă— 6.31
= -93/235.994 = -0.394 Alternatively By the method of regression coefficients we have, rxy = = =
đ?‘?đ?‘Ľđ?‘Ś Ă— đ?‘?đ?‘Śđ?‘Ľ đ?‘Ľđ?‘Ś đ?‘Ś
2
Ă—
đ?‘Ľđ?‘Ś đ?‘Ľ
2
−0.2337 Ă— −0.6643
= 0.1552 = -0.394 Note. Since the regression coefficients are negative, the correlation coefficient has been negative. (c) Probable value of Y when X = 30 This will be determined by the regression equation of Y on X as follows : We have,
Y = 59.2576 – 0.6643X
Thus, when X = 30, Y = 59.257 – 0.6643(30) = 59.2576 – 19.929 = 39.3286.
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