Problem on Multiple non-linear Regressions

Multiple non-linear Regressions:

            Suppose you found out from the scatter diagram that the relation between the number of apartments demanded and the independent variables is a non-linear relationship, in the power functional form:

You can still use the LS method to run the regression, and estimate the demand function in the following log-linear form:

            log Q = log B₀ + B₁ log P + B₂ log AD + B₃ log Dist

In the log-linear form, the coefficients of logP and LogAD and logDIS measure the elasticity with respect to each of the three independent variables.

[(∆logQ/∆logP) = B₁= (%∆Q/%∆P) = E_P].

To run this regression, copy the data of the example solved in the last lecture in a new excel file and call it Example 2, then convert the data into logarithm. To do that, click in an empty cell where you want your fist cell of the logarithmic data to appear, type the formula for the logarithm [for example if the first data value fall in a2 cell, just choose an empty cell where you want your logarithmic values to appear and write  in [ =log10(a2)] and hit enter. The log of the first entry in Q column will show in the chosen cell, copy the content of that cell down to get the log of all values in Q, and horizontally to get the logarithmic values for the rest of the variables. Now, proceed in the same steps you have learned in the last lecture to get regression estimates. Make sure you have a printout similar to that presented on next page. Now, using the information in the output print answer the following questions:

1.      Write the demand equation.

2.      Check the signs of the three variables, which signs do not conform to the theory of demand?

3.      What is the value of the price elasticity of demand, what does it mean?

4.      What other elasticities can you find in the results?

5.      Explain the meaning of the R square.

6.      Comment on the significance of the effect of the explanatory variables.

7.      What does the significance level of F statistic tells you?

8.      The firm owning this apartment complex suffered for years from low profit rates, as a result of an average vacancy rate of nearly 40%. In light of the estimates obtained, what is your advice to the firm. Should the firm move its apartments closer to the university? I’m sure you have better ideas.

9.      What other variables do you think should be added to this model and which variables should be omitted?

Q	P	AD	Dis	log Q	log P	log AD	log Dis
28	250	11	12	1.44715803	2.39794	1.041393	1.079181
69	400	24	6	1.83884909	2.60205	1.380211	0.778151
43	450	15	5	1.63346846	2.65321	1.176091	0.69897
32	550	31	7	1.50514998	2.74036	1.491362	0.845098
42	575	34	4	1.62324929	2.75966	1.531479	0.60206
72	375	22	2	1.8573325	2.57403	1.342423	0.30103
66	375	12	5	1.81954394	2.57403	1.079181	0.69897
49	450	24	7	1.69019608	2.65321	1.380211	0.845098
70	400	22	4	1.84509804	2.60205	1.342423	0.60206
60	375	10	5	1.77815125	2.57403	1	0.69897
SUMMARY OUTPUT
Regression Statistics
Multiple R	0.727198
R Square	0.528817
Adjusted R Square	0.293226
Standard Error	0.124585
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	3	0.104519	0.03484	2.24463564	0.183571
Residual	6	0.093128	0.015521
Total	9	0.197647
	Coefficients	Standard Error	t Stat	P-value
Intercept	2.828655	1.411635	2.003815	0.09193898
log P	-0.2344	0.631537	-0.37116	0.72327068
log AD	-0.08786	0.333757	-0.26324	0.80117571
log Dis	-0.55973	0.215898	-2.59255	0.04107075

The answer:

1 . The demand equation :

Log Q = 2.83 – 0.23 log P  -0.088 log AD – 0.56Log Dis

2 .  AD was found to be the only variable that has a wrong sign. Advertising  is expected to have a positive effect on Q. In other words, AD should have a positive sign.

3 .   The value of the price elasticity  of demand 0.23

          It means that a 1% increase in P will decrease Q by 0.23%.

4 .   The AD elasticity of demand 0.9  and the Dis elasticity of demand 0.56

5.  The R² of 52% indicate that, % 52 of the variation in the number of demanded apartment is explained by variations in P , AD and Dis ,while 48% of these variation are due to other variables not included in the model .

6 .  From the results, P- value is greater than 5% for P and AD, which means that both variables have no significant effect on Q, therefore, the researcher may not reject the null hypothesis for these two vsriables. DIS has a P-value of 4%, less than the acceptable probability of error in the estimation. Therefore, we conclude that DIS has a significant negative effect on Q. So, the researcher may reject the null hypothesis (H₀: B₃ = 0) and accept the alternative hypothesis

(H₁: B₃ < 0)

7 .  F significance of 0.1836 (18%) is greater than 5%, the acceptable probability of error in the estimated coefficients. We conclude that ( P ,  AD , and Dis ) together have no significant effect on Q. The null hypothesis (H₀: B₁ = B₂= B₃ 0)should not be rejected.

8 .   We might increase the profits by two ways either by increasing the ( P ) since Ep<1   or  by providing other services .

9 .  We should add ( Income & Complement ) and we should omit the non significant variable ( AD ) .