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Formula Essay

The following demand for housing function was estimated using ordinary least squares techniques with data on 250 households.

Q = A + aY + bP + u

where A is the regression constant, Q is quantity demanded, P is price, Y is personal income and u is the error term.  The results of double log estimation were:

In Q=19.21 + 0.98 In Y – 1.13 In P,  R2 = 0.83

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a.    What are the price and income elasticities estimated?

On the basis of the estimated regression equation (ln Q) shown above, the estimated price elasticity is 1.13 and the estimated income elasticity is 0.98.  These are notably the regression coefficients of the predictor variables income and price in the estimated regression equation.

b.    Are these values significantly different from zero? How do you know?

Income

For income, the corresponding estimated elasticity and standard error of the said estimate are 0.98 and 0.11 respectively.  To determine if 0.98 is significantly different from zero, use the t-test procedure.  The form of the test statistic is as follows:  .

In the case of income,   .  With the number of households equal to 250, one can assume that this sample size is sufficiently large and therefore the critical value to which the computed test statistic value will be compared will be approximated by a Z critical value instead of the a t critical value.  If we use a level of significance equal to 5%, the critical value is 1.96.

To determine if the estimated income elasticity is significantly different from zero, compare the computed test statistic value with the Z critical value.  If the former is higher than the latter, then one can safely conclude that the estimate is significantly different from zero.  Hence, with income elasticity, it can be concluded that the estimate is significantly different from zero since 8.91 is higher than 1.96.

Price

For price elasticity, the estimate and the corresponding standard error are 1.13 and 0.52, respectively.  Computing the value of the t test statistic, .  Since 2.17 is higher than 1.96, the data provides sufficient evidence to conclude that the estimated price elasticity is significantly different from zero.  This conclusion stands for a level of significance equal to 5%.

Note that for both t tests to determine if price and income elasticity are significantly different from zero, if for instance the level of significance set by a researcher or an economist is 10% instead of 5%, the critical value becomes 1.645, not 1.96.  Therefore, in this case once can still safely say that both estimates are significantly different from zero.

c.    It is often important for policy purposes to know if demand is inelastic, unitary elastic, or elastic, i.e., is the elasticity greater than, equal to, or less than one?

Based from the sample regression equation, the income elasticity of demand is 0.98 which is less than one.  This implies that demand is inelastic.  A 1 percent income increase will result to a 0.98 increase in quantity demanded.

The price elasticity of demand, on the other hand is equal to -1.13 which means that price is also inelastic.  A one percent increase in price will result to a 1.13 percent decrease in quantity demanded.

d.    For the above equation, test this question for the price elasticity and tell the level of significance of the result using Normal Distribution.

Answer for this section is already expounded in letter b.

e.    Do the same for the income elasticity.

Answer for this section is already expounded in letter b.

f.     What is the assumed expected value (mean) of the error term as the model is estimated?

The assumed expected value of the error term in a regression model is zero.

g.    What is the assumed variance of the error term?

The assumed variance of the error term in a regression model is .

h.    What is the interpretation of R-squared value?

The R2 value is equal to 0.83.  This implies that 83 percent of the total variation in quantity demanded is explained by price and income.  The other 17 percent has not been accounted for by the model because there could be other economic variables significantly affecting quantity demanded that are not included in the model.

i.      Tell why the original demand function was converted into logarithms before estimation.

The original demand function was converted into logarithms because the OLS technique for estimation of regression parameters requires that the function is linear.  Since the demand function given is non-linear, then conversion to log is imperative.

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