Question one

  1. The equation is given by:

Ln (?????) = ?1 + ?2????? + ?3???? + ?4ln (???) + ?5??????? + ?6ln (???????) + ?7?????? + ?8??????? + ?9????? + ?

The regression results are shown in the table below


From the results, the estimated equation will be given by;

LPRICE = -0.00899022863382*LOWSTAT + 0.176450565448*LNOX 0.00119235090003*DIST — 0.00667964959875*CRIME + 0.231670438192*LPROPTAX — 0.037923718021*NOX + 3.62441426754e-05*PRICE — 0.00822618311505*PROPTAX + 0.00479484711547*RADIAL — 0.0462807286603*ROOMS — 0.00114227351782*STRATIO + 8.4075273902.

From the results, it is clear that distance, lnox, nox and stratio are not statistically significance while the rest of the variables are statistically with p-value < 0.05.

  1. Using significance level of 0.15 on rooms, the regression equation is given below

econometrics 1

From the results, p-value is 0.00<0.15 meaning that the null hypothesis is rejected and accepting alternative hypothesis that it adds less than 15% with coefficient of 0.368664 indicating positive relationship of the variables.

  1. Plot the residuals of your regression against dist and comment

econometrics 2

From the graph, there is strong correlation between the regression residual and the distance. The correlations also shows positive correlation. This is shown by highly condense dotted line in the graph (Wooldridge, 2010).

  1. White’s test (without cross products) to test for heteroskedasticity errors

Dependent Variable: LPRICE

Method: Least Squares

Sample: 1 506

Included observations: 506

White heteroskedasticity-consistent standard errors & covariance

econometrics 3

From the estimates, lowstat gives negative relationship with price, while lnox, distance, nox and protax are not statistically significance in this case. Crime, LPROPTAX, PRICE and ROOMS are statistically significance with most of them negatively correlated. R-squared is 0.953864 meaning that null hypothesis is rejected since there is presence of heteroskedasticity (Wooldridge, 2010).

  1. Conduct a Goldfeld-Quandt test of the null hypothesis of no heteroskedasticity against the alternative that the variances of the errors are changing with dist

econometrics 4

From the table above, the probability is given as 0.00 indicating that it is statistically significance with F-test of 83.27 and observation times R-square gives 71.75396 indicating presence of heteroskedasticity principles in the data. It further shows that the variances of the errors are changing with distance as clearly shown (Wooldridge, 2010). However, the relationship between the two is negative as distance gives a coefficient of -0.01182.

  1. Perform a Breusch-Pagan test of the null hypothesis of no heteroskedasticity against the alternative hypothesis that ??2 = exp(?1 + ?2????)

econometrics 5

The results shows that F-test 42.43338 while Obs*R-squared is 39.2. The probability has already been calculated and it gives 0.00 < 0.05 indicating that it is statistically significance. R-square is 0.077655 indicating that a good percentage of the variables were included in the analysis with high presence of heteroskedasticity in the data hence rejecting null hypothesis of no heteroskedasticity and accepting the alternative hypothesis that there is heteroskedasticity (Wooldridge, 2010).

??2 = exp (?1 + ?2????)

??2 = exp (0.017970
— 0.002703????)

  1. Re-estimate the model and obtain White’s standard errors and write out the estimated equation. How do the coefficient estimates and standard errors compare with those obtained in part (a)?

econometrics 6

From the results, standard error in the first case is positive (0.001800) while in the second case it is positive (1.024342) which is larger compared to the first instance. It simply shows higher error in the second estimate that the first estimate (Wooldridge, 2010).

  1. Estimate the model using generalized least squares under the assumption ??2 = exp(?1 + ?2????) and write out the estimated equation

econometrics 7

In this case the error is much larger compared to the first two indicating high level of error in the data re-estimated using the normal regression.??2 = exp (3.795751 -2.47E-12 Resid.)


Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press.