Gauss Markov Assumption

Gauss Markov Assumption 

Question 1

The file Wage 2017.sav contains data collected from a sample of 500workers. The variables are:

Lwagei Natural log of monthly earnings of worker i
IQi IQ score of worker i
educi Years of education of worker i
experi Years of experience of worker i
tenurei Years with current employer of worker i
whitei = 1 if the worker is white; = 0 otherwise
marriedi = 1 if the worker is married/living as married; = 0 otherwise

 

  • Estimate the following model of monthly earnings:

Report the results of your estimated model, indicating the level of significance using stars (** denotes strong significance p<0.01 and * denotes significance p<0.05).

  • Interpret the estimated coefficients.
  • Comment on the explanatory power of this model.
  • Add dummy variables for white and married to the model in (a). Estimate the new model; report the results and interpret the estimated coefficients associated with the dummy variables only.  Test for their individual significance.
  • Using the Park test, investigate whether there isheteroskedasticity in the model you estimated in (d). Assume that tenure is the variable causing the problem of heteroskedasticity. Provide an intuitive explanation for your findings.
  • Does the presence of heteroskedasticity violate one of the Gauss Markov assumptions? Explain your answer.  What implications would heteroskedasticity have for the results you reported in (d)?
Question 2

Nitrogen Dioxide (NO2) is a pollutant that attacks the human respiratory system and increases the likelihood of respiratory illness.  One common cause of nitrogen dioxide is car exhaust.

The file pollution.sav contains data from 500 observations made from October 2001 to August 2003 in the US (data from CarnegieMellonUniversity archive).

The variables are:

LNO2: Natural log of the concentration of NO2 (particles)
LCARS: Natural log of the number of cars per hour
TEMP: Temperature 2 metres above the ground (degrees C)
TCHNG23: Temperature difference between 25 metres and 2 metres above the ground (degrees C)
WNDSPD: Wind Speed (metres per second)
WNDDIR: Wind direction (degrees between 0 and 360)
HOUR: Hour of Day
DAYS: Number of the day in the sequence of 500 days

 

  • Estimate the following model:

Report the results of your estimated model, indicating the level of significance using stars (** denotes strong significance p<0.01 and * denotes significance p<0.05).

  • Interpret the estimated coefficients associated with the explanatory variables and conduct an appropriate test of their individual significance.
  • Calculate the value of wind direction that optimises LNO2.
  • Is there evidence that temperature is an important determinant of pollution?
  • Using the model you estimated in (a), conduct a Durbin Watson test for serial correlation.If serial correlation were detected, what implications would it have for your results?
  • Create a one period lag of LNO2and estimate the following model:

Report the results of your estimated model.

(g)     Using the model you estimated in (f), conduct a Breusch-Godfrey test for serial correlation.

Solution 

Question 1
  • Estimate the following model of monthly earnings:

Report the results of your estimated model, indicating the level of significance using stars (** denotes strong significance p<0.01 and * denotes significance p<0.05).

Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 (Constant) 5.092 .170 30.012 .0005**
IQ .005 .001 .192 4.013 .0005**
educ .061 .010 .304 5.928 .0005**
exper .020 .005 .191 3.860 .0005**
tenure .013 .005 .126 2.786 .006*
a. Dependent Variable: Lwage

 

  • Interpret the estimated coefficients.

The coefficients of the model are all significant, four coefficients are significant under p<0.01, hence the model was under 90% level of significance. The model can be explained to mean for any unit change in the natural log of wage , there equates a change in IQ score by a factor 0.005, education by 0.061, year of experience by 0.020 and tenure by 0.013.

  • Comment on the explanatory power of this model.
Model Summary
Model R R Square Adjusted R Square Std. Error of the Estimate
1 .427a .182 .176 .38433
a. Predictors: (Constant), exper, IQ, tenure, educ

The model is weak based on the r-squared value of 0.182, meaning only 18.2% of the variability can be explained by the model.

  • Add dummy variables for white and married to the model in (a). Estimate the new model; report the results and interpret the estimated coefficients associated with the dummy variables only.
Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 (Constant) 4.979 .169 29.377 .000
IQ .004 .001 .149 3.030 .003
educ .061 .010 .304 6.025 .000
tenure .013 .005 .121 2.727 .007
exper .018 .005 .169 3.450 .001
white .118 .053 .095 2.220 .027**
married .183 .052 .144 3.530 .000**
a. Dependent Variable: Lwage

The coefficients with the dummy variables as statistically significant viewed on the basis of the p values represented by **.

Using the Park test, investigate whether there is heteroskedasticity in the model you estimated in (d).  Assume that tenure is the variable causing the problem of heteroskedasticity. Provide an intuitive explanation for your findings.

The model displayed in (b) experience a heteroskedasticity characteristic with regards to the circumstance in which the variability of the variables is unequal across the range of values of a other variables variable that predicts it.

  • Does the presence of heteroskedasticity violate one of the Gauss Markov assumptions? Explain your answer.  What implications would heteroskedasticity have for the results you reported in (d)?

The error term in the model is heteroskedastic because the variance isn’t constant. Instead, the variance of the error term in the model depends on the value of the independent variable(s).

The presence of heteroscedacity implies that the estimated SE is wrong. Because of this, confidence intervals and hypotheses tests cannot be relied on.

Question 2

Nitrogen Dioxide (NO2) is a pollutant that attacks the human respiratory system and increases the likelihood of respiratory illness.  One common cause of nitrogen dioxide is car exhaust.

The file pollution.sav contains data from 500 observations made from October 2001 to August 2003 in the US (data from Carnegie Mellon University archive).

The variables are:

  • Estimate the following model:

Report the results of your estimated model, indicating the level of significance using stars (** denotes strong significance p<0.01 and * denotes significance p<0.05).

Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
(Constant) .768 .188 4.094 .0005**
LCARS .427 .023 .619 18.619 .0005**
TEMP -.024 .004 -.211 -5.595 .0005**
TCHNG23 .121 .025 .171 4.737 .0005**
WNDSPD -.124 .014 -.295 -8.781 .0005**
WNDDIR .005 .001 .545 3.500 .001**
Winddir2 -1.239E-005 .000 -.470 -3.059 .002**
a. Dependent Variable: LNO2

  • Interpret the estimated coefficients associated with the explanatory variables and conduct an appropriate test of their individual significance.

The coefficients of the model are all significant, four coefficients are significant under p<0.01, hence the model was under 90% level of significance. The model can be explained to mean for any unit change in Natural log of the concentration of NO2, there equates a change in LCARS score by a factor 0.427, TEMP by -0.024, TCHNG23 by 0.121 , WNDSPD by -0.124, WNDDR by 0.005 and WNDDR2 by (-1.239E-005).

  • Is there evidence that temperature is an important determinant of pollution?

This variable carries a small weight in being a determinant of pollution based on the coefficient allotted to it of –0.024, which in essence is indirectly proportional to the determinant of pollution.

  • Using the model you estimated in (a), conduct a Durbin Watson test for serial correlation. If serial correlation were detected, what implications would it have for your results?
Model Summaryb
Model R R Square Adjusted R Square Std. Error of the Estimate Durbin-Watson
1 .383a .147 .138 .697 1.352
a. Predictors: (Constant), Winddir2, TCHNG23, WNDSPD, TEMP, WNDDIR
b. Dependent Variable: LNO2

Residuals in this set of variables are increasingly positive correlated with a value of 1.352 per the Durbin Watson.

  • Create a one period lag of LNO2 and estimate the following model: