Question

1. (5 marks) The following equations were estimated using the data in “bwght": bwght=4.682 -0.005cigs, +0.155parity;+0.026male;+0.062white; (0.016) (0.0008) (0.006) (0.010) (0.012) n=1,388, R2=0.045, RP=0.043 and bwght=4.668 - 0.005cigs, +0.171parity;+0.033male;+0.049white;-0.002motheduc;+0.004fatheduci (0.036) (0.001) (0.006) (0.011) (0.015) (0.003) (0.003) n=1,191, R=0.048, RP=0.043 The variables' definition can be found in the link: http://fmwww.bc.edu/ec-p/data/wooldridge/bwght.des. i (2 marks) Considering the second regression results, comment on the estimated effect and statistical significance of fatheduc.(i.e., comment on the practical and statistical significance of fatheduc's partial effect separately) (3 marks) Recall the content we learned in Chapter 4 (4-5f Testing General Linear Restrictions). If two linear models are nested, it means that the restricted model is obtained from the full model by setting some constraints on the parameters). In this question, the first regression is the restricted model. It is obtained by restricting motheque =0 and Bfatheduc=0 in the full model. Usually we can compare nested models with the F test to see whether the restrictions are valid. In this questions, it means we need to compute the F statistic for joint significance of motheduc and fatheduc by using the formula 4.37 in the textbook. (SSR, SSRur)/q SSRur/(n - k-1) FE However, from the given information, why are you unable to compute the F statistic? What would you have to do to compute the F statistic?

Answer 1

I) Looking at the full model, the independent variable fatheduc is a significant variable as the p- value <0.05 (0.003), we can safely reject the null hypothesis that says no association exist. The coefficient is 0.004, we can conclude that every one unit increase in fatheduc, there is 0.004 increase in the dependent variable bwght, However the coefficient of determination i.e adjusted R^2, is the same in the full model and the reduced model, that explains how the model improves with every new addition in the variable, so again it depends on the researched whether to include this new variable or not, also F statistics can be studied to take a thorough decision.

ii) we basically have the full model to check if the additional Betas does any value addition to check the association between y and the predictors.

To compute the F statistic , we firstly have to fit both the reduced model and then the complete model and then get the values for SSE for the reduced model and the SSE for the complete model. And then at a certain level of alpha, we compare the F−statistic to an Fν1,ν2 from F table, where ν1 = p and ν2 = n − (k + p + 1)

In the formula given above , q = no of additional betas

k = no of parameters in the reduced model ( k+ p in the full model)

n = no of observations

Once we have the value from the F table, we can decide to accept or reject the Null Hypothesis

If F ≥ Fα,ν1,ν2 , reject H0 [Hypotheses: H0 : βk+1 = βk+2 = ... = βk+p = 0 versus Ha : At least one β not equal to 0]

To compute the F statistics, we would require all the individual observations for the reduced (n = 1388) as well for the full (n = 1191) model, then once we calculate the mean value, we can calculate the deviation of individual values from the mean, then square the deviation, and then add all the deviation^2 ( for 1388 ) , likewise for 1191 values . this is the SSE value.

Else, if we had known the S.D value then using the below formula SSE could have been calculated as we know the n.

SSE • Standard Deviation = 1 V n-1

We can still calculate F statistics with the help of R^2 value, but for an individual model not for a nested model.

F =- R? / k (1– R?)/(n-k-1)

Thanks,