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Econometrics I Homework 2 Author May 24, 2014 Using the adjusted sample about wages, we are expected to run the Regression 1:

log(wageNew ) = β 0 + β 1 hours + β 2 IQ + β 3 black + β 4 married + ε

(1)

Moreover, there are 7 questions we need to solve. In order to do this, we divide our solution into 7 sections.

1

White test

Using ε from Regression 1 we generate εˆ2 and run Regression 2:

ˆε2 =α0 + α1 hours + α2 IQ + α3 black + α4 married + α5 hours 2 +

(2)

2

+α6 IQ + α7 hours ∗ IQ + α8 hours ∗ black + α9 hours ∗ married+ +α10 IQ ∗ black + α11 IQ ∗ married + α12 black ∗ married + ζ 2 Using the White test, we know that LM = nR2 follows the χ12 . The coefficient of determination of Regression 2 is 0.0379 and we have 935 observations thus the LM is approximately 35.43. Comparing this value 2 with χ12 , we obtain that the p-value is around 0.0004 and the H0 of homoscedasticity is strongly rejected. Figure 1 plots squared residuals on Y axis and wa geNew on X axis.

1

0

.005

resid2

.01

.015

Figure 1

1.75

1.8

1.85

1.9

1.95

2

lwageNew

2

Weighted Least Squares

There are two common methods how to deal with the heteroscedasticity. Firstly, instead of OLS one can use WLS estimator, where the weights suppress the heteroscedasticity problem, however, there are two issues related to this process - complicated interpretation and the fact that the weights are usually unknown. Secondly, one can employ robust standard errors. In this case, we are supposed to use WLS with unknown weights, therefore we first need to estimate these weights. The process how we estimate the weights can be summarized as follows. 1. we run regression 1 and save residuals εˆ 2. we create new variable log(ǫˆ2 ) 3. we regress log(ǫˆ2 ) on hours IQ black married and obtain the fitted values 4. exponentiate the fitted values and obtain estimate of the weights hˆ Having found the weighs, we can either run regression with weights in STATA or divide all variables by ˆh and run the standard OLS.

3

Results in standard form

We try both methods and obtain slightly different results. However, we believe that these differences are caused only by a rounding and different form of the model, but they do not affect the conclusion.

2

4

Diagonal elements Weighting matrix 1 9127.12 2 7000.32 3 9636.03 4 9226.71 5 8520.89

5

Black & Married

Using heteroscedasticity robust LM-test, we test whether coefficients on black and married are jointly significant. Unlike the usual LM-statistic, the heteroscedasticity LM-test requires several regressions. We run restricted regression as well as both black and married regress on the included variables. Thus we obtain three different residuals and their products are used as explanatory variables in regression with constant 1 as the dependent variable. Finally, LM-statistic (n − SSR) is distributed according to the χq2. In our case the SSR is 859 and n is 935, hence the LM-statistic is 76. TO DO !! CONCLUSION podle mne strongly reject the Null Hypotheses

3

6 OLS

Comparing all three models- OLS and both WLS - the coefficients seem to be similar. Moreover, all coefficient are significant as all p-values equal 0.000. While the first model (WLS with weighted variables) has SSR around 3 ∗ 107 , the other WLS as well as OLS model have SSR around 0.505. As mentioned above, this is thought to be caused by the form of the first model. Unlike the OLS model which can be easily interpreted, coefficients from both WLS models are rather complicated to interpret.

7 LM-test LM-statistic (n ∗ R2 ) follows χq2 distribution, where R2 is from model in which residuals from restricted model are regressed on all explanatory variables. n is number of observations. In this case, R2 is 0.215 and n is 935; hence the LM-statistic is 201.02. This leads to rejection the null hypothesis, therefore both married and black are believed to be significant.

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