Introduction to Econometrics Chapter 6

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Introduction to Econometrics Chapter 6 Ezequiel Uriel Jiménez University of Valencia Valencia, September 013

6 Relaxing the assumptions in the linear classical 6.1 Relaxing the assumptions in the linear classical : an overview 6. Misspecification 6.3 Multicollinearity 6.4 Normality test 6.5 Heteroskedasticity 6.6 Autocorrelation Exercises Appendix

6. Misspecification Corr(x,x 3)>0 Corr(x,x 3)<0 TABLE 6.1. Summary of bias in when x is omitted in estimating equation. 3>0 Positive bias Negative bias 3<0 Negative bias Positive bias [3]

6. Misspecification EXAMPLE 6.1 Misspecification in a for determination of wages (file wage06sp) Initial wage = b + b educ + b tenure + u Augmented 1 3 wage = 4.679 + 0.681educ + 0.93tenure i i i (1.55) (0.146) (0.071) R init = 0.49 n= 150 3 wage = b + b educ + b tenure + a wage + a wage + u F 1 3 1 1 Raugm = 0.89 R R r 4.18 (1 R ) / ( nh) ( augm init )/ augm [4]

6.3 Multicollinearity EXAMPLE 6. Analyzing multicollinearity in the case of labor absenteeism (file absent) TABLE 6.. Tolerance and VIF. Collinearity statistics Tolerance VIF age 0.346 4.634 tenure 0.104 47.53 wage 0.7891 1.673 [5]

6.3 Multicollinearity EXAMPLE 6.3 Analyzing the multicollinearity of factors determining time devoted to housework (file timuse03) houswork educ hhinc age paidwork u 1 3 4 5 54.14 max min 7.06E 06 878 TABLE 6.3. Eigenvalues and variance decomposition proportions. Eigenvalues 7.03E-06 0.000498 0.05701 1.861396 54.1400 Variance decomposition proportions Associated Eigenvalue Variable 1 3 4 5 C 0.999995 4.7E-06 8.36E-09 1.3E-13 1.90E-15 EDUC 0.9574 0.70416 4.E-05.3E-09 3.7E-11 HHINC 0.064857 0.3850 0.09016 0.100193 0.40913 AGE 0.651909 0.08485 0.63805 5.85E-07 1.86E-08 PAIDWORK 0.015405 0.03183 0.007178 0.945516 7.80E-05 [6]

6.4 Normality test EXAMPLE 6.4 Is the hypothesis of normality acceptable in the to analyze the efficiency of the Madrid Stock Exchange? (file bolmadef) TABLE 6.4. Normality test in the on the Madrid Stock Exchange. skewness coefficient n=47 kurtosis coefficient Bera and Jarque statistic -0.041 4.468 1.03 [7]

[8] 6.5 Heteroskedasticity FIGURE 6.1. Scatter diagram corresponding to a with homoskedastic disturbances. FIGURE 6.. Scatter diagram corresponding to a with heteroskedastic disturbances. y x y x

6.5 Heteroskedasticity [9] EXAMPLE 6.5 Application of the Breusch-Pagan-Godfrey test TABLE 6.5. Hostel and inc data. i hostel inc 1 17 500 4 700 3 7 50 4 17 430 5 31 810 6 3 00 7 8 300 8 4 760 9 30 650 10 9 30 Step 1. Applying OLS to the, hostel b1+ binc + u using data from table 6.5, the following estimated is obtained: hostel i =- 7.47+ 0.0533inc (3.48) (0.0065) The residuals corresponding to this fitted appear in table 6.6. i

6.5 Heteroskedasticity EXAMPLE 6.5 Application of the Breusch-Pagan-Godfrey test. (Cont.) TABLE 6.6. Residuals of the regression of hostel on inc. i 1 3 4 5 6 7 8 9 10 uˆi -.6-5.888 1.1 1.505-4.751-0.34-0.565 8.913.777-0.631 Step. The auxiliary regression i Step 3. The BPG statistics is: uˆ i 1 inci i uˆ 3.93 0.0799inc R 0. 5045 (0.05) 1 BPG nr ar ( ) = = 10 0.56 = 5.05 Step 4. Given that =3.84, the null hypothesis of homoskedasticity is rejected for a significance level of 5%, but not for the significance level of 1%. [10]

6.5 Heteroskedasticity [11] EXAMPLE 6.6 Application of the White test Step 1. This step is the same as in the Breusch-Pagan-Godfrey test. Step. The regressors of the auxiliary regression will be Step 3. The W statistic: W 1i i 3i 1 i 1inc inc i i 1 i 3 i i uˆ inc inc i i i uˆ 14.9 0.10inc 0.00018inc R 0. 56 (0.10) i ( ) = nr = = 10 0.56 5.60 Step 4. Given that =4.61, the null hypothesis of homoskedasticity is rejected for a 10% significance level because W=nR >4.61, but not for significance levels of 5% and 1%.

6.5 Heteroskedasticity [1] EXAMPLE 6.7 Heteroskedasticity tests in s explaining the market value of the Spanish banks (file bolmad95) Heteroskedasticity in the linear marktval = b + b bookval + u marktval 9.4+ 1.19 bookval n = 0 1 Residuals in absolute value (30.85) (0.17) 400 350 300 50 00 150 100 50 0 0 100 00 300 400 500 600 700 bookval GRAPHIC 6.1. Scatter plot between the residuals in absolute value and the variable bookval in the linear. (0.01) 1 As =6.64<10.44, the null hypothesis of homoskedasticity is rejected for a significance level of 1%, and therefore for =0.05 and for =0.10.l (0.01) As =9.1<1.03, the null hypothesis of homoskedasticity is rejected for a significance level of 1%. BPG nr ar 00.50 10.44 W nr ar 00.6017 1.03

6.5 Heteroskedasticity EXAMPLE 6.7 Heteroskedasticity tests in s explaining the market value of the Spanish banks (Cont.) Heteroskedasticity in the log-log ln( marktval) 0.676+ 0.9384ln( bookval) Residuals in absolute value 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 (0.65) (0.06) 0.0 1.5.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 ln(bookval) GRAPHIC 6.. Scatter plot between the residuals in absolute value and the variable bookval in the log-log. TABLE 6.7. Tests of heteroskedasticity on the log-log to explain the market value of Spanish banks. Test Statistic Table values nr ra (0.10) Breusch-Pagan BP = =1.05 =4.61 [13] nr ra (0.10) White W= =.64 =4.61

6.5 Heteroskedasticity EXAMPLE 6.8 Is there heteroskedasticity in demand of hostel services? (file hostel) [14] ( ) ln hostel b1+ bln( inc ) + b3secstud + b4terstud + b5hhsize + u ln( hostel) - 16.37 +.73ln( inc ) + 1.398secstud +.97terstud -0.444hhsize i i i i i (.6) (0.34) (0.58) (0.333) (0.088) Residuals in absolute value 1.6 1.4 1. 1 0.8 0.6 0.4 0. R = 0.91 n= 40 0 6.4 6.6 6.8 7 7. 7.4 7.6 7.8 8 ln(inc) GRAPHIC 6.3. Scatter plot between the residuals in absolute value and the variable ln(inc) in the hostel. TABLE 6.8. Tests of heteroskedasticity in the of demand for hostel services. Breusch-Pagan- Godfrey Test Statistic Table values nr ra (0.05) BPG= =7.83 =5.99 nr ra (0.01) White W= =1.4 =9.1

[15] 6.5 Heteroskedasticity EXAMPLE 6.9 Heteroskedasticity consistent standard errors in the s explaining the market value of Spanish banks (Continuation of example 6.7) (file bolmad95) Non consistent marktval 9.4+ 1.19 bookval ln( marktval) 0.676+ 0.9384ln( bookval) White procedure (30.85) (0.17) (0.65) (0.06) marktval 9.4+ 1.19 bookval ln( marktval) 0.676+ 0.9384ln( bookval) (18.67) (0.49) (0.318) (0.0698)

[16] 6.5 Heteroskedasticity EXAMPLE 6.10 Application of weighted least squares in the demand of hotel services (Continuation of example 6.8) (file hostel) WLS estimation uˆ 0.039+ 0.0003 inc R 0.1638 i (0.143) (.73) uˆ - 0.4198+ 0.035 inc R 0.1733 i (-1.34) (.8) 1 uˆ i 0.8857-53.1 R 0.1780 (5.39) (-.87) inc uˆ -.7033+ 0.4389ln( inc) R 0.1788 i (-.46) (.88) ln( hostel) - 16.1+.709ln( inc) + 1.401secstud +.98terstud -0.445hhsize i i i i i (.15) (0.309) (0.47) (0.36) (0.085) R = 0.914 n= 40

6.6 Autocorrelation 3 u 1 0 time -1 - -3 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 FIGURE 6.3. Plot of non-autocorrelated disturbances. [17]

6.6 Autocorrelation [18] 4 u 3 1 00 time 1 3 4 5 6 7 8 9 1011113141516171819013456789-1 - -3-4 FIGURE 6.4. Plot of positive autocorrelated disturbances. 5 u 4 3 1 0 time 1 3 4 5 6 7 8 9 1011113141516171819013456789-1 - -3-4 -5 FIGURE 6.5. Plot of negative autocorrelated disturbances.

6.6 Autocorrelation y FIGURE 6.6. Autocorrelated disturbances due to a specification bias. x [19]

[0] 6.6 Autocorrelation EXAMPLE 6.11 Autocorrelation in the to determine the efficiency of the Madrid Stock Exchange (file bolmadef) d L =1.664; d U =1.684 Since DW=.04>d U, we do not reject the null hypothesis that the disturbances are not autocorrelated for a significance level of =0.01, i.e. of 1%. 4 3 1 0-1 - -3-4 GRAPHIC 6.4. Standardized residuals in the estimation of the to determine the efficiency of the Madrid Stock Exchange.

6.6 Autocorrelation [1] EXAMPLE 6.1 Autocorrelation in the for the demand for fish (file fishdem) For n=8 and k'=3, and for a significance level of 1%: Since d L <1.0<d U, there is not enough evidence to accept the null hypothesis, or to reject it. 3 1 0-1 - d L =0.969; d U =1.415 4 6 8 10 1 14 16 18 0 4 6 8 GRAPHIC 6.5. Standardized residuals in the on the demand for fish.

6.6 Autocorrelation [] EXAMPLE 6.13 Autocorrelation in the case of Lydia E. Pinkham (file pinkham) n é dù n é 1.01ù 53 h rˆ 1-1- 1-nvar êë úû 1- var êë úû 1-53 0.0814 ( bˆ ) ( ˆ j n bj) Given this value of h, the null hypothesis of no autocorrelation is rejected for =0.01 or, even, for =0.001, according to the table of the normal distribution. 5,0 4,0 3,0,0 1,0 0,0-1,0 -,0-3,0-4,0-5,0 8 13 18 3 8 33 38 43 48 53 58 GRAPHIC 6.6. Standardized residuals in the estimation of the of the Lydia E. Pinkham case.

[3] 6.6 Autocorrelation EXAMPLE 6.14 Autocorrelation in a to explain the expenditures of residents abroad (file qnatacsp) ln( turimp ) =- 17.31+.0155ln( gdp ) t (3.43) (0.76) R DW n = 0.531 =.055 = 49.5.0 1.5 1.0 0.5 0.0-0.5-1.0-1.5 -.0 5 10 15 0 5 30 35 40 45 GRAPHIC 6.7. Standardized residuals in the estimation of the explaining the expenditures of residents abroad. For a AR(4) scheme, is equal to BG = nr ar =36.35. Given this value of BG, the ( ) null hypothesis of no autocorrelation is rejected for =0.01, since =15.09. t 5

6.6.4 HAC standard errors EXAMPLE 6.15 HAC standard errors in the case of Lydia E. Pinkham (Continuation of example 6.13) (file pinkham) TABLE 6.9.The t statistics, conventional and HAC, in the case of Lydia E. Pinkham. regressor t conventional t HAC ratio intercept.644007 1.779151 1.49 advexp 3.98965 5.73763 0.69 sales (-1) 7.45915 6.9457 1.07 d 1-1.49905-1.50571 1 d 3.5871.7431 1.4 d 3-3.01993 -.65891 1.14 [4]

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