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Dependent Variable: INCOME Included observations: 30 Variable Coefficient Std. Error t-Statistic Prob. C -1001.869 520.7067 -1.924056 0.0654 AGE 8.846979 5.453205 1.622345 0.1168 EDU 95.16914 38.53614 2.469607 0.0204 INHER 1.514066 0.464237 3.261411 0.0031 R-squared 0.540692 S.E. of regression 353.2761 Sum squared resid 3244903. F-statistic 10.20230 Prob(F-statistic)0.000128 Dependent Variable: INCOME Sample: 1 30 Included observations: 30 Variable Coefficient Std. Error t-Statistic C 884.5000 90.11339 9.815412 R-squared0.000000 S.E. of regression 493.5713 Sum squared resid 7064768. Prob. 0.0000 F = [(RSSR - RSSU)/m] [RSSU/ (n-k)] where RSSR RSS of the regression performed under the restrictions imposed RSSU RSS of the regression performed without any restriction m Number of restrictions n Total number of observations k no. of parameters to be estimated The numerator of the F-statistic has m degrees of freedom. The denominator of the F-statistic has (n-k) degrees of freedom. We reject H0 if F exceeds the critical value corresponding to the level of significance. Otherwise we retain H0. Series: INCOME Sample 1 30 Observations 30 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis 884.5000 705.0000 2000.000 250.0000 493.5713 0.948059 2.811966 The Chow Test Dependent Variable: QTYBEFORE Variable Coefficient Std. Error t-Statistic Prob. C 2.093668 2.375737 0.881271 0.3818 PRICE 2.997041 0.152499 19.65284 0.0000 BEFORE R-squared 0.869438 S.E. of regression 3.644955 Sum squared resid 770.5704 Dependent Variable: QTYAFTER Variable Coefficient Std. Error t-Statistic Prob. C 19.73596 2.172561 9.084191 0.0000 PRICE 3.036940 0.291535 10.41708 0.0000 AFTER R-squared0.651684 S.E. of regression 3.619314 Sum squared resid 759.7673 Dependent Variable: QTYALL Variable Coefficient Std. Error t-Statistic Prob. C 29.38339 1.417390 20.73063 0.0000 PRICEALL 1.371748 0.116072 11.81805 0.0000 R-squared0.542043 S.E. of regression 5.965826 Sum squared resid 4199.747 F = [(RSSR - RSSU)/k] [RSSU/ (n1+n2-2k)] where RSSR RSS from the pooled data regression RSS1 RSS from the first regression RSS2 RSS from the second regression RSSU RSS1+ RSS2 n1 Total number of observations in the first dataset n1 Total number of observations in the first dataset k no. of parameters to be estimated The F-statistic constructed above has an F-distribution with the numerator having k degrees of freedom and the denominator having (n1+n2-2k) degrees of freedom. If the null hypothesis is wrong, then F will be a “large” value and we shall accept that the structural breakdown has taken place. Otherwise, we do not reject H0. PRICEALL vs. QTYALL 20 15 PRICEALL 10 5 0 20 30 40 QTYALL 50 60 70 • Qualitative (indicator or dummy) Variables • The number of dummy variables needed for a qualitative variable is the number of categories less one. • For dichotomous variables, such as gender, only one dummy variable is needed. There are two categories (female, male); c = 1; c - 1 = 0. • Your office is located in which region of the country? ___Northeast___ Midwest ___South___West Number of dummy variables = c - 1 = 4 - 1 =3 Dependent Variable: QTYALL Variable Coefficient Std. Error t-Statistic C 19.96464 1.083132 Prob. 18.43233 0.0000 PRICEALL 3.005518 0.134288 22.38111 0.0000 DUMMY -14.28446 0.0000 -18.00042 1.260140 R-squared 0.833105 Sum squared resid S.E. of regression 3.616830 1530.531 F-statistic 292.0197 Prob(F-statistic) 0.000000 The Classical Linear Regression Model Some Procedural Problems 1. Heteroskedasticity: The different random terms do not have the same variance s2(as was assumed) 2. Autocorrelation amongst the random terms means that Cov(ei ej) 0 for some i and j Presence of either of these factors means that E(ee’) s2In 3. Multicollinearity: Some explanatory variables are not linearly independent and so the matrix (X’X)-1 does not exist 4. Explanatory variables correlated with the random term:The OLS estimators are no longer unbiased 5. Some relevant explanatory variables are omitted:The OLS estimators are no longer unbiased