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1 CHOW TEST AND DUMMY VARIABLE GROUP TEST In the dummy variable sequences and in the Chow test sequence we investigated whether the cost functions for occupational and regular schools are different. COST N

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2 In each case we performed tests that showed that the functions are significantly different. Could the two approaches have led to different conclusions? CHOW TEST AND DUMMY VARIABLE GROUP TEST COST N

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3 The answer is no. The Chow test is equivalent to an F test testing the explanatory power of the dummy variables as a group. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST N

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4 With both approaches the starting point is a simple regression of annual recurrent expenditure on the number of students enrolled, using the entire sample. We make a note of RSS. CHOW TEST AND DUMMY VARIABLE GROUP TEST. reg COST N Source | SS df MS Number of obs = F( 1, 72) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.1e COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | _cons |

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5 The regression line is shown graphically. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST N

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6 We now make a distinction between occupational schools and regular schools. CHOW TEST AND DUMMY VARIABLE GROUP TEST occupational school regular school COST N

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7 With the dummy variable approach, we take one type of school as the reference category. We will choose regular schools for this category, but it makes no difference to the test. CHOW TEST AND DUMMY VARIABLE GROUP TEST. reg COST N OCC NOCC Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | NOCC | _cons |

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8 We add an intercept dummy and a slope dummy to allow the overhead and marginal costs of the occupational schools to be different. Again we make a note of RSS, which is smaller than before. CHOW TEST AND DUMMY VARIABLE GROUP TEST. reg COST N OCC NOCC Source | SS df MS Number of obs = F( 3, 70) = Model | e e+11 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+10 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | OCC | NOCC | _cons |

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9 Here are the regression lines for the two subsamples. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST occupational school regular school N

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10 To see if the cost functions are significantly different, we investigate whether there is a significant reduction in RSS when the dummy variables are added. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11

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^ ^ Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x We perform the F test described in the sequence on slope dummy variables. The numerator of the test statistic is the reduction in RSS on adding the dummy variables, divided by the cost in terms of degrees of freedom. CHOW TEST AND DUMMY VARIABLE GROUP TEST

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^ ^ 12 The denominator is the RSS remaining after adding the dummy variables, divided by the number of degrees of freedom remaining. CHOW TEST AND DUMMY VARIABLE GROUP TEST Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11

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^ ^ 13 The critical value of F at the 0.1% level with 2 and 70 degrees of freedom is 7.6. Hence we conclude that the dummy variables do have significant explanatory power and the cost functions are different. CHOW TEST AND DUMMY VARIABLE GROUP TEST Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11

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14 With the Chow test approach we also start by running a regression using the whole sample, and make a note of the RSS. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST N

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15 We then split the sample into occupational and regular schools, and run separate regressions, again making a note of RSS. This is the regression output when COST is regressed on N for the subsample of 40 regular schools. CHOW TEST AND DUMMY VARIABLE GROUP TEST. reg COST N if OCC==0 Source | SS df MS Number of obs = F( 1, 38) = Model | e e+10 Prob > F = Residual | e e+09 R-squared = Adj R-squared = Total | e e+09 Root MSE = COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | _cons |

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16 And this is the regression output when COST is regressed on N using the subsample of 34 occupational schools. CHOW TEST AND DUMMY VARIABLE GROUP TEST. reg COST N if OCC==1 Source | SS df MS Number of obs = F( 1, 32) = Model | e e+11 Prob > F = Residual | e e+10 R-squared = Adj R-squared = Total | e e+10 Root MSE = 1.0e COST | Coef. Std. Err. t P>|t| [95% Conf. Interval] N | _cons |

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17 The graph shows the regression lines. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST occupational school regular school N

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18 The regression equations are as shown. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11 ^ Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11

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19 The cost functions are identical to those implicit in the dummy variable regression with both intercept and slope dummies. This is because the dummy variable regression has a dummy variable for each component of the original model (here, the constant and N). CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11 ^ Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11

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^ ^ ^ Occupational schools, subsample regression COST = 47, NRSS = 3.49x The intercept and the coefficient of N in the dummy variable regression are chosen so as to minimize the residual sum of squares for the reference category, the regular schools. Hence they must be the same as for the regression with regular schools only. CHOW TEST AND DUMMY VARIABLE GROUP TEST Implicit cost function for regular schools COST = 51, N ^ Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11

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^ ^ Implicit cost function for regular schools COST = 51, N ^ Regular schools, subsample regression COST = 51, NRSS = 1.22x The intercept and slope dummies then allow the intercept and slope coefficient to be modified so as to give the best possible fit for the occupational schools. Hence the implicit cost function must be the same as for the regression with occupational schools only. CHOW TEST AND DUMMY VARIABLE GROUP TEST Implicit cost function for occupational schools COST = 47, N ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11 ^

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22 The cost function for regular schools implicit in the dummy variable regression must coincide with the regression line for the regular schools only. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST occupational school regular school N

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23 Similarly, the cost function for occupational schools implicit in the dummy variable regression must coincide with the regression line for the occupational schools only. CHOW TEST AND DUMMY VARIABLE GROUP TEST COST occupational school regular school N

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24 Since the cost functions implicit in the dummy variable regression coincide with those in the separate regressions, the residuals will be the same. It follows that RSS for the dummy variable regression must be equal the sum of RSS for the separate regressions. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Implicit cost function for regular schools COST = 51, N ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Implicit cost function for occupational schools COST = 47, N Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x Hence the F statistics for the F tests will be the same. The starting point for both approaches is the residual sum of squares for the basic regression making no distinction between types of school. CHOW TEST AND DUMMY VARIABLE GROUP TEST Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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26 In the Chow test approach, RSS is reduced by splitting the sample. In the dummy variable approach, RSS is reduced by adding the intercept and slope dummies. RSS after making the change will be the same because the residuals will be the same. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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27 This also means that the first part of the denominator of the F statistic will be the same. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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28 The cost of the improvement in the fit is the same, since either way two extra parameters have to be estimated. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 ` Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 ` 29 And either way, the number of degrees of freedom remaining will be 70, since the number of observations is 74 and 4 parameters have to be estimated. CHOW TEST AND DUMMY VARIABLE GROUP TEST Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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30 Thus all the components of the F statistics are the same, and the outcome of the test must be the same. In this case, the null hypothesis of identical cost functions for the two types of school was rejected at the 0.1% level. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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31 What are the advantages and disadvantages of the two approaches? CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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32 The Chow test is quick. You just run the three regressions and compute the test statistic. But it does not tell you how the functions differ, if they do. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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33 The dummy variable approach involves more preparation because you have to define a dummy variable for the intercept and for each slope coefficient. CHOW TEST AND DUMMY VARIABLE GROUP TEST ^ ^ ^ Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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34 ^ ^ ^ However, it is more informative because you can perform t tests on the individual dummy coefficients and find out where the functions differ, if they do. CHOW TEST AND DUMMY VARIABLE GROUP TEST Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Regular schools, subsample regression COST = 51, NRSS = 1.22x10 11 Occupational schools, subsample regression COST = 47, NRSS = 3.49x10 11

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35 A final note. The Chow test and the dummy variable group test are equivalent only if there is a full set of dummy variables. CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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36 By this is meant an intercept dummy (here D) and a slope dummy variable for every X (here DX 2, DX 3, … DX K ). CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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37 If there is a full set of dummy variables, OLS will choose the intercept b 1 and the b coefficients of X 2 … X K so as to optimise the fit for the D = 0 observations. The coefficients will be exactly the same as if the regression has been run with only the subsample of D = 0 observations. CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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38 The coefficient of the intercept dummy D and the slope dummy variables will then be chosen so as to optimise the fit for the D = 1 observations. (b 1 +d), (b 2 +l 2 ), …, (b K +l K ) will be the same as the coefficients in a regression using only the subsample of D = 1 observations. CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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39 Thus with a full set of intercept and slope dummy variables, the improvement in fit on adding the dummy variables to the basic equation is the same as that obtained by splitting the sample and running separate subsample regressions. CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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40 It follows that the F statistic for the test of the joint explanatory power of the intercept and slope dummy variables is equivalent to the F statistic for the Chow test. CHOW TEST AND DUMMY VARIABLE GROUP TEST Basic model Model with dummy variables Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u Y = 1 + 2 X 2 + 3 X 3 + … + K X K + D + 2 DX DX 3 + … + K DX K + u D = 0 Y = 1 + 2 X 2 + 3 X 3 + … + K X K + u D = 1 Y = ( 1 + ) + ( )X 2 + ( )X 3 + … + ( K + K )X K + u

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 5.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics

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