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Christopher Dougherty EC220 - Introduction to econometrics (chapter 5) Slideshow: Chow test and dummy variable group test Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 5). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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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.

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

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

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. 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 | CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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

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

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. 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 | CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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. 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 | CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 9 Here are the regression lines for the two subsamples.

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Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 11 ^ ^ 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.

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

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Whole sample COST = 24, NRSS = 8.91x10 11 Whole sample COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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

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. 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 | CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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. 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 | CHOW TEST AND DUMMY VARIABLE GROUP TEST 16 And this is the regression output when COST is regressed on N using the subsample of 34 occupational schools.

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 17 The graph shows the regression lines.

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

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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). ^ ^ ^

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Implicit cost function for regular schools COST = 51, N CHOW TEST AND DUMMY VARIABLE GROUP TEST 20 The intercept and the coefficient of N in the dummy variable regression are then 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. ^ ^ ^ ^

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51,000 – 4,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Implicit cost function for regular schools COST = 51, N Implicit cost function for occupational schools COST = 47, N CHOW TEST AND DUMMY VARIABLE GROUP TEST 21 ^ ^ ^ ^ ^ 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.

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

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 Implicit cost function for regular schools COST = 51, N Implicit cost function for occupational schools COST = 47, N CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 25 ^ ^ ^ 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.

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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

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

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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

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Regular schools only COST = 51, NRSS = 1.22x10 11 Occupational schools only COST = 47, NRSS = 3.49x10 11 Whole sample, with dummy variables COST = 51, ,000OCC + 152N + 284NOCCRSS = 4.71x10 11 CHOW TEST AND DUMMY VARIABLE GROUP TEST 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.

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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. 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 + (b )X 3 + … + ( K + K )X K + u

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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 ). 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 + (b )X 3 + … + ( K + K )X K + u

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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. 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 + (b )X 3 + … + ( K + K )X K + u

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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. 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 + (b )X 3 + … + ( K + K )X K + u

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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. 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 + (b )X 3 + … + ( K + K )X K + u

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CHOW TEST AND DUMMY VARIABLE GROUP TEST 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. 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 + (b )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 and 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 20 Elements of Econometrics

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