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Module II Lecture 4:F-Tests

Graduate School 2004/2005 Quantitative Research Methods Gwilym Pryce g.pryce@socsci.gla.ac.uk. Module II Lecture 4:F-Tests. Plan:. (1) Testing a set of linear restrictions – the general case (2) Testing homogenous Restrictions

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Module II Lecture 4:F-Tests

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  1. Graduate School 2004/2005 Quantitative Research Methods Gwilym Pryce g.pryce@socsci.gla.ac.uk Module II Lecture 4:F-Tests

  2. Plan: • (1) Testing a set of linear restrictions – the general case • (2) Testing homogenous Restrictions • (3) Testing for a relationship – Special Case of Homogenous Restrictions • (4) Testing for Structural Breaks

  3. (1) Testing a set of linear Restrictions - The General Procedure • Suppose we want to test whether there are any country specific effects in the relationship between inflation and the money supply: INFL = a + b MS + g1 COUNTRY1 + …. + g42 COUNTRY2 • I.e. we want to test the following null hypothesis: • H0: g1 = g2 = g3 =…. = g42 = 0 • Then we can think of this as being equivalent to comparing two regressions, one restricted and one unrestricted:

  4. The Unrestricted regression is: INFL = a + b MS + g1 COUNTRY1 + …. + g42 COUNTRY2 • The Restricted regression is: INFL = a + b MS • We can test whether all the g coefficients equal zero using the F-test:

  5. The General formula for F: NB RSSR is always greater than RSSU since imposing a restriction on an equation can never reduce the RSS Where: RSSU = restricted residual sum of squares = RSS under H1 RSSR= unrestricted residual sum of squares = RSS under H0 r = number of restrictions = diff. in no. parameters between restricted and unrestricted equations dfu = df from unrestricted regression = n - k where k is all coefficients including the intercept.

  6. Using the F-test: • If the null hypothesis is true (i.e. restrictions are satisfied) then we would expect the restricted and unrestricted regressions to give similar results • I.e. RSSR and RSSU will be similar • so we accept H0 when the test statistic gives a small value for F. • But if one of the restrictions does not hold, then the restricted regression will have had an invalid restriction imposed upon it and will be mispecified. •  higher residual variation  higher RSSR • so we reject H0 when the test stat. gives a large value

  7. Test Procedure: • (i) Compute RSSU • Run the unrestricted form of the regression in SPSS and take a note of the residual sum of squares = RSSU • (ii) Compute RSSR • Run the restricted form of the regression in SPSS and take a note of the residual sum of squares = RSSR • (iii) Calculate r and dfU • (iv) Substitute RSSU, RSSR, r and dfU in the equation for F and find the significance level associated with the value of F you have calculated.

  8. Example 1: Ho: no country effects(R and U regressions have the same dependent variable) Step (i) RSSU = 1835.811

  9. Step (ii) RSSR = 2097.722

  10. Step (iii) r and dfu • r = number of restrictions = difference in no. of parameters between the restricted and unrestricted equations = 3 • dfu = df from unrestricted regression = nU - kU where k is total number of all coefficients including the intercept • = 516 - 5 = 511

  11. (iv) Substitute RSSU, RSSR, r and dfU in the formula for F • F = (RSSR - RSSU) / r = (2097.722 - 1835.811)/3 RSSU/dfU 1835.811 / 511 = 87.304 3.593 = 24.298 • dfnumerator = r = 3 • dfdenominator = dfU = 511 • From Tables, we know that at P = 0.01, the value for F[3,511] would be 3.88 (I.e. Prob(F > 3.88) = 0.01) • F we have calculated is > 3.88, so we know that P < 0.01 (I.e. Prob(F > 24.298) <0.01) Reject Ho

  12. Alternatively use Excel calculator:F-Tests.xls First Paste ANOVA tables of U and R models:

  13. Second, check cell formulas, & let Excel do the rest:

  14. Example 2: Ho: b2 + b3 = 1(R and U regressions have different dependent variables) • (i) Compute RSSU • Run the unrestricted form of the regression in SPSS and take a note of the residual sum of squares = RSSU • (ii) Compute RSSR • Run the restricted form of the regression in SPSS by: • substituting the restrictions into the equation • rearrange the equation so that each parameter appears only once • create new variables where necessary and estimate by OLS • and take a note of the residual sum of squares = RSSR • (iii) Calculate r and dfU • (iv) Calculate F and find the significance level

  15. Unrestricted regression: y = b1 + b2x2 + b3x3 + u • H0: b2 + b3 = 1; • If H0 is true, then: b3 = 1 - b2 and: y = b1 + b2x2 + (1-b2)x3 + u = b1 + b2x2 + x3- b2x3 + u = b1 + b2(x2 - x3)+ x3 + u y - x3= b1 + b2(x2 - x3)+ u • Restricted regression: z = b1 + b2(v)+ u where z = y - x3; v = x2 - x3

  16. Example 3: Ho: b2 = b3(R and U regressions have the same dependent variable) • Unrestricted regression: y = b1 + b2x2 + b3x3 + u • H0: b2 = b3; H1: b2b3 • If H0 is true, then: y = b1 + b2x2 + b2x3 + u = b1 + b2(x2 + x3) + u • Restricted regression: y = b1 + b2(w)+ uwhere w = x2 + x3;

  17. Example 4: Ho: b3 = b2 + 1(R and U regressions have the different dependent variables) • Unrestricted regression: Infl = b1 + b2MS_GDP + b3MP_GDP + u • Ho: b3 = b2 + 1 • If H0 is true, then: Infl = b1 + b2MS_GDP + (b2+1)MP_GDP + u = b1 + b2MS_GDP + b2MP_GDP + 1MP_GDP + u = b1 + b2(MS_GDP + MP_GDP) + MP_GDP + u Infl - MP_GDP = b1 + b2(MS_GDP + MP_GDP) + u • Restricted regression: z = b1 + b2(v)+ uwhere z = Infl - MP_GDP; v = MS_GDP + MP_GDP

  18. Step (i) RSSU = 2069.060

  19. Step (ii) RSSR = 2070.305 • SPSS syntax for creating Z and V COMPUTE Z = Infl - MP_GDP. EXECUTE. COMPUTE V = MS_GDP + MP_GDP. EXECUTE. • SPSS syntax for Restricted Regression: REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /NOORIGIN /DEPENDENT Z /METHOD=ENTER V . • SPSS ANOVA Output:

  20. Step (iii) r and dfu • r = number of restrictions = difference in no. of parameters between the restricted and unrestricted equations = 1 • dfu = df from unrestricted regression = nU - kU where k is total number of all coefficients including the intercept • = 516 - 3 = 513

  21. (iv) Substitute RSSU, RSSR, r and dfU in the formula for F • F = (RSSR - RSSU) / r = (2070.305 - 2069.060)/1 RSSU/dfU 2069.060 / 513 = 1.245 / 4.033 = 0.309 • dfnumerator = r = 1 • dfdenominator = dfU = 513 • From Excel =FDIST(0.309,1,513), we know that Prob(F > 0.309) = 0.58 (I.e. 58% chance of Type I Error) • I.e. if we reject H0 then there is more than a one in two chance that we have rejected H0 incorrectly •  Accept H0 that b3 = b2 + 1

  22. Using F-Tests.xls:

  23. (2) Testing a set of linear Restrictions - When the Restrictions are Homogenous • When linear restrictions are homogenous: • e.g. H0: b2 = b3 = 0 • e.g. H0: b2 = b3 we do not need to transform the dependent variable of the restricted equation. • For restrictions of this type • I.e. where the dependent variable is the same in the restricted and unrestricted regressions we can re-write our F-ratio test statistic in terms of R2s:

  24. F-ratio test statistic for homogenous restrictions: Where: RSSU = unrestricted residual sum of squares = RSS under H1 RSSR= unrestricted residual sum of squares = RSS under H0 r = number of restrictions = diff. in no. parameters between restricted and unrestricted equations dfu = df from unrestricted regression = n - k where k is all coefficients including the intercept.

  25. Proof of simpler formula for homogenous restrictions: If the dependent variable is the same in both the restricted and unrestricted equations, then the TSS will be the same We can then make use of the fact that RSS = (1 - R2) TSS, which implies that: RSSR = (1- RR2) TSS RSSU = (1- RU2) TSS

  26. Proof continued... • Substituting RSSR = (1- RR2) TSS and RSSU = (1- RU2) TSS into our original formula for the F-ratio, we find that:

  27. Example 1: Ho: no country effects(R and U regressions have the same dependent variable) • Our approach to this restriction when we tested it above was to use the RSSs as follows: • Since it is a homogenous restriction (I.e. dep var is same in restricted and unrestricted models), we shall now attempt the same test but using the R2 formulation of the F-ratio formula: • F = (RSSR - RSSU) / r = (2097.722 - 1835.811)/3 • RSSU/dfU 1835.811 / 511 • = 24.301 • Prob(F > F[3,511] 24.298) = 1.028E-14  Reject H0

  28. Unrestricted model: RU2 = 0.139 • Restricted model: RR2 = 0.016

  29. F = (RU2 - RR2) / r = (0.139 - 0.016)/3 = 0.041 = 24.301 (1-RU2)/dfU (1-0.139) / 511 0.0017

  30. (3) Testing a set of linear Restrictions - When the Restrictions say that bi = 0 i • A special case of homogenous restrictions is where we test for the existence of a relationship • I.e. H0: all slope coefficients are zero: • Unrestricted regression: y = b1 + b2x2 + b3x3 + u • H0: b2 = b3 = 0; • If H0 is true, then y = b1 • In this case, Restricted regression does no explaining at all and so RR2 = 0

  31. And the homogenous restriction F-ratio test statistic reduces to: where, r = k -1 dfU = n - k • This is the F-test we came across in MII Lecture 2, and is the one automatically calculated in the SPSS ANOVA table

  32. (4) Testing for Structural Breaks • The F-test also comes into play when we want to test whether the estimated coefficients change significantly if we split the sample in two at a given point • These tests are sometimes called “Chow Tests” after one of its proponents. • There are actually two versions of the test: • Chow’s first test • Chow’s second test

  33. (a) Chow’s First TestUse where n2 > k • (1) Run the regression on the first set of data (i = 1, 2, 3, … n1) & let its RSS be RSSn1 • (2) Run the regression on the second set of data (i = n1+1, n1+2, …, end of data) & let its RSS be RSSn2 • (3) Run the regression on the two sets of data combined (i = 1, …, end of data) & let its RSS be RSSn1 + n2

  34. (4) Compute RSSU, RSSR, r and dfU: • RSSU = RSSn1 + RSSn2 • RSSR= RSSn1 + n2 • r = k = total no. of coeffts including the constant • dfU = n1 + n2 -2k • (5) Use RSSU, RSSR,r and dfU to calculate F using the general formula for F and find the sig. Level:

  35. (b) Chow’s Second TestUse where n2 < k(I.e. when you have insufficient observations on 2nd subsample to do Chow’s 1st test) • (1) Run the regression on the first set of data (i = 1, 2, 3, … n1) & let its RSS be RSSn1 • (2) Run the regression on the two sets of data combined (i = 1, …, end of data) & let its RSS be RSSn1 + n2

  36. (3) Compute RSSU, RSSR, r and dfU: • RSSU = RSSn1 • RSSR= RSSn1 + n2 • r = n2 • dfU = n1 - k • (4) Use RSSU, RSSR,r and dfU to calculate F using the general formula for F and find the sig.:

  37. Example of Chow’s 1st Test: n1: before 1986: n2: 1986 and after

  38. Summary: • (1) Testing a set of linear restrictions – the general case • (2) Testing homogenous Restrictions • (3) Testing for a relationship – Special Case of Homogenous Restrictions • (4) Testing for Structural Breaks

  39. Reading • Kennedy (1998) “A Guide to Econometrics”, Chapters 4 and 6 • Maddala, G.S. (1992) “Introduction to Econometrics” p. 170-177

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