Lecture 4:FTests

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SSSII Gwilym Pryce www.gpryce.com

• Lecture 4F-Tests

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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

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(1) Testing a set of linear Restrictions - The
General Procedure

• E.g. Does Monetarism Explain Everything about
  inflation?
• 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
• i.e. idiosyncrasies of countries (their culture,
  history, economic structure, level of development
  etc) have no effect on inflation. The money
  supply explains everything about inflation.
• Then we can think of this as being equivalent to
  comparing two regressions, one restricted and one
  unrestricted

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• The Unrestricted regression (qualified
  monetarism) is
• INFL a b MS g1 COUNTRY1 . g42
  COUNTRY2
• The Restricted regression (pure monetarism) is
• INFL a b MS
• We can test whether all the g coefficients
  (country specific effects) equal zero using the
  F-test

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The General formula for F
NB RSS is a measure of the total amount of error
in a model. RSSR is always greater than RSSU
since imposing a restriction on an equation can
never reduce the RSS. Question is whether theres
a large increase in RSS from imposing a
restriction.
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.
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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

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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.

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Example 1 Ho no country effects (R and U
regressions have the same dependent variable)
Step (i) RSSU 1835.811
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Step (ii) RSSR 2097.722
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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

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(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 F3,511 would be 3.88 (I.e. Prob(F gt
  3.88) 0.01)
• F we have calculated is gt 3.88, so we know that P
  lt 0.01
• (I.e. Prob(F gt 24.298) lt0.01)
  ?Reject Ho

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Alternatively use Excel calculatorF-Tests.xls
First Paste ANOVA tables of U and R models
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• Second, check cell formulas, let Excel do the
  rest

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Example 2 Ho b2 b3 1 y b1 b2x2 b3x3
u (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

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• If the restriction is
• b2 b3 1
• How would you incorporate this information into
• y b1 b2x2 b3x3 u
• to derive the restricted model?

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• 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

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Example 3 Ho b2 b3 (R and U regressions have
the same dependent variable)

• If the unrestricted regression is
• y b1 b2x2 b3x3 u
• How would you derive the restricted regression?

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• 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) u where w x2
  x3

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Example 4 Ho b3 b2 1 (R and U regressions
have the different dependent variables)

• If the unrestricted regression is
• Infl b1 b2MS_GDP b3MP_GDP u
• How would you derive the restricted regression?

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• Unrestricted regression
• Infl b1 b2MS_GDP b3MP_GDP u
• Ho b3 b2 1
• If H0 is true, then
• Infl b1 b2MS_GDP (b21)MP_GDP u
• b1 b2MS_GDP b2 MP_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) u where z Infl -
  MP_GDP
• 
  v MS_GDP MP_GDP

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• Step (i) RSSU 2069.060

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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
• /METHODENTER V .
• SPSS ANOVA Output

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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

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(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 gt 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

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Using F-Tests.xls
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(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

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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.
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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
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Proof continued...

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

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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 gt F3,511
24.298) 1.028E-14 ? Reject H0
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• Unrestricted model RU2 0.139
• Restricted model RR2 0.016

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

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(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

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And the homogenous restriction F-ratio test
statistic reduces to

• This is the F-test we came across in MII Lecture
  2, and is the one automatically calculated in the
  SPSS ANOVA table

where, r k -1 dfU n - k
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(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
• Chows first test
• Chows second test

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(a) Chows First TestUse where n2 gt 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 n11, n12, , 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

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• (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

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(b) Chows Second TestUse where n2 lt k (I.e.
when you have insufficient observations on 2nd
subsample to do Chows 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

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• (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.

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Example of Chows 1st Test

• n1 before 1986 n2 1986 and after

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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

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Reading

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