1 Lecture 4:F-Tests SSSII Gwilym Pryce .

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Lecture 4:F-Tests

SSSIIGwilym Prycewww.gpryce.com

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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:

5

The General formula for F:

UU

URrdf

dfdf dfRSS

rRSSRSSFF

U /

/)(numerator

rdenominato

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. parametersbetween restricted and unrestricted equations

dfu = df from unrestricted regression

= n - k where k is all coefficients including the

intercept.

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 there’s a large increase in RSS from imposing a restriction.

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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)

Model Summary

.373a .139 .132 1.895Model1

R R SquareAdjustedR Square

Std. Errorof the

Estimate

Predictors: (Constant), CNTRY_3, CNTRY_2,CNTRY_1, money supply

a.

ANOVAb

296.772 4 74.193 20.652 .000a

1835.811 511 3.593

2132.583 515

Regression

Residual

Total

Model1

Sum ofSquares df

MeanSquare F Sig.

Predictors: (Constant), CNTRY_3, CNTRY_2, CNTRY_1, money supplya.

Dependent Variable: inflationb.

Coefficientsa

3.281 1.219 2.692 .007

3.787E-02 .055 .037 .693 .489

-1.716 .692 -.127 -2.479 .014

-4.446 .562 -.330 -7.914 .000

-1.384 .573 -.103 -2.413 .016

(Constant)

money supply

CNTRY_1

CNTRY_2

CNTRY_3

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: inflationa.

Step (i) RSSU = 1835.811

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Step (ii) RSSR = 2097.722

Model Summary

.128a .016 .014 2.020Model1


Std. Errorof the

Estimate

Predictors: (Constant), money supplya.

ANOVAb

34.860 1 34.860 8.542 .004a

2097.722 514 4.081

2132.583 515

Regression

Residual

Total

Model1

Sum ofSquares df

MeanSquare F Sig.



Coefficientsa

1.031 1.000 1.031 .303

.132 .045 .128 2.923 .004

(Constant)

money supply

Model1

B Std. Error


Beta

Standardized

Coefficients

t Sig.


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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

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

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F Critical ValuesDegrees of freedom in the numerator

p 1 2 3 4 5 6 7

0.100 2.73 2.33 2.11 1.97 1.88 1.80 1.75200 0.050 3.89 3.04 2.65 2.42 2.26 2.14 2.06

0.025 5.10 3.76 3.18 2.85 2.63 2.47 2.350.010 6.76 4.71 3.88 3.41 3.11 2.89 2.730.001 11.15 7.15 5.63 4.81 4.29 3.92 3.65

1000 0.100 2.71 2.31 2.09 1.95 1.85 1.78 1.720.050 3.85 3.00 2.61 2.38 2.22 2.11 2.020.025 5.04 3.70 3.13 2.80 2.58 2.42 2.300.010 6.66 4.63 3.80 3.34 3.04 2.82 2.660.001 10.89 6.96 5.46 4.65 4.14 3.78 3.51D

egre

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edom

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F Critical ValuesDegrees of freedom in the numerator

p 1 2 3 4 5 6 7

0.100 2.73 2.33 2.11 1.97 1.88 1.80 1.75200 0.050 3.89 3.04 2.65 2.42 2.26 2.14 2.06

0.025 5.10 3.76 3.18 2.85 2.63 2.47 2.350.010 6.76 4.71 3.88 3.41 3.11 2.89 2.730.001 11.15 7.15 5.63 4.81 4.29 3.92 3.65

1000 0.100 2.71 2.31 2.09 1.95 1.85 1.78 1.720.050 3.85 3.00 2.61 2.38 2.22 2.11 2.020.025 5.04 3.70 3.13 2.80 2.58 2.42 2.300.010 6.66 4.63 3.80 3.34 3.04 2.82 2.660.001 10.89 6.96 5.46 4.65 4.14 3.78 3.51D

egre

es o

f fre

edom

in th

e de

nom

inat

or

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Alternatively use Excel calculator:F-Tests.xls

Restricted ModelANOVAModel Sum of Squaresdf Mean SquareF Sig.

1 Regression 34.86042 1 34.86042 8.541767 0.003624Residual 2097.722 514 4.081172Total 2132.583 515

a Predictors: (Constant), money supplyb Dependent Variable: inflation

Unrestricted ModelANOVAModel Sum of Squaresdf Mean SquareF Sig.

1 Regression 296.7719 4 74.19296 20.65169 0Residual 1835.811 511 3.592585Total 2132.583 515

a Predictors: (Constant), CNTRY_3, CNTRY_2, CNTRY_1, money supplyb Dependent Variable: inflation

First Paste ANOVA tables of U and R models:

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

F Test

r = k U - k R = 3

k u = 5

n = 516

df u = n - k u = 511

= 511

F = 24.30111338

Sig F = 1.02857E-14

UU

URrdf dfRSS

rRSSRSSF

U /

/)(

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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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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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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 + b2 MP_GDP + 1MP_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 = 2069.060ANOVAb

63.523 2 31.761 7.875 .000a

2069.060 513 4.033

2132.583 515

Regression

Residual

Total

Model1

Sum ofSquares df

MeanSquare F Sig.

Predictors: (Constant), MP_GDP, MS_GDPa.


Model Summary

.173a .030 .026 2.008Model1


Std. Errorof the

Estimate

Predictors: (Constant), MP_GDP, MS_GDPa.

Coefficientsa

-6.706 3.419 -1.961 .050

3.836 1.573 .116 2.438 .015

7.543 4.018 .089 1.877 .061

(Constant)

MS_GDP

MP_GDP

Model1

B Std. Error


Beta

Standardized

Coefficients

t Sig.


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Step (ii) RSSR = 2070.305

SPSS syntax for creating Z and VCOMPUTE 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:

ANOVAb

55.701 1 55.701

2070.305 514 4.028

2126.006 515

Regression

Residual

Total

Model1

Sum ofSquares df

MeanSquare

Predictors: (Constant), Va.

Dependent Variable: Zb.

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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 >

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:



a Predictors: (Constant), Vb Dependent Variable: Z



a Predictors: (Constant), MP_GDP, MS_GDPb Dependent Variable: inflation

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F Test

r = k U - k R = 1

k u = 3

n = 516

df u = n - k u = 513

= 513

F = 0.308674039

Sig F = 0.578737192

UU

URrdf dfRSS

rRSSRSSF

U /

/)(

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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:

UU

RUrdf

dfdf dfR

rRRFF

U /1

/)(2

22numerator

rdenominato

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. parametersbetween restricted and unrestricted equations

dfu = df from unrestricted regression = n - k where k is allcoefficients 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:

UU

RU

UU

UR

UU

URrdf

dfR

rRR

dfR

rRR

dfTSSR

rTSSRTSSRF

U

/)1(

/)(

/)1(

/)11(

,/])1[(

/])1()1[(

2

22

2

22

2

22

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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

= 24.301Prob(F > F[3,511] 24.298) = 1.028E-14 Reject H0

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Unrestricted model: RU2 = 0.139

Restricted model: RR2 = 0.016

Model Summary

.373a .139 .132 1.895Model1


Std. Errorof the

Estimate

Predictors: (Constant), CNTRY_3, CNTRY_2,CNTRY_1, money supply

a.

Model Summary

.128a .016 .014 2.020Model1


Std. Errorof the

Estimate


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F = (RU2 - RR

2) / r = (0.139 - 0.016)/3 = 0.041 = 24.301

(1-RU2)

/dfU (1- 0.139) / 511 0.0017

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a Predictors: (Constant), money supplyb Dependent Variable: inflation

Model SummaryModel R R Square Adjusted R SquareStd. Error of the Estimate

1 0.127854 0.016347 0.014433 2.020191a Predictors: (Constant), money supply



a Predictors: (Constant), CNTRY_3, CNTRY_2, CNTRY_1, money supplyb Dependent Variable: inflation

Model SummaryModel R R Square Adjusted R SquareStd. Error of the Estimate

1 0.373043 0.139161 0.132422 1.895412a Predictors: (Constant), CNTRY_3, CNTRY_2, CNTRY_1, money supply

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F Test Statistic for homogenous restrictions:

r = k U - k R = 3

k u = 5

n = 516

df u = n - k u = 511

= 511

F = 24.30111

Sig F = 1.03E-14

UU

RUrdf

dfdf dfR

rRRFF

U /1

/)(2

22numerator

rdenominato

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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:


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 RR

2 = 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

UU

U

UU

Urdf

dfdf

dfR

rR

dfR

rRFF

U

/1

/)(

/1

/)0(

2

2

2

2numerator

rdenominato

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:– Chow’s first test– Chow’s second test

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(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

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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:

UU

URrdf

dfdf dfRSS

rRSSRSSFF

U /

/)(numerator

rdenominato

43

(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

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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.:

UU

URrdf

dfdf dfRSS

rRSSRSSFF

U /

/)(numerator

rdenominato

45

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

-8.183 4.140

3.648 4.043

8.767 3.617

-3.406 1.547

-7.164 .551

-3.585 .544

.214 .536

.320 .538

.873 .567

-7.85E-02 .539

9.764E-02 .542

7.878E-02 .550

(Constant)

MS_GDP

MP_GDP

CNTRY_1

CNTRY_2

CNTRY_3

CNTRY_4

CNTRY_5

CNTRY_6

CNTRY_7

CNTRY_8

CNTRY_9

Model1

B Std. Error



Coefficients

16.393 4.718

-5.806 4.969

-6.240 5.993

-.167 1.857

-.633 .632

1.943 .642

-.195 .601

-.400 .609

-.559 .624

-.130 .607

-.111 .615

6.658E-02 .624

(Constant)

MS_GDP

MP_GDP

CNTRY_1

CNTRY_2

CNTRY_3

CNTRY_4

CNTRY_5

CNTRY_6

CNTRY_7

CNTRY_8

CNTRY_9

Model1

B Std. Error



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ANOVA from n 1

ANOVAModel Sum of Squaresdf Mean SquareF Sig.

1 Regression 670.1102 11 60.91911 31.2705 4.53E-43Residual 563.0107 289 1.948134Total 1233.121 300

a Predictors: (Constant), CNTRY_9, CNTRY_8, CNTRY_7, CNTRY_6, CNTRY_5, CNTRY_4, CNTRY_3, CNTRY_2, CNTRY_1, MP_GDP, MS_GDPb Dependent Variable: inflation

ANOVA from n 2

ANOVAModel Sum of Squaresdf Mean SquareF Sig.


a Predictors: (Constant), CNTRY_9, CNTRY_8, CNTRY_7, CNTRY_6, CNTRY_5, CNTRY_4, CNTRY_3, CNTRY_2, CNTRY_1, MP_GDP, MS_GDPb Dependent Variable: inflation

ANOVA from n1 + n2ANOVAModel Sum of Squaresdf Mean SquareF Sig.


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k = 12

r = k = 12

n1 = 301n2 = 215df u = n - k = 492

RSS R = RSSn1+n2 = 1831.730825

RSSU = RSSn1 + RSSn2 = 918.2283021

F = 40.78898826

Sig F = 4.18611E-66

UU

URrdf dfRSS

rRSSRSSF

U /

/)(

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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

49

Reading

Kennedy (1998) “A Guide to Econometrics”, Chapters 4 and 6

Maddala, G.S. (1992) “Introduction to Econometrics” p. 170-177

1 Lecture 4:F-Tests SSSII Gwilym Pryce .

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Transcript of 1 Lecture 4:F-Tests SSSII Gwilym Pryce .