6.1 The F-Test 6.2 Testing the Significance of the Model 6.3 An Extended Model 6.4 Testing Some Economic Hypotheses 6.5 The Use of Nonsample

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2 6.1 The F-Test 6. Testing the Significance of the Model 6.3 An Extended Model 6.4 Testing Some Economic Hypotheses 6.5 The Use of Nonsample Information 6.6 Model Specification 6.7 Poor Data, Collinearity and Insignificance 6.8 Prediction

3 S = β+β P +β A + e i 1 i 3 i i H0: β = 0 H : β 0 1 S = β+β A + e i 1 3 i i

4 F = SSE SSE J ( ) R U SSE ( N K ) U If the null hypothesis is not true, then the difference between SSE R and SSE U becomes large, implying that the constraints placed on the model by the null hypothesis have a large effect on the ability of the model to fit the data.

5 Hypothesis testing steps: 1. Specify the null and alternative hypotheses: H0: β = 0 H1: β 0. Specify the test statistic and its distribution if the null hypothesis is true F = ( SSER SSEU) SSE ( 75 3) U (1, 7) 3. Set and determine the rejection region Using α=.05, the critical value from the -distribution is c. Thus, H 0 is rejected if F F F = F = (.95,1,7) 3.97

6 4. Calculate the sample value of the test statistic and, if desired, the p-value F ( R U) SSE ( N K ) ( ) ( 75 3) SSE SSE J = = = 5.06 U p = P[ F 5.06] =.0000 (1,7) 5. State your conclusion Since F = F, we reject the null hypothesis and conclude that price does 5.06 c have a significant effect on sales revenue. Alternatively, we reject H 0 because p =

7 The elements of an F-test : 1. The null hypothesis consists of one or more equality restrictions J. The null hypothesis may not include any greater than or equal to or less than or equal to hypotheses.. The alternative hypothesis states that one or more of the equalities in the null hypothesis is not true. The alternative hypothesis may not include any greater than or less than options. 3. The test statistic is the F-statistic F = ( R U) SSE ( N K ) SSE SSE J U

8 4. If the null hypothesis is true, F has the F-distribution with J numerator degrees of freedom and N-K denominator degrees of freedom. The null hypothesis is rejected if F > Fc, where Fc = F(1 α, J, N K). 5. When testing a single equality null hypothesis it is perfectly correct to use either the t-or F-test procedure; they are equivalent.

9 y =β + x β + x β + L+ x β + e i 1 i i3 3 ik K i H H : β = 0, β = 0, L, β = 0 : At least one of the β is nonzero for k =,3, KK K k y i = β+ e 1 i

10 N N * R ( i 1 ) ( i ) i= 1 i= 1 SSE y b y y SST = = = F = ( SST SSE)/( K 1) SSE /( N K)

11 Example: Big Andy s sales revenue S =β +β P +β A + e i 1 i 3 i i 1. H0: β = 0, β 3 = 0 H : β 0 or β 0, or both are nonzero 1 3. If the null is true F ( SST SSE)/(3 1) = SSE /(75 3) F (,7) 3. H 0 is rejected if F F ( SST SSE) / ( K 1) ( ) / = = = SSE / ( N K) / (75 3) Since 9.95>3.1 we reject the null and conclude that price or advertising expenditure or both have an influence on sales.

12 Figure 6.1 A Model Where Sales Exhibits Diminishing Returns to Advertising Expenditure Principles of Econometrics, 3rd Edition Slide 6-1

13 S = β+β 1 P+β 3A+ e S = β+β 1 P+β 3A+β 4A + e ΔES ( ) ES ( ) = =β + β ΔA ( P held constant) A 3 4 A

14 y = β+β x +β x +β x + e i 1 i 3 i3 4 i4 i y = S, x = P, x = A, x = A i i i i i3 i i4 i Sˆ = P A.768 A i i i i (se) (6.80) (1.046) (3.556) (.941)

15 6.4.1 The Significance of Advertising S =β +β P +β A +β A + e i 1 i 3 i 4 i i S = β+β P + e i 1 i i H0: β 3 = 0, β 4 = 0 H : β 0 or β

16 F = SSE ( ) R U SSE ( 75 4) U SSE F (,71) Fc = F = (.95,,71) 3.16 PF [ (,71) > 8.44] =.0005 Since F = 8.44 > F c = 3.16, we reject the null hypothesis and conclude that advertising does have a significant effect upon sales revenue.

17 Economic theory tells us that we should undertake all those actions for which the marginal benefit is greater than the marginal cost. This optimizing principle applies to Big Andy s Burger Barn as it attempts to choose the optimal level of advertising expenditure. ΔES ( ) ΔA ( P held constant) = β + β 3 4 A β + β = 1 3 4A o (.76796) ˆ = 1 A o

18 Big Andy has been spending $1,900 per month on advertising. He wants to know whether this amount could be optimal. The null and alternative hypotheses for this test are H : β + β 1.9= 1 H : β + β H : β + 3.8β = 1 H : β + 3.8β

19 t = ( b b4) 1 se( b b ) 3 4 var( b b ) = var( b ) var( b ) cov( b, b ) = =.47967

20 t = = = Because <.9676 < 1.994, we cannot reject the null hypothesis that the optimal level of advertising is $1,900 per month. There is insufficient evidence to suggest Andy should change his advertising strategy.

21 S =β +β P + (1 3.8 β ) A +β A + e i 1 i 4 i 4 i i ( Si Ai) =β 1+β Pi +β4( Ai 3.8 Ai) + ei F ( ) /1 = = / 71 F c = = t = (1.994) c p-value = P[ F >.936] (1,71) = Pt [ >.9676] + Pt [ <.9676] =.3365 (71) (71)

22 H H : β + 3.8β : β + 3.8β > Reject H 0 if t t =.9676 Because.9676 < 1.667, we do not reject H 0. There is not enough evidence in the data to suggest the optimal level of advertising expenditure is greater than $1900.

23 ES ( ) =β +β P+β A+β A = β+ 6β+ 1.9β+ 1.9 β = H : β + 3.8β = 1 and β + 6β + 1.9β β = F = 5.74 and the p-value =.0049

24 ln( Q) =β +β ln( PB) +β ln( PL) +β ln( PR) +β ln( I) ( ) ( ) ( ) ( ) ln( Q) =β +β ln λ PB +β ln λ PL +β ln λ PR +β ln λi =β +β ln( PB) +β ln( PL) +β ln( PR) +β ln( I) ( ) + β +β +β +β ln( λ) 3 4 5

25 β +β 3+β 4+β 5 = 0 ln( Q ) =β +β ln( PB ) +β ln( PL) +β ln( PR ) +β ln( I ) + e t 1 t 3 t 4 t 5 t t

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27 β 4 = β β3 β5 ln( Q ) =β +β ln( PB ) +β ln( PL) t 1 t 3 t ( ) + β β β ln( PR ) +β ln( I ) + e t 5 t t ( ln( PB ) ln( PR )) =β +β t ( ln( ) ln( )) ( ln( ) ln( )) +β PL PR + β I PR + e t 3 t t 5 t t t PB t PL t I t =β 1+β ln +β 3ln +β 5ln + e PRt PRt PRt t

28 PB t PL t I t ln( Qt ) = ln ln ln PRt PRt PRt (se) (0.166) (0.84) (0.47) b = b b b * * * * = ( 1.994) =

29 6.6.1 Omitted Variables FAMINC = HEDU + 453WEDU i i i (se) (1130) (803) (1066) ( p-value) (.6) (.000) (.000)

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31 FAMINC i = HEDU (se) (8541) (658) ( p-value) (.00) (.000) i y = β+β x +β x + e i 1 i 3 i3 i

32 bias( b ) = E( b ) β =β cov( x, x ) * * 3 3 var( x ) FAMINC = HEDU WEDU 14311KL6 i i i i (se) (11163) (797) (1061) (5004) ( p-value) (.488) (.000) (.000) (.004)

33 FAMINC = HEDU WEDU 1400 KL X 1067X i i i i i5 i6 (se) (11195) (150) (78) (5044) (4) (198) ( p-value) (.500) (.008) (.010) (.005) (.69) (.591)

34 1. Choose variables and a functional form on the basis of your theoretical and general understanding of the relationship. where β = cβ and x = x c * *. If an estimated equation has coefficients with unexpected signs, or unrealistic magnitudes, they could be caused by a misspecification such as the omission of an important variable.

35 3. One method for assessing whether a variable or a group of variables should be included in an equation is to perform significance tests. That is, t-tests for hypotheses such as or F-tests for hypotheses such as H0: β 3 =β 4 = 0. Failure to reject hypotheses such as these can be an indication that the variable(s) are irrelevant. 4. The adequacy of a model can be tested using a general specification test known as RESET. H0: β 3 = 0

36 y = β+β x +β x + e i 1 i 3 i3 i y = b + b x + b x ˆi 1 i 3 i 3

37 y x x y e i = β+β 1 i +β 3 i3+γ 1ˆ i + i y =β +β x +β x +γ y垐 +γ y + e 3 i 1 i 3 i3 1 i i i

38 H : γ = 0 F = p-value = H : γ =γ = 0 F = 3.13 p-value =

39 6.7.1 The Consequences of Collinearity y = β+β x +β x + e i 1 i 3 i3 i var( b ) = σ N 3 i i= 1 (1 r ) ( x x )

40 The effects of imprecise information: 1. When estimator standard errors are large, it is likely that the usual t- tests will lead to the conclusion that parameter estimates are not significantly different from zero. This outcome occurs despite possibly high or F-values indicating significant explanatory power of the model as a whole.

41 . The estimators may be very sensitive to the addition or deletion of a few observations, or the deletion of an apparently insignificant variable. 3. Despite the difficulties in isolating the effects of individual variables from such a sample, accurate forecasts may still be possible if the nature of the collinear relationship remains the same within the new (future) sample observations.

42 MPG = miles per gallon CYL = number of cylinders ENG = engine displacement in cubic inches WGT = vehicle weight in pounds MPG i = CYL (se) (.83) (.146) ( p-value) (.000) (.000) i

43 MPG = CYL.017 ENG.00571WGT i i i i (se) (1.5) (.413) (.0083) (.0071) ( p-value) (.000) (.517) (.15) (.000)

44 Identifying Collinearity 1. Examining pairwise correlations.. Using auxiliary regression x = ax + ax + L+ a x + error i 1 i1 3 i3 K ik If the R from this artificial model is high, above.80 say, the implication is that a large portion of the variation in is explained by variation in the other explanatory variables.

45 Mitigating Collinearity 1. Obtain more information and include it in the analysis.. Introduce nonsample information in the form of restrictions on the parameters.

46 y = β+ x β+ x β+ e i 1 i i3 3 i y0 = β+ 1 x0β+ x03β+ 3 e0 ŷ0 = b1+ x0b + x03b3

47 var( f) = var[( β+β x +β x + e ) ( b + b x + b x )] = var( e b b x b x ) = var( e ) + var( b) + x var( b ) + x var( b ) x cov( b, b ) + x cov( b, b ) + x x cov( b, b )

48 Principles of Econometrics, 3rd Edition Slide 6-48

49 Principles of Econometrics, 3rd Edition Slide 6-49

50 F = ( SSER SSEU)/ J SSE /( N K) U (6A.1) V Vˆ V Principles of Econometrics, 3rd Edition ( SSE SSE ) = χ σ R U 1 ( J ) ( SSE SSE ) = χ σˆ R U 1 ( J ) ( N K)ˆ σ σ = χ ( N K) (6A.) (6A.3) (6A.4) Slide 6-50

51 F = V V / m / m 1 1 F( m, m ) 1 SSE ( ) N ( ) SSE R U [ ] σ SSER SSEU J = F, K σˆ σˆ ( N K) σ J ( J N K) (6A.5) Principles of Econometrics, 3rd Edition Slide 6-51

52 F = Vˆ 1 J F H0: β 3 =β 4 = 0 S = β+β P +β A +β A + e i 1 i 3 i 4 i i = 8.44 p-value =.0005 χ = p = value.000 Principles of Econometrics, 3rd Edition Slide 6-5

53 H0: β β 4 = 1 F =.936 p-value =.3365 χ = p =.936 -value.3333 Principles of Econometrics, 3rd Edition Slide 6-53

54 y = β+β x +β x + e i 1 i 3 i3 i y = β+β x + v i 1 i i Principles of Econometrics, 3rd Edition Slide 6-54

55 b ( x x )( y y) = =β + * i i ( xi x) wv i i (6B.1) w i = ( xi x) ( x x ) i b =β +β wx + we * 3 i i3 i i Principles of Econometrics, 3rd Edition Slide 6-55

56 Eb ( ) =β +β wx * 3 i i3 =β +β ( x x ) x i i3 3 ( xi x) =β +β ( x x )( x x ) i i3 3 3 ( xi x) cov( x, x ) =β +β β var( x ) 3 3 Principles of Econometrics, 3rd Edition Slide 6-56