2. 6.1 Lecture #6 Studenmund(2006) Chapter 7 1. Suppressing the intercept 2. Alternative Functional forms 3. Scaling and units of measurement Objectives:

3. 6.2 True relation Y X Estimated relationship Suppressing the intercept Such an effect potentially biases the βs and inflates their t-values ^

4. 6.3 Regression through the origin The intercept term is absent or zero. i.e., iii  XY  1  i Y i X 1 1  ^ ii X  YSRF 1 : ^^  0

5. 6.4 The estimated model: i 1 XY  ~ ~  or ii XY  ~ ~  1 Regression through the origin Applied OLS method:    2 1 i ii X Y X  ~    2 2 1 i X Var   ^ ~ and ~ N -1 2 2     ^ and

6. 6.5 Some feature of no-intercept model 1.  i  ~ need not be zero2. R2R2 can be occasions turn out to be negative. may not be appropriate for the summary of statistics. 3. df does not include the constant term, i.e., (n-k) In practice: 1. A very strong priori or theoretical expectation, otherwise stick to the conventional intercept-present model. 2. If intercept is included in the regression model but it turns out to be statistically insignificant, then we have to drop the intercept to re-run the regression.

7. 6.6 i  XY  10  Regression through origin i ’’ X Y  1   ^ 2 2 2   n  ^  1 2 2    n ’’ ~ ^          22 2 2 YY X X YYXX R or     22 2 2 yx xy R     2 2 Y2Y2 X2X2 ( XY ) R raw    1 x2x2 xy ^    1 X2X2 XY  ~     1 X2X2 Var    ~ ^   2   2 1 x Var ^ ^ N-k-1 N-k

8. 6.7 i Example 1: Capital Asset Pricing Model (CAPM) security i’s expect risk premium=expected market risk premium  fm 1 fi rERr  expected rate of return on security i risk free of return expected rate of return on market portfolio 1  as a measure of systematic risk. 1  >1 ==> implies a volatile or aggressive security. 1  implies a defensive security.

9. 6.8 Example 1:(cont.) fi rER  f m  1 1  Security market line

10. 6.9 Example 2:Covered Interest Parity International interest rate differentials equal exchange rate forward premium. i.e.,  )( * e eF ii 1   N N f e eF ii    )( * 1 1  Covered interest parity line

11. 6.10 Example 2:(Cont.) in regression: i10 u e eF ii    )()( *  0)(  0 E  If covered interest parity holds, 0  is expected to be zero. Use the t-test to test the intercept to be zero

12. 6.11 y: Return on A Future Fund, % X: Return on Fisher Index, % XY0899.1  ^ (5.689) Formal report: R 2 =0.714 SEE=19.54 N=10

13. 6.12 The t-value shows that b 0 is statistically insignificant different from zero XY0691.12797.1  ^ (0.166)(4.486) R 2 =0.715 SEE=20.69 N=10 1.279 - 0 7.668 H 0 :  0 = 0

14. 6.13 Functional Forms of Regression The term linear in a simple regression model means that there are linear in the parameters; variables in the regression model may or may not be linear.

15. 6.14 True model is nonlinear Y X Income Age 60 15 PRF SRF But run the wrong linear regression model and makes a wrong prediction

16. 6.15 Y i =  0 +  1 X i +  i Examples of Linear Statistical Models ln(Y i ) =  0 +  1 X i +  i Y i =  0 +  1 ln(X i ) +  i Y i =  0 +  1 X i +  i 2 Examples of Non-linear Statistical Models Y i =  0 +  1 X i +  i 22 Y i =  0 +  1 X i + exp(  2 X i ) +  i Y i =  0 +  1 X i +  i 22 Linear vs. Nonlinear

17. 6.16 Different Functional Forms 5. Reciprocal (or inverse) Attention to each form’s slope and elasticity 1. Linear 2. Log-Log 3. Semilog Linear-Log or Log-Linear 4. Polynomial

18. 6.17 Functional Forms of Regression models Transform into linear log-form: i  X ln Y  1  i  XY  * * 1 * 0 *  i  X Y  1 * 0  ==> 1 * 1  where * * * ln   X dX Y dY Xd Yd dX dY elasticity coefficient 2. Log-log model: ii eXY 0     This is a non- linear model

19. 6.18 Functional Forms of Regression models Quantity Demand Y X price 1 0     XY ln Y lnX XYln 10  ln Y lnX XYln 10  Quantity Demand price Y X 1 0    XY

20. 6.19 Functional Forms of Regression models 3. Semi log model: Log-lin model or lin-log model: iii  XY  10 ln  iii  XY  10  or and  1  relative change in Y absolute change in X YdX dY dX Y dY dX Yd1ln   1  absolute change in Y relative change in X 1ln X dX dY Xd 

21. 6.20 5. Reciprocal (or inverse) transformations i i i  X Y  ) 1 ( 10  Functional Forms of Regression models(Cont.) iii  XY  )( * 10  ==> Where i i X X 1 *  4. Polynomial: Quadratic term to capture the nonlinear pattern Y i =  0 +  1 X i +  2 X 2 i +  i Yi XiXi  1 >0,  2 <0 Yi XiXi  1 0

22. 6.21 Some features of reciprocal model X Y 1    Y 0  X 0 0   and 0 1  Y X 0  0 + - X Y 1    0 0  and 0 1  Y 0  X 0 01 /  0 0  0 1  Y 0  X 0 01 /  0 0  and 0 1 

23. 6.22 Two conditions for nonlinear, non-additive equation transformation. 1. Exist a transformation of the variable. 2. Sample must provide sufficient information. Example 1: Suppose 213 2 12110 XXXXY  transforming X 2 * = X 1 2 X 3 * = X 1 X 2 rewrite * 33 * 22110 XXXY 

24. 6.23 Example 2: 2 1 0      X Y transforming 2 * 1 1   X X * 110 XY  rewrite However, X 1 * cannot be computed, because  is unknown. 2

25. 6.24 Application of functional form regression 1. Cobb-Douglas Production function:  eKLY 0    Transforming:  KLY  KLY   ln 210 210   ==> 1 ln  Ld Yd 2  Kd Yd : elasticity of output w.r.t. labor input : elasticity of output w.r.t. capital input. 1 21  > < Information about the scale of returns.

26. 6.25 2. Polynomial regression model: Marginal cost function or total cost function costs y MC i.e. costs y  XXY  2 210  (MC) or costs y TC  XXXY  3 3 2 210  (TC)

27. 6.26 linear XY 10  1  dX dYX )( 1 Y  XYln 10  Log-log 1 ln  X dX Y dY Xd Yd 1  )( 1 X Y dX dY  ==> SlopeElasticity Summary Model Equation )( dX dY  )( X dX Y dY 

28. 6.27 Summary(Count.) Reciprocal X Y 1 10  1 2 ) 1 ( 1    dX X dY X d X2X2 dX dY 1  ) ( 1 XY  ==> Lin-log XYln 10  1  X dX dY Xd Y 1 1  XdX dY1 1  ==> Y dX dY 1  ==> XY 10 ln  Log-lin X 1  1 ln  dX Y dY dX Yd SlopeElasticity

29. 6.28 2 5325.1304.100MPNG  ^ (1.368)(39.20) Linear model

30. 6.29 GNP = -1.6329.21 + 2584.78 lnM 2 (-23.44)(27.48) ^ Lin-log model

31. 6.30 lnGNP = 6.8612 + 0.00057 M 2 (100.38)(15.65) ^ Log-lin model

32. 6.31 2 ln9882.05529.0lnMNPG  ^ (3.194)(42.29) Log-log model

33. 6.32 Wage(y) unemp.(x) SRF 10.43 wage=10.343-3.808(unemploy) (4.862)(-2.66) ^

34. 6.33 ) 1 ( x y SRF -1.428 uNuN u N : natural rate of unemployment Reciprocal Model (1/unemploy) Wage = -1.4282+8.7243 ) 1 ( x (-.0690)(3.063) ^ The  0 is statistically insignificant Therefore, -1.428 is not reliable

35. 6.34 lnwage = 1.9038 - 1.175ln(unemploy) (10.375)(-2.618) ^

36. 6.35 Lnwage = 1.9038 + 1.175 ln ) 1 ( X (10.37)(2.618) ^ Antilog(1.9038) = 6.7113, therefore it is a more meaningful and statistically significant bottom line for min. wage Antilog(1.175) = 3.238, therefore it means that one unit X increase will have 3.238 unit decrease in wage

37. 6.36 (MacKinnon, White, Davidson) MWD Test for the functional form (Wooldridge, pp.203) Procedures: 1. Run OLS on the linear model, obtain Y ^ Y =  0 +  1 X 1 +  2 X 2 ^ ^ ^ ^ 2. Run OLS on the log-log model and obtain lnY ^ lnY =  0 +  1 ln X 1 +  2 ln X 2 ^ ^ ^ ^ 3. Compute Z 1 = ln(Y) - lnY ^ ^ 4. Run OLS on the linear model by adding z 1 Y =  0 ’ +  1 ’ X 1 +  2 ’ X 2 +  3 ’ Z 1 ^ ^ ^ ^ ^ and check t-statistic of  3 ’ If t *  3 > t c ==> reject H 0 : linear model ^ If t *  3 not reject H 0 : linear model ^

38. 6.37 MWD test for the functional form (Cont.) 5. Compute Z 2 = antilog (lnY) - Y ^ ^ 6. Run OLS on the log-log model by adding Z 2 lnY =  0 ’ +  1 ’ ln X 1 +  2 ’ ln X 2 +  3 ’ Z 2 ^ ^ ^ ^ ^ If t *  3 > t c ==> reject H 0 : log-log model ^ If t *  3 not reject H 0 : log-log model ^ and check t-statistic of  ’ 3 ^

39. 6.38 MWD TEST: TESTING the Functional form of regression CV 1 =  Y _ = 1583.279 24735.33 = 0.064 ^ Y ^ Example:(Table 7.3) Step 1: Run the linear model and obtain C X1 X2

40. 6.39 lnY ^ fitted or estimated Step 2: Run the log-log model and obtain C LNX1 LNX2 CV 2 =  Y _ = 0.07481 10.09653 = 0.0074 ^

41. 6.40 MWD TEST t c 0.05, 11 = 1.796 t c 0.10, 11 = 1.363 t * < t c at 5% => not reject H 0 t * > t c at 10% => reject H 0 Step 4: H 0 : true model is linear C X1 X2 Z1

42. 6.41 MWD Test t c 0.025, 11 = 2.201 t c 0.05, 11 = 1.796 t c 0.10, 11 = 1.363 Since t * < t c => not reject H 0 Comparing the C.V. = C.V. 1 C.V. 2 = 0.064 0.0074 Step 6: H 0 : true model is log-log model C LNX1 LNX2 Z2

43. 6.42  Y ^ coefficient of variation The coefficient of variation: C.V. = It measures the average error of the sample regression function relative to the mean of Y. Linear, log-linear, and log-log equations can be meaningfully compared. smaller C.V The smaller C.V. of the model, more preferredequation the more preferred equation (functional model). Criterion for comparing two different functional models:

44. 6.43 = 4.916 means that model 2 is better Coefficient Variation (C.V.)  / Y of model 1 ^  / Y of model 2 ^ = 2.1225/89.612 0.0217/4.4891 = 0.0236 0.0048 Compare two different functional form models: Model 1 linear model Model 2 log-log model

45. 6.44 Scaling and units of measurement X +  i Y 10  1  : the slope of the regression line. 1  = Units of change of y Units of change of x = dXdX dYdY or X Y   if Y * = 1000Y X * = 1000X then * 10 *  XY ^ ^^ ** i  X Y ^ ^^ 1000 1 0    ==> *

46. 6.45 Changing the scale of X and Y Y i /k = (  0 /k)+(  1 )X i /k +  i /k Y i =  0 +  1 X i +  i R 2 and the t-statistics are no change in regression results for  1 but all other statistics are change.  0 =  0 /k * and Y i =  0 +  1 X i +  i * * * * X i = X i /k * *  i =  i /k Y i = Y i /k where *

47. 6.46 00 ^ 0*0* ^ 5 10 Y X 25 50

48. 6.47 Changing the scale of x Y i =  0 + (k  1 )(X i /k) +  i Y i =  0 +  1 X i +  i * *  1 = k  1 * X i = X i /k * where and The estimated coefficient and standard error change but the other statistics are unchanged.

49. 6.48 00 ^ 5 10 Y X 50

50. 6.49 Changing the scale of Y Y i /k = (  1 /k) + (  1 /k)X i +  i /k Y i =  0 +  1 X i +  i All statistics are changed except for the t-statistics and R 2.  0 =  0 /k * and Y i =  0 +  1 X i +  i * * * *  1 =  1 /k * *  i =  i /k Y i = Y i /k where *

51. 6.50 5 10 Y X 00 ^ 25

52. 6.51 Effects of scaling and units change Butt-statistic F-statisticwill not be affected. R 2 All properties of OLS estimations are also unaffected. The values of  i, SEE, RSS will be affected.

53. 6.52 GNPBIBDGP1739.0001.37  ^ (-0.485)(3.217) Both in billion measure: Billion …B: Billion of 1972 dollar

54. 6.53 GNPMIMDGP1739.052.37001  ^ (-0.485)(3.217) Both in million measure: Million …M: Million of 1972 dollar

55. 6.54 GNPBIMDGP9491.17352.37001  ^ (-0.485)(3.217)

56. 6.55 GNPMIBDGP00017.00015.37  ^ (-0.485)(3.217)

57. 6.56The “ex-post” and “ex ante” forecasting: For example:Suppose you have data of C and Y from 1947–1999. And the estimated consumption expenditures for 1947-1995 is Given values of Y 96 = 10,419; Y 97 = 10,625; … Y 99 = 11,286 ex post The calculated predictions or the “ ex post ” forecasts are : 1996: C 96 = 238.4 + 0.87(10,149) = 9.355 1997: C 97 = 238.4 + 0.87(10,625) = 9.535.50 ….. 1999: C 99 = 238.4 + 0.87(11285) = 10,113.70 ^ ^ ^ C t = 238.4 + 0.87Y t ^ 1947 – 1995: ex ante The calculated predictions or the “ ex ante ” forecasts base on the assumed value of Y 2000 =12000 : 2000: C 2000 = 238.4 + 0.87(12,000) = 10678.4 ^