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Understanding Regression Modeling: Functional Forms and Intercept Suppression

This lecture, based on Studenmund's work, explores critical concepts in regression analysis, including the implications of suppressing intercepts, alternative functional forms, and the scaling of measurement units. The absence of an intercept can bias coefficient estimates (βs), affecting t-values and overall model validity. We’ll discuss scenarios for using regression through the origin, examine linear and nonlinear models, and highlight practical guidelines, such as when to include or exclude an intercept in models based on statistical significance.

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Understanding Regression Modeling: Functional Forms and Intercept Suppression

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

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

  3. The intercept term is absent or zero. = b + Y X  i.e., i 1 i i Y ^ ^ i = SRF : Y  X i 1 i ^ b 1 1 X 0 i Regression through the origin

  4. The estimated model: ~ ~ = b Y X 1 i ~ ~ or = b +  Y X 1 i i Applied OLS method: ^ ( ) 2 s ~ å X Y ~ = b Var and i i å b = 1 X 2 1 i 2 å X i ~ å 2  and ^ 2 s = N -1 Regression through the origin

  5. ~ å 1. need not be zero  i can be occasions turn out to be negative. may not be appropriate for the summary of statistics. R2 2. df 3. does not include the constant term, i.e., (n-k) Some feature of no-intercept model • 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.

  6. Regression through origin = b + Y X ’ = b + b + Y X  1 i 0 1 i å ^ xy å XY b = ~ b = 1 x2 å å X2 1 s ^ 2 ^ ( ) s 2 ^ ) b = ( ~ Var b = Var å å 1 1 2 x X2 ~ å 2 ’ s ^ ^ å = 2 2 N-k-1  N-k ^ - s = 2 n 1 [ ] ( ) ( ) - n 2 2 - - å X X Y Y = 2 R å 2 ( ) ( ) ( XY ) 2 2 - - å å X raw X Y Y = 2 R å å X2 Y2 ( ) 2 å xy or = 2 R å å 2 2 x y

  7. ( ) ( ) - = b - ER r ER r 1 i f m f risk free of return expected rate of return on security i expected rate of return on market portfolio b as a measure of systematic risk. 1 b >1 ==> implies a volatile or aggressive security. 1 b <1 ==> implies a defensive security. 1 Example 1: Capital Asset Pricing Model (CAPM) security i’s expect risk premium=expected market risk premium

  8. Example 1:(cont.) Security market line - ER r i f b 1 1 - ER f m

  9. - F e - = = * ( i i ) f N N e International interest rate differentials equal exchange rate forward premium. - F e ( ) - = b * i i ( ) i.e., 1 e Covered interest parity line b 1 1 Example 2:Covered Interest Parity

  10. in regression: a = E ( ) 0 0 - F e - = a + b + * ( i i ) ( ) u 0 1 i e a If covered interest parity holds, is expected to be zero. 0 Example 2:(Cont.) Use the t-test to test the intercept to be zero

  11. y: Return on A Future Fund, % X: Return on Fisher Index, % ^ Formal report: = R2=0.714 Y 1 . 0899 X N=10 SEE=19.54 (5.689)

  12. H0: 0 = 0 1.279 - 0 7.668 ^ = + Y 1 . 2797 1 . 0691 X R2=0.715 N=10 (0.166) (4.486) SEE=20.69 The t-value shows that b0 is statistically insignificant different from zero

  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.

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

  15. Linear vs. Nonlinear Examples of Linear Statistical Models Yi = 0 + 1Xi + i Yi = 0 + 1 ln(Xi)+ i 2 ln(Yi) = 0 + 1Xi + i Yi = 0 + 1Xi + i Examples of Non-linear Statistical Models 2 2 Yi = 0 + 1Xi + i Yi = 0 + 1Xi + i Yi = 0 + 1Xi + exp(2Xi)+ i

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

  17. 2. Log-log model: This is a non- linear model = b - b1 i Y X e 0 Transform into linear log-form: X = b0 - b + ln ln Y ln  1 i X = b - b + * ln Y ln  ==> 0 1 i = b + b + * * ==> * * Y X  b = - b * where 0 1 i 1 1 dY * elasticity coefficient dY d ln Y Y * = = = b1 dX * dX d ln X X Functional Forms of Regression models

  18. Functional Forms of Regression models Quantity Demand lnY Y = b - b ln Y ln ln X = b - b Y X 0 1 1 0 lnY Quantity Demand Y lnX X = b + b ln Y ln ln X = b b Y X price 0 1 1 0 lnX X price

  19. 3.Semi log model: Log-lin model or lin-log model: = a + a + ln Y X  i 0 1 i i or = b + b + Y ln X  i 0 1 i i dY relative change in Y d ln Y dY 1 Y a = = = = absolute change in X dX dX dX Y 1 and b = absolute change in Y 1 dY dY X = = relative change in X dX d ln X 1 Functional Forms of Regression models

  20. Functional Forms of Regression models(Cont.) 4. Polynomial: Quadratic term to capture the nonlinear pattern Yi= 0 + 1 Xi +2X2i + i 1<0, 2>0 1 Yi Yi = b + b + Y ( )  i 0 1 i X 1>0, 2<0 i 1 Xi Xi = b + b + * = Where * Y ( X )  X ==> i i 0 1 i i 5. Reciprocal (or inverse) transformations X i

  21. Some features of reciprocal model Y Y 1 = b0 + b1 Y X and and and b b b < < > b0 b b > > < 0 0 0 0 0 0 0 0 1 1 1 -b b -b b 0 0 0 0 X X 0 0 - b b / 1 0 Y 1 = b0 + b1 Y Y X + and X b > b < 0 0 0 0 1 0 X - - b b / 1 0

  22. Example 1: = b + b + b + b 2 Suppose Y X X X X 0 1 1 2 1 3 1 2 X2* = X12 transforming X3* = X1X2 = b + b + b + b * * rewrite Y X X X 0 1 1 2 2 3 3 Two conditions for nonlinear, non-additive equation transformation. 1. Exist a transformation of the variable. 2. Sample must provide sufficient information.

  23. Example 2: b = b + Y 1 + b 0 X 2 1 transforming = * X + b 1 X 2 = b + b * rewrite Y X 0 1 1 However, X1* cannot be computed, because b is unknown. 2

  24. Application of functional form regression 1. Cobb-Douglas Production function: = b b2 b1  Y L K e 0 Transforming: = b + b + b + ln Y ln ln L ln K  0 1 2 = b + b + b + ln Y ln L ln K  ==> 0 1 2 d ln Y : elasticity of output w.r.t. labor input = b 1 d ln L d ln Y : elasticity of output w.r.t. capital input. = b 2 d ln K > Information about the scale of returns. b + b = 1 < 1 2

  25. 2. Polynomial regression model: Marginal cost function or total cost function costs costs (MC) = b + b + b + i.e. Y X X 2  0 1 2 MC y y or (TC) = b + b + b + b + costs Y X X 2 X 3  0 1 2 3 TC y

  26. Slope Elasticity dY dY Model Equation Y = ( ) = ( ) dX dX X dY X = b + b Y X = b b linear ( ) 0 1 1 1 dX Y dY Log-log = b + b ln Y ln X d ln Y Y b = = b 0 1 dX 1 1 d ln X X dY Y = b ==> ( ) 1 dX X Summary

  27. Slope Elasticity dY Log-lin = b + b ln Y X d ln Y Y b X = = b 0 1 1 1 dX dX dY = b ==> Y 1 dX 1 dY dY = b + b b = = b Y ln X Lin-log dX 1 1 Y d ln X 0 1 X dY 1 = b ==> 1 dX X 1 dY dY -1 = = b = b + b Y b ( ) Reciprocal - 1 1 1 0 1 1 XY X d ( ) dX 2 X X dY -1 = b ==> 1 dX X2 Summary(Count.)

  28. Linear model ^ = + G N P 100 . 304 1 . 5325 M 2 (1.368) (39.20)

  29. ^ GNP = -1.6329.21 + 2584.78 lnM2 (-23.44) (27.48) Lin-log model

  30. ^ lnGNP = 6.8612 + 0.00057 M2 (100.38) (15.65) Log-lin model

  31. ^ = + ln G NP 0 . 5529 0 . 9882 ln M 2 (3.194) (42.29) Log-log model

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

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

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

  35. ^ 1 Lnwage = 1.9038 + 1.175 ln ( ) 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

  36. Procedures: 1. Run OLS on the linear model, obtain Y ^ Y = 0 + 1 X1 + 2 X2 ^ ^ ^ ^ 2. Run OLS on the log-log model and obtain lnY ^ ^ lnY = 0 + 1 lnX1 + 2 lnX2 ^ ^ ^ ^ 3. Compute Z1 = ln(Y) - lnY ^ 4. Run OLS on the linear model by adding z1 ^ Y = 0’ + 1’X1 + 2’X2 + 3’ Z1 ^ ^ ^ ^ and check t-statistic of 3’ If t*3 > tc ==> reject H0 : linear model ^ If t*3 < tc ==> not reject H0 : linear model ^ (MacKinnon, White, Davidson) MWDTest for the functional form (Wooldridge, pp.203)

  37. 5. Compute Z2 = antilog (lnY) - Y ^ ^ 6. Run OLS on the log-log model by adding Z2 ^ lnY = 0’ + 1’ ln X1 + 2’ ln X2 + 3’ Z2 ^ ^ ^ ^ ^ and check t-statistic of b’3 If t*3 > tc ==> reject H0 : log-log model ^ If t*3 < tc ==> not reject H0 : log-log model ^ MWD test for the functional form (Cont.)

  38. Example:(Table 7.3) Step 1: Run the linear model and obtain Y ^ C X1 X2  1583.279 ^ CV1 = = _ Y 24735.33 = 0.064 MWD TEST: TESTING the Functional form of regression

  39. Step 2: Run the log-log model and obtain lnY ^ C LNX1 LNX2 fitted or estimated  _ Y ^ 0.07481 CV2 = = = 0.0074 10.09653

  40. Step 4: H0 : true model is linear C X1 X2 Z1 MWD TEST tc0.05, 11 = 1.796 tc0.10, 11 = 1.363 t* < tc at 5% => not reject H0 t* > tc at 10% => reject H0

  41. Step 6: H0 : true model is log-log model C LNX1 LNX2 Z2 C.V.1 Comparing the C.V. = 0.064 = 0.0074 C.V.2 MWD Test tc0.025, 11 = 2.201 tc0.05, 11 = 1.796 tc0.10, 11 = 1.363 Since t* < tc => not reject H0

  42. Criterion for comparing two different functional models: 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. ^  Y The smaller C.V. of the model, the more preferredequation (functional model).

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

  44. = b + b Y X + i 0 1 b : the slope of the regression line. 1 D Y dY Units of change of y b = = or 1 D X dX Units of change of x if Y* = 1000Y X* = 1000X ^ ^ ^ then b = b + X + 1000 Y 1000 1000  1000 1 0 i ^ ^ * ^ = * + b + * * Y X  ==> 0 1 Scaling and units of measurement

  45. Yi = 0 + 1Xi + i Yi/k = (0/k)+(1)Xi/k+ i/k * * * Yi = 0 + 1Xi+ i * * i = i/k * Yi = Yi/k where * * 0 = 0/k Xi = Xi/k and Changing the scale of X and Y R2 and the t-statistics are no change in regression results for 1 but all other statistics are change.

  46. Y 25 ^ 0* 5 ^ 0 X 10 50

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

  48. Y 5 ^ 0 X 10 50 50

  49. Yi = 0 + 1Xi + i Yi/k = (1/k)+ (1/k)Xi + i/k * * * * Yi = 0 + 1Xi + i * i = i/k * Yi = Yi/k where * * 0 = 0/k 1 = 1/k and Changing the scale of Y All statistics are changed except for the t-statistics and R2.

  50. Y 25 5 ^ 0 X 10

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