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

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

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

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  ~    i X Var   ^ ~ and ~ N     ^ and

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.

6.6 i  XY  10  Regression through origin i ’’ X Y  1   ^   n  ^     n ’’ ~ ^          YY X X YYXX R or     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

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.

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

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

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

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

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

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.

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

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

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

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

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

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 

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

6.21 Some features of reciprocal model X Y 1    Y 0  X 0 0   and 0 1  Y X 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 

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 XXXXY  transforming X 2 * = X 1 2 X 3 * = X 1 X 2 rewrite * 33 * XXXY 

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

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

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

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 

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

MPNG  ^ (1.368)(39.20) Linear model

6.29 GNP = lnM 2 (-23.44)(27.48) ^ Lin-log model

6.30 lnGNP = M 2 (100.38)(15.65) ^ Log-lin model

ln lnMNPG  ^ (3.194)(42.29) Log-log model

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

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

6.34 lnwage = ln(unemploy) (10.375)(-2.618) ^

6.35 Lnwage = ln ) 1 ( X (10.37)(2.618) ^ Antilog(1.9038) = , 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 unit decrease in wage

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 ^

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 ^

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

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

6.40 MWD TEST t c 0.05, 11 = t c 0.10, 11 = 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

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

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:

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

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 ^ ^^    ==> *

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 *

6.46 00 ^ 0*0* ^ 5 10 Y X 25 50

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.

6.48 00 ^ 5 10 Y X 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 *

Y X 00 ^ 25

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.

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

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

6.54 GNPBIMDGP  ^ (-0.485)(3.217)

6.55 GNPMIBDGP  ^ (-0.485)(3.217)

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 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 = (10,149) = : C 97 = (10,625) = … : C 99 = (11285) = 10, ^ ^ ^ C t = Y 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 = (12,000) = ^