Lecture Applied econometrics course - Chapter 1: Simple regression model

Lecture Applied econometrics course - Chapter 1: Simple regression model has content: What is simple regression model, how to estimate simple regression model, R – Square, assumption, variance and standard error of parameters,... and other contents.

APPLIED ECONOMETRICS COURSE CHAPTER I SIMPLE REGRESSION MODEL NGUYEN BA TRUNG - 2016 TODAY’S TALK  What is simple regression model  How to estimate simple regression model  R – Square  Assumption  Variance and Standard Error of Parameters  Measurement Unit  Function Form  Illustration by Computer NGUYEN BA TRUNG - 2016 I WHAT IS SIMPLE REGRESSION MODEL?  Linear simple regression: Y  i   1 X i  ui (1.1) Y: Dependent variable, Explained variable X: Independent variable, Explanatory variable U: Error term, disturbance β: Parameters need to be estimated  SIMPLE regression model = AN independent variable (X)  Why does error term (U) exist? NGUYEN BA TRUNG - 2016 I WHAT IS SIMPLE REGRESSION MODEL?  An important assumption: E(u/ x)  (1.2)  Take the expectation of (1.1) and use equation (1.2), we have: E (Y/ X)    1 X (1.3)  Equation (1.3) is so called population regression function (PRF)  Attention: distinction between PRF and SRF NGUYEN BA TRUNG - 2016 II ESTIMATION: ORDINARY LEAST SQUARE (OLS)  OLS method:   2 ˆ ˆ ˆ ˆ f  , 1   uˆ   (Y    1 X )  MIN (1.4)  Take derivative (1.4) respect to the parameters, we have:  f      f    ( ˆ0 , ˆ1 ) 0 ˆ  ( ˆ0 , ˆ1 ) 0 ˆ 1 ➨ n n  ˆ  ˆ n   X i   Yi   i 1 i 1  n n n  ˆ ˆ X   0 i  X i   X i Yi  i 1 i 1 i 1 NGUYEN BA TRUNG - 2016 II ESTIMATION: ORDINARY LEAST SQUARE (OLS)  we have: n ˆ1   X Y  nXY i i i 1 n X i   n X i 1 (1.5) ˆ0  Y  ˆ1 X (1.6)  Equation (1.5) can be expressed as: ˆ1 xy   x i i i (1.7) where yi  Yi  Y xi  X i  X NGUYEN BA TRUNG - 2016 II ESTIMATION: ORDINARY LEAST SQUARE (OLS)  The fitted Y is computed as: Yˆi  ˆ0  ˆ1 X i (1.8)  The disturbance is followed by : uˆi  Y  ˆ0  ˆ1 X i  Note that: S XY ˆ 1  SX (1.9) (1.10) NGUYEN BA TRUNG - 2016 EXAMPLE 1: EXPENDITURE.wf Suppose that we have the data expenditure (Y: $/week) and income (X: $/week) of 10 families as the table below: Yi 70 65 90 95 110 115 120 140 155 150 Xi 80 100 120 140 160 180 200 220 240 260 Let’s estimate a linear regression model describing the relationship between expenditure and income? NGUYEN BA TRUNG - 2016 SOLVE THE EXAMPLE BY HAND NGUYEN BA TRUNG - 2016 SOLVE THE EXAMPLE BY HAND Solution: ˆ xy   x i i i 16800   0.5091 33000 ˆ0  Y  ˆ1 X  111  (170 * 0.5091)  24.45 The linear regression model is expressed by: Yˆi  24.45  0.5091 X i Explain the meaning of your estimated parameters? NGUYEN BA TRUNG - 2016 ESTIMATION BY COMPUTER NGUYEN BA TRUNG - 2016 III R - SQUARE  n TSS   Yi  Y i 1 n    Y ESS   Yˆi  Y i 1 n i    n Y  ( ˆ1 ) (1.11) n 2 x  i (1.12) i 1 n RSS   uˆ   i i 1 i 1  Yi  Yˆi  (1.13) TSS = ESS + RSS (1.14) ESS RSS R   1 TSS TSS (1.15)  R2  (1.16) NGUYEN BA TRUNG - 2016 R – SQUARE BY HAND  n TSS   Yi  Y i 1 n    Y ESS   Yˆi  Y i 1 i  2   n Y = 132100 – 10*(111)2 = 8890 n  ( ˆ1 )  xi2 = (0.5091)2*33000 = 8553.0327 i 1 ESS RSS R   1 TSS TSS 2 = 8553.0327/ 8890 = 0.9620     xi yi  R  n i 1 n   rX2 ,Y = (16800)2/(33000*8890)= 0.9620 2 x y  i i n i 1 i 1 NGUYEN BA TRUNG - 2016 R – SQUARE BY COMPUTER NGUYEN BA TRUNG - 2016 IV ASSUMPTION  Assumption 1: Yi    1 X i  ui  Assumption : X,Y are random variables  Assumption 3:  X X 0 i  Assumption 4: E(u/ x)  Theorem 1: Unbiased estimator of the OLS method Under the assumptions from to 4, estimators of the OLS are unbiased, this is: E ( ˆ )   0 E ( ˆ1 )  1 NGUYEN BA TRUNG - 2016 EXAMPLE OF VIOLATION  The free lunch policy in US: math10    1luchprog  u  what is the signal of 1 that you expect?  Using MEAP93.wf, we estimate the above model, and we have: math10  32.14  0.319luchprog n  408, R  0.171  Explain the result and give the reason for that? NGUYEN BA TRUNG - 2016 V VARIANCE AND STANDARD ERROR OF PARAMETERS  Assumption 5: Homoskedasticity Var(ui )   Theorem 2: The Unbiasedness of estimator’s variance  Under the assumptions 1- 5, the variance of the estimators is unbiased, this is: E (ˆ )   NGUYEN BA TRUNG - 2016 V VARIANCE AND STANDARD ERROR OF PARAMETER n var( ˆ0 )  X  i i 1 n n  xi2 2 (2.17) se( ˆ0 )  var( ˆ0 ) (2.18) se( ˆ1 )  var( ˆ1 ) (2.20) i 1 var( ˆ1 )  2 (2.19) n x i i 1  where 2 is substituted by ˆ n ˆ  2 ˆ u  i i 1 n2 (2.21) ˆ  ˆ NGUYEN BA TRUNG - 2016 (2.22) V COMPUTE VARIANCE AND STANDARD ERROR BY HAND n ˆ  ˆ u  i i 1 n2   var ˆ0  se( ˆ0 )  var( ˆ1 )  = 337.27/(10-2)= 42.1591 X  i n  xi2  = (322000*42.1591)/ (10*33000)= 41.1371 var( ˆ0 ) 2 x  i = 6.4138 = 42.1591/33000= 0.00127 se( ˆ1 )  var( ˆ1 ) = 0.0357 NGUYEN BA TRUNG - 2016 STANDARD ERROR BY COMPUTER NGUYEN BA TRUNG - 2016 VI MEASUREMENT UNIT  Consider the two models below: Unit: 1000 $ NGUYEN BA TRUNG - 2016 Unit: $ VII FUNCTION FORM LOG - LIN  Consider the model:  Meaning:  Explain your estimated result? NGUYEN BA TRUNG - 2016 (1.23) VII FUNCTION FORM LOG - LOG  Consider the model: Log(sale) = 0 + 1log(price) (2.24)  Meaning: % sale  1 % price  Explain your estimated result? NGUYEN BA TRUNG - 2016 VII FUNCTION FORM LIN - LOG  Consider the model: GDP = 0 + 1log (M2) (2.25) GDP: Gross domestic Product (million $) M2: Money Supply (million $) GDPi = -2547585 + 283143.4 lnM2i + ei  Meaning: GDP  ( 1 / 100)% M  Explain your estimated result? NGUYEN BA TRUNG - 2016 APPLIED ECONOMETRICS COURSE END OF CHAPTER I NGUYEN BA TRUNG - 2016 ... Parameters need to be estimated  SIMPLE regression model = AN independent variable (X)  Why does error term (U) exist? NGUYEN BA TRUNG - 2016 I WHAT IS SIMPLE REGRESSION MODEL?  An important assumption:... Measurement Unit  Function Form  Illustration by Computer NGUYEN BA TRUNG - 2016 I WHAT IS SIMPLE REGRESSION MODEL?  Linear simple regression: Y  i   1 X i  ui (1.1) Y: Dependent variable, Explained...TODAY’S TALK  What is simple regression model  How to estimate simple regression model  R – Square  Assumption  Variance and Standard Error of Parameters