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 n2 (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 n2 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