Eco Basic 1 8

Basic Econometrics Course Leader Prof. Dr.Sc VuThieu May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Introduction : What is Econometrics? May 2004 Prof.VuThieu ,[object Object]

Introduction What is Econometrics ? ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Introduction What is Econometrics ? ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

May 2004 Prof.VuThieu ,[object Object],Econometrics Economic Theory Mathematical Economics Economic Statistics Mathematic Statistics

Introduction Why a separate discipline ? ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (2) Specification of the mathematical model of the theory Y = ß 1 + ß 2 X ; 0 < ß 2 < 1 Y= consumption expenditure X= income ß 1 and ß 2 are parameters; ß 1 is intercept, and ß 2 is slope coefficients May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (3) Specification of the econometric model of the theory Y = ß 1 + ß 2 X + u ; 0 < ß 2 < 1; Y = consumption expenditure; X = income; ß 1 and ß 2 are parameters; ß 1 is intercept and ß 2 is slope coefficients; u is disturbance term or error term. It is a random or stochastic variable May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (4) Obtaining Data (See Table 1.1, page 6) Y= Personal consumption expenditure X= Gross Domestic Product all in Billion US Dollars May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (4) Obtaining Data May 2004 Prof.VuThieu ,[object Object],Year X Y 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 2447.1 2476.9 2503.7 2619.4 2746.1 2865.8 2969.1 3052.2 3162.4 3223.3 3260.4 3240.8 3776.3 3843.1 3760.3 3906.6 4148.5 4279.8 4404.5 4539.9 4718.6 4838.0 4877.5 4821.0

Introduction Methodology of Econometrics (5) Estimating the Econometric Model Y^ = - 231.8 + 0.7194 X (1.3.3) MPC was about 0.72 and it means that for the sample period when real income increases 1 USD, led ( on average ) real consumption expenditure increases of about 72 cents Note : A hat symbol (^) above one variable will signify an estimator of the relevant population value May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (6) Hypothesis Testing Are the estimates accord with the expectations of the theory that is being tested? Is MPC < 1 statistically? If so, it may support Keynes’ theory. Confirmation or refutation of economic theories based on sample evidence is object of Statistical Inference (hypothesis testing) May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics (8) Using model for control or policy purposes Y=4000= -231.8+0.7194 X  X  5882 MPC = 0.72, an income of $5882 Bill will produce an expenditure of $4000 Bill. By fiscal and monetary policy, Government can manipulate the control variable X to get the desired level of target variable Y May 2004 Prof.VuThieu ,[object Object]

Introduction Methodology of Econometrics ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

May 2004 Prof.VuThieu ,[object Object],Economic Theory Mathematic Model Econometric Model Data Collection Estimation Hypothesis Testing Forecasting Application in control or policy studies

Basic Econometrics Chapter 1 : THE NATURE OF REGRESSION ANALYSIS May 2004 Prof.VuThieu ,[object Object]

1-1. Historical origin of the term “Regression” ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

1-2. Modern Interpretation of Regression Analysis ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Dependent Variable Y; Explanatory Variable Xs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

1-3. Statistical vs. Deterministic Relationships ,[object Object],May 2004 Prof.VuThieu ,[object Object]

1-4. Regression vs. Causation: ,[object Object],May 2004 Prof.VuThieu ,[object Object]

1-5. Regression vs. Correlation ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

1-6. Terminology and Notation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

1-7. The Nature and Sources of Data for Econometric Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

1-8. Summary and Conclusions ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 2 : TWO-VARIABLE REGRESSION ANALYSIS: Some basic Ideas May 2004 Prof.VuThieu ,[object Object]

2-1. A Hypothetical Example ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

May 2004 Prof.VuThieu ,[object Object],Table 2-2: Weekly family income X ($), and consumption Y ($) X Y 80 100 120 140 160 180 200 220 240 260 Weekly family consumption expenditure Y ($) 55 65 79 80 102 110 120 135 137 150 60 70 84 93 107 115 136 137 145 152 65 74 90 95 110 120 140 140 155 175 70 80 94 103 116 130 144 152 165 178 75 85 98 108 118 135 145 157 175 180 -- 88 -- 113 125 140 -- 160 189 185 -- -- -- 115 -- -- -- 162 -- 191 Total 325 462 445 707 678 750 685 1043 966 1211 Mean 65 77 89 101 113 125 137 149 161 173

2-2. The concepts of population regression function (PRF) ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-4. Stochastic Specification of PRF ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-5. The Significance of the Stochastic Disturbance Term ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-5. The Significance of the Stochastic Disturbance Term ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-6. The Sample Regression Function (SRF) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

May 2004 Prof.VuThieu ,[object Object],SRF1 SRF2 Weekly Consumption Expenditure (Y) Weekly Income (X)

2-6. The Sample Regression Function (SRF) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-6. The Sample Regression Function (SRF) ,[object Object],May 2004 Prof.VuThieu ,[object Object]

2-6. The Sample Regression Function (SRF) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

2-7. Summary and Conclusions ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 3 : TWO-VARIABLE REGRESSION MODEL: The problem of Estimation May 2004 Prof.VuThieu ,[object Object]

3-1. The method of ordinary least square (OLS) ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-2. The assumptions underlying the method of least squares ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-2. The assumptions underlying the method of least squares ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-3. Precision or standard errors of least-squares estimates ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-3. Precision or standard errors of least-squares estimates ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-4. Properties of least-squares estimators: The Gauss-Markov Theorem ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-4. Properties of least-squares estimators: The Gauss-Markov Theorem ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-5. The coefficient of determination r 2 : A measure of “Goodness of fit” ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-5. The coefficient of determination r 2 : A measure of “Goodness of fit” ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

3-5. The coefficient of determination r 2 : A measure of “Goodness of fit” 3-6. A numerical Example (pages 80-83) 3-7. Illustrative Examples (pages 83-85) 3-8. Coffee demand Function 3-9. Monte Carlo Experiments (page 85) 3-10. Summary and conclusions (pages 86-87) May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 4 : THE NORMALITY ASSUMPTION: Classical Normal Linear Regression Model (CNLRM) May 2004 Prof.VuThieu ,[object Object]

4-2.The normality assumption ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

4-2.The normality assumption ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

4-3. Properties of OLS estimators under the normality assumption ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

4-3. Properties of OLS estimators under the normality assumption ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

4-3. Properties of OLS estimators under the normality assumption ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Some last points of chapter 4 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Some last points of chapter 4 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 5 : TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object],Type of Hypothesis H 0 H 1 Reject H 0 if Two-tail  2 =  2 *  2 #  2 * |t| > t  /2,df Right-tail  2   2 *  2 >  2 * t > t  ,df Left-tail  2   2 *  2 <  2 * t < - t  ,df

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object],H 0 H 1 Reject H 0 if  2 =  2 0  2 >  2 0 Df.(  ^ 2 )/  2 0 >  2  ,df  2 =  2 0  2 <  2 0 Df.(  ^ 2 )/  2 0 <  2 ( 1-  ),df  2 =  2 0  2 #  2 0 Df.(  ^ 2 )/  2 0 >  2  /2,df or <  2 ( 1-  /2), df

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object],Source of Variation Sum of square ( SS) Degree of Freedom - (Df) Mean sum of square ( MSS) ESS (due to regression)  y^ i 2 =  2 ^ 2  x i 2 1  2 ^ 2  x i 2 RSS (due to residuals)  u^ i 2 n-2  u^ i 2 /(n-2)=  ^ 2 TSS  y i 2 n-1

Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin Comparison two types of regressions:  ^ 2 =  X i Y i /  X 2 i (6.1.6) O   ^ 2 =  x i y i /  x 2 i (3.1.6) I var(  ^ 2 ) =  2 /  X 2 i (6.1.7) O var(  ^ 2 ) =  2 /  x 2 i (3.3.1) I  ^ 2 =  u^ i ) 2 /(n-1) (6.1.8) O   ^ 2 =  u^ i ) 2 /(n-2) (3.3.5) I  May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-3. Functional form of regression model ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-4. How to measure elasticity ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-4. How to measure elasticity ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-5. Semi-log model : Log-lin and Lin-log Models ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-5. Semi-log model : Log-lin and Lin-log Models ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-6. Reciprocal Models : Log-lin and Lin-log Models ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

6-7. Summary of Functional Forms Table 6.5 (page 178) May 2004 Prof.VuThieu ,[object Object],Model Equation Slope = dY/dX Elasticity = (dY/dX).(X/Y) Linear Y =      X     (X/Y) */ Log-linear (log-log) lnY =      lnX    (Y  X)   Log-lin lnY =      X    Y     X */ Lin-log Y =      lnX  2 (1/X)    Y) */ Reciprocal Y =      X) -  2 (1/X 2 ) -    XY) */

6-7. Summary of Functional Forms ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 7 MULTIPLE REGRESSION ANALYSIS: The Problem of Estimation May 2004 Prof.VuThieu ,[object Object]

7-1. The three-Variable Model: Notation and Assumptions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-2. Interpretation of Multiple Regression ,[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-3. The meaning of partial regression coefficients ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-4. OLS and ML estimation of the partial regression coefficients ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-5. The multiple coefficient of determination R 2 and the multiple coefficient of correlation R ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-6. Example 7.1: The expectations-augmented Philips Curve for the US (1970-1982) ,[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-7. Simple regression in the context of multiple regression: Introduction to specification bias ,[object Object],May 2004 Prof.VuThieu ,[object Object]

7-8. R 2 and the Adjusted-R 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-8. R 2 and the Adjusted-R 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-9. Partial Correlation Coefficients ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-10. Example 7.3: The Cobb-Douglas Production function More on functional form ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

7-11 Polynomial Regression Models ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Basic Econometrics Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-4. Hypothesis testing about individual partial regression coefficients With the assumption that u i ~ N(0,  2 ) we can use t-test to test a hypothesis about any individual partial regression coefficient. H 0 :  2 = 0 H 1 :  2  0 If the computed t value > critical t value at the chosen level of significance, we may reject the null hypothesis; otherwise, we may not reject it May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference ,[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference ,[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-5. Testing the overall significance of a multiple regression [R 2 new - R 2 old ] / Df 1 F com = ---------------------- (8.5.18) [1 - R 2 new ] / Df 2 Where Df 1 = number of new regressors Df 2 = n – number of parameters in the new model R 2 new is standing for coefficient of determination of the new regression (by adding  X); R 2 old is standing for coefficient of determination of the old regression (before adding  X). May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-6. Testing the equality of two regression coefficients Y i =  1 +  2 X 2i +  3 X 3i +  4 X 4i + u i (8.6.1) Test the hypotheses: H 0 :  3 =  4 or  3 -  4 = 0 (8.6.2) H 1 :  3   4 or  3 -  4  0 Under the classical assumption it can be shown: t = [(  ^ 3 -  ^ 4 ) – (  3 -  4 )] / se(  ^ 3 -  ^ 4 ) follows the t distribution with (n-4) df because (8.6.1) is a four-variable model or, more generally, with (n-k) df. where k is the total number of parameters estimated, including intercept term. se(  ^ 3 -  ^ 4 ) =  [var( (  ^ 3 ) + var(  ^ 4 ) – 2cov (  ^ 3 ,  ^ 4 )] (8.6.4) (see appendix) May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-7. Restricted least square: Testing linear equality restrictions Y i =  1 X   2 2i X   3 3i e u i (7.10.1) and (8.7.1) Y = output X 2 = labor input X 3 = capital input In the log-form: lnY i =  0 +  2 lnX 2i +  3 lnX 3i + u i (8.7.2) with the constant return to scale:  2 +  3 = 1 (8.7.3) May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-7. Restricted least square: Testing linear equality restrictions  u^ 2 UR =RSS UR of unrestricted regression (8.7.2) and  u^ 2 R = RSS R of restricted regression (8.7.7), m = number of linear restrictions, k = number of parameters in the unrestricted regression, n = number of observations. R 2 UR and R 2 R are R 2 values obtained from unrestricted and restricted regressions respectively. Then F=[(RSS R – RSS UR )/m]/[RSS UR /(n-k)] = = [(R 2 UR – R 2 R ) / m] / [1 – R 2 UR / (n-k)] (8.7.10) follows F distribution with m, (n-k) df . Decision rule: If F > F m, n-k , reject H 0 :  2 +  3 = 1 May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-8. Comparing two regressions: Testing for structural stability of regression models + The structural change may mean that the two intercept are different, or the two slopes are different, or both are different, or any other suitable combination of the parameters. If there is no structural change we can combine all the n 1 , n 2 and just estimate one savings function as: Y t =  1 +  2 X t + U t (t = 1,2,...,n 1 , 1,....n 2 ). (8.8.3) How do we find out whether there is a structural change in the savings-income relationship between the two period? A popular test is Chow-Test, it is simply the F Test discussed earlier H O :  i =  i  i Vs H 1 :  i that  i   i May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-8. Comparing two regressions: Testing for structural stability of regression models + The assumptions underlying the Chow test u 1t and u 2t ~ N(0,s 2 ), two error terms are normally distributed with the same variance u 1t and u 2t are independently distributed Step 1 : Estimate (8.8.3), get RSS, say, S 1 with df = (n 1 +n 2 – k); k is number of parameters estimated ) Step 2 : Estimate (8.8.1) and (8.8.2) individually and get their RSS, say, S 2 and S 3 , with df = (n 1 – k) and (n 2 -k) respectively. Call S 4 = S 2 +S 3 ; with df = (n 1 +n 2 – 2k) Step 3 : S 5 = S 1 – S 4 ; May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-8. Comparing two regressions: Testing for structural stability of regression models Step 4 : Given the assumptions of the Chow Test, it can be show that F = [S 5 / k] / [S 4 / (n 1 +n 2 – 2k)] (8.8.4) follows the F distribution with Df = (k, n 1 +n 2 – 2k) Decision Rule : If F computed by (8.8.4) > F- critical at the chosen level of significance a => reject the hypothesis that the regression (8.8.1) and (8.8.2) are the same, or reject the hypothesis of structural stability; One can use p-value of the F obtained from (8.8.4) to reject H 0 if p-value low reasonably. + Apply for the data in Table 8.8 May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-9. Testing the functional form of regression: Choosing between linear and log-linear regression models: MWD Test (MacKinnon, White and Davidson) H 0 : Linear Model Y is a linear function of regressors, the X s ; H 1 : Log-linear Model Y is a linear function of logs of regressors, the lnX s ; May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference 8-9. Testing the functional form of regression: Step 1 : Estimate the linear model and obtain the estimated Y values. Call them Yf (i.e.,Y^). Take lnYf. Step 2 : Estimate the log-linear model and obtain the estimated lnY values, call them lnf (i.e., ln^Y ) Step 3 : Obtain Z 1 = (lnYf – lnf) Step 4 : Regress Y on X s and Z 1 . Reject H 0 if the coefficient of Z 1 is statistically significant, by the usual t - test Step 5 : Obtain Z 2 = antilog of (lnf – Yf) Step 6 : Regress lnY on lnX s and Z 2 . Reject H 1 if the coefficient of Z 2 is statistically significant, by the usual t-test May 2004 Prof.VuThieu ,[object Object]

Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference Example 8.5 : The demand for Roses (page 266-267). Data in exercise 7.20 (page 225) 8-10. Prediction with multiple regression Follow the section 5-10 and the illustration in pages 267-268 by using data set in the Table 8.1 (page 241) 8-11. The troika of hypothesis tests: The likelihood ratio (LR), Wald (W) and Lagarange Multiplier (LM) Tests 8-12. Summary and Conclusions May 2004 Prof.VuThieu ,[object Object]

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