eco basic 1 8

1. Basic Econometrics Course Leader Prof. Dr.Sc VuThieu May 2004 Prof.VuThieu

2. Basic Econometrics Introduction : What is Econometrics? May 2004 Prof.VuThieu

3. Introduction What is Econometrics ?

Definition 1 : Economic Measurement

Definition 2 : Application of the mathematical statistics to economic data in order to lend empirical support to the economic mathematical models and obtain numerical results ( Gerhard Tintner, 1968 )

May 2004 Prof.VuThieu


Definition 3 : The quantitative analysis of actual economic phenomena based on concurrent development of theory and observation, related by appropriate methods of inference

( P.A.Samuelson, T.C.Koopmans and J.R.N.Stone, 1954 )



Definition 4 : The social science

which applies economics, mathematics and statistical inference to the analysis of economic phenomena ( By Arthur S. Goldberger, 1964 )

Definition 5 : The empirical determination of economic laws ( By H. Theil, 1971 )



Definition 6 : A conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier ( By T.Haavelmo, 1944 )

And the others


7. May 2004 Prof.VuThieu Econometrics Economic Theory Mathematical Economics Economic Statistics Mathematic Statistics

8. Introduction Why a separate discipline ?

Economic theory makes statements that are mostly qualitative in nature, while econometrics gives empirical content to most economic theory

Mathematical economics is to express economic theory in mathematical form without empirical verification of the theory, while econometrics is mainly interested in the later


9. Introduction Why a separate discipline ?

Economic Statistics is mainly concerned with collecting, processing and presenting economic data. It does not being concerned with using the collected data to test economic theories

Mathematical statistics provides many of tools for economic studies, but econometrics supplies the later with many special methods of quantitative analysis based on economic data


10. May 2004 Prof.VuThieu Econometrics Economic Theory Mathematical Economics Economic Statistics Mathematic Statistics

11. Introduction Methodology of Econometrics

Statement of theory or hypothesis:

Keynes stated: Consumption increases as income increases, but not as much as the increase in income. It means that The marginal propensity to consume (MPC) for a unit change in income is grater than zero but less than unit


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

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

14. 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

15. Introduction Methodology of Econometrics (4) Obtaining Data May 2004 Prof.VuThieu 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

16. 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

17. 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


(7) Forecasting or Prediction

With given future value(s) of X, what is the future value(s) of Y?

GDP=$6000Bill in 1994, what is the forecast consumption expenditure?

Y^= - 231.8+0.7196(6000) = 4084.6

Income Multiplier M = 1/(1 MPC) (=3.57). decrease (increase) of $1 in investment will eventually lead to $3.57 decrease (increase) in income


19. 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


Figure 1.4: Anatomy of economic modelling

1) Economic Theory

2) Mathematical Model of Theory

3) Econometric Model of Theory

4) Data

5) Estimation of Econometric Model

6) Hypothesis Testing

7) Forecasting or Prediction

8) Using the Model for control or policy purposes


21. May 2004 Prof.VuThieu Economic Theory Mathematic Model Econometric Model Data Collection Estimation Hypothesis Testing Forecasting Application in control or policy studies

22. Basic Econometrics Chapter 1 : THE NATURE OF REGRESSION ANALYSIS May 2004 Prof.VuThieu

23. 1-1. Historical origin of the term Regression

The term REGRESSION was introduced by Francis Galton

Tendency for tall parents to have tall children and for short parents to have short children, but the average height of children born from parents of a given height tended to move (or regress) toward the average height in the population as a whole (F. Galton, Family Likeness in Stature )


24. 1-1. Historical origin of the term Regression

Galtons Law was confirmed by Karl Pearson: The average height of sons of a group of tall fathers < their fathers height. And the average height of sons of a group of short fathers > their fathers height. Thus regressing tall and short sons alike toward the average height of all men. (K. Pearson and A. Lee, On the law of Inheritance )

By the words of Galton, this was Regression to mediocrity


25. 1-2. Modern Interpretation of Regression Analysis

The modern way in interpretation of Regression: Regression Analysis is concerned with the study of the dependence of one variable ( The Dependent Variable ) , on one or more other variable(s) ( The Explanatory Variable ) , with a view to estimating and/or predicting the (population) mean or average value of the former in term of the known or fixed (in repeated sampling) values of the latter.

Examples : (pages 16-19)


26. Dependent Variable Y; Explanatory Variable Xs

1. Y = Sons Height; X = Fathers Height

2. Y = Height of boys; X = Age of boys

3. Y = Personal Consumption Expenditure

X = Personal Disposable Income

4. Y = Demand; X = Price

5. Y = Rate of Change of Wages

X = Unemployment Rate

6. Y = Money/Income; X = Inflation Rate

7. Y = % Change in Demand; X = % Change in the

advertising budget

8. Y = Crop yield; Xs = temperature, rainfall, sunshine,

fertilizer


27. 1-3. Statistical vs. Deterministic Relationships

In regression analysis we are concerned with STATISTICAL DEPENDENCE among variables (not Functional or Deterministic), we essentially deal with RANDOM or STOCHASTIC variables (with the probability distributions)


28. 1-4. Regression vs. Causation:

Regression does not necessarily imply causation . A statistical relationship cannot logically imply causation. A statistical relationship, however strong and however suggestive, can never establish causal connection: our ideas of causation must come from outside statistics, ultimately from some theory or other ( M.G. Kendal and A. Stuart, The Advanced Theory of Statistics )


29. 1-5. Regression vs. Correlation

Correlation Analysis: the primary objective is to measure the strength or degree of linear association between two variables (both are assumed to be random)

Regression Analysis: we try to estimate or predict the average value of one variable (dependent, and assumed to be stochastic) on the basis of the fixed values of other variables (independent, and non-stochastic)


30. 1-6. Terminology and Notation

Dependent Variable

Explained Variable

Predictand

Regressand

Response

Endogenous

Explanatory Variable(s)

Independent Variable(s)

Predictor(s)

Regressor(s)

Stimulus or control variable(s)

Exogenous(es)


31. 1-7. The Nature and Sources of Data for Econometric Analysis

1) Types of Data :

Time series data;

Cross-sectional data;

Pooled data

2) The Sources of Data

3) The Accuracy of Data


32. 1-8. Summary and Conclusions

1) The key idea behind regression analysis is the statistic dependence of one variable on one or more other variable(s)

2) The objective of regression analysis is to estimate and/or predict the mean or average value of the dependent variable on basis of known (or fixed) values of explanatory variable(s)



3) The success of regression depends on the available and appropriate data

4) The researcher should clearly state the sources of the data used in the analysis, their definitions, their methods of collection, any gaps or omissions and any revisions in the data


34. Basic Econometrics Chapter 2 : TWO-VARIABLE REGRESSION ANALYSIS: Some basic Ideas May 2004 Prof.VuThieu

35. 2-1. A Hypothetical Example

Total population: 60 families

Y=Weekly family consumption expenditure

X=Weekly disposable family income

60 families were divided into 10 groups of approximately the same income level

( 80, 100, 120, 140, 160, 180, 200, 220, 240, 260 )



Table 2-1 gives the conditional distribution

of Y on the given values of X

Table 2-2 gives the conditional probabilities of Y: p(Y X)

Conditional Mean

(or Expectation): E(Y X=X i )


37. May 2004 Prof.VuThieu 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


Figure 2-1 shows the population regression line (curve). It is the

regression of Y on X

Population regression curve is the

locus of the conditional means or expectations of the dependent variable

for the fixed values of the explanatory variable X (Fig.2-2)


39. 2-2. The concepts of population regression function (PRF)

E(Y X=X i ) = f(X i ) is Population Regression Function (PRF) or

Population Regression (PR)

In the case of linear function we have linear population regression function (or equation or model)

E(Y X=X i ) = f(X i ) = 1 + 2 X i


40. 2-2. The concepts of population regression function (PRF)

E(Y X=X i ) = f(X i ) = 1 + 2 X i

1 and 2 are regression coefficients, 1 is intercept and 2 is slope coefficient

Linearity in the Variables

Linearity in the Parameters


41. 2-4. Stochastic Specification of PRF

U i = Y - E(Y X=X i ) or Y i = E(Y X=X i ) + U i

U i = Stochastic disturbance or stochastic error term. It is nonsystematic component

Component E(Y X=X i ) is systematic or deterministic. It is the mean consumption expenditure of all the families with the same level of income

The assumption that the regression line passes through the conditional means of Y implies that E(U i X i ) = 0


42. 2-5. The Significance of the Stochastic Disturbance Term

U i = Stochastic Disturbance Term is a surrogate for all variables that are omitted from the model but they collectively affect Y

Many reasons why not include such variables into the model as follows:


43. 2-5. The Significance of the Stochastic Disturbance Term

Why not include as many as variable into the model (or the reasons for using u i )

+ Vagueness of theory

+ Unavailability of Data

+ Core Variables vs. Peripheral Variables

+ Intrinsic randomness in human behavior

+ Poor proxy variables

+ Principle of parsimony

+ Wrong functional form


44. 2-6. The Sample Regression Function (SRF)

Table 2-4: A random sample from the population

Y X

------------------

70 80

65 100

90 120

95 140

110 160

115 180

120 200

140 220

155 240

150 260

------------------

Table 2-5: Another random sample from the population

Y X

-------------------

55 80

88 100

90 120

80 140

118 160

120 180

145 200

135 220

145 240

175 260

--------------------


45. May 2004 Prof.VuThieu SRF1 SRF2 Weekly Consumption Expenditure (Y) Weekly Income (X)


Fig.2-3: SRF1 and SRF 2

Y^ i = ^ 1 + ^ 2 X i (2.6.1)

Y^ i = estimator of E(Y X i )

^ 1 = estimator of 1

^ 2 = estimator of 2

Estimate = A particular numerical value obtained by the estimator in an application

SRF in stochastic form: Y i = ^ 1 + ^ 2 X i + u^ i

or Y i = Y^ i + u^ i (2.6.3)



Primary objective in regression analysis is to estimate the PRF Y i = 1 + 2 X i + u i on the basis of the SRF Y i = ^ 1 + ^ 2 X i + e i and how to construct SRF so that ^ 1 close to 1 and ^ 2 close to 2 as much as possible



Population Regression Function PRF

Linearity in the parameters

Stochastic PRF

Stochastic Disturbance Term u i plays a critical role in estimating the PRF

Sample of observations from population

Stochastic Sample Regression Function SRF used to estimate the PRF



The key concept underlying regression analysis is the concept of the population regression function (PRF).

This book deals with linear PRFs: linear in the unknown parameters. They may or may not linear in the variables.



For empirical purposes, it is the stochastic PRF that matters. The stochastic disturbance term u i plays a critical role in estimating the PRF.

The PRF is an idealized concept, since in practice one rarely has access to the entire population of interest. Generally, one has a sample of observations from population and use the stochastic sample regression (SRF) to estimate the PRF.


51. Basic Econometrics Chapter 3 : TWO-VARIABLE REGRESSION MODEL: The problem of Estimation May 2004 Prof.VuThieu

52. 3-1. The method of ordinary least square (OLS)

Least-square criterion:

Minimizing U^ 2 i = (Y i Y^ i ) 2

= (Y i - ^ 1 - ^ 2 X) 2 (3.1.2)

Normal Equation and solving it for ^ 1 and ^ 2 = Least-square estimators [ See (3.1.6)(3.1.7) ]

Numerical and statistical properties of OLS are as follows:


53. 3-1. The method of ordinary least square (OLS)

OLS estimators are expressed solely in terms of observable quantities. They are point estimators

The sample regression line passes through sample means of X and Y

The mean value of the estimated Y^ is equal to the mean value of the actual Y: E(Y) = E(Y^)

The mean value of the residuals U^ i is zero: E(u^ i )=0

u^ i are uncorrelated with the predicted Y^ i and with X i : That are u^ i Y^ i = 0; u^ i X i = 0


54. 3-2. The assumptions underlying the method of least squares

Ass 1: Linear regression model

(in parameters)

Ass 2: X values are fixed in repeated

sampling

Ass 3: Zero mean value of u i : E(u i X i )=0

Ass 4: Homoscedasticity or equal

variance of u i : Var (u i X i ) = 2

[VS. Heteroscedasticity]

Ass 5: No autocorrelation between the

disturbances: Cov(u i ,u j X i ,X j ) = 0

with i # j [VS. Correlation, + or - ]


55. 3-2. The assumptions underlying the method of least squares

Ass 6: Zero covariance between u i and X i

Cov(u i , X i ) = E(u i , X i ) = 0

Ass 7: The number of observations n must be greater than the number of parameters to be estimated

Ass 8: Variability in X values. They must not all be the same

Ass 9: The regression model is correctly specified

Ass 10: There is no perfect multicollinearity between Xs


56. 3-3. Precision or standard errors of least-squares estimates

In statistics the precision of an

estimate is measured by its standard

error (SE)

var( ^ 2 ) = 2 / x 2 i (3.3.1)

se( ^ 2 ) = Var( ^ 2 ) (3.3.2)

var( ^ 1 ) = 2 X 2 i / n x 2 i (3.3.3)

se( ^ 1 ) = Var( ^ 1 ) (3.3.4)

^ 2 = u^ 2 i / (n - 2) (3.3.5)

^ = ^ 2 is standard error of the

estimate


57. 3-3. Precision or standard errors of least-squares estimates

Features of the variance:

+ var( ^ 2 ) is proportional to 2 and inversely proportional to x 2 i

+ var( ^ 1 ) is proportional to 2 and X 2 i but inversely proportional to x 2 i and the sample size n.

+ cov ( ^ 1 , ^ 2 ) = - var( ^ 2 ) shows the independence between ^ 1 and ^ 2


58. 3-4. Properties of least-squares estimators: The Gauss-Markov Theorem

An OLS estimator is said to be BLUE if :

+ It is linear , that is, a linear function of a random variable, such as the dependent variable Y in the regression model

+ It is unbiased , that is, its average or expected value, E( ^ 2 ), is equal to the true value 2

+ It has minimum variance in the class of all such linear unbiased estimators

An unbiased estimator with the least variance is known as an efficient estimator


59. 3-4. Properties of least-squares estimators: The Gauss-Markov Theorem

Gauss- Markov Theorem:

Given the assumptions of the classical linear regression model, the least-squares estimators, in class of unbiased linear estimators, have minimum variance, that is, they are BLUE


60. 3-5. The coefficient of determination r 2 : A measure of Goodness of fit

Y i = i + i or

Y i - = i - i + i or

y i = i + i (Note: = )

Squaring on both side and summing =>

y i 2 = 2 x 2 i + 2 i ; or

TSS = ESS + RSS



TSS = y i 2 = Total Sum of Squares

ESS = Y^ i 2 = ^ 2 2 x 2 i =

Explained Sum of Squares

RSS = u^ 2 I = Residual Sum of

Squares

ESS RSS

1 = -------- + -------- ; or

TSS TSS

RSS RSS

1 = r 2 + ------- ; or r 2 = 1 - -------

TSS TSS



r 2 = ESS/TSS

is coefficient of determination, it measures the proportion or percentage of the total variation in Y explained by the regression

Model

0 r 2 1;

r = r 2 is sample correlation coefficient

Some properties of r


63. 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

64. Basic Econometrics Chapter 4 : THE NORMALITY ASSUMPTION: Classical Normal Linear Regression Model (CNLRM) May 2004 Prof.VuThieu

65. 4-2.The normality assumption

CNLR assumes that each u i is distributed normally u i N(0, 2 ) with:

Mean = E(u i ) = 0 Ass 3

Variance = E(u 2 i ) = 2 Ass 4

Cov(u i , u j ) = E(u i , u j ) = 0 (i#j) Ass 5

Note : For two normally distributed variables, the zero covariance or correlation means independence of them, so u i and u j are not only uncorrelated but also independently distributed. Therefore u i NID(0, 2 ) is Normal and

Independently Distributed



Why the normality assumption?

With a few exceptions, the distribution of sum of a large number of independent and identically distributed random variables tends to a normal distribution as the number of such variables increases indefinitely

If the number of variables is not very large or they are not strictly independent, their sum may still be normally distributed



Why the normality assumption?

Under the normality assumption for u i , the OLS estimators ^ 1 and ^ 2 are also normally distributed

The normal distribution is a comparatively simple distribution involving only two parameters (mean and variance)


68. 4-3. Properties of OLS estimators under the normality assumption

With the normality assumption the OLS estimators ^ 1 , ^ 2 and ^ 2 have the following properties:

1. They are unbiased

2. They have minimum variance. Combined 1 and 2, they are efficient estimators

3. Consistency, that is, as the sample size increases indefinitely, the estimators converge to their true population values



4. ^ 1 is normally distributed

N( 1 , ^ 1 2 )

And Z = ( ^ 1 - 1 )/ ^ 1 is N(0,1)

5. ^ 2 is normally distributed N( 2 , ^ 2 2 )

And Z = ( ^ 2 - 2 )/ ^ 2 is N(0,1)

6. (n-2) ^ 2 / 2 is distributed as the

2 (n-2)



7. ^ 1 and ^ 2 are distributed independently of ^ 2 . They have minimum variance in the entire class of unbiased estimators, whether linear or not. They are best unbiased estimators (BUE)

8. Let u i is N(0, 2 ) then Y i is

N[E(Y i ); Var(Y i )] = N[ 1 + 2 X i ; 2 ]


71. Some last points of chapter 4

4-4. The method of Maximum likelihood (ML)

ML is point estimation method with some

stronger theoretical properties than OLS

(Appendix 4.A on pages 110-114)

The estimators of coefficients s by OLS and ML are

identical. They are true estimators of the s

(ML estimator of 2 ) = u^ i 2 /n (is biased estimator)

(OLS estimator of 2 ) = u^ i 2 /n-2 (is unbiased estimator)

When sample size (n) gets larger the two estimators tend to be equal


72. Some last points of chapter 4

4-5. Probability distributions related

to the Normal Distribution: The t, 2 ,

and F distributions

See section (4.5) on pages 107-108

with 8 theorems and Appendix A, on

pages 755-776

4-6. Summary and Conclusions

See 10 conclusions on pages 109-110


73. Basic Econometrics Chapter 5 : TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing May 2004 Prof.VuThieu

74. Chapter 5 TWO-VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing

5-1. Statistical Prerequisites

S ee Appendix A with key concepts such as probability, probability distributions, Type I Error, Type II Error,level of significance, power of a statistic test, and confidence interval



5-2. Interval estimation: Some basic Ideas

How close is, say, ^ 2 to 2 ?

Pr ( ^ 2 - 2 ^ 2 + ) = 1 - (5.2.1)

Random interval ^ 2 - 2 ^ 2 +

if exits, it known as confidence interval

^ 2 - is lower confidence limit

^ 2 + is upper confidence limit




(1 - ) is confidence coefficient,

0 < < 1 is significance level

Equation (5.2.1) does not mean that the Pr of 2 lying between the given limits is (1 - ), but the Pr of constructing an interval that contains 2 is (1 - )

( ^ 2 - , ^ 2 + ) is random interval




In repeated sampling, the intervals will enclose, in (1 - )*100 of the cases, the true value of the parameters

For a specific sample, can not say that the probability is (1 - ) that a given fixed interval includes the true 2

If the sampling or probability distributions of the estimators are known, one can make confidence interval statement like (5.2.1)



5-3. Confidence Intervals for Regression

Coefficients

Z= ( ^ 2 - 2 )/se( ^ 2 ) = ( ^ 2 - 2 ) x 2 i / ~N(0,1)

(5.3.1)

We did not know and have to use ^ instead, so:

t= ( ^ 2 - 2 )/se( ^ 2 ) = ( ^ 2 - 2 ) x 2 i / ^ ~ t(n-2)

(5.3.2)

=> Interval for 2

Pr [ -t /2 t t /2 ] = 1- (5.3.3)



5-3. Confidence Intervals for Regression

Coefficients

Or confidence interval for 2 is

Pr [ ^ 2 -t /2 se( ^ 2 ) 2 ^ 2 +t /2 se( ^ 2 )] = 1-

(5.3.5)

Confidence Interval for 1

Pr [ ^ 1 -t /2 se( ^ 1 ) 1 ^ 1 +t /2 se( ^ 1 )] = 1-

(5.3.7)



5-4. Confidence Intervals for 2

Pr [(n-2) ^ 2 / 2 /2 2 (n-2) ^ 2 / 2 1- /2 ] = 1-

(5.4.3)

The interpretation of this interval is: If we establish (1- ) confidence limits on 2 and if we maintain a priori that these limits will include true 2 , we shall be right in the long run (1- ) percent of the time



5-5. Hypothesis Testing: General Comments

The stated hypothesis is known as the

null hypothesis: H o

The H o is tested against and alternative

hypothesis: H 1

5-6. Hypothesis Testing: The confidence interval approach

One-sided or one-tail Test

H 0 : 2 * versus H 1 : 2 > *



Two-sided or two-tail Test

H 0 : 2 = * versus H 1 : 2 # *

^ 2 - t /2 se( ^ 2 ) 2 ^ 2 + t /2 se( ^ 2 ) values of 2 lying in this interval are plausible under H o with 100*(1- )% confidence.

If 2 lies in this region we do not reject H o (the finding is statistically insignificant )

If 2 falls outside this interval, we reject H o (the finding is statistically significant )



5-7. Hypothesis Testing:

The test of significance approach

A test of significance is a procedure by which sample results are used to verify the truth or falsity of a null hypothesis

Testing the significance of regression coefficient: The t-test

Pr [ ^ 2 -t /2 se( ^ 2 ) 2 ^ 2 +t /2 se( ^ 2 )]= 1-

(5.7.2)



5-7. Hypothesis Testing: The test of significance approach

Table 5-1: Decision Rule for t-test of significance

May 2004 Prof.VuThieu 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



Testing the significance of 2 : The 2 Test

Under the Normality assumption we have:

^ 2

2 = (n-2) ------- ~ 2 (n-2) (5.4.1)

2

From (5.4.2) and (5.4.3) on page 520 =>




Table 5-2: A summary of the 2 Test

May 2004 Prof.VuThieu 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



Some practical aspects

1) The meaning of Accepting or Rejecting a Hypothesis

2) The Null Hypothesis and the Rule of

Thumb

3) Forming the Null and Alternative

Hypotheses

4) Choosing , the Level of Significance




Some practical aspects

5) The Exact Level of Significance:

The p-Value [ See page 132 ]

6) Statistical Significance versus

Practical Significance

7) The Choice between Confidence-

Interval and Test-of-Significance

Approaches to Hypothesis Testing

[Warning: Read carefully pages 117-134 ]



5-9. Regression Analysis and Analysis

of Variance

TSS = ESS + RSS

F=[MSS of ESS] /[MSS of RSS] =

= 2 ^ 2 x i 2 / ^ 2 (5.9.1)

If u i are normally distributed; H 0 : 2 = 0 then F follows the F distribution with 1 and n-2 degree of freedom



5-9. Regression Analysis and Analysis of Variance

F provides a test statistic to test the null hypothesis that true 2 is zero by compare this F ratio with the F-critical obtained from F tables at the chosen level of significance, or obtain the p-value of the computed F statistic to make decision



5-9. Regression Analysis and Analysis of Variance

Table 5-3. ANOVA for two-variable regression model

May 2004 Prof.VuThieu 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


5-10. Application of Regression

Analysis: Problem of Prediction

By the data of Table 3-2, we obtained the sample regression (3.6.2) :

Y^ i = 24.4545 + 0.5091X i , where

Y^ i is the estimator of true E(Y i )

There are two kinds of prediction as

follows:



5-10. Application of Regression

Analysis: Problem of Prediction

Mean prediction : Prediction of the conditional mean value of Y corresponding to a chosen X, say X 0 , that is the point on the population regression line itself (see pages 137-138 for details)

Individual prediction : Prediction of an individual Y value corresponding to X 0 (see pages 138-139 for details)



5-11. Reporting the results of

regression analysis

An illustration:

Y^ I = 24.4545 + 0.5091X i (5.1.1)

Se = (6.4138) (0.0357) r 2 = 0.9621

t = (3.8128) (14.2405) df= 8

P = (0.002517) (0.000000289) F 1,2 =2202.87



5-12. Evaluating the results of regression analysis:

Normality Test: The Chi-Square ( 2 ) Goodness of fit Test

2 N-1-k = (O i E i ) 2 /E i (5.12.1)

O i is observed residuals (u^ i ) in interval i

E i is expected residuals in interval i

N is number of classes or groups; k is number of

parameters to be estimated. If p-value of

obtaining 2 N-1-k is high (or 2 N-1-k is small) =>

The Normality Hypothesis can not be rejected




Normality Test: The Chi-Square ( 2 ) Goodness of fit Test

H 0 : u i is normally distributed

H 1 : u i is un-normally distributed

Calculated- 2 N-1-k = (O i E i ) 2 /E i (5.12.1)

Decision rule:

Calculated- 2 N-1-k > Critical- 2 N-1-k then H 0 can

be rejected




The Jarque-Bera (JB) test of normality

This test first computes the Skewness (S)

and Kurtosis (K) and uses the following

statistic:

JB = n [S 2 /6 + (K-3) 2 /24] (5.12.2)

Mean= x bar = x i /n ; SD 2 = (x i - x bar ) 2 /(n-1)

S=m 3 /m 2 3/2 ; K=m 4 /m 2 2 ; m k = (x i - x bar ) k /n



5-12. (Continued)

Under the null hypothesis H 0 that the residuals are normally distributed Jarque and Bera show that in large sample (asymptotically) the JB statistic given in (5.12.12) follows the Chi-Square distribution with 2 df. If the p-value of the computed Chi-Square statistic in an application is sufficiently low, one can reject the hypothesis that the residuals are normally distributed. But if p-value is reasonable high, one does not reject the normality assumption.




1. Estimation and Hypothesis testing constitute the two main branches of classical statistics

2. Hypothesis testing answers this question: Is a given finding compatible with a stated hypothesis or not?

3. There are two mutually complementary approaches to answering the preceding question: Confidence interval and test of significance.




4. Confidence-interval approach has a specified probability of including within its limits the true value of the unknown parameter. If the null-hypothesized value lies in the confidence interval, H 0 is not rejected, whereas if it lies outside this interval, H 0 can be rejected




5. Significance test procedure develops a test statistic which follows a well-defined probability distribution (like normal, t, F, or Chi-square). Once a test statistic is computed, its p-value can be easily obtained.

The p-value The p-value of a test is the lowest significance level, at which we would reject H 0 . It gives exact probability of obtaining the estimated test statistic under H 0 . If p-value is small, one can reject H 0 , but if it is large one may not reject H 0 .




6. Type I error is the error of rejecting a true hypothesis. Type II error is the error of accepting a false hypothesis. In practice, one should be careful in fixing the level of significance , the probability of committing a type I error (at arbitrary values such as 1%, 5%, 10%). It is better to quote the p-value of the test statistic.




7. This chapter introduced the normality test to find out whether u i follows the normal distribution. Since in small samples, the t, F,and Chi-square tests require the normality assumption, it is important that this assumption be checked formally



5-13. Summary and Conclusions (ended)

8. If the model is deemed practically adequate, it may be used for forecasting purposes. But should not go too far out of the sample range of the regressor values. Otherwise, forecasting errors can increase dramatically.


105. Basic Econometrics Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL May 2004 Prof.VuThieu

106. Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS

6-1. Regression through the origin

The SRF form of regression:

Y i = ^ 2 X i + u^ i (6.1.5)

Comparison two types of regressions:

* Regression through-origin model and

* Regression with intercept


107. 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



r 2 for regression through-origin model

Raw r 2 = ( X i Y i ) 2 / X 2 i Y 2 i (6.1.9 )

Note: Without very strong a priory expectation, well advise is sticking to the conventional, intercept-present model. If intercept equals to zero statistically, for practical purposes we have a regression through the origin. If in fact there is an intercept in the model but we insist on fitting a regression through the origin, we would be committing a specification error




Illustrative Examples:

1) Capital Asset Pricing Model - CAPM (page 156)

2) Market Model (page 157)

3) The Characteristic Line of Portfolio Theory (page 159)



6-2. Scaling and units of measurement

Let Y i = ^ 1 + ^ 2 X i + u^ i (6.2.1)

Define Y* i =w 1 Y i and X* i =w 2 X i then:

^ 2 = (w 1 /w 2 ) ^ 2 (6.2.15)

^ 1 = w 1 ^ 1 (6.2.16)

*^ 2 = w 1 2 ^ 2 (6.2.17)

Var( ^ 1 ) = w 2 1 Var( ^ 1 ) (6.2.18)

Var( ^ 2 ) = (w 1 /w 2 ) 2 Var( ^ 2 ) (6.2.19)

r 2 xy = r 2 x*y* (6.2.20)



6-2. Scaling and units of measurement

From one scale of measurement, one can derive the results

based on another scale of measurement. If w 1 = w 2 the

intercept and standard error are both multiplied by w 1 . If

w 2 =1 and scale of Y changed by w 1 , then all coefficients and

standard errors are all multiplied by w 1 . If w 1 =1 and scale of

X changed by w 2 , then only slope coefficient and its standard

error are multiplied by 1/w 2 . Transformation from (Y,X) to

(Y*,X*) scale does not affect the properties of OLS

Estimators

A numerical example : (pages 161, 163-165)


112. 6-3. Functional form of regression model

The log-linear model

Semi-log model

Reciprocal model


113. 6-4. How to measure elasticity


Exponential regression model:

Y i = 1 X i e u i (6.4.1)

By taking log to the base e of both side:

lnY i = ln 1 + 2 lnX i + u i , by setting ln 1 =

lnY i = + 2 lnX i + u i (6.4.3)

(log-log, or double-log, or log-linear model)

This can be estimated by OLS by letting

Y* i = + 2 X* i + u i , where Y* i =lnY i , X* i =lnX i ;

2 measures the ELASTICITY of Y respect to X, that is, percentage change in Y for a given (small) percentage change in X.


114. 6-4. How to measure elasticity


The elasticity E of a variable Y with

respect to variable X is defined as:

E=dY/dX=(% change in Y)/(% change in X)

~ [( Y/Y) x 100] / [( X/X) x100]=

= ( Y/ X)x (X/Y) = slope x (X/Y)

An illustrative example : The coffee

demand function (pages 167-168)


115. 6-5. Semi-log model : Log-lin and Lin-log Models

How to measure the growth rate: The log-lin model

Y t = Y 0 (1+r) t (6.5.1)

lnY t = lnY 0 + t ln(1+r) (6.5.2)

lnY t = + 2 t , called constant growth model (6.5.5)

where 1 = lnY 0 ; 2 = ln(1+r)

lnY t = + 2 t + u i (6.5.6)

It is Semi-log model, or log-lin model. The slope coefficient measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor (t)

2 = (Relative change in regressand)/(Absolute change in regressor) (6.5.7)



Instantaneous Vs. compound rate of growth

2 is instantaneous rate of growth

antilog( 2 ) 1 is compound rate of growth

The linear trend model

Y t = + 2 t + u t (6.5.9)

If 2 > there is an upward trend in Y

If 2 < there is an downward trend in Y

Note: (i) Cannot compare the r 2 values of models (6.5.5) and (6.5.9) because the regressands in the two models are different, (ii) Such models may be appropriate only if a time series is stationary.



The lin-log model:

Y i = 1 + 2 lnX i + u i (6.5.11)

2 = (Change in Y) / Change in lnX = (Change in Y)/(Relative change in X) ~ ( Y)/( X/X) (6.5.12)

or Y = 2 ( X/X) (6.5.13)

That is, the absolute change in Y equal to 2 times the relative change in X.


118. 6-6. Reciprocal Models : Log-lin and Lin-log Models

The reciprocal model:

Y i = 1 + 2 ( 1/X i ) + u i (6.5.14)

As X increases definitely, the term

2 ( 1/X i ) approaches to zero and Y i

approaches the limiting or asymptotic value 1 ( See figure 6.5 in page 174 )

An Illustrative example : The Phillips Curve for the United Kingdom 1950-1966


119. 6-7. Summary of Functional Forms Table 6.5 (page 178) May 2004 Prof.VuThieu 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) */

120. 6-7. Summary of Functional Forms

Note : */ indicates that the elasticity coefficient is variable, depending on the value taken by X or Y or both. when no X and Y values are specified, in practice, very often these elasticities are measured at the mean values E(X) and E(Y).

-----------------------------------------------

6 -8. A note on the stochastic error term

6-9. Summary and conclusions

(pages 179-180)


121. Basic Econometrics Chapter 7 MULTIPLE REGRESSION ANALYSIS: The Problem of Estimation May 2004 Prof.VuThieu

122. 7-1. The three-Variable Model: Notation and Assumptions

Y i = 1 + 2 X 2i + 3 X 3i + u i (7.1.1)

2 , 3 are partial regression coefficients

With the following assumptions:

+ Zero mean value of U i: : E(u i |X 2i ,X 3i ) = 0. i (7.1.2)

+ No serial correlation: Cov(u i ,u j ) = 0, i # j (7.1.3)

+ Homoscedasticity: Var(u i ) = 2 (7.1.4)

+ Cov(u i ,X 2i ) = Cov(u i ,X 3i ) = 0 (7.1.5)

+ No specification bias or model correct specified (7.1.6)

+ No exact collinearity between X variables (7.1.7)

(no multicollinearity in the cases of more explanatory

vars. If there is linear relationship exits, X vars. Are said

to be linearly dependent)

+ Model is linear in parameters


123. 7-2. Interpretation of Multiple Regression

E(Y i | X 2i ,X 3i ) = 1 + 2 X 2i + 3 X 3i (7.2.1)

(7.2.1) gives conditional mean or expected value of Y conditional upon the given or fixed value of the X 2 and X 3


124. 7-3. The meaning of partial regression coefficients

Y i = 1 + 2 X 2i + 3 X 3 +.+ s X s + u i

k measures the change in the mean value of Y per unit change in X k , holding the rest explanatory variables constant. It gives the direct effect of unit change in X k on the E(Y i ), net of X j (j # k)

How to control the true effect of a unit change in X k on Y? (read pages 195-197)


125. 7-4. OLS and ML estimation of the partial regression coefficients

This section (pages 197-201) provides:

1. The OLS estimators in the case of three-variable regression

Y i = 1 + 2 X 2i + 3 X 3 + u i

2. Variances and standard errors of OLS estimators

3. 8 properties of OLS estimators (pp 199-201)

4. Understanding on ML estimators


126. 7-5. The multiple coefficient of determination R 2 and the multiple coefficient of correlation R

This section provides:

1. Definition of R 2 in the context of multiple regression like r 2 in the case of two-variable regression

2. R = R 2 is the coefficient of multiple regression, it measures the degree of association between Y and all the explanatory variables jointly

3. Variance of a partial regression coefficient

Var(^ k ) = 2 / x 2 k (1/(1-R 2 k )) (7.5.6)

Where ^ k is the partial regression coefficient of regressor X k and R 2 k is the R 2 in the regression of X k on the rest regressors


127. 7-6. Example 7.1: The expectations-augmented Philips Curve for the US (1970-1982)

This section provides an illustration for the ideas introduced in the chapter

Regression Model (7.6.1)

Data set is in Table 7.1


128. 7-7. Simple regression in the context of multiple regression: Introduction to specification bias

This section provides an understanding on Simple regression in the context of multiple regression. It will cause the specification bias which will be discussed in Chapter 13


129. 7-8. R 2 and the Adjusted-R 2

R 2 is a non-decreasing function of the number of explanatory variables. An additional X variable will not decrease R 2

R 2 = ESS/TSS = 1- RSS/TSS = 1- u^ 2 I / y^ 2 i (7.8.1)

This will make the wrong direction by adding more irrelevant variables into the regression and give an idea for an adjusted-R 2 (R bar ) by taking account of degree of freedom

R 2 bar = 1- [ u^ 2 I /(n-k)] / [ y^ 2 i /(n-1) ] , or (7.8.2)

R 2 bar = 1- ^ 2 / S 2 Y (S 2 Y is sample variance of Y)

K= number of parameters including intercept term

By substituting (7.8.1) into (7.8.2) we get

R 2 bar = 1- (1-R 2 ) (n-1)/(n- k) (7.8.4)

For k > 1, R 2 bar < R 2 thus when number of X variables increases R 2 bar increases less than R 2 and R 2 bar can be negative



R 2 is a non-decreasing function of the number of explanatory variables. An additional X variable will not decrease R 2

R 2 = ESS/TSS = 1- RSS/TSS = 1- u^ 2 I / y^ 2 i (7.8.1)

This will make the wrong direction by adding more irrelevant variables into the regression and give an idea for an adjusted-R 2 (R bar ) by taking account of degree of freedom

R 2 bar = 1- [ u^ 2 I /(n-k)] / [ y^ 2 i /(n-1) ] , or (7.8.2)

R 2 bar = 1- ^ 2 / S 2 Y (S 2 Y is sample variance of Y)

K= number of parameters including intercept term

By substituting (7.8.1) into (7.8.2) we get

R 2 bar = 1- (1-R 2 ) (n-1)/(n- k) (7.8.4)

For k > 1, R 2 bar < R 2 thus when number of X variables increases R 2 bar increases less than R 2 and R 2 bar can be negative



Comparing Two R 2 Values:

To compare, the size n and the dependent variable must be the same

Example 7-2: Coffee Demand Function Revisited (page 210)

The game of maximizing adjusted-R 2 : Choosing the model that gives the highest R 2 bar may be dangerous, for in regression our objective is not for that but for obtaining the dependable estimates of the true population regression coefficients and draw statistical inferences about them

Should be more concerned about the logical or theoretical relevance of the explanatory variables to the dependent variable and their statistical significance


132. 7-9. Partial Correlation Coefficients

This section provides:

1. Explanation of simple and partial correlation coefficients

2. Interpretation of simple and partial correlation coefficients

(pages 211-214)


133. 7-10. Example 7.3: The Cobb-Douglas Production function More on functional form

Y i = 1 X 2 2i X 3 3i e U i (7.10.1)

By log-transform of this model:

lnY i = ln 1 + 2 ln X 2i + 3 ln X 3i + U i = 0 + 2 ln X 2i + 3 ln X 3i + U i (7.10.2)


Report of results is in page 216


134. 7-11 Polynomial Regression Models

Y i = 0 + 1 X i + 2 X 2 i ++ k X k i + U i

(7.11.3)

Example 7.4: Estimating the Total Cost Function


Empirical results is in page 221

--------------------------------------------------------------


(page 221)


135. Basic Econometrics Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference May 2004 Prof.VuThieu

136. Chapter 8 MULTIPLE REGRESSION ANALYSIS: The Problem of Inference

8-3. Hypothesis testing in multiple regression:

Testing hypotheses about an individual partial regression coefficient

Testing the overall significance of the estimated multiple regression model, that is, finding out if all the partial slope coefficients are simultaneously equal to zero

Testing that two or more coefficients are equal to one another

Testing that the partial regression coefficients satisfy certain restrictions

Testing the stability of the estimated regression model over time or in different cross-sectional units

Testing the functional form of regression models


137. 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


8-5. Testing the overall significance of a multiple

regression: The F-Test

For Y i = 1 + 2 X 2i + 3 X 3i + ........+ k X ki + u i

To test the hypothesis H 0 : 2 = 3 =....= k = 0 ( all slope coefficients are simultaneously zero ) versus H 1 : Not at all slope coefficients are simultaneously zero, compute

F=(ESS/df)/(RSS/df)=(ESS/(k-1))/(RSS/(n-k)) (8.5.7) ( k = total number of parameters to be estimated including intercept )

If F > F critical = F (k-1,n-k), reject H 0

Otherwise you do not reject it



8-5. Testing the overall significance of a multiple regression

Alternatively, if the p-value of F obtained from (8.5.7) is sufficiently low, one can reject H 0

An important relationship between R 2 and F:

F=(ESS/(k-1))/(RSS/(n-k)) or

R 2 /(k-1)

F = ---------------- (8.5.1)

(1-R 2 ) / (n-k)

( see prove on page 249)



8-5. Testing the overall significance of a multiple regression in terms of R 2

For Y i = 1 + 2 X 2i + 3 X 3i + ........+ k X ki + u i

To test the hypothesis H 0 : 2 = 3 = .....= k = 0 ( all slope coefficients are simultaneously zero ) versus H 1 : Not at all slope coefficients are simultaneously zero, compute

F = [R 2 /(k-1)] / [(1-R 2 ) / (n-k)] (8.5.13) ( k = total number of parameters to be estimated including intercept )

If F > F critical = F (k-1,n-k), reject H 0




Alternatively, if the p-value of F obtained from (8.5.13) is sufficiently low, one can reject H 0

The Incremental or Marginal contribution of an explanatory variable:

Let X is the new (additional) term in the right hand of a regression. Under the usual assumption of the normality of u i and the H O : = 0, it can be shown that the following F ratio will follow the F distribution with respectively degree of freedom


142. 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



Decision Rule:

If F com > F Df1 , Df2 one can reject the Ho that = 0 and conclude that the addition of X to the model significantly increases ESS and hence the R 2 value

When to Add a New Variable? If |t| of coefficient of X > 1 (or F= t 2 of that variable exceeds 1)

When to Add a Group of Variables? If adding a group of variables to the model will give F value greater than 1;


144. 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


t = ( ^ 3 - ^ 4 ) / [var( ( ^ 3 ) + var( ^ 4 ) 2cov ( ^ 3 , ^ 4 )] (8.6.5)

Steps for testing :

1.Estimate ^ 3 and ^ 4

2.Compute se( ^ 3 - ^ 4 ) through (8.6.4)

3.Obtain t- ratio from (8.6.5) with H 0 : 3 = 4

4.If t-computed > t-critical at designated level of significance for given df, then reject H 0 . Otherwise do not reject it. Alternatively, if the p-value of t statistic from (8.6.5) is reasonable low, one can reject H 0 .

Example 8.2: The cubic cost function revisited


146. 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


8-7 . Restricted least square: Testing linear equality restrictions

How to test (8.7.3)

The t Test approach (unrestricted): test of the hypothesis H 0 : 2 + 3 = 1 can be conducted by t- test:

t = [( ^ 2 + ^ 3 ) ( 2 + 3 )] / se( ^ 2 - ^ 3 ) (8.7.4)

The F Test approach (restricted least square -RLS): Using, say, 2 = 1- 3 and substitute it into (8.7.2) we get: ln(Y i /X 2i ) = 0 + 3 ln(X 3i /X 2i ) + u i (8.7.8). Where (Y i /X 2i ) is output/labor ratio, and (X 3i / X 2i ) is capital/labor ratio


148. 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


8-7. Restricted least square: Testing linear equality restrictions

Note: R 2 UR R 2 R (8.7.11)

and u^ 2 UR u^ 2 R (8.7.12)

Example 8.3 : The Cobb-Douglas Production

function for Taiwanese Agricultural Sector,

1958-1972. (pages 259-260). Data in Table 7.3

(page 216)

General F Testing (page 260)

Example 8.4 : The demand for chicken in the US,

1960-1982. Data in exercise 7.23 (page 228)



8-8. Comparing two regressions: Testing for structural stability of regression models

Table 8.8: Personal savings and income data, UK, 1946-1963 (millions of pounds)

Savings function:

Reconstruction period:

Y t = 1 + 2 X t + U 1t (t = 1,2,...,n 1 )

Post-Reconstruction period:

Y t = 1 + 2 X t + U 2t (t = 1,2,...,n 2 )

Where Y is personal savings, X is personal income, the u s are disturbance terms in the two equations and n 1 , n 2 are the number of observations in the two period


151. 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

152. 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

153. 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

154. 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

155. 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

156. 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

eco basic 1 8

Education

partial correlation

partial regression

population

stochastic

sample regression

population

population

type ii error