Basic Econometrics

1
Basic Econometrics

• Course Leader
• Prof. Dr.Sc VuThieu

2
Basic Econometrics

• Introduction
• What is Econometrics?

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)

4
Introduction What is Econometrics?

• 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)

5
Introduction What is Econometrics?

• 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)

6
Introduction What is Econometrics?

• 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
Economic Theory
Mathematical Economics
Econometrics
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
Economic Theory
Mathematical Economics
Econometrics
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 ß1 ß2X 0 lt ß2lt 1
• Y consumption expenditure
• X income
• ß1 and ß2 are parameters ß1 is
• intercept, and ß2 is slope coefficients

13
Introduction Methodology of Econometrics

• (3) Specification of the econometric model of the
  theory
• Y ß1 ß2X u 0 lt ß2lt 1
• Y consumption expenditure
• X income
• ß1 and ß2 are parameters ß1is intercept and ß2
  is slope coefficients u is disturbance term or
  error term. It is a random or stochastic variable

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

15
Introduction Methodology of Econometrics

• (4) Obtaining Data

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

17
Introduction Methodology of Econometrics

• (6) Hypothesis Testing
• Are the estimates accord with the
• expectations of the theory that is being
• tested? Is MPC lt 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)

18
Introduction Methodology of Econometrics

• (7) Forecasting or Prediction
• With given future value(s) of X, what is the
  future value(s) of Y?
• GDP6000Bill in 1994, what is the forecast
  consumption expenditure?
• Y - 231.80.7196(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
• Y4000 -231.80.7194 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

20
Introduction Methodology of Econometrics

• 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
Economic Theory
Mathematic Model
Econometric Model
Data Collection
Estimation
Hypothesis Testing
Application in control or policy studies
Forecasting
22
Basic Econometrics

• Chapter 1
• THE NATURE OF REGRESSION ANALYSIS

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
  lt their fathers height. And the average height
  of sons of a group of short fathers gt 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)

33
1-8. Summary and Conclusions

• 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

35
2-1. A Hypothetical Example

• Total population 60 families
• YWeekly family consumption expenditure
• XWeekly 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)

36
2-1. A Hypothetical Example

• 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?XXi )

37
Table 2-2 Weekly family income X (), and
consumption Y ()
38
2-1. A Hypothetical Example

• 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?XXi ) f(Xi) 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?XXi ) f(Xi) ß1 ß2Xi

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

• E(Y?XXi ) f(Xi) ß1 ß2Xi
• ß1 and ß2 are regression coefficients, ß1is
  intercept and ß2 is slope coefficient
• Linearity in the Variables
• Linearity in the Parameters

41
2-4. Stochastic Specification of PRF

• Ui Y - E(Y?XXi ) or Yi E(Y?XXi ) Ui
• Ui Stochastic disturbance or stochastic error
  term. It is nonsystematic component
• Component E(Y?XXi ) 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(Ui?Xi ) 0

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

• Ui 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 ui)
• 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
Weekly Consumption Expenditure (Y)
SRF1
SRF2
Weekly Income (X)
46
2-6. The Sample Regression Function (SRF)

• Fig.2-3 SRF1 and SRF 2
• Yi ?1 ?2Xi (2.6.1)
• Yi estimator of E(Y?Xi)
• ?1 estimator of ?1
• ?2 estimator of ?2
• Estimate A particular numerical value obtained
  by the estimator in an application
• SRF in stochastic form Yi ?1 ?2Xi ui
• or Yi Yi ui (2.6.3)

47
2-6. The Sample Regression Function
(SRF)

• Primary objective in regression analysis is to
  estimate the PRF Yi ?1 ?2Xi ui on the basis
  of the SRF Yi ?1 ?2Xi ei and how to
  construct SRF so that ?1 close to ?1 and ?2
  close to ?2 as much as possible

48
2-6. The Sample Regression Function (SRF)

• Population Regression Function PRF
• Linearity in the parameters
• Stochastic PRF
• Stochastic Disturbance Term ui plays a critical
  role in estimating the PRF
• Sample of observations from population
• Stochastic Sample Regression Function SRF used to
  estimate the PRF

49
2-7. Summary and Conclusions

• 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.

50
2-7. Summary and Conclusions

• For empirical purposes, it is the stochastic PRF
  that matters. The stochastic disturbance term ui
  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

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

• Least-square criterion
• Minimizing ?U2i ?(Yi Yi) 2
• ?(Yi- ?1 - ?2X)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 Ui is zero
  E(ui )0
• ui are uncorrelated with the predicted Yi and
  with Xi That are ?uiYi 0 ?uiXi 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 ui E(ui?Xi)0
• Ass 4 Homoscedasticity or equal
• variance of ui Var (ui?Xi) ?2
  
• VS. Heteroscedasticity
• Ass 5 No autocorrelation between the
• disturbances Cov(ui,uj?Xi,Xj )
  0
• with i j VS. Correlation, or
  -

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

• Ass 6 Zero covariance between ui and Xi
• Cov(ui, Xi) E(ui, Xi) 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 / ?x2i (3.3.1)
• se(?2) ? Var(?2) (3.3.2)
• var( ?1) ?2 ?X2i / n ?x2i (3.3.3)
• se(?1) ? Var(?1) (3.3.4)
• ? 2 ?u2i / (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 ?x2i
• var( ?1) is proportional to ?2 and ?X2i but
  inversely proportional to ?x2i 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 r2 A
measure of Goodness of fit

• Yi i i or
• Yi - i - i i or
• yi i i (Note )
• Squaring on both side and summing gt
• ? yi2 2 ?x2i ? 2i or
• TSS ESS RSS

61
3-5. The coefficient of determination r2 A
measure of Goodness of fit

• TSS ? yi2 Total Sum of Squares
• ESS ? Y i2 ?22 ?x2i
• Explained Sum of Squares
• RSS ? u2I Residual Sum of
• Squares
• ESS RSS
• 1 -------- -------- or
• TSS TSS
  
• RSS
  RSS
• 1 r2 ------- or
  r2 1 - -------
• TSS
  TSS

62
3-5. The coefficient of determination r2 A
measure of Goodness of fit

• r2 ESS/TSS
• is coefficient of determination, it measures the
  proportion or percentage of the total variation
  in Y explained by the regression
• Model
• 0 ? r2 ? 1
• r ?? r2 is sample correlation coefficient
• Some properties of r

63
3-5. The coefficient of determination r2 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)

64
Basic Econometrics

• Chapter 4
• THE NORMALITY ASSUMPTION
• Classical Normal Linear
• Regression Model
• (CNLRM)

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(u2i) ?2 Ass 4
• Cov(u i , u j ) E(u i , u j) 0
  (ij) 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

66
4-2.The normality assumption

• 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

67
4-2.The normality assumption

• Why the normality assumption?
• Under the normality assumption for ui , 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

69
4-3. Properties of OLS estimators under the
normality assumption

• 4. ?1 is normally distributed ?
• N(?1, ??12)
• And Z (?1- ?1)/ ??1 is ? N(0,1)
• 5. ?2 is normally distributed ?N(?2 ,??22)
• And Z (?2- ?2)/ ??2 is ? N(0,1)
• 6. (n-2) ?2/ ?2 is distributed as the
• ?2(n-2)

70
4-3. Properties of OLS estimators under the
normality assumption

• 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 ui is ? N(0, ?2 ) then Yi is ?
• NE(Yi) Var(Yi) N?1 ?2X 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) ?ui2/n (is biased
  estimator)
• (OLS estimator of ?2) ?ui2/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

74
Chapter 5 TWO-VARIABLE REGRESSIONInterval
Estimation and Hypothesis Testing

• 5-1. Statistical Prerequisites
• See 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

75
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 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

76
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-2. Interval estimation Some basic Ideas
• (1 - ?) is confidence coefficient,
• 0 lt ? lt 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

77
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-2. Interval estimation Some basic Ideas
• 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)

78
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-3. Confidence Intervals for Regression
• Coefficients
• Z (?2 - ?2)/se(?2) (?2 - ?2) ??x2i /?
  N(0,1)
• (5.3.1)
• We did not know ? and have to use ? instead,
  so
• t (?2 - ?2)/se(?2) (?2 - ?2) ??x2i /?
  t(n-2)
• (5.3.2)
• gt Interval for ?2
• Pr -t ?/2 ? t ? t ?/2 1- ?
  (5.3.3)

79
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-3. Confidence Intervals for Regression
• Coefficients
• Or confidence interval for ?2 is
• Pr ?2-t ?/2se(?2) ? ?2 ? ?2t ?/2se(?2)
  1- ?
• (5.3.5)
• Confidence Interval for ?1
• Pr ?1-t ?/2se(?1) ? ?1 ? ?1t ?/2se(?1)
  1- ?
• (5.3.7)

80
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-4. Confidence Intervals for ?2
• Pr (n-2)?2/ ?2?/2 ? ?2 ?(n-2)?2/ ?21- ?/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

81
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-5. Hypothesis Testing General Comments
• The stated hypothesis is known as the
• null hypothesis Ho
• The Ho is tested against and alternative
• hypothesis H1
• 5-6. Hypothesis Testing The confidence interval
  approach
• One-sided or one-tail Test
• H0 ?2 ? ? versus H1 ?2 gt ?

82
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• Two-sided or two-tail Test
• H0 ?2 ? versus H1 ?2 ?
• ?2 - t ?/2se(?2) ? ?2 ? ?2 t
  ?/2se(?2) values of ?2 lying in this interval
  are plausible under Ho with 100(1- ?)
  confidence.
• If ?2 lies in this region we do not reject Ho
  (the finding is statistically insignificant)
• If ?2 falls outside this interval, we reject Ho
  (the finding is statistically significant)

83
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 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 ?/2se(?2) ? ?2 ? ?2t ?/2se(?2) 1-
  ?
• (5.7.2)

84
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-7. Hypothesis Testing The test of
  significance approach
• Table 5-1 Decision Rule for t-test of
  significance

85
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-7. Hypothesis Testing The test of
  significance approach
• 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 gt

86
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-7. Hypothesis Testing The test of
  significance approach
• Table 5-2 A summary of the ?2 Test

87
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-8. Hypothesis Testing
• 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

88
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-8. Hypothesis Testing
• 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

89
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-9. Regression Analysis and Analysis
• of Variance
• TSS ESS RSS
• FMSS of ESS/MSS of RSS
• ?22 ?xi2/ ?2 (5.9.1)
• If ui are normally distributed H0 ?2 0 then F
  follows the F distribution with 1 and n-2 degree
  of freedom

90
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 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

91
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-9. Regression Analysis and Analysis of Variance
  
• Table 5-3. ANOVA for two-variable regression model

92
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-10. Application of Regression
• Analysis Problem of Prediction
• By the data of Table 3-2, we obtained the sample
  regression (3.6.2)
• Yi 24.4545 0.5091Xi , where
• Yi is the estimator of true E(Yi)
• There are two kinds of prediction as
• follows

93
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 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
  X0, 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 X0 (see pages
  138-139 for details)

94
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-11. Reporting the results of
• regression analysis
• An illustration
• YI 24.4545 0.5091Xi (5.1.1)
• Se (6.4138) (0.0357) r2 0.9621
• t (3.8128) (14.2405) df 8
• P (0.002517) (0.000000289) F1,22202.87

95
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-12. Evaluating the results of regression
  analysis
• Normality Test The Chi-Square (?2) Goodness of
  fit Test
• ?2N-1-k ? (Oi Ei)2/Ei
  (5.12.1)
• Oi is observed residuals (ui) in interval i
• Ei 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 ?2N-1-k is high (or ?2N-1-k is small)
  gt
• The Normality Hypothesis can not be rejected

96
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-12. Evaluating the results of regression
  analysis
• Normality Test The Chi-Square (?2) Goodness of
  fit Test
• H0 ui is normally distributed
• H1 ui is un-normally distributed
• Calculated-?2N-1-k ? (Oi Ei)2/Ei
  (5.12.1)
• Decision rule
• Calculated-?2N-1-k gt Critical-?2N-1-k then H0 can
• be rejected

97
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-12. Evaluating the results of regression
  analysis
• The Jarque-Bera (JB) test of normality
• This test first computes the Skewness (S)
• and Kurtosis (K) and uses the following
• statistic
• JB n S2/6 (K-3)2/24 (5.12.2)
• Mean xbar ?xi/n SD2 ?(xi-xbar)2/(n-1)
• Sm3/m2 3/2 Km4/m22 mk ?(xi-xbar)k/n

98
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-12. (Continued)
• Under the null hypothesis H0 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.

99
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-13. Summary and Conclusions
• 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.

100
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-13. Summary and Conclusions
• 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, H0 is not rejected, whereas if it lies
  outside this interval, H0 can be rejected

101
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-13. Summary and Conclusions
• 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 H0.
  It gives exact probability of obtaining the
  estimated test statistic under H0. If p-value is
  small, one can reject H0, but if it is large one
  may not reject H0.

102
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-13. Summary and Conclusions
• 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.

103
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 5-13. Summary and Conclusions
• 7. This chapter introduced the normality test to
  find out whether ui 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

104
Chapter 5 TWO-VARIABLE REGRESSION Interval
Estimation and Hypothesis Testing

• 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

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

• 6-1. Regression through the origin
• The SRF form of regression
• Yi b2X 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
• b2 SXiYi/SX2i (6.1.6) O
• b2 Sxiyi/Sx2i (3.1.6) I
• var(b2) s2/ SX2i (6.1.7) O
• var(b2) s2/ Sx2i (3.3.1) I
• s2 S(ui)2/(n-1) (6.1.8) O
• s2 S(ui)2/(n-2) (3.3.5) I
•  

108
Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS

• 6-1. Regression through the origin
• r2 for regression through-origin model
• Raw r2 (SXiYi)2 /SX2i SY2i (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

109
Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS

• 6-1. Regression through the origin
• Illustrative Examples
• 1) Capital Asset Pricing Model - CAPM (page 156)
• 2) Market Model (page 157)
• 3) The Characteristic Line of Portfolio Theory
  (page 159)

110
Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS

• 6-2. Scaling and units of measurement
• Let Yi b1 b2Xi u i (6.2.1)
• Define Yiw 1 Y i and Xiw 2 X i then
• b2 (w1/w2) b2 (6.2.15)
• b1 w1b1 (6.2.16)
• s2 w12s2 (6.2.17)
• Var(b1) w21 Var(b1) (6.2.18)
• Var(b2) (w1/w2)2 Var(b2) (6.2.19)
• r2xy r2xy (6.2.20)

111
Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS

• 6-2. Scaling and units of measurement
• From one scale of measurement, one can derive the
  results
• based on another scale of measurement. If w1 w2
  the
• intercept and standard error are both multiplied
  by w1. If
• w21 and scale of Y changed by w1, then all
  coefficients and
• standard errors are all multiplied by w1. If
  w11 and scale of
• X changed by w2, then only slope coefficient and
  its standard
• error are multiplied by 1/w2. 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

• The log-linear model
• Exponential regression model
• Yi b1Xi b2 e u i (6.4.1)
• By taking log to the base e of both side
• lnYi lnb1 b2lnXi ui , by setting lnb1 a gt
• lnYi a b2lnXi ui (6.4.3)
• (log-log, or double-log, or log-linear
  model)
• This can be estimated by OLS by letting
• Yi a b2Xi ui , where YilnYi, XilnXi
• b2 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 log-linear model
• The elasticity E of a variable Y with
• respect to variable X is defined as
• EdY/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 Y0 (1r) t (6.5.1)
• lnYt lnY0 t ln(1r) (6.5.2)
• lnYt b1 b2t , called constant growth model
  (6.5.5)
• where b1 lnY0 b2 ln(1r)
• lnYt b1 b2t ui (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)
• b2 (Relative change in regressand)/(Absolute
  change in regressor) (6.5.7)

116
6-5. Semi-log model Log-lin and Lin-log
Models

• Instantaneous Vs. compound rate of growth
• b2 is instantaneous rate of growth
• antilog(b2) 1 is compound rate of growth
• The linear trend model
• Yt b1 b2t ut (6.5.9)
• If b2 gt 0, there is an upward trend in Y
• If b2 lt 0, there is an downward trend in Y
• Note (i) Cannot compare the r2 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.

117
6-5. Semi-log model Log-lin and Lin-log
Models

• The lin-log model
• Yi b1 b2lnXi ui (6.5.11)
• b2 (Change in Y) / Change in lnX (Change in
  Y)/(Relative change in X) (?Y)/(?X/X)
  (6.5.12)
• or ?Y b2 (?X/X) (6.5.13)
• That is, the absolute change in Y equal to b2
  times the relative change in X. 

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

• The reciprocal model
• Yi b1 b2( 1/Xi ) ui (6.5.14)
• As X increases definitely, the term
• b2( 1/Xi ) approaches to zero and Yi
• approaches the limiting or asymptotic value b1
  (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)

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

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

• Yi ß1 ß2X2i ß3X3i u i
  (7.1.1)
• ß2 , ß3 are partial regression coefficients
• With the following assumptions
• Zero mean value of U i E(u iX2i,X3i) 0. ?i
  (7.1.2)
• No serial correlation Cov(ui,uj) 0, ?i j
  (7.1.3)
• Homoscedasticity Var(u i) ?2
  (7.1.4)
• Cov(ui,X2i) Cov(ui,X3i) 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(Yi X2i ,X3i) ß1 ß2X2i ß3X3i (7.2.1)
• (7.2.1) gives conditional mean or expected value
  of Y conditional upon the given or fixed value of
  the X2 and X3

124
7-3. The meaning of partial regression
coefficients

• Yi ß1 ß2X2i ß3X3 . ßsXs ui
• ßk measures the change in the mean value of Y per
  unit change in Xk, holding the rest explanatory
  variables constant. It gives the direct effect
  of unit change in Xk on the E(Yi), net of Xj (j
  k)
• How to control the true effect of a unit change
  in Xk 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
• Yi ß1 ß2X2i ß3X3 ui
• 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 R2
and the multiple coefficient of correlation R

• This section provides
• 1. Definition of R2 in the context of multiple
  regression like r2 in the case of two-variable
  regression
• 2. R ??R2 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(ßk) ?2/ ?x2k (1/(1-R2k)) (7.5.6)
• Where ßk is the partial regression coefficient
  of regressor Xk and R2k is the R2 in the
  regression of Xk 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. R2 and the Adjusted-R2

• R2 is a non-decreasing function of the number of
  explanatory variables. An additional X variable
  will not decrease R2
• R2 ESS/TSS 1- RSS/TSS 1-?u2I / ?y2i
  (7.8.1)
• This will make the wrong direction by adding more
  irrelevant variables into the regression and give
  an idea for an adjusted-R2 (R bar) by taking
  account of degree of freedom
• R2bar 1- ?u2I /(n-k) / ?y2i /(n-1) , or
  (7.8.2)
• R2bar 1- ?2 / S2Y (S2Y is sample variance of
  Y)
• K number of parameters including intercept
  term
• By substituting (7.8.1) into (7.8.2) we get
• R2bar 1- (1-R2) (n-1)/(n- k)
  (7.8.4)
• For k gt 1, R2bar lt R2 thus when number of X
  variables increases R2bar increases less than R2
  and R2bar can be negative

130
7-8. R2 and the Adjusted-R2

• R2 is a non-decreasing function of the number of
  explanatory variables. An additional X variable
  will not decrease R2
• R2 ESS/TSS 1- RSS/TSS 1-?u2I / ?y2i
  (7.8.1)
• This will make the wrong direction by adding more
  irrelevant variables into the regression and give
  an idea for an adjusted-R2 (R bar) by taking
  account of degree of freedom
• R2bar 1- ?u2I /(n-k) / ?y2i /(n-1) , or
  (7.8.2)
• R2bar 1- ?2 / S2Y (S2Y is sample variance of
  Y)
• K number of parameters including intercept
  term
• By substituting (7.8.1) into (7.8.2) we get
• R2bar 1- (1-R2) (n-1)/(n- k)
  (7.8.4)
• For k gt 1, R2bar lt R2 thus when number of X
  variables increases R2bar increases less than R2
  and R2bar can be negative

131
7-8. R2 and the Adjusted-R2

• Comparing Two R2 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-R2 Choosing
  the model that gives the highest R2bar 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
functionMore on functional form

• Yi ?1X?22i X?33ieUi (7.10.1)
• By log-transform of this model
• lnYi ln?1 ?2ln X2i ?3ln X3i Ui
  ?0 ?2ln X2i ?3ln X3i Ui
  (7.10.2)
• Data set is in Table 7.3
• Report of results is in page 216

134
7-11 Polynomial Regression Models

• Yi ?0 ?1 Xi ?2 X2i ?k Xki Ui
• (7.11.3)
• Example 7.4 Estimating the Total Cost Function
• Data set is in Table 7.4
• Empirical results is in page 221
• --------------------------------------------------
  ------------
• 7-12. Summary and Conclusions
• (page 221)

135
Basic Econometrics

• Chapter 8
• MULTIPLE REGRESSION ANALYSIS
• The Problem of Inference

136
Chapter 8MULTIPLE 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 8MULTIPLE 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.
• H0 ?2 0
• H1 ?2 ? 0
• If the computed t value gt critical t value at the
  chosen level of significance, we may reject the
  null hypothesis otherwise, we may not reject it

138
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-5. Testing the overall significance of a
  multiple
• regression The F-Test
• For Yi ?1 ?2X2i ?3X3i ........ ?kXki
  ui
• To test the hypothesis H0 ?2 ?3 .... ?k 0
  (all slope coefficients are simultaneously zero)
  versus H1 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 gt F critical F?(k-1,n-k), reject H0
• Otherwise you do not reject it

139
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 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 H0
• An important relationship between R2 and F
• F(ESS/(k-1))/(RSS/(n-k)) or
• R2/(k-1)
• F ---------------- (8.5.1)
• (1-R2) / (n-k)
• ( see prove on page 249)

140
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-5. Testing the overall significance of a
  multiple regression in terms of R2
• For Yi b1 b2X2i b3X3i ........ bkXki
  ui
• To test the hypothesis H0 b2 b3 ..... bk
  0 (all slope coefficients are simultaneously
  zero) versus H1 Not at all slope coefficients
  are simultaneously zero, compute
• F R2/(k-1) / (1-R2) / (n-k) (8.5.13) (k
  total number of parameters to be estimated
  including intercept)
• If F gt F critical F a, (k-1,n-k), reject H0

141
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-5. Testing the overall significance of a
  multiple regression
• Alternatively, if the p-value of F obtained from
  (8.5.13) is sufficiently low, one can reject H0
• 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 ui and the HO ?
  0, it can be shown that the following F ratio
  will follow the F distribution with respectively
  degree of freedom

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Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-5. Testing the overall significance of a
  multiple regression
• R2new - R2old / Df1
• F com ---------------------- (8.5.18)
• 1 - R2new / Df2
• Where Df1 number of new regressors
• Df2 n number of parameters in the
  new model
• R2new is standing for coefficient of
  determination of the new regression (by adding
  bX)
• R2old is standing for coefficient of
  determination of the old regression (before
  adding bX).

143
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-5. Testing the overall significance of a
  multiple regression
• Decision Rule
• If F com gt F a, Df1 , Df2 one can reject the Ho
  that b 0 and conclude that the addition of X to
  the model significantly increases ESS and hence
  the R2 value
• When to Add a New Variable? If t of
  coefficient of X gt 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 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• 8-6. Testing the equality of two regression
  coefficients
• Yi b1 b2X2i b3X3i b4X4i ui
  (8.6.1)
• Test the hypotheses
• H0 b3 b4 or b3 - b4 0
  (8.6.2)
• H1 b3 ? b4 or b3 - b4 ? 0
• Under the classical assumption it can be shown
• t (b3 - b4) (b3 - b4) / se(b3 - b4)
• 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(b3 - b4) ? var((b3) var( b4)
  2cov(b3, b4) (8.6.4)
• (see appendix)

145
Chapter 8MULTIPLE REGRESSION ANALYSIS The
Problem of Inference

• t (b3 - b4) / ? var((b3) var( b4)
  2cov(b3, b4) (8.6.5)
• Steps for testing
• 1. Estimate b3 and b4
• 2. Compute se(b3 - b4) through (8.6.4)
• 3. Obtain t- ratio from (8.6.5) with H0 b3 b4
• 4. If t-computed gt t-critical at designated level
  of significance for given df, then r