The document discusses the basics of econometrics and regression analysis. It provides 6 definitions for econometrics, highlighting that econometrics applies economic theory, mathematics, and statistical methods to empirical economic data. Regression analysis is introduced as studying the dependence of one variable on others, with the goal of estimating or predicting the average value of the dependent variable based on the independent variables. The methodology of regression involves specifying a theory, mathematical model, econometric model, obtaining data, estimation, hypothesis testing, forecasting, and using the model for policy purposes.
Linear Models and Econometrics Chapter 4 Econometrics.pptfaisal960287
This document provides an overview of econometrics and regression analysis. It defines econometrics as the application of statistical methods to economic data and theory. Regression analysis studies the dependence of one variable on one or more other variables to estimate or predict average values. The document outlines the methodology of econometrics, including specifying economic theories, models, collecting data, estimation, hypothesis testing, forecasting and policy applications. It also discusses key concepts like dependent and independent variables, and the nature and sources of data used in econometric analysis.
This document provides an introduction to econometrics and regression analysis. It defines econometrics as the application of statistical methods to economic data and models. The document outlines the methodology of econometrics, including specifying economic theories as mathematical and econometric models, obtaining data, estimating models, hypothesis testing, forecasting, and using models for policy purposes. It also discusses key concepts in regression analysis such as the dependent and explanatory variables, and distinguishes regression from correlation and causation.
This document provides an introduction to econometrics. It defines econometrics as the application of statistical and mathematical techniques to economic data in order to test economic theories and models. The document outlines the methodology of econometrics, including stating an economic theory, specifying mathematical and econometric models, obtaining data, estimating models, hypothesis testing, forecasting, and using models for policy purposes. It also discusses the structure of economic data such as time series, cross-sectional, and panel data. Finally, it covers key econometric concepts like the categories of variables and the differences between ratio and interval scales.
This document discusses the methodology of econometrics. It involves 8 steps: 1) stating an economic theory, 2) specifying a mathematical model, 3) specifying an econometric model, 4) obtaining data, 5) estimating parameters, 6) hypothesis testing, 7) forecasting, and 8) using the model for policy purposes. An example is provided to estimate a model that relates GDP to receipts based on data from 1990-2009. The model finds GDP increases with receipts, supporting the initial economic theory.
This document discusses the key concepts and applications of econometrics. It provides an overview of econometrics methodology, including statement of theory, specification of mathematical and statistical models, obtaining data, estimation of parameters, hypothesis testing, forecasting and using models for policy purposes. It also discusses regression analysis and the classical normal linear regression model, addressing topics like interval estimation, hypothesis testing, and issues like multicollinearity.
The document introduces econometrics and its methodology. Econometrics is defined as the quantitative analysis of economic phenomena based on concurrent development of economic theory and observation. It differs from economic theory, mathematics economics, and economic statistics by empirically testing economic theories. The methodology of econometrics involves: (1) stating an economic theory or hypothesis, (2) specifying its mathematical model, (3) specifying the econometric model, (4) obtaining data, (5) estimating the model, (6) testing hypotheses, (7) forecasting, and (8) using the model for policy purposes.
Econometrics is the application of statistical and mathematical methods to economic data in order to test economic theories and estimate relationships between economic variables. The methodology of econometrics involves stating an economic theory or hypothesis, specifying the theory mathematically and as an econometric model, obtaining data, estimating the model, testing hypotheses, making forecasts, and using the model for policy purposes. Regression analysis is a key tool in econometrics that relates a dependent variable to one or more independent variables, with an error term included to account for the inexact nature of economic relationships.
Linear Models and Econometrics Chapter 4 Econometrics.pptfaisal960287
This document provides an overview of econometrics and regression analysis. It defines econometrics as the application of statistical methods to economic data and theory. Regression analysis studies the dependence of one variable on one or more other variables to estimate or predict average values. The document outlines the methodology of econometrics, including specifying economic theories, models, collecting data, estimation, hypothesis testing, forecasting and policy applications. It also discusses key concepts like dependent and independent variables, and the nature and sources of data used in econometric analysis.
This document provides an introduction to econometrics and regression analysis. It defines econometrics as the application of statistical methods to economic data and models. The document outlines the methodology of econometrics, including specifying economic theories as mathematical and econometric models, obtaining data, estimating models, hypothesis testing, forecasting, and using models for policy purposes. It also discusses key concepts in regression analysis such as the dependent and explanatory variables, and distinguishes regression from correlation and causation.
This document provides an introduction to econometrics. It defines econometrics as the application of statistical and mathematical techniques to economic data in order to test economic theories and models. The document outlines the methodology of econometrics, including stating an economic theory, specifying mathematical and econometric models, obtaining data, estimating models, hypothesis testing, forecasting, and using models for policy purposes. It also discusses the structure of economic data such as time series, cross-sectional, and panel data. Finally, it covers key econometric concepts like the categories of variables and the differences between ratio and interval scales.
This document discusses the methodology of econometrics. It involves 8 steps: 1) stating an economic theory, 2) specifying a mathematical model, 3) specifying an econometric model, 4) obtaining data, 5) estimating parameters, 6) hypothesis testing, 7) forecasting, and 8) using the model for policy purposes. An example is provided to estimate a model that relates GDP to receipts based on data from 1990-2009. The model finds GDP increases with receipts, supporting the initial economic theory.
This document discusses the key concepts and applications of econometrics. It provides an overview of econometrics methodology, including statement of theory, specification of mathematical and statistical models, obtaining data, estimation of parameters, hypothesis testing, forecasting and using models for policy purposes. It also discusses regression analysis and the classical normal linear regression model, addressing topics like interval estimation, hypothesis testing, and issues like multicollinearity.
The document introduces econometrics and its methodology. Econometrics is defined as the quantitative analysis of economic phenomena based on concurrent development of economic theory and observation. It differs from economic theory, mathematics economics, and economic statistics by empirically testing economic theories. The methodology of econometrics involves: (1) stating an economic theory or hypothesis, (2) specifying its mathematical model, (3) specifying the econometric model, (4) obtaining data, (5) estimating the model, (6) testing hypotheses, (7) forecasting, and (8) using the model for policy purposes.
Econometrics is the application of statistical and mathematical methods to economic data in order to test economic theories and estimate relationships between economic variables. The methodology of econometrics involves stating an economic theory or hypothesis, specifying the theory mathematically and as an econometric model, obtaining data, estimating the model, testing hypotheses, making forecasts, and using the model for policy purposes. Regression analysis is a key tool in econometrics that relates a dependent variable to one or more independent variables, with an error term included to account for the inexact nature of economic relationships.
This document analyzes monetary policy and output-inflation volatility interaction in Nigeria using a bivariate GARCH-M model. It finds:
1) There is evidence of a short-run tradeoff relationship between output growth and inflation within and across monetary policy regimes in Nigeria from 1981-2007. However, no strong evidence of a long-run relationship was found.
2) Monetary policy regime changes affected the magnitude of policy effects on output and inflation. Policy had a stronger effect on output during direct control, while it has a larger impact on inflation now under indirect control.
3) Volatility of output and inflation became more persistent during the period of indirect monetary control compared to direct control.
This document discusses the methodology of econometrics. It begins by defining econometrics as applying economic theory, mathematics and statistical inference to analyze economic phenomena. It then outlines the typical steps in an econometric analysis: 1) stating an economic theory or hypothesis, 2) specifying a mathematical model, 3) specifying an econometric model, 4) collecting data, 5) estimating parameters, 6) hypothesis testing, 7) forecasting, and 8) using the model for policy purposes. As an example, it walks through Keynes' consumption theory using U.S. consumption and GDP data to estimate the marginal propensity to consume.
This document provides an introduction to econometrics. It defines econometrics as the application of statistical and mathematical tools to economic data and theory. The document outlines the methodology of econometrics, including specifying a theoretical model, collecting data, estimating model parameters, testing hypotheses, forecasting, and using models for policy purposes. It provides the example of estimating the parameters of Keynes' consumption function to illustrate these steps.
This document provides an introduction to the subject of basic econometrics. It defines econometrics as applying economic theory, mathematics, and statistical inference to analyze economic phenomena. The document outlines the steps in econometric research methodology, including specifying a mathematical model, collecting data, estimating parameters, hypothesis testing, and forecasting. It also discusses different types of economic data like cross-sectional data, time series data, and panel data.
This document provides an introduction to financial econometrics. It defines econometrics as the application of statistical techniques to economic and financial problems. The key aspects of econometrics discussed include establishing mathematical models of economic theories, collecting and testing data, and using models for forecasting, prediction, and policy purposes. The document also distinguishes between econometrics and financial econometrics, noting that the latter focuses more on financial data and variables like stock and index prices and returns. It outlines some common financial data characteristics and approaches to modeling financial data.
Relationship Between Monetary Policy And The Stock MarketCasey Hudson
The document discusses the relationship between monetary policy and stock markets. It introduces the Taylor Rule as the core theory, which states that monetary policy is mainly affected by inflation and output gaps. It also examines debates around including asset price volatility in the Taylor Rule equation and whether asset prices should influence monetary policy decisions.
The document analyzes the relationship between stock market returns in Pakistan (KSE 100 index) and five macroeconomic variables (inflation, GDP growth, exchange rate, money supply, interest rates) using monthly data from 1991 to 2008. It finds that in the long run, inflation, GDP growth, and exchange rate have a positive impact on stock returns, while money supply and interest rates have a negative impact. In the short run, the model shows it takes over four months for disequilibriums from the previous period to adjust. Inflation explains the most variance in forecast errors of stock returns among the macroeconomic variables.
This document provides an overview of using Stata for data management and reproducible research. It describes the Stata environment including the toolbar, command panel, review panel, results panel and variables panel. It demonstrates loading sample data using sysuse and viewing metadata about the data using describe and summary statistics using summarize. Reproducible research is facilitated by writing commands in a do-file that can be executed from the do-file editor.
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
The document provides an introduction to econometrics methodology. It outlines the typical steps as: 1) stating an economic theory or hypothesis, 2) specifying a mathematical model, 3) specifying an econometric model, 4) collecting data, 5) estimating model parameters, 6) hypothesis testing, 7) forecasting, and 8) using the model for policy purposes. It then gives an example applying these steps to Keynes' consumption theory and the relationship between consumption and income.
This document provides an introduction to the subject of basic econometrics. It defines econometrics as applying tools of economic theory, mathematics and statistical inference to analyze economic phenomena. Econometrics is concerned with empirically testing economic laws and considering economic theory, collecting data on economic variables, and applying statistical analysis to test hypotheses and draw conclusions to inform policymaking. Studying econometrics allows for empirical testing of economic theories and converting qualitative economic relationships into quantitative models that can be used to describe economic systems, test hypotheses, and forecast future economic activity.
1. The document discusses econometrics and the linear regression model. It outlines the methodology of econometric research which includes stating a theory or hypothesis, specifying a mathematical model, specifying an econometric model, obtaining data, estimating parameters, hypothesis testing, forecasting, and using the model for policy purposes.
2. It provides an example of specifying Keynes' consumption function as the mathematical model C= β1 + β2X where C is consumption and X is income. For the econometric model, an error term is added to allow for inexact relationships.
3. Assumptions of the classical linear regression model are discussed including the error term being uncorrelated with X, having a mean of zero,
Econometrics combines economic theory, mathematics, and statistical methods to analyze economic data and test hypotheses. It allows economists to quantify economic relationships and forecast future trends. Some key points covered in the document include:
- Econometrics uses statistical methods and economic theory to develop and test economic models and hypotheses about economic relationships using real-world data.
- Important founders of econometrics include Jan Tinbergen and Ragnar Frisch.
- Econometric models specify statistical relationships between economic variables based on economic theory and allow testing of theories and forecasting.
- Data sources include time series data, cross-sectional data, and panel data. Econometrics is useful for
This document reviews the literature on the relationship between monetary policy and economic growth. It begins with an overview of the evolution of theories from classical quantity theory to modern New Keynesian and New Consensus models. While theories differ in their assumptions around price flexibility and market clearing, most support some short-run effect of monetary policy on output. Empirically, studies find mixed results, with some supporting and others finding no relationship, depending on factors like country development and institutional quality. Overall, the literature suggests monetary policy can impact growth in developed markets with independent central banks, while the relationship is weaker in developing economies.
This document presents a novel approach for combining individual realized volatility measures to form new estimators of asset price variability. It analyzes 30 different realized measures estimated from high frequency IBM stock price data from 1996-2007. It finds that a simple equally-weighted average of the realized measures is not outperformed by any individual measure and that combining measures provides benefits by incorporating information from different estimators. Optimal linear and multiplicative combination estimators are estimated and none of the individual measures are found to encompass all the information in other measures, further supporting the use of combination estimators.
T. Dergiades, C. Milas, T. Panagioditis
IHU, Greece, University of Liverpool, UK , University of Macedonia, Greece, LSE, UK and RCEA, Italy.
November 2015. Open Seminar at Eesti Pank
This document provides an overview and introduction to the scope and method of economics. It discusses the following key points in 3 sentences:
Economics is the study of how individuals and societies make choices with scarce resources. The document outlines why economics is studied, including to learn a way of thinking, understand society and global affairs, and be an informed voter. It also describes the scope of economics in terms of microeconomics, macroeconomics, and diverse fields, as well as the method which involves theories, models, and empirical testing of economic concepts.
This document provides an overview and introduction to economics. It discusses the scope of economics, including microeconomics and macroeconomics. It also covers the reasons to study economics, such as to learn a way of thinking and to understand society and global affairs. Additionally, it summarizes the method of economics, including theories, models, and empirical testing. Economic policy goals like efficiency, equity, growth and stability are also briefly outlined.
This document discusses econometrics and economic data. It defines econometrics as the combined study of economic models, mathematical statistics, and economic data. There are three main steps in developing an econometric model: specification, estimation, and validation. Specification involves defining the economic and econometric models and assumptions. Estimation obtains numerical values for the model coefficients. Validation assesses if the estimates are acceptable based on economic theory and statistics. The document also describes different types of economic data that can be used: time series data, cross-sectional data, and panel data.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
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2) Monetary policy regime changes affected the magnitude of policy effects on output and inflation. Policy had a stronger effect on output during direct control, while it has a larger impact on inflation now under indirect control.
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The paper studies the implication of initial beliefs and associated confidence on the system’s
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2. It provides an example of specifying Keynes' consumption function as the mathematical model C= β1 + β2X where C is consumption and X is income. For the econometric model, an error term is added to allow for inexact relationships.
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- Important founders of econometrics include Jan Tinbergen and Ragnar Frisch.
- Econometric models specify statistical relationships between economic variables based on economic theory and allow testing of theories and forecasting.
- Data sources include time series data, cross-sectional data, and panel data. Econometrics is useful for
This document reviews the literature on the relationship between monetary policy and economic growth. It begins with an overview of the evolution of theories from classical quantity theory to modern New Keynesian and New Consensus models. While theories differ in their assumptions around price flexibility and market clearing, most support some short-run effect of monetary policy on output. Empirically, studies find mixed results, with some supporting and others finding no relationship, depending on factors like country development and institutional quality. Overall, the literature suggests monetary policy can impact growth in developed markets with independent central banks, while the relationship is weaker in developing economies.
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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)
3
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)
4
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)
5
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
6
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
8
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
9
11. Introduction
Methodology of Econometrics
(1) 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”
11
12. Introduction
Methodology of Econometrics
(2) Specification of the
mathematical model of the
theory
Y = ß1+ ß2X ; 0 < ß2< 1
Y= consumption expenditure
X= income
ß1 and ß2 are parameters; ß1 is
intercept, and ß2 is slope coefficients
12
13. Introduction
Methodology of Econometrics
(3) Specification of the
econometric model of the
theory
Y = ß1+ ß2X + u ; 0 < ß2< 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
13
15. Introduction
Methodology of Econometrics
(4) Obtaining Data
May 2004 Prof.VuThieu 15
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 16
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 17
18. Introduction
Methodology of Econometrics
(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
May 2004 Prof.VuThieu 18
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 19
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
May 2004 Prof.VuThieu 20
21. May 2004 Prof.VuThieu 21
Economic Theory
Mathematic Model Econometric Model Data Collection
Estimation
Hypothesis Testing
Forecasting
Application
in control or
policy
studies
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”)
May 2004 Prof.VuThieu 23
24. 1-1. Historical origin of the term “Regression”
• Galton’s 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”
May 2004 Prof.VuThieu 24
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)
May 2004 Prof.VuThieu 25
26. Dependent Variable Y; Explanatory Variable Xs
1. Y = Son’s Height; X = Father’s 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
May 2004 Prof.VuThieu 26
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)
May 2004 Prof.VuThieu 27
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”)
May 2004 Prof.VuThieu 28
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)
May 2004 Prof.VuThieu 29
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)
May 2004 Prof.VuThieu 30
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
May 2004 Prof.VuThieu 31
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)
May 2004 Prof.VuThieu 32
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
May 2004 Prof.VuThieu 33
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)
May 2004 Prof.VuThieu 35
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(YX)
• Conditional Mean
(or Expectation): E(YX=Xi )
May 2004 Prof.VuThieu 36
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)
May 2004 Prof.VuThieu 38
39. 2-2. The concepts of population
regression function (PRF)
• E(YX=Xi ) = 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(YX=Xi ) = f(Xi) = ß1 + ß2Xi
May 2004 Prof.VuThieu 39
40. 2-2. The concepts of population
regression function (PRF)
E(YX=Xi ) = 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
May 2004 Prof.VuThieu 40
41. 2-4. Stochastic Specification of PRF
•Ui = Y - E(YX=Xi ) or Yi = E(YX=Xi ) + Ui
•Ui = Stochastic disturbance or stochastic error
term. It is nonsystematic component
•Component E(YX=Xi ) 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(UiXi ) = 0
May 2004 Prof.VuThieu 41
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:
May 2004 Prof.VuThieu 42
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
May 2004 Prof.VuThieu 43
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
--------------------
May 2004 Prof.VuThieu 44
45. May 2004 Prof.VuThieu 45
SRF1
SRF2
Weekly Consumption
Expenditure (Y)
Weekly Income (X)
46. 2-6. The Sample Regression
Function (SRF)
•Fig.2-3: SRF1 and SRF 2
•Y^i = ^1 + ^2Xi (2.6.1)
•Y^i = estimator of E(YXi)
•^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 + u^i
or Yi= Y^i + u^i (2.6.3)
May 2004 Prof.VuThieu 46
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
May 2004 Prof.VuThieu 47
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
May 2004 Prof.VuThieu 48
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.
May 2004 Prof.VuThieu 49
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.
May 2004 Prof.VuThieu 50
52. 3-1. The method of ordinary least
square (OLS)
Least-square criterion:
Minimizing U^2
i = (Yi – Y^i) 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:
May 2004 Prof.VuThieu 52
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 Xi : That are u^iY^i = 0; u^iXi = 0
May 2004 Prof.VuThieu 53
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(uiXi)=0
Ass 4: Homoscedasticity or equal
variance of ui : Var (uiXi) = 2
[VS. Heteroscedasticity]
Ass 5: No autocorrelation between the
disturbances: Cov(ui,ujXi,Xj ) = 0
with i # j [VS. Correlation, + or - ]
May 2004 Prof.VuThieu 54
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
May 2004 Prof.VuThieu 55
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 / x2
i (3.3.1)
se(^2) = Var(^2) (3.3.2)
var( ^1) = 2 X2
i / n x2
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
May 2004 Prof.VuThieu 56
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 x2
i
+ var( ^1) is proportional to 2 and X2
i but inversely proportional to
x2
i and the sample size n.
+ cov ( ^1 , ^2) = - var( ^2) shows the independence between ^1 and
^2
May 2004 Prof.VuThieu 57
X
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
May 2004 Prof.VuThieu 58
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
May 2004 Prof.VuThieu 59
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 =>
yi
2 = 2 x2
i + 2
i ; or
TSS = ESS + RSS
May 2004 Prof.VuThieu 60
Y
Y
Ŷ
Ŷ Ŷ
Ŷ
Û
Û
Û
ŷ Û
2
β̂
2
β̂
61. 3-5. The coefficient of determination r2:
A measure of “Goodness of fit”
TSS = yi
2 = Total Sum of Squares
ESS = Y^ i
2 = ^2
2 x2
i =
Explained Sum of Squares
RSS = u^2
I = Residual Sum of
Squares
ESS RSS
1 = -------- + -------- ; or
TSS TSS
RSS RSS
1 = r2 + ------- ; or r2 = 1 - -------
TSS TSS
May 2004 Prof.VuThieu 61
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
May 2004 Prof.VuThieu 62
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)
May 2004 Prof.VuThieu 63
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(u2
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
May 2004 Prof.VuThieu 65
66. 4-2.The normality assumption
• Why the normality assumption?
(1) 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
(2) If the number of variables is not very large or
they are not strictly independent, their sum
may still be normally distributed
May 2004 Prof.VuThieu 66
67. 4-2.The normality assumption
• Why the normality assumption?
(3) Under the normality assumption for ui , the OLS
estimators ^1 and ^2 are also normally distributed
(4) The normal distribution is a comparatively simple
distribution involving only two parameters (mean and
variance)
May 2004 Prof.VuThieu 67
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
May 2004 Prof.VuThieu 68
69. 4-3. Properties of OLS estimators
under the normality assumption
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)
May 2004 Prof.VuThieu 69
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
N[E(Yi); Var(Yi)] = N[1+ 2X i ; 2]
May 2004 Prof.VuThieu 70
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
May 2004 Prof.VuThieu 71
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
May 2004 Prof.VuThieu 72
74. Chapter 5 TWO-VARIABLE REGRESSION:
Interval 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
May 2004 Prof.VuThieu 74
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
May 2004 Prof.VuThieu 75
76. Chapter 5 TWO-VARIABLE REGRESSION:
Interval Estimation and Hypothesis Testing
5-2. Interval estimation: Some basic Ideas
•(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
May 2004 Prof.VuThieu 76
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)
May 2004 Prof.VuThieu 77
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) x2
i / ~N(0,1)
(5.3.1)
We did not know and have to use ^ instead, so:
•t= (^2 - 2)/se(^2) = (^2 - 2) x2
i /^ ~ t(n-2)
(5.3.2)
• => Interval for 2
Pr [ -t /2 t t /2] = 1- (5.3.3)
May 2004 Prof.VuThieu 78
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 ^2+t /2se(^2)] = 1-
(5.3.5)
• Confidence Interval for 1
Pr [^1-t /2se(^1) 1 ^1+t /2se(^1)] = 1-
(5.3.7)
May 2004 Prof.VuThieu 79
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/ 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
May 2004 Prof.VuThieu 80
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 > *
May 2004 Prof.VuThieu 81
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)
May 2004 Prof.VuThieu 82
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 ^2+t /2se(^2)]= 1-
(5.7.2)
May 2004 Prof.VuThieu 83
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
May 2004 Prof.VuThieu 84
Type of
Hypothesis
H0 H1 Reject H0
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
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 =>
May 2004 Prof.VuThieu 85
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
May 2004 Prof.VuThieu 86
H0 H1 Reject H0 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
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
May 2004 Prof.VuThieu 87
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 ]
May 2004 Prof.VuThieu 88
89. Chapter 5 TWO-VARIABLE REGRESSION:
Interval Estimation and Hypothesis Testing
5-9. Regression Analysis and Analysis
of Variance
• TSS = ESS + RSS
• F=[MSS of ESS]/[MSS of RSS] =
= 2^2 xi
2/ ^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
May 2004 Prof.VuThieu 89
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
May 2004 Prof.VuThieu 90
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
May 2004 Prof.VuThieu 91
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 xi
2 1 2^2 xi
2
RSS (due to
residuals)
u^i
2 n-2 u^i
2 /(n-2)=^2
TSS y i
2 n-1
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) :
Y^i = 24.4545 + 0.5091Xi , where
Y^i is the estimator of true E(Yi)
• There are two kinds of prediction as
follows:
May 2004 Prof.VuThieu 92
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)
May 2004 Prof.VuThieu 93
94. Chapter 5 TWO-VARIABLE REGRESSION:
Interval Estimation and Hypothesis Testing
5-11. Reporting the results of
regression analysis
• An illustration:
Y^I= 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) F1,2=2202.87
May 2004 Prof.VuThieu 94
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
2
N-1-k = (Oi – Ei)2/Ei (5.12.1)
Oi is observed residuals (u^i) 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 2
N-1-k is high (or 2
N-1-k is small) =>
The Normality Hypothesis can not be rejected
May 2004 Prof.VuThieu 95
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-2
N-1-k = (Oi – Ei)2/Ei (5.12.1)
Decision rule:
Calculated-2
N-1-k > Critical-2
N-1-k then H0 can
be rejected
May 2004 Prof.VuThieu 96
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)
S=m3/m2
3/2 ; K=m4/m2
2 ; mk= (xi-xbar)k/n
May 2004 Prof.VuThieu 97
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
May 2004 Prof.VuThieu 98
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.
May 2004 Prof.VuThieu 99
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
May 2004 Prof.VuThieu 100
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
May 2004 Prof.VuThieu 101
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.
May 2004 Prof.VuThieu 102
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
May 2004 Prof.VuThieu 103
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.
May 2004 Prof.VuThieu 104
106. Chapter 6
EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS
6-1. Regression through the origin
The SRF form of regression:
Yi = ^2X i + u^ i (6.1.5)
Comparison two types of regressions:
* Regression through-origin model and
* Regression with intercept
May 2004 Prof.VuThieu 106
107. Chapter 6
EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS
6-1. Regression through the origin
Comparison two types of regressions:
^2 = XiYi/X2
i (6.1.6) O
^2 = xiyi/x2
i (3.1.6) I
var(^2) = 2/ X2
i (6.1.7) O
var(^2) = 2/ x2
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 107
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 = (XiYi)2 /X2
i Y2
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
May 2004 Prof.VuThieu 108
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)
May 2004 Prof.VuThieu 109
110. Chapter 6
EXTENSIONS OF THE TWO-VARIABLE LINEAR
REGRESSION MODELS
6-2. Scaling and units of measurement
Let Yi = ^1 + ^2Xi + u^ i (6.2.1)
Define Y*i=w 1 Y i and X*i=w 2 X i then:
*^2 = (w1/w2) ^2 (6.2.15)
*^1 = w1^1 (6.2.16)
*^2
= w1
2^2 (6.2.17)
Var(*^1) = w2
1 Var(^1) (6.2.18)
Var(*^2) = (w1/w2)2 Var(^2) (6.2.19)
r2
xy = r2
x*y* (6.2.20)
May 2004 Prof.VuThieu 110
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
w2=1 and scale of Y changed by w1, then all coefficients and
standard errors are all multiplied by w1. If w1=1 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)
May 2004 Prof.VuThieu 111
112. 6-3. Functional form of regression model
The log-linear model
Semi-log model
Reciprocal model
May 2004 Prof.VuThieu 112
113. 6-4. How to measure elasticity
The log-linear model
Exponential regression model:
Yi= 1Xi
2 e u
i (6.4.1)
By taking log to the base e of both side:
lnYi = ln1 +2lnXi + ui , by setting ln1 = =>
lnYi = +2lnXi + ui (6.4.3)
(log-log, or double-log, or log-linear model)
This can be estimated by OLS by letting
Y*i = +2X*i + ui , where Y*i=lnYi, X*i=lnXi ;
2 measures the ELASTICITY of Y respect to X, that is,
percentage change in Y for a given (small) percentage
change in X.
May 2004 Prof.VuThieu 113
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:
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)
May 2004 Prof.VuThieu 114
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 (1+r) t (6.5.1)
lnYt = lnY0 + t ln(1+r) (6.5.2)
lnYt = 1 + 2t , called constant growth model (6.5.5)
where 1 = lnY0 ; 2 = ln(1+r)
lnYt = 1 + 2t + 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)
2 = (Relative change in regressand)/(Absolute change in
regressor) (6.5.7)
May 2004 Prof.VuThieu 115
116. 6-5. Semi-log model:
Log-lin and Lin-log Models
Instantaneous Vs. compound rate of growth
2 is instantaneous rate of growth
antilog(2) – 1 is compound rate of growth
The linear trend model
Yt = 1 + 2t + ut (6.5.9)
If 2 > 0, there is an upward trend in Y
If 2 < 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.
May 2004 Prof.VuThieu 116
117. 6-5. Semi-log model:
Log-lin and Lin-log Models
The lin-log model:
Yi = 1 +2lnXi + ui (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.
May 2004 Prof.VuThieu 117
118. 6-6. Reciprocal Models:
Log-lin and Lin-log Models
The reciprocal model:
Yi = 1 + 2( 1/Xi ) + ui (6.5.14)
As X increases definitely, the term
2( 1/Xi ) approaches to zero and Yi
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
May 2004 Prof.VuThieu 118
119. 6-7. Summary of Functional Forms
Table 6.5 (page 178)
May 2004 Prof.VuThieu 119
Model Equation Slope =
dY/dX
Elasticity =
(dY/dX).(X/Y)
Linear Y = 1 + 2 X 2 2(X/Y) */
Log-linear
(log-log)
lnY = 1 + 2 lnX 2 (Y/X) 2
Log-lin lnY = 1 + 2 X 2 (Y) 2 X */
Lin-log Y = 1 + 2 lnX 2(1/X) 2 (1/Y) */
Reciprocal Y = 1 + 2 (1/X) - 2(1/X2) - 2 (1/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)
May 2004 Prof.VuThieu 120
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 Ui:: E(u i|X2i,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
May 2004 Prof.VuThieu 122
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
May 2004 Prof.VuThieu 123
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)
May 2004 Prof.VuThieu 124
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
May 2004 Prof.VuThieu 125
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(ß^k) = 2/ x2
k (1/(1-R2
k)) (7.5.6)
Where ß^k is the partial regression coefficient of
regressor Xk and R2
k is the R2 in the regression of
Xk on the rest regressors
May 2004 Prof.VuThieu 126
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
May 2004 Prof.VuThieu 127
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
May 2004 Prof.VuThieu 128
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-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-R2 (R bar) by taking account of degree of
freedom
• R2
bar= 1- [ u^2
I /(n-k)] / [y^2
i /(n-1) ] , or (7.8.2)
R2
bar= 1- ^2 / S2
Y (S2
Y is sample variance of Y)
K= number of parameters including intercept term
• By substituting (7.8.1) into (7.8.2) we get
R2
bar = 1- (1-R2) (n-1)/(n- k) (7.8.4)
• For k > 1, R2
bar < R2 thuswhen number of X variables increases
R2
bar increases less than R2 and R2
bar can be negative
May 2004 Prof.VuThieu 129
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-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-R2 (R bar) by taking account of degree of
freedom
• R2
bar= 1- [ u^2
I /(n-k)] / [y^2
i /(n-1) ] , or (7.8.2)
R2
bar= 1- ^2 / S2
Y (S2
Y is sample variance of Y)
K= number of parameters including intercept term
• By substituting (7.8.1) into (7.8.2) we get
R2
bar = 1- (1-R2) (n-1)/(n- k) (7.8.4)
• For k > 1, R2
bar < R2 thuswhen number of X variables increases
R2
bar increases less than R2 and R2
bar can be negative
May 2004 Prof.VuThieu 130
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 R2
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
May 2004 Prof.VuThieu 131
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)
May 2004 Prof.VuThieu 132
133. 7-10. Example 7.3: The Cobb-Douglas
Production function
More on functional form
• Yi = 1X2
2i X3
3ieU
i (7.10.1)
By log-transform of this model:
• lnYi = ln1 + 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
May 2004 Prof.VuThieu 133
134. 7-11 Polynomial Regression Models
• Yi = 0 + 1 Xi + 2 X2
i +…+ k Xk
i + 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)
May 2004 Prof.VuThieu 134
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
May 2004 Prof.VuThieu 136
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.
H0: 2 = 0
H1: 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 137
138. Chapter 8
MULTIPLE 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 > F critical = F(k-1,n-k), reject H0
Otherwise you do not reject it
May 2004 Prof.VuThieu 138
139. Chapter 8
MULTIPLE 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)
May 2004 Prof.VuThieu 139
140. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-5. Testing the overall significance of a multiple
regression in terms of R2
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 = [R2/(k-1)] / [(1-R2) / (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 H0
May 2004 Prof.VuThieu 140
141. Chapter 8
MULTIPLE 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
May 2004 Prof.VuThieu 141
142. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-5. Testing the overall significance of a multiple
regression
[R2
new - R2
old] / Df1
F com = ---------------------- (8.5.18)
[1- R2
new] / Df2
Where Df1 = number of new regressors
Df2 = n – number of parameters in the
new model
R2
new is standing for coefficient of determination of the
new regression (by adding X);
R2
old is standing for coefficient of determination of the old
regression (before adding X).
May 2004 Prof.VuThieu 142
143. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-5. Testing the overall significance of a multiple
regression
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 R2 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;
May 2004 Prof.VuThieu 143
144. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-6. Testing the equality of two regression coefficients
Yi = 1 + 2X2i + 3X3i + 4X4i + ui (8.6.1)
Test the hypotheses:
H0: 3 = 4 or 3 - 4 = 0 (8.6.2)
H1: 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 144
145. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
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 H0: 3 = 4
4. If t-computed > t-critical at designated level of
significance for given df, then reject H0. Otherwise do
not reject it. Alternatively, if the p-value of t statistic
from (8.6.5) is reasonable low, one can reject H0.
Example 8.2: The cubic cost function revisited
May 2004 Prof.VuThieu 145
146. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-7. Restricted least square: Testing linear
equality restrictions
Yi = 1X 2
2i X 3
3i eu
i (7.10.1) and (8.7.1)
Y = output
X2 = labor input
X3 = capital input
In the log-form:
lnYi = 0 + 2lnX2i + 3lnX3i + ui (8.7.2)
with the constant return to scale: 2 + 3 = 1
(8.7.3)
May 2004 Prof.VuThieu 146
147. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-7. Restricted least square: Testing linear
equality restrictions
How to test (8.7.3)
The t Test approach (unrestricted): test of the hypothesis
H0: 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(Yi /X2i)
= 0 + 3 ln(X3i /X2i) + ui (8.7.8). Where (Yi /X2i) is
output/labor ratio, and (X3i / X2i) is capital/labor ratio
May 2004 Prof.VuThieu 147
148. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-7. Restricted least square: Testing linear equality restrictions
u^2
UR=RSSUR of unrestricted regression (8.7.2)
and u^2
R = RSSR of restricted regression (8.7.7),
m = number of linear restrictions,
k = number of parameters in the unrestricted regression,
n = number of observations.
R2
UR and R2
R are R2 values obtained from unrestricted and
restricted regressions respectively. Then
F=[(RSSR – RSSUR)/m]/[RSSUR/(n-k)] =
= [(R2
UR – R2
R) / m] / [1 – R2
UR / (n-k)] (8.7.10)
follows F distribution with m, (n-k) df.
Decision rule: If F > F m, n-k , reject H0: 2 + 3 = 1
May 2004 Prof.VuThieu 148
149. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
8-7. Restricted least square: Testing linear equality restrictions
Note: R2
UR R2
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)
May 2004 Prof.VuThieu 149
150. Chapter 8
MULTIPLE REGRESSION ANALYSIS:
The Problem of Inference
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+ 2X t + U1t (t = 1,2,...,n1)
Post-Reconstruction period:
Y t = 1 + 2X t + U2t (t = 1,2,...,n2)
Where Y is personal savings, X is personal income, the us
are disturbance terms in the two equations and n1, n2 are
the number of observations in the two period
May 2004 Prof.VuThieu 150
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 n1, n2 and just estimate one savings function as:
Y t = l1 + l2X t + Ut (t = 1,2,...,n1, 1,....n2). (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
HO: i = i i Vs H1: i that i i
May 2004 Prof.VuThieu 151
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
u1t and u2t ~ N(0,s2), two error terms are normally
distributed with the same variance
u1t and u2t are independently distributed
Step 1: Estimate (8.8.3), get RSS, say, S1 with df =
(n1+n2 – k); k is number of parameters estimated )
Step 2: Estimate (8.8.1) and (8.8.2) individually and
get their RSS, say, S2 and S3 , with df = (n1 – k) and
(n2-k) respectively. Call S4 = S2+S3; with df = (n1+n2 –
2k)
Step 3: S5 = S1 – S4;
May 2004 Prof.VuThieu 152
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 = [S5 / k] / [S4 / (n1+n2 – 2k)] (8.8.4)
follows the F distribution with Df = (k, n1+n2 – 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 H0 if p-value low reasonably.
+ Apply for the data in Table 8.8
May 2004 Prof.VuThieu 153
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)
H0: Linear Model Y is a linear function of regressors, the Xs;
H1: Log-linear Model Y is a linear function of logs of regressors, the lnXs;
May 2004 Prof.VuThieu 154
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 Z1 = (lnYf – lnf)
Step 4: Regress Y on Xs and Z1. Reject H0 if the
coefficient of Z1 is statistically significant, by the
usual t - test
Step 5: Obtain Z2 = antilog of (lnf – Yf)
Step 6: Regress lnY on lnXs and Z2. Reject H1 if the
coefficient of Z2 is statistically significant, by the
usual t-test
May 2004 Prof.VuThieu 155
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 156