econometrics notes

Chapter1Introduction§1.1Econometrics?1.What is econometrics?•Literally interpreted,econometrics means economic measurement •Econometrics may be defined as the quantitative analysis of actual economic phenomena based on economic theory,mathematics,and statistical inference.2.Relationships with Mathematics,Statistics and Economics •There are several aspects of the quantitative approach to economics,and no single one of these aspects taken by itself,should be confounded with econometrics.•Thus,econometrics is–by no means the same as economic statistics.–Nor is it identical with what we call general economic theory,al-though a considerable portion of this theory has a definitely quanti-tative character.–Nor should econometrics be taken as synonomous with the applica-tion of mathematics to economics.•Experience has shown that each of these three viewpoints,that of statis-tics,economic theory,and mathematics,is a necessary,but not by itself a sufficient,condition for a real understanding of the quantitative relations in modern economic life.It is the unification of all three that is powerful.And it is this unification that constitutes econometrics.3.Econometrics’History1Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008•–1933,thefirst issue of Econometrica,Ragnar Frisch;–Nobel1969,Jan Tinbergen,Ragnar Frisch;–Nobel1980,Lawrence Robert Klein;–Nobel1989,Trygve Haavelmo;–Nobel1993,Douglass North,Robert Fogel;–Nobel2000,James Heckman,Daniel McFaddan;–Nobel2003,Robert Engle,Clive Granger.4.Types of Econometricstheoretical econometrics and applied econometrics.5.Methodology of Econometrics1.(a)Statement of theory or hypothesis;(b)Specification of the mathematical model of the theory;(c)Specification of the statistical,or econometric,model;(d)Obtaining the data;(e)Estimation of the parameters of the econometric model;(f)Hypothesis testing;(g)Forecasting or prediction;(h)Using the model for control or policy purposes.6.Software for Econometrics•–EViews:QMS Software–GAUSS:Aptech System,Inc.–LIMDEP:Econometric Software–MATLAB:The Math Works,Inc.–RATS:Estima–SHAZAM:Department of Economics,University of British ColumbiaEconometric Analysis Fall2008–SPSS:SPSS,Inc.–SAS:SAS,Inc.7.References•–Damodar N.Gujarati,Basic Econometrics(4th edition),Mc Graw Hill.–Jeffrey M.Wooldridge,Introductory Econometrics:a Modern Ap-proach(2nd edition,South-western College Publishing.–Robert S.Pindyck,Daniel L.Rubinfeld,Econometric Modelsand Economic Forecasts(4th edition),Mc Graw Hill.–Greene,W.,Econometric Analysis(5th edition),2003.–Hamilton,J.,Time Series Analysis,1994,Princeton UniversityPress.–Brockwell,P.J.and R.A.Davis,Time Series:Theory andMethods,2nd Edition,1991,Springer-Verlag.–Campbell,J.Y.,A.W.Lo,and A.C.MacKinlay,The Econo-metrics of Financial Markets,1997,Princeton University Press.–Tsay,R.,Analysis of Financial Time Series,2nd Edition,2005,Wiley-Interscience.Econometric Analysis Fall2008§1.2Example:Keynes’s Consumption Function1.Statement of Theory or HypothesisFrom Keynes’s(1936)General Theory of Employment,Interest and Money: The amount that the community spends on consumption depends(i) partly on the amount of its income,(ii)partly on other objective attendant circumstances,and(iii)partly on the subjective needs and the psychological propensities and habits of the individuals composing it.Men are disposed, as a rule and on the average,to increase their consumption as their income increases,but not by as much as the increase in their income.In short,Keynes postulated that the marginal propensity to con-sume(MPC),is greater than zero but less than1.2.Specification of the Mathematical Model of ConsumptionFor simplicity,a mathematical economist might suggest the following form of the Keynesian consumption function:Y=β1+β2X,0<β2<1where Y=consumption expenditure and X=income,and whereβ1and β2,known as the parameters of the model,are,respectively,the intercept and slope coefficients.3.Specification of the Econometric Model of ConsumptionIn addition to income,other variables affect consumption expenditure.For example,size of family,ages of the members in the family,family religion,etc., are likely to exert some influence on consumption.Modify the deterministic consumption function as follows:Y=β1+β2X+εwhereε,known as the disturbance,or error term,is a random(stochastic) variable that has well-defined probabilistic properties.4.Obtaining Data5.Estimation of the Econometric ModelRegression analysis is the main tool used to obtain the estimates.6.Hypothesis TestingAs noted earlier,Keynes expected the MPC to be positive but less than1. In our example we found the MPC to be about0.72.But we want to know that,is0.72statistically less than1?7.Forecasting or PredictionIf the chosen model does not refute the hypothesis or theory under consid-eration,we may use it to predict the future value(s)of the dependent,or forecast,variable Y on the basis of known or expected future value(s)ofthe explanatory,or predictor,variable X.e of the Model for Control or Policy PurposesAn estimated model may be used for control,or policy,purposes.By appro-priatefiscal and monetary policy mix,the government can manipulate the control variable X to produce the desired level of the target variable Y.§1.3Some applications of econometricsEconometrics can play the following roles economics:•Examine how well an economic theory can explain historical economic data(particularly the important stylized facts);•Test validity of economic theories and economic hypotheses;•Predict the future evolution of the economy.To appreciate the roles of modern econometrics in economic analysis,we now discuss a number of illustrative econometric examples in variousfields of economics andfinance.1.The Keynes Model,the Multiplier and Policy Recommendation The simplest Keynes model can be described by the system of equationsY t=C t+I t+G tC t=α+βY t+εtwhere Y t is aggregate income,C t is private consumption,I t is private invest-ment,G t is government spending,andεt is consumption shock.The param-etersαandβcan have appealing economic interpretations:αis survival level consumption,andβis the marginal propensity to consume.The multiplier of the income with respect to government spending is∂Y t1−βwhich depends on the marginal propensity to consumeβ.To assess the effect offiscal policies on the economy,it is important to know the magnitude ofβ.Economic theory can only suggest a positive qual-itative relationship between income and consumption.It never tells exactly whatβshould be for a given economy.Problems:—βdiffers from country to country;—βdepends on the stage of economic development in an economy.Methods:Econometrics offers a feasible way to estimateβfrom observed data.In fact,economic theory even does not suggest a specific functional form for the consumption function.Econometrics can provide a consistent estima-tion procedure for the unknown consumption function.This is called the nonparametric method.•Hardle,W.(1990),Applied Nonparametric Regression,Cambridge Uni-versity Press,Cambridge.•Pagan,A.and A.Ullah(1999),Nonparametric Econometrics,Cam-bridge University Press,Cambridge.2.The Production Function and the Hypothesis on Constant Re-turn to ScaleSuppose that for some industry,there are two inputs,labor L i and capital stock K i,and one output Y i,where i is the index forfirm i.The production function offirm i is a mapping from inputs(L i,K i)to output Y i:Y i=exp(εi)F(L i,K i)whereεi is a stochastic factor.An important economic hypothesis is that the production technology displays a constant return to scale(CRS),which is defined as follows:λF(L i,K i)=F(λL i,λK i)for allλ>0.CRS is a necessary condition for the existence of a long-run equilibrium of a competitive market economy,and testing CRS versus IRS has important policy implication,namely whether regulation is necessary.A conventional approach to testing CRS is to assume that the production function is a Cob-Douglas function:F(L i,K i)=A exp(εi)Lαi Kβi.Then CRS becomes a mathematical restriction on parameters(α,β):H0:α+β=1Ifα+β>1,the technology display IRS.Problems:T test procedure is not suitable for many cross-sectional economic data,which usually displays conditional heteroskedasticity.Methods:One need to use a robust,heteroskedasticity-consistent test procedure,origi-nally proposed in White(1980).•White,H.(1980),A Heteroskedasticity-Consistent Covariance matrix Es-timator and a Direct Test for Heterokedasticity,Econometrica48,817-838.3.Effect of Economic Reforms on a Transitional EconomyWe now consider an extended Cob-Dauglas production function(after taking a logarithmic operation)ln Y it=ln A it+αln L it+βln K it+γBonus it+δContract it+εitwhere i is the index forfirm i∈{1,2,···,N},and t is the index for year t∈{1,2,···,T};Bonus it is the proportion of bonus out of total wage bill, and Contract it is the proportion of workers who have signed afixed-term contract.This is an example of the so-called panel data model.To examine the effects of these incentive reforms,we consider the null statistical hypothesisH0:γ=δ=0.Problems:Conventional t-tests or F-tests cannot be used because there may well exist the other way of causation from Y it to Bonus it.This will cause correlation between the bonuses and the error termεit,rendering the OLS estimator inconsistent and invalidating the conventional t-tests or F-tests.Methods:Econometricians have developed an important estimation procedure called Instrumental Variables estimation,which can effectivelyfilter out the impact of the causation from output to bonus and obtain a consistent estimator for the bonus parameter.•Hsiao,C.(2003),Panel Data Analysis,2nd Edition,Cambridge University Press,Cambridge.4.The Efficient Market Hypothesis and Predictability of Finan-cial ReturnsLet Y t be the stock return in period t,and let I t−1={Y t−1,Y t−2,···}be the information set containing the history of past stock returns.The weak form of efficient market hypothesis(EMH)states that it is impossible to predictfuture stock returns using the history of past stock returns:E(Y t|I t−1)=E(Y t)An important implication of EMH is that mutual fund managers will have no informational advantage over layman investors.One simple way to test EMH is to consider the following autoregression AR(p)modelY t=α0+pi=1αi Y t−i+εt,t=1,2,···,Twhere p is a pre-selected number of lags,andεt is a random disturbance. EMH impliesH0:α1=α2=···=αp=0.Any nonzero coefficientαi,i=1,2,···,p,is evidence against EMH.Thus,to test EMH,one can test whether theαi are jointly zero.Problems:EMH may coexist with volatility clustering,which is one of the most im-portant empirical stylized facts offinancial markets.This implies that the standard F-test statistic cannot be used here.Similarly,the popular Box and Pierce’s(1970)portmanteau Q test also cannot be used.Methods:One has to use procedures that are robust to conditional heteroskedasticity.5.Volatility Clustering and ARCH ModelsIn economics,volatility is a key instrument for measuring uncertainty and risk infinance.This concept is important to investigate informationflows andvolatility spillover,financial contagions betweenfinancial markets,options pricing,and calculation of Value at Risk.Volatility can be measured by the conditional variance of asset return Y t given the information available at time t−1.An example of the condi-tional variance is the AutoRegressive Conditional Heteroskdeasticity(ARCH) model,originally proposed by Engle(1982).An ARCH(q)model assumes thatY t=µt+εt,εt=σ2t z t,{z t}∼i.i.d.N(0,1)µt=E(Y t|I t−1),σ2t=α+ q j=1βjε2t−j,α>0,β>0.This model can explain a well-known stylized fact infinancial markets-volatility clustering:a high volatility tends to be followed by another high volatility, and a small volatility tends to be followed by another small volatility.It can also explain the non-Gaussian heavy tail of asset returns.More sophisticated volatility models,such as Bollerslev’s(1986)Generalized ARCH or GARCH model,have been developed in time series econometrics.Problems:In practice,an important issue is how to estimate a volatility model.Methods:Although{z t}is not necessarily i.i.d.N(0,1)and we know this,the estimator obtained this way is still consistent for the true model parameters.However, the asymptotic variance of this estimator is larger than that of the MLE(i.e., when the true distribution of{z t}is known),due to the effect of not knowing the true distribution of{z t}.This method is called the quasi-MLE,or QMLE.•Bollerslev,T.(1986),Generalized Autoregressive Conditional Heteroskedastc-ity,Journal of Econometrics,31,307-327.•Engle,R.(1982),Autoregressive Conditional Hetersokedasticity with Es-timates of the Variance of United Kingdom Inflation,,Econometrica50, 987-2008.•White,H.(1994),Estimation,Inference and Specification Analysis,Cam-bridge University Press:Cambridge.Chapter2The Classical Multiple LinearRegression Model§2.1Regression•Historical origin of the term regression.The term regression was introduced by Francis Galton.•Example of father-son-height.•Example of60families’weekly income(x)and weekly expenditure(y).It is clear that each conditional mean E(y|x i)is a function of x i,that isE(y|x i)=f(x i),i=1,2,···,60We assume that PRF E(y|x i)is a linear function of x i,E(y|x i)=β1+β2x iεi=y i−E(y|x i)y i=E(y|x i)+εi=β1+β2x i+εi.§2.2The Linear Regression ModelThe generic form of the linear regression model isy=f(x1,x2,···,x K)+ε=x1β1+x2β2+···+x KβK+ε—y:dependent variables();—x1,x2,···,x K:independent variables();—In the literature the terms dependent variable and independent variable are described variously.A representative list is:∗Explained variable,Explanatory variable∗Dependent variable,Independent variable21Econometric Analysis Fall2008∗Predictand,Predictor∗Regressand,Regressor∗Response,Stimulus∗Endogenous,Exogenous∗Outcome,Covariate∗Controlled variable,Control variable—ε:random disturbance().—β1,β2,···,βK are unknown butfixed parameters()knownas the regression coefficients(intercept coefficient(),slope coeffi-cient()).—Population Regression Function(PRF)()of y on x1,x2,···,x K.Each observation in a sample(y i,x i1,x i2,···,x iK)i=1,2,···,n is gener-ated by an underlying process described byy i=x i1β1+x i2β2+···+x iKβK+εi(i=1,2,···,n)(1)—Sample Regression Function(SRF)()ˆy i=x i1ˆβ1+x i2ˆβ2+···+x iKˆβK(i=1,2,···,n)where(ˆβ1,ˆβ2,···,ˆβK)are some estimations()ofβ1,β2,···,βK.Our objective is—estimate the unknown parameters of the model;—use the data to study the validity of theoretical propositions();—use the model to predict()the variable y.Example1:Keynes’s Consumption FunctionExample2:Earning and EducationEconometric Analysis Fall2008§2.3Assumptions of the Classical Linear Regression Model •Linearity()•Full Rank()•Exogeneity of the independent variables()•Homoscedasticity and nonautocorrelation()•Exogenously generated data()•Normal distribution()Firstly,we give the following notations:—x k is a column vector(n×1),x k=(x1k,x2k,···,x nk) k=1,2,···,K.—X is a data matrix(n×K)X=x11x12···x1Kx21x22···x2K.........x n1x n2···x nK=(x1,x2,···,x K)—y is a column observation vector(n×1),y=(y1,y2,···,y n) —εis a column random vector(n×1)ε=(ε1,ε2,···,εn) —βis a column coefficient vector(K×1)β=(β1,β2,···,βK) Using above notations,the model(1)can be written asEconometric Analysis Fall2008y=x1β1+x2β2+···+x KβK+εor in form ofy=Xβ+ε.When referring to a single observation,we would writey i=x iβ+εiAssumption1:LinearityWe can write the multiple regression model asy=Xβ+εFor the regression to be a linear,it must be of the form of model(1) either in the original variables or after some suitable transformation.—loglinear model()—semilog model()—translog model()Assumption2:Full RankX has full column rank(),rank(X)=K.—columns of X are linearly independent();—size of observation()n is at least as large as K.—identification condition()—short rank()cause nearly multicollinearity().Assumption3:Conditional Mean()is ZeroThis conditional mean assumption stats that no observations on X conveyEconometric AnalysisFall 2008information about the expected value of the disturbance.(X)E (εi |X )=0E (ε|X )=E (ε1|X )=0E (ε2|X )=0...E (εn |X )=0=0—assume that the disturbances convey no information about each other.E [εi |ε1,···,εi −1,εi +1,···,εn ]=0—E (εi )=0,Cov (εi ,X )=0,i =1,2,···,nAssumption4:Homoscedasticity and NonautocorrelationHomoscedasticity :V ar [εi |X ]=σ2,for all i =1,···,n.—heteroscedasticity:(1)Profits of firms (size);(2)Household expendi-ture patterns (income).Nonautocorrelation :the assumption is that deviations of observations from their expected value are uncorrelated().Cov [εi ,εj |X ]=0,for alli =j.Econometric AnalysisFall 2008V ar [ε|X ]=E [εε |X ]=E (ε1ε1|X )E (ε1ε2|X )···E (ε1εn |X )E (ε2ε1|X )E (ε2ε2|X )···E (ε2εn |X )...E (εn ε1|X )E (εn ε2|X )···E (εn εn |X )=σ20···00σ2...0 (00)···σ2=σ2I—V ar [ε]=σ2I—Disturbances that meet the twin assumptions of homoscedasticity and nonautocorrelation are sometimes called spherical disturbances ().Assumption5:Exogenously Generated DataX may be fixed or random,but it is generated by a mechanism that is unrelated to ε.Assumption6:Normalityε|X ∼N [0,σ2I ]Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Econometric Analysis Fall2008Appendix for Chapter2Conditional DistributionSuppose X and Y are continuous random variable,and the joint probability density function is f(x,y).The conditional densities over y for each value of x isf(x,y)f Y|X(y|x)=31Econometric Analysis Fall2008•Decomposition of VarianceV ar[y]=V ar X[E[y|x]]+E X[V ar[y|x]].Econometric Analysis Fall2008§3.4Goodness of Fit and the Analysis of Variance()Consider a multiple regression modely=Xb+e=ˆy+e,For an individual observation,we havey i=ˆy i+e i=x i b+e i,y i−¯y=ˆy i−¯y+e i=(x i−¯x) b+e i.(2) where¯x=133Econometric Analysis Fall2008Econometric Analysis Fall2008 deviations from sample means,M0=I n−1n−1n−1n···−1n−1n,M0y=y1−¯yy2−¯y...y n−¯ywhere i=(1,1,···,1) ,then equation(2)could use the following formM0y=M0Xb+M0e.The total sum of squares isy M0y=b X M0Xb+e e.(4) Definition:Coefficient of determination()R2=SSRy M0y=1−e e35Econometric Analysis Fall2008Theorem3.6Change in R2When Adding Variable to a Regression Let R2Xz be the coefficient of determination in the regression of y on X and an additional variable z,let R2X be the same for the regression of y on X alone,and let r∗2yz be the partial correlation between y and z,controlling for X.ThenR2Xz=R2X+(1−R2X)r∗2yz.Two problems:•Thefirst concerns the number of the explanatory variable.•The second concerns the constant term in the model.The adjusted R2()which incorporates a penalty for these results is computed as follows:¯R2=1−e e/(n−K)(1−R2).n−KWhether¯R2rises or falls depends on actually computed as the contribution of the new variable to thefit of the regression more than offsets the correction for the loss of an additional degree of freedom.(¯R2)Econometric Analysis Fall2008Appendix-B for Chapter31SPSSSPSS Analyze→Correlatio()→Par-tial()→”Partial Correlation”→”Controlling for”→”Variables”→OK.2EViews(1)New File Work File()(2)Quick Empty Group,(3)ls c12...K.Chapter4Finite-Sample Properties of the LeastSquares Estimator§4.1Gauss Markov Theorem1.Assumptions of the Classical Linear Regression Model:A1.Linearity().A2.Full Rank().A3.Exogeneity of the Independent Variables().A4.Homoscedasticity and nonautocorrelation().A5.Exogenously generated data().A6.Normal Distribution().2.Unbiased Estimation():Writeb=(X X)−1X y=β+(X X)−1X ε.Take expectations,iterating over X,E(b|X)=β+E{(X X)−1X ε|X}=β+(X X)−1X E(ε|X)=β.Therefore,E(b)=E X{E(b|X)}=E X(β)=β.38Econometric Analysis Fall2008V ar(b|X)=E{(b−E(b|X))(b−E(b|X)) |X}=E{(b−β)(b−β) |X}=E{(X X)−1X ε·ε X(X X)−1|X}=(X X)−1X ·E{εε |X}·X(X X)−1=σ2(X X)−1.Use the decomposition of varianceV ar(b)=E X{Var(b|X)}+Var X{E(b|X)}=σ2E X{(X X)−1}.3.Minimum Variance():Since b has been proved that it is unbiased estimation,in the following we are going to show that b has minimum variance in the class of unbiased estimations(b).Let b∗be a another linear unbiased estimator ofβ,b∗=[(X X)−1X +C]ywhere C is a constant matrix.b∗=[(X X)−1X +C](Xβ+ε)=β+(X X)−1X ε+CXβ+Cε.Since b∗is a unbiased estimation ofβ,the following equation must be hold (b∗)CX=0.Therefore,b∗−β=(X X)−1X ε+Cε.Econometric Analysis Fall2008 V ar(b∗|X)=E{[(X X)−1X ε+Cε]·[(X X)−1X ε+Cε] |X}=σ2(X X)−1+σ2(X X)−1(CX) +σ2(CX)(X X)−1+σ2(CC )=σ2(X X)−1+σ2(CC ).Note that,in the above equation CC is a nonnegative definite matrix( ).Therefore,every quadratic form in V ar(b∗|X)is larger than the corresponding quadratic form in V ar(b|X),which implies a very important conclusion of the least squares.Theorem4.2Gauss-Markov TheoremIn the classical linear regression model with regressor matrix X,the least squares estimator b is the minimum variance linear unbiased estimator of β.For any vector of constants w,the minimum variance linear unbiased estimator of w βin the classical regression model is w b,where b is the least squares estimator.(X bβw,w βw b,b)Theorem4.3Gauss-Markov Theorem(Concluded)In the classical linear regression model,the least squares estimator b is the minimum variance linear unbiased estimator ofβwhether X is stochastic or nonstochastic,so long as the other assumptions of the model continue to hold.§4.2Sampling Distributions of the Least Squares1.Unbiased estimator ofσ2:We can use the following to estimate the variance of stochastic error term()ˆσ2=1n.Note thatE(e e|X)=E{tr(ε Mε)|X}=E{tr(Mεε )|X}=tr{M·E(εε |X)}=σ2·tr(M)=σ2·tr{I n−X(X X)−1X }=(n−K)σ2andE(e e)=E X{E(e e|X)}=(n−K)σ2.An unbiased estimator ofσ2iss2=e e41Theorem B.8Distribution of an Idempotent Quadratic Form in a Standard Normal VectorIf x∼N(0,I)and A is idempotent,then x Ax has a chi-squared distribution with degrees of freedom equal to the number of unit roots of A,which is equal to the rank of A.(x∼N(0,I)A x AxA A)Applying the above theorem,s2follows theχ2distribution in the following form(n−K)s2σ2=(εσ)∼χ2(n−K).3.A very important theorem about the independence of estimations:()Theorem4.4Independence of b and s2Ifεis normally distributed,then the least squares coefficient estimator b is statistically independent of the vector e and therefore,all functions of e, including s2.§4.3Testing Hypothesis of Regression Model1.Testing a hypothesis about a coefficient():Considering the significance of the k th explanatory variable,that is testing the following hypothesis:H0:βk=0←→H1:βk=0.With the normality assumption forε,the LS estimator b are themselves normally distributed with means and variance.(1)Whenσ2is given,the z-statisticz k=b k−βk V ar(b k)follows a standard normal distribution,where V ar(b k)=σ2(X X)−1kkis the variance of the b k.(2)Whenσ2is unknown,by using s2instead ofσ2,we can derive a statistic(n−K)s2V ar(b k)[(n−K)s243Notes:•Under the null hypothesis;If the value of trueβk is specified under the null hypothesis,the t ratio can readily be computed from the available sample,and therefore it can serve as a test statistic.•Given significance levelα=5%;The confidence-interval statements such as the following can be made:Prob −tα/2≤b k−β∗k V ar(b k)≤tα/2 =1−αwhereβ∗k is the ture value ofβk under H0and where−tα/2and tα/2are the values of t(the critical t values)obtained from the t table for(α/2)level of significance.•Acceptance region of t−test(t)Rearranging confidence-interval,we obtainProb β∗k−tα/2 V ar(b k) =1−α. which gives the interval in which b k will fall with(1−α)probability,given βk=β∗k.acceptance region(of the null hypothesis);rejection region(of H0)or the critical region;critical values.•Two sides test and one side test;•Simple test and composite test;•p−value.2.Confidence interval for parameters():Under the confidence level1−α,a confidence interval forβk isProb b k−tα/2 V ar(b k) =1−α.3.Testing the significance of the regression(): Considering the significance of the regression equation as a whole,that is testing the following hypothesisH0:β=0←→H1:β=0orH0:β1=β2=···=βK=0←→H1:Not all ofβ1,β2,···,βK equals zero.We have the following results:(1)SST=1σ2ni=1(y i−ˆy i)2∼χ2(n−K)(3)SSR=1SSE/(n−K)=R2/(K−1)45Appendix-A for Chapter4Finite Sample Properties of Statistics()1.Unbiasedness.An estimatorˆθis said to be an unbiased estimator ofθif the expected value ofˆθis equal to the trueθ,that isE(ˆθ)=θ.2.Minimum Variance.ˆθis said to be a minimum-variance estimator ofθif the variance ofˆθ1is 1smaller than or at most equal to the variance ofˆθ2,which is any other estimator ofθ.3.Best Unbiased,or Efficient,Estimator.Ifˆθ1andˆθ2are two unbiased estimators ofθ,and the variance ofˆθ1is smaller than or at most equal to the variance ofˆθ2,thenˆθ1is a minimum-variance unbiased,or best unbiased,or efficient,estimator.Appendix-B for Chapter4Large Sample Properties of Statistics()It often happens that an estimator does not satisfy one or more of the desir-able statistical properties in small samples.But as the sample size increases indefinitely,the estimator possesses several desirable statistical properties. These properties are known as the large-sample,or asymptotic,proper-ties.1.Asymptotic Unbiasedness.An estimatorˆθis said to be an asymptotically unbiased estimator ofθifE(ˆθ)=θ.limn→∞2.Consistency.ˆθis said to be a consistent estimator if it approaches the true valueθas the sample size gets larger and larger.3.Asymptotic EfficiencyLetˆθbe an estimator ofθ.The variance of the asymptotic distribution ofˆθis called the asymptotic variance ofˆθ.Ifˆθis consistent and its asymptotic variance is smaller than the asymptotic variance of all other consistent esti-mators ofθ.ˆθis called asymptotically efficient.4.Asymptotic Normality.An estimatorˆθis said to be asymptotically normally distributed if its sam-pling distribution tends to approach the normal distribution as the sample size n increases indefinitely.。