Lecture 0 Motivating Examples

外教英语讲座作文模板范文外教英语讲座作文模板。

Introduction:As English becomes increasingly important in our globalized world, many students are seeking opportunities to improve their English skills. One effective way to do this is through attending English lectures given by foreign teachers. In this article, we will discuss the benefits of attending such lectures and provide a template for writing a composition based on a lecture.Benefits of attending foreign English lectures:1. Exposure to native speakers: One of the greatest benefits of attending foreign English lectures is the opportunity to interact with native English speakers. This can greatly improve students' listening and speaking skills, as well as their understanding of different accents and dialects.2. Cultural exchange: Foreign English lectures often provide insights into the culture and customs of English-speaking countries. This can help students gain a better understanding of the context in which English is used, and improve their overall language proficiency.3. Motivation and inspiration: Foreign English lectures can be highly motivating and inspiring for students, as they provide a real-life example of fluent English speakers. This can encourage students to work harder and strive for better language skills.4. Practical language use: In foreign English lectures, students are exposed to authentic language use in a real-life context. This can help them improve their vocabulary, grammar, and overall language proficiency.Template for writing a composition based on a lecture:1. Introduction: Begin by introducing the topic of the lecture and the speaker. Provide some background information on the speaker and the context in which the lecture was given.2. Summary of the lecture: Summarize the main points and key ideas presented in the lecture. This may include a discussion of the topic, examples, and supporting evidence provided by the speaker.3. Personal reflection: Share your personal thoughts and reactions to the lecture. Discuss how the lecture has impacted your understanding of the topic and your English language skills.4. Conclusion: Conclude your composition by summarizing the key points made in the lecture and expressing your overall impression of the experience.Example composition:Title: A Foreign English Lecture on Environmental Sustainability。

Lecture Notes of Bus41202(Spring2010)Analysis of Financial Time SeriesRuey S.TsaySimple AR models:(Regression with lagged variables.) Motivating example:The growth rate of U.S.quarterly real GNP from1947to1991.Recall that the model discussed before isr t=0.005+0.35r t−1+0.18r t−2−0.14r t−3+a t,ˆσa=0.01.This is called an AR(3)model because the growth rate r t depends on the growth rates of the past three quarters.How do we specify this model from the data?Is it adequate for the data?What are the implications of the model?These are the questions we shall address in this lecture.Another example:U.S.monthly unemployment rate.AR(1)model:1.Form:r t=φ0+φ1r t−1+a t,whereφ0andφ1are real numbers,which are referred to as“parameters”(to be estimated from the data in an application).For example,r t=0.005+0.2r t−1+a t2.Stationarity:necessary and sufficient condition|φ1|<1.Why?3.Mean:E(r t)=φ01−φ1Timeg n p19501960197019801990−0.02−0.010.000.010.020.030.04U.S. quarterly real GNP growth rate: 1947.II to 1991.IFigure 1:U.S.quarterly growth rate of real GNP:1947-19914.Alternative representation:Let E (r t )=µbe the mean of r t so that µ=φ0/(1−φ1).Equivalently,φ0=µ(1−φ1).Plugging in the model,we have(r t −µ)=φ1(r t −1−µ)+a t .(1)This model also has two parameters (µand φ1).It explicitly uses the mean of the series.It is less commonly used in the literature,but is the model representation used in R.5.Variance:Var(r t )=σ2a 1−φ21.6.Autocorrelations:ρ1=φ1,ρ2=φ21,etc.In general,ρk =φk1and ACF ρk decays exponentially as k increases,Timey19501960197019801990200020104681U.S. monthly unemployment rate 1948.1 to 2010.2Figure 2:U.S.monthly unemployment rate (total civilian,16and older)from January 1948to February 20107.Forecast(minimum squared error):Suppose the forecast origin is n.For simplicity,we shall use the model representation in(1) and write x t=r t−µ.The model then becomes x t=φ1x t−1+a t. Note that forecast of r t is simply the forecast of x t plusµ.(a)1-step ahead forecast at time n:ˆx n(1)=φ1x n(b)1-step ahead forecast error:e n(1)=x n+1−ˆx n(1)=a n+1Thus,a n+1is the un-predictable part of x n+1.It is the shock at time n+1!(c)Variance of1-step ahead forecast error:Var[e n(1)]=Var(a n+1)=σ2a.(d)2-step ahead forecast:ˆx n(2)=φ1ˆx n(1)=φ21x n.(e)2-step ahead forecast error:e n(2)=x n+2−ˆx n(2)=a n+2+φ1a n+1(f)Variance of2-step ahead forecast error:Var[e n(2)]=(1+φ21)σ2awhich is greater than or equal to Var[e n(1)],implying that uncertainty in forecasts increases as the number of steps in-creases.(g)Behavior of multi-step ahead forecasts.In general,for the-step ahead forecast at n,we haveˆx n( )=φ 1x n,the forecast errore n( )=a n+ +φ1a n+ −1+···+φ −11a n+1,and the variance of forecast errorVar[e n( )]=(1+φ21+···+φ2( −1)1)σ2a.In particular,as →∞,ˆx n( )→0,i.e.,ˆr n( )→µ.This is called the mean-reversion of the AR(1)process.The variance of forecast error approachesVar[e n( )]=11−φ21σ2a=Var(r t).In practice,it means that for the long-term forecasts serialdependence is not important.The forecast is just the samplemean and the uncertainty is simply the uncertainty about theseries.8.A compact form:(1−φ1B)r t=φ0+a t.Half-life:A common way to quantify the speed of mean reversion is the half-life,which is defined as the number of periods needed sothat the magnitude of the forecast becomes half of that of the forecast origin.For an AR(1)model,this meanx n (k )=12x n .Thus,φk 1x n =12x n .Consequently,the half-life of the AR(1)modelis k =ln(0.5)ln(|φ1|).For example,if φ1=0.5,the k =1.If φ1=0.9,thenk ≈6.58.AR(2)model:1.Form:r t =φ0+φ1r t −1+φ2r t −2+a t ,or(1−φ1B −φ2B 2)r t =φ0+a t .2.Stationarity condition:(factor of polynomial)3.Characteristic equation:(1−φ1x −φ2x 2)=04.Mean:E (r t )=φ01−φ1−φ25.ACF:ρ0=1,ρ1=φ11−φ2,ρ =φ1ρ −1+φ2ρ −1,≥2.6.Stochastic business cycle:if φ21+4φ2<0,then r t shows char-acteristics of business cycles with average lengthk =2πcos −1[φ1/(2√−φ2)],where the cosine inverse is stated in radian.If we denote thesolutions of the polynomial as a ±bi ,where i =√−1,then wehaveφ1=2a andφ2=−(a2+b2)so thatk=2πcos−1(a/√a2+b2).In R or S-Plus,one can obtain √a2+b2using the commandMod.7.Forecasts:Similar to AR(1)modelsSimulation in R:Use the command arima.sim1.y1=arima.sim(model=list(ar=c(1.3,-.4)),1000)2.y2=arima.sim(model=list(ar=c(.8,-.7)),1000) Check the ACF and PACF of the above two simulated series. Building an AR model•Order specification1.Partial ACF:(naive,but effective)–Use consecutivefittings–See Text(p.40)for details–Key feature:PACF cuts offat lag p for an AR(p)model.–Illustration:See the PACF of the U.S.quarterly growthrate of GNP.2.Akaike information criterionAIC( )=ln(˜σ2 )+2 T ,for an AR( )model,where˜σ2 is the MLE of residual vari-ance.Find the AR order with minimum AIC for ∈[0,···,P].3.BIC criterion:BIC( )=ln(˜σ2 )+ ln(T) T.•Needs a constant term?Check the sample mean.•Estimation:least squares method or maximum likelihood method •Model checking:1.Residual:obs minus thefit,i.e.1-step ahead forecast errorsat each time point.2.Residual should be close to white noise if the model is ade-e Ljung-Box statistics of residuals,but degrees offreedom is m−g,where g is the number of AR coefficientsused in the model.Example:Analysis of U.S.GNP growth rate series.R demonstration:>setwd("C:/Users/rst/teaching/bs41202/sp2010")>library(fBasics)>da=read.table("dgnp82.dat")>x=da[,1]>par(mfcol=c(2,2))%put4plots on a page>plot(x,type=’l’)%first plot>plot(x[1:175],x[2:176])%2nd plot>plot(x[1:174],x[3:176])%3rd plot>acf(x,lag=12)%4th plot>pacf(x,lag.max=12)%Compute PACF(not shown in this handout)>Box.test(x,lag=10,type=’Ljung’)%Compute Q(10)statistics Box-Ljung testdata:xX-squared=43.2345,df=10,p-value=4.515e-06>m1=ar(x,method=’mle’)%Automatic AR fitting using AIC criterion. >m1Call:ar(x=x,method="mle")Coefficients:123%An AR(3)is specified.0.34800.1793-0.1423Order selected3sigma^2estimated as9.427e-05>names(m1)[1]"order""ar""var.pred""x.mean""aic" [6]"ed""order.max""partialacf""resid""method"[11]"series""frequency""call""asy.var.coef">plot(m1$resid,type=’l’)%Plot residuals of the fitted model(not shown) >Box.test(m1$resid,lag=10,type=’Ljung’)%Model checkingBox-Ljung testdata:m1$residX-squared=7.0808,df=10,p-value=0.7178>m2=arima(x,order=c(3,0,0))%Another approach with order given.>m2Call:arima(x=x,order=c(3,0,0))Coefficients:ar1ar2ar3intercept%Fitted model is0.34800.1793-0.14230.0077%y(t)=0.348y(t-1)+0.179y(t-2) s.e.0.07450.07780.07450.0012%-0.142y(t-3)+a(t),%where y(t)=x(t)-0.0077sigma^2estimated as9.427e-05:log likelihood=565.84,aic=-1121.68 >names(m2)[1]"coef""sigma2""var.coef""mask""loglik""aic" [7]"arma""residuals""call""series""code""n.cond"[13]"model">Box.test(m2$residuals,lag=10,type=’Ljung’)Box-Ljung testdata:m2$residualsX-squared=7.0169,df=10,p-value=0.7239>plot(m2$residuals,type=’l’)%Residual plot>tsdiag(m2)%obtain3plots of model checking(not shown in handout).>>p1=c(1,-m2$coef[1:3])%Further analysis of the fitted model.>roots=polyroot(p1)>roots[1] 1.590253+1.063882e+00i-1.920152-3.530887e-17i 1.590253-1.063882e+00i>Mod(roots)[1]1.9133081.9201521.913308>k=2*pi/acos(1.590253/1.913308)>k[1]10.65638>predict(m2,8)%Prediction1-step to8-step ahead.$predTime Series:Start=177End=184Frequency=1[1]0.0012362540.0045555190.0074549060.007958518[5]0.0081814420.0079368450.0078200460.007703826$seTime Series:Start=177End=184Frequency=1[1]0.0097093220.010*******.010*******.010688994[5]0.010*******.010*******.010*******.010696190Another example:Monthly U.S.unemployment rate from Jan-uary1948to February2010.Demonstration:in class,including the R scripts fore,foreplot, and backtest.>source(‘‘fore.R’’)>source(‘‘foreplot.R’’)>source(‘‘backtest.R’’)>x=read.table("m-unrate.txt",header=T)>dim(x)[1]7464>head(x)Mon Day Year VALUE1111948 3.42211948 3.83311948 4.04411948 3.95511948 3.56611948 3.6>y=x$VALUE>plot(y)>y1=ts(y,frequency=12,start=c(1948,1))>plot(y1)>acf(diff(y))>m1=ar(diff(y),method=’mle’)>m1Call:ar(x=diff(y),method="mle")Coefficients:123456780.01000.22540.15910.10000.1274-0.0034-0.03220.008191011120.0015-0.10540.0223-0.1208Order selected12sigma^2estimated as0.03849>length(y)[1]746>t.test(diff(y))One Sample t-testdata:diff(y)t=1.0673,df=744,p-value=0.2862alternative hypothesis:true mean is not equal to095percent confidence interval:-0.0070982570.024011008sample estimates:mean of x0.008456376>m1=arima(y,order=c(12,1,0))>m1Call:arima(x=y,order=c(12,1,0))Coefficients:ar1ar2ar3ar4ar5ar6ar7ar8ar90.01050.22590.15940.10040.1276-0.0032-0.03210.00840.0018 s.e.0.03650.03660.03730.03790.03810.03840.03840.03830.0381ar10ar11ar12-0.10510.0226-0.1206s.e.0.03760.03690.0369sigma^2estimated as0.03851:log likelihood=155.65,aic=-285.29>tsdiag(m1,gof=24)>m2=arima(y,order=c(2,1,1),seasonal=list(order=c(1,0,1),period=12))>m2Call:arima(x=y,order=c(2,1,1),seasonal=list(order=c(1,0,1),period=12)) Coefficients:ar1ar2ma1sar1sma10.57930.2467-0.58070.5624-0.8199s.e.0.06220.03950.05750.07330.0536sigma^2estimated as0.03715:log likelihood=167.76,aic=-323.53>tsdiag(m2,gof=24)>p2=fore(m2,y,740,6)Time Series:Start=741End=746Frequency=1[1]9.7831779.8410139.8648439.7974719.8199089.802541Time Series:Start=741End=746Frequency=1[1]0.19288600.27282860.36345510.45141650.53947100.6263706>foreplot(p2,y,740,720)>p1=fore(m1,y,740,6)Time Series:Start=741End=746Frequency=1[1]9.7354539.7550069.7512179.6593909.6393419.545857Time Series:Start=741End=746Frequency=1[1]0.19637340.27921720.36979860.46142790.55377860.6530188>foreplot(p1,y,740,720)>>backtest(m1,y,740,1)[1]"RMSE of out-of-sample forecasts"[1]0.181902>backtest(m2,y,740,1)[1]"RMSE of out-of-sample forecasts"[1]0.1752202>backtest(m1,y,720,1)[1]"RMSE of out-of-sample forecasts"[1]0.2149663>backtest(m2,y,720,1)[1]"RMSE of out-of-sample forecasts"[1]0.2048295Moving-average(MA)modelModel withfinite memory!Some daily stock returns have minor serial correlations and can be modeled as MA or AR models.MA(1)model•Form:r t=µ+a t−θa t−1•Stationarity:always stationary.•Mean(or expectation):E(r t)=µ•Variance:Var(r t)=(1+θ2)σ2a.•Autocovariance:g1:Cov(r t,r t−1)=−θσ2ag :Cov(r t,r t− )=0for >1.Thus,r t is not related to r t−2,r t−3,···.•ACF:ρ1=−θ,ρ =0for >1.1+θ2Finite memory!MA(1)models do not remember what happen two time periods ago.•Forecast(at origin t=n):1.1-step ahead:ˆr n(1)=µ−θa n.Why?Because at time n,a n is known,but a n+1is not.2.1-step ahead forecast error:e n(1)=a n+1with varianceσ2a.3.Multi-step ahead:ˆr n( )=µfor >1.Thus,for an MA(1)model,the multi-step ahead forecasts are just the mean of the series.Why?Because the model has memory of1time period.4.Multi-step ahead forecast error:e n( )=a n+ −θa n+ −15.Variance of multi-step ahead forecast error:(1+θ2)σ2a=variance of r t.•Invertibility:–Concept:r t is a proper linear combination of a t and the past observations{r t−1,r t−2,···}.–Why is it important?It provides a simple way to obtain the shock a t.For an invertible model,the dependence of r t on r t− con-verges to zero as increases.–Condition:|θ|<1.–Invertibility of MA models is the dual property of stationarityfor AR models.MA(2)model•Form:r t=µ+a t−θ1a t−1−θ2a t−2.orr t=µ+(1−θ1B−θ2B2)a t.•Stationary with E(r t)=µ.•Variance:Var(r t)=(1+θ21+θ22)σ2a.•ACF:ρ2=0,butρ =0for >2.•Forecasts go the the mean after2periods.Building an MA model•Specification:Use sample ACFSample ACFs are all small after lag q for an MA(q)series.(See test of ACF.)•Constant term?Check the sample mean.•Estimation:use maximum likelihood method–Conditional:Assume a t=0for t≤0–Exact:Treat a t with t≤0as parameters,estimate them toobtain the likelihood function.Exact method is preferred,but it is more computing intensive.•Model checking:examine residuals(to be white noise)•Forecast:use the residuals as{a t}(which can be obtained from the data andfitted parameters)to perform forecasts.Model form in R:R parameterizes the MA(q)model asr t=µ+a t+θ1a t−1+···+θq a t−q,instead of the usual minus sign inθ.Consequently,care needs to be exercised in writing down afitted MA parameter in R.For instance, an estimateˆθ1=−0.5of an MA(1)in R indicates the model is r t=a t−0.5a t−1.Example:Daily log return of the value-weighted indexR demonstration>setwd("C:/Users/rst/teaching/bs41202/sp2010")>library(fBasics)>da=read.table("d-ibmvwew6202.txt")>dim(da)[1]101944>vw=log(1+da[,3])*100%Compute percentage log returns of the vw index.>acf(vw,lag.max=10)%ACF plot is not shon in this handout.>m1=arima(vw,order=c(0,0,1))%fits an MA(1)model>m1Call:arima(x=vw,order=c(0,0,1))Coefficients:ma1intercept0.14650.0396%The model is vw(t)=0.0396+a(t)+0.1465a(t-1).s.e.0.00990.0100sigma^2estimated as0.7785:log likelihood=-13188.48,aic=26382.96>tsdiag(m1)>predict(m1,5)$predTime Series:Start=10195End=10199Frequency=1[1]0.050362980.039608870.039608870.039608870.03960887$seTime Series:Start=10195End=10199Frequency=1[1]0.88232900.89175230.89175230.89175230.8917523Mixed ARMA model:A compact form forflexible models. Focus on the ARMA(1,1)model for1.simplicityeful for understanding GARCH models in Ch.3for volatilitymodeling.ARMA(1,1)model•Form:(1−φ1B)r t=φ0+(1−θB)a t orr t=φ1r t−1+φ0+a t−θ1a t−1.A combination of an AR(1)on the LHS and an MA(1)on theRHS.•Stationarity:same as AR(1)•Invertibility:same as MA(1)•Mean:as AR(1),i.e.E(r t)=φ01−φ1•Variance:given in the text•ACF:Satisfiesρk=φ1ρk−1for k>1,butρ1=φ1−[θ1σ2a/Var(r t)]=φ1.This is the difference between AR(1)and ARMA(1,1)models.•PACF:does not cut offatfinite lags.Building an ARMA(1,1)model•Specification:use EACF or AIC•What is EACF?How to use it?[See text].•Estimation:cond.or exact likelihood method•Model checking:as before•Forecast:MA(1)affects the1-step ahead forecast.Others are similar to those of AR(1)models.Three model representations:•ARMA form:compact,useful in estimation and forecasting•AR representation:(by long division)r t=φ0+a t+π1r t−1+π2r t−2+···It tells how r t depends on its past values.•MA representation:(by long division)r t=µ+a t+ψ1a t−1+ψ2a t−2+···It tells how r t depends on the past shocks.For a stationary series,ψi converges to zero as i→∞.Thus,the effect of any shock is transitory.The MA representation is particularly useful in computing variances of forecast errors.For a -step ahead forecast,the forecast error ise n( )=a n+ +ψ1a n+ −1+···+ψ −1a n+1.The variance of forecast error isVar[e n( )]=(1+ψ21+···+ψ2 −1)σ2a.Unit-root NonstationarityRandom walk•Form p t=p t−1+a t•Unit root?It is an AR(1)model with coefficientφ1=1.•Nonstationary:Why?Because the variance of r t diverges to infinity as t increases.•Strong memory:sample ACF approaches1for anyfinite lag.•Repeated substitution showsp t=∞i=0a t−i=∞i=0ψi a t−iwhereψi=1for all i.Thus,ψi does not converge to zero.The effect of any shock is permanent.Random walk with drift•Form:p t=µ+p t−1+a t,µ=0.•Has a unit root•Nonstationary•Strong memory•Has a time trend with slopeµ.Why?differencing•1st difference:r t=p t−p t−1If p t is the log price,then the1st difference is simply the log return.Typically,1st difference means the“change”or“incre-ment”of the original series.•Seasonal difference:y t=p t−p t−s,where s is the periodicity,e.g.s=4for quarterly series and s=12for monthly series.If p t denotes quarterly earnings,then y t is the change in earning from the same quarter one year before.Meaning of the constant term in a model•MA model:mean•AR model:related to mean•1st differenced:time slope,etc.Practical implication infinancial time seriesExample:Monthly log returns of General Electrics(GE)from1926 to1999(74years)Sample mean:1.04%,std(ˆµ)=0.26Very significant!is about12.45%a year$1investment in the beginning of1926is worth•annual compounded payment:$5907•quarterly compounded payment:$8720•monthly compounded payment:$9570•Continuously compounded?Unit-root testLet p t be the log price of an asset.To test that p t is not predictable (i.e.has a unit root),two models are commonly employed:p t=φ1p t−1+e tp t=φ0+φ1p t−1+e t.The hypothesis of interest is H o:φ1=1vs H a:φ1<1.Dickey-Fuller test is the usual t-ratio of the OLS estimate ofφ1being 1.This is the DF unit-root test.The t-ratio,however,has a non-standard limiting distribution.Let∆p t=p t−p t−1.Then,the augmented DF unit-root test for an AR(p)model is based on∆p t=c t+βp t−1+p−1i=1φi∆p t−i+e t.The t-ratio of the OLS estimate ofβis the ADF unit-root test statis-tic.Again,the statistic has a non-standard limiting distribution. Example:Consider the log series of U.S.quaterly real GDP se-ries from1947.I to2009.IV.(data from Federal Reserve Bank of St. Louis).See q-gdpc96.txt on the course web.R demonstration>library(fUnitRoots)>help(UnitrootTests)%See the tests available>da=read.table(‘‘q-gdpc96.txt’’,header=T)>gdp=log(da[,4])>adfTest(gdp,lag=4,type=c("c"))#Assume an AR(4)model for the series.Title:Augmented Dickey-Fuller TestTest Results:PARAMETER:Lag Order:4STATISTIC:Dickey-Fuller:-1.7433P VALUE:0.4076#cannot reject the null hypothesis of a unit root.***A more careful analysis>x=diff(gdp)>ord=ar(x)#identify an AR model for the differenced series.>ordCall:ar(x=x)Coefficients:1230.34290.1238-0.1226Order selected3sigma^2estimated as8.522e-05>#An AR(3)for the differenced data is confirmed.#Our previous analysis is justified.Discussion:The command arima on R.1.dealing with the constant term.If there is any differencing,noconstant is used.2.fixing some e subcommand fixed in arima.S-Plus demonstration>da=read.table("q-gdp05.txt")>dim(da)[1]2364>plot(da[,4],type=’l’)>module(finmetrics)>gdp=log(da[,4])>plot(gdp,type=’l’)>x=diff(gdp)%take the first difference>ord=ar(x)>ord$order:[1]4>adf=unitroot(gdp,trend=’c’,lags=5,method=’adf’)>adfTest for Unit Root:Augmented DF TestNull Hypothesis:there is a unit rootType of Test:t-testTest Statistic:-1.12P-value:0.7083Coefficients:lag1lag2lag3lag4lag5constant-0.00120.29540.1358-0.0864-0.11080.0168Degrees of freedom:231total;225residualResidual standard error:0.009283S-Plus demonstration>module(finmetrics)>gnp=scan(file=’dgnp82.dat’)>plot(gnp,type=’l’)>acf(gnp,lag.max=12)Call:acf(x=gnp,lag.max=12)%Plot not shown in the handout. Autocorrelation matrix:lag gnp10 1.0000210.3769320.2539430.012554-0.085965-0.107176-0.057587-0.018298-0.0772109-0.070211100.01041211-0.02301312-0.0967>acf(gnp,lag.max=12,type=’partial’)%Compute PACFCall:acf(x=gnp,lag.max=12,type="partial")Partial Correlation matrix:lag gnp110.3769220.130433-0.142144-0.098855-0.0199660.0325770.012088-0.110699-0.041510100.09811111-0.03701212-0.1533>ord=ar(gnp,order.max=10)%Perform order selection via AIC>ord$aic[1]27.5691310 2.6081086 1.58955500.00000000.2734771 2.2034466[7] 4.0171066 5.9916210 5.82648337.52300257.8223499>ord$order[1]3>m1=arima.mle(gnp,model=list(order=c(3,0,0)))%This fit misses the mean.>summary(m1)Call:arima.mle(x=gnp,model=list(order=c(3,0,0)))Method:Maximum Likelihood with likelihood conditional on3observationsARIMA order:300Value Std.Error t-value%No intercept because the program assumes it is zero. ar(1)0.454200.07597 5.9780ar(2)0.266800.08095 3.2960ar(3)-0.038170.07597-0.5024Variance-Covariance Matrix:ar(1)ar(2)ar(3)ar(1)0.005771926-0.002566306-0.001441892ar(2)-0.0025663060.006552753-0.002566306ar(3)-0.001441892-0.0025663060.005771926Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-1085.0397>x=gnp-mean(gnp)%Remove sample mean.>m1=arima.mle(x,model=list(order=c(3,0,0)))>summary(m1)Call:arima.mle(x=x,model=list(order=c(3,0,0)))Method:Maximum Likelihood with likelihood conditional on3observationsARIMA order:300Value Std.Error t-valuear(1)0.35090.07523 4.664%Fitted model isar(2)0.18090.07863 2.301%x(t)=0.351x(t-1)+0.181x(t-2)-0.144x(t-3)+a(t). ar(3)-0.14430.07523-1.919Variance-Covariance Matrix:ar(1)ar(2)ar(3)ar(1)0.0056599161-0.001877448-0.0007529176ar(2)-0.00187744800.006182526-0.0018774480ar(3)-0.0007529176-0.0018774480.0056599161Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-1104.1574>names(m1)[1]"model""var.coef""method""series""aic"[6]"loglik""sigma2""ed""n.cond""converged"[11]"conv.type""call">names(m1$model)[1]"order""ar""ndiff">m1$model$ar[1]0.35091070.1809056-0.1443412>>arima.diag(m1)%Model checking,plots not shown.>p1=c(1,-m1$model$ar)%Further analysis of the fitted model. >roots=polyroot(p1)>roots[1] 1.582837+1.057071e+000i-1.912355-6.609277e-017i[3] 1.582837-1.057071e+000i>Mod(roots)[1]1.9033591.9123551.903359>k=2*pi/acos(1.582837/1.903359)>k[1]10.67098>>arima.forecast(x,m1$model,8)%prediction$mean:[1]-0.00651901645-0.00317061250-0.000236329850.00028445018 [5]0.000514713150.000266189120.000145465240.00002490612 $std.err:[1]0.0097793140.010*******.010*******.010*******.010785783 [6]0.010*******.010*******.010792592S-Plus demonstration>vw=d6202[,3]%Identify the vw-index returns.>lnvw=log(1+vw)%compute log returns.>acf(lnvw,lag.max=10)%ACF plot i snot shown in this handout. Call:acf(x=lnvw,lag.max=10)Autocorrelation matrix:lag lnvw10 1.0000210.140232-0.012043-0.0027540.0029650.007576-0.014987-0.006698-0.0034109-0.00851110-0.0074>length(lnvw)[1]10194>x1=rep(1,10194)%Create a constant to handle non-zero mean>m1=arima.mle(lnvw,xreg=x1,model=list(order=c(0,0,1)))>summary(m1)Call:arima.mle(x=lnvw,model=list(order=c(0,0,1)),xreg=x1) Method:Maximum Likelihood with likelihood conditional on0observationsARIMA order:001Value Std.Error t-valuema(1)-0.14650000.009797-14.96x10.0003962NA NA%Model is vw=.000396+a(t)+0.1465a(t-1) Variance-Covariance Matrix:ma(1)ma(1)0.00009599039Estimated innovations variance:0.0001Optimizer has convergedConvergence Type:relative function convergenceAIC:-67509.2476>arima.diag(m1)%Plots not shown in this handout.>arima.forecast(lnvw,model=m1$model,6)$mean:[1]0.00015816540.00000000000.00000000000.00000000000.0000000000[6]0.0000000000%Need to add the constant0.000396to the forecast. $std.err:[1]0.0088300560.0089243610.0089243610.0089243610.008924361[6]0.008924361。

难忘的讲座英语作文600字An Unforgettable Lecture.Nestled amidst the hallowed halls of academia, I had the privilege of attending an extraordinary lecture that left an indelible mark on my intellectual tapestry. The esteemed speaker, Professor Emily Carter, embarked on a mesmerizing journey through the labyrinthine complexities of artificial intelligence (AI).From the outset, Professor Carter captivated her audience with her erudite yet engaging delivery. Her articulation was impeccable, her voice lilting with an infectious enthusiasm that ignited a spark of curiosity within each of us. She deftly navigated the intricacies of AI, elucidating its potential to revolutionize countless aspects of human endeavor.As she delved deeper into the subject matter, Professor Carter deftly wove together theoretical concepts with real-world examples. She demonstrated how AI is already transforming industries ranging from healthcare to finance, empowering us to solve heretofore intractable problems. She also addressed the ethical considerations surrounding AI, urging us to approach its development with a responsible and forward-looking mindset.Particularly thought-provoking was her exploration of the potential impact of AI on the workforce. Professor Carter acknowledged that while AI could automate certain tasks, it also had the potential to create new jobs and industries. She emphasized the importance of investing in education and training to ensure that workers are equipped with the skills necessary to thrive in the rapidly evolving landscape.Beyond its intellectual merits, the lecture also left a profound emotional impact on me. Professor Carter's passion for her subject matter was palpable, inspiring me to delve deeper into the field of AI. She challenged us to think critically about the implications of this transformative technology and to embrace its potential while mitigatingits risks.In the days and weeks that followed, I found myself revisiting Professor Carter's insights and reflecting on their profound implications. The lecture ignited within me an unyielding desire to learn more about AI and to contribute to its responsible development. I am eternally grateful for the opportunity to have attended such an unforgettable educational experience.Years later, the memory of Professor Carter's lecture continues to linger in the recesses of my mind, shaping my perspective on AI and motivating me to pursue a career in this field. It was an encounter that not only expanded my knowledge but also kindled a passion that has become an integral part of my life.。

【导语】托福⼝语共分为六⼤题型，其中前两个部分为独⽴⼝语任务，后四个部分为综合⼝语任务。

以下是⽆忧考整理的托福⼝语的六⼤题型解析，欢迎阅读！1.托福⼝语的六⼤题型解析 1、Task1 题⽬为⾮限制性问题，考试内容⼀定和学校或者⽇常⽣活有关，考⽣在答题时⼀定要注意回答thereason&detail，对这些还要有相对应的细节来⽀持。

答题时间只有45s。

考⽣在平时复习的时候就要注意这⽅⾯内容的积累和运⽤，考试的时候才能够使⽤起来得⼼应⼿。

2、Task2 ⼀般为⼆选⼀的题型，形式如问A和B哪个好?为什么好?考⽣要通过这两个⽅⾯来回答问题，答题时间为45s。

在托福⼝语中前⾯两题的题⽬都是⼀开始就出现的，电脑也会读，考⽣只要看到题⽬就可以进⾏构思了。

在准备时间过完之后就可以开始答题，不必等到电脑读完提⽰准备时间时才去答题，为⾃⼰多争取⼀些思考的时间，对答题也是很不错的。

3、Task3 托福⼝语从第三题开始有听⼒和阅读两个部分，主要是以校园事件为话题，语⾔并不学术，通常是两个对话，有习语出现。

如果你听到对话中有70%都是重复其中⼀个⼈的观点，那么这个观点就是该对话的⼀致观点，另外的30%就是不⼀致的观点。

4、Task4 这⼀类题型为学术类的⽂章。

听⼒和阅读之间存在⼀定的关系，⼀般不是承接关系就是驳斥关系。

⼤部分情况下就是让你⽤听⼒讲座中的⼀个例⼦来解释，即细节解释，这个时候就是体现你听⼒能⼒的时候。

第3&4题都是有听⼒和阅读两部分的内容，其中阅读部分第3题较短，第4题较长，都是要求在45s读完，⽂章长度⼀般为75~120词(⼀般是5~7⾏)。

考⽣要注意的是第4题的阅读部分⽐较重要，第3题还好，因为第4题要知道听⼒内容和阅读内容之间有什么观点，这是答题需要答的。

这两题的答题时间都是1分钟。

提醒⼤家，如果考⽣在回答时要⽤到Take下的Notes⾥的词，做同义替换，并且在答题时绝对不能出现reading passage ⾥⾯的词。

人教版高中英语必修第一册一词一句Welcome Unit1. exchangeThe cultural exchange program allowed students to immerse themselves in a new environment and gain a deeper understanding of different traditions.文化交流项目让学生能够沉浸在新的环境中，更深入地了解不同的传统。

2. lectureThe professor delivered an inspiring lecture on environmental sustainability, motivating the students to take action for a greener future.教授就环境可持续性发表了一场激励人心的讲座，激励学生为绿色未来采取行动。

3. registrationThe online registration process for the conference was smooth and efficient, allowing participants to sign up with ease.会议的在线注册流程顺利高效，使参与者可以轻松报名。

4. registerStudents need to register for their courses before the semester begins to secure their spots in the classes.学生需要在学期开始前注册他们的课程，以确保能够上好自己想要的课程。

5. sexGender equality aims to create a society where individuals are respected regardless of their sex.性别平等旨在创造一个社会，无论性别，个体都受到尊重。

老师工作十分认真的事例初一英语作文全文共3篇示例，供读者参考篇1A Teacher's Dedication: Mrs. Roberts' Tireless EffortsAs an eighth grader, I've had my fair share of teachers over the years – some good, some not so good. However, one teacher stands out from the rest as being extraordinarily dedicated to her job and her students – Mrs. Roberts, my English teacher this year. Through her tireless work ethic, creative lessons, and genuine care for each one of us, Mrs. Roberts has made a lasting impact. I'll never forget the great lengths she goes to in order to ensure we are learning.To start, Mrs. Roberts gets to school extremely early each morning, often arriving before 6am. I know this because a few times I've gotten to school early for basketball practice and her car is already there in the parking lot. She uses those early hours to meticulously plan out her lessons for the day, making sure everything is prepared just right. Her room is always decorated with educational posters, examples of excellent student work,and boxes of books she's purchased with her own money. You can tell a lot of thought goes into the setup.Once the school day begins, Mrs. Roberts teaches with an energy and passion unlike any teacher I've ever had. She doesn't just lecture at us; she gets us actively involved through group activities, debates, acting out literary scenes, and creative games and projects. Her lessons feel more like fun than work. For example, when we studied poetry, she had us go outside and utilize nature for inspiration by writing poems about anything we observed. This Freedom allowed us to express ourselves in a low-pressure environment. To wrap it up, she posted our poems around the room, with positive feedback written beside them.Mrs. Roberts' creativity and enthusiasm are contagious. You can't help but get engaged and stay focused in her class. I've learned more this year about writing, analyzing literature, and becoming a better communicator than I did in all my previous years combined. It's clear how knowledgeable and skilled she is at teaching English and inspiring a love of language in her students.Her dedication extends far beyond just class time too. Mrs. Roberts is constantly giving up her lunch period and staying late after school to provide extra help for any students who need it.Multiple days a week, you'll find her room filled with kids getting one-on-one tutoring, feedback on their essays, or just extra practice. She is extremely patient and never makes anyone feel inadequate for asking questions. To Mrs. Roberts, there is no such thing as a dumb question – she wants us to understand and is willing to explain concepts over and over until we get it.In addition to academics, Mrs. Roberts is also heavily involved in extracurricular activities. She runs the school's writing club, where students can workshop their creative pieces and provide feedback to one another. At the meetings, she goes above and beyond by providing healthy snacks, fun prompts to inspire our writing, and plenty of support and encouragement. Mrs. Roberts also coaches the school's literary competition team, helping students prepare for regional and state tournaments through weekly practice sessions. I'm on the team and she has worked tirelessly researching strategies, holding mock competitions, and doing whatever it takes to make us competitive.Beyond the clubs, Mrs. Roberts is also the advisor for our school's literary magazine and sits on several school improvement committees. She never says no when asked to be involved in something that will benefit students. You can find herat every school event, from dances and sporting events to concerts and plays, always willing to lend a hand. It's amazing she has any free time with her endless commitments, yet she still manages to make time for her family and her own hobbies like reading, crafting, and running.What has impacted me most about Mrs. Roberts is simply her genuine care for every single one of her students. She has taken the time to get to know each of us – our personalities, strengths, weaknesses, interests outside of school, etc. Because of this, she can tailor her instruction, advice, and approach to what works best for each individual. She pushes us when we need motivating and offers compassion when we need understanding.I've witnessed firsthand how she goes out of her way to look out for struggling students and make sure nobody falls through the cracks. For kids having a hard time personally or academically, she meets with them one-on-one, talks to their other teachers, communicates with their parents, points them toward helpful resources, tutors them on her own time, or whatever else is required. It's like she has a sixth sense for knowing when someone needs extra support. I have a friend who was dealing with some mental health issues and going through avery dark time, but Mrs. Roberts' caring words, patience, and referral to the school counselor helped get her through it.Overall, Mrs. Roberts is the epitome of what a world-class teacher should be. She leads by example through her strong work ethic, creative and engaging teaching methods, willingness to help any student who needs it, and deep devotion to ensuring we grow academically and personally. I feel so fortunate to have had her class this year. Looking ahead to high school and beyond, Mrs. Roberts has instilled in me a newfound appreciation for the written and spoken word. More importantly, she's modeled the qualities of dedication, compassion, and integrity that I hope to emulate in whatever path I choose in life. Teachers like her are the ones who change lives and leave篇2My English Teacher's Unwavering DedicationAs an 8th grade student, I've had the privilege of being taught by some truly exceptional teachers over the years. However, one educator stands out as the embodiment of dedication and commitment to their craft - my English teacher, Mrs. Johnson. Her unwavering work ethic and genuine passion for her subject are truly inspiring, and I can confidently say thather diligence has had a profound impact on my academic journey.From the moment I stepped into her classroom, it was evident that Mrs. Johnson approached her role with utmost seriousness and professionalism. Her meticulous lesson plans were always well-structured, and she ensured that every minute of class time was utilized effectively. Whether it was dissecting a literary masterpiece, honing our writing skills, or delving into the intricacies of grammar, her lessons were engaging, interactive, and tailored to our diverse learning needs.What struck me most about Mrs. Johnson was her willingness to go above and beyond for her students. She would often arrive at school well before the first bell, dedicating countless hours to preparing engaging materials and activities. Her attention to detail was remarkable, as she would carefully review and provide comprehensive feedback on every assignment we submitted. Her comments were always constructive, encouraging us to improve while simultaneously acknowledging our efforts.Mrs. Johnson's dedication extended far beyond the classroom walls. She frequently stayed late after school, offering extra help sessions for students who needed additional supportor clarification. Her door was always open, and she welcomed us with a warm smile, ready to patiently guide us through any difficulties we faced. It was truly inspiring to witness her commitment to ensuring that no student was left behind.One particular instance that left a lasting impression on me was when our class was preparing for the annual school spelling bee. Mrs. Johnson went out of her way to organize weekly practice sessions, meticulously selecting challenging words and providing strategies to help us improve our spelling skills. She even created personalized study guides for each student, tailoring them to our individual strengths and weaknesses. Her dedication paid off, as several students from our class advanced to the district-level competition, a testament to her tireless efforts.Mrs. Johnson's commitment to professional development was equally impressive. She was a lifelong learner, consistently seeking opportunities to enhance her teaching methods and stay current with the latest educational trends and research. During summer breaks, she eagerly attended workshops and seminars, bringing back innovative ideas and techniques to enrich our learning experiences.Moreover, Mrs. Johnson's dedication extended beyond academics. She genuinely cared about each student's well-being and personal growth. She took the time to get to know us individually, fostering a classroom environment that was not only intellectually stimulating but also emotionally supportive. Her open-door policy encouraged us to share our thoughts, concerns, and aspirations, and she offered valuable guidance and mentorship.Mrs. Johnson's impact on my life cannot be overstated. Her dedication and passion for teaching have instilled in me a deep appreciation for the English language and a love for literature. More importantly, she has taught me the value of perseverance, hard work, and the pursuit of excellence in all endeavors. As I embark on my academic journey beyond middle school, I carry with me the invaluable lessons and inspiration I've gained from Mrs. Johnson's unwavering dedication.In a world where teaching is often undervalued and underappreciated, Mrs. Johnson stands as a shining example of what it truly means to be an exceptional educator. Her commitment to her craft, her students, and her personal growth is a testament to the profound impact a dedicated teacher can have on young minds. As I reflect on my educational journeythus far, I am filled with gratitude for having had the opportunity to learn from such an extraordinary individual who has left an indelible mark on my life.篇3A Truly Dedicated TeacherTeachers play a vital role in shaping the minds and futures of their students. While many teachers are hardworking and committed, there are some who go above and beyond, leaving a lasting impact on the lives of their pupils. Mrs. Jennifer Smith, my 8th grade English teacher, is one such exceptional educator whose dedication and passion for teaching have inspired me in countless ways.From the moment I stepped into her classroom, I could sense the positive and engaging atmosphere she had created. The walls were adorned with thought-provoking quotes, colorful displays of student work, and educational posters that sparked curiosity. Mrs. Smith's classroom was a vibrant haven of learning, where every student felt welcomed and valued.One of the most remarkable qualities of Mrs. Smith was her unwavering commitment to ensuring that each student grasped the material thoroughly. She understood that every child learnsdifferently, and she took the time to tailor her teaching methods to accommodate various learning styles. For visual learners, she prepared engaging presentations and visual aids. For auditory learners, she employed storytelling and interactive discussions. And for kinesthetic learners, she incorporated hands-on activities and group projects.Mrs. Smith's dedication extended far beyond the classroom walls. She was always available before and after school to provide extra help to students who needed it. I vividly remember the countless times I sought her guidance, whether it was for clarification on a challenging concept or for feedback on an essay. She would patiently listen, offer insightful explanations, and provide constructive criticism that helped me improve my writing skills.Her commitment to her students' success was truly remarkable. She meticulously graded our assignments, providing detailed feedback and suggestions for improvement. Her comments were never harsh or demoralizing; instead, they were thoughtful and encouraging, motivating us to strive for excellence.One particular instance that stands out in my mind occurred during our study of Shakespeare's "Romeo and Juliet." Mrs.Smith recognized the complexity of the language and the potential challenges students might face in comprehending the play. To facilitate our understanding, she organized an interactive activity where we reenacted key scenes from the play. She assigned roles to each student, provided guidance on character development, and encouraged us to immerse ourselves in the Elizabethan era. This hands-on approach not only made the learning process more engaging but also fostered a deeper appreciation for Shakespeare's work.Mrs. Smith's dedication extended beyond academics as well. She was a compassionate mentor who truly cared about the well-being of her students. Whenever she noticed a student struggling with personal issues or facing challenges outside the classroom, she would make herself available to listen and offer support. Her door was always open, and her kindness and empathy created a safe space for students to confide in her.Furthermore, Mrs. Smith was actively involved in extracurricular activities, serving as the faculty advisor for the school's literary club. She dedicated countless hours to organizing meetings, coordinating events, and providing guidance to club members. Her passion for literature was contagious, and her unwavering support inspired many students,including myself, to explore the world of creative writing and literary analysis.One particular memory that stands out is the annual poetry slam organized by the literary club. Mrs. Smith worked tirelessly to ensure its success, coordinating with local poets and artists to participate as judges and guest speakers. She encouraged us to express ourselves through poetry, providing feedback and guidance throughout the writing process. The poetry slam not only showcased our talents but also fostered a sense of community and appreciation for the arts.Mrs. Smith's impact extended far beyond the walls of our school. She was actively involved in professional development programs and workshops, constantly seeking new and innovative teaching methods. She collaborated with fellow educators, sharing her knowledge and experiences, and embracing a lifelong learning approach. Her dedication to staying up-to-date with the latest educational trends and best practices ensured that her students received a top-notch education.In addition to her professional commitments, Mrs. Smith found time to give back to the community. She organized book drives and reading programs for underprivileged children,instilling in them the love of literature from an early age. Her selfless acts inspired many of her students, including myself, to participate in community service and contribute positively to society.As I reflect on my time with Mrs. Smith, I am filled with gratitude and admiration for her unwavering dedication. Her passion for teaching, her commitment to her students' success, and her genuine care for our well-being have left an indelible mark on my life. She has taught me valuable lessons that extend far beyond the classroom, such as the importance of perseverance, empathy, and continuous learning.Mrs. Smith's influence has been a driving force in my own pursuit of education and personal growth. Her example has inspired me to approach my studies with enthusiasm and dedication, and to strive for excellence in all my endeavors. As I embark on the next chapter of my academic journey, I carry with me the invaluable lessons and memories she has imparted.In a world where teaching is often undervalued and underappreciated, educators like Mrs. Smith shine as beacons of hope and inspiration. Their dedication and commitment to shaping young minds and nurturing the leaders of tomorrow areinvaluable contributions to society. Mrs. Smith's unwavering passion for her craft and her genuine care for her。

动机和学习方法的重要性,英语作文The Importance of Motivation and Learning MethodsIntroductionMotivation and learning methods are essential components of education. They play a crucial role in shaping a student's academic success and overall learning experience. In this essay, we will explore the significance of motivation and learning methods in the educational process.Importance of MotivationMotivation is often described as the driving force behind our actions and behaviors. In the context of education, motivation refers to the desire and willingness to learn, succeed, and achieve personal goals. The importance of motivation in learning cannot be overstated. Motivation is the fuel that keeps students engaged, focused, and committed to their studies. It helps students overcome challenges, stay determined, and persevere through difficult times.Motivation also plays a significant role in shaping a student's attitude towards learning. Students who are motivated are more likely to exhibit positive attitudes, curiosity, and a growth mindset. They are willing to take risks, explore new ideas, andpush themselves out of their comfort zones. Motivated students are also more likely to set ambitious goals, work hard to achieve them, and strive for continuous improvement.In contrast, students who lack motivation may struggle to stay engaged, focused, and committed to their studies. They may exhibit negative attitudes, procrastinate, and struggle to stay motivated when faced with challenges. Without motivation, students may find it difficult to persevere through difficult times, overcome setbacks, and achieve their full potential.Ways to Motivate StudentsThere are several ways in which educators can motivate students to learn and succeed. Some effective strategies include:1. Setting clear goals and expectations: Providing students with clear goals, objectives, and expectations can help motivate them to strive for success.2. Celebrating achievements: Recognizing and celebrating students' achievements, no matter how small, can boost their confidence and motivation.3. Providing meaningful feedback: Offering constructive feedback that is specific, actionable, and focused on improvement can help students stay motivated and engaged.4. Cultivating a positive learning environment: Creating a supportive, inclusive, and nurturing learning environment can help foster students' motivation and enthusiasm for learning.Importance of Learning MethodsLearning methods refer to the strategies, techniques, and approaches that students use to acquire new knowledge, skills, and competencies. Effective learning methods are essential for helping students understand complex concepts, retain information, and apply what they have learned in real-world situations.The importance of learning methods lies in their ability to enhance students' understanding, retention, and application of knowledge. Different students have different learning styles, preferences, and strengths. Some students may learn best through visual aids, such as diagrams, charts, and videos, while others may prefer auditory or kinesthetic methods.By using a variety of learning methods, educators can cater to students' diverse learning needs and preferences, making learning more engaging, interactive, and effective. For example, incorporating hands-on activities, group projects, and simulations can help students apply theoretical concepts in practical, real-world contexts.Effective learning methods also play a crucial role in promoting deeper learning, critical thinking, andproblem-solving skills. By encouraging students to think critically, analyze information, and draw connections between concepts, educators can help students develop higher-order thinking skills that enable them to succeed in academic and professional settings.Ways to Enhance Learning MethodsThere are several ways in which educators can enhance learning methods and promote effective learning. Some strategies include:1. Incorporating active learning strategies: Encouraging students to actively engage with course material through discussions, debates, and group activities can promote deeper understanding and retention.2. Providing opportunities for hands-on experiential learning: Allowing students to apply theoretical concepts in practical, real-world contexts can help enhance their learning experience and promote the development of critical thinking skills.3. Using technology: Integrating technology tools, such as online resources, educational apps, and multimediapresentations, can help enhance students' learning experiences and make learning more interactive and engaging.4. Personalizing learning experiences: Tailoring learning activities, assessments, and feedback to students' individual needs, interests, and learning styles can help enhance their motivation, engagement, and success.ConclusionIn conclusion, motivation and learning methods are essential components of education that play a crucial role in shaping students' academic success and overall learning experience. Motivation helps students stay focused, engaged, and committed to their studies, while effective learning methods enhance students' understanding, retention, and application of knowledge. By understanding the importance of motivation and learning methods and implementing strategies to enhance them, educators can help students achieve their full potential and succeed in their academic and professional pursuits.。

Lecture Notes onADV ANCED ECONOMETRICSY ongmiao HongDepartment of Economics,Department of Statistical Science andCenter for Financial EngineeringCornell UniversityEmail:yh20@F ALL2006c 2006Yongmiao Hong.All rights reserved.T ables of ContentsChapter1Introduction to Econometrics1.1Quantitative Features of Modern Economics1.2Mathematical Modeling1.3Econometric Analysis1.4Motivating ExamplesChapter2General Regression Analysis2.1Conditional Probability Distribution2.2Regression Analysis2.3Linear Regression Modeling2.4Correct Model Speci…cationChapter3Classical Linear Regression Models3.1Framework and Assumptions3.2OLS Estimation3.3Goodness of Fit and Model Selection Criteria3.4Consistency and E¢ciency of OLS3.5Sampling Distribution of OLS3.6Variance Matrix Estimator for OLS3.7Hypothesis Testing3.8Generalized Least Squares(GLS)Estimation3.9Empirical Applications3.9.1Foreign Applications3.9.2Chinese ApplicationsChapter4Linear Regression Models with I.I.D.Observations 4.1Introduction to Asymptotic Analysis4.2Framework and Assumptions4.3Consistency of OLS4.4Asymptotic Normality of OLS4.5Asymptotic Variance Estimator for OLS4.5.1The Case of Conditional Homoskedasticity4.5.2The Case of Conditional Heteroskedasticity4.6Hypothesis Testing4.7Testing for Conditional Homoskedasticity4.8Empirical Applications to Cross-Sectional Data4.8.1Foreign Applications4.8.2Chinese ApplicationsChapter5Linear Regression Models with Dependent Observations5.1Basic Concepts in Time Series Analysis5.2Framework and Assumptions5.3Consistency of OLS5.4Asymptotic Normality of OLS5.5Asymptotic Variance Estimator for OLS5.5.1The Case of Conditional Homoskedasticity5.5.2The Case of Conditional Heteroskedasticity5.6Hypothesis Testing5.7Testing for Serial Correlation5.8Empirical Applications to Time Series Data5.8.1Foreign Applications5.8.2Chinese ApplicationsChapter6Linear Regression Models with Heteroskedastic and Autocorrelated Disturbances6.1Framework,Assumptions and Motivating Examples6.2Long-run Variance Estimation6.3Consistency of OLS6.4Asymptotic Normality of OLS6.5Hypothesis Testing6.6Empirical Applications6.6.1Foreign Applications6.6.2Chinese ApplicationsChapter7Instrumental Variable Regressions7.1Framework,Assumptions,and Motivating Examples7.2Two-Stage Least Squares(2SLS)Estimation7.3Consistency of2SLS7.4Asymptotic Normality of2SLS7.5Interpretation and Estimation of the2SLS Asymptotic Variance7.6Hypothesis Testing7.7Empirical Applications7.7.1Foreign Applications7.7.2Chinese ApplicationsChapter8Generalized Method of Moments Estimation8.1Method of Moments Estimation8.2Generalized Method of Moments(GMM)Estimation8.3Consistency of GMM8.4Asymptotic Normality of GMM8.5Asymptotic E¢ciency of GMM8.6Asymptotic Variance Estimation8.7Hypothesis Testing8.8Model Speci…cation Testing8.9Empirical Applications8.9.1Foreign Applications8.9.2Chinese ApplicationsChapter9Maximum Likelihood Estimation and Quasi-Maximum Likelihood Estimation9.1Motivating Examples9.2Maximum Likelihood Estimation(MLE)and Quasi-MLE9.3Asymptotic Properties of MLE/QMLE9.3.1Consistency9.3.2Asymptotic Normality9.3.3E¢ciency9.4MLE-based Hypothesis Testing9.4.1Wald Test9.4.2Lagrange Multiplier Test9.4.3Likelihood Ratio Test9.5QMLE-based Hypothesis Testing9.5.1Impact of Model Misspeci…cation9.5.2Wald Test9.5.3Lagarnge Multiplier Test9.6Model Speci…cation Testing9.7Empirical Applications3.7.1Foreign Applications3.7.2Chinese ApplicationsChapter10ConclusionAppendix A Introduction to Linear AlgebraAppendix B Asymptotic Tools in EconometricsReferencesPrefaceModern economies are full of uncertainties and risk.Economics studies resource allocations in an uncertain market environment.As a generally applicable analytic tool for uncertain events, probability and statistics have been playing an important role in economic research.Econometrics is statistical analysis of economic and…nancial data.It has become an integral of training in modern economics and business.This book develops a coherent set of econometric theory and methods for economic models.It is written for an advanced econometrics course for doctoral students in economics,business and management.Chapter1is a general introduction to econometrics.It…rst describes the two most important features of modern economics,namely mathematical modeling and empirical veri…cation,and then discusses the role of econometrics as a methodology in empirical studies.A few motivating economic motivated examples are given to illustrate how econometrics can be used in empirical studies.Finally,it points out the limitations of econometrics and economics due to the fact that an economy is not a repeatedly controlled experiment.Assumptions and careful interpretations are needed when conducting empirical studies in economics and…nance.Chapter2introduces a general regression analysis.Regression analysis is modeling,esti-mation,inference,and speci…cation analysis of the conditional mean of economic variables of interest given a set of explanatory variables.It is most widely applied in economics.Among other things,this chapter interprets the mean squared error and its optimizer,which lays down the probability-theoretic foundation for least squares estimation.In particular,it provides an interpretation for the least squares estimator and its relationship with the true model parameter.Chapter3introduces the classical linear regression analysis.A set of classical assumptions are given and discussed,and conventional statistical procedures for estimation,inference,and hypothesis testing are introduced.We also discuss the generalized least squares estimation as an e¢cient estimation method of a linear regression model when the variance-covariance matrix is known up to a constant.In particular,the generalized least squares estimation is embedded as an ordinary least squares estimation of a suitably transformed regression model via conditional variance scaling.The subsequent chapters4–7are the generalizations of classical linear regression analysis when various classical assumptions fail.Chapter4…rst relaxes the normality and conditional homoskedasticity assumptions,two key conditions assumed in classical linear regression modeling. An large sample theoretic approach is taken.For simplicity,it is assumed that the observed data is generated from an independent and identically distributed random sample.It is shown that while the…nite distributional theory is no longer valid,the classical statistical procedures are stillapproximately applicable when the sample size is large,if conditional homoskedasticity holds.In contrast,if the data displays conditional heteroskedasticity holds,classical statistical procedures are not applicable even for large samples.In contrast,heteroskedasticity-robust procedures are called for.Chapter5extends the linear regression theory to time series data.First,it introduces a variety of basic concepts in time series analysis.Then it shows that the large sample theory for i.i.d.random samples carries over to stationary ergodic time series data if the regression error follows a martingale di¤erence sequence.Chapter6extends the large sample theory to a very general case where there exist conditional heteroskedasticity and autocorrelation.In this case,the classical regression theory cannot be used,and a long-run variance-covariance estimator is called for to validate statistical inferences.Chapter7is the instrumental variable estimation for linear regression models,where the regression error is correlated with the regressors.This can arise due to measurement errors and simultaneous equation biases.Two stage-least squares estimation method and related statistical inference procedures are fully exploited.Chapter8introduces the generalized method of moments,which is a popular estimation method for possibly nonlinear econometric models characterized as a moment condition.Indeed, most economic theories,such as rational expectations,can be formulated by a moment condi-tion.The generalized method of moments is particularly suitable to estimate model parameters contained in the moment condition.Chapter9introduces the maximum likelihood estimation and the quasi-maximum likelihood estimation methods for conditional probability models and other nonlinear econometric models.Chapter10concludes the book by summarizing the main econometric theory and methods covered in this book,and pointing out directions for further buildup in econometrics.The appendix contains a relatively comprehensive review of asymptotic analytic tools that are needed for this course.By going through this book,students will learn how to do asymptotic analysis for econometric models.Such skills are useful not only for those students who intend to work on theoretical econometrics,but also for those who intend to work on applied subjects in economics because with such analytic skills,students will be able to understand more specialized or more advanced econometrics textbooks.This book is based on my lecture notes taught at Cornell University,Renmin University of China,Shandong University,Shanghai Jiao Tong University,Tsinghua University,and Xiamen University,where the graduate students provide rather detailed comments on my lecture notes.CHAPTER1INTRODUCTION TOECONOMETRICSKey W ords:Data generating process,Econometrics,Probability law,Statistics.Abstract:Econometrics has become an integral part of training in modern economics and business.Together with microeconomics and macroeconomics,econometrics has been taught as one of the three core courses in most undergraduate and graduate economic programs in North America.In China,the importance of econometrics has been increasingly recognized and econometric tools and methods have been widely employed in empirical studies on Chinese economy.This chapter discusses the philosophy and methodology of econometrics in economic research,the roles and limitations of econometrics,and the di¤erences between econometrics and mathematical economics as well as mathematical statistics.A variety of illustrative econometric examples are given,which cover various…elds of economics and…nance.1.1IntroductionEconometrics has become an integrated part of teaching and research in modern economics and business.The importance of econometrics has been increasingly recognized in China.In this chapter,we will discuss the philosophy and methodology of econometrics in economic re-search.First,we will discuss the qualitative feature of modern economics,and the di¤erences between econometrics and mathematical economics as well as mathematical statistics.Then we will focus on the important roles of econometrics as a fundamental methodology in economic research via a variety of illustrative economic examples including the consumption function, marginal propensity to consume and multipliers,rational expectations models and dynamic as-set pricing,the constant return to scale and regulations,evaluation of e¤ects of economic reforms in a transitional economy,the e¢cient market hypothesis,modeling uncertainty and volatility, and duration analysis in labor economics and…nance.These examples range from econometric analysis of the conditional mean to the conditional variance to the conditional distribution of economic variables of interest.we will also discuss the limitations of econometrics,due to the nonexperimental nature of economic data and the time-varying nature of econometric structures. Finally,problems in current econometric teaching and research in China are pointed out,and possible solutions are proposed.1.2Quantitative Features of Modern EconomicsModern market economies are full of uncertainties and risk.When economic agents make a decision,the outcome is usually unknown in advance and economic agents will take this uncer-tainty into account in their decision-making.Modern economics is a study on scarce resourceallocations in an uncertain market environment.Generally speaking,modern economics can be roughly classi…ed into four categories:macroeconomics,microeconomics,…nancial economics, and econometrics.Of them,macroeconomics,microeconomics and econometrics now consti-tute the core courses for most economic doctoral programs in North America,while…nancial economics is now mainly being taught in business and management schools.Most doctoral programs in economics in the U.S.emphasize quantitative analysis.Quantita-tive analysis consists of mathematical modeling and empirical studies.To understand the roles of quantitative analysis,it may be useful to…rst describe the general process of modern economic research.Like most natural science,the general methodology of modern economic research can be roughly summarized as follows:Step1:Data collection and summary of empirical stylized facts.The so-called stylized facts are often summarized from observed economic data.For example,in microeconomics,a well-known stylized fact is the Engel’s curve,which characterizes that the share of a con-sumer’s expenditure on a commodity out of her or his total income will eventually decline as the income increases;in macroeconomics,a well-known stylized fact is the Phillips Curve, which characterizes a negative correlation between the in‡ation rate and the unemployment rate in an aggregate economy;and in…nance,a well-known stylized fact about…nancial markets is volatility clustering,that is,a high volatility today tends to be followed by another high volatility tomorrow,a low volatility today tends to be followed by another low volatility tomorrow,and both alternate over time.The empirical stylized facts often serve as a starting point for economic research.For example,the development of unit root and cointegration econometrics was mainly motivated by the empirical study of Nelson and Plossor(1982)who found that most macroeconomic time series are unit root processes. Step2:Development of economic theories/models.With the empirical stylized facts in mind,economists then develop an economic theory or model in order to explain them.This usually calls for specifying a mathematical model of economic theory.In fact,the objective of economic modelling is not merely to explain the stylized facts,but to understand the mechanism governing the economy and to forecast the future evolution of the economy. Step3:Empirical veri…cation of economic models.Economic theory only suggests a qualita-tive economic relationship.It does not o¤er any concrete functional form.In the process of transforming a mathematical model into a testable empirical econometric model,one often has to assume some functional form,up to some unknown model parameters.One need to estimate unknown model parameters based on the observed data,and check whether the econometric model is adequate.An adequate model should be at least consistent with the empirical stylized facts.Step4:Applications.After an econometric model passes the empirical evaluation,it can then be used to test economic theory or hypotheses,to forecast future evolution of the economy,and to make policy recommendations.For an excellent example highlighting these four steps,see Gujarati(2006,Section1.3)on labor force participation.We note that not every economist or every research paper has to complete these four steps.In fact,it is not uncommon that each economist may only work on research belonging to certain stage in his/her entire academic lifetime.From the general methodology of economic research,we see that modern economics has two important features:one is mathematical modeling for economic theory,and the other is empirical analysis for economic phenomena.These two features arise from the e¤ort of several generations of economists to make economics a"science".To be a science,any theory must ful…ll two criteria: one is logical consistency and coherency in theory itself,and the other is consistency between theory and stylized facts.Mathematics and econometrics serve to help ful…ll these two criteria respectively.This has been the main objective of the econometric society.The setup of the Nobel Memorial Prize in economics in1969may be viewed as the recognition of economics as a science in the academic profession.1.3Mathematical ModellingWe…rst discuss the role of mathematical modeling in economics.Why do we need mathe-matics and mathematical models in economics?It should be pointed out that there are many ways or tools(e.g.,graphical methods,verbal discussions,mathematical models)to describe eco-nomic theory.Mathematics is just one of them.To ensure logical consistency of the theory,it is not necessary to use mathematics.Chinese medicine is an excellent example of science without using mathematical modeling.However,mathematics is well-known as the most rigorous logical language.Any theory,when it can be represented by the mathematical language,will ensure log-ical consistency and coherency of economic theory,thus indicating that it has achieved a rather sophisticated level.Indeed,as Karl Marx pointed out,the use of mathematics is an indication of the mature development of a science.It has been a long history to use mathematics in economics.In his Mathematical Principles of the Wealth Theory,Cournot(1838)was among the earliest to use mathematics in economic analysis.Although the marginal revolution,which provides a cornerstone for modern economics, was not proposed using mathematics,it was quickly found in the economic profession that the marginal concepts,such as marginal utility,marginal productivity,and marginal cost,correspond to the derivative concepts in calculus.Walras(1874),a mathematical economist,heavily used mathematics to develop his general equilibrium theory.The game theory,which was proposed by Von Neumann and Morgenstern(1944)and now becomes a core in modern microeconomics, originated from a branch in mathematics.Why does economics need mathematics?Brie‡y speaking,mathematics plays a number of important roles in economics.First,the mathematical language can summarize the essence of a theory in a very concise manner.For example,macroeconomics studies relationships between aggregate economic variables(e.g.,DGP,consumption,unemployment,in‡ation,interest rate, exchange rate,and etc.)A very important macroeconomic theory was proposed by Keynes (1936).The classical Keynesian theory can be summarized by two simple mathematical equa-tions:National Income identity:Y=C+I+G;Consumption function:C= + Y;where Y is income,C is consumption,I is private investment,G is government spending, is the“survival level”consumption,and is the marginal propensity to consume.Substituting the consumption function into the income identity,arranging terms,and taking a partial derivative, we can obtain the multiplier e¤ect of(e.g.)government spending@Y @G =11:Thus,the Keynesian theory can be e¤ectively summarized by two mathematical equations.Second,complicated logical analysis in economics can be greatly simpli…ed by using math-ematics.In introductory economics,economic analysis can be done by verbal descriptions or graphical methods.These methods are very intuitive and easy to grasp.One example is the partial equilibrium analysis where a market equilibrium can be characterized by the intersection of the demand curve and the supply curve.However,in many cases,economic analysis cannot be done easily by verbal languages or graphical methods.One example is the general equilib-rium theory…rst proposed by Walras(1874).This theory addresses a fundamental problem in economics,namely whether the market force can achieve an equilibrium for a competitive mar-ket economy where there exist many markets and when there exist mutual interactions between di¤erent markets.Suppose there are n goods,with demand D i(P);supply S i(P)for good i; where P=(P1;P2;:::;P n)0is a price vector for n goods.Then the general equilibrium analysis addresses whether there exists an equilibrium price vector P such that all markets are clear simultaneously:D i(P )=S i(P )for all i2f1;:::;n g:Conceptually simple,it is rather challenging to give a de…nite answer because both the demand and supply functions could be highly nonlinear.Indeed,Walras was unable to establish this theory formally.It was satisfactorily solved by Arrow and Debreu many years later,when they used the…xed point theorem in mathematics to prove the existence of an equilibrium price vector.The power and magic of mathematics was clearly demonstrated in the development of the general equilibrium theory.Third,mathematical modeling is a necessary path to empirical veri…cation of an economic theory.Most economic and…nancial phenomena are in form of data(indeed we are in a digital era!).We need“digitalize”economic theory so as to link the economic theory to data.In particular,one need to formulate economic theory into a testable mathematical model whose functional form or important structural model parameters will be estimated from observed data.1.4Empirical V eri…cationWe now turn to discuss the second feature of modern economics:empirical analysis of an economic theory.Why is empirical analysis of an economic theory important?The use of mathematics,although it can ensure logical consistency of a theory itself,cannot ensure that economics is a science.An economic theory would be useless from a practical point of view if the underlying assumptions are incorrect or unrealistic.This is the case even if the mathematical treatment is free of errors and elegant.As pointed out earlier,to be a science,an economic theory must be consistent with reality.That is,it must be able to explain historical stylized facts and predict future economic phenomena.How to check a theory or model empirically?Or how to validate an economic theory?In practice,it is rather di¢cult or even impossible to check whether the underlying assumptions of an economic theory or model are correct.Nevertheless,one can confront the implications of an economic theory with the observed data to check if they are consistent.In the early stage of economics,empirical veri…cation was often conducted by case studies or indirect veri…cations. For example,in his well-known Wealth of Nations,Adam Smith(1776)explained the advantage of specialization using a case study example.Such a method is still useful nowadays,but is no longer su¢cient for modern economic analysis,because economic phenomena is much more com-plicated while data may be limited.For rigorous empirical analysis,we need to use econometrics. Econometrics is the…eld of economics that concerns itself with the application of mathemati-cal statistics and the tools of statistical inference to the empirical measurement of relationships postulated by economic theory.It was founded as a scienti…c discipline around1930as marked by the founding of the econometric society and the creation of the most in‡uential economic journal—Econometrica in1933.Econometrics has witnessed a rather rapid development in the past several decades,for a number of reasons.First,there is a need for empirical veri…cation of economic theory,and for forecasting using economic models.Second,there is more and more high-quality economic data available.Third,advance in computing technology has made the cost of computation cheaper and cheaper over time.The speed of computing grows faster than the speed of data accumulation.Although not explicitly stated in most of the econometric literature,modern econometrics is essentially built upon on the following fundamental axioms:Any economy can be viewed as a stochastic process governed by some probability law. Economic phenomenon,as often summarized in form of data,can be reviewed as a realiza-tion of this stochastic data generating process.There is no way to verify these axioms.They are the philosophic views of econometricians toward an economy.Not every economist or even econometrician agrees with this view.For example,some economists view an economy as a deterministic chaotic process which can generate seemingly random numbers.However,most economists and econometricians(e.g.,Granger and Terasvirta1993,Lucas1977)view that there are a lot of uncertainty in an economy,and they are best described by stochastic factors rather than deterministic systems.For instance,the multiplier-accelerator model of Samuelson(1939)is characterized by a deterministic second-order di¤erence equation for aggregate output.Over a certain range of parameters,this equation produces deterministic cycles with a constant period of business cycles.Without doubt this model sheds deep insight into macroeconomic‡uctuations.Nevertheless,a stochastic framework will provide a more realistic basis for analysis of periodicity in economics,because the observed periods of business cycles never occur evenly in any economy.Frisch(1933)demonstrates that a structural propagation mechanism can convert uncorrelated stochastic impulses into cyclical outputs with uneven,stochastic periodicity.Indeed,although not all uncertainties can be well characterized by probability theory,probability is the best quantitative analytic tool to describe uncertainties.The probability law of this stochastic economic system,which characterizes the evolution of the economy,can be viewed as the“law of economic motions.”Accordingly,the tools and methods of mathematical statistics will provide the operating principles.One important implication of the fundamental axioms is that one should not hope to de-termine precise,deterministic economic relationships,as do the models of demand,production, and aggregate consumption in standard micro-and macro-economic textbooks.No model could encompass the myriad essentially random aspects of economic life(i.e.,no precise point forecast is possible,using a statistical terminology).Instead,one can only postulate some stochastic economic relationships.The purpose of econometrics is to infer the probability law of the eco-nomic system using observed data.Economic theory usually takes a form of imposing certain restrictions on the probability law.Thus,one can test economic theory or economic hypotheses by checking the validity of these restrictions.It should be emphasized that the role of mathematics is di¤erent from the role of econometrics. The main task of mathematical economics is to express economic theory in the mathematical form of equations(or models)without regard to measurability or empirical veri…cation of economic theory.Mathematics can check whether the reasoning process of an economic theory is correct and sometime can give surprising results and conclusions.However,it cannot check whether an economic theory can explain reality.To check whether a theory is consistent with reality, one needs econometrics.Econometrics is a fundamental methodology in the process of economicanalysis.Like the development of a natural science,the development of economic theory is a process of refuting the existing theories which cannot explain newly arising empirical stylized facts and developing new theories which can explain them.Econometrics rather than mathematics plays a crucial role in this process.There is no absolutely correctly and universally applicable economic theory.Any economic theory can only explain the reality at certain stage,and therefore, is a“relative truth”in the sense that it is consistent with historical data available at that time. An economic theory may not be rejected due to limited data information.It is possible that more than one economic theory or model coexist simultaneously,because data does not contain su¢cient information to distinguish the true one(if any)from false ones.When new data becomes available,a theory that can explain the historical data well may not explain the new data well and thus will be refuted.In many cases,new econometric methods can lead to new discovery and call for new development of economic theory.Econometrics is not simply an application of a general theory of mathematical statistics. Although mathematical statistics provides many of the operating tools used in econometrics, econometrics often needs special methods because of the unique nature of economic data,and the unique nature of economic problems at hand.One example is the generalized method of moment estimation(Hansen1982),which was proposed by econometricians aiming to estimate rational expectations models which only impose certain conditional moment restrictions charac-terized by the Euler equation and the conditional distribution of economic processes is unknown (thus,the classical maximum likelihood estimation cannot be used).The development of unit root and cointegration(e.g.,Engle and Granger1987,Phillips1987),which is a core in modern time series econometrics,has been mainly motivated from Nelson and Plosser’s(1982)empirical documentation that most macroeconomic time series display unit root behaviors.Thus,it is necessary to provide an econometric theory for unit root and cointegrated systems because the standard statistical inference theory is no longer applicable.The emergence of…nancial econo-metrics is also due to the fact that…nancial time series display some unique features such as persistent volatility clustering,heavy tails,infrequent but large jumps,and serially uncorrelated but not independent asset returns.Financial applications,such as…nancial risk management, hedging and derivatives pricing,often call for modeling for volatilities and the entire conditional probability distributions of asset returns.The features of…nancial data and the objectives of …nancial applications make the use of standard time series analysis quite limited,and therefore, call for the development of…nancial bor economics is another example which shows how labor economics and econometrics have bene…ted from each bor economics has advanced quickly over the last few decades because of availability of high-quality labor data and rigorous empirical veri…cation of hypotheses and theories on labor economics.On the other hand,microeconometrics,particularly panel data econometrics,has also advanced quickly due to the increasing availability of microeconomic data and the need to develop econometric theory。