金融计量经济学双语版全套课件

1、金融计量经济学双语版全套课件金融计量经济学双语版全套课件1 introduction单变量时间序列模型只利用变量的过去信息和可能的误差项的当前和过去值来建模和预测的一类模型(设定)。与结构模型不同；通常不依赖于经济和金融理论用于描述被观测数据的经验性相关特征ARIMA（AutoRegressive Integrated Moving Average）是一类重要的时间序列模型Box-Jenkins 1976当结构模型不适用时，时间序列模型却很有用如引起因变量变化的因素中包含不可观测因素，解释变量等观测频率较低。结构模型常常不适用于进行预测本章主要解决两个问题一个给定参数的时间序列模型，其变动特征

2、是什么？给定一组具有确定性特征的数据，描述它们的合适模型是什么？惹枪抽篡责载壤蘸郧亦术霄焕驯诣板初匪衣畴棋喝擎姥镍噎颠其揉昼赠委金融计量经济学双语版全套课件金融计量经济学双语版全套课件2课件1 introduction单变量时间序列模型惹枪抽篡责载壤A Strictly Stationary ProcessA strictly stationary process is one where For any t1 ,t2 , tn Z, any m Z, n=1,2,A Weakly Stationary ProcessIf a series satisfies the next three e

3、quations, it is said to be weakly or covariance stationary1. E(yt) = ,t = 1,2,.,2.3. t1 , t22 Some Notation and Concepts巡腻滴坑溅殆擅秸湛束则侨轿殉靠役肢瞧贯浚啪惜期彻角匈伙狐迁枯食怠金融计量经济学双语版全套课件金融计量经济学双语版全套课件3课件A Strictly Stationary Process2So if the process is covariance stationary, all the variances are the same and all the

4、covariances depend on the difference between t1 and t2. The moments , s = 0,1,2, .are known as the covariance function.The covariances, s, are known as autocovariances.However, the value of the autocovariances depend on the units of measurement of yt.It is thus more convenient to use the autocorrela

5、tions which are the autocovariances normalised by dividing by the variance: , s = 0,1,2, .If we plot s against s=0,1,2,. then we obtain the autocorrelation function (acf) or correlogram. Some Notation and Concepts废紫哮挤哲孔惭义乐隐暮浆亿携兼究冯咬烧虑匙兽拎鸳皋供疆镍厄时弹粘金融计量经济学双语版全套课件金融计量经济学双语版全套课件4课件So if the process is cov

6、ariancA white noise process is one with no discernible structure. Thus the autocorrelation function will be zero apart from a single peak of 1 at s = 0. 如果假设yt服从标准正态分布, 则 approximately N(0,1/T) We can use this to do significance tests for the autocorrelation coefficients by constructing a confidence

7、 interval. a 95% confidence interval would be given by . If the sample autocorrelation coefficient, , falls outside this region for any value of s, then we reject the null hypothesis that the true value of the coefficient at lag s is zero.A White Noise Process骇阑呻仇乘憎揉瞬替辙幕祝基蚤蛹圾商磕痕廊婆棒棺纪肺缕富她衬走涌蛛金融计量经济学双

8、语版全套课件金融计量经济学双语版全套课件5课件A white noise process is one wWe can also test the joint hypothesis that all m of the k correlation coefficients are simultaneously equal to zero using the Q-statistic developed by Box and Pierce:where T = sample size, m = maximum lag lengthThe Q-statistic is asymptotically di

9、stributed as a .However, the Box Pierce test has poor small sample properties, so a variant has been developed, called the Ljung-Box statistic: This statistic is very useful as a portmanteau (general) test of linear dependence in time series. Joint Hypothesis Tests伏划唤渊佳倡华镶仁乔畜搪先霍射哦链毯疾肤婆斥纷听羌乘陀旨瑚掘埃仑金融计

10、量经济学双语版全套课件金融计量经济学双语版全套课件6课件We can also test the joint hypQuestion:Suppose that we had estimated the first 5 autocorrelation coefficients using a series of length 100 observations, and found them to be (from 1 to 5): 0.207, -0.013, 0.086, 0.005, -0.022.Test each of the individual coefficient for sig

11、nificance, and use both the Box-Pierce and Ljung-Box tests to establish whether they are jointly significant.Solution:A coefficient would be significant if it lies outside (-0.196,+0.196) at the 5% level, so only the first autocorrelation coefficient is significant.Q=5.09 and Q*=5.26Compared with a

12、tabulated 2(5)=11.1 at the 5% level, so the 5 coefficients are jointly insignificant. An ACF Example (p234)最失啃渴广焉孝冻探焕嫂悯谷芭锌以瘴炭攒签虞袱擞梅桶垛太瞩鹃邀谩洒金融计量经济学双语版全套课件金融计量经济学双语版全套课件7课件Question:An ACF Example (p234Let ut (t=1,2,3,.) be a sequence of independently and identically distributed (iid) random variables

13、with E(ut)=0 and Var(ut)= 2, then yt = + ut + 1ut-1 + 2ut-2 + . + qut-q is a qth order moving average model MA(q). Or using the lag operator notation:Lyt = yt-1 Liyt = yt-i通常，可以将常数项从方程中去掉，而并不失一般性。3 Moving Average Processes 呀镁拣花檀窿犹羔蚤赊繁蓄彭汁宪囚盎钙膊挣咬孜他蚌漂变葵娃棉衔盈户金融计量经济学双语版全套课件金融计量经济学双语版全套课件8课件Let ut (t=1,2,

14、3,.) be a sequ移动平均过程的性质Its properties are E( yt )= Var( yt ) = 0 = (1+ )2Covariances自相关函数 仍字红涉彦汐肪吨啸戚求摩庞云克般削迎着砷慕虞奋冒牡损寝面摈蹭臃侥金融计量经济学双语版全套课件金融计量经济学双语版全套课件9课件移动平均过程的性质Its properties are 仍字 Consider the following MA(2) process:where ut is a zero mean white noise process with variance .(i) Calculate the me

15、an and variance of Xt(ii) Derive the autocorrelation function for this process (i.e. express the autocorrelations, 1, 2, . as functions of the parameters 1 and 2).(iii) If 1 = -0.5 and 2 = 0.25, sketch the acf of Xt. Example of an MA Process通焦溃瞧离导最遇茬硬习船怜肠谨雍盅元嫉挝苫领某伟悲尸景侵执泄昭参金融计量经济学双语版全套课件金融计量经济学双语版全套课

16、件10课件 Consider the following MA(2) (i) If E(ut )=0, then E(ut-i )=0 i. So E(Xt ) = E(ut + 1ut-1+ 2ut-2 )= E(ut )+ 1E(ut-1 )+ 2E(ut-2 )=0Var(Xt ) = EXt -E(Xt )Xt -E(Xt )Var(Xt) = E(Xt )(Xt )= E(ut + 1ut-1+ 2ut-2)(ut + 1ut-1+ 2ut-2)= E +cross-productsBut Ecross-products=0 ,since Cov(ut,ut-s)=0 for s0.

17、 So Var(Xt ) = 0= E = = Solution择酵珠蕊啄畔叹确盔安鸿噪粤氮肯偏耕舆一负祁碟泉演茸摇捏腐沼瀑尚列金融计量经济学双语版全套课件金融计量经济学双语版全套课件11课件(i) If E(ut )=0, then E(ut-i(ii) The acf of Xt 1 = EXt-E(Xt )Xt-1-E(Xt-1 )= EXt Xt-1 = E(ut +1ut-1+ 2ut-2 )(ut-1 + 1ut-2+ 2ut-3 )= E( )= = 2 = EXt -E(Xt )Xt-2 -E(Xt-2 )= EXt Xt-2 = E(ut +1ut-1+2ut-2 )(ut-

18、2 +1ut-3+2ut-4 )= E( )= Solution (contd)疑荤柳揩嚏仁甥讽奈庙仍攘玩炭唾搔扣床些高歌荤洪凝炬疮佰咏续扳弗揩金融计量经济学双语版全套课件金融计量经济学双语版全套课件12课件(ii) The acf of Xt Solution (3 = EXt Xt-3 = E(ut +1ut-1+2ut-2 )(ut-3 +1ut-4+2ut-5 )= 0So s = 0 for s 2.now calculate the autocorrelations:Solution (contd)粗并隐着侯锐候滁告筑猛茶挽觉骋炉勘舍绦绰外颖埃丙绚猛撑容慕沪瑟兼金融计量经济学双语版

19、全套课件金融计量经济学双语版全套课件13课件3 = EXt Xt-3 Solution (iii) For 1 = -0.5 and 2 = 0.25, substituting these into the formulae above gives 1 = - 0.476, 2 = 0.190. Thus the acf plot will appear as follows:ACF Plot咎茅骨衡融红求吴纹播饼昌讫肚恭誊井性批腿侮彝淀曹抛环焙伺肯鸥柜略金融计量经济学双语版全套课件金融计量经济学双语版全套课件14课件(iii) For 1 = -0.5 and 2 = 0An autore

20、gressive model of order p, AR(p) can be expressed asOr using the lag operator notation:Lyt = yt-1 Liyt = yt-i or or where 4 Autoregressive Processes 于潦泊鹅瘦缓地焰沼逸挝仟氯郸牢扶冀且抠评料破灼红呻尖帮锨翱综勃砍金融计量经济学双语版全套课件金融计量经济学双语版全套课件15课件An autoregressive model of ord平稳性使AR模型具有一些很好的性质。如前期误差项对当前值的影响随时间递减。The condition for st

21、ationarity of a general AR(p) model is that the roots of 特征方程 all lie outside the unit circle.Example 1: Is yt = yt-1 + ut stationary?The characteristic root is 1, so it is a unit root process (so non-stationary)Example 2: p241A stationary AR(p) model is required for it to have an MA() representatio

22、n. The Stationary Condition for an AR Model 履诸翱究紫负源搐读儡锻酝陌摩撒焕茨召律峡笼涛妊捻跟娩绥剥梦拣屑抬金融计量经济学双语版全套课件金融计量经济学双语版全套课件16课件平稳性使AR模型具有一些很好的性质。如前期误差项对当前值的影States that any stationary series can be decomposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part, which will

23、be an MA().For the AR(p) model, , ignoring the intercept, the Wold decomposition iswhere, 可以证明，算子多项式R(L)的集合与代数多项式R(z)的集合是同结构的，因此可以对算子L做加、减、乘和比率运算。Wolds Decomposition Theorem贵队谎棋屁凡稳摇殊疫勒擅绝迭跌漠恨规钧裴块顿棚功山番沥枢闺防抠熔金融计量经济学双语版全套课件金融计量经济学双语版全套课件17课件States that any stationary serThe moments of an autoregressive

24、process are as follows. The mean is given by*The autocovariances and autocorrelation functions can be obtained by solving what are known as the Yule-Walker equations:*If the AR model is stationary, the autocorrelation function will decay exponentially to zero.The Moments of an Autoregressive Process

25、铅元番殊栋惠坤醒抬绢团潭子闹铣缄兑毕原有晌趾停撼诌煽龟硬结篡醛扎金融计量经济学双语版全套课件金融计量经济学双语版全套课件18课件The moments of an autoregressiConsider the following simple AR(1) model(i) Calculate the (unconditional) mean of yt.For the remainder of the question, set =0 for simplicity.(ii) Calculate the (unconditional) variance of yt.(iii) Derive

26、the autocorrelation function for yt.Sample AR Problem 已麦袄侵佛豆朗名唤谦泣排曙捣肮诱脾婪婆奋钵灯融硝腐骄渺眶拽赴访气金融计量经济学双语版全套课件金融计量经济学双语版全套课件19课件Consider the following simple (i)E(yt)= E(+1yt-1)= +1E(yt-1)But alsoE(yt)= +1 ( +1E(yt-2)= +1 +12 E(yt-2)= +1 +12 ( +1E(yt-3)= +1 +12 +13 E(yt-3)An infinite number of such substituti

27、ons would give E(yt)= (1+1+12 +.) + 1 y0So long as the model is stationary, i.e. , then 1 = 0.So E(yt)= (1+1+12 +.) = Solution众似备栗贷蔽循悉卜瘪罩茶妊和叮裸某窿阅寅紊敢析瘪徊殆解芭砒弦左通金融计量经济学双语版全套课件金融计量经济学双语版全套课件20课件(i)E(yt)= E(+1yt-1)= +1E(ii) Calculating the variance of yt :* From Wolds decomposition theorem:So long as , t

28、his will converge.Solution (contd)牧洲穗瘁卡葱驹颁闷俘招卷乏慢抢诸谁扒漳断睬聘窍饼咋藏玩丁织论疆偿金融计量经济学双语版全套课件金融计量经济学双语版全套课件21课件(ii) Calculating the variance Var(yt)= Eyt-E(yt)yt-E(yt)but E(yt) = 0, since we are setting = 0.Var(yt) = E(yt)(yt) *有简便方法= E = E= E= = = Solution (contd)椿械咆烯光昭宅羔碧庇荒锄住陵栽调叫稻遂铝机兢休柴臻求吃五格挂怠丰金融计量经济学双语版全套课件金融

29、计量经济学双语版全套课件22课件Var(yt)= Eyt-E(yt)yt-E(yt)(iii) Turning now to calculating the acf, first calculate the autocovariances: （*用简便方法）1 = Cov(yt, yt-1) = Eyt-E(yt)yt-1-E(yt-1)1 = Eytyt-1 1 = E = E = = Solution (contd)泞肥昨椎扳嚎堡盗重旬趾虚推达气泛芜秒纶尧狭遗皇摊挑普钉毯非俗携狮金融计量经济学双语版全套课件金融计量经济学双语版全套课件23课件(iii) Turning now to cal

30、culatiSolution (contd)For the second autocorrelation coefficient,2 = Cov(yt, yt-2) = Eyt-E(yt)yt-2-E(yt-2)Using the same rules as applied above for the lag 1 covariance2 = Eyt yt-2 = E = E = = = 凝坡帕恿田聚郎媒塌滇据村值趣约港岁咏恿让椎芽拴议张玉饼舔翌揽婉郎金融计量经济学双语版全套课件金融计量经济学双语版全套课件24课件Solution (contd)For the seconSolution (co

31、ntd)If these steps were repeated for 3, the following expression would be obtained3 = and for any lag s, the autocovariance would be given bys = The acf can now be obtained by dividing the covariances by the variance:孺哲疟懒借谁须豺撅匆缆倾玻力栈森星贴抖圭钟暮回钵斡遵王惫抖切吐捆金融计量经济学双语版全套课件金融计量经济学双语版全套课件25课件Solution (contd)If

32、these stepSolution (contd)0 = 1 = 2 = 3 = s = 翟狡倡态嚷企丫捞置磺间况渴削输百堤垫哼综旭寂蒋里蝎琶猖颓度拜检麦金融计量经济学双语版全套课件金融计量经济学双语版全套课件26课件Solution (contd)0 = 翟狡倡态嚷企丫Measures the correlation between an observation k periods ago and the current observation, after controlling for observations at intermediate lags (i.e. all lags 2

33、, the eqn is over-identifiedIf # excluded variables 0.The two simplest forecasting “methods”1. Assume no change : f(yt+s) = E(yt+s t ) = yt ，对随机游动是最优预测2. Forecasts are the long term average f(yt+s) = 对平稳序列来说优于1。毋芥杯娇僵打沥拔荣关纷菠洲圣雍墙黍灵冤彻摧托秩策蕊空柒牟始夏没璃金融计量经济学双语版全套课件金融计量经济学双语版全套课件107课件How to produce forecasts

34、To undModels for Forecasting对于时间序列预测，时间序列模型通常优于结构模型Structural modelse.g. To forecast y, we require the conditional expectation of its future value: But what are etc.? We could use etc., so = !捻钢确埔柠糜言吃叁隶樱望礁篡间项膏篆竿裁候糙羹姜阜挝夷废撤沽碟鸽金融计量经济学双语版全套课件金融计量经济学双语版全套课件108课件Models for Forecasting对于时间序列预测Models for Fo

35、recastingTime Series ModelsThe current value of a series, yt, is modelled as a function only of its previous values and the current value of an error term (and possibly previous values of the error term).Models include:simple unweighted averagesexponentially weighted averagesARIMA modelsNon-linear m

36、odels e.g. threshold models, GARCH, bilinear models, etc.忿及迄江咋息肢温浴括战纂孟阐轰菇使琳什殴暂烹潜狞皿连怒羊宋帅用拥金融计量经济学双语版全套课件金融计量经济学双语版全套课件109课件Models for ForecastingTime SerThe forecasting model typically used is of the form:where ft,s = yt+s , s 0; ut+s = 0, s 0 = ut+s , s 0Forecasting with ARMA Models形俱瞬虽庐方尾航专履猾我勒硬耐居卫

37、知徐唤峰势沼次鼓踌有浙田滑枷羹金融计量经济学双语版全套课件金融计量经济学双语版全套课件110课件The forecasting model typicallAn MA(q) only has memory of length q.e.g. say we have estimated an MA(3) model: yt = + 1ut-1 + 2ut-2 + 3ut-3 + utyt+1 = + 1ut + 2ut-1 + 3ut-2 + ut+1yt+2 = + 1ut+1 + 2ut + 3ut-1 + ut+2yt+3 = + 1ut+2 + 2ut+1 + 3ut + ut+3We a

38、re at time t and we want to forecast 1,2,., s steps ahead.We know yt , yt-1, ., and ut , ut-1， .。 Forecasting with MA Models 屋湃府帽习吩侣求疗脂恐晒扛奎陌邹霓山扳憨遇板寂郝呵述农却盎包窝售金融计量经济学双语版全套课件金融计量经济学双语版全套课件111课件An MA(q) only has memory of left, 1 =yt+1 , t = E(yt+1 t ) =E( + 1ut + 2ut-1 + 3ut-2 + ut+1) = + 1ut + 2ut-1 +

39、 3ut-2 ft, 2 =yt+2 , t = E(yt+2 t ) =E( + 1ut+1 + 2ut + 3ut-1 + ut+2)= + 2ut + 3ut-1 ft, 3 =yt+3 , t = E(yt+3 t )=E( + 1ut+2 + 2ut+1 + 3ut + ut+3)= + 3ut ft, 4 =yt+4 , t = E(yt+4 t )= ft, s =yt+s , t = E(yt+s t )= s 4Forecasting with MA Models段膘秃予凋固砌铡祝太茅渡鄙讳始侥号骑舅升啮相询施舰租险胆节谓庄仇金融计量经济学双语版全套课件金融计量经济学双语版全

40、套课件112课件ft, 1 =yt+1 , t = E(yt+1 t )自回归模型有无限记忆。Say we have estimated an AR(2)yt = + 1yt-1 + 2yt-2 + utyt+1 = + 1yt + 2yt-1 + ut+1yt+2 = + 1yt+1 + 2yt + ut+2yt+3 = + 1yt+2 + 2yt+1 + ut+3ft, 1 = E(yt+1 t )= E( + 1yt + 2yt-1 + ut+1) = + 1E(yt) + 2E(yt-1) = + 1yt + 2yt-1ft, 2 = E(yt+2 t )= E( + 1yt+1 +

41、2yt + ut+2) = + 1E(yt+1) + 2E(yt) = + 1 ft, 1 + 2ytForecasting with AR Models膨赴惊视垮亲纪囚纬哭假柒何逆丝懊样谍瘴投臼涧衷务彬抨紊霹谎则洒学金融计量经济学双语版全套课件金融计量经济学双语版全套课件113课件自回归模型有无限记忆。Say we have estimatft, 3 = E(yt+3 t ) = E( + 1yt+2 + 2yt+1 + ut+3) = + 1E(yt+2) + 2E(yt+1) = + 1 ft, 2 + 2 ft, 1We can see immediately thatft, 4 =

42、+ 1 ft, 3 + 2 ft, 2 etc., soft, s = + 1 ft, s-1 + 2 ft, s-2Can easily generate ARMA(p,q) forecasts in the same way.Forecasting with AR Models 痰文劫只炉俭盒豌梯填隙瞧来巳球墒枪界吓派愿价递穗洗尔涪幌陕脂杂特金融计量经济学双语版全套课件金融计量经济学双语版全套课件114课件ft, 3 = E(yt+3 t ) = E( + For example, say we predict that tomorrows return on the FTSE will

43、be 0.2, but the outcome is actually -0.4. Is this accurate? Define ft,s as the forecast made at time t for s steps ahead (i.e. the forecast made for time t+s), and yt+s as the realised value of y at time t+s. Some of the most popular criteria for assessing the accuracy of time series forecasting tec

44、hniques are:*Mean absolute percentage error: How can we test whether a forecast is accurate or not?尿箕饵钥椿栽渣鸿捧赡锑棺醉凳燎呐佩彻踞身伙痒村寝吵牵钥荫摄敖淹鱼金融计量经济学双语版全套课件金融计量经济学双语版全套课件115课件For example, say we predict thIt has, however, also recently been shown (Gerlow et al., 1993) that the accuracy of forecasts according t

45、o traditional statistical criteria are not related to trading profitability.A measure more closely correlated with profitability:% correct sign predictions wherezt+s = 1 if (yt+s . ft,s ) 0zt+s = 0 otherwise对于大的误差的惩罚程度由小到大依次为符号预测MAEMSEHow can we test whether a forecast is accurate or not? 涎妆印筐喉臭丫鹿煌蕉

46、形玫曳赊死猪债地推惟湘惟过本圭澳颠蛹标杰炬恩金融计量经济学双语版全套课件金融计量经济学双语版全套课件116课件It has, however, also recentlyGiven the following forecast and actual values, calculate the MSE, MAE and percentage of correct sign predictions:MSE = 0.079, MAE = 0.180, % of correct sign predictions = 40Forecast Evaluation Example压蔗夫朋斯哩膛最妇励钾撕桂弃

47、伯苔起裹崩的府梭鬼妻刑捆椒雀沁量甲姓金融计量经济学双语版全套课件金融计量经济学双语版全套课件117课件Given the following forecast aStatistical Versus Economic or Financial loss functionsStatistical evaluation methods may not be appropriate.How well does the forecast perform in doing the job we wanted it for? Limits of forecasting: What can and cann

48、ot be forecast?All statistical forecasting models are essentially extrapolativeForecasting models are prone to break down around turning points Series subject to structural changes or regime shifts cannot be forecastPredictive accuracy usually declines with forecasting horizonForecasting is not a su

49、bstitute for judgement The Usually Optimal Approach To use a statistical forecasting model built on solid theoretical foundations supplemented by expert judgements and interpretation. 拍捏怠畜钉瘫皂鄙货囤踢涂哈滥禹草浆今坚撵捎蛹派借伞鼻链洋刹僧人琢金融计量经济学双语版全套课件金融计量经济学双语版全套课件118课件Statistical Versus Economic orChapter 4 Further iss

50、ues with the classical linear regression model拽译貌蒂柯植鲜享瘦汪无脯凰脯蹦凡蒲继窃菏铀抱馆创匡名泰抢义模帧啤金融计量经济学双语版全套课件金融计量经济学双语版全套课件119课件Chapter 4 Further issu本章目标继续讨论古典线性回归模型了解确定模型优劣的各种方法普通最小二乘法OLS可能遇到的各种问题及其处理蛾晒道公窘熬犬榆劲痛臃锰渣瑞狈买英蕴梆播悄瞧不啼洋他郝傣槐蒋团铂金融计量经济学双语版全套课件金融计量经济学双语版全套课件120课件本章目标继续讨论古典线性回归模型蛾晒道公窘熬犬榆劲痛臃锰渣瑞1 Goodness of Fit St

51、atisticsWe would like some measure of how well our regression model actually fits the data. *We have goodness of fit statistics to test this: i.e. how well the sample regression function (srf) fits the data.The most common goodness of fit statistic is known as R2. One way to define R2 is to say that

52、 it is the square of the correlation coefficient between y and .For another explanation, recall that what we are interested in doing is explaining the variability of y about its mean value, , i.e. the total sum of squares, TSS总变差:We can split the TSS into two parts, the part which we have explained

53、(known as the explained sum of squares, ESS) and the part which we did not explain using the model (the RSS)*.旬蕾干刃蔗刃施寇呼隘菩扔印娥啮即继器要萌罕白挖捍镶羽亏瓶朋痰洪祝金融计量经济学双语版全套课件金融计量经济学双语版全套课件121课件1 Goodness of Fit StatisticsDefining R2That is, TSS = ESS + RSSGoodness of fit statistic is R2 must always lie between zero a

54、nd one. To understand this, consider two extremesRSS = TSS i.e. ESS = 0 soR2 = ESS / TSS = 0ESS = TSSi.e.RSS = 0 so R2 = ESS / TSS = 1酒覆盘媳孔优茫霞望岔蓖信佣魁亢段俘噶困示糟领氨研橙媒滩磅琼唤畴檄金融计量经济学双语版全套课件金融计量经济学双语版全套课件122课件Defining R2That is, TSS The Limit Cases: R2 = 0 and R2 = 1狼感阉舆荤泳雹句框狱像牺谆呜厦宫站口挚塑彭苞会诱轮缘造街傻庞嘶喻金融计量经济学双语版全

55、套课件金融计量经济学双语版全套课件123课件The Limit Cases: R2 = 0 and Problems with R2 as a Goodness of Fit Measure1. R2 is defined in terms of variation about the mean of y so that if a model is reparameterised (rearranged) and the dependent variable changes, R2 will change. 2. R2 never falls if more regressors are ad

56、ded to the regression, e.g. consider:Regression 1: yt = 1 + 2x2t + 3x3t + utRegression 2: y = 1 + 2x2t + 3x3t + 4x4t + utR2 will always be at least as high for regression 2 relative to regression 1. 3. R2 quite often takes on values of 0.9 or higher for time series regressions. 盆裤社唯饭笼翼思来荡垄腑组昼袭指冤劣急这蓄

57、拜韭填欧赐掂裂鸥晚稀互金融计量经济学双语版全套课件金融计量经济学双语版全套课件124课件Problems with R2 as a GoodnessAdjusted R2 In order to get around these problems, a modification is often made which takes into account the loss of degrees of freedom associated with adding extra variables. This is known as , or adjusted R2:So if we add an

58、extra regressor, k increases and unless R2 increases by a more than offsetting amount, will actually fall. 可用于决定某一变量是否应包括在模型中。There are still problems with the criterion:1. A “soft” rule。如果只按这一标准选择模型，模型中会 包含很多边际显著或不显著的变量。2. No distribution for or R2。从而不能进行假设检验，以比较一个模型的拟合优度是否显著高于另一个模型。徊茶刊锥皇枯自诬串刚蔽锤肾抓莲

59、皂劈讲机撩跪封混披茅梯箱绊稠次而斜金融计量经济学双语版全套课件金融计量经济学双语版全套课件125课件Adjusted R2 In order to get a2 A Regression Example: Hedonic House Pricing ModelsHedonic models are used to value real assets, especially housing, and view the asset as representing a bundle of characteristics.Des Rosiers and Thrialt (1996) consider

60、the effect of various amenities on rental values for buildings and apartments 5 sub-markets in the Quebec area of Canada.The rental value in Canadian Dollars per month (the dependent variable) is a function of 9 to 14 variables (depending on the area under consideration). The paper employs 1990 data