计量经济学英文课件共35页
计量经济学(英文版)精品PPT课件
(4.3a)
Expand and multiply top and bottom by n:
b2
=
nSxiyi - Sxi Syi nSxi2-(Sxi) 2
(4.3b)
Variance of b2
4.12
Given that both yi and ei have variance s2,
the variance of the estimator b2 is:
4. cov(ei,ej) = cov(yi,yj) = 0 5. xt c for every observation
6. et~N(0,s 2) <=> yt~N(b1+ b2xt,
The population parameters b1 and b2 4.4 are unknown population constants.
b2
+
nSxiEei - Sxi SEei nSxi2-(Sxi) 2
Since Eei = 0, then Eb2 = b2 .
An Unbiased Estimator
4.8
The result Eb2 = b2 means that the distribution of b2 is centered at b2.
4.6
The Expected Values of b1 and b2
The least squares formulas (estimators) in the simple regression case:
b2 =
nSxiyi - Sxi Syi nSxi22 -(Sxi) 2
b1 = y - b2x
计量经济学课件英文版 伍德里奇
Two methods to estimate
2016/9/22 Department of Statistics-Zhaoliqin 17
Some Terminology, cont.
β0 :intercept parameter β1 :slope parameter means that a one-unit increase in x changes the expected value of y by the amount β1,holding the other factors in u fixed.
Department of Statistics-Zhaoliqin 16
2016/9/22
Some Terminology, cont.
y = β0 + β1x + u, E [u|x] = 0. β0 + β1x is the systematic part of y. u is the unsystematic part of y. u is denoted the error term. Other terms for u : error shock disturbance residual (sometimes in reference to fitted error term)
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2016/9/22
Department of Statistics-Zhaoliqin
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Hence, we are interested in studying is the mean of wages given the years of Education that will be denoted as E [Wages|Education] Following economic theory, we assume a specific model for E[Wages|Education]
《计量经济学英文版》课件
2 ARMA模型
介绍ARMA模型的基本原 理和参数估计方法，以及 ARMA模型在经济数据分 析中的应用。
3 GARCH模型
讲解GARCH模型的原理 和估计方法，以及 GARCH模型在金融市场 波动预测中的应用。
计量经济模型的应用
金融市场分析
应用计量经济模型进行金融市 场的预测和分析。
政策评估
利用计量经济模型评估政策的 效果和影响。
面板数据模型
1
面板数据概述
介绍面板数据的特点和应用领域，以及面板数据模型的基本概念。
2
固定效应模型
讲解固定效应模型的估计和推断方法，以及固定效应模型的优缺点。
3
随机效应模型
介绍随机效应模型的估计和推断方法，以及随机效应模型与固定效应模型的比较。
时间序列模型
1 时间序列数据基本特
征
描述时间序列数据的基本 特征，如趋势、季节性和 周期性。
《计量经济学英文版》 PPT课件
本课程介绍了计量经济学的基本概念和分析方法。涵盖了经济数据与计量分 析、经典线性回归模型、面板数据模型、时间序列模型和计量经济模型的应 用。
课程概述
本节将概述本课程的内容，包括学习目标和涵盖的主要内容。通过本课程的学习，您将掌握计量经济学的基本 理论和实际应用。
经济数据与计量分析
企业ቤተ መጻሕፍቲ ባይዱ策
运用计量经济模型辅助企业的 决策制定。
总结
通过本课程的学习，您将获得计量经济学分析的基本知识和技能，能够应用计量经济模型进行实证研究，并在 实际问题中运用计量经济学的方法和工具。
• 经济数据的概念与特点 • 统计概率基础 • 经济计量分析的目的与意义
经典线性回归模型
普通最小二乘法
计量经济学导论第四版英文完整教学课件
Economics 20 - Prof. Anderson
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The Question of Causality
Simply establishing a relationship between variables is rarely sufficient Want to the effect to be considered causal If we’ve truly controlled for enough other variables, then the estimated ceteris paribus effect can often be considered to be causal Can be difficult to establish causality
Need to use nonexperimental, or observational, data to make inferences
Important to be able to apply economic theory to real world data
Economics 20 - Prof. Anderson
3
Why study Econometrics?
An empirical analysis uses data to test a theory or to estimate a relationship
A formal economic model can be tested
Theory may be ambiguous as to the effect of some policy change – can use econometrics to evaluate the program
计量经济学(第五版)课件 ets1
计量经济学的三个要素
计量经济学的三个要素是经济理论、经济数据和统 计方法。对于解释经济现象来说，“没有计量的理论 ”和“没有理论的计量”都是不够的，正如计量经济 学创始人之一的弗里希所强调的那样，它们的结合是 计量经济学的发展能够取得成功的关键。
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计量经济学是经济预测的科学
计量经济学从根上说，是对经验规律的认识以及将 这些规律推广为经济学“定律”的系统性努力，这些 “定律”被用来进行预测，即关于什么可能发生或者 什么将会发生的预测。因此，广义地说，计量经济学 可以称为经济预测的科学。
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二．计量经济学的产生和发展 1.产生年代
计量经济学产生于上世纪三十年代。 • 1930年12月，弗里希（R. Frisch）、丁伯根 (J. Tinbergen)和费歇尔(I. Fisher)等经济学家在美国克利 夫兰成立计量经济学会。
• 1933 年 起 ， 定 期 出 版 《 计 量 经 济 学 》 (Econometrica)杂志。弗里希在该杂志发刊词中明确 提出计量经济学的范围和方法，指出计量经济学是 经济理论、数学和统计学的综合，但它又完全不同 于这三个学科中的每一个。
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3. 学科发展环境
同时，随着科学技术的发展，各门学科相互渗透，数 学、系统论、信息论、控制论等相继进入经济研究领 域，使经济科学进一步数量化，有助于计量经济学的 发展。高速电子计算机的出现和发展，为计量经济技 术的广泛应用铺平了道路。
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4. 发展过程
上世纪三十年代，侧重于个别商品供给与需求的计 量，基本上属于个量分析或微观分析。
自四十年代起，计量经济研究的范围扩大到整个经 济体系，其特征是宏观经济总量的计量分析，亦即总 量分析或宏观分析。
五十年代起，在计量经济学的理论和方法得到迅速 发展的同时，宏观计量经济模型在计量经济学的应用 中开始占重要地位。50年代末至60年代初是宏观计量 经济模型蓬勃发展的时期。
计量经济学(英文PPT)Chapter 2 TWO-VARIABLE REGRESSION ANALYSIS SOME BASIC IDEAS
is a linear function of X i ,say, of the type
E(Y | X i ) 1 2 X i (2.2.2)
1 and 2 are known as the regression coefficients. Equation（2.2.2）itself is known as the linear population regression function or simply linear population regression.
If we want to get the relationship between weekly family consumption expenditure (Y) and weekly family income (X).
In the hypothetical community, there is a total population of 60 families. The 60 families are divided into 10 income groups (from $80 to $260) . And assume that we get the observations given in Table 2.1.
Therefore, we can express the deviation of an individual Yi around its expected value as follows:
计量经济学(英文PPT)Chapter 0 Introduction
is：2
,
namely,
－231.8
and
0.7194.
Thus,
the
estimated
Yˆ 231.8 0.7194 X
We see that，for the period 1980—1992, MPC≈0.72 in America, suggesting that for the sample period an increase in real income of 1 dollar led, on average, to an increase of about 72 cents in real
However, the establishment of World Econometric Society in December 29th, 1930 and the publication of its academic journal Econometrics in 1933 are generally acknowledged as a landmark of econometrics as a separate discipline.
Why a separate discipline?
Economic theory makes statements or hypotheses that are mostly qualitative in nature; It is the job of the econometrician to provide such numerical estimates.
Example: Keynesian theory of consumption
计量经济学(英文PPT)Chapter 3 TWO-VARIABLE REGRESSION MODEL-THE PROBLEM OF ESTIMATION
THE PROBLEM OF ESTIMATION
§3.1 THE METHOD OF ORDINARY LEAST SQUARES
The Method of ordinary least squares is attributed to Carl Friedrich Gauss, a German mathematician.
Ⅱ．The estimators which are point estimators are different from interval estimators.
Ⅲ．Once the OLS estimators are obtained from the sample data, the sample regression line can be easily obtained. The regression line thus obtained has the following properties:
Yˆi ˆ1 ˆ2 X i (Y ˆ2 X ) ˆ2 X i Y ˆ2 ( X i X )
while ∵ (Xi X ) 0
∴sum the equation above for the sample value on both sides and divide the result through by n( sum for i，then dived by n),
least-squares estimators, for they are derived from the least-
squares principle.
note：
计量经济学英文课件 (3)
Z
b2 2
2
xi x
2
~ N 0,1
(3.1)
The standardized random variable Z is normally distributed with mean 0 and variance 1.
Slide 3-4
Principles of Econometrics, 3rd Edition
and its standard error
se(b2 ) var(b2 ) 4.38 2.09
Principles of Econometrics, 3rd Edition
Slide 3-12
A “95% confidence interval estimate” for 2:
The critical value tc for degrees of freedom m is the percentile value t1 2,m .
Principles of Econometrics, 3rd Edition
Slide 3-9
Figure 3.1 Critical Values from a t-distribution
Principles of Econometrics, 3rd Edition
Slide 3-16
The Null Hypothesis
parameter.
The null hypothesis, which is denoted H0 (H-naught), specifies a value for a regression The null hypothesis is stated H 0 : k c, where c is a constant, and is an important value in the context of a specific regression model.
计量经济学课件
WEEK 10: MACROECONOMETRICS Introduction1.The concept of stationarity2.Spurious regressions3.Testing for unit roots4.Cointegration analysis1. S TATIONARITYConditions for t y to be a stationary time series process i. t E y constant t ii. t Var y constant tiii. ,t t k Cov y y constant t and all k≠0 Autoregressive time series1t t t y y- Notice no constant and t is a white noise error term.- AR(1) model – time series behaviour of t y is largely explained by its value in the previous period.- Necessary condition for stationarity 1 , if , 1 series is explosive and if 1 have a unit root.Example 1 – Stationary AR(1) ModelSTATA codeset obs 500 /*set number of observations*/gen time=_n /*create time trend*/gen y=0 if time==1 /* first observation set y=0*/gen e=rnormal(0, 1) /*create a random number*/replace y=(0.67*y[_n-1])+e if time~=1 /*AR(1) model =0.67*/ twoway (line y time) /*line plot*/Example 2 – Explosive AR(1) ModelSTATA codeset obs 500 /*set number of observations*/gen time=_n /*create time trend*/gen y=0 if time==1 /* first observation set y=0*/gen e=rnormal(0, 1) /*create a random number*/replace y=(1.16*y[_n-1])+e if time~=1 /*AR(1) model =1.16*/ twoway (line y time) /*line plot*/Example 3 – Non-stationary AR(1) ModelSTATA codeset obs 500 /*set number of observations*/gen time=_n /*create time trend*/gen y=0 if time==1 /* first observation set y=0*/ gen e=rnormal(0, 1) /*create a random number*/ replace y=y[_n-1]+e if time~=1 /*AR(1) model =1*/ twoway (line x time) /*line plot*/ Noticety is not mean reverting. Random walk =1In the model:1t t t y yif 1 then t y is said to contain a UNIT ROOT i.e. is non-stationarySo 1t t t y y subtract 1t y from the LHS and RHS:111t t t t t y y y yt t y and because t is white noise t y is a stationary series.Example 3 (continued) – Non-stationary AR(1) Model and First Difference- A series t y is integrated of order one, i.e. t y I (1), and contains a unit root if t y is non-stationary but t y is a stationary series.- Possible that the series t y needs to be differenced more than once to achieve a stationary process.- A series t y is integrated of order d , i.e. t y I (d) if t y is non-stationary but d t y is a stationary series: Note: 211t t t t t t t y y y y y y y2.S PURIOUS REGRESSIONWhy worry whether t y is stationary?Most macroeconomic time series are trended and in most cases non-stationary processes.Using OLS to model non-stationary data can lead to problems and incorrect conclusions.a.high R squared often >0.95b.high t valuesc.theoretically variables in the analysis have no interrelationship Why does non-stationarity arise in macro data?Economic time series e.g. GDP, money supply, employment, all tend to grow at an annual rate.Such series non-stationary as the mean is continually rising. Even after differencing the series cannot be made stationary.So, usually take logarithms of time series data before undertaking econometric analysis.Take logarithm of a series which exhibits an average growth rate it will follow a linear trend and become an integrated series, i.e. one which is stationarity after differencing.Consider t y which grows by 10% per period, thus11.1t t y yTake the log of both the LHS and RHS, then1log log 1.1log t t y yThe lagged dependent variable has a unit coefficient and log 1.1 is a constant. The series would now be I(1), see example 3.Consider the model01t t t y xCLRM assumptions require that both variables have zero mean and constant variance (i.e. stationary, see (1i and 1ii)).If these assumption are violated and the series are non-stationary Granger and Newbold (1974) proved that results obtained are totally spurious. Granger, C. and P. Newbold (1974) Spurious regressions in econometrics. Journal of Econometrics, 2, 111-120.‘Rule of thumb’ for detecting a spurious regression:or,a.2R DWRb.12Logic behind spurious regression.Consider two unrelated series that are non-stationary, then:– both either together, or one will whilst the other .– Either way likely to find a +ve or –ve significant relationship.PROFESSOR KARL TAYLOR ECN6540 2017-18 Example 4 – Spurious Regression: Artificial DataSource | SS df MS Number of obs = 500-------------+------------------------------ F( 1, 498) = 143.04Model | 9096.19185 1 9096.19185 Prob > F = 0.0000Residual | 31668.4706 498 63.5913065 R-squared = 0.2231-------------+------------------------------ Adj R-squared = 0.2216Total | 40764.6625 499 81.6927104 Root MSE = 7.9744------------------------------------------------------------------------------y | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------x | .9443085 .0789556 11.96 0.000 .7891813 1.099436_cons | 3.505432 .4156699 8.43 0.000 2.688749 4.322114------------------------------------------------------------------------------ Durbin-Watson d-statistic (2, 500) = 0.0316917Example 5 – Spurious Regression: Economic DataRegress by OLS the logarithm of GDP against the logarithm of M2. Quarterly time series data over the period 1975Q1 until 1997Q4.01log log tt t gdp muse "C:\Karl's files\2016-17\ECN6540\LECTURES\spurious1.dta", cleargen date=q(1975q1)+_n-1format date %tq tsset dategen lgdp=log(gdp) gen lm=log(m) reg lgdp lm estat dwatsonSource | SS df MS Number of obs = 92 -------------+------------------------------ F( 1, 90) = 547.56 Model | 1.78659606 1 1.78659606 Prob > F = 0.0000 Residual | .29365627 90 .003262847 R-squared = 0.8588 -------------+------------------------------ Adj R-squared = 0.8573 Total | 2.08025233 91 .022859916 Root MSE = .05712 ------------------------------------------------------------------------------ lgdp | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lm | .219892 .0093971 23.40 0.000 .201223 .2385611 _cons | 3.075618 .0571457 53.82 0.000 2.962088 3.189147 ------------------------------------------------------------------------------ Durbin-Watson d-statistic( 2, 92) = 0.032836Regression fits the data well – both variables trended.Why is this regression spurious?In the model 01t t t y x there are four possible scenarios:a. Both t y and t x are stationary processes and so the CLRM is appropriate and OLS is BLUE;b. t y and t x are integrated of difference orders, e.g. I(0) and I(1), – the regression is now meaningless, since t y has a constant mean and t x drifts over time;c. t y and t x are integrated of the same order, e.g. I(1), and t contains a stochastic trend, i.e. I(1). This is the case of a spurious regression. Could re-estimate the model in first differences.d. t y and t x are integrated of the same order,e.g. I(1), and t is a stationary process, i.e. I(0). In this special case t y and t x are said to be cointegrated .Hence testing for non-stationarity is extremely important.3. T ESTING FOR UNIT ROOTSTesting the order of integration is a test for the number of unit roots.i. Test t y to see if stationary. If stationary then t y I (0); if not stationary then t y I (d); d >0.ii. Take first differences of t y i.e. 1t t t y y y then test t y to see ifstationary. If stationary then t y I (1); if not stationary then t y I (d); d >0.iii. Take the second difference of t y i.e. 21t t t t y y y y then test 2t yto see if stationary. If stationary then t y I (2); if not stationary then t y I (d); d >0.Continue process until stationary.Dickey-Fuller Test for Unit RootsDickey and Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association , 74, 427-431.Dickey and Fuller (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica , 49, 1057-1072.Test based on testing for the existence of a unit root.Start with an AR(1) model:1t t t y yTo test for a unit root 0:1H 1:1HCan re-write the above model by subtracting 1t y from both sides:111111t t t t t tt t t t y y y y y y y(1)1To test for a unit root the null and alternative hypotheses are 0:0H 1:0HDickey and Fuller proposed two alternative regression equations which can be used for testing the presence of a unit root:1t t t y y (2)Testing for a unit root based upon eq. (1) is only valid if the d.g.p has a zero mean and no trend. So eq. (2) includes a constant in the random walk process.1t t t y t y (3)Allows for a non zero mean and trend component in the series.The DF test for a unit root is based upon a conventional t test on from one of the three models.Critical values (based upon Mackinnon, 1991): MODEL 1% 5% 10% 1t t t y y 1t t t y y -3.43 -2.86 -2.57 1t t t y t y -3.96 -3.41 -3.13 Standard critical values -2.33 -1.65 -1.28If t statistic is greater than the critical value then the null hypothesis of a unit root is rejected and can conclude that t y is a stationary process.3.1 Augmented Dickey-Fuller Test for Unit RootsUnlikely that the error term t in eqs. (1) to (3) is white noise – auto/serial correlation.Dickey and Fuller proposed augmenting the DF test by including extra lagged terms of the dependent variable to eliminate the presence of autocorrelation.11p k t k k t t t y y y (4)11p k t k k t t t y y y (5)11p k t k k t t t y y y t (6)Again the difference between eqs. (4) to (6) concerns the inclusion of a constant and trend.An important consideration is the optimal lag length p.-If p is too small then the remaining autocorrelation will bias the test.-If p is too large then the power of the test will suffer.Ng and Perron (1995) suggest firstly, set an upper bound for p i.e. p . Then estimate the ADF test based on the p lag length.If the absolute value of the t-statistic for testing the significance of the last lagged difference is greater than 1.6 then set p=p and perform the unit root test. Otherwise, reduce the lag length by one and repeat the process.Rule of thumb (Schwert, 1989),0.25int12100Tp3.2 Kwiatkowski-Phillips-Schmidt-Shin (1992)Testing the null hypothesis of stationarity against the alternative of a unit root. How sure are we that economic time series have a unit root? Journal of Econometrics , 54, 159-178.KPSS test differs to the DF and ADF null hypothesis is stationarity. Model decomposed into trend, random walk (t r ) and a stationary error term:21,0u t tt t tt t IID u u r r r t yThe initial value of t r is fixed and serves the role of the intercept 0r . Thestationary null hypothesis is that 20u since tis stationary, hence under the null hypothesis t y is trend stationary.To undertake the test:i. Regress t y of on an intercept and time trend, i.e. t t y t ;ii. Save the OLS residuals from (i) ˆt t eand compute the partial sum process, i.e. 1t s t s S e ;iii. Test statistic is LM, given by:221ˆT tt KPSS S .2ˆis an estimate of the error variance RSS/T (may be corrected for autocorrelation);iv. Critical value at 5% level 0.145. If trend omitted from (i) then the criticalvalue at the 5% level is 0.463.Example 6 – Unit Root Tests – ADF test: Nelson & Plosser U.S. Data Nelson, C.R. and Plosser, C.I. (1982), Trends and Random Walks in Macroeconomic Time Series, Journal of Monetary Economics, 10, 139–162.clear all /*clear memory*/use /ec-p/data/macro/nelsonplosser.dta /*load data*/ keep year lip lsp500 /*keep a subset of Nelson and Plosser variables*/drop if lip==. | lsp500==. /*drop any missing observation*/tsset year /*set year as time identifier*/twoway (line lip year) (line lsp500 year, yaxis(2)) /*plot data over time*//*ADF tests on industrial production*/dfuller lip, regress noconstant lags(3) /*ADF test no constant or trend eq.(4) */ dfuller lip, regress lags(3) /*ADF test constant no trend eq.(5) */dfuller lip, regress trend lags(3) /*ADF test constant and trend eq.(6) *//*ADF tests on S&P 500 index*/dfuller lsp500, regress noconstant lags(3) /*ADF test no constant or trend eq.(4) */ dfuller lsp500, regress lags(3) /*ADF test constant no trend eq.(5) */dfuller lsp500, regress trend lags(3) /*ADF test constant and trend eq.(6) *//*ADF tests on industrial production*/dfuller lip, regress noconstant lags(3) /*ADF test no constant or trend eq.(4) */ Augmented Dickey-Fuller test for unit root Number of obs = 96---------- Interpolated Dickey-Fuller ---------Test 1% Critical 5% Critical 10% CriticalStatistic Value Value Value------------------------------------------------------------------------------Z(t) 2.640 -2.602 -1.950 -1.610------------------------------------------------------------------------------D.lip | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------lip |L1. | .0121854 .0046149 2.64 0.010 .0030198 .0213511LD. | .0855473 .1060297 0.81 0.422 -.1250368 .2961313L2D. | -.0727104 .1059196 -0.69 0.494 -.2830758 .1376551L3D. | .0177574 .1045445 0.17 0.865 -.189877 .2253918------------------------------------------------------------------------------dfuller lip, regress lags(3) /*ADF test constant no trend eq.(5) */Augmented Dickey-Fuller test for unit root Number of obs = 96 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -0.687 -3.516 -2.893 -2.582 ------------------------------------------------------------------------------------------------------------------------------------------------------------ D.lip | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lip |L1. | -.006546 .0095294 -0.69 0.494 -.025475 .0123831 LD. | .05638 .1046247 0.54 0.591 -.1514442 .2642041 L2D. | -.0932085 .1041037 -0.90 0.373 -.2999977 .1135807 L3D. | -.0161771 .1034743 -0.16 0.876 -.2217161 .1893618 _cons | .0627725 .0281174 2.23 0.028 .0069208 .1186242 ------------------------------------------------------------------------------dfuller lip, regress trend lags(3) /*ADF test constant and trend eq.(6) */Augmented Dickey-Fuller test for unit root Number of obs = 96---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -3.298 -4.049 -3.454 -3.152 ------------------------------------------------------------------------------------------------------------------------------------------------------------ D.lip | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lip |L1. | -.2207902 .0669447 -3.30 0.001 -.3537874 -.0877929 LD. | .168619 .1054778 1.60 0.113 -.040931 .378169 L2D. | .0151678 .10462 0.14 0.885 -.1926782 .2230138 L3D. | .0831198 .1031807 0.81 0.423 -.1218666 .2881062 _trend | .0088867 .0027512 3.23 0.002 .0034209 .0143524 _cons | .1611592 .0405474 3.97 0.000 .0806046 .2417137 ------------------------------------------------------------------------------Note can find given 10.220810.220810.7792/*ADF tests on S&P 500 index*/dfuller lsp500, regress noconstant lags(3) /*ADF test no constant & trend eq.(4) */ Augmented Dickey-Fuller test for unit root Number of obs = 96---------- Interpolated Dickey-Fuller ---------Test 1% Critical 5% Critical 10% CriticalStatistic Value Value Value------------------------------------------------------------------------------Z(t) 1.567 -2.602 -1.950 -1.610------------------------------------------------------------------------------D.lsp500 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------lsp500 |L1. | .0106279 .0067804 1.57 0.120 -.0028386 .0240944LD. | .2531512 .1064478 2.38 0.019 .0417367 .4645658L2D. | -.1932357 .107425 -1.80 0.075 -.406591 .0201196L3D. | -.017031 .1064838 -0.16 0.873 -.228517 .194455------------------------------------------------------------------------------dfuller lsp500, regress lags(3) /*ADF test constant no trend eq.(5) */ Augmented Dickey-Fuller test for unit root Number of obs = 96 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) 0.059 -3.516 -2.893 -2.582 ------------------------------------------------------------------------------------------------------------------------------------------------------------ D.lsp500 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lsp500 |L1. | .0011734 .0200279 0.06 0.953 -.0386096 .0409564 LD. | .2612615 .1080976 2.42 0.018 .046539 .475984 L2D. | -.1856923 .1089062 -1.71 0.092 -.4020212 .0306366 L3D. | -.0079754 .1084307 -0.07 0.942 -.2233596 .2074089 _cons | .0252099 .0502234 0.50 0.617 -.0745527 .1249725 ------------------------------------------------------------------------------dfuller lsp500, regress trend lags(3) /*ADF test constant and trend eq.(6) */ Augmented Dickey-Fuller test for unit root Number of obs = 96---------- Interpolated Dickey-Fuller ---------Test 1% Critical 5% Critical 10% CriticalStatistic Value Value Value------------------------------------------------------------------------------Z(t) -2.121 -4.049 -3.454 -3.152------------------------------------------------------------------------------------------------------------------------------------------------------------D.lsp500 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------lsp500 |L1. | -.0969978 .0457218 -2.12 0.037 -.1878321 -.0061635LD. | .3015991 .1068021 2.82 0.006 .0894181 .5137802L2D. | -.1405117 .1079217 -1.30 0.196 -.3549171 .0738936L3D. | .0396776 .1076543 0.37 0.713 -.1741965 .2535517_trend | .0032803 .0013813 2.37 0.020 .0005362 .0060245_cons | .0942689 .0569704 1.65 0.101 -.0189127 .2074505------------------------------------------------------------------------------So both industrial production and the S&P 500 index definitely not stationary over the period, i.e. not I(0).Order of integration?Take first difference then undertake test again.In STATA the first difference operator is D, second difference operator is D2 /*ADF tests on industrial production first differenced*/dfuller D.lip, lags(3) /*ADF test constant no trend eq.(5) */dfuller D.lip, trend lags(3) /*ADF test constant and trend eq.(6) *//*ADF tests on S&P 500 index first differenced*/dfuller D.lsp500, lags(3) /*ADF test constant no trend eq.(5) */dfuller D.lsp500, trend lags(3) /*ADF test constant and trend eq.(6) *//*ADF tests on industrial production first differenced*/dfuller D.lip, lags(3) /*ADF test constant no trend eq.(5) */Augmented Dickey-Fuller test for unit root Number of obs = 95 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -5.624 -3.517 -2.894 -2.582 ------------------------------------------------------------------------------dfuller D.lip, trend lags(3) /*ADF test constant and trend eq.(6) */ Augmented Dickey-Fuller test for unit root Number of obs = 95---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -5.600 -4.051 -3.455 -3.153 ------------------------------------------------------------------------------/*ADF tests on S&P 500 index first differenced*/dfuller D.lsp500, lags(3) /*ADF test constant no trend eq.(5) */Augmented Dickey-Fuller test for unit root Number of obs = 95 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -5.996 -3.517 -2.894 -2.582 ------------------------------------------------------------------------------dfuller D.lsp500, trend lags(3) /*ADF test constant and trend eq.(6) */ Augmented Dickey-Fuller test for unit root Number of obs = 95---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -6.149 -4.051 -3.455 -3.153 ------------------------------------------------------------------------------Hence both industrial production and the S&P 500 index are I(1).4.C OINTEGRATION-Trended data can create problems due to spurious regressions.-Most macro variables are trended and hence non-stationary.-Hence the problem of spurious regressions is likely in macro models. Solution?Difference the data until it becomes stationary.Problemsi.If the model is correctly specified in both y and x then differencing bothvariables means the error process is also differenced non-invertibleMA process, estimation difficulties;ii.Once the model is differenced it can no longer have a unique long run solution.4.1 Cointegration DefinitionsWhere the regression of one non-stationary variable y on one or more non-stationary variables ,,,12k x x x results in a non spurious regression.Then a long run equilibrium relationship exists between the variables.Hence cointegration should only occur where there is a relationship linking the variables.We will only consider the case of two variables t y and t x i.e. not multivariate cointegration.If there is a long run equilibrium relationship between t y and t x , despite them both rising over time (trended), then a linear combination of the two variables must be I(0).A linear combination can be taken from the model:01t t t y xConsider the residuals:01ˆˆˆt t t t e y xIf t e I (0) then t y and t x are said to be cointegrated.10, is known as the cointegrating vector.Definitions:i. Time series t y and t x cointegrated of order d , b where d ≥b ≥0, which canbe written as t t x y , CI (d,b ), if both series are I(d ) and a linear combinationexists between the variables integrated of order I(d-b ). Then ,12 is the cointegrating vector (there is only one).ii. Generalisation, multivariate cointegration: let t Z denote an n×1 vector ofthe series ,,,,123nt t t t Z Z Z Z , if each it Z is I(d ) and an n×1 vector exists such that ' t Z I (d-b ) then it Z CI (d,b ). Can be more than one cointegrating vector.4.2Testing for Cointegration in single equationsEngle, R. and C. Granger (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica, 35, 251-276.Step 1-By definition cointegration requires that the variables are integrated of the same order I(d).-So apply ADF tests to infer the number of unit roots in each variable.i.If both t y and t x are I(0) then OLS and appropriate.ii.If t y and t x are integrated to different orders, e.g. I(0) and I(1), then they can-not be cointegrated.iii.If both t y and t x are integrated to the same order, e.g. I(1), then go to step 2.Step 2- If both t y and t x are integrated to the same order, in macro data usually I(1), then estimate the long run equilibrium relationship:01t t t y x- Obtain the residuals from the model, t e .- If there is no cointegration then the results will be spurious (see section 2).- If the variables are cointegrated then OLS yields super-consistent estimatesfor the cointegrating parameter 1ˆ .Step 3- Test the residuals from step 2 for stationarity using the ADF test:11p k t k k t t t u e e e- If t e I (0) then t y and t x are cointegrated.- Note critical value differ to those standard ADF tests:Critical values (based upon Engle-Granger, 1987): MODEL 1% 5% 10%Lags (ADF) -3.73 -3.17-2.91Example 7 – Engle Granger Cointegration Use King et al. data for the U.S.:King, R. G., C. I. Plosser, J. H. Stock, and M. W. Watson. 1991. Stochastic trends and economic fluctuations. American Economic Review , 81, 819–840.Model logarithm of consumption (c) as a function of the logarithm of gdp (y).01log log t t t c yclear all /*clear memory*/use /data/r11/balance2 /*load data*/ tsset time /*set year as time identifier*/ twoway (line y time) (line c time, yaxis(2))/*STEP 1 - order of integration*//*ADF tests on real gdp*/dfuller y, regress trend lags(4) /*ADF test constant and trend eq.(6)*/ /*ADF tests on consumption*/dfuller c, regress trend lags(4) /*ADF test constant and trend eq.(6)*/ /*STEP 2 - estimate long run relationship*/regress c ypredict e, resid /*gain residuals*/estat dwatson /*durbin watson statistic*//*STEP 3 - test residuals for stationarity*/dfuller e, regress noconstant lags(4) /*ADF test on residual*//*step 1 - order of integration*//*ADF tests on real gdp*/Dfuller y, regress trend lags(4) /*ADF test constant and trend eq.(6)*/Augmented Dickey-Fuller test for unit root Number of obs = 91---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -2.133 -4.060 -3.459 -3.155 ------------------------------------------------------------------------------------------------------------------------------------------------------------ D.y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- y |L1. | -.0089805 .0042109 -2.13 0.036 -.0173543 -.0006067 LD. | .6394239 .1096428 5.83 0.000 .4213873 .8574606 L2D. | -.0055359 .1296175 -0.04 0.966 -.2632945 .2522227 L3D. | .0906288 .1302257 0.70 0.488 -.1683393 .3495969 L4D. | .0008952 .1130575 0.01 0.994 -.2239321 .2257225 _trend | .0001668 .0000632 2.64 0.010 .0000411 .0002926 _cons | .0260383 .0118528 2.20 0.031 .0024678 .0496088 ------------------------------------------------------------------------------/*ADF tests on consumption*/dfuller c, regress trend lags(4) /*ADF test constant and trend eq.(6)*/ Augmented Dickey-Fuller test for unit root Number of obs = 91 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -1.622 -4.060 -3.459 -3.155 ------------------------------------------------------------------------------------------------------------------------------------------------------------ D.c | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- c |L1. | -.0063104 .0038897 -1.62 0.108 -.0140456 .0014248 LD. | .6579478 .1068026 6.16 0.000 .4455591 .8703365 L2D. | .1038069 .1289307 0.81 0.423 -.152586 .3601998 L3D. | .2031445 .1317853 1.54 0.127 -.0589249 .465214 L4D. | -.2282238 .1125153 -2.03 0.046 -.4519729 -.0044747 _trend | .0001326 .0000526 2.52 0.014 .0000279 .0002372 _cons | .0180606 .0109068 1.66 0.101 -.0036288 .0397499 ------------------------------------------------------------------------------。
计量经济学英文课件 (1)
1.2 What is Econometrics About
1.2.1 Some Examples
•
Every day, decision-makers face ‘‘how much’’ questions
: (Continued)
– A real estate developer must predict by how much population and income will increase to the south of Baton Rouge, Louisiana, over the next few years, and whether it will be profitable to begin construction of a gambling casino and golf course
– You must decide how much of your savings will go into a stock fund, and how much into the money market. This requires you to make predictions of the level of economic activity, the rate of inflation, and interest rates over your planning horizon
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One-Sided Alternatives (cont)
Having picked a significance level, a, we look up the (1 – a)th percentile in a t distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null
Under the CLM assumptions, conditional on the sample values of the independent variable s
bˆ j ~ Normal b j ,Var bˆ j , so that
bˆ j b j sd bˆ j ~ Normal 0,1
7
t Test: One-Sided Alternatives
Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided
because we have to estimate s 2by sˆ 2
Note the degrees of freedom : n k 1
5
The t Test (cont)
Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis
9
One-Sided Alternatives (cont)
yi = b0 + b1xi1 + … + bkxik + ui
H0: bj = 0
H1ห้องสมุดไป่ตู้ bj > 0
Fail to reject
H1: bj > 0 and H1: bj < 0 are one-sided H1: bj 0 is a two-sided alternative
If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%
"the"t statistic for bˆj :tbˆ j bˆ j se bˆ j
We will then useour t statistic along with a rejection rule to determine whether ot accept thenull hypothesis, H0
mean and variance s2: u ~ Normal(0,s2)
1
CLM Assumptions (cont)
Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows
bˆj is distributed normally becauseit
is a linear combinatio n of the errors
4
The t Test
Under the CLM assumptions
bˆ j b j se bˆ j ~ tnk1
Note this is a t distributi on (vs normal)
For example, H0: bj=0
If accept null, then accept that xj has no effect on y, controlling for other x’s
6
The t Test (cont)
To performour test we first need to form
2
The homoskedastic normal distribution with a single explanatory variable
y
f(y|x)
.
.
E(y|x) = b0 + b1x
Normal distributions
x1
x2
3
Normal Sampling Distributions
Assumptions of the Classical Linear Model (CLM)
So far, we know that given the GaussMarkov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2,…, xk and u is normally distributed with zero
y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality