回归分析讲义

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北大暑期课程《回归分析》(Linear-Regression-Analysis)讲义3

Class 3: Multiple regressionI. Linear Regression Model in Matrices For a sample of fixed size y n i,,1 is the dependent variable; 11,,1 p x x Xare independent variables. We can write the model in the following way:(1) X y ,wheren y y y (1))1()1(1211211...1......1p n p n n x x x x x x Xn (1)and110....p[expand from the matrix form into the element form]Assumption A0 (model specification assumption):X y R )(We call R(Y) the regression function. That is, the regression function of y is a linear function of the x variables. Also, we assume nonsingularity of X'X . That is, we have meaningful X 's..........21 n y y yII. Least Squares Estimator in Matrices Pre-multiply (1) by X ' (2),'''1 X X X y X pAssumption A1 (orthogonality assumption): we assume that is uncorrelated with each and every vector in X . That is,(3).0)(0),(0)(0),(0)(0),(0)(112111 p p x E x Cov x E x Cov x E x Cov ESample analog of expectation operator is n1. Thus, we have(4)01010101)1(21 i p i i i i i i x nx n x n n That is, there are a total of p restriction conditions, necessary for solving p linear equations to identify p parameters. In matrix format, this is equivalent to:(5).][or,][1o X o X nSubstitute (5) into (2), we have (6))()(X X y XThe LS estimator is then:(7),)(1y X X X bwhich is the same as the least squares estimator. Note: A1 assumption is needed for avoiding biases. III. Properties of the LS Estimator For the mode X y ,A1]using result, [important ][)(][)(])[(])[()]()[(])[()()(1111111o X E X X E X X X X X X X E X X X X E X X X X E y X X X E b E y X X X bthat is, b is unbiased.V (b ) is a symmetric matrix, called variance and covariance matrix of b .)(......)...(),(...),()()(1110100p b V b V b b Cov b b Cov b V b V 1111111)(][)()on al (condition ])[()])()[()]()[(])[()( X X X V X X X X X X X V O X X X X X X X V X X X X V y X X X V b V(after assuming I V 2][ non-serial correlation and homoscedasticity)21)( X X [important result, using A2][blackboard ]22001. Assumption A2 (iid assumption): independent and identically distributed errors. Two implications:1. Independent disturbances, j i E j i ,0),(Obtaining neat v (b ).2. Homoscedasticity, j i v E i j i ,)()(2Obtaining neat v (b ).I V 2)( , scalar matrix.IV. Fitted Values and Residualsy H y X X X X b X y 1)(ˆ X X X X H n n 1)(is called H matrix, or hat matrix. H is an idempotent matrix:H HHFor residuals:y H I y H y yy e )(ˆ (I-H ) is also an idempotent matrix.V. Estimation of the Residual Variance A. Sample Analog (8))()]([)(222i i i i E E E Vis unknown but can be estimated by e , where e is residual. Some of you may have noticedthat I have intentionally distinguished from e . is called disturbance, and e is called residual. Residual is defined by the difference between observed and predicted values.The sample analog of (8) is2)1()1(2211022)]([1)ˆ(11 p i p i i i i i i x b x b x b b y ny y n e nIn matrix:e e e i 2The sample analog is thenn e e /B. Degrees of FreedomAs a general rule, the correct degrees of freedom equals the number of totalobservations minus the number of parameters used in estimation.In multiple regression, there are p parameters to be estimated. Therefore, theremaining degrees of freedom for estimating disturbance variance is n-p . C. MSE as the EstimatorMSE is the unbiased estimator. It is unbiased because it corrects for the loss ofdegrees of freedom in estimating the parameters.ee p n MSE e pn MSE i112D. Statistical InferencesNow that we have point estimates (b ) and the variance-covariance matrix of b . But wecannot do formal statistical tests yet. The question, then, is how to make statistical inferences, such as testing hypotheses and constructing confidence intervals. Well, the only remaining thing we need is the ability to use some tests, say t , Z , or F tests.Statistical theory tells us that we can conduct such tests if e is not only iid, but iid in anormal distribution. That is, we assumeAssumption A3 (normality assumption): i is distributed as ),0(2NWith this assumption, we can look up tables for small samples.However, A3 is not necessary for large samples. For large samples, central limit theoryassures that we can still make the same statistical inferences based on t, z , or F tests if the sample is large enough.A Summary of Assumptions for the LS Estimator 1. A0: Specification assumptionX X y )|(EIncluding nonsingularity of X X .Meaningful X 's.With A0, we can computey X X X b 1)(2. A1:orthoganality assumption0)(k x E , for k = 0, .... p-1, x 0 = 1.Meaning: 0)( E is needed for the identification of 0 .All other column vectors in X are orthogonal with respect to .A1 is needed for avoiding biases. With A1, b is unbiased and consistent estimator of . Unbiasedness means that)(b EConsistency: n b as .For large samples, consistency is the most important criterion for evaluating estimators.3. A2. iid independent and identically distributed errors. Two implications:1. Independent disturbances, j i Cov j i ,0),(Obtaining neat v(b).2. Homoscedasticity, j i v Cov i j i ,)(),(2Obtaining neat v(b).I V 2)( , scalar matrix.With A2, b is an efficient estimator.Efficiency: an efficient estimator has the smallest sampling variance among all unbiased estimators. That is),ˆ()(Var somehow b Var whereˆ denotes any unbiased estimator. Roughly, for efficient estimators, imprecision [i.e., SD(b )] decreases by the inverse of the square root of n . That is, if you wish to increase precision by 10 times, (i.e., reduce S.E. by a factor of ten), you would need to increase the sample size by 100 times.A1 + A2 make OLS a BLUE estimator, where BLUE means the best, linear, unbiased estimator. That is, no other unbiased linear estimator has a smaller sampling variance than b .This result is called "Gauss-Markov theorem."4. A3. Normality, i is distributed as ),0(2NInferences: looking up tables for small samples.A1 + A2 + A3 make OLS a maximum likelihood (ML) estimator. Like all other ML estimators, OLS in this case is BUE (best unbiased estimator). That is, no other unbiased estimator can have a smaller sampling variance than OLS.Note that ML is always the most efficient estimator among all unbiased estimators. The cost of ML is really the requirement of we know the true parametric distribution of the residual. If you can afford the assumption, ML is always the best. Very often, we don't make the assumption because we don't know the parametric family of the disturbance. In general, the following tradeoff is true:More information == more efficiency. Less assumption == less efficiency.It is not correct to call certain models OLS models and other ML models. Theoretically, a same model can be estimated by OLS or ML. Model specification is different from estimation procedure.VI. ML for linear model under normality assumption (A1+A2+A3) , i :i :d N(0， 2), i = 1, … nObservations y i are independently distributed as y i ~ N(x i ’ 2); i = 1, … nUnder the normal errors assumption, the joint pdf of y’s isL = f (y 1…y n | 2) = ∏ f (y i | 2)= (2π 2)-n/2 exp{-(2 2)-1∑(y i - x i ’ }Log transformation is a monotone transformation. Maximizing L is equivalent to maximizing logL below:l = logL = (-n/2) log(2π 2) - (2 2)-1 ∑(y i - x i ’It is easy to see that what maximizes l (Maximum Likelihood Estimator) is the same as the LS estimator.。

回归分析法(精品PPT课件)

b0
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W 2 n yi b0 b1xi xi 0
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求解上述方程组得：
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补充内容：回归分析法
回归分析是计量经济学中最为基础的一部份内容。在这里我们简单地介绍回归分析中估计模型具体参数值的方法。
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一、一元线性回归与最小二乘法
Y=b0+b1x+ε，其中y 为应变量，x为自变量， b0为模型的截距，b1为x变量的系数， ε为随机误差项。
如果现在有一系列的y与x的值，我们可以用很多方法来找到一个线性的方程，例如任意连接两个特定的点，但这种方法显然不能给出一条最好的拟合直线。另一种方法是找出一条直线，使得直线与已有的点之间的距离的和最小，但由于这条直线与点之间的距离有时为正有时为负，求和时会相互抵消，所以用这种方法找到的直线也并不一定最好。于是我们想到要找到一条这样的直线，使得直线与点之间的距离的平方和最小：
xi
n i1
n i1
9
例1：
某地区人均收入与某耐用消费品销售额的资料如下表所示：请求出其一元回归模型。
年份 1991
人均收入x/元
680
耐用消
费品销售额y/
164
万元
1992 760
180
1993 900
200
1994 940
228

北大暑期课程《回归分析报告》(Linear Regression Analysis)讲义1

实用文案Class 1: Expectations, variances, and basics of estimationBasics of matrix (1)I. Organizational Matters(1)Course requirements:1)Exercises: There will be seven (7) exercises, the last of which is optional. Eachexercise will be graded on a scale of 0-10. In addition to the graded exercise, ananswer handout will be given to you in lab sections.2)Examination: There will be one in-class, open-book examination.(2)Computer software: StataII. Teaching Strategies(1) Emphasis on conceptual understanding.Yes, we will deal with mathematical formulas, actually a lot of mathematical formulas. But, I do not want you to memorize them. What I hope you will do, is to understand the logic behind the mathematical formulas.(2) Emphasis on hands-on research experience.Yes, we will use computers for most of our work. But I do not want you to become a computer programmer. Many people think they know statistics once they know how to run astatistical package. This is wrong. Doing statistics is more than running computer programs. What I will emphasize is to use computer programs to your advantage in research settings. Computer programs are like automobiles. The best automobile is useless unless someone drives it. You will be the driver of statistical computer programs.(3) Emphasis on student-instructor communication.I happen to believe in students' judgment about their own education. Even though I willbe ultimately responsible if the class should not go well, I hope that you will feel part of the class and contribute to the quality of the course. If you have questions, do not hesitate toask in class. If you have suggestions, please come forward with them. The class is as muchyours as mine.Now let us get to the real business.III(1). Expectation and VarianceRandom Variable: A random variable is a variable whose numerical value is determined by the outcome of a random trial.Two properties: random and variable.A random variable assigns numeric values to uncertain outcomes. In a common language, "give a number". For example, income can be a random variable. There are many ways to do it. You can use the actual dollar amounts.In this case, you have a continuous random variable. Or you can use levels of income, such as high, median, and low. In this case, you have an ordinal random variable [1=high,2=median, 3=low]. Or if you are interested in the issue of poverty, you can have a dichotomous variable: 1=in poverty, 0=not in poverty.In sum, the mapping of numeric values to outcomes of events in this way is the essenceof a random variable.Probability Distribution: The probability distribution for a discrete random variable X associates with each of the distinct outcomes x i(i = 1, 2,..., k) a probability P(X = x i). Cumulative Probability Distribution: The cumulative probability distribution for a discrete random variable X provides the cumulative probabilities P(X x) for all values x.Expected Value of Random Variable: The expected value of a discrete random variable X is denoted by E{X} and defined:E{X}= P(x i)where: P(x i) denotes P(X = x i). The notation E{ } (read “expectation of”) is called the expectation operator.In common language, expectation is the mean. But the difference is that expectation is a concept for the entire population that you never observe. It is the result of the infinite number of repetitions. For example, if you toss a coin, the proportion of tails should be .5 in the limit. Or the expectation is .5. Most of the times you do not get the exact .5, but a number close to it.Conditional ExpectationIt is the mean of a variable conditional on the value of another random variable.Note the notation: E(Y|X).In 1996, per-capita average wages in three Chinese cities were (in RMB):Shanghai: 3,778Wuhan: 1,709Xi’an: 1,155Variance of Random Variable: The variance of a discrete random variable X is denoted by V{X} and defined:V{X}=(x i - E{X})2 P(x i)where: P(x i) denotes P(X = x i). The notation V{ } (read “variance of”) is called the variance operator.Since the variance of a random variable X is a weighted average of the squared deviations, (X - E{X})2 , it may be defined equivalently as an expected value: V{X} = E{(X - E{X})2}. An algebraically identical expression is: V{X} = E{X2} - (E{X})2.Standard Deviation of Random Variable: The positive square root of the variance of X is called the standard deviation of X and is denoted by σ{X}:σ{X} =The notation σ{ } (read “standard deviation of”) is called the standard deviation operator. Standardized Random Variables: If X is a random variable with expected value E{X} and standard deviation σ{X}, then:Y=}{}{ X XEXσ-is known as the standardized form of random variable X.Covariance: The covariance of two discrete random variables X and Y is denoted by Cov{X,Y} and defined:Cov{X,Y} =where: P(x i, y j) denotes )The notation of Cov{ , } (read “covariance of”) is called the covariance operator.When X and Y are independent, Cov {X,Y} = 0.Cov {X,Y} = E{(X - E{X})(Y - E{Y})}; Cov {X,Y} = E{XY} - E{X}E{Y}(Variance is a special case of covariance.)Coefficient of Correlation: The coefficient of correlation of two random variables X and Y is denoted by ρ{X,Y} (Greek rho) and defined:where: σ{X} is the standard deviation of X; σ{Y} is the standard deviation of Y; Cov is the covariance of X and Y.Sum and Difference of Two Random Variables: If X and Y are two random variables, then the expected value and the variance of X + Y are as follows:Expected Value: E{X+Y} = E{X} + E{Y};Variance: V{X+Y} = V{X} + V{Y}+ 2 Cov(X,Y).If X and Y are two random variables, then the expected value and the variance of X - Y are as follows:Expected Value : E {X - Y } = E {X } - E {Y };Variance : V {X - Y } = V {X } + V {Y } - 2 Cov (X,Y ).Sum of More Than Two Independent Random Variables: If T = X 1 + X 2 + ... + X s is the sum of sindependent random variables, then the expected value and the variance of T are as follows:Expected Value: ; Variance:III(2). Properties of Expectations and Covariances:(1) Properties of Expectations under Simple Algebraic Operations)()(x bE a bX a E +=+This says that a linear transformation is retained after taking an expectation.bX a X +=*is called rescaling: a is the location parameter, b is the scale parameter.Special cases are:For a constant: a a E =)(For a different scale: )()(X E b bX E =, e.g., transforming the scale of dollars intothe scale of cents.(2) Properties of Variances under Simple Algebraic Operations)()(2X V b bX a V =+This says two things: (1) Adding a constant to a variable does not change the varianceof the variable; reason: the definition of variance controls for the mean of the variable[graphics]. (2) Multiplying a constant to a variable changes the variance of the variable by a factor of the constant squared; this is to easy prove, and I will leave it to you. This is the reason why we often use standard deviation instead of variance2x x σσ=is of the same scale as x.(3) Properties of Covariance under Simple Algebraic OperationsCov(a + bX, c + dY) = bd Cov(X,Y).Again, only scale matters, location does not.(4) Properties of Correlation under Simple Algebraic OperationsI will leave this as part of your first exercise:),(),(Y X dY c bX a ρρ=++That is, neither scale nor location affects correlation.IV: Basics of matrix.1. DefinitionsA. MatricesToday, I would like to introduce the basics of matrix algebra. A matrix is a rectangular array of elements arranged in rows and columns:11121211.......m n nm x x x x X x x ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦Index: row index, column index.Dimension: number of rows x number of columns (n x m)Elements: are denoted in small letters with subscripts.An example is the spreadsheet that records the grades for your home work in the following way:Name 1st 2nd ....6thA 7 10 (9)B 6 5 (8)... ... ......Z 8 9 (8)This is a matrix.Notation: I will use Capital Letters for Matrices.B. VectorsVectors are special cases of matrices:If the dimension of a matrix is n x 1, it is a column vector:⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=n x x x x (21)If the dimension is 1 x m, it is a row vector: y' = | 1y 2y .... m y |Notation: small underlined letters for column vectors (in lecture notes)C. TransposeThe transpose of a matrix is another matrix with positions of rows and columns being exchanged symmetrically.For example: if⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⨯nm n m m n x x x x x x X 12111211)( (1121112)()1....'...n m n m nm x x x x X x x ⨯⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦It is easy to see that a row vector and a column vector are transposes of each other. 2. Matrix Addition and SubtractionAdditions and subtraction of two matrices are possible only when the matrices have the same dimension. In this case, addition or subtraction of matrices forms another matrix whoseelements consist of the sum, or difference, of the corresponding elements of the two matrices.⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡±±±±±=Y ±X mn nm n n m m y x y x y x y x y x (11)2121111111 Examples:⎥⎦⎤⎢⎣⎡=A ⨯4321)22(⎥⎦⎤⎢⎣⎡=B ⨯1111)22(⎥⎦⎤⎢⎣⎡=B +A =⨯5432)22(C 3. Matrix MultiplicationA. Multiplication of a scalar and a matrixMultiplying a scalar to a matrix is equivalent to multiplying the scalar to each of the elements of the matrix.11121211Χ...m n nm cx c cx cx ⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦ B. Multiplication of a Matrix by a Matrix (Inner Product)The inner product of matrix X (a x b) and matrix Y (c x d) exists if b is equal to c. The inner product is a new matrix with the dimension (a x d). The element of the new matrix Z is:c∑=kj ik ij y x zk=1Note that XY and YX are very different. Very often, only one of the inner products (XY and YX) exists.Example:⎥⎦⎤⎢⎣⎡=4321)22(x A⎥⎦⎤⎢⎣⎡=10)12(x BBA does not exist. AB has the dimension 2x1⎥⎦⎤⎢⎣⎡=42ABOther examples:If )53(x A , )35(x B , what is the dimension of AB? (3x3)If )53(x A , )35(x B , what is the dimension of BA? (5x5)If )51(x A , )15(x B , what is the dimension of AB? (1x1, scalar)If )53(x A , )15(x B , what is the dimension of BA? (nonexistent)4. Special MatricesA. Square Matrix)(n n A ⨯B. Symmetric MatrixA special case of square matrix.For )(n n A ⨯, ji ij a a =. All i, j .A' = AC. Diagonal MatrixA special case of symmetric matrix⎥⎥⎥⎥⎦⎢⎢⎢⎢⎣=X nn x x 0 (2211)D. Scalar Matrix0....0c c c c ⎡⎤⎢⎥⎢⎥=I ⎢⎥⎢⎥⎣⎦E. Identity MatrixA special case of scalar matrix⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=I 10 (101)Important: for r r A ⨯AI = IA = AF. Null (Zero) MatrixAnother special case of scalar matrix⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=O 00 (000)From A to E or F, cases are nested from being more general towards being more specific.G. Idempotent MatrixLet A be a square symmetric matrix. A is idempotent if....32=A =A =AH. Vectors and Matrices with elements being oneA column vector with all elements being 1,⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⨯1......111r A matrix with all elements being 1, ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⨯1...1...111...11rr J Examples let 1 be a vector of n 1's: )1(1⨯n 1'1 = )11(⨯n11' = )(n n J ⨯I. Zero Vector A zero vector is⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⨯0....001r 5. Rank of a MatrixThe maximum number of linearly independent rows is equal to the maximum number of linearly independent columns. This unique number is defined to be the rank of the matrix.For example,⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=B 542211014321 Because row 3 = row 1 + row 2, the 3rd row is linearly dependent on rows 1 and 2. The maximum number of independent rows is 2. Let us have a new matrix:⎥⎦⎤⎢⎣⎡=B 11014321* Singularity: if a square matrix A of dimension ()n n ⨯has rank n, the matrix is nonsingular. If the rank is less than n, the matrix is then singular.。

第十二讲回归分析

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散点图
此即非线性回归或曲线回归问题（需要配曲线）
配曲线的一般方法是：
先对两个变量 x 和 y 作 n 次试验观察得 (xi , yi ), i 1,2,..., n 画出散点图，
根据散点图确定须配曲线的类型.然后由 n 对试验数据确定每一类曲线的未知
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10.49 10.59 10.60 10.80 10.60 10.90 10.76
2019/12/18
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yi ˆ0 ˆ1xi 2 n ( yi yˆi )2
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2 的无偏估计为
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其中 r 1 1 n 2 F 1 1 1 , n 2
2019/12/18
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2、回归系数的置信区间

应用统计方法第四章-回归分析PPT课件

应用统计方法第四章-回归分析ppt课件
• 回归分析概述 • 线性回归分析 • 非线性回归分析 • 多元回归分析 • 回归分析的注意事项
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回归分析概述
回归分析的定义
回归分析是一种统计学方法，用于研究自变量和因变量之间的相关关系，并建立数学模型来描述这种关系。
它通过分析因变量对自变量的依赖程度，来预测因变量的未来值或解释因变量的变异。
影响
共线性会导致回归系数不稳定，降低模型的预测精度和可靠性。
解决方法
通过剔除不必要的自变量、使用主成分分析等方法来降低共线性的影响。
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回归分析的注意事项
数据质量与预处理数据完整性源自确保数据集中的所有必要信息都已收集，没有遗漏或缺失值。
数据准确性
核实数据的准确性，并处理任何错误或异常值。
回归分析的分类
线性回归分析
研究自变量和因变量之间线性关系的回归分析。
多元回归分析
研究多个自变量与一个因变量之间关系的回归分析。
ABCD
非线性回归分析
研究自变量和因变量之间非线性关系的回归分析，如多项式回归、指数回归、对数回归等。
一元回归分析
研究一个自变量与一个因变量之间关系的回归分析。
回归分析的应用场景
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线性回归分析
线性回归模型
线性回归模型
描述因变量与自变量之间线性关系的数学模型。
模型形式
(Y = beta_0 + beta_1X_1 + beta_2X_2 + ldots + beta_pX_p + epsilon)
最小二乘法估计
最小二乘法
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通过最小化预测值与实际值之间的残差平方和来估计回归参数

第7章回归分析第一节`回归分析意义

yˆ a bx
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（四）回归方程（概念要点）
1. 描述 y 的平均值或期望值如何依赖于 x 的方程称为回归方程。
2. 简单线性回归方程的形式如下
▪
yˆ a bx
方程的图示是一条直线，因此也称为直线回归
方程
▪ a是回归直线在 y 轴上的截距，是当 x=0 时 y 的
第二节、回归分析的种类
按照自变量和因变量之间的关系类型，可分为线性回归分析和非线性回归分析。
回归分析按照涉及的自变量的多少，可分为一元回归分析和多元回归分析；
如果在回归分析中，只包括一个自变量和一个因变量，且二者的关系可用一条直线近似表示，这种回归分析称为一元线性回归分析。
如果回归分析中包括两个或两个以上的自变量，且因变量和自变量之间是线性关系，则称为多元线性回归分析。
❖ ｅt是因变量的实际值和估计值的离差：当给定X一数值时，y的实际值可以看作由两部分组成：
❖ 一部分是X对y均值的线性影响而形成的系统部分，由回归量Y=a+bX来决定；
❖ 另一部分是内ｅt所代表的各种偶然因素、观察误差以及被忽略的其他影响因素所带来的随机误差。
（三）总体回归线与随机误差项
误差项ut是一个期望值为0的随机变量，即E(ut)=0。对于一个给定的 x 值，y 的期望值为
xy 120 156 168 195 182 224 270 340 1655
建立直线回归方程：（结果保留一位小数）
（2）试预测当居民收入增加1亿元时，商品零售额平均增加多少？（3）试预测当居民收入增加到30亿元时，商品零售额是多少？
公式1

回归分析法PPT课件

线性回归模型的参数估计
最小二乘法
通过最小化误差平方和的方法来估计模型参数。
最大似然估计
通过最大化似然函数的方法来估计模型参数。
参数估计的步骤
包括数据收集、模型设定、参数初值、迭代计算等步骤。
参数估计的注意事项
包括异常值处理、多重共线性、自变量间的交互作用等。
线性回归模型的假设检验
假设检验的基本原理
回归分析法的历史与发展
总结词
回归分析法自19世纪末诞生以来，经历了多个发展阶段，不断完善和改进。
VS
详细描述
19世纪末，英国统计学家Francis Galton 在研究遗传学时提出了回归分析法的概念。后来，统计学家R.A. Fisher对其进行了改进和发展，提出了线性回归分析和方差分析的方法。随着计算机技术的发展，回归分析法的应用越来越广泛，并出现了多种新的回归模型和技术，如多元回归、岭回归、套索回归等。
回归分析法的应用场景
总结词
回归分析法广泛应用于各个领域，如经济学、金融学、生物学、医学等。
详细描述
在经济学中，回归分析法用于研究影响经济发展的各种因素，如GDP、消费、投资等；在金融学中，回归分析法用于股票价格、收益率等金融变量的预测；在生物学和医学中，回归分析法用于研究疾病发生、药物疗效等因素与结果之间的关系。
梯度下降法
基于目标函数对参数的偏导数，通过不断更新参数值来最小化目标函数，实现参数的迭代优化。
非线性回归模型的假设检验
1 2
模型检验
对非线性回归模型的适用性和有效性进行检验，包括残差分析、正态性检验、异方差性检验等。
参数检验
通过t检验、z检验等方法对非线性回归模型的参数进行假设检验，以验证参数的显著性和可信度。

回归分析讲义

1 是总体回归直线的斜率，而0 为截距。为方便，将1和 0 统称为回归参数
§2.1 总体与总体回归模型
一、总体与总体回归模型的含义
1.总体回归模型
Yi E(Y Xi ) Ui
0 1 Xi Ui
上式是参数和变量的线性函数。
计量经济学中的线性模型或线性总体，一般是指对参数而言的线性模型，或者是可以转换为关于参数是线性的模型。
为此，我们首先应准确理解总体回归模型。
§2.1 总体与总体回归模型
一、总体与总体回归模型的含义
1.总体回归模型
X 80
100
120 140
160 180 200 220 240 260
Y
55 65
79 80
102 110 120 135 137 150
60 70
84 93
107 115 136 137 145 152
第二章回归分析
§2.1 总体与总体回归模型
§2.2样本与样本回归模型
§2.3总体回归模型和样本回归模型 ——基于蒙特卡罗实验的再认识
§2.1 总体与总体回归模型
一、总体与总体回归模型的含义 1. 总体 2. 总体回归模型
二、总体回归模型中 u i所包含的内容 1.从数量上看 ui 2.从实际经济行为看 ui 3.从回归关系看 ui
X2)
12 20
0.6
也就是说，当X变化一个单位，对应的Y的总
体均值变化0.6个单位。显然，基于总体信息，
我们还不能计算截距
§2.1 总体与总体回归模型
二．总体回归模型中的 Ui 所包含的内容
1.从数量上看Ui
Yi E(Y Xi ) Ui
0 1 Xi Ui
总体回归模型中的随机扰动Ui，从数量上看，它等于在给定X的条件下，第i个样本点Yi与对应的总体均值的“距离”。如，对于X=200, 共有5户家庭，条件期望为137，其中消费为 120的家庭与条件期望的“距离”等于-17，即对应的U=-17, 而消费为140的家庭与其条件期望的“距离”为3

第二章回归分析ppt课件

U和Q的相对大小反映了因子x对y的影响程度, 在n固定的情况下，如果回归
方差所占y方差的比重越大，剩余方差所占的比重越小，就表明回归的效果
越好，即：x的变化对y的变化起主要作用，利用回归方程所估计出的ŷ也会
越接近观测值y。
ŷ的方差占y的方差的比重（U/(U+Q)）可作为衡量回归模型效果的标准：
ŷ
y -y
ŷ -y
y
x
syy
1 n
n t 1
( yt
y)2
1 n
n t 1
( yt
y)2
1 n
n t 1
( yt
yt )2
“回归平方和”与“剩余平方和”
对上式两边分别乘以n，研究各变量的离差平方和的关系。为避免过多数学符
号，等号左边仍采用方差的记号syy。
n
n
syy ( yt y)2 ( yt yt )2 U Q
回忆前文所讲， y的第i个观测值yi服从怎样的分布？
yi ~ N (β0 +βxi , σ2)
e=yi- (β0 +βxi ) 服从N(0, σ2)
于是， yi (0 xi ) 服从标准正态分布N (0,1)
0.4
在95%的置信概率下：
因为定理：若有z ~ N (, 2 )，则有 z ~ N (0,1)
通过方差分析可知，可用“回归平方和”U与“剩余平方和”Q的比值来衡量回归效果的好坏。可以证明，假设总体的回归系数为0的条件下，统计量：
U
F=
1 Q
注意Q的自由度为n-2, 即：残差e的方差的无偏估计为：Q/(n-2)
n2 服从分子自由度为1，分母自由度为n - 2的F分布
上式可以用相关系数的平方来表示：

第二讲回归分析

1 拟合优度检验所谓拟合程度，是指样本观测值聚集在样本回归直线周围的紧密程度。判断回归模型拟合程度优劣最常用的数量指标是判定系数（Coefficient of Determination）
总的离差平方和：在回归分析中，表示y的n次观测值之间的差异，记为
S 总 L yy
可以证明
( yi y ) 2
a 1
n
(i 1,2,, k )
正规方程组也可以写成
L11b1 L12b2 L1k bk L1 y L21b1 L22b2 L2 k bk L2 y L b L b L b L kk k ky k1 1 k 2 2 b0 y b1 x1 b2 x2 bk xk
n
（二）一元线性回归模型的显著性检验
一元线性回归模型检验的种类实际意义检验参数估计值的符号和取值范围统计检验检验样本回归方程的可靠性消费支出与可支配收入：如果估计出来的 b小于 0 或大于 1，
ˆ ˆ ˆ y a bx
收入
支出 •拟合程度检验； •相关系数检验； •参数显著性检验(t检验)； •回归方程显著性检验（F 检验）计量检验假定条件是否满足序列相关检验异方差性检验
则正规方程组（2.15）式可以进一步写成矩阵形式
Ab B
求解得
b A1B ( X T X ) 1 X T Y
（2.16）
引入记号
Lij L ji ( xia xi )(x j x j )
a 1 n
(i, j 1,2,, k )
Liy ( xia xi )( ya y )
i 1 n
可以证明

回归分析学习课件PPT课件

03 网格搜索
为了找到最优的参数组合，可以使用网格搜索方法对参数空间进行穷举或随机搜索，通过比较不同参数组合下的预测性能来选择最优的参数。
非线性回归模型的假设检验与评估
假设检验
与线性回归模型类似，非线性回归模型也需要进行假设检验，以检验模型是否满足某些统计假设，如误差项的独立性、同方差性等。
整估计。
最大似然法
03
基于似然函数的最大值来估计参数，能够同时估计参数和模型
选择。
多元回归模型的假设检验与评估
线性假设检验
检验回归模型的线性关系是否成立，通常使用F检验或t检验。
异方差性检验
检验回归模型残差的异方差性，常用的方法有图检验、White检验和 Goldfeld-Quandt检验。
多重共线性检验
检验回归模型中自变量之间的多重共线性问题，常用的方法有VIF、条件指数等。
模型评估指标
包括R方、调整R方、AIC、 BIC等指标，用于评估模型的拟合优度和预测能力。
05
回归分析的实践应用
案例一：股票价格预测
总结词
通过历史数据建立回归模型，预测未来股票价格走势。
详细描述
利用股票市场的历史数据，如开盘价、收盘价、成交量等，通过回归分析方法建立模型，预测未来股票价格的走势。
描述因变量与自变量之间的非线性关系，通过变换或使用其他方法来适应非线性关系。
03 混合效应回归模型
同时考虑固定效应和随机效应，适用于面板数据或重复测量数据。
多元回归模型的参数估计
最小二乘法
01
通过最小化残差平方和来估计参数，是最常用的参数估计方法。
加权最小二乘法
02
适用于异方差性数据，通过给不同观测值赋予不同的权重来调

回归分析(第一讲)

例如：研究产品的销量与用于产品宣传的广告费之间的关系；

因变量——销售量自变量——广告费

我们用Y代表因变量， X代表自变量。如果有多个解释变量，我们将用适当的下标，表示各个不同的X。

例如，X1，X2，X3等等。
概念:总体回归线

下面通过一个例子予以说明。

某城市A产品生产企业共有5 5个（总体），下表给出了这些企业产品价格（元）与A 产品月销量（万件）的有关数据。

例如，当X＝10.1时，有7个Y值与之对应当X＝10.4时，相应地有6个Y值，等等。

对每个X，计算出一个Y的均值。将这些均值点连起来，构成一条直线。我们称该直线为总体回归直线 (Population Regression Line，PRL)。
（销量）
（各平均值连成的直线）
（售价）
概念要点：总体回归线

总体回归线： Y =β0+β1X 它描述的是X与Y的均值之间的关系。
概念：随机误差

每个个体的Y值与总体回归线之间的距离（可正可负）
（销量）
每个点都有一个随机误差，以该点为例。
ε
i
（售价）
概念：回归模型（一元线性回归）
总体 Y的截距总体斜率随机误差
i
Yi 0 1Xi ε
因变量 Dependent Variable
自变量 Independent Variables
概念：回归模型（多元线性回归）
总体 Y的截距总体斜率随机误差
Y 0 1X1 2 X2 P X P
因变量 Dependent Variable 自变量 Independent Variables

第10讲--回归分析

位时y 的平均变化量。
包含了p个解释变量之外的对y的变异有影响的其
它因素。
建立回归模型的步骤： Step 3：模型估计
针对具体样本应该选择一条什么直线？
办法：最小二乘估计
找
使得如下的平方和最小
n
2
y i01 x i12 x i2px ip
i 1
当p=1时
问题：估计出来的回归系数是样本观测值的函数？它们会随着
50 0 0Leabharlann 51015
20
25
30
学生人数（千人）
9
相关系数为0.95。相关系数可以告诉你什么？不能告诉你什么？
10
变异性的分解
为什么不同连锁店的季度销售收入存在差异？
一种理解模式：
误差项 ε，包含了x之外的对y的变异有影响的其它因素。
简单线性回归模型
这个模型表达了y与x之间的什么关系？
线性回归
1
据英国媒体2008年2月18日报道，通过对过去 20年里定期在东京和大阪街头进行的随机调查发现。当日本经济迅速发展时，女性更愿意留长头发；而当经济出现停滞时，她们更愿意更多地剪短发…
环球时报，2008年2月20日
管理决策，经常取决于对两个或更多个变量的分析。例如：一位销售部经理在考虑了广告费和销售收入之间的关系后，才能尝试去预测一定水平的广告费可能带来多少销售收入。通常，一位管理人员要依靠直觉或经验去判断两个变量的关系。但是，如果能取得数据，我们就能利用统计模型（如回归分析）去建立一个表示变量间相互关系的方程，来做预测。
残差（residual）：对数据拟合回归线后剩余的部分，记为
样本数据可以分解成拟合值加残差。
24

第二讲回归分析概述

Λ Λ Λ
随机形式
Yi = β1 + β 2 X i + ui
Λ Λ
Λ
Λ
Λ
（2-6）（2-7）
（2-8） Yi = Y i + u i Λ u i ：残差项,代表了样本中影响 Yi 的随机因素的组合
SRF
Yi = β1 + β 2 X i
Λ
Λ
Λ
Yi = β1 + β 2 X i + ui
Λ
Λ
Λ
作物的收成对气温、降雨、阳光以及施肥的依赖关系是统计性质的。这个性质的意义在于：这些解释变量固然重要，但并不能使农业经济学家准确预测作物的收成。
对随机变量的假定贯穿整个课程
回归和因果关系回归分析依赖关系
因果关系
其他条件不变——意味着“其他因素保持不变”——的概念在因果分析中有着重要作用。关键问题：对于作出因果推断，是否有足够多的其他因素被保持不变？对一项计量经济研究进行评价时，都要面临这个问题。
Yi = β1 + β 2 X i + ui
表 2.2.1 某社区家庭每月收入与消费支出统计表每月家庭可支配收入X（元） 800 每月家庭消费支出 Y （元） 561 594 627 638 1100 638 748 814 847 935 968 1400 869 913 924 979 1012 1045 1078 1122 1155 1188 1210 1700 1023 1100 1144 1155 1210 1243 1254 1298 1331 1364 1408 1430 1485 共计 2420 2000 1254 1309 1364 1397 1408 1474 1496 1496 1562 1573 1606 1650 1716 2300 1408 1452 1551 1595 1650 1672 1683 1716 1749 1771 1804 1870 1947 2600 1650 1738 1749 1804 1848 1881 1925 1969 2013 2035 2101 2112 2200 21450 21285 15510 2900 1969 1991 2046 2068 2101 2189 2233 2244 2299 2310 3200 2090 2134 2178 2266 2354 2486 2552 2585 2640 3500 2299 2321 2530 2629 2860 2871

北大暑期课程《回归分析》(Linear-Regression-Analysis)讲义PKU5

Class 5: ANOVA (Analysis of Variance) andF-testsI.What is ANOVAWhat is ANOVA? ANOVA is the short name for the Analysis of Variance. The essence ofANOVA is to decompose the total variance of the dependent variable into two additivecomponents, one for the structural part, and the other for the stochastic part, of a regression. Today we are going to examine the easiest case.II.ANOVA: An Introduction Let the model beεβ+= X y .Assuming x i is a column vector (of length p) of independent variable values for the i th'observation,i i i εβ+='x y .Then is the predicted value. sum of squares total:[]∑-=2Y y SST i[]∑-+-=2'x b 'x y Y b i i i[][][][]∑∑∑-+-+-=Y -b 'x b 'x y 2Y b 'x b 'x y 22i i i i i i[][]∑∑-+=22Y b 'x e i ibecause .This is always true by OLS. = SSE + SSRImportant: the total variance of the dependent variable is decomposed into two additive parts: SSE, which is due to errors, and SSR, which is due to regression. Geometric interpretation: [blackboard ]Decomposition of VarianceIf we treat X as a random variable, we can decompose total variance to the between-group portion and the within-group portion in any population:()()()i i i x y εβV 'V V +=Prove:()()i i i x y εβ+='V V()()()i i i i x x εβεβ,'Cov 2V 'V ++=()()iix εβV 'V +=(by the assumption that ()0 ,'Cov =εβk x , for all possible k.)The ANOVA table is to estimate the three quantities of equation (1) from the sample.As the sample size gets larger and larger, the ANOVA table will approach the equation closer and closer.In a sample, decomposition of estimated variance is not strictly true. We thus need toseparately decompose sums of squares and degrees of freedom. Is ANOVA a misnomer?III.ANOVA in MatrixI will try to give a simplied representation of ANOVA as follows:[]∑-=2Y y SST i ()∑-+=i i y Y 2Y y 22∑∑∑-+=i i y Y 2Y y 22∑-+=222Y n 2Y n y i (because ∑=Y n y i )∑-=22Y n y i2Y n y 'y -=y J 'y n /1y 'y -= (in your textbook, monster look)SSE = e'e[]∑-=2Y b 'x SSR i()()[]∑-+=Y b 'x 2Y b 'x 22i i()[]()∑∑-+=b 'x Y 2Y n b 'x 22i i()[]()∑∑--+=i i ie y Y 2Y n b 'x 22()[]∑-+=222Y n 2Y n b 'x i(because ∑∑==0e ,Y n y i i , as always)()[]∑-=22Yn b 'x i2Y n Xb X'b'-=y J 'y n /1y X'b'-= (in your textbook, monster look)IV.ANOVA TableLet us use a real example. Assume that we have a regression estimated to be y = - 1.70 + 0.840 xANOVA TableSOURCE SS DF MS F with Regression 6.44 1 6.44 6.44/0.19=33.89 1, 18Error 3.40 18 0.19 Total 9.8419We know , , , , . If we know that DF for SST=19, what is n?n= 205.220/50Y ==84.95.25.22084.134Y n y SST 22=⨯⨯-=-=∑i()[]∑-+=0.1250.84x 1.7-SSR 2i[]∑-⨯⨯⨯-⨯+⨯=0.125x 84.07.12x 84.084.07.17.12i i= 20⨯1.7⨯1.7+0.84⨯0.84⨯509.12-2⨯1.7⨯0.84⨯100- 125.0 = 6.44SSE = SST-SSR=9.84-6.44=3.40DF (Degrees of freedom): demonstration. Note: discounting the intercept when calculating SST. MS = SS/DFp = 0.000 [ask students].What does the p-value say?V.F-TestsF-tests are more general than t-tests, t-tests can be seen as a special case of F-tests.If you have difficulty with F-tests, please ask your GSIs to review F-tests in the lab. F-tests takes the form of a fraction of two MS's.MSR/MSE F ,=df2df1An F statistic has two degrees of freedom associated with it: the degree of freedom inthe numerator, and the degree of freedom in the denominator.An F statistic is usually larger than 1. The interpretation of an F statistics is thatwhether the explained variance by the alternative hypothesis is due to chance. In other words, the null hypothesis is that the explained variance is due to chance, or all the coefficients are zero.The larger an F-statistic, the more likely that the null hypothesis is not true. There is atable in the back of your book from which you can find exact probability values.In our example, the F is 34, which is highly significant. VI.R2R 2 = SSR / SSTThe proportion of variance explained by the model. In our example, R-sq = 65.4%VII.What happens if we increase more independent variables. 1.SST stays the same. 2.SSR always increases. 3.SSE always decreases. 4.R2 always increases. 5.MSR usually increases. 6.MSE usually decreases.7.F-test usually increases.Exceptions to 5 and 7: irrelevant variables may not explain the variance but take up degrees of freedom. We really need to look at the results.VIII.Important: General Ways of Hypothesis Testing with F-Statistics.All tests in linear regression can be performed with F-test statistics. The trick is to run"nested models."Two models are nested if the independent variables in one model are a subset or linearcombinations of a subset （子集）of the independent variables in the other model.That is to say. If model A has independent variables (1, , ), and model B has independent variables (1, , , ), A and B are nested. A is called the restricted model; B is called less restricted or unrestricted model. We call A restricted because A implies that . This is a restriction.Another example: C has independent variable (1, , + ), D has (1, + ). C and A are not nested.C and B are nested.One restriction in C: . C andD are nested.One restriction in D: . D and A are not nested.D and B are nested: two restriction in D: 32ββ=; 0=1β.We can always test hypotheses implied in the restricted models. Steps: run tworegression for each hypothesis, one for the restricted model and one for the unrestricted model. The SST should be the same across the two models. What is different is SSE and SSR. That is, what is different is R2. Let()()df df SSE ,df df SSE u u r r ==;df df ()()0u r u r r u n p n p p p -=---=-<Use the following formulas:()()()()(),SSE SSE /df SSE df SSE F SSE /df r u r u dfr dfu dfu u u---=or()()()()(),SSR SSR /df SSR df SSR F SSE /df u r u r dfr dfu dfu u u---=(proof: use SST = SSE+SSR)Note, df(SSE r )-df(SSE u ) = df(SSR u )-df(SSR r ) =df ∆,is the number of constraints (not number of parameters) implied by the restricted modelor()()()22,2R R /df F 1R /dfur dfr dfu dfuuu--∆=- Note thatdf 1df ,2F t =That is, for 1df tests, you can either do an F-test or a t-test. They yield the same result. Another way to look at it is that the t-test is a special case of the F test, with the numerator DF being 1.IX.Assumptions of F-testsWhat assumptions do we need to make an ANOVA table work?Not much an assumption. All we need is the assumption that (X'X) is not singular, so that the least square estimate b exists.The assumption of =0 is needed if you want the ANOVA table to be an unbiased estimate of the true ANOVA (equation 1) in the population. Reason: we want b to be an unbiased estimator of , and the covariance between b and to disappear.For reasons I discussed earlier, the assumptions of homoscedasticity and non-serial correlation are necessary for the estimation of .The normality assumption that (i is distributed in a normal distribution is needed for small samples.X.The Concept of IncrementEvery time you put one more independent variable into your model, you get an increase in . We sometime called the increase "incremental ." What is means is that more variance is explained, or SSR is increased, SSE is reduced. What you should understand is that the incremental attributed to a variable is always smaller than the when other variables are absent.XI.Consequences of Omitting Relevant Independent VariablesSay the true model is the following:0112233i i i i i y x x x ββββε=++++.But for some reason we only collect or consider data on . Therefore, we omit in the regression. That is, we omit in our model. We briefly discussed this problem before. The short story is that we are likely to have a bias due to the omission of a relevant variable in the model. This is so even though our primary interest is to estimate the effect of or on y. Why? We will have a formal presentation of this problem.XII.Measures of Goodness-of-FitThere are different ways to assess the goodness-of-fit of a model.A. R2R2 is a heuristic measure for the overall goodness-of-fit. It does not have an associated test statistic.R 2 measures the proportion of the variance in the dependent variable that is “explained” by the model: R 2 =SSESSR SSRSST SSR +=B.Model F-testThe model F-test tests the joint hypotheses that all the model coefficients except for the constant term are zero.Degrees of freedoms associated with the model F-test: Numerator: p-1Denominator: n-p.C.t-tests for individual parametersA t-test for an individual parameter tests the hypothesis that a particular coefficient is equal to a particular number (commonly zero).tk = (bk- (k0)/SEk, where SEkis the (k, k) element of MSE(X ’X)-1, with degree of freedom=n-p. D.Incremental R2Relative to a restricted model, the gain in R 2 for the unrestricted model: ∆R 2= R u 2- R r 2E.F-tests for Nested ModelIt is the most general form of F-tests and t-tests.()()()()(),SSE SSE /df SSE df SSE F SSE /df r u r dfu dfr u dfu u u---=It is equal to a t-test if the unrestricted and restricted models differ only by one single parameter.It is equal to the model F-test if we set the restricted model to the constant-only model.[Ask students] What are SST, SSE, and SSR, and their associated degrees of freedom, for the constant-only model?Numerical ExampleA sociological study is interested in understanding the social determinants of mathematical achievement among high school students. You are now asked to answer a series of questions. The data are real but have been tailored for educational purposes. The total number of observations is 400. The variables are defined as: y: math scorex1: father's education x2: mother's educationx3: family's socioeconomic status x4: number of siblings x5: class rankx6: parents' total education (note: x6 = x1 + x2) For the following regression models, we know: Table 1 SST SSR SSE DF R 2 (1) y on (1 x1 x2 x3 x4) 34863 4201 (2) y on (1 x6 x3 x4) 34863 396 .1065 (3) y on (1 x6 x3 x4 x5) 34863 10426 24437 395 .2991 (4) x5 on (1 x6 x3 x4) 269753 396 .02101.Please fill the missing cells in Table 1.2.Test the hypothesis that the effects of father's education (x1) and mother's education (x2) on math score are the same after controlling for x3 and x4.3.Test the hypothesis that x6, x3 and x4 in Model (2) all have a zero effect on y.4.Can we add x6 to Model (1)? Briefly explain your answer.5.Test the hypothesis that the effect of class rank (x5) on math score is zero after controlling for x6, x3, and x4.Answer: 1. SST SSR SSE DF R 2 (1) y on (1 x1 x2 x3 x4) 34863 4201 30662 395 .1205 (2) y on (1 x6 x3 x4) 34863 3713 31150 396 .1065 (3) y on (1 x6 x3 x4 x5) 34863 10426 24437 395 .2991 (4) x5 on (1 x6 x3 x4) 275539 5786 269753 396 .0210Note that the SST for Model (4) is different from those for Models (1) through (3). 2.Restricted model is 01123344()y b b x x b x b x e =+++++Unrestricted model is ''''''011223344y b b x b x b x b x e =+++++(31150 - 30662)/1F 1,395 = -------------------- = 488/77.63 = 6.29 30662 / 395 3.3713 / 3F 3,396 = --------------- = 1237.67 / 78.66 = 15.73 31150 / 3964.No. x6 is a linear combination of x1 and x2. X'X is singular.5.(31150 - 24437)/1F 1,395 = -------------------- = 6713 / 61.87 = 108.50 24437/395t = 10.42t ===。

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