洪永淼计量经济学讲义_ch03
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Abstract: In this chapter, we will introduce the classical linear regression theory, including the classical model assumptions, the statistical properties of the OLS estimator, the t-test and the F -test, as well as the GLS estimator and related statistical procedures. This chapter will serve as a starting point from which we will develop the modern econometric theory.
As Kendall and Stuart (1961, Vol.2, Ch. 26, p.279) point out, “a statistical relationship,
however strong and however suggestive, can never establish causal connection. our ideas
1
coe¢ cient vector. When the linear model is correctly for the conditional mean E(YtjXt),
i.e., when E("tjXt) = 0; the parameter
0
=
@ @Xt
E
(Yt
jXt
)
is
the
marginal
Notations: (i) K k + 1 is the number of regressors for which there are k economic variables
and an intercept. (ii) The index t may denote an individual unit (e.g., a …rm, a household, a country)
e¤ect
of
Xt
on
Yt:
(ii) The key notion of linearity in the classical linear regression model is that the
regression model is linear in 0 rather than in Xt:
(iii) Does Assumption 3.1 imply a causal relationship from Xt to Yt? Not necessarily.
for cross-sectional data, or denote a time period (e.g., day, weak, month, year) in a time series context.
We …rst list and discuss the assumptions of the classical linear regression theory.
(iv) Matrix Notation: Denote
Y = (Y1; :::; Yn)0; " = ("1; :::; "n)0; X = (X1; :::; Xn)0;
of causation must come from outside statistics ultimately, from some theory or other.”
Assumption 3.1 only implies a predictive relationship: Given Xt, can we predict Yt linearly?
3.1 AssumptionsΒιβλιοθήκη Baidu
Suppose we have a random sample fZtgnt=1 of size n; where Zt = (Yt; Xt0)0; Yt is a scalar, Xt = (1; X1t; X2t; :::; Xkt)0 is a (k + 1) 1 vector, t is an index (either crosssectional unit or time period) for observations, and n is the sample size. We are interested in modelling the conditional mean E(YtjXt) using an observed realization (i.e., a data set) of the random sample fYt; Xt0g0; t = 1; :::; n:
Assumption 3.1 [Linearity]:
Yt = Xt0 0 + "t; t = 1; :::; n;
where 0 is a K 1 unknown parameter vector, and "t is an unobservable disturbance.
Remarks:
(i) In Assumption 3.1, Yt is the dependent variable (or regresand), Xt is the vector of regressors (or independent variables, or explanatory variables), and 0 is the regression
CHAPTER 3 CLASSICAL LINEAR REGRESSION MODELS
Key words: Classical linear regression, Conditional heteroskedasticity, Conditional homoskedasticity, F -test, GLS, Hypothesis testing, Model selection criterion, OLS, R2; t-test
As Kendall and Stuart (1961, Vol.2, Ch. 26, p.279) point out, “a statistical relationship,
however strong and however suggestive, can never establish causal connection. our ideas
1
coe¢ cient vector. When the linear model is correctly for the conditional mean E(YtjXt),
i.e., when E("tjXt) = 0; the parameter
0
=
@ @Xt
E
(Yt
jXt
)
is
the
marginal
Notations: (i) K k + 1 is the number of regressors for which there are k economic variables
and an intercept. (ii) The index t may denote an individual unit (e.g., a …rm, a household, a country)
e¤ect
of
Xt
on
Yt:
(ii) The key notion of linearity in the classical linear regression model is that the
regression model is linear in 0 rather than in Xt:
(iii) Does Assumption 3.1 imply a causal relationship from Xt to Yt? Not necessarily.
for cross-sectional data, or denote a time period (e.g., day, weak, month, year) in a time series context.
We …rst list and discuss the assumptions of the classical linear regression theory.
(iv) Matrix Notation: Denote
Y = (Y1; :::; Yn)0; " = ("1; :::; "n)0; X = (X1; :::; Xn)0;
of causation must come from outside statistics ultimately, from some theory or other.”
Assumption 3.1 only implies a predictive relationship: Given Xt, can we predict Yt linearly?
3.1 AssumptionsΒιβλιοθήκη Baidu
Suppose we have a random sample fZtgnt=1 of size n; where Zt = (Yt; Xt0)0; Yt is a scalar, Xt = (1; X1t; X2t; :::; Xkt)0 is a (k + 1) 1 vector, t is an index (either crosssectional unit or time period) for observations, and n is the sample size. We are interested in modelling the conditional mean E(YtjXt) using an observed realization (i.e., a data set) of the random sample fYt; Xt0g0; t = 1; :::; n:
Assumption 3.1 [Linearity]:
Yt = Xt0 0 + "t; t = 1; :::; n;
where 0 is a K 1 unknown parameter vector, and "t is an unobservable disturbance.
Remarks:
(i) In Assumption 3.1, Yt is the dependent variable (or regresand), Xt is the vector of regressors (or independent variables, or explanatory variables), and 0 is the regression
CHAPTER 3 CLASSICAL LINEAR REGRESSION MODELS
Key words: Classical linear regression, Conditional heteroskedasticity, Conditional homoskedasticity, F -test, GLS, Hypothesis testing, Model selection criterion, OLS, R2; t-test